(*  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 Sledgehammer.

 *)

 signature ATP_TRANSLATE =

 sig

   type 'a fo_term = 'a ATP_Problem.fo_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

   type name = string * string

   datatype type_literal =

     TyLitVar of name * name |

     TyLitFree of name * name

   datatype arity_literal =

     TConsLit of name * name * name list |

     TVarLit of name * name

   type arity_clause =

     {name: string,

      prem_lits: arity_literal list,

      concl_lits: arity_literal}

   type class_rel_clause =

     {name: string,

      subclass: name,

      superclass: name}

   datatype combterm =

     CombConst of name * typ * typ list |

     CombVar of name * typ |

     CombApp of combterm * combterm

   datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained

   datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

   datatype type_level =

     All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types

   datatype type_heaviness = Heavyweight | Lightweight

   datatype type_sys =

     Simple_Types of type_level |

     Preds of polymorphism * type_level * type_heaviness |

     Tags of polymorphism * type_level * type_heaviness

   val bound_var_prefix : 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 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 preds_sym_formula_prefix : string

   val lightweight_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_tag_name : string

   val type_pred_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 make_bound_var : string -> string

   val make_schematic_var : string * int -> string

   val make_fixed_var : string -> string

   val make_schematic_type_var : string * int -> string

   val make_fixed_type_var : string -> string

   val make_fixed_const : string -> string

   val make_fixed_type_const : string -> string

   val make_type_class : string -> string

   val new_skolem_var_name_from_const : string -> string

   val num_type_args : theory -> string -> int

   val atp_irrelevant_consts : string list

   val atp_schematic_consts_of : term -> typ list Symtab.table

   val make_arity_clauses :

     theory -> string list -> class list -> class list * arity_clause list

   val make_class_rel_clauses :

     theory -> class list -> class list -> class_rel_clause list

   val combtyp_of : combterm -> typ

   val strip_combterm_comb : combterm -> combterm * combterm list

   val atyps_of : typ -> typ list

   val combterm_from_term :

     theory -> (string * typ) list -> term -> combterm * typ list

   val is_locality_global : locality -> bool

   val type_sys_from_string : string -> type_sys

   val polymorphism_of_type_sys : type_sys -> polymorphism

   val level_of_type_sys : type_sys -> type_level

   val is_type_sys_virtually_sound : type_sys -> bool

   val is_type_sys_fairly_sound : type_sys -> bool

   val choose_format : format list -> type_sys -> format * type_sys

   val mk_aconns :

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

   val unmangled_const_name : string -> string

   val unmangled_const : string -> string * string fo_term list

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

   val should_specialize_helper : type_sys -> term -> bool

   val tfree_classes_of_terms : term list -> string list

   val tvar_classes_of_terms : term list -> string list

   val type_constrs_of_terms : theory -> term list -> string list

   val prepare_atp_problem :

     Proof.context -> format -> formula_kind -> formula_kind -> type_sys

     -> bool -> 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

 (* experimental *)

 val generate_useful_info = false

 fun useful_isabelle_info s =

   if generate_useful_info then

     SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))

   else

     NONE

 val intro_info = useful_isabelle_info "intro"

 val elim_info = useful_isabelle_info "elim"

 val simp_info = useful_isabelle_info "simp"

 val bound_var_prefix = "B_"

 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 class_prefix = "cl_"

 val skolem_const_prefix = "Sledgehammer" ^ 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 preds_sym_formula_prefix = "psy_"

 val lightweight_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 typed_helper_suffix = "_T"

 val untyped_helper_suffix = "_U"

 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"

 val predicator_name = "hBOOL"

 val app_op_name = "hAPP"

 val type_tag_name = "ti"

 val type_pred_name = "is"

 val simple_type_prefix = "ty_"

 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 sledgehammer_weak_prefix = "Sledgehammer:"

 (*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}))))]

 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 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

 (*Remove the initial ' character from a type variable, if it is present*)

 fun trim_type_var s =

   if s <> "" andalso String.sub(s,0) = #"'" then String.extract(s,1,NONE)

   else raise Fail ("trim_type: Malformed type variable encountered: " ^ s)

 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_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 (trim_type_var x, i))

 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (trim_type_var x))

 (* HOL.eq MUST BE "equal" because it's built into ATPs. *)

 fun make_fixed_const @{const_name HOL.eq} = "equal"

   | make_fixed_const c = const_prefix ^ lookup_const c

 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

 (* The number of type arguments of a constant, zero if it's monomorphic. For

    (instances of) Skolem pseudoconstants, this information is encoded in the

    constant name. *)

 fun num_type_args thy s =

   if String.isPrefix skolem_const_prefix s then

     s |> space_explode Long_Name.separator |> List.last |> Int.fromString |> the

   else

     (s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length

 (* 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 =

   TConsLit of name * name * name list |

   TVarLit of name * name

 fun gen_TVars 0 = []

   | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)

 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 = gen_TVars (length args)

     val tvars_srts = ListPair.zip (tvars, args)

   in

     {name = name,

      prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,

      concl_lits = TConsLit (`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 ^ "_" ^ class ^ "_" ^ string_of_int n, ar) ::

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

       else

           make_axiom_arity_clause (tcons, lookup_const tcons ^ "_" ^ 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)

 datatype combterm =

   CombConst of name * typ * typ list |

   CombVar of name * typ |

   CombApp of combterm * combterm

 fun combtyp_of (CombConst (_, T, _)) = T

   | combtyp_of (CombVar (_, T)) = T

   | combtyp_of (CombApp (t1, _)) = snd (dest_funT (combtyp_of t1))

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

 fun strip_combterm_comb u =

     let fun stripc (CombApp(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

 (* Converts a term (with combinators) into a combterm. Also accumulates sort

    infomation. *)

 fun combterm_from_term thy bs (P $ Q) =

     let

       val (P', P_atomics_Ts) = combterm_from_term thy bs P

       val (Q', Q_atomics_Ts) = combterm_from_term thy bs Q

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

   | combterm_from_term thy _ (Const (c, T)) =

     let

       val tvar_list =

         (if String.isPrefix old_skolem_const_prefix c then

            [] |> Term.add_tvarsT T |> map TVar

          else

            (c, T) |> Sign.const_typargs thy)

       val c' = CombConst (`make_fixed_const c, T, tvar_list)

     in (c', atyps_of T) end

   | combterm_from_term _ _ (Free (v, T)) =

     (CombConst (`make_fixed_var v, T, []), atyps_of T)

   | combterm_from_term _ _ (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 CombConst (`make_fixed_const s', T, Ts) end

      else

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

   | combterm_from_term _ bs (Bound j) =

     nth bs j

     |> (fn (s, T) => (CombConst (`make_bound_var s, T, []), atyps_of T))

   | combterm_from_term _ _ (Abs _) = raise Fail "HOL clause: Abs"

 datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained

 (* (quasi-)underapproximation of the truth *)

 fun is_locality_global Local = false

   | is_locality_global Assum = false

   | is_locality_global Chained = false

   | is_locality_global _ = true

 datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

 datatype type_level =

   All_Types | Nonmonotonic_Types | Finite_Types | Const_Arg_Types | No_Types

 datatype type_heaviness = Heavyweight | Lightweight

 datatype type_sys =

   Simple_Types of type_level |

   Preds of polymorphism * type_level * type_heaviness |

   Tags of polymorphism * type_level * type_heaviness

 fun try_unsuffixes ss s =

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

 fun type_sys_from_string s =

   (case try (unprefix "poly_") s of

      SOME s => (SOME Polymorphic, s)

    | NONE =>

      case try (unprefix "mono_") s of

        SOME s => (SOME Monomorphic, s)

      | NONE =>

        case try (unprefix "mangled_") s of

          SOME s => (SOME Mangled_Monomorphic, s)

        | NONE => (NONE, s))

   ||> (fn s =>

           (* "_query" and "_bang" are for the ASCII-challenged Mirabelle. *)

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

             SOME s => (Nonmonotonic_Types, s)

           | NONE =>

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

               SOME s => (Finite_Types, s)

             | NONE => (All_Types, s))

   ||> apsnd (fn s =>

                 case try (unsuffix "_heavy") s of

                   SOME s => (Heavyweight, s)

                 | NONE => (Lightweight, s))

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

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

            ("simple", (NONE, _, Lightweight)) => Simple_Types level

          | ("preds", (SOME poly, _, _)) => Preds (poly, level, heaviness)

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

            Tags (Polymorphic, level, heaviness)

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

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

            Preds (poly, Const_Arg_Types, Lightweight)

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

            Preds (Polymorphic, No_Types, Lightweight)

          | _ => raise Same.SAME)

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

 fun polymorphism_of_type_sys (Simple_Types _) = Mangled_Monomorphic

   | polymorphism_of_type_sys (Preds (poly, _, _)) = poly

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

 fun level_of_type_sys (Simple_Types level) = level

   | level_of_type_sys (Preds (_, level, _)) = level

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

 fun heaviness_of_type_sys (Simple_Types _) = Heavyweight

   | heaviness_of_type_sys (Preds (_, _, heaviness)) = heaviness

   | heaviness_of_type_sys (Tags (_, _, heaviness)) = heaviness

 fun is_type_level_virtually_sound level =

   level = All_Types orelse level = Nonmonotonic_Types

 val is_type_sys_virtually_sound =

   is_type_level_virtually_sound o level_of_type_sys

 fun is_type_level_fairly_sound level =

   is_type_level_virtually_sound level orelse level = Finite_Types

 val is_type_sys_fairly_sound = is_type_level_fairly_sound o level_of_type_sys

 fun is_setting_higher_order THF (Simple_Types _) = true

   | is_setting_higher_order _ _ = false

 fun choose_format formats (Simple_Types level) =

     if member (op =) formats THF then (THF, Simple_Types level)

     else if member (op =) formats TFF then (TFF, Simple_Types level)

     else choose_format formats (Preds (Mangled_Monomorphic, level, Heavyweight))

   | choose_format formats type_sys =

     (case hd formats of

        CNF_UEQ =>

        (CNF_UEQ, case type_sys of

                    Preds stuff =>

                    (if is_type_sys_fairly_sound type_sys then Tags else Preds)

                        stuff

                  | _ => type_sys)

      | format => (format, type_sys))

 type translated_formula =

   {name: string,

    locality: locality,

    kind: formula_kind,

    combformula: (name, typ, combterm) formula,

    atomic_types: typ list}

 fun update_combformula f ({name, locality, kind, combformula, atomic_types}

                           : translated_formula) =

   {name = name, locality = locality, kind = kind, combformula = f combformula,

    atomic_types = atomic_types} : translated_formula

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

 val type_instance = Sign.typ_instance o Proof_Context.theory_of

 fun insert_type ctxt get_T x xs =

   let val T = get_T x in

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

     else x :: filter_out (curry (type_instance ctxt o swap) 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 (* since TFF doesn't support overloading *)

   | should_drop_arg_type_args type_sys =

     level_of_type_sys type_sys = All_Types andalso

     heaviness_of_type_sys type_sys = Heavyweight

 fun general_type_arg_policy (Tags (_, All_Types, Heavyweight)) = No_Type_Args

   | general_type_arg_policy type_sys =

     if level_of_type_sys type_sys = No_Types then

       No_Type_Args

     else if polymorphism_of_type_sys type_sys = Mangled_Monomorphic then

       Mangled_Type_Args (should_drop_arg_type_args type_sys)

     else

       Explicit_Type_Args (should_drop_arg_type_args type_sys)

 fun type_arg_policy type_sys s =

   if s = @{const_name HOL.eq} orelse

      (s = app_op_name andalso level_of_type_sys type_sys = Const_Arg_Types) then

     No_Type_Args

   else if s = type_tag_name then

     Explicit_Type_Args false

   else

     general_type_arg_policy type_sys

 (*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_sys add_sorts_on_typ Ts =

   [] |> level_of_type_sys type_sys <> 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 combterm_vars (CombApp (tm1, tm2)) = fold combterm_vars [tm1, tm2]

   | combterm_vars (CombConst _) = I

   | combterm_vars (CombVar (name, T)) = insert (op =) (name, SOME T)

 fun close_combformula_universally phi = close_universally combterm_vars phi

 fun term_vars (ATerm (name as (s, _), tms)) =

   is_tptp_variable s ? insert (op =) (name, NONE) #> fold term_vars tms

 fun close_formula_universally phi = close_universally term_vars phi

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

 val homo_infinite_type = Type (homo_infinite_type_name, [])

 fun fo_term_from_typ format type_sys =

   let

     fun term (Type (s, Ts)) =

       ATerm (case (is_setting_higher_order format type_sys, s) of

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

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

              | _ => if s = homo_infinite_type_name andalso

                        (format = TFF orelse format = THF) 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 as (s, _)), _)) =

       ATerm ((make_schematic_type_var x, s), [])

   in term end

 (* 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)

     ^ ")"

 val bool_atype = AType (`I tptp_bool_type)

 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_fo_term format type_sys pred_sym ary =

   let

     fun to_atype ty =

       AType ((make_simple_type (generic_mangled_type_name fst ty),

               generic_mangled_type_name snd ty))

     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

     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

   in if is_setting_higher_order format type_sys then to_ho else to_fo ary end

 fun mangled_type format type_sys pred_sym ary =

   ho_type_from_fo_term format type_sys pred_sym ary

   o fo_term_from_typ format type_sys

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

   let

     val ty_args = map (fo_term_from_typ format type_sys) T_args

     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 format type_sys =

   let

     fun intro top_level (CombApp (tm1, tm2)) =

         CombApp (intro top_level tm1, intro false tm2)

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

         (case proxify_const s of

            SOME proxy_base =>

            if top_level orelse is_setting_higher_order format type_sys then

              case (top_level, s) of

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

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

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

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

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

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

              | (false, s) =>

                if is_tptp_equal s then (`I tptp_equal, [])

                else (proxy_base |>> prefix const_prefix, T_args)

              | _ => (name, [])

            else

              (proxy_base |>> prefix const_prefix, T_args)

           | NONE => (name, T_args))

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

       | intro _ tm = tm

   in intro true end

 fun combformula_from_prop thy format type_sys eq_as_iff =

   let

     fun do_term bs t atomic_types =

       combterm_from_term thy bs (Envir.eta_contract t)

       |>> (introduce_proxies format type_sys #> AAtom)

       ||> union (op =) atomic_types

     fun do_quant bs q s T t' =

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

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

         #>> mk_aquant q [(`make_bound_var s, SOME T)]

       end

     and do_conn bs c t1 t2 =

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

     and do_formula bs t =

       case t of

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

       | @{const Not} $ t1 => do_formula bs t1 #>> mk_anot

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

         do_quant bs AForall s T t'

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

         do_quant bs AExists s T t'

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

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

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

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

         if eq_as_iff then do_conn bs AIff t1 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 = sledgehammer_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 introduce_combinators_in_term ctxt kind 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 |> conceal_bounds Ts

                      |> Envir.eta_contract

                      |> cterm_of thy

                      |> Meson_Clausify.introduce_combinators_in_cterm

                      |> prop_of |> Logic.dest_equals |> snd

                      |> reveal_bounds Ts

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

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

       handle THM _ =>

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

              if kind = Conjecture then HOLogic.false_const

              else HOLogic.true_const

   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 (sledgehammer_weak_prefix ^ s ^ "_" ^ string_of_int i, T)

       | aux t = t

   in t |> exists_subterm is_Var t ? aux end

 fun preprocess_prop ctxt presimp_consts kind 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))

       |> introduce_combinators_in_term ctxt kind

   end

 (* making fact and conjecture formulas *)

 fun make_formula thy format type_sys eq_as_iff name loc kind t =

   let

     val (combformula, atomic_types) =

       combformula_from_prop thy format type_sys eq_as_iff t []

   in

     {name = name, locality = loc, kind = kind, combformula = combformula,

      atomic_types = atomic_types}

   end

 fun make_fact ctxt format type_sys eq_as_iff preproc presimp_consts

               ((name, loc), t) =

   let val thy = Proof_Context.theory_of ctxt in

     case t |> preproc ? preprocess_prop ctxt presimp_consts Axiom

            |> make_formula thy format type_sys (eq_as_iff andalso format <> CNF)

                            name loc Axiom of

       formula as {combformula = AAtom (CombConst ((s, _), _, _)), ...} =>

       if s = tptp_true then NONE else SOME formula

     | formula => SOME formula

   end

 fun make_conjecture ctxt format prem_kind type_sys preproc presimp_consts ts =

   let

     val thy = Proof_Context.theory_of ctxt

     val last = length ts - 1

   in

     map2 (fn j => fn t =>

              let

                val (kind, maybe_negate) =

                  if j = last then

                    (Conjecture, I)

                  else

                    (prem_kind,

                     if prem_kind = Conjecture then update_combformula mk_anot

                     else I)

               in

                 t |> preproc ?

                      (preprocess_prop ctxt presimp_consts kind #> freeze_term)

                   |> make_formula thy format type_sys (format <> CNF)

                                   (string_of_int j) Local kind

                   |> maybe_negate

               end)

          (0 upto last) ts

   end

 (** Finite and infinite type inference **)

 fun deep_freeze_atyp (TVar (_, S)) = TFree ("v", S)

   | deep_freeze_atyp T = T

 val deep_freeze_type = map_atyps deep_freeze_atyp

 (* 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 ctxt (nonmono_Ts as _ :: _) _ T =

     exists (curry (type_instance ctxt) (deep_freeze_type T)) nonmono_Ts

   | should_encode_type _ _ All_Types _ = true

   | should_encode_type ctxt _ Finite_Types T = is_type_surely_finite ctxt T

   | should_encode_type _ _ _ _ = false

 fun should_predicate_on_type ctxt nonmono_Ts (Preds (_, level, heaviness))

                              should_predicate_on_var T =

     (heaviness = Heavyweight orelse should_predicate_on_var ()) andalso

     should_encode_type ctxt nonmono_Ts level T

   | should_predicate_on_type _ _ _ _ _ = false

 fun is_var_or_bound_var (CombConst ((s, _), _, _)) =

     String.isPrefix bound_var_prefix s

   | is_var_or_bound_var (CombVar _) = true

   | is_var_or_bound_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 nonmono_Ts (Tags (poly, level, heaviness)) site

                          u T =

     (case heaviness of

        Heavyweight => should_encode_type ctxt nonmono_Ts level T

      | Lightweight =>

        case (site, is_var_or_bound_var u) of

          (Eq_Arg pos, true) =>

          (* The first disjunct prevents a subtle soundness issue explained in

             Blanchette's Ph.D. thesis. See also

             "formula_lines_for_lightweight_tags_sym_decl". *)

          (pos <> SOME false andalso poly = Polymorphic andalso

           level <> All_Types andalso heaviness = Lightweight andalso

           exists (fn T' => type_instance ctxt (T', T)) nonmono_Ts) orelse

          should_encode_type ctxt nonmono_Ts level T

        | _ => false)

   | should_tag_with_type _ _ _ _ _ _ = false

 fun homogenized_type ctxt nonmono_Ts level =

   let

     val should_encode = should_encode_type ctxt nonmono_Ts level

     fun homo 0 T = if should_encode T then T else homo_infinite_type

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

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

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

   in homo end

 (** "hBOOL" and "hAPP" **)

 type sym_info =

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

 fun add_combterm_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_var_or_bound_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_combterm_comb tm in

         (case head of

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

            if String.isPrefix bound_var_prefix s then

              add_var_or_bound_var T 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

          | CombVar (_, T) => add_var_or_bound_var T accum

          | _ => accum)

         |> fold (add false) args

       end

   in add true end

 fun add_fact_syms_to_table ctxt explicit_apply =

   fact_lift (formula_fold NONE

                           (K (add_combterm_syms_to_table ctxt explicit_apply)))

 val default_sym_tab_entries : (string * sym_info) list =

   (prefixed_predicator_name,

    {pred_sym = true, min_ary = 1, max_ary = 1, 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_pred_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 =

   CombConst (`make_fixed_const predicator_name, @{typ "bool => bool"}, [])

 fun predicator tm = CombApp (predicator_combconst, tm)

 fun introduce_predicators_in_combterm sym_tab tm =

   case strip_combterm_comb tm of

     (CombConst ((s, _), _, _), _) =>

     if is_pred_sym sym_tab s then tm else predicator tm

   | _ => predicator tm

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

 val app_op = `make_fixed_const app_op_name

 fun explicit_app arg head =

   let

     val head_T = combtyp_of head

     val (arg_T, res_T) = dest_funT head_T

     val explicit_app =

       CombConst (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_combterm sym_tab =

   let

     fun aux tm =

       case strip_combterm_comb tm of

         (head as CombConst ((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

       (* will throw "TYPE" for pseudo-constants *)

       val U = if s = app_op_name then

                 @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global

               else

                 s |> Sign.the_const_type thy

     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_filter (fn (U, T) =>

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

                               SOME T

                             else

                               NONE)

         end

     end

     handle TYPE _ => T_args

 fun enforce_type_arg_policy_in_combterm ctxt format type_sys =

   let

     val thy = Proof_Context.theory_of ctxt

     fun aux arity (CombApp (tm1, tm2)) =

         CombApp (aux (arity + 1) tm1, aux 0 tm2)

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

         (case strip_prefix_and_unascii const_prefix s of

            NONE => (name, 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_sys s'' of

                Explicit_Type_Args drop_args =>

                (name, filtered_T_args drop_args)

              | Mangled_Type_Args drop_args =>

                (mangled_const_name format type_sys (filtered_T_args drop_args)

                                    name, [])

              | No_Type_Args => (name, [])

            end)

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

       | aux _ tm = tm

   in aux 0 end

 fun repair_combterm ctxt format type_sys sym_tab =

   not (is_setting_higher_order format type_sys)

   ? (introduce_explicit_apps_in_combterm sym_tab

      #> introduce_predicators_in_combterm sym_tab)

   #> enforce_type_arg_policy_in_combterm ctxt format type_sys

 fun repair_fact ctxt format type_sys sym_tab =

   update_combformula (formula_map

       (repair_combterm ctxt format type_sys sym_tab))

 (** Helper facts **)

 (* 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}),

    (("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]}),

    (("fFalse", false), [@{lemma "~ fFalse" by (unfold fFalse_def) fast}]),

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

    (("fTrue", false), [@{lemma "fTrue" by (unfold fTrue_def) fast}]),

    (("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+}),

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

   |> map (apsnd (map zero_var_indexes))

 val type_tag = `make_fixed_const type_tag_name

 fun type_tag_idempotence_fact () =

   let

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

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

     val tagged_a = tag (var "A")

   in

     Formula (type_tag_idempotence_helper_name, Axiom,

              AAtom (ATerm (`I tptp_equal, [tag tagged_a, tagged_a]))

              |> close_formula_universally, simp_info, NONE)

   end

 fun should_specialize_helper type_sys t =

   case general_type_arg_policy type_sys of

     Mangled_Type_Args _ => not (null (Term.hidden_polymorphism t))

   | _ => false

 fun helper_facts_for_sym ctxt format type_sys (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_and_inst needs_fairly_sound (th, j) =

         ((unmangled_s ^ "_" ^ string_of_int j ^

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

           (if needs_fairly_sound then typed_helper_suffix

            else untyped_helper_suffix),

           General),

          let val t = th |> prop_of in

            t |> should_specialize_helper type_sys t

                 ? (case types of

                      [T] => specialize_type thy (invert_const unmangled_s, T)

                    | _ => I)

          end)

       val make_facts =

         map_filter (make_fact ctxt format type_sys false false [])

       val fairly_sound = is_type_sys_fairly_sound type_sys

     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)

                     |> map (dub_and_inst needs_fairly_sound)

                     |> make_facts)

     end

   | NONE => []

 fun helper_facts_for_sym_table ctxt format type_sys sym_tab =

   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_sys) 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 OldTerm.term_tfrees

 val tvar_classes_of_terms = classes_of_terms OldTerm.term_tvars

 (*fold type constructors*)

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

     fold (fold_type_constrs f) Ts (f (a,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 (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_sys preproc hyp_ts concl_t

                        facts =

   let

     val thy = Proof_Context.theory_of ctxt

     val fact_ts = facts |> map snd

     val presimp_consts = Meson.presimplified_consts ctxt

     val make_fact = make_fact ctxt format type_sys true preproc presimp_consts

     val (facts, fact_names) =

       facts |> map (fn (name, t) => (name, t) |> make_fact |> rpair name)

             |> map_filter (try (apfst the))

             |> ListPair.unzip

     (* 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 goal_t = Logic.list_implies (hyp_ts, concl_t)

     val all_ts = goal_t :: 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 conjs =

       hyp_ts @ [concl_t]

       |> make_conjecture ctxt format prem_kind type_sys preproc presimp_consts

     val (supers', arity_clauses) =

       if level_of_type_sys type_sys = 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, (conjs, facts, class_rel_clauses, arity_clauses))

   end

 fun fo_literal_from_type_literal (TyLitVar (class, name)) =

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

   | fo_literal_from_type_literal (TyLitFree (class, name)) =

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

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

 val type_pred = `make_fixed_const type_pred_name

 fun type_pred_combterm ctxt format type_sys T tm =

   CombApp (CombConst (type_pred, T --> @{typ bool}, [T])

            |> enforce_type_arg_policy_in_combterm ctxt format type_sys, tm)

 fun var_occurs_positively_naked_in_term _ (SOME false) _ accum = accum

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

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

 fun is_var_nonmonotonic_in_formula _ _ (SOME false) _ = false

   | is_var_nonmonotonic_in_formula pos phi _ name =

     formula_fold pos (var_occurs_positively_naked_in_term name) phi false

 fun mk_const_aterm format type_sys x T_args args =

   ATerm (x, map (fo_term_from_typ format type_sys) T_args @ args)

 fun tag_with_type ctxt format nonmono_Ts type_sys pos T tm =

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

   |> enforce_type_arg_policy_in_combterm ctxt format type_sys

   |> term_from_combterm ctxt format nonmono_Ts type_sys (Top_Level pos)

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

 and term_from_combterm ctxt format nonmono_Ts type_sys =

   let

     fun aux site u =

       let

         val (head, args) = strip_combterm_comb u

         val (x as (s, _), T_args) =

           case head of

             CombConst (name, _, T_args) => (name, T_args)

           | CombVar (name, _) => (name, [])

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

         val (pos, arg_site) =

           case site of

             Top_Level pos =>

             (pos, if is_tptp_equal s then Eq_Arg pos else Elsewhere)

           | Eq_Arg pos => (pos, Elsewhere)

           | Elsewhere => (NONE, Elsewhere)

         val t = mk_const_aterm format type_sys x T_args

                     (map (aux arg_site) args)

         val T = combtyp_of u

       in

         t |> (if should_tag_with_type ctxt nonmono_Ts type_sys site u T then

                 tag_with_type ctxt format nonmono_Ts type_sys pos T

               else

                 I)

       end

   in aux end

 and formula_from_combformula ctxt format nonmono_Ts type_sys

                              should_predicate_on_var =

   let

     fun do_term pos =

       term_from_combterm ctxt format nonmono_Ts type_sys (Top_Level pos)

     val do_bound_type =

       case type_sys of

         Simple_Types level =>

         homogenized_type ctxt nonmono_Ts level 0

         #> mangled_type format type_sys false 0 #> SOME

       | _ => K NONE

     fun do_out_of_bound_type pos phi universal (name, T) =

       if should_predicate_on_type ctxt nonmono_Ts type_sys

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

         CombVar (name, T)

         |> type_pred_combterm ctxt format type_sys T

         |> do_term pos |> AAtom |> SOME

       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 o SOME end

 fun bound_tvars type_sys Ts =

   mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)

                 (type_literals_for_types type_sys add_sorts_on_tvar Ts))

 fun formula_for_fact ctxt format nonmono_Ts type_sys

                      ({combformula, atomic_types, ...} : translated_formula) =

   combformula

   |> close_combformula_universally

   |> formula_from_combformula ctxt format nonmono_Ts type_sys

                               is_var_nonmonotonic_in_formula true

   |> bound_tvars type_sys atomic_types

   |> close_formula_universally

 (* 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 nonmono_Ts type_sys

                           (j, formula as {name, locality, kind, ...}) =

   Formula (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^

            encode name,

            kind, formula_for_fact ctxt format nonmono_Ts type_sys formula, NONE,

            case locality of

              Intro => intro_info

            | Elim => elim_info

            | Simp => simp_info

            | _ => NONE)

 fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}

                                        : class_rel_clause) =

   let val ty_arg = ATerm (`I "T", []) in

     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,

              AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),

                                AAtom (ATerm (superclass, [ty_arg]))])

              |> close_formula_universally, intro_info, NONE)

   end

 fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =

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

   | fo_literal_from_arity_literal (TVarLit (c, sort)) =

     (false, ATerm (c, [ATerm (sort, [])]))

 fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}

                                    : arity_clause) =

   Formula (arity_clause_prefix ^ ascii_of name, Axiom,

            mk_ahorn (map (formula_from_fo_literal o apfst not

                           o fo_literal_from_arity_literal) prem_lits)

                     (formula_from_fo_literal

                          (fo_literal_from_arity_literal concl_lits))

            |> close_formula_universally, intro_info, NONE)

 fun formula_line_for_conjecture ctxt format nonmono_Ts type_sys

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

   Formula (conjecture_prefix ^ name, kind,

            formula_from_combformula ctxt format nonmono_Ts type_sys

                is_var_nonmonotonic_in_formula false

                (close_combformula_universally combformula)

            |> bound_tvars type_sys atomic_types

            |> close_formula_universally, NONE, NONE)

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

   atomic_types |> type_literals_for_types type_sys add_sorts_on_tfree

                |> map fo_literal_from_type_literal

 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_sys facts =

   let

     val litss = map (free_type_literals type_sys) facts

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

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

 (** Symbol declarations **)

 fun should_declare_sym type_sys pred_sym s =

   is_tptp_user_symbol s andalso not (String.isPrefix bound_var_prefix s) andalso

   (case type_sys of

      Simple_Types _ => true

    | Tags (_, _, Lightweight) => true

    | _ => not pred_sym)

 fun sym_decl_table_for_facts ctxt type_sys repaired_sym_tab (conjs, facts) =

   let

     fun add_combterm in_conj tm =

       let val (head, args) = strip_combterm_comb tm in

         (case head of

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

            let val pred_sym = is_pred_sym repaired_sym_tab s in

              if should_declare_sym type_sys pred_sym s then

                Symtab.map_default (s, [])

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

                                          in_conj))

              else

                I

            end

          | _ => I)

         #> fold (add_combterm in_conj) args

       end

     fun add_fact in_conj =

       fact_lift (formula_fold NONE (K (add_combterm in_conj)))

   in

     Symtab.empty

     |> is_type_sys_fairly_sound type_sys

        ? (fold (add_fact true) conjs #> fold (add_fact false) facts)

   end

 (* 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"}]

 (* 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_combterm_nonmonotonic_types _ _ _ (SOME false) _ = I

   | add_combterm_nonmonotonic_types ctxt level locality _

         (CombApp (CombApp (CombConst ((s, _), Type (_, [T, _]), _), tm1),

                            tm2)) =

     (is_tptp_equal s andalso exists is_var_or_bound_var [tm1, tm2] andalso

      (case level of

         Nonmonotonic_Types =>

         not (is_locality_global locality) orelse

         not (is_type_surely_infinite ctxt known_infinite_types T)

       | Finite_Types => is_type_surely_finite ctxt T

       | _ => true)) ? insert_type ctxt I (deep_freeze_type T)

   | add_combterm_nonmonotonic_types _ _ _ _ _ = I

 fun add_fact_nonmonotonic_types ctxt level ({kind, locality, combformula, ...}

                                             : translated_formula) =

   formula_fold (SOME (kind <> Conjecture))

                (add_combterm_nonmonotonic_types ctxt level locality) combformula

 fun nonmonotonic_types_for_facts ctxt type_sys facts =

   let val level = level_of_type_sys type_sys in

     if level = Nonmonotonic_Types orelse level = Finite_Types then

       [] |> fold (add_fact_nonmonotonic_types ctxt level) facts

          (* We must add "bool" in case the helper "True_or_False" is added

             later. In addition, several places in the code rely on the list of

             nonmonotonic types not being empty. *)

          |> insert_type ctxt I @{typ bool}

     else

       []

   end

 fun decl_line_for_sym ctxt format nonmono_Ts type_sys s

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

   let

     val (T_arg_Ts, level) =

       case type_sys of

         Simple_Types level => ([], level)

       | _ => (replicate (length T_args) homo_infinite_type, No_Types)

   in

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

           (T_arg_Ts ---> (T |> homogenized_type ctxt nonmono_Ts level ary))

           |> mangled_type format type_sys pred_sym (length T_arg_Ts + ary))

   end

 fun is_polymorphic_type T = fold_atyps (fn TVar _ => K true | _ => I) T false

 fun formula_line_for_preds_sym_decl ctxt format conj_sym_kind nonmono_Ts

         type_sys 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 bound_names =

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

     val bounds =

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

     val bound_Ts =

       arg_Ts |> map (fn T => if n > 1 orelse is_polymorphic_type T then SOME T

                              else NONE)

   in

     Formula (preds_sym_formula_prefix ^ s ^

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

              CombConst ((s, s'), T, T_args)

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

              |> type_pred_combterm ctxt format type_sys res_T

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

              |> formula_from_combformula ctxt format nonmono_Ts type_sys

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

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

              |> close_formula_universally

              |> maybe_negate,

              intro_info, NONE)

   end

 fun formula_lines_for_lightweight_tags_sym_decl ctxt format conj_sym_kind

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

   let

     val ident_base =

       lightweight_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_const_aterm format type_sys (s, s') T_args

     val atomic_Ts = atyps_of T

     fun eq tms =

       (if pred_sym then AConn (AIff, map AAtom tms)

        else AAtom (ATerm (`I tptp_equal, tms)))

       |> bound_tvars type_sys atomic_Ts

       |> close_formula_universally

       |> maybe_negate

     (* See also "should_tag_with_type". *)

     fun should_encode T =

       should_encode_type ctxt nonmono_Ts All_Types T orelse

       (case type_sys of

          Tags (Polymorphic, level, Lightweight) =>

          level <> All_Types andalso Monomorph.typ_has_tvars T

        | _ => false)

     val tag_with = tag_with_type ctxt format nonmono_Ts type_sys 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],

                        simp_info, 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],

                            simp_info, NONE))

           | _ => raise Fail "expected nonempty tail"

         else

           I

       end

   in

     [] |> not pred_sym ? add_formula_for_res

        |> 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 nonmono_Ts type_sys

                                 (s, decls) =

   case type_sys of

     Simple_Types _ =>

     decls |> map (decl_line_for_sym ctxt format nonmono_Ts type_sys s)

   | Preds _ =>

     let

       val decls =

         case decls of

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

           let val T = result_type_of_decl decl in

             if forall (curry (type_instance ctxt o swap) T

                        o result_type_of_decl) decls' then

               [decl]

             else

               decls

           end

         | _ => decls

       val n = length decls

       val decls =

         decls

         |> filter (should_predicate_on_type ctxt nonmono_Ts type_sys (K true)

                    o result_type_of_decl)

     in

       (0 upto length decls - 1, decls)

       |-> map2 (formula_line_for_preds_sym_decl ctxt format conj_sym_kind

                                                 nonmono_Ts type_sys n s)

     end

   | Tags (_, _, heaviness) =>

     (case heaviness of

        Heavyweight => []

      | Lightweight =>

        let val n = length decls in

          (0 upto n - 1 ~~ decls)

          |> maps (formula_lines_for_lightweight_tags_sym_decl ctxt format

                       conj_sym_kind nonmono_Ts type_sys n s)

        end)

 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind nonmono_Ts

                                      type_sys sym_decl_tab =

   sym_decl_tab

   |> Symtab.dest

   |> sort_wrt fst

   |> rpair []

   |-> fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind

                                                      nonmono_Ts type_sys)

 fun needs_type_tag_idempotence (Tags (poly, level, heaviness)) =

     poly <> Mangled_Monomorphic andalso

     ((level = All_Types andalso heaviness = Lightweight) orelse

      level = Nonmonotonic_Types orelse level = Finite_Types)

   | 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 experimental purposes *)

 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_sys

         freshen_facts readable_names preproc hyp_ts concl_t facts =

   let

     val (format, type_sys) = choose_format [format] type_sys

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

       translate_formulas ctxt format prem_kind type_sys preproc hyp_ts concl_t

                          facts

     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply

     val nonmono_Ts = conjs @ facts |> nonmonotonic_types_for_facts ctxt type_sys

     val repair = repair_fact ctxt format type_sys 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_sys

                        |> map repair

     val lavish_nonmono_Ts =

       if null nonmono_Ts orelse nonmono_Ts = [@{typ bool}] orelse

          polymorphism_of_type_sys type_sys <> Polymorphic then

         nonmono_Ts

       else

         [TVar (("'a", 0), HOLogic.typeS)]

     val sym_decl_lines =

       (conjs, helpers @ facts)

       |> sym_decl_table_for_facts ctxt type_sys repaired_sym_tab

       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind

                                           lavish_nonmono_Ts type_sys

     val helper_lines =

       0 upto length helpers - 1 ~~ helpers

       |> map (formula_line_for_fact ctxt format helper_prefix I false

                                     lavish_nonmono_Ts type_sys)

       |> (if needs_type_tag_idempotence type_sys then

             cons (type_tag_idempotence_fact ())

           else

             I)

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

        FLOTTER hack. *)

     val problem =

       [(explicit_declsN, sym_decl_lines),

        (factsN,

         map (formula_line_for_fact ctxt format fact_prefix ascii_of

                                    freshen_facts nonmono_Ts type_sys)

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

        (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),

        (aritiesN, map formula_line_for_arity_clause arity_clauses),

        (helpersN, helper_lines),

        (conjsN,

         map (formula_line_for_conjecture ctxt format nonmono_Ts type_sys)

             conjs),

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

     val problem =

       problem

       |> (case format of

             CNF => ensure_cnf_problem

           | CNF_UEQ => filter_cnf_ueq_problem

           | _ => I)

       |> (if is_format_typed format then

             declare_undeclared_syms_in_atp_problem type_decl_prefix

                                                    implicit_declsN

           else

             I)

     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

 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;