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

     Author:     Fabian Immler, TU Muenchen

     Author:     Makarius

     Author:     Jasmin Blanchette, TU Muenchen

 Translation of HOL to FOL for Metis and Sledgehammer.

 *)

 signature ATP_TRANSLATE =

 sig

   type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term

   type connective = ATP_Problem.connective

   type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula

   type format = ATP_Problem.format

   type formula_kind = ATP_Problem.formula_kind

   type 'a problem = 'a ATP_Problem.problem

   datatype locality =

     General | Helper | Induction | Extensionality | Intro | Elim | Simp |

     Local | Assum | Chained

   datatype order = First_Order | Higher_Order

   datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic

   datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound

   datatype type_level =

     All_Types |

     Noninf_Nonmono_Types of soundness |

     Fin_Nonmono_Types |

     Const_Arg_Types |

     No_Types

   datatype type_uniformity = Uniform | Nonuniform

   datatype type_enc =

     Simple_Types of order * polymorphism * type_level |

     Guards of polymorphism * type_level * type_uniformity |

     Tags of polymorphism * type_level * type_uniformity

   val type_tag_idempotence : bool Config.T

   val type_tag_arguments : bool Config.T

   val no_lambdasN : string

   val concealedN : string

   val liftingN : string

   val combinatorsN : string

   val hybridN : string

   val lambdasN : string

   val smartN : string

   val schematic_var_prefix : string

   val fixed_var_prefix : string

   val tvar_prefix : string

   val tfree_prefix : string

   val const_prefix : string

   val type_const_prefix : string

   val class_prefix : string

   val polymorphic_free_prefix : string

   val skolem_const_prefix : string

   val old_skolem_const_prefix : string

   val new_skolem_const_prefix : string

   val type_decl_prefix : string

   val sym_decl_prefix : string

   val guards_sym_formula_prefix : string

   val tags_sym_formula_prefix : string

   val fact_prefix : string

   val conjecture_prefix : string

   val helper_prefix : string

   val class_rel_clause_prefix : string

   val arity_clause_prefix : string

   val tfree_clause_prefix : string

   val typed_helper_suffix : string

   val untyped_helper_suffix : string

   val type_tag_idempotence_helper_name : string

   val predicator_name : string

   val app_op_name : string

   val type_guard_name : string

   val type_tag_name : string

   val simple_type_prefix : string

   val prefixed_predicator_name : string

   val prefixed_app_op_name : string

   val prefixed_type_tag_name : string

   val ascii_of : string -> string

   val unascii_of : string -> string

   val strip_prefix_and_unascii : string -> string -> string option

   val proxy_table : (string * (string * (thm * (string * string)))) list

   val proxify_const : string -> (string * string) option

   val invert_const : string -> string

   val unproxify_const : string -> string

   val new_skolem_var_name_from_const : string -> string

   val atp_irrelevant_consts : string list

   val atp_schematic_consts_of : term -> typ list Symtab.table

   val type_enc_from_string : soundness -> string -> type_enc

   val is_type_enc_higher_order : type_enc -> bool

   val polymorphism_of_type_enc : type_enc -> polymorphism

   val level_of_type_enc : type_enc -> type_level

   val is_type_enc_quasi_sound : type_enc -> bool

   val is_type_enc_fairly_sound : type_enc -> bool

   val adjust_type_enc : format -> type_enc -> type_enc

   val mk_aconns :

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

   val unmangled_const : string -> string * (string, 'b) ho_term list

   val unmangled_const_name : string -> string

   val helper_table : ((string * bool) * thm list) list

   val factsN : string

   val prepare_atp_problem :

     Proof.context -> format -> formula_kind -> formula_kind -> type_enc

     -> bool -> string -> bool -> bool -> term list -> term

     -> ((string * locality) * term) list

     -> string problem * string Symtab.table * int * int

        * (string * locality) list vector * int list * int Symtab.table

   val atp_problem_weights : string problem -> (string * real) list

 end;

 structure ATP_Translate : ATP_TRANSLATE =

 struct

 open ATP_Util

 open ATP_Problem

 type name = string * string

 val type_tag_idempotence =

   Attrib.setup_config_bool @{binding atp_type_tag_idempotence} (K false)

 val type_tag_arguments =

   Attrib.setup_config_bool @{binding atp_type_tag_arguments} (K false)

 val no_lambdasN = "no_lambdas"

 val concealedN = "concealed"

 val liftingN = "lifting"

 val combinatorsN = "combinators"

 val hybridN = "hybrid"

 val lambdasN = "lambdas"

 val smartN = "smart"

 val generate_info = false (* experimental *)

 fun isabelle_info s =

   if generate_info then SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))

   else NONE

 val introN = "intro"

 val elimN = "elim"

 val simpN = "simp"

 (* TFF1 is still in development, and it is still unclear whether symbols will be

    allowed to have types like "$tType > $o" (i.e., "![A : $tType] : $o"), with

    "dummy" type variables. *)

 val exploit_tff1_dummy_type_vars = false

 val bound_var_prefix = "B_"

 val all_bound_var_prefix = "BA_"

 val exist_bound_var_prefix = "BE_"

 val schematic_var_prefix = "V_"

 val fixed_var_prefix = "v_"

 val tvar_prefix = "T_"

 val tfree_prefix = "t_"

 val const_prefix = "c_"

 val type_const_prefix = "tc_"

 val simple_type_prefix = "s_"

 val class_prefix = "cl_"

 val polymorphic_free_prefix = "poly_free"

 val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"

 val old_skolem_const_prefix = skolem_const_prefix ^ "o"

 val new_skolem_const_prefix = skolem_const_prefix ^ "n"

 val type_decl_prefix = "ty_"

 val sym_decl_prefix = "sy_"

 val guards_sym_formula_prefix = "gsy_"

 val tags_sym_formula_prefix = "tsy_"

 val fact_prefix = "fact_"

 val conjecture_prefix = "conj_"

 val helper_prefix = "help_"

 val class_rel_clause_prefix = "clar_"

 val arity_clause_prefix = "arity_"

 val tfree_clause_prefix = "tfree_"

 val lambda_fact_prefix = "ATP.lambda_"

 val typed_helper_suffix = "_T"

 val untyped_helper_suffix = "_U"

 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"

 val predicator_name = "pp"

 val app_op_name = "aa"

 val type_guard_name = "gg"

 val type_tag_name = "tt"

 val prefixed_predicator_name = const_prefix ^ predicator_name

 val prefixed_app_op_name = const_prefix ^ app_op_name

 val prefixed_type_tag_name = const_prefix ^ type_tag_name

 (* Freshness almost guaranteed! *)

 val atp_weak_prefix = "ATP:"

 (*Escaping of special characters.

   Alphanumeric characters are left unchanged.

   The character _ goes to __

   Characters in the range ASCII space to / go to _A to _P, respectively.

   Other characters go to _nnn where nnn is the decimal ASCII code.*)

 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "

 fun stringN_of_int 0 _ = ""

   | stringN_of_int k n =

     stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)

 fun ascii_of_char c =

   if Char.isAlphaNum c then

     String.str c

   else if c = #"_" then

     "__"

   else if #" " <= c andalso c <= #"/" then

     "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))

   else

     (* fixed width, in case more digits follow *)

     "_" ^ stringN_of_int 3 (Char.ord c)

 val ascii_of = String.translate ascii_of_char

 (** Remove ASCII armoring from names in proof files **)

 (* We don't raise error exceptions because this code can run inside a worker

    thread. Also, the errors are impossible. *)

 val unascii_of =

   let

     fun un rcs [] = String.implode(rev rcs)

       | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)

         (* Three types of _ escapes: __, _A to _P, _nnn *)

       | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs

       | un rcs (#"_" :: c :: cs) =

         if #"A" <= c andalso c<= #"P" then

           (* translation of #" " to #"/" *)

           un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs

         else

           let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in

             case Int.fromString (String.implode digits) of

               SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))

             | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)

           end

       | un rcs (c :: cs) = un (c :: rcs) cs

   in un [] o String.explode end

 (* If string s has the prefix s1, return the result of deleting it,

    un-ASCII'd. *)

 fun strip_prefix_and_unascii s1 s =

   if String.isPrefix s1 s then

     SOME (unascii_of (String.extract (s, size s1, NONE)))

   else

     NONE

 val proxy_table =

   [("c_False", (@{const_name False}, (@{thm fFalse_def},

        ("fFalse", @{const_name ATP.fFalse})))),

    ("c_True", (@{const_name True}, (@{thm fTrue_def},

        ("fTrue", @{const_name ATP.fTrue})))),

    ("c_Not", (@{const_name Not}, (@{thm fNot_def},

        ("fNot", @{const_name ATP.fNot})))),

    ("c_conj", (@{const_name conj}, (@{thm fconj_def},

        ("fconj", @{const_name ATP.fconj})))),

    ("c_disj", (@{const_name disj}, (@{thm fdisj_def},

        ("fdisj", @{const_name ATP.fdisj})))),

    ("c_implies", (@{const_name implies}, (@{thm fimplies_def},

        ("fimplies", @{const_name ATP.fimplies})))),

    ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},

        ("fequal", @{const_name ATP.fequal})))),

    ("c_All", (@{const_name All}, (@{thm fAll_def},

        ("fAll", @{const_name ATP.fAll})))),

    ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},

        ("fEx", @{const_name ATP.fEx}))))]

 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)

 (* Readable names for the more common symbolic functions. Do not mess with the

    table unless you know what you are doing. *)

 val const_trans_table =

   [(@{type_name Product_Type.prod}, "prod"),

    (@{type_name Sum_Type.sum}, "sum"),

    (@{const_name False}, "False"),

    (@{const_name True}, "True"),

    (@{const_name Not}, "Not"),

    (@{const_name conj}, "conj"),

    (@{const_name disj}, "disj"),

    (@{const_name implies}, "implies"),

    (@{const_name HOL.eq}, "equal"),

    (@{const_name All}, "All"),

    (@{const_name Ex}, "Ex"),

    (@{const_name If}, "If"),

    (@{const_name Set.member}, "member"),

    (@{const_name Meson.COMBI}, "COMBI"),

    (@{const_name Meson.COMBK}, "COMBK"),

    (@{const_name Meson.COMBB}, "COMBB"),

    (@{const_name Meson.COMBC}, "COMBC"),

    (@{const_name Meson.COMBS}, "COMBS")]

   |> Symtab.make

   |> fold (Symtab.update o swap o snd o snd o snd) proxy_table

 (* Invert the table of translations between Isabelle and ATPs. *)

 val const_trans_table_inv =

   const_trans_table |> Symtab.dest |> map swap |> Symtab.make

 val const_trans_table_unprox =

   Symtab.empty

   |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table

 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)

 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)

 fun lookup_const c =

   case Symtab.lookup const_trans_table c of

     SOME c' => c'

   | NONE => ascii_of c

 fun ascii_of_indexname (v, 0) = ascii_of v

   | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i

 fun make_bound_var x = bound_var_prefix ^ ascii_of x

 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x

 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x

 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v

 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x

 fun make_schematic_type_var (x, i) =

   tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))

 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))

 (* "HOL.eq" and Choice are mapped to the ATP's equivalents *)

 local

   val choice_const = (fst o dest_Const o HOLogic.choice_const) Term.dummyT

   fun default c = const_prefix ^ lookup_const c

 in

   fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal

     | make_fixed_const (SOME (THF0 THF_With_Choice)) c =

         if c = choice_const then tptp_choice else default c

     | make_fixed_const _ c = default c

 end

 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c

 fun make_type_class clas = class_prefix ^ ascii_of clas

 fun new_skolem_var_name_from_const s =

   let val ss = s |> space_explode Long_Name.separator in

     nth ss (length ss - 2)

   end

 (* These are either simplified away by "Meson.presimplify" (most of the time) or

    handled specially via "fFalse", "fTrue", ..., "fequal". *)

 val atp_irrelevant_consts =

   [@{const_name False}, @{const_name True}, @{const_name Not},

    @{const_name conj}, @{const_name disj}, @{const_name implies},

    @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]

 val atp_monomorph_bad_consts =

   atp_irrelevant_consts @

   (* These are ignored anyway by the relevance filter (unless they appear in

      higher-order places) but not by the monomorphizer. *)

   [@{const_name all}, @{const_name "==>"}, @{const_name "=="},

    @{const_name Trueprop}, @{const_name All}, @{const_name Ex},

    @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]

 fun add_schematic_const (x as (_, T)) =

   Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x

 val add_schematic_consts_of =

   Term.fold_aterms (fn Const (x as (s, _)) =>

                        not (member (op =) atp_monomorph_bad_consts s)

                        ? add_schematic_const x

                       | _ => I)

 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty

 (** Definitions and functions for FOL clauses and formulas for TPTP **)

 (* The first component is the type class; the second is a "TVar" or "TFree". *)

 datatype type_literal =

   TyLitVar of name * name |

   TyLitFree of name * name

 (** Isabelle arities **)

 datatype arity_literal =

   AryLitConst of name * name * name list |

   AryLitVar of name * name

 val type_class = the_single @{sort type}

 fun add_packed_sort tvar =

   fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))

 type arity_clause =

   {name : string,

    prem_lits : arity_literal list,

    concl_lits : arity_literal}

 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)

 fun make_axiom_arity_clause (tcons, name, (cls, args)) =

   let

     val tvars = map (prefix tvar_prefix o string_of_int) (1 upto length args)

     val tvars_srts = ListPair.zip (tvars, args)

   in

     {name = name,

      prem_lits =

        [] |> fold (uncurry add_packed_sort) tvars_srts |> map AryLitVar,

      concl_lits =

        AryLitConst (`make_type_class cls, `make_fixed_type_const tcons,

                     tvars ~~ tvars)}

   end

 fun arity_clause _ _ (_, []) = []

   | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)

     arity_clause seen n (tcons, ars)

   | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =

     if member (op =) seen class then

       (* multiple arities for the same (tycon, class) pair *)

       make_axiom_arity_clause (tcons,

           lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,

           ar) ::

       arity_clause seen (n + 1) (tcons, ars)

     else

       make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^

                                ascii_of class, ar) ::

       arity_clause (class :: seen) n (tcons, ars)

 fun multi_arity_clause [] = []

   | multi_arity_clause ((tcons, ars) :: tc_arlists) =

       arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists

 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in

    theory thy provided its arguments have the corresponding sorts. *)

 fun type_class_pairs thy tycons classes =

   let

     val alg = Sign.classes_of thy

     fun domain_sorts tycon = Sorts.mg_domain alg tycon o single

     fun add_class tycon class =

       cons (class, domain_sorts tycon class)

       handle Sorts.CLASS_ERROR _ => I

     fun try_classes tycon = (tycon, fold (add_class tycon) classes [])

   in map try_classes tycons end

 (*Proving one (tycon, class) membership may require proving others, so iterate.*)

 fun iter_type_class_pairs _ _ [] = ([], [])

   | iter_type_class_pairs thy tycons classes =

       let

         fun maybe_insert_class s =

           (s <> type_class andalso not (member (op =) classes s))

           ? insert (op =) s

         val cpairs = type_class_pairs thy tycons classes

         val newclasses =

           [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs

         val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses

       in (classes' @ classes, union (op =) cpairs' cpairs) end

 fun make_arity_clauses thy tycons =

   iter_type_class_pairs thy tycons ##> multi_arity_clause

 (** Isabelle class relations **)

 type class_rel_clause =

   {name : string,

    subclass : name,

    superclass : name}

 (* Generate all pairs (sub, super) such that sub is a proper subclass of super

    in theory "thy". *)

 fun class_pairs _ [] _ = []

   | class_pairs thy subs supers =

       let

         val class_less = Sorts.class_less (Sign.classes_of thy)

         fun add_super sub super = class_less (sub, super) ? cons (sub, super)

         fun add_supers sub = fold (add_super sub) supers

       in fold add_supers subs [] end

 fun make_class_rel_clause (sub, super) =

   {name = sub ^ "_" ^ super, subclass = `make_type_class sub,

    superclass = `make_type_class super}

 fun make_class_rel_clauses thy subs supers =

   map make_class_rel_clause (class_pairs thy subs supers)

 (* intermediate terms *)

 datatype iterm =

   IConst of name * typ * typ list |

   IVar of name * typ |

   IApp of iterm * iterm |

   IAbs of (name * typ) * iterm

 fun ityp_of (IConst (_, T, _)) = T

   | ityp_of (IVar (_, T)) = T

   | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))

   | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm

 (*gets the head of a combinator application, along with the list of arguments*)

 fun strip_iterm_comb u =

   let

     fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)

       | stripc x = x

   in stripc (u, []) end

 fun atyps_of T = fold_atyps (insert (op =)) T []

 fun new_skolem_const_name s num_T_args =

   [new_skolem_const_prefix, s, string_of_int num_T_args]

   |> space_implode Long_Name.separator

 fun robust_const_type thy s =

   if s = app_op_name then Logic.varifyT_global @{typ "('a => 'b) => 'a => 'b"}

   else s |> Sign.the_const_type thy

 (* This function only makes sense if "T" is as general as possible. *)

 fun robust_const_typargs thy (s, T) =

   (s, T) |> Sign.const_typargs thy

   handle TYPE _ => [] |> Term.add_tvarsT T |> rev |> map TVar

 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.

    Also accumulates sort infomation. *)

 fun iterm_from_term thy format bs (P $ Q) =

     let

       val (P', P_atomics_Ts) = iterm_from_term thy format bs P

       val (Q', Q_atomics_Ts) = iterm_from_term thy format bs Q

     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end

   | iterm_from_term thy format _ (Const (c, T)) =

     (IConst (`(make_fixed_const (SOME format)) c, T,

              robust_const_typargs thy (c, T)),

      atyps_of T)

   | iterm_from_term _ _ _ (Free (s, T)) =

     (IConst (`make_fixed_var s, T,

              if String.isPrefix polymorphic_free_prefix s then [T] else []),

      atyps_of T)

   | iterm_from_term _ format _ (Var (v as (s, _), T)) =

     (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then

        let

          val Ts = T |> strip_type |> swap |> op ::

          val s' = new_skolem_const_name s (length Ts)

        in IConst (`(make_fixed_const (SOME format)) s', T, Ts) end

      else

        IVar ((make_schematic_var v, s), T), atyps_of T)

   | iterm_from_term _ _ bs (Bound j) =

     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))

   | iterm_from_term thy format bs (Abs (s, T, t)) =

     let

       fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string

       val s = vary s

       val name = `make_bound_var s

       val (tm, atomic_Ts) = iterm_from_term thy format ((s, (name, T)) :: bs) t

     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end

 datatype locality =

   General | Helper | Induction | Extensionality | Intro | Elim | Simp |

   Local | Assum | Chained

 datatype order = First_Order | Higher_Order

 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic

 datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound

 datatype type_level =

   All_Types |

   Noninf_Nonmono_Types of soundness |

   Fin_Nonmono_Types |

   Const_Arg_Types |

   No_Types

 datatype type_uniformity = Uniform | Nonuniform

 datatype type_enc =

   Simple_Types of order * polymorphism * type_level |

   Guards of polymorphism * type_level * type_uniformity |

   Tags of polymorphism * type_level * type_uniformity

 fun try_unsuffixes ss s =

   fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE

 fun type_enc_from_string soundness s =

   (case try (unprefix "poly_") s of

      SOME s => (SOME Polymorphic, s)

    | NONE =>

      case try (unprefix "raw_mono_") s of

        SOME s => (SOME Raw_Monomorphic, s)

      | NONE =>

        case try (unprefix "mono_") s of

          SOME s => (SOME Mangled_Monomorphic, s)

        | NONE => (NONE, s))

   ||> (fn s =>

           (* "_query" and "_bang" are for the ASCII-challenged Metis and

              Mirabelle. *)

           case try_unsuffixes ["?", "_query"] s of

             SOME s => (Noninf_Nonmono_Types soundness, s)

           | NONE =>

             case try_unsuffixes ["!", "_bang"] s of

               SOME s => (Fin_Nonmono_Types, s)

             | NONE => (All_Types, s))

   ||> apsnd (fn s =>

                 case try (unsuffix "_uniform") s of

                   SOME s => (Uniform, s)

                 | NONE => (Nonuniform, s))

   |> (fn (poly, (level, (uniformity, core))) =>

          case (core, (poly, level, uniformity)) of

            ("simple", (SOME poly, _, Nonuniform)) =>

            (case poly of

               Raw_Monomorphic => raise Same.SAME

             | _ => Simple_Types (First_Order, poly, level))

          | ("simple_higher", (SOME poly, _, Nonuniform)) =>

            (case (poly, level) of

               (Raw_Monomorphic, _) => raise Same.SAME

             | (_, Noninf_Nonmono_Types _) => raise Same.SAME

             | _ => Simple_Types (Higher_Order, poly, level))

          | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)

          | ("tags", (SOME Polymorphic, _, _)) =>

            Tags (Polymorphic, level, uniformity)

          | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)

          | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>

            Guards (poly, Const_Arg_Types, Nonuniform)

          | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>

            Guards (Polymorphic, No_Types, Nonuniform)

          | _ => raise Same.SAME)

   handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")

 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _, _)) = true

   | is_type_enc_higher_order _ = false

 fun polymorphism_of_type_enc (Simple_Types (_, poly, _)) = poly

   | polymorphism_of_type_enc (Guards (poly, _, _)) = poly

   | polymorphism_of_type_enc (Tags (poly, _, _)) = poly

 fun level_of_type_enc (Simple_Types (_, _, level)) = level

   | level_of_type_enc (Guards (_, level, _)) = level

   | level_of_type_enc (Tags (_, level, _)) = level

 fun uniformity_of_type_enc (Simple_Types _) = Uniform

   | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity

   | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity

 fun is_type_level_quasi_sound All_Types = true

   | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true

   | is_type_level_quasi_sound _ = false

 val is_type_enc_quasi_sound =

   is_type_level_quasi_sound o level_of_type_enc

 fun is_type_level_fairly_sound level =

   is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types

 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc

 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true

   | is_type_level_monotonicity_based Fin_Nonmono_Types = true

   | is_type_level_monotonicity_based _ = false

 fun adjust_type_enc (THF0 _) (Simple_Types (order, Polymorphic, level)) =

     Simple_Types (order, Mangled_Monomorphic, level)

   | adjust_type_enc (THF0 _) type_enc = type_enc

   | adjust_type_enc (TFF (TFF_Monomorphic, _)) (Simple_Types (_, _, level)) =

     Simple_Types (First_Order, Mangled_Monomorphic, level)

   | adjust_type_enc (TFF (_, _)) (Simple_Types (_, poly, level)) =

     Simple_Types (First_Order, poly, level)

   | adjust_type_enc format (Simple_Types (_, poly, level)) =

     adjust_type_enc format (Guards (poly, level, Uniform))

   | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =

     (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff

   | adjust_type_enc _ type_enc = type_enc

 fun lift_lambdas ctxt type_enc =

   map (close_form o Envir.eta_contract) #> rpair ctxt

   #-> Lambda_Lifting.lift_lambdas

           (if polymorphism_of_type_enc type_enc = Polymorphic then

              SOME polymorphic_free_prefix

            else

              NONE)

           Lambda_Lifting.is_quantifier

   #> fst

 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =

     intentionalize_def t

   | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =

     let

       fun lam T t = Abs (Name.uu, T, t)

       val (head, args) = strip_comb t ||> rev

       val head_T = fastype_of head

       val n = length args

       val arg_Ts = head_T |> binder_types |> take n |> rev

       val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))

     in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end

   | intentionalize_def t = t

 type translated_formula =

   {name : string,

    locality : locality,

    kind : formula_kind,

    iformula : (name, typ, iterm) formula,

    atomic_types : typ list}

 fun update_iformula f ({name, locality, kind, iformula, atomic_types}

                        : translated_formula) =

   {name = name, locality = locality, kind = kind, iformula = f iformula,

    atomic_types = atomic_types} : translated_formula

 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula

 fun insert_type ctxt get_T x xs =

   let val T = get_T x in

     if exists (type_instance ctxt T o get_T) xs then xs

     else x :: filter_out (type_generalization ctxt T o get_T) xs

   end

 (* The Booleans indicate whether all type arguments should be kept. *)

 datatype type_arg_policy =

   Explicit_Type_Args of bool |

   Mangled_Type_Args of bool |

   No_Type_Args

 fun should_drop_arg_type_args (Simple_Types _) = false

   | should_drop_arg_type_args type_enc =

     level_of_type_enc type_enc = All_Types andalso

     uniformity_of_type_enc type_enc = Uniform

 fun type_arg_policy type_enc s =

   if s = type_tag_name then

     (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then

        Mangled_Type_Args

      else

        Explicit_Type_Args) false

   else case type_enc of

     Tags (_, All_Types, Uniform) => No_Type_Args

   | _ =>

     let val level = level_of_type_enc type_enc in

       if level = No_Types orelse s = @{const_name HOL.eq} orelse

          (s = app_op_name andalso level = Const_Arg_Types) then

         No_Type_Args

       else

         should_drop_arg_type_args type_enc

         |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then

               Mangled_Type_Args

             else

               Explicit_Type_Args)

     end

 (* Make literals for sorted type variables. *)

 fun generic_add_sorts_on_type (_, []) = I

   | generic_add_sorts_on_type ((x, i), s :: ss) =

     generic_add_sorts_on_type ((x, i), ss)

     #> (if s = the_single @{sort HOL.type} then

           I

         else if i = ~1 then

           insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))

         else

           insert (op =) (TyLitVar (`make_type_class s,

                                    (make_schematic_type_var (x, i), x))))

 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)

   | add_sorts_on_tfree _ = I

 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z

   | add_sorts_on_tvar _ = I

 fun type_literals_for_types type_enc add_sorts_on_typ Ts =

   [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts

 fun mk_aconns c phis =

   let val (phis', phi') = split_last phis in

     fold_rev (mk_aconn c) phis' phi'

   end

 fun mk_ahorn [] phi = phi

   | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])

 fun mk_aquant _ [] phi = phi

   | mk_aquant q xs (phi as AQuant (q', xs', phi')) =

     if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)

   | mk_aquant q xs phi = AQuant (q, xs, phi)

 fun close_universally atom_vars phi =

   let

     fun formula_vars bounds (AQuant (_, xs, phi)) =

         formula_vars (map fst xs @ bounds) phi

       | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis

       | formula_vars bounds (AAtom tm) =

         union (op =) (atom_vars tm []

                       |> filter_out (member (op =) bounds o fst))

   in mk_aquant AForall (formula_vars [] phi []) phi end

 fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]

   | iterm_vars (IConst _) = I

   | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)

   | iterm_vars (IAbs (_, tm)) = iterm_vars tm

 fun close_iformula_universally phi = close_universally iterm_vars phi

 fun term_vars type_enc =

   let

     fun vars bounds (ATerm (name as (s, _), tms)) =

         (if is_tptp_variable s andalso not (member (op =) bounds name) then

            (case type_enc of

               Simple_Types (_, Polymorphic, _) =>

               if String.isPrefix tvar_prefix s then SOME atype_of_types

               else NONE

             | _ => NONE)

            |> pair name |> insert (op =)

          else

            I)

         #> fold (vars bounds) tms

       | vars bounds (AAbs ((name, _), tm)) = vars (name :: bounds) tm

   in vars end

 fun close_formula_universally type_enc =

   close_universally (term_vars type_enc [])

 val fused_infinite_type_name = @{type_name ind} (* any infinite type *)

 val fused_infinite_type = Type (fused_infinite_type_name, [])

 fun tvar_name (x as (s, _)) = (make_schematic_type_var x, s)

 fun ho_term_from_typ format type_enc =

   let

     fun term (Type (s, Ts)) =

       ATerm (case (is_type_enc_higher_order type_enc, s) of

                (true, @{type_name bool}) => `I tptp_bool_type

              | (true, @{type_name fun}) => `I tptp_fun_type

              | _ => if s = fused_infinite_type_name andalso

                        is_format_typed format then

                       `I tptp_individual_type

                     else

                       `make_fixed_type_const s,

              map term Ts)

     | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])

     | term (TVar (x, _)) = ATerm (tvar_name x, [])

   in term end

 fun ho_term_for_type_arg format type_enc T =

   if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)

 (* This shouldn't clash with anything else. *)

 val mangled_type_sep = "\000"

 fun generic_mangled_type_name f (ATerm (name, [])) = f name

   | generic_mangled_type_name f (ATerm (name, tys)) =

     f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)

     ^ ")"

   | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"

 fun mangled_type format type_enc =

   generic_mangled_type_name fst o ho_term_from_typ format type_enc

 fun make_simple_type s =

   if s = tptp_bool_type orelse s = tptp_fun_type orelse

      s = tptp_individual_type then

     s

   else

     simple_type_prefix ^ ascii_of s

 fun ho_type_from_ho_term type_enc pred_sym ary =

   let

     fun to_mangled_atype ty =

       AType ((make_simple_type (generic_mangled_type_name fst ty),

               generic_mangled_type_name snd ty), [])

     fun to_poly_atype (ATerm (name, tys)) = AType (name, map to_poly_atype tys)

       | to_poly_atype _ = raise Fail "unexpected type abstraction"

     val to_atype =

       if polymorphism_of_type_enc type_enc = Polymorphic then to_poly_atype

       else to_mangled_atype

     fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))

     fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty

       | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys

       | to_fo _ _ = raise Fail "unexpected type abstraction"

     fun to_ho (ty as ATerm ((s, _), tys)) =

         if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty

       | to_ho _ = raise Fail "unexpected type abstraction"

   in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end

 fun ho_type_from_typ format type_enc pred_sym ary =

   ho_type_from_ho_term type_enc pred_sym ary

   o ho_term_from_typ format type_enc

 fun mangled_const_name format type_enc T_args (s, s') =

   let

     val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)

     fun type_suffix f g =

       fold_rev (curry (op ^) o g o prefix mangled_type_sep

                 o generic_mangled_type_name f) ty_args ""

   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end

 val parse_mangled_ident =

   Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode

 fun parse_mangled_type x =

   (parse_mangled_ident

    -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")

                     [] >> ATerm) x

 and parse_mangled_types x =

   (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x

 fun unmangled_type s =

   s |> suffix ")" |> raw_explode

     |> Scan.finite Symbol.stopper

            (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^

                                                 quote s)) parse_mangled_type))

     |> fst

 val unmangled_const_name = space_explode mangled_type_sep #> hd

 fun unmangled_const s =

   let val ss = space_explode mangled_type_sep s in

     (hd ss, map unmangled_type (tl ss))

   end

 fun introduce_proxies type_enc =

   let

     fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])

       | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))

                        _ =

         (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser

            limitation. This works in conjuction with special code in

            "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever

            possible. *)

         IAbs ((`I "P", p_T),

               IApp (IConst (`I ho_quant, T, []),

                     IAbs ((`I "X", x_T),

                           IApp (IConst (`I "P", p_T, []),

                                 IConst (`I "X", x_T, [])))))

       | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"

     fun intro top_level args (IApp (tm1, tm2)) =

         IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)

       | intro top_level args (IConst (name as (s, _), T, T_args)) =

         (case proxify_const s of

            SOME proxy_base =>

            if top_level orelse is_type_enc_higher_order type_enc then

              case (top_level, s) of

                (_, "c_False") => IConst (`I tptp_false, T, [])

              | (_, "c_True") => IConst (`I tptp_true, T, [])

              | (false, "c_Not") => IConst (`I tptp_not, T, [])

              | (false, "c_conj") => IConst (`I tptp_and, T, [])

              | (false, "c_disj") => IConst (`I tptp_or, T, [])

              | (false, "c_implies") => IConst (`I tptp_implies, T, [])

              | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args

              | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args

              | (false, s) =>

                if is_tptp_equal s andalso length args = 2 then

                  IConst (`I tptp_equal, T, [])

                else

                  (* Use a proxy even for partially applied THF0 equality,

                     because the LEO-II and Satallax parsers complain about not

                     being able to infer the type of "=". *)

                  IConst (proxy_base |>> prefix const_prefix, T, T_args)

              | _ => IConst (name, T, [])

            else

              IConst (proxy_base |>> prefix const_prefix, T, T_args)

           | NONE => if s = tptp_choice then

                       tweak_ho_quant tptp_choice T args

                     else

                       IConst (name, T, T_args))

       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)

       | intro _ _ tm = tm

   in intro true [] end

 fun iformula_from_prop thy format type_enc eq_as_iff =

   let

     fun do_term bs t atomic_types =

       iterm_from_term thy format bs (Envir.eta_contract t)

       |>> (introduce_proxies type_enc #> AAtom)

       ||> union (op =) atomic_types

     fun do_quant bs q pos s T t' =

       let

         val s = singleton (Name.variant_list (map fst bs)) s

         val universal = Option.map (q = AExists ? not) pos

         val name =

           s |> `(case universal of

                    SOME true => make_all_bound_var

                  | SOME false => make_exist_bound_var

                  | NONE => make_bound_var)

       in

         do_formula ((s, (name, T)) :: bs) pos t'

         #>> mk_aquant q [(name, SOME T)]

       end

     and do_conn bs c pos1 t1 pos2 t2 =

       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)

     and do_formula bs pos t =

       case t of

         @{const Trueprop} $ t1 => do_formula bs pos t1

       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot

       | Const (@{const_name All}, _) $ Abs (s, T, t') =>

         do_quant bs AForall pos s T t'

       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>

         do_quant bs AExists pos s T t'

       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2

       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2

       | @{const HOL.implies} $ t1 $ t2 =>

         do_conn bs AImplies (Option.map not pos) t1 pos t2

       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>

         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t

       | _ => do_term bs t

   in do_formula [] end

 fun presimplify_term _ [] t = t

   | presimplify_term ctxt presimp_consts t =

     t |> exists_Const (member (op =) presimp_consts o fst) t

          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)

             #> Meson.presimplify ctxt

             #> prop_of)

 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j

 fun conceal_bounds Ts t =

   subst_bounds (map (Free o apfst concealed_bound_name)

                     (0 upto length Ts - 1 ~~ Ts), t)

 fun reveal_bounds Ts =

   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))

                     (0 upto length Ts - 1 ~~ Ts))

 fun is_fun_equality (@{const_name HOL.eq},

                      Type (_, [Type (@{type_name fun}, _), _])) = true

   | is_fun_equality _ = false

 fun extensionalize_term ctxt t =

   if exists_Const is_fun_equality t then

     let val thy = Proof_Context.theory_of ctxt in

       t |> cterm_of thy |> Meson.extensionalize_conv ctxt

         |> prop_of |> Logic.dest_equals |> snd

     end

   else

     t

 fun simple_translate_lambdas do_lambdas ctxt t =

   let val thy = Proof_Context.theory_of ctxt in

     if Meson.is_fol_term thy t then

       t

     else

       let

         fun aux Ts t =

           case t of

             @{const Not} $ t1 => @{const Not} $ aux Ts t1

           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>

             t0 $ Abs (s, T, aux (T :: Ts) t')

           | (t0 as Const (@{const_name All}, _)) $ t1 =>

             aux Ts (t0 $ eta_expand Ts t1 1)

           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>

             t0 $ Abs (s, T, aux (T :: Ts) t')

           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>

             aux Ts (t0 $ eta_expand Ts t1 1)

           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2

           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2

           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2

           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))

               $ t1 $ t2 =>

             t0 $ aux Ts t1 $ aux Ts t2

           | _ =>

             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t

             else t |> Envir.eta_contract |> do_lambdas ctxt Ts

         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single

       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end

   end

 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =

     do_cheaply_conceal_lambdas Ts t1

     $ do_cheaply_conceal_lambdas Ts t2

   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =

     Free (polymorphic_free_prefix ^ serial_string (),

           T --> fastype_of1 (T :: Ts, t))

   | do_cheaply_conceal_lambdas _ t = t

 fun do_introduce_combinators ctxt Ts t =

   let val thy = Proof_Context.theory_of ctxt in

     t |> conceal_bounds Ts

       |> cterm_of thy

       |> Meson_Clausify.introduce_combinators_in_cterm

       |> prop_of |> Logic.dest_equals |> snd

       |> reveal_bounds Ts

   end

   (* A type variable of sort "{}" will make abstraction fail. *)

   handle THM _ => t |> do_cheaply_conceal_lambdas Ts

 val introduce_combinators = simple_translate_lambdas do_introduce_combinators

 fun preprocess_abstractions_in_terms trans_lambdas facts =

   let

     val (facts, lambda_ts) =

       facts |> map (snd o snd) |> trans_lambdas

             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts

     val lambda_facts =

       map2 (fn t => fn j =>

                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))

            lambda_ts (1 upto length lambda_ts)

   in (facts, lambda_facts) end

 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the

    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)

 fun freeze_term t =

   let

     fun aux (t $ u) = aux t $ aux u

       | aux (Abs (s, T, t)) = Abs (s, T, aux t)

       | aux (Var ((s, i), T)) =

         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)

       | aux t = t

   in t |> exists_subterm is_Var t ? aux end

 fun presimp_prop ctxt presimp_consts t =

   let

     val thy = Proof_Context.theory_of ctxt

     val t = t |> Envir.beta_eta_contract

               |> transform_elim_prop

               |> Object_Logic.atomize_term thy

     val need_trueprop = (fastype_of t = @{typ bool})

   in

     t |> need_trueprop ? HOLogic.mk_Trueprop

       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []

       |> extensionalize_term ctxt

       |> presimplify_term ctxt presimp_consts

       |> perhaps (try (HOLogic.dest_Trueprop))

   end

 (* making fact and conjecture formulas *)

 fun make_formula thy format type_enc eq_as_iff name loc kind t =

   let

     val (iformula, atomic_types) =

       iformula_from_prop thy format type_enc eq_as_iff (SOME (kind <> Conjecture)) t []

   in

     {name = name, locality = loc, kind = kind, iformula = iformula,

      atomic_types = atomic_types}

   end

 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =

   let val thy = Proof_Context.theory_of ctxt in

     case t |> make_formula thy format type_enc (eq_as_iff andalso format <> CNF)

                            name loc Axiom of

       formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>

       if s = tptp_true then NONE else SOME formula

     | formula => SOME formula

   end

 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t

   | s_not_trueprop t = s_not t

 fun make_conjecture thy format type_enc =

   map (fn ((name, loc), (kind, t)) =>

           t |> kind = Conjecture ? s_not_trueprop

             |> make_formula thy format type_enc (format <> CNF) name loc kind)

 (** Finite and infinite type inference **)

 type monotonicity_info =

   {maybe_finite_Ts : typ list,

    surely_finite_Ts : typ list,

    maybe_infinite_Ts : typ list,

    surely_infinite_Ts : typ list,

    maybe_nonmono_Ts : typ list}

 (* These types witness that the type classes they belong to allow infinite

    models and hence that any types with these type classes is monotonic. *)

 val known_infinite_types =

   [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]

 fun is_type_kind_of_surely_infinite ctxt soundness cached_Ts T =

   soundness <> Sound andalso

   is_type_surely_infinite ctxt (soundness <> Unsound) cached_Ts T

 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are

    dangerous because their "exhaust" properties can easily lead to unsound ATP

    proofs. On the other hand, all HOL infinite types can be given the same

    models in first-order logic (via Löwenheim-Skolem). *)

 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true

   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,

                              maybe_nonmono_Ts, ...}

                        (Noninf_Nonmono_Types soundness) T =

     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso

     not (exists (type_instance ctxt T) surely_infinite_Ts orelse

          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso

           is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts T))

   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,

                              maybe_nonmono_Ts, ...}

                        Fin_Nonmono_Types T =

     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso

     (exists (type_generalization ctxt T) surely_finite_Ts orelse

      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso

       is_type_surely_finite ctxt T))

   | should_encode_type _ _ _ _ = false

 fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var

                       T =

     (uniformity = Uniform orelse should_guard_var ()) andalso

     should_encode_type ctxt mono level T

   | should_guard_type _ _ _ _ _ = false

 fun is_maybe_universal_var (IConst ((s, _), _, _)) =

     String.isPrefix bound_var_prefix s orelse

     String.isPrefix all_bound_var_prefix s

   | is_maybe_universal_var (IVar _) = true

   | is_maybe_universal_var _ = false

 datatype tag_site =

   Top_Level of bool option |

   Eq_Arg of bool option |

   Elsewhere

 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false

   | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =

     (case uniformity of

        Uniform => should_encode_type ctxt mono level T

      | Nonuniform =>

        case (site, is_maybe_universal_var u) of

          (Eq_Arg _, true) => should_encode_type ctxt mono level T

        | _ => false)

   | should_tag_with_type _ _ _ _ _ _ = false

 fun fused_type ctxt mono level =

   let

     val should_encode = should_encode_type ctxt mono level

     fun fuse 0 T = if should_encode T then T else fused_infinite_type

       | fuse ary (Type (@{type_name fun}, [T1, T2])) =

         fuse 0 T1 --> fuse (ary - 1) T2

       | fuse _ _ = raise Fail "expected function type"

   in fuse end

 (** predicators and application operators **)

 type sym_info =

   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}

 fun add_iterm_syms_to_table ctxt explicit_apply =

   let

     fun consider_var_arity const_T var_T max_ary =

       let

         fun iter ary T =

           if ary = max_ary orelse type_instance ctxt var_T T orelse

              type_instance ctxt T var_T then

             ary

           else

             iter (ary + 1) (range_type T)

       in iter 0 const_T end

     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =

       if explicit_apply = NONE andalso

          (can dest_funT T orelse T = @{typ bool}) then

         let

           val bool_vars' = bool_vars orelse body_type T = @{typ bool}

           fun repair_min_arity {pred_sym, min_ary, max_ary, types} =

             {pred_sym = pred_sym andalso not bool_vars',

              min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,

              max_ary = max_ary, types = types}

           val fun_var_Ts' =

             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T

         in

           if bool_vars' = bool_vars andalso

              pointer_eq (fun_var_Ts', fun_var_Ts) then

             accum

           else

             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)

         end

       else

         accum

     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =

       let val (head, args) = strip_iterm_comb tm in

         (case head of

            IConst ((s, _), T, _) =>

            if String.isPrefix bound_var_prefix s orelse

               String.isPrefix all_bound_var_prefix s then

              add_universal_var T accum

            else if String.isPrefix exist_bound_var_prefix s then

              accum

            else

              let val ary = length args in

                ((bool_vars, fun_var_Ts),

                 case Symtab.lookup sym_tab s of

                   SOME {pred_sym, min_ary, max_ary, types} =>

                   let

                     val pred_sym =

                       pred_sym andalso top_level andalso not bool_vars

                     val types' = types |> insert_type ctxt I T

                     val min_ary =

                       if is_some explicit_apply orelse

                          pointer_eq (types', types) then

                         min_ary

                       else

                         fold (consider_var_arity T) fun_var_Ts min_ary

                   in

                     Symtab.update (s, {pred_sym = pred_sym,

                                        min_ary = Int.min (ary, min_ary),

                                        max_ary = Int.max (ary, max_ary),

                                        types = types'})

                                   sym_tab

                   end

                 | NONE =>

                   let

                     val pred_sym = top_level andalso not bool_vars

                     val min_ary =

                       case explicit_apply of

                         SOME true => 0

                       | SOME false => ary

                       | NONE => fold (consider_var_arity T) fun_var_Ts ary

                   in

                     Symtab.update_new (s, {pred_sym = pred_sym,

                                            min_ary = min_ary, max_ary = ary,

                                            types = [T]})

                                       sym_tab

                   end)

              end

          | IVar (_, T) => add_universal_var T accum

          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm

          | _ => accum)

         |> fold (add false) args

       end

   in add true end

 fun add_fact_syms_to_table ctxt explicit_apply =

   K (add_iterm_syms_to_table ctxt explicit_apply)

   |> formula_fold NONE |> fact_lift

 val tvar_a_str = "'a"

 val tvar_a = TVar ((tvar_a_str, 0), HOLogic.typeS)

 val tvar_a_name = (make_schematic_type_var (tvar_a_str, 0), tvar_a_str)

 val itself_name = `make_fixed_type_const @{type_name itself}

 val TYPE_name = `(make_fixed_const NONE) @{const_name TYPE}

 val tvar_a_atype = AType (tvar_a_name, [])

 val a_itself_atype = AType (itself_name, [tvar_a_atype])

 val default_sym_tab_entries : (string * sym_info) list =

   (prefixed_predicator_name,

    {pred_sym = true, min_ary = 1, max_ary = 1, types = []})

        (* FIXME: needed? *) ::

   (make_fixed_const NONE @{const_name undefined},

    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::

   ([tptp_false, tptp_true]

    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @

   ([tptp_equal, tptp_old_equal]

    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))

 fun sym_table_for_facts ctxt explicit_apply facts =

   ((false, []), Symtab.empty)

   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd

   |> fold Symtab.update default_sym_tab_entries

 fun min_arity_of sym_tab s =

   case Symtab.lookup sym_tab s of

     SOME ({min_ary, ...} : sym_info) => min_ary

   | NONE =>

     case strip_prefix_and_unascii const_prefix s of

       SOME s =>

       let val s = s |> unmangled_const_name |> invert_const in

         if s = predicator_name then 1

         else if s = app_op_name then 2

         else if s = type_guard_name then 1

         else 0

       end

     | NONE => 0

 (* True if the constant ever appears outside of the top-level position in

    literals, or if it appears with different arities (e.g., because of different

    type instantiations). If false, the constant always receives all of its

    arguments and is used as a predicate. *)

 fun is_pred_sym sym_tab s =

   case Symtab.lookup sym_tab s of

     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>

     pred_sym andalso min_ary = max_ary

   | NONE => false

 val predicator_combconst =

   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])

 fun predicator tm = IApp (predicator_combconst, tm)

 fun introduce_predicators_in_iterm sym_tab tm =

   case strip_iterm_comb tm of

     (IConst ((s, _), _, _), _) =>

     if is_pred_sym sym_tab s then tm else predicator tm

   | _ => predicator tm

 fun list_app head args = fold (curry (IApp o swap)) args head

 val app_op = `(make_fixed_const NONE) app_op_name

 fun explicit_app arg head =

   let

     val head_T = ityp_of head

     val (arg_T, res_T) = dest_funT head_T

     val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])

   in list_app explicit_app [head, arg] end

 fun list_explicit_app head args = fold explicit_app args head

 fun introduce_explicit_apps_in_iterm sym_tab =

   let

     fun aux tm =

       case strip_iterm_comb tm of

         (head as IConst ((s, _), _, _), args) =>

         args |> map aux

              |> chop (min_arity_of sym_tab s)

              |>> list_app head

              |-> list_explicit_app

       | (head, args) => list_explicit_app head (map aux args)

   in aux end

 fun chop_fun 0 T = ([], T)

   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =

     chop_fun (n - 1) ran_T |>> cons dom_T

   | chop_fun _ _ = raise Fail "unexpected non-function"

 fun filter_type_args _ _ _ [] = []

   | filter_type_args thy s arity T_args =

     let val U = robust_const_type thy s in

       case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of

         [] => []

       | res_U_vars =>

         let val U_args = (s, U) |> Sign.const_typargs thy in

           U_args ~~ T_args

           |> map (fn (U, T) =>

                      if member (op =) res_U_vars (dest_TVar U) then T

                      else dummyT)

         end

     end

     handle TYPE _ => T_args

 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =

   let

     val thy = Proof_Context.theory_of ctxt

     fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)

       | aux arity (IConst (name as (s, _), T, T_args)) =

         (case strip_prefix_and_unascii const_prefix s of

            NONE =>

            (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice

                   then [] else T_args)

          | SOME s'' =>

            let

              val s'' = invert_const s''

              fun filtered_T_args false = T_args

                | filtered_T_args true = filter_type_args thy s'' arity T_args

            in

              case type_arg_policy type_enc s'' of

                Explicit_Type_Args drop_args =>

                (name, filtered_T_args drop_args)

              | Mangled_Type_Args drop_args =>

                (mangled_const_name format type_enc (filtered_T_args drop_args)

                                    name, [])

              | No_Type_Args => (name, [])

            end)

         |> (fn (name, T_args) => IConst (name, T, T_args))

       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)

       | aux _ tm = tm

   in aux 0 end

 fun repair_iterm ctxt format type_enc sym_tab =

   not (is_type_enc_higher_order type_enc)

   ? (introduce_explicit_apps_in_iterm sym_tab

      #> introduce_predicators_in_iterm sym_tab)

   #> enforce_type_arg_policy_in_iterm ctxt format type_enc

 fun repair_fact ctxt format type_enc sym_tab =

   update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))

 (** Helper facts **)

 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}

 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}

 (* The Boolean indicates that a fairly sound type encoding is needed. *)

 val helper_table =

   [(("COMBI", false), @{thms Meson.COMBI_def}),

    (("COMBK", false), @{thms Meson.COMBK_def}),

    (("COMBB", false), @{thms Meson.COMBB_def}),

    (("COMBC", false), @{thms Meson.COMBC_def}),

    (("COMBS", false), @{thms Meson.COMBS_def}),

    ((predicator_name, false), [not_ffalse, ftrue]),

    (("fFalse", false), [not_ffalse]),

    (("fFalse", true), @{thms True_or_False}),

    (("fTrue", false), [ftrue]),

    (("fTrue", true), @{thms True_or_False}),

    (("fNot", false),

     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]

            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),

    (("fconj", false),

     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"

         by (unfold fconj_def) fast+}),

    (("fdisj", false),

     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"

         by (unfold fdisj_def) fast+}),

    (("fimplies", false),

     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"

         by (unfold fimplies_def) fast+}),

    (("fequal", true),

     (* This is a lie: Higher-order equality doesn't need a sound type encoding.

        However, this is done so for backward compatibility: Including the

        equality helpers by default in Metis breaks a few existing proofs. *)

     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]

            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),

    (* Partial characterization of "fAll" and "fEx". A complete characterization

       would require the axiom of choice for replay with Metis. *)

    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),

    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),

    (("If", true), @{thms if_True if_False True_or_False})]

   |> map (apsnd (map zero_var_indexes))

 fun type_class_aterm type_enc class arg =

   case type_enc of

     Simple_Types (_, Polymorphic, _) =>

     if exploit_tff1_dummy_type_vars then ATerm (class, [arg])

     else ATerm (class, [arg, ATerm (TYPE_name, [arg])])

   | _ => ATerm (class, [arg])

 fun fo_literal_from_type_literal type_enc (TyLitVar (class, name)) =

     (true, type_class_aterm type_enc class (ATerm (name, [])))

   | fo_literal_from_type_literal type_enc (TyLitFree (class, name)) =

     (true, type_class_aterm type_enc class (ATerm (name, [])))

 (* FIXME: Really "true" in both cases? If so, merge "TyLitVar" and

    "TyLitFree". *)

 fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot

 fun bound_tvars type_enc Ts =

   mk_ahorn (map (formula_from_fo_literal

                  o fo_literal_from_type_literal type_enc)

                 (type_literals_for_types type_enc add_sorts_on_tvar Ts))

 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =

   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])

    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))

   |> bound_tvars type_enc atomic_Ts

   |> close_formula_universally type_enc

 val type_tag = `(make_fixed_const NONE) type_tag_name

 fun type_tag_idempotence_fact type_enc =

   let

     fun var s = ATerm (`I s, [])

     fun tag tm = ATerm (type_tag, [var "A", tm])

     val tagged_var = tag (var "X")

   in

     Formula (type_tag_idempotence_helper_name, Axiom,

              eq_formula type_enc [] false (tag tagged_var) tagged_var,

              isabelle_info simpN, NONE)

   end

 fun should_specialize_helper type_enc t =

   polymorphism_of_type_enc type_enc <> Polymorphic andalso

   level_of_type_enc type_enc <> No_Types andalso

   not (null (Term.hidden_polymorphism t))

 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =

   case strip_prefix_and_unascii const_prefix s of

     SOME mangled_s =>

     let

       val thy = Proof_Context.theory_of ctxt

       val unmangled_s = mangled_s |> unmangled_const_name

       fun dub needs_fairly_sound j k =

         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^

          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^

          (if needs_fairly_sound then typed_helper_suffix

           else untyped_helper_suffix),

          Helper)

       fun dub_and_inst needs_fairly_sound (th, j) =

         let val t = prop_of th in

           if should_specialize_helper type_enc t then

             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)

                 types

           else

             [t]

         end

         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1

       val make_facts = map_filter (make_fact ctxt format type_enc false)

       val fairly_sound = is_type_enc_fairly_sound type_enc

     in

       helper_table

       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>

                   if helper_s <> unmangled_s orelse

                      (needs_fairly_sound andalso not fairly_sound) then

                     []

                   else

                     ths ~~ (1 upto length ths)

                     |> maps (dub_and_inst needs_fairly_sound)

                     |> make_facts)

     end

   | NONE => []

 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =

   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab

                   []

 (***************************************************************)

 (* Type Classes Present in the Axiom or Conjecture Clauses     *)

 (***************************************************************)

 fun set_insert (x, s) = Symtab.update (x, ()) s

 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)

 (* Remove this trivial type class (FIXME: similar code elsewhere) *)

 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset

 fun classes_of_terms get_Ts =

   map (map snd o get_Ts)

   #> List.foldl add_classes Symtab.empty

   #> delete_type #> Symtab.keys

 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees

 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars

 fun fold_type_constrs f (Type (s, Ts)) x =

     fold (fold_type_constrs f) Ts (f (s, x))

   | fold_type_constrs _ _ x = x

 (* Type constructors used to instantiate overloaded constants are the only ones

    needed. *)

 fun add_type_constrs_in_term thy =

   let

     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I

       | add (t $ u) = add t #> add u

       | add (Const (x as (s, _))) =

         if String.isPrefix skolem_const_prefix s then I

         else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)

       | add (Free (s, T)) =

         if String.isPrefix polymorphic_free_prefix s then

           T |> fold_type_constrs set_insert

         else

           I

       | add (Abs (_, _, u)) = add u

       | add _ = I

   in add end

 fun type_constrs_of_terms thy ts =

   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)

 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc

                        hyp_ts concl_t facts =

   let

     val thy = Proof_Context.theory_of ctxt

     val presimp_consts = Meson.presimplified_consts ctxt

     val fact_ts = facts |> map snd

     (* Remove existing facts from the conjecture, as this can dramatically

        boost an ATP's performance (for some reason). *)

     val hyp_ts =

       hyp_ts

       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)

     val facts = facts |> map (apsnd (pair Axiom))

     val conjs =

       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]

       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)

     val ((conjs, facts), lambdas) =

       if preproc then

         conjs @ facts

         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))

         |> preprocess_abstractions_in_terms trans_lambdas

         |>> chop (length conjs)

         |>> apfst (map (apsnd (apsnd freeze_term)))

       else

         ((conjs, facts), [])

     val conjs = conjs |> make_conjecture thy format type_enc

     val (fact_names, facts) =

       facts

       |> map_filter (fn (name, (_, t)) =>

                         make_fact ctxt format type_enc true (name, t)

                         |> Option.map (pair name))

       |> ListPair.unzip

     val lambdas =

       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)

     val all_ts = concl_t :: hyp_ts @ fact_ts

     val subs = tfree_classes_of_terms all_ts

     val supers = tvar_classes_of_terms all_ts

     val tycons = type_constrs_of_terms thy all_ts

     val (supers, arity_clauses) =

       if level_of_type_enc type_enc = No_Types then ([], [])

       else make_arity_clauses thy tycons supers

     val class_rel_clauses = make_class_rel_clauses thy subs supers

   in

     (fact_names |> map single, union (op =) subs supers, conjs, facts @ lambdas,

      class_rel_clauses, arity_clauses)

   end

 val type_guard = `(make_fixed_const NONE) type_guard_name

 fun type_guard_iterm ctxt format type_enc T tm =

   IApp (IConst (type_guard, T --> @{typ bool}, [T])

         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)

 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum

   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =

     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))

   | is_var_positively_naked_in_term _ _ _ _ = true

 fun should_guard_var_in_formula pos phi (SOME true) name =

     formula_fold pos (is_var_positively_naked_in_term name) phi false

   | should_guard_var_in_formula _ _ _ _ = true

 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false

   | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =

     should_encode_type ctxt mono level T

   | should_generate_tag_bound_decl _ _ _ _ _ = false

 fun mk_aterm format type_enc name T_args args =

   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)

 fun tag_with_type ctxt format mono type_enc pos T tm =

   IConst (type_tag, T --> T, [T])

   |> enforce_type_arg_policy_in_iterm ctxt format type_enc

   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)

   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])

        | _ => raise Fail "unexpected lambda-abstraction")

 and ho_term_from_iterm ctxt format mono type_enc =

   let

     fun aux site u =

       let

         val (head, args) = strip_iterm_comb u

         val pos =

           case site of

             Top_Level pos => pos

           | Eq_Arg pos => pos

           | Elsewhere => NONE

         val t =

           case head of

             IConst (name as (s, _), _, T_args) =>

             let

               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere

             in

               mk_aterm format type_enc name T_args (map (aux arg_site) args)

             end

           | IVar (name, _) =>

             mk_aterm format type_enc name [] (map (aux Elsewhere) args)

           | IAbs ((name, T), tm) =>

             AAbs ((name, ho_type_from_typ format type_enc true 0 T),

                   aux Elsewhere tm)

           | IApp _ => raise Fail "impossible \"IApp\""

         val T = ityp_of u

       in

         t |> (if should_tag_with_type ctxt mono type_enc site u T then

                 tag_with_type ctxt format mono type_enc pos T

               else

                 I)

       end

   in aux end

 and formula_from_iformula ctxt format mono type_enc should_guard_var =

   let

     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level

     val do_bound_type =

       case type_enc of

         Simple_Types (_, _, level) => fused_type ctxt mono level 0

         #> ho_type_from_typ format type_enc false 0 #> SOME

       | _ => K NONE

     fun do_out_of_bound_type pos phi universal (name, T) =

       if should_guard_type ctxt mono type_enc

              (fn () => should_guard_var pos phi universal name) T then

         IVar (name, T)

         |> type_guard_iterm ctxt format type_enc T

         |> do_term pos |> AAtom |> SOME

       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then

         let

           val var = ATerm (name, [])

           val tagged_var = var |> tag_with_type ctxt format mono type_enc pos T

         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end

       else

         NONE

     fun do_formula pos (AQuant (q, xs, phi)) =

         let

           val phi = phi |> do_formula pos

           val universal = Option.map (q = AExists ? not) pos

         in

           AQuant (q, xs |> map (apsnd (fn NONE => NONE

                                         | SOME T => do_bound_type T)),

                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))

                       (map_filter

                            (fn (_, NONE) => NONE

                              | (s, SOME T) =>

                                do_out_of_bound_type pos phi universal (s, T))

                            xs)

                       phi)

         end

       | do_formula pos (AConn conn) = aconn_map pos do_formula conn

       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)

   in do_formula end

 (* Each fact is given a unique fact number to avoid name clashes (e.g., because

    of monomorphization). The TPTP explicitly forbids name clashes, and some of

    the remote provers might care. *)

 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc

                           (j, {name, locality, kind, iformula, atomic_types}) =

   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,

    iformula

    |> close_iformula_universally

    |> formula_from_iformula ctxt format mono type_enc

                             should_guard_var_in_formula

                             (if pos then SOME true else NONE)

    |> bound_tvars type_enc atomic_types

    |> close_formula_universally type_enc,

    NONE,

    case locality of

      Intro => isabelle_info introN

    | Elim => isabelle_info elimN

    | Simp => isabelle_info simpN

    | _ => NONE)

   |> Formula

 fun formula_line_for_class_rel_clause type_enc

         ({name, subclass, superclass, ...} : class_rel_clause) =

   let val ty_arg = ATerm (tvar_a_name, []) in

     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,

              AConn (AImplies,

                     [AAtom (type_class_aterm type_enc subclass ty_arg),

                      AAtom (type_class_aterm type_enc superclass ty_arg)])

              |> close_formula_universally type_enc, isabelle_info introN, NONE)

   end

 fun fo_literal_from_arity_literal type_enc (AryLitConst (class, t, args)) =

     (true, type_class_aterm type_enc class

                             (ATerm (t, map (fn arg => ATerm (arg, [])) args)))

   | fo_literal_from_arity_literal type_enc (AryLitVar (class, var)) =

     (false, type_class_aterm type_enc class (ATerm (var, [])))

 fun formula_line_for_arity_clause type_enc

         ({name, prem_lits, concl_lits, ...} : arity_clause) =

   Formula (arity_clause_prefix ^ name, Axiom,

            mk_ahorn (map (formula_from_fo_literal o apfst not

                           o fo_literal_from_arity_literal type_enc) prem_lits)

                     (formula_from_fo_literal

                          (fo_literal_from_arity_literal type_enc concl_lits))

            |> close_formula_universally type_enc, isabelle_info introN, NONE)

 fun formula_line_for_conjecture ctxt format mono type_enc

         ({name, kind, iformula, atomic_types, ...} : translated_formula) =

   Formula (conjecture_prefix ^ name, kind,

            formula_from_iformula ctxt format mono type_enc

                should_guard_var_in_formula (SOME false)

                (close_iformula_universally iformula)

            |> bound_tvars type_enc atomic_types

            |> close_formula_universally type_enc, NONE, NONE)

 fun free_type_literals type_enc ({atomic_types, ...} : translated_formula) =

   atomic_types |> type_literals_for_types type_enc add_sorts_on_tfree

                |> map (fo_literal_from_type_literal type_enc)

 fun formula_line_for_free_type j lit =

   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,

            formula_from_fo_literal lit, NONE, NONE)

 fun formula_lines_for_free_types type_enc facts =

   let

     val litss = map (free_type_literals type_enc) facts

     val lits = fold (union (op =)) litss []

   in map2 formula_line_for_free_type (0 upto length lits - 1) lits end

 (** Symbol declarations **)

 fun decl_line_for_class s =

   let val name as (s, _) = `make_type_class s in

     Decl (sym_decl_prefix ^ s, name,

           if exploit_tff1_dummy_type_vars then

             AFun (atype_of_types, bool_atype)

           else

             ATyAbs ([tvar_a_name], AFun (a_itself_atype, bool_atype)))

   end

 fun decl_lines_for_classes type_enc classes =

   case type_enc of

     Simple_Types (_, Polymorphic, _) => map decl_line_for_class classes

   | _ => []

 fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab

                              (conjs, facts) =

   let

     fun add_iterm_syms in_conj tm =

       let val (head, args) = strip_iterm_comb tm in

         (case head of

            IConst ((s, s'), T, T_args) =>

            let

              val pred_sym = is_pred_sym repaired_sym_tab s

              val decl_sym =

                (case type_enc of

                   Guards _ => not pred_sym

                 | _ => true) andalso

                is_tptp_user_symbol s

            in

              if decl_sym then

                Symtab.map_default (s, [])

                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,

                                          in_conj))

              else

                I

            end

          | IAbs (_, tm) => add_iterm_syms in_conj tm

          | _ => I)

         #> fold (add_iterm_syms in_conj) args

       end

     fun add_fact_syms in_conj =

       K (add_iterm_syms in_conj) |> formula_fold NONE |> fact_lift

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

         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs

         #> add_formula_var_types phi

       | add_formula_var_types (AConn (_, phis)) =

         fold add_formula_var_types phis

       | add_formula_var_types _ = I

     fun var_types () =

       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]

       else fold (fact_lift add_formula_var_types) (conjs @ facts) []

     fun add_undefined_const T =

       let

         val (s, s') =

           `(make_fixed_const NONE) @{const_name undefined}

           |> (case type_arg_policy type_enc @{const_name undefined} of

                 Mangled_Type_Args _ => mangled_const_name format type_enc [T]

               | _ => I)

       in

         Symtab.map_default (s, [])

                            (insert_type ctxt #3 (s', [T], T, false, 0, false))

       end

     fun add_TYPE_const () =

       let val (s, s') = TYPE_name in

         Symtab.map_default (s, [])

             (insert_type ctxt #3

                          (s', [tvar_a], @{typ "'a itself"}, false, 0, false))

       end

   in

     Symtab.empty

     |> is_type_enc_fairly_sound type_enc

        ? (fold (add_fact_syms true) conjs

           #> fold (add_fact_syms false) facts

           #> (case type_enc of

                 Simple_Types (_, poly, _) =>

                 if poly = Polymorphic then add_TYPE_const () else I

               | _ => fold add_undefined_const (var_types ())))

   end

 (* We add "bool" in case the helper "True_or_False" is included later. *)

 val default_mono =

   {maybe_finite_Ts = [@{typ bool}],

    surely_finite_Ts = [@{typ bool}],

    maybe_infinite_Ts = known_infinite_types,

    surely_infinite_Ts = known_infinite_types,

    maybe_nonmono_Ts = [@{typ bool}]}

 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it

    out with monotonicity" paper presented at CADE 2011. *)

 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono

   | add_iterm_mononotonicity_info ctxt level _

         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))

         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,

                   surely_infinite_Ts, maybe_nonmono_Ts}) =

     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then

       case level of

         Noninf_Nonmono_Types soundness =>

         if exists (type_instance ctxt T) surely_infinite_Ts orelse

            member (type_aconv ctxt) maybe_finite_Ts T then

           mono

         else if is_type_kind_of_surely_infinite ctxt soundness

                                                 surely_infinite_Ts T then

           {maybe_finite_Ts = maybe_finite_Ts,

            surely_finite_Ts = surely_finite_Ts,

            maybe_infinite_Ts = maybe_infinite_Ts,

            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,

            maybe_nonmono_Ts = maybe_nonmono_Ts}

         else

           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,

            surely_finite_Ts = surely_finite_Ts,

            maybe_infinite_Ts = maybe_infinite_Ts,

            surely_infinite_Ts = surely_infinite_Ts,

            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}

       | Fin_Nonmono_Types =>

         if exists (type_instance ctxt T) surely_finite_Ts orelse

            member (type_aconv ctxt) maybe_infinite_Ts T then

           mono

         else if is_type_surely_finite ctxt T then

           {maybe_finite_Ts = maybe_finite_Ts,

            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,

            maybe_infinite_Ts = maybe_infinite_Ts,

            surely_infinite_Ts = surely_infinite_Ts,

            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}

         else

           {maybe_finite_Ts = maybe_finite_Ts,

            surely_finite_Ts = surely_finite_Ts,

            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,

            surely_infinite_Ts = surely_infinite_Ts,

            maybe_nonmono_Ts = maybe_nonmono_Ts}

       | _ => mono

     else

       mono

   | add_iterm_mononotonicity_info _ _ _ _ mono = mono

 fun add_fact_mononotonicity_info ctxt level

         ({kind, iformula, ...} : translated_formula) =

   formula_fold (SOME (kind <> Conjecture))

                (add_iterm_mononotonicity_info ctxt level) iformula

 fun mononotonicity_info_for_facts ctxt type_enc facts =

   let val level = level_of_type_enc type_enc in

     default_mono

     |> is_type_level_monotonicity_based level

        ? fold (add_fact_mononotonicity_info ctxt level) facts

   end

 fun add_iformula_monotonic_types ctxt mono type_enc =

   let

     val level = level_of_type_enc type_enc

     val should_encode = should_encode_type ctxt mono level

     fun add_type T = not (should_encode T) ? insert_type ctxt I T

     fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2

       | add_args _ = I

     and add_term tm = add_type (ityp_of tm) #> add_args tm

   in formula_fold NONE (K add_term) end

 fun add_fact_monotonic_types ctxt mono type_enc =

   add_iformula_monotonic_types ctxt mono type_enc |> fact_lift

 fun monotonic_types_for_facts ctxt mono type_enc facts =

   [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso

          is_type_level_monotonicity_based (level_of_type_enc type_enc))

         ? fold (add_fact_monotonic_types ctxt mono type_enc) facts

 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =

   Formula (guards_sym_formula_prefix ^

            ascii_of (mangled_type format type_enc T),

            Axiom,

            IConst (`make_bound_var "X", T, [])

            |> type_guard_iterm ctxt format type_enc T

            |> AAtom

            |> formula_from_iformula ctxt format mono type_enc

                                     (K (K (K (K true)))) (SOME true)

            |> bound_tvars type_enc (atyps_of T)

            |> close_formula_universally type_enc,

            isabelle_info introN, NONE)

 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =

   let val x_var = ATerm (`make_bound_var "X", []) in

     Formula (tags_sym_formula_prefix ^

              ascii_of (mangled_type format type_enc T),

              Axiom,

              eq_formula type_enc (atyps_of T) false

                         (tag_with_type ctxt format mono type_enc NONE T x_var)

                         x_var,

              isabelle_info simpN, NONE)

   end

 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =

   case type_enc of

     Simple_Types _ => []

   | Guards _ =>

     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts

   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts

 fun decl_line_for_sym ctxt format mono type_enc s

                       (s', T_args, T, pred_sym, ary, _) =

   let

     val thy = Proof_Context.theory_of ctxt

     val (T, T_args) =

       if null T_args then

         (T, [])

       else case strip_prefix_and_unascii const_prefix s of

         SOME s' =>

         let

           val s' = s' |> invert_const

           val T = s' |> robust_const_type thy

         in (T, robust_const_typargs thy (s', T)) end

       | NONE =>

         case strip_prefix_and_unascii fixed_var_prefix s of

           SOME s' =>

           if String.isPrefix polymorphic_free_prefix s' then (tvar_a, [tvar_a])

           else raise Fail "unexpected type arguments to free variable"

         | NONE => raise Fail "unexpected type arguments"

   in

     Decl (sym_decl_prefix ^ s, (s, s'),

           T |> fused_type ctxt mono (level_of_type_enc type_enc) ary

             |> ho_type_from_typ format type_enc pred_sym ary

             |> not (null T_args)

                ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))

   end

 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s

                                      j (s', T_args, T, _, ary, in_conj) =

   let

     val (kind, maybe_negate) =

       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)

       else (Axiom, I)

     val (arg_Ts, res_T) = chop_fun ary T

     val num_args = length arg_Ts

     val bound_names =

       1 upto num_args |> map (`I o make_bound_var o string_of_int)

     val bounds =

       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))

     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args

     fun should_keep_arg_type T =

       sym_needs_arg_types andalso

       should_guard_type ctxt mono type_enc (K true) T

     val bound_Ts =

       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)

   in

     Formula (guards_sym_formula_prefix ^ s ^

              (if n > 1 then "_" ^ string_of_int j else ""), kind,

              IConst ((s, s'), T, T_args)

              |> fold (curry (IApp o swap)) bounds

              |> type_guard_iterm ctxt format type_enc res_T

              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)

              |> formula_from_iformula ctxt format mono type_enc

                                       (K (K (K (K true)))) (SOME true)

              |> n > 1 ? bound_tvars type_enc (atyps_of T)

              |> close_formula_universally type_enc

              |> maybe_negate,

              isabelle_info introN, NONE)

   end

 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono

         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =

   let

     val ident_base =

       tags_sym_formula_prefix ^ s ^

       (if n > 1 then "_" ^ string_of_int j else "")

     val (kind, maybe_negate) =

       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)

       else (Axiom, I)

     val (arg_Ts, res_T) = chop_fun ary T

     val bound_names =

       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)

     val bounds = bound_names |> map (fn name => ATerm (name, []))

     val cst = mk_aterm format type_enc (s, s') T_args

     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym

     val should_encode =

       should_encode_type ctxt mono (level_of_type_enc type_enc)

     val tag_with = tag_with_type ctxt format mono type_enc NONE

     val add_formula_for_res =

       if should_encode res_T then

         cons (Formula (ident_base ^ "_res", kind,

                        eq (tag_with res_T (cst bounds)) (cst bounds),

                        isabelle_info simpN, NONE))

       else

         I

     fun add_formula_for_arg k =

       let val arg_T = nth arg_Ts k in

         if should_encode arg_T then

           case chop k bounds of

             (bounds1, bound :: bounds2) =>

             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,

                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))

                               (cst bounds),

                            isabelle_info simpN, NONE))

           | _ => raise Fail "expected nonempty tail"

         else

           I

       end

   in

     [] |> not pred_sym ? add_formula_for_res

        |> Config.get ctxt type_tag_arguments

           ? fold add_formula_for_arg (ary - 1 downto 0)

   end

 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd

 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc

                                 (s, decls) =

   case type_enc of

     Simple_Types _ =>

     decls |> map (decl_line_for_sym ctxt format mono type_enc s)

   | Guards (_, level, _) =>

     let

       val decls =

         case decls of

           decl :: (decls' as _ :: _) =>

           let val T = result_type_of_decl decl in

             if forall (type_generalization ctxt T o result_type_of_decl)

                       decls' then

               [decl]

             else

               decls

           end

         | _ => decls

       val n = length decls

       val decls =

         decls |> filter (should_encode_type ctxt mono level

                          o result_type_of_decl)

     in

       (0 upto length decls - 1, decls)

       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono

                                                  type_enc n s)

     end

   | Tags (_, _, uniformity) =>

     (case uniformity of

        Uniform => []

      | Nonuniform =>

        let val n = length decls in

          (0 upto n - 1 ~~ decls)

          |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format

                       conj_sym_kind mono type_enc n s)

        end)

 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc

                                      mono_Ts sym_decl_tab =

   let

     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst

     val mono_lines =

       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts

     val decl_lines =

       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind

                                                      mono type_enc)

                syms []

   in mono_lines @ decl_lines end

 fun needs_type_tag_idempotence ctxt (Tags (poly, level, uniformity)) =

     Config.get ctxt type_tag_idempotence andalso

     poly <> Mangled_Monomorphic andalso

     ((level = All_Types andalso uniformity = Nonuniform) orelse

      is_type_level_monotonicity_based level)

   | needs_type_tag_idempotence _ _ = false

 fun offset_of_heading_in_problem _ [] j = j

   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =

     if heading = needle then j

     else offset_of_heading_in_problem needle problem (j + length lines)

 val implicit_declsN = "Should-be-implicit typings"

 val explicit_declsN = "Explicit typings"

 val factsN = "Relevant facts"

 val class_relsN = "Class relationships"

 val aritiesN = "Arities"

 val helpersN = "Helper facts"

 val conjsN = "Conjectures"

 val free_typesN = "Type variables"

 val explicit_apply = NONE (* for experiments *)

 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter

         lambda_trans readable_names preproc hyp_ts concl_t facts =

   let

     val type_enc = type_enc |> adjust_type_enc format

     val lambda_trans =

       if lambda_trans = smartN then

         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN

       else if lambda_trans = lambdasN andalso

               not (is_type_enc_higher_order type_enc) then

         error ("Lambda translation method incompatible with first-order \

                \encoding.")

       else

         lambda_trans

     val trans_lambdas =

       if lambda_trans = no_lambdasN then

         rpair []

       else if lambda_trans = concealedN then

         lift_lambdas ctxt type_enc ##> K []

       else if lambda_trans = liftingN then

         lift_lambdas ctxt type_enc

       else if lambda_trans = combinatorsN then

         map (introduce_combinators ctxt) #> rpair []

       else if lambda_trans = hybridN then

         lift_lambdas ctxt type_enc

         ##> maps (fn t => [t, introduce_combinators ctxt

                                   (intentionalize_def t)])

       else if lambda_trans = lambdasN then

         map (Envir.eta_contract) #> rpair []

       else

         error ("Unknown lambda translation method: " ^

                quote lambda_trans ^ ".")

     val (fact_names, classes, conjs, facts, class_rel_clauses, arity_clauses) =

       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc

                          hyp_ts concl_t facts

     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply

     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc

     val repair = repair_fact ctxt format type_enc sym_tab

     val (conjs, facts) = (conjs, facts) |> pairself (map repair)

     val repaired_sym_tab =

       conjs @ facts |> sym_table_for_facts ctxt (SOME false)

     val helpers =

       repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc

                        |> map repair

     val mono_Ts =

       helpers @ conjs @ facts

       |> monotonic_types_for_facts ctxt mono type_enc

     val class_decl_lines = decl_lines_for_classes type_enc classes

     val sym_decl_lines =

       (conjs, helpers @ facts)

       |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab

       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono

                                                type_enc mono_Ts

     val helper_lines =

       0 upto length helpers - 1 ~~ helpers

       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono

                                     type_enc)

       |> (if needs_type_tag_idempotence ctxt type_enc then

             cons (type_tag_idempotence_fact type_enc)

           else

             I)

     (* Reordering these might confuse the proof reconstruction code or the SPASS

        FLOTTER hack. *)

     val problem =

       [(explicit_declsN, class_decl_lines @ sym_decl_lines),

        (factsN,

         map (formula_line_for_fact ctxt format fact_prefix ascii_of

                                    (not exporter) (not exporter) mono type_enc)

             (0 upto length facts - 1 ~~ facts)),

        (class_relsN,

         map (formula_line_for_class_rel_clause type_enc) class_rel_clauses),

        (aritiesN, map (formula_line_for_arity_clause type_enc) arity_clauses),

        (helpersN, helper_lines),

        (conjsN,

         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),

        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]

     val problem =

       problem

       |> (case format of

             CNF => ensure_cnf_problem

           | CNF_UEQ => filter_cnf_ueq_problem

           | FOF => I

           | TFF (_, TFF_Implicit) => I

           | _ => declare_undeclared_syms_in_atp_problem type_decl_prefix

                                                         implicit_declsN)

     val (problem, pool) = problem |> nice_atp_problem readable_names

     val helpers_offset = offset_of_heading_in_problem helpersN problem 0

     val typed_helpers =

       map_filter (fn (j, {name, ...}) =>

                      if String.isSuffix typed_helper_suffix name then SOME j

                      else NONE)

                  ((helpers_offset + 1 upto helpers_offset + length helpers)

                   ~~ helpers)

     fun add_sym_arity (s, {min_ary, ...} : sym_info) =

       if min_ary > 0 then

         case strip_prefix_and_unascii const_prefix s of

           SOME s => Symtab.insert (op =) (s, min_ary)

         | NONE => I

       else

         I

   in

     (problem,

      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,

      offset_of_heading_in_problem conjsN problem 0,

      offset_of_heading_in_problem factsN problem 0,

      fact_names |> Vector.fromList,

      typed_helpers,

      Symtab.empty |> Symtab.fold add_sym_arity sym_tab)

   end

 (* FUDGE *)

 val conj_weight = 0.0

 val hyp_weight = 0.1

 val fact_min_weight = 0.2

 val fact_max_weight = 1.0

 val type_info_default_weight = 0.8

 fun add_term_weights weight (ATerm (s, tms)) =

     is_tptp_user_symbol s ? Symtab.default (s, weight)

     #> fold (add_term_weights weight) tms

   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm

 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =

     formula_fold NONE (K (add_term_weights weight)) phi

   | add_problem_line_weights _ _ = I

 fun add_conjectures_weights [] = I

   | add_conjectures_weights conjs =

     let val (hyps, conj) = split_last conjs in

       add_problem_line_weights conj_weight conj

       #> fold (add_problem_line_weights hyp_weight) hyps

     end

 fun add_facts_weights facts =

   let

     val num_facts = length facts

     fun weight_of j =

       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j

                         / Real.fromInt num_facts

   in

     map weight_of (0 upto num_facts - 1) ~~ facts

     |> fold (uncurry add_problem_line_weights)

   end

 (* Weights are from 0.0 (most important) to 1.0 (least important). *)

 fun atp_problem_weights problem =

   let val get = these o AList.lookup (op =) problem in

     Symtab.empty

     |> add_conjectures_weights (get free_typesN @ get conjsN)

     |> add_facts_weights (get factsN)

     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)

             [explicit_declsN, class_relsN, aritiesN]

     |> Symtab.dest

     |> sort (prod_ord Real.compare string_ord o pairself swap)

   end

 end;