(*  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 | Extensionality | Intro | Elim | Simp | Local | Assum |

     Chained

   datatype order = First_Order | Higher_Order

   datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

   datatype type_level =

     All_Types | Noninf_Nonmono_Types | Fin_Nonmono_Types | Const_Arg_Types |

     No_Types

   datatype type_heaviness = Heavyweight | Lightweight

   datatype type_enc =

     Simple_Types of order * 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 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 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 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 is_locality_global : locality -> bool

   val type_enc_from_string : 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_virtually_sound : type_enc -> bool

   val is_type_enc_fairly_sound : type_enc -> bool

   val choose_format : format list -> type_enc -> format * 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 conceal_lambdas : Proof.context -> term -> term

   val introduce_combinators : Proof.context -> term -> term

   val prepare_atp_problem :

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

     -> bool -> (term list -> term list * term list) -> 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 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"

 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 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 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 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 = "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 atp_weak_prefix = "ATP:"

 val concealed_lambda_prefix = atp_weak_prefix ^ "lambda_"

 (*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_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" is mapped to the ATP's equality. *)

 fun make_fixed_const @{const_name HOL.eq} = tptp_old_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

 (* 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 ^ "___" ^ 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

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

    Also accumulates sort infomation. *)

 fun iterm_from_term thy bs (P $ Q) =

     let

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

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

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

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

     (IConst (`make_fixed_const c, T,

              if String.isPrefix old_skolem_const_prefix c then

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

              else

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

      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 _ _ (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 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 (s, T) => (IConst (`make_bound_var s, T, []), atyps_of T))

   | iterm_from_term thy 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 (tm, atomic_Ts) = iterm_from_term thy ((s, T) :: bs) t

     in

       (IAbs ((`make_bound_var s, T), tm),

        union (op =) atomic_Ts (atyps_of T))

     end

 datatype locality =

   General | Helper | Extensionality | 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 order = First_Order | Higher_Order

 datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic

 datatype type_level =

   All_Types | Noninf_Nonmono_Types | Fin_Nonmono_Types | Const_Arg_Types |

   No_Types

 datatype type_heaviness = Heavyweight | Lightweight

 datatype type_enc =

   Simple_Types of order * 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_enc_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 Metis and

              Mirabelle. *)

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

             SOME s => (Noninf_Nonmono_Types, s)

           | NONE =>

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

               SOME s => (Fin_Nonmono_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 (First_Order, level)

          | ("simple_higher", (NONE, _, Lightweight)) =>

            if level = Noninf_Nonmono_Types then raise Same.SAME

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

   | is_type_enc_higher_order _ = false

 fun polymorphism_of_type_enc (Simple_Types _) = Mangled_Monomorphic

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

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

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

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

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

 fun heaviness_of_type_enc (Simple_Types _) = Heavyweight

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

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

 fun is_type_level_virtually_sound level =

   level = All_Types orelse level = Noninf_Nonmono_Types

 val is_type_enc_virtually_sound =

   is_type_level_virtually_sound o level_of_type_enc

 fun is_type_level_fairly_sound level =

   is_type_level_virtually_sound level orelse level = Fin_Nonmono_Types

 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc

 fun choose_format formats (Simple_Types (order, level)) =

     if member (op =) formats THF then

       (THF, Simple_Types (order, level))

     else if member (op =) formats TFF then

       (TFF, Simple_Types (First_Order, level))

     else

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

   | choose_format formats type_enc =

     (case hd formats of

        CNF_UEQ =>

        (CNF_UEQ, case type_enc of

                    Preds stuff =>

                    (if is_type_enc_fairly_sound type_enc then Tags else Preds)

                        stuff

                  | _ => type_enc)

      | format => (format, type_enc))

 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

 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_enc =

     level_of_type_enc type_enc = All_Types andalso

     heaviness_of_type_enc type_enc = Heavyweight

 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, Heavyweight) => No_Type_Args

   | _ =>

     if level_of_type_enc type_enc = No_Types orelse

        s = @{const_name HOL.eq} orelse

        (s = app_op_name andalso

         level_of_type_enc type_enc = 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)

 (* 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 bounds (ATerm (name as (s, _), tms)) =

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

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

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

 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 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 = 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

 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"

 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_ho_term type_enc 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

       | 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 intro top_level (IApp (tm1, tm2)) =

         IApp (intro top_level tm1, intro false tm2)

       | intro top_level (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") => (`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, "c_All") => (`I tptp_ho_forall, [])

              | (false, "c_Ex") => (`I tptp_ho_exists, [])

              | (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) => 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 type_enc eq_as_iff =

   let

     fun do_term bs t atomic_types =

       iterm_from_term thy bs (Envir.eta_contract t)

       |>> (introduce_proxies type_enc #> 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 = 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_conceal_lambdas Ts (t1 $ t2) =

     do_conceal_lambdas Ts t1 $ do_conceal_lambdas Ts t2

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

     (* slightly unsound because of hash collisions *)

     Free (concealed_lambda_prefix ^ string_of_int (hash_term t),

           T --> fastype_of1 (Ts, t))

   | do_conceal_lambdas _ t = t

 val conceal_lambdas = simple_translate_lambdas (K do_conceal_lambdas)

 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_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 type_enc eq_as_iff name loc kind t =

   let

     val (iformula, atomic_types) =

       iformula_from_prop thy type_enc eq_as_iff 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 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 make_conjecture ctxt format type_enc ps =

   let

     val thy = Proof_Context.theory_of ctxt

     val last = length ps - 1

   in

     map2 (fn j => fn ((name, loc), (kind, t)) =>

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

                |> (j <> last) = (kind = Conjecture) ? update_iformula mk_anot)

          (0 upto last) ps

   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 _ Fin_Nonmono_Types T =

     is_type_surely_finite ctxt false 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 (IConst ((s, _), _, _)) =

     String.isPrefix bound_var_prefix s

   | is_var_or_bound_var (IVar _) = 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_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_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_iterm_comb tm in

         (case head of

            IConst ((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

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

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

          | _ => 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_iterm_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 =

   IConst (`make_fixed_const 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 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

       (* 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 (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, 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 **)

 (* 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}),

    (("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+}),

    (("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]}),

    (("fAll", false), []), (*TODO: add helpers*)

    (("fEx", false), []), (*TODO: add helpers*)

    (("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, isabelle_info simpN, NONE)

   end

 fun should_specialize_helper type_enc t =

   polymorphism_of_type_enc type_enc = Mangled_Monomorphic 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 OldTerm.term_tfrees

 val tvar_classes_of_terms = classes_of_terms OldTerm.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, 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 ctxt 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,

      (conjs, facts @ lambdas, 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_iterm ctxt format type_enc T tm =

   IApp (IConst (type_pred, 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_predicate_on_var_in_formula pos phi (SOME true) name =

     formula_fold pos (is_var_positively_naked_in_term name) phi false

   | should_predicate_on_var_in_formula _ _ _ _ = true

 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 nonmono_Ts 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 nonmono_Ts 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 nonmono_Ts 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 nonmono_Ts type_enc site u T then

                 tag_with_type ctxt format nonmono_Ts type_enc pos T

               else

                 I)

       end

   in aux end

 and formula_from_iformula ctxt format nonmono_Ts type_enc

                           should_predicate_on_var =

   let

     val do_term = ho_term_from_iterm ctxt format nonmono_Ts type_enc o Top_Level

     val do_bound_type =

       case type_enc of

         Simple_Types (_, level) =>

         homogenized_type ctxt nonmono_Ts 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_predicate_on_type ctxt nonmono_Ts type_enc

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

         IVar (name, T)

         |> type_pred_iterm ctxt format type_enc 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 end

 fun bound_tvars type_enc Ts =

   mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)

                 (type_literals_for_types type_enc add_sorts_on_tvar Ts))

 (* 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 nonmono_Ts

         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 nonmono_Ts type_enc

                             should_predicate_on_var_in_formula

                             (if pos then SOME true else NONE)

    |> bound_tvars type_enc atomic_types

    |> close_formula_universally,

    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 ({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, isabelle_info introN, 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 ^ 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, isabelle_info introN, NONE)

 fun formula_line_for_conjecture ctxt format nonmono_Ts type_enc

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

   Formula (conjecture_prefix ^ name, kind,

            formula_from_iformula ctxt format nonmono_Ts type_enc

                should_predicate_on_var_in_formula (SOME false)

                (close_iformula_universally iformula)

            |> bound_tvars type_enc atomic_types

            |> close_formula_universally, 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

 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 should_declare_sym type_enc pred_sym s =

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

   (case type_enc of

      Simple_Types _ => true

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

    | _ => not pred_sym)

 fun sym_decl_table_for_facts ctxt type_enc repaired_sym_tab (conjs, facts) =

   let

     fun add_iterm 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 in

              if should_declare_sym type_enc 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

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

          | _ => I)

         #> fold (add_iterm in_conj) args

       end

     fun add_fact in_conj = fact_lift (formula_fold NONE (K (add_iterm in_conj)))

   in

     Symtab.empty

     |> is_type_enc_fairly_sound type_enc

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

   end

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

   | add_iterm_nonmonotonic_types ctxt level sound locality _

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

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

      (case level of

         Noninf_Nonmono_Types =>

         not (is_locality_global locality) orelse

         not (is_type_surely_infinite ctxt sound T)

       | Fin_Nonmono_Types => is_type_surely_finite ctxt false T

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

   | add_iterm_nonmonotonic_types _ _ _ _ _ _ = I

 fun add_fact_nonmonotonic_types ctxt level sound

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

   formula_fold (SOME (kind <> Conjecture))

                (add_iterm_nonmonotonic_types ctxt level sound locality)

                iformula

 fun nonmonotonic_types_for_facts ctxt type_enc sound facts =

   let val level = level_of_type_enc type_enc in

     if level = Noninf_Nonmono_Types orelse level = Fin_Nonmono_Types then

       [] |> fold (add_fact_nonmonotonic_types ctxt level sound) 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_enc s

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

   let

     val (T_arg_Ts, level) =

       case type_enc 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))

           |> ho_type_from_typ format type_enc pred_sym (length T_arg_Ts + ary))

   end

 fun formula_line_for_preds_sym_decl ctxt format conj_sym_kind nonmono_Ts

         poly_nonmono_Ts 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 orelse

       not (should_predicate_on_type ctxt nonmono_Ts type_enc (K false) T)

     val bound_Ts =

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

   in

     Formula (preds_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_pred_iterm ctxt format type_enc res_T

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

              |> formula_from_iformula ctxt format poly_nonmono_Ts type_enc

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

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

              |> close_formula_universally

              |> maybe_negate,

              isabelle_info introN, NONE)

   end

 fun formula_lines_for_lightweight_tags_sym_decl ctxt format conj_sym_kind

         poly_nonmono_Ts type_enc 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_aterm format type_enc (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_enc atomic_Ts

       |> close_formula_universally

       |> maybe_negate

     (* See also "should_tag_with_type". *)

     fun should_encode T =

       should_encode_type ctxt poly_nonmono_Ts All_Types T orelse

       (case type_enc of

          Tags (Polymorphic, level, Lightweight) =>

          level <> All_Types andalso Monomorph.typ_has_tvars T

        | _ => false)

     val tag_with = tag_with_type ctxt format poly_nonmono_Ts 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

        |> 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

                                 poly_nonmono_Ts type_enc (s, decls) =

   case type_enc of

     Simple_Types _ =>

     decls |> map (decl_line_for_sym ctxt format nonmono_Ts type_enc 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 poly_nonmono_Ts type_enc

                                                   (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 poly_nonmono_Ts type_enc 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 poly_nonmono_Ts type_enc n s)

        end)

 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind nonmono_Ts

                                      poly_nonmono_Ts type_enc 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 poly_nonmono_Ts type_enc)

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

     poly <> Mangled_Monomorphic andalso

     ((level = All_Types andalso heaviness = Lightweight) orelse

      level = Noninf_Nonmono_Types orelse level = Fin_Nonmono_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 experiments *)

 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc sound

         exporter trans_lambdas readable_names preproc hyp_ts concl_t facts =

   let

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

     val (fact_names, (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 nonmono_Ts =

       conjs @ facts |> nonmonotonic_types_for_facts ctxt type_enc sound

     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 poly_nonmono_Ts =

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

          polymorphism_of_type_enc type_enc <> Polymorphic then

         nonmono_Ts

       else

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

     val sym_decl_lines =

       (conjs, helpers @ facts)

       |> sym_decl_table_for_facts ctxt type_enc repaired_sym_tab

       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind nonmono_Ts

                                                poly_nonmono_Ts type_enc

     val helper_lines =

       0 upto length helpers - 1 ~~ helpers

       |> map (formula_line_for_fact ctxt format helper_prefix I false true

                                     poly_nonmono_Ts type_enc)

       |> (if needs_type_tag_idempotence type_enc 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

                                    (not exporter) (not exporter) nonmono_Ts

                                    type_enc)

             (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_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

           | _ => 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

   | 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;