src/HOL/Tools/ATP/atp_translate.ML
author nik
Tue Aug 30 14:12:55 2011 +0200 (2011-08-30)
changeset 44586 eeba1eedf32d
parent 44585 cfe7f4a68e51
child 44587 0f50f158eb57
permissions -rw-r--r--
improved handling of induction rules in Sledgehammer
     1 (*  Title:      HOL/Tools/ATP/atp_translate.ML
     2     Author:     Fabian Immler, TU Muenchen
     3     Author:     Makarius
     4     Author:     Jasmin Blanchette, TU Muenchen
     5 
     6 Translation of HOL to FOL for Metis and Sledgehammer.
     7 *)
     8 
     9 signature ATP_TRANSLATE =
    10 sig
    11   type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
    12   type connective = ATP_Problem.connective
    13   type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
    14   type format = ATP_Problem.format
    15   type formula_kind = ATP_Problem.formula_kind
    16   type 'a problem = 'a ATP_Problem.problem
    17 
    18   datatype locality =
    19     General | Helper | Induction | Extensionality | Intro | Elim | Simp |
    20     Local | Assum | Chained
    21 
    22   datatype order = First_Order | Higher_Order
    23   datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
    24   datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
    25   datatype type_level =
    26     All_Types |
    27     Noninf_Nonmono_Types of soundness |
    28     Fin_Nonmono_Types |
    29     Const_Arg_Types |
    30     No_Types
    31   datatype type_uniformity = Uniform | Nonuniform
    32 
    33   datatype type_enc =
    34     Simple_Types of order * type_level |
    35     Guards of polymorphism * type_level * type_uniformity |
    36     Tags of polymorphism * type_level * type_uniformity
    37 
    38   val type_tag_idempotence : bool Config.T
    39   val type_tag_arguments : bool Config.T
    40   val no_lambdasN : string
    41   val concealedN : string
    42   val liftingN : string
    43   val combinatorsN : string
    44   val hybridN : string
    45   val lambdasN : string
    46   val smartN : string
    47   val schematic_var_prefix : string
    48   val fixed_var_prefix : string
    49   val tvar_prefix : string
    50   val tfree_prefix : string
    51   val const_prefix : string
    52   val type_const_prefix : string
    53   val class_prefix : string
    54   val polymorphic_free_prefix : string
    55   val skolem_const_prefix : string
    56   val old_skolem_const_prefix : string
    57   val new_skolem_const_prefix : string
    58   val type_decl_prefix : string
    59   val sym_decl_prefix : string
    60   val guards_sym_formula_prefix : string
    61   val tags_sym_formula_prefix : string
    62   val fact_prefix : string
    63   val conjecture_prefix : string
    64   val helper_prefix : string
    65   val class_rel_clause_prefix : string
    66   val arity_clause_prefix : string
    67   val tfree_clause_prefix : string
    68   val typed_helper_suffix : string
    69   val untyped_helper_suffix : string
    70   val type_tag_idempotence_helper_name : string
    71   val predicator_name : string
    72   val app_op_name : string
    73   val type_guard_name : string
    74   val type_tag_name : string
    75   val simple_type_prefix : string
    76   val prefixed_predicator_name : string
    77   val prefixed_app_op_name : string
    78   val prefixed_type_tag_name : string
    79   val ascii_of : string -> string
    80   val unascii_of : string -> string
    81   val strip_prefix_and_unascii : string -> string -> string option
    82   val proxy_table : (string * (string * (thm * (string * string)))) list
    83   val proxify_const : string -> (string * string) option
    84   val invert_const : string -> string
    85   val unproxify_const : string -> string
    86   val new_skolem_var_name_from_const : string -> string
    87   val atp_irrelevant_consts : string list
    88   val atp_schematic_consts_of : term -> typ list Symtab.table
    89   val type_enc_from_string : soundness -> string -> type_enc
    90   val is_type_enc_higher_order : type_enc -> bool
    91   val polymorphism_of_type_enc : type_enc -> polymorphism
    92   val level_of_type_enc : type_enc -> type_level
    93   val is_type_enc_quasi_sound : type_enc -> bool
    94   val is_type_enc_fairly_sound : type_enc -> bool
    95   val adjust_type_enc : format -> type_enc -> type_enc
    96   val mk_aconns :
    97     connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
    98   val unmangled_const : string -> string * (string, 'b) ho_term list
    99   val unmangled_const_name : string -> string
   100   val helper_table : ((string * bool) * thm list) list
   101   val factsN : string
   102   val prepare_atp_problem :
   103     Proof.context -> format -> formula_kind -> formula_kind -> type_enc
   104     -> bool -> string -> bool -> bool -> term list -> term
   105     -> ((string * locality) * term) list
   106     -> string problem * string Symtab.table * int * int
   107        * (string * locality) list vector * int list * int Symtab.table
   108   val atp_problem_weights : string problem -> (string * real) list
   109 end;
   110 
   111 structure ATP_Translate : ATP_TRANSLATE =
   112 struct
   113 
   114 open ATP_Util
   115 open ATP_Problem
   116 
   117 type name = string * string
   118 
   119 val type_tag_idempotence =
   120   Attrib.setup_config_bool @{binding atp_type_tag_idempotence} (K false)
   121 val type_tag_arguments =
   122   Attrib.setup_config_bool @{binding atp_type_tag_arguments} (K false)
   123 
   124 val no_lambdasN = "no_lambdas"
   125 val concealedN = "concealed"
   126 val liftingN = "lifting"
   127 val combinatorsN = "combinators"
   128 val hybridN = "hybrid"
   129 val lambdasN = "lambdas"
   130 val smartN = "smart"
   131 
   132 val generate_info = false (* experimental *)
   133 
   134 fun isabelle_info s =
   135   if generate_info then SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
   136   else NONE
   137 
   138 val introN = "intro"
   139 val elimN = "elim"
   140 val simpN = "simp"
   141 
   142 val bound_var_prefix = "B_"
   143 val all_bound_var_prefix = "BA_"
   144 val exist_bound_var_prefix = "BE_"
   145 val schematic_var_prefix = "V_"
   146 val fixed_var_prefix = "v_"
   147 val tvar_prefix = "T_"
   148 val tfree_prefix = "t_"
   149 val const_prefix = "c_"
   150 val type_const_prefix = "tc_"
   151 val simple_type_prefix = "s_"
   152 val class_prefix = "cl_"
   153 
   154 val polymorphic_free_prefix = "poly_free"
   155 
   156 val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"
   157 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
   158 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
   159 
   160 val type_decl_prefix = "ty_"
   161 val sym_decl_prefix = "sy_"
   162 val guards_sym_formula_prefix = "gsy_"
   163 val tags_sym_formula_prefix = "tsy_"
   164 val fact_prefix = "fact_"
   165 val conjecture_prefix = "conj_"
   166 val helper_prefix = "help_"
   167 val class_rel_clause_prefix = "clar_"
   168 val arity_clause_prefix = "arity_"
   169 val tfree_clause_prefix = "tfree_"
   170 
   171 val lambda_fact_prefix = "ATP.lambda_"
   172 val typed_helper_suffix = "_T"
   173 val untyped_helper_suffix = "_U"
   174 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"
   175 
   176 val predicator_name = "pp"
   177 val app_op_name = "aa"
   178 val type_guard_name = "gg"
   179 val type_tag_name = "tt"
   180 
   181 val prefixed_predicator_name = const_prefix ^ predicator_name
   182 val prefixed_app_op_name = const_prefix ^ app_op_name
   183 val prefixed_type_tag_name = const_prefix ^ type_tag_name
   184 
   185 (* Freshness almost guaranteed! *)
   186 val atp_weak_prefix = "ATP:"
   187 
   188 (*Escaping of special characters.
   189   Alphanumeric characters are left unchanged.
   190   The character _ goes to __
   191   Characters in the range ASCII space to / go to _A to _P, respectively.
   192   Other characters go to _nnn where nnn is the decimal ASCII code.*)
   193 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
   194 
   195 fun stringN_of_int 0 _ = ""
   196   | stringN_of_int k n =
   197     stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
   198 
   199 fun ascii_of_char c =
   200   if Char.isAlphaNum c then
   201     String.str c
   202   else if c = #"_" then
   203     "__"
   204   else if #" " <= c andalso c <= #"/" then
   205     "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
   206   else
   207     (* fixed width, in case more digits follow *)
   208     "_" ^ stringN_of_int 3 (Char.ord c)
   209 
   210 val ascii_of = String.translate ascii_of_char
   211 
   212 (** Remove ASCII armoring from names in proof files **)
   213 
   214 (* We don't raise error exceptions because this code can run inside a worker
   215    thread. Also, the errors are impossible. *)
   216 val unascii_of =
   217   let
   218     fun un rcs [] = String.implode(rev rcs)
   219       | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
   220         (* Three types of _ escapes: __, _A to _P, _nnn *)
   221       | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
   222       | un rcs (#"_" :: c :: cs) =
   223         if #"A" <= c andalso c<= #"P" then
   224           (* translation of #" " to #"/" *)
   225           un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
   226         else
   227           let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
   228             case Int.fromString (String.implode digits) of
   229               SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
   230             | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
   231           end
   232       | un rcs (c :: cs) = un (c :: rcs) cs
   233   in un [] o String.explode end
   234 
   235 (* If string s has the prefix s1, return the result of deleting it,
   236    un-ASCII'd. *)
   237 fun strip_prefix_and_unascii s1 s =
   238   if String.isPrefix s1 s then
   239     SOME (unascii_of (String.extract (s, size s1, NONE)))
   240   else
   241     NONE
   242 
   243 val proxy_table =
   244   [("c_False", (@{const_name False}, (@{thm fFalse_def},
   245        ("fFalse", @{const_name ATP.fFalse})))),
   246    ("c_True", (@{const_name True}, (@{thm fTrue_def},
   247        ("fTrue", @{const_name ATP.fTrue})))),
   248    ("c_Not", (@{const_name Not}, (@{thm fNot_def},
   249        ("fNot", @{const_name ATP.fNot})))),
   250    ("c_conj", (@{const_name conj}, (@{thm fconj_def},
   251        ("fconj", @{const_name ATP.fconj})))),
   252    ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
   253        ("fdisj", @{const_name ATP.fdisj})))),
   254    ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
   255        ("fimplies", @{const_name ATP.fimplies})))),
   256    ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
   257        ("fequal", @{const_name ATP.fequal})))),
   258    ("c_All", (@{const_name All}, (@{thm fAll_def},
   259        ("fAll", @{const_name ATP.fAll})))),
   260    ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
   261        ("fEx", @{const_name ATP.fEx}))))]
   262 
   263 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
   264 
   265 (* Readable names for the more common symbolic functions. Do not mess with the
   266    table unless you know what you are doing. *)
   267 val const_trans_table =
   268   [(@{type_name Product_Type.prod}, "prod"),
   269    (@{type_name Sum_Type.sum}, "sum"),
   270    (@{const_name False}, "False"),
   271    (@{const_name True}, "True"),
   272    (@{const_name Not}, "Not"),
   273    (@{const_name conj}, "conj"),
   274    (@{const_name disj}, "disj"),
   275    (@{const_name implies}, "implies"),
   276    (@{const_name HOL.eq}, "equal"),
   277    (@{const_name All}, "All"),
   278    (@{const_name Ex}, "Ex"),
   279    (@{const_name If}, "If"),
   280    (@{const_name Set.member}, "member"),
   281    (@{const_name Meson.COMBI}, "COMBI"),
   282    (@{const_name Meson.COMBK}, "COMBK"),
   283    (@{const_name Meson.COMBB}, "COMBB"),
   284    (@{const_name Meson.COMBC}, "COMBC"),
   285    (@{const_name Meson.COMBS}, "COMBS")]
   286   |> Symtab.make
   287   |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
   288 
   289 (* Invert the table of translations between Isabelle and ATPs. *)
   290 val const_trans_table_inv =
   291   const_trans_table |> Symtab.dest |> map swap |> Symtab.make
   292 val const_trans_table_unprox =
   293   Symtab.empty
   294   |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
   295 
   296 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
   297 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
   298 
   299 fun lookup_const c =
   300   case Symtab.lookup const_trans_table c of
   301     SOME c' => c'
   302   | NONE => ascii_of c
   303 
   304 fun ascii_of_indexname (v, 0) = ascii_of v
   305   | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
   306 
   307 fun make_bound_var x = bound_var_prefix ^ ascii_of x
   308 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
   309 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
   310 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
   311 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
   312 
   313 fun make_schematic_type_var (x, i) =
   314       tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
   315 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
   316 
   317 (* "HOL.eq" is mapped to the ATP's equality. *)
   318 fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
   319   | make_fixed_const (SOME (THF With_Choice)) "Hilbert_Choice.Eps"(*FIXME*) =
   320       tptp_choice
   321   | make_fixed_const _ c = const_prefix ^ lookup_const c
   322 
   323 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
   324 
   325 fun make_type_class clas = class_prefix ^ ascii_of clas
   326 
   327 fun new_skolem_var_name_from_const s =
   328   let val ss = s |> space_explode Long_Name.separator in
   329     nth ss (length ss - 2)
   330   end
   331 
   332 (* These are either simplified away by "Meson.presimplify" (most of the time) or
   333    handled specially via "fFalse", "fTrue", ..., "fequal". *)
   334 val atp_irrelevant_consts =
   335   [@{const_name False}, @{const_name True}, @{const_name Not},
   336    @{const_name conj}, @{const_name disj}, @{const_name implies},
   337    @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
   338 
   339 val atp_monomorph_bad_consts =
   340   atp_irrelevant_consts @
   341   (* These are ignored anyway by the relevance filter (unless they appear in
   342      higher-order places) but not by the monomorphizer. *)
   343   [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
   344    @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
   345    @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
   346 
   347 fun add_schematic_const (x as (_, T)) =
   348   Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
   349 val add_schematic_consts_of =
   350   Term.fold_aterms (fn Const (x as (s, _)) =>
   351                        not (member (op =) atp_monomorph_bad_consts s)
   352                        ? add_schematic_const x
   353                       | _ => I)
   354 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
   355 
   356 (** Definitions and functions for FOL clauses and formulas for TPTP **)
   357 
   358 (* The first component is the type class; the second is a "TVar" or "TFree". *)
   359 datatype type_literal =
   360   TyLitVar of name * name |
   361   TyLitFree of name * name
   362 
   363 
   364 (** Isabelle arities **)
   365 
   366 datatype arity_literal =
   367   TConsLit of name * name * name list |
   368   TVarLit of name * name
   369 
   370 fun gen_TVars 0 = []
   371   | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)
   372 
   373 val type_class = the_single @{sort type}
   374 
   375 fun add_packed_sort tvar =
   376   fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))
   377 
   378 type arity_clause =
   379   {name : string,
   380    prem_lits : arity_literal list,
   381    concl_lits : arity_literal}
   382 
   383 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
   384 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
   385   let
   386     val tvars = gen_TVars (length args)
   387     val tvars_srts = ListPair.zip (tvars, args)
   388   in
   389     {name = name,
   390      prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,
   391      concl_lits = TConsLit (`make_type_class cls,
   392                             `make_fixed_type_const tcons,
   393                             tvars ~~ tvars)}
   394   end
   395 
   396 fun arity_clause _ _ (_, []) = []
   397   | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)
   398     arity_clause seen n (tcons, ars)
   399   | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
   400     if member (op =) seen class then
   401       (* multiple arities for the same (tycon, class) pair *)
   402       make_axiom_arity_clause (tcons,
   403           lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
   404           ar) ::
   405       arity_clause seen (n + 1) (tcons, ars)
   406     else
   407       make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
   408                                ascii_of class, ar) ::
   409       arity_clause (class :: seen) n (tcons, ars)
   410 
   411 fun multi_arity_clause [] = []
   412   | multi_arity_clause ((tcons, ars) :: tc_arlists) =
   413       arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
   414 
   415 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
   416    theory thy provided its arguments have the corresponding sorts. *)
   417 fun type_class_pairs thy tycons classes =
   418   let
   419     val alg = Sign.classes_of thy
   420     fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
   421     fun add_class tycon class =
   422       cons (class, domain_sorts tycon class)
   423       handle Sorts.CLASS_ERROR _ => I
   424     fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
   425   in map try_classes tycons end
   426 
   427 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
   428 fun iter_type_class_pairs _ _ [] = ([], [])
   429   | iter_type_class_pairs thy tycons classes =
   430       let
   431         fun maybe_insert_class s =
   432           (s <> type_class andalso not (member (op =) classes s))
   433           ? insert (op =) s
   434         val cpairs = type_class_pairs thy tycons classes
   435         val newclasses =
   436           [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
   437         val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
   438       in (classes' @ classes, union (op =) cpairs' cpairs) end
   439 
   440 fun make_arity_clauses thy tycons =
   441   iter_type_class_pairs thy tycons ##> multi_arity_clause
   442 
   443 
   444 (** Isabelle class relations **)
   445 
   446 type class_rel_clause =
   447   {name : string,
   448    subclass : name,
   449    superclass : name}
   450 
   451 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
   452    in theory "thy". *)
   453 fun class_pairs _ [] _ = []
   454   | class_pairs thy subs supers =
   455       let
   456         val class_less = Sorts.class_less (Sign.classes_of thy)
   457         fun add_super sub super = class_less (sub, super) ? cons (sub, super)
   458         fun add_supers sub = fold (add_super sub) supers
   459       in fold add_supers subs [] end
   460 
   461 fun make_class_rel_clause (sub, super) =
   462   {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
   463    superclass = `make_type_class super}
   464 
   465 fun make_class_rel_clauses thy subs supers =
   466   map make_class_rel_clause (class_pairs thy subs supers)
   467 
   468 (* intermediate terms *)
   469 datatype iterm =
   470   IConst of name * typ * typ list |
   471   IVar of name * typ |
   472   IApp of iterm * iterm |
   473   IAbs of (name * typ) * iterm
   474 
   475 fun ityp_of (IConst (_, T, _)) = T
   476   | ityp_of (IVar (_, T)) = T
   477   | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
   478   | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
   479 
   480 (*gets the head of a combinator application, along with the list of arguments*)
   481 fun strip_iterm_comb u =
   482   let
   483     fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
   484       | stripc x = x
   485   in stripc (u, []) end
   486 
   487 fun atyps_of T = fold_atyps (insert (op =)) T []
   488 
   489 fun new_skolem_const_name s num_T_args =
   490   [new_skolem_const_prefix, s, string_of_int num_T_args]
   491   |> space_implode Long_Name.separator
   492 
   493 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
   494    Also accumulates sort infomation. *)
   495 fun iterm_from_term thy format bs (P $ Q) =
   496     let
   497       val (P', P_atomics_Ts) = iterm_from_term thy format bs P
   498       val (Q', Q_atomics_Ts) = iterm_from_term thy format bs Q
   499     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
   500   | iterm_from_term thy format _ (Const (c, T)) =
   501     (IConst (`(make_fixed_const (SOME format)) c, T,
   502              if String.isPrefix old_skolem_const_prefix c then
   503                [] |> Term.add_tvarsT T |> map TVar
   504              else
   505                (c, T) |> Sign.const_typargs thy),
   506      atyps_of T)
   507   | iterm_from_term _ _ _ (Free (s, T)) =
   508     (IConst (`make_fixed_var s, T,
   509              if String.isPrefix polymorphic_free_prefix s then [T] else []),
   510      atyps_of T)
   511   | iterm_from_term _ format _ (Var (v as (s, _), T)) =
   512     (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
   513        let
   514          val Ts = T |> strip_type |> swap |> op ::
   515          val s' = new_skolem_const_name s (length Ts)
   516        in IConst (`(make_fixed_const (SOME format)) s', T, Ts) end
   517      else
   518        IVar ((make_schematic_var v, s), T), atyps_of T)
   519   | iterm_from_term _ _ bs (Bound j) =
   520     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))
   521   | iterm_from_term thy format bs (Abs (s, T, t)) =
   522     let
   523       fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
   524       val s = vary s
   525       val name = `make_bound_var s
   526       val (tm, atomic_Ts) = iterm_from_term thy format ((s, (name, T)) :: bs) t
   527     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end
   528 
   529 datatype locality =
   530   General | Helper | Induction | Extensionality | Intro | Elim | Simp |
   531   Local | Assum | Chained
   532 
   533 datatype order = First_Order | Higher_Order
   534 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
   535 datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
   536 datatype type_level =
   537   All_Types |
   538   Noninf_Nonmono_Types of soundness |
   539   Fin_Nonmono_Types |
   540   Const_Arg_Types |
   541   No_Types
   542 datatype type_uniformity = Uniform | Nonuniform
   543 
   544 datatype type_enc =
   545   Simple_Types of order * type_level |
   546   Guards of polymorphism * type_level * type_uniformity |
   547   Tags of polymorphism * type_level * type_uniformity
   548 
   549 fun try_unsuffixes ss s =
   550   fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
   551 
   552 fun type_enc_from_string soundness s =
   553   (case try (unprefix "poly_") s of
   554      SOME s => (SOME Polymorphic, s)
   555    | NONE =>
   556      case try (unprefix "raw_mono_") s of
   557        SOME s => (SOME Raw_Monomorphic, s)
   558      | NONE =>
   559        case try (unprefix "mono_") s of
   560          SOME s => (SOME Mangled_Monomorphic, s)
   561        | NONE => (NONE, s))
   562   ||> (fn s =>
   563           (* "_query" and "_bang" are for the ASCII-challenged Metis and
   564              Mirabelle. *)
   565           case try_unsuffixes ["?", "_query"] s of
   566             SOME s => (Noninf_Nonmono_Types soundness, s)
   567           | NONE =>
   568             case try_unsuffixes ["!", "_bang"] s of
   569               SOME s => (Fin_Nonmono_Types, s)
   570             | NONE => (All_Types, s))
   571   ||> apsnd (fn s =>
   572                 case try (unsuffix "_uniform") s of
   573                   SOME s => (Uniform, s)
   574                 | NONE => (Nonuniform, s))
   575   |> (fn (poly, (level, (uniformity, core))) =>
   576          case (core, (poly, level, uniformity)) of
   577            ("simple", (NONE, _, Nonuniform)) =>
   578            Simple_Types (First_Order, level)
   579          | ("simple_higher", (NONE, _, Nonuniform)) =>
   580            (case level of
   581               Noninf_Nonmono_Types _ => raise Same.SAME
   582             | _ => Simple_Types (Higher_Order, level))
   583          | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)
   584          | ("tags", (SOME Polymorphic, _, _)) =>
   585            Tags (Polymorphic, level, uniformity)
   586          | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)
   587          | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>
   588            Guards (poly, Const_Arg_Types, Nonuniform)
   589          | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>
   590            Guards (Polymorphic, No_Types, Nonuniform)
   591          | _ => raise Same.SAME)
   592   handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
   593 
   594 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _)) = true
   595   | is_type_enc_higher_order _ = false
   596 
   597 fun polymorphism_of_type_enc (Simple_Types _) = Mangled_Monomorphic
   598   | polymorphism_of_type_enc (Guards (poly, _, _)) = poly
   599   | polymorphism_of_type_enc (Tags (poly, _, _)) = poly
   600 
   601 fun level_of_type_enc (Simple_Types (_, level)) = level
   602   | level_of_type_enc (Guards (_, level, _)) = level
   603   | level_of_type_enc (Tags (_, level, _)) = level
   604 
   605 fun uniformity_of_type_enc (Simple_Types _) = Uniform
   606   | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity
   607   | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity
   608 
   609 fun is_type_level_quasi_sound All_Types = true
   610   | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
   611   | is_type_level_quasi_sound _ = false
   612 val is_type_enc_quasi_sound =
   613   is_type_level_quasi_sound o level_of_type_enc
   614 
   615 fun is_type_level_fairly_sound level =
   616   is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types
   617 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
   618 
   619 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
   620   | is_type_level_monotonicity_based Fin_Nonmono_Types = true
   621   | is_type_level_monotonicity_based _ = false
   622 
   623 fun adjust_type_enc (THF _) type_enc = type_enc
   624   | adjust_type_enc (TFF _) (Simple_Types (_, level)) =
   625     Simple_Types (First_Order, level)
   626   | adjust_type_enc format (Simple_Types (_, level)) =
   627     adjust_type_enc format (Guards (Mangled_Monomorphic, level, Uniform))
   628   | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
   629     (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
   630   | adjust_type_enc _ type_enc = type_enc
   631 
   632 fun lift_lambdas ctxt type_enc =
   633   map (close_form o Envir.eta_contract) #> rpair ctxt
   634   #-> Lambda_Lifting.lift_lambdas
   635           (if polymorphism_of_type_enc type_enc = Polymorphic then
   636              SOME polymorphic_free_prefix
   637            else
   638              NONE)
   639           Lambda_Lifting.is_quantifier
   640   #> fst
   641 
   642 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
   643     intentionalize_def t
   644   | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
   645     let
   646       fun lam T t = Abs (Name.uu, T, t)
   647       val (head, args) = strip_comb t ||> rev
   648       val head_T = fastype_of head
   649       val n = length args
   650       val arg_Ts = head_T |> binder_types |> take n |> rev
   651       val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
   652     in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
   653   | intentionalize_def t = t
   654 
   655 type translated_formula =
   656   {name : string,
   657    locality : locality,
   658    kind : formula_kind,
   659    iformula : (name, typ, iterm) formula,
   660    atomic_types : typ list}
   661 
   662 fun update_iformula f ({name, locality, kind, iformula, atomic_types}
   663                        : translated_formula) =
   664   {name = name, locality = locality, kind = kind, iformula = f iformula,
   665    atomic_types = atomic_types} : translated_formula
   666 
   667 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
   668 
   669 fun insert_type ctxt get_T x xs =
   670   let val T = get_T x in
   671     if exists (type_instance ctxt T o get_T) xs then xs
   672     else x :: filter_out (type_generalization ctxt T o get_T) xs
   673   end
   674 
   675 (* The Booleans indicate whether all type arguments should be kept. *)
   676 datatype type_arg_policy =
   677   Explicit_Type_Args of bool |
   678   Mangled_Type_Args of bool |
   679   No_Type_Args
   680 
   681 fun should_drop_arg_type_args (Simple_Types _) =
   682     false (* since TFF doesn't support overloading *)
   683   | should_drop_arg_type_args type_enc =
   684     level_of_type_enc type_enc = All_Types andalso
   685     uniformity_of_type_enc type_enc = Uniform
   686 
   687 fun type_arg_policy type_enc s =
   688   if s = type_tag_name then
   689     (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   690        Mangled_Type_Args
   691      else
   692        Explicit_Type_Args) false
   693   else case type_enc of
   694     Tags (_, All_Types, Uniform) => No_Type_Args
   695   | _ =>
   696     let val level = level_of_type_enc type_enc in
   697       if level = No_Types orelse s = @{const_name HOL.eq} orelse
   698          (s = app_op_name andalso level = Const_Arg_Types) then
   699         No_Type_Args
   700       else
   701         should_drop_arg_type_args type_enc
   702         |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   703               Mangled_Type_Args
   704             else
   705               Explicit_Type_Args)
   706     end
   707 
   708 (* Make literals for sorted type variables. *)
   709 fun generic_add_sorts_on_type (_, []) = I
   710   | generic_add_sorts_on_type ((x, i), s :: ss) =
   711     generic_add_sorts_on_type ((x, i), ss)
   712     #> (if s = the_single @{sort HOL.type} then
   713           I
   714         else if i = ~1 then
   715           insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))
   716         else
   717           insert (op =) (TyLitVar (`make_type_class s,
   718                                    (make_schematic_type_var (x, i), x))))
   719 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
   720   | add_sorts_on_tfree _ = I
   721 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
   722   | add_sorts_on_tvar _ = I
   723 
   724 fun type_literals_for_types type_enc add_sorts_on_typ Ts =
   725   [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
   726 
   727 fun mk_aconns c phis =
   728   let val (phis', phi') = split_last phis in
   729     fold_rev (mk_aconn c) phis' phi'
   730   end
   731 fun mk_ahorn [] phi = phi
   732   | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
   733 fun mk_aquant _ [] phi = phi
   734   | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
   735     if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
   736   | mk_aquant q xs phi = AQuant (q, xs, phi)
   737 
   738 fun close_universally atom_vars phi =
   739   let
   740     fun formula_vars bounds (AQuant (_, xs, phi)) =
   741         formula_vars (map fst xs @ bounds) phi
   742       | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
   743       | formula_vars bounds (AAtom tm) =
   744         union (op =) (atom_vars tm []
   745                       |> filter_out (member (op =) bounds o fst))
   746   in mk_aquant AForall (formula_vars [] phi []) phi end
   747 
   748 fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]
   749   | iterm_vars (IConst _) = I
   750   | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)
   751   | iterm_vars (IAbs (_, tm)) = iterm_vars tm
   752 fun close_iformula_universally phi = close_universally iterm_vars phi
   753 
   754 fun term_vars bounds (ATerm (name as (s, _), tms)) =
   755     (is_tptp_variable s andalso not (member (op =) bounds name))
   756     ? insert (op =) (name, NONE) #> fold (term_vars bounds) tms
   757   | term_vars bounds (AAbs ((name, _), tm)) = term_vars (name :: bounds) tm
   758 fun close_formula_universally phi = close_universally (term_vars []) phi
   759 
   760 val homo_infinite_type_name = @{type_name ind} (* any infinite type *)
   761 val homo_infinite_type = Type (homo_infinite_type_name, [])
   762 
   763 fun ho_term_from_typ format type_enc =
   764   let
   765     fun term (Type (s, Ts)) =
   766       ATerm (case (is_type_enc_higher_order type_enc, s) of
   767                (true, @{type_name bool}) => `I tptp_bool_type
   768              | (true, @{type_name fun}) => `I tptp_fun_type
   769              | _ => if s = homo_infinite_type_name andalso
   770                        is_format_typed format then
   771                       `I tptp_individual_type
   772                     else
   773                       `make_fixed_type_const s,
   774              map term Ts)
   775     | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
   776     | term (TVar ((x as (s, _)), _)) =
   777       ATerm ((make_schematic_type_var x, s), [])
   778   in term end
   779 
   780 fun ho_term_for_type_arg format type_enc T =
   781   if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)
   782 
   783 (* This shouldn't clash with anything else. *)
   784 val mangled_type_sep = "\000"
   785 
   786 fun generic_mangled_type_name f (ATerm (name, [])) = f name
   787   | generic_mangled_type_name f (ATerm (name, tys)) =
   788     f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
   789     ^ ")"
   790   | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
   791 
   792 fun mangled_type format type_enc =
   793   generic_mangled_type_name fst o ho_term_from_typ format type_enc
   794 
   795 val bool_atype = AType (`I tptp_bool_type)
   796 
   797 fun make_simple_type s =
   798   if s = tptp_bool_type orelse s = tptp_fun_type orelse
   799      s = tptp_individual_type then
   800     s
   801   else
   802     simple_type_prefix ^ ascii_of s
   803 
   804 fun ho_type_from_ho_term type_enc pred_sym ary =
   805   let
   806     fun to_atype ty =
   807       AType ((make_simple_type (generic_mangled_type_name fst ty),
   808               generic_mangled_type_name snd ty))
   809     fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
   810     fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
   811       | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
   812       | to_fo _ _ = raise Fail "unexpected type abstraction"
   813     fun to_ho (ty as ATerm ((s, _), tys)) =
   814         if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
   815       | to_ho _ = raise Fail "unexpected type abstraction"
   816   in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
   817 
   818 fun ho_type_from_typ format type_enc pred_sym ary =
   819   ho_type_from_ho_term type_enc pred_sym ary
   820   o ho_term_from_typ format type_enc
   821 
   822 fun mangled_const_name format type_enc T_args (s, s') =
   823   let
   824     val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)
   825     fun type_suffix f g =
   826       fold_rev (curry (op ^) o g o prefix mangled_type_sep
   827                 o generic_mangled_type_name f) ty_args ""
   828   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
   829 
   830 val parse_mangled_ident =
   831   Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
   832 
   833 fun parse_mangled_type x =
   834   (parse_mangled_ident
   835    -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
   836                     [] >> ATerm) x
   837 and parse_mangled_types x =
   838   (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
   839 
   840 fun unmangled_type s =
   841   s |> suffix ")" |> raw_explode
   842     |> Scan.finite Symbol.stopper
   843            (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
   844                                                 quote s)) parse_mangled_type))
   845     |> fst
   846 
   847 val unmangled_const_name = space_explode mangled_type_sep #> hd
   848 fun unmangled_const s =
   849   let val ss = space_explode mangled_type_sep s in
   850     (hd ss, map unmangled_type (tl ss))
   851   end
   852 
   853 fun introduce_proxies type_enc =
   854   let
   855     fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
   856       | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
   857                        _ =
   858         (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
   859            limitation. This works in conjuction with special code in
   860            "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
   861            possible. *)
   862         IAbs ((`I "P", p_T),
   863               IApp (IConst (`I ho_quant, T, []),
   864                     IAbs ((`I "X", x_T),
   865                           IApp (IConst (`I "P", p_T, []),
   866                                 IConst (`I "X", x_T, [])))))
   867       | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
   868     fun intro top_level args (IApp (tm1, tm2)) =
   869         IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
   870       | intro top_level args (IConst (name as (s, _), T, T_args)) =
   871         (case proxify_const s of
   872            SOME proxy_base =>
   873            if top_level orelse is_type_enc_higher_order type_enc then
   874              case (top_level, s) of
   875                (_, "c_False") => IConst (`I tptp_false, T, [])
   876              | (_, "c_True") => IConst (`I tptp_true, T, [])
   877              | (false, "c_Not") => IConst (`I tptp_not, T, [])
   878              | (false, "c_conj") => IConst (`I tptp_and, T, [])
   879              | (false, "c_disj") => IConst (`I tptp_or, T, [])
   880              | (false, "c_implies") => IConst (`I tptp_implies, T, [])
   881              | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
   882              | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
   883              | (false, s) =>
   884                if is_tptp_equal s andalso length args = 2 then
   885                  IConst (`I tptp_equal, T, [])
   886                else
   887                  (* Use a proxy even for partially applied THF equality, because
   888                     the LEO-II and Satallax parsers complain about not being
   889                     able to infer the type of "=". *)
   890                  IConst (proxy_base |>> prefix const_prefix, T, T_args)
   891              | _ => IConst (name, T, [])
   892            else
   893              IConst (proxy_base |>> prefix const_prefix, T, T_args)
   894           | NONE => if s = tptp_choice then
   895                       tweak_ho_quant tptp_choice T args
   896                     else IConst (name, T, T_args))
   897       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
   898       | intro _ _ tm = tm
   899   in intro true [] end
   900 
   901 fun iformula_from_prop thy format type_enc eq_as_iff =
   902   let
   903     fun do_term bs t atomic_types =
   904       iterm_from_term thy format bs (Envir.eta_contract t)
   905       |>> (introduce_proxies type_enc #> AAtom)
   906       ||> union (op =) atomic_types
   907     fun do_quant bs q pos s T t' =
   908       let
   909         val s = singleton (Name.variant_list (map fst bs)) s
   910         val universal = Option.map (q = AExists ? not) pos
   911         val name =
   912           s |> `(case universal of
   913                    SOME true => make_all_bound_var
   914                  | SOME false => make_exist_bound_var
   915                  | NONE => make_bound_var)
   916       in
   917         do_formula ((s, (name, T)) :: bs) pos t'
   918         #>> mk_aquant q [(name, SOME T)]
   919       end
   920     and do_conn bs c pos1 t1 pos2 t2 =
   921       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
   922     and do_formula bs pos t =
   923       case t of
   924         @{const Trueprop} $ t1 => do_formula bs pos t1
   925       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
   926       | Const (@{const_name All}, _) $ Abs (s, T, t') =>
   927         do_quant bs AForall pos s T t'
   928       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
   929         do_quant bs AExists pos s T t'
   930       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
   931       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
   932       | @{const HOL.implies} $ t1 $ t2 =>
   933         do_conn bs AImplies (Option.map not pos) t1 pos t2
   934       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
   935         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
   936       | _ => do_term bs t
   937   in do_formula [] end
   938 
   939 fun presimplify_term _ [] t = t
   940   | presimplify_term ctxt presimp_consts t =
   941     t |> exists_Const (member (op =) presimp_consts o fst) t
   942          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
   943             #> Meson.presimplify ctxt
   944             #> prop_of)
   945 
   946 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
   947 fun conceal_bounds Ts t =
   948   subst_bounds (map (Free o apfst concealed_bound_name)
   949                     (0 upto length Ts - 1 ~~ Ts), t)
   950 fun reveal_bounds Ts =
   951   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
   952                     (0 upto length Ts - 1 ~~ Ts))
   953 
   954 fun is_fun_equality (@{const_name HOL.eq},
   955                      Type (_, [Type (@{type_name fun}, _), _])) = true
   956   | is_fun_equality _ = false
   957 
   958 fun extensionalize_term ctxt t =
   959   if exists_Const is_fun_equality t then
   960     let val thy = Proof_Context.theory_of ctxt in
   961       t |> cterm_of thy |> Meson.extensionalize_conv ctxt
   962         |> prop_of |> Logic.dest_equals |> snd
   963     end
   964   else
   965     t
   966 
   967 fun simple_translate_lambdas do_lambdas ctxt t =
   968   let val thy = Proof_Context.theory_of ctxt in
   969     if Meson.is_fol_term thy t then
   970       t
   971     else
   972       let
   973         fun aux Ts t =
   974           case t of
   975             @{const Not} $ t1 => @{const Not} $ aux Ts t1
   976           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
   977             t0 $ Abs (s, T, aux (T :: Ts) t')
   978           | (t0 as Const (@{const_name All}, _)) $ t1 =>
   979             aux Ts (t0 $ eta_expand Ts t1 1)
   980           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
   981             t0 $ Abs (s, T, aux (T :: Ts) t')
   982           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
   983             aux Ts (t0 $ eta_expand Ts t1 1)
   984           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   985           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   986           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   987           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
   988               $ t1 $ t2 =>
   989             t0 $ aux Ts t1 $ aux Ts t2
   990           | _ =>
   991             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
   992             else t |> Envir.eta_contract |> do_lambdas ctxt Ts
   993         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
   994       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
   995   end
   996 
   997 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
   998     do_cheaply_conceal_lambdas Ts t1
   999     $ do_cheaply_conceal_lambdas Ts t2
  1000   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
  1001     Free (polymorphic_free_prefix ^ serial_string (),
  1002           T --> fastype_of1 (T :: Ts, t))
  1003   | do_cheaply_conceal_lambdas _ t = t
  1004 
  1005 fun do_introduce_combinators ctxt Ts t =
  1006   let val thy = Proof_Context.theory_of ctxt in
  1007     t |> conceal_bounds Ts
  1008       |> cterm_of thy
  1009       |> Meson_Clausify.introduce_combinators_in_cterm
  1010       |> prop_of |> Logic.dest_equals |> snd
  1011       |> reveal_bounds Ts
  1012   end
  1013   (* A type variable of sort "{}" will make abstraction fail. *)
  1014   handle THM _ => t |> do_cheaply_conceal_lambdas Ts
  1015 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
  1016 
  1017 fun preprocess_abstractions_in_terms trans_lambdas facts =
  1018   let
  1019     val (facts, lambda_ts) =
  1020       facts |> map (snd o snd) |> trans_lambdas
  1021             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
  1022     val lambda_facts =
  1023       map2 (fn t => fn j =>
  1024                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
  1025            lambda_ts (1 upto length lambda_ts)
  1026   in (facts, lambda_facts) end
  1027 
  1028 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
  1029    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
  1030 fun freeze_term t =
  1031   let
  1032     fun aux (t $ u) = aux t $ aux u
  1033       | aux (Abs (s, T, t)) = Abs (s, T, aux t)
  1034       | aux (Var ((s, i), T)) =
  1035         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
  1036       | aux t = t
  1037   in t |> exists_subterm is_Var t ? aux end
  1038 
  1039 fun presimp_prop ctxt presimp_consts t =
  1040   let
  1041     val thy = Proof_Context.theory_of ctxt
  1042     val t = t |> Envir.beta_eta_contract
  1043               |> transform_elim_prop
  1044               |> Object_Logic.atomize_term thy
  1045     val need_trueprop = (fastype_of t = @{typ bool})
  1046   in
  1047     t |> need_trueprop ? HOLogic.mk_Trueprop
  1048       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
  1049       |> extensionalize_term ctxt
  1050       |> presimplify_term ctxt presimp_consts
  1051       |> perhaps (try (HOLogic.dest_Trueprop))
  1052   end
  1053 
  1054 (* making fact and conjecture formulas *)
  1055 fun make_formula thy format type_enc eq_as_iff name loc kind t =
  1056   let
  1057     val (iformula, atomic_types) =
  1058       iformula_from_prop thy format type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
  1059   in
  1060     {name = name, locality = loc, kind = kind, iformula = iformula,
  1061      atomic_types = atomic_types}
  1062   end
  1063 
  1064 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  1065   let val thy = Proof_Context.theory_of ctxt in
  1066     case t |> make_formula thy format type_enc (eq_as_iff andalso format <> CNF)
  1067                            name loc Axiom of
  1068       formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
  1069       if s = tptp_true then NONE else SOME formula
  1070     | formula => SOME formula
  1071   end
  1072 
  1073 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
  1074   | s_not_trueprop t = s_not t
  1075 
  1076 fun make_conjecture thy format type_enc =
  1077   map (fn ((name, loc), (kind, t)) =>
  1078           t |> kind = Conjecture ? s_not_trueprop
  1079             |> make_formula thy format type_enc (format <> CNF) name loc kind)
  1080 
  1081 (** Finite and infinite type inference **)
  1082 
  1083 type monotonicity_info =
  1084   {maybe_finite_Ts : typ list,
  1085    surely_finite_Ts : typ list,
  1086    maybe_infinite_Ts : typ list,
  1087    surely_infinite_Ts : typ list,
  1088    maybe_nonmono_Ts : typ list}
  1089 
  1090 (* These types witness that the type classes they belong to allow infinite
  1091    models and hence that any types with these type classes is monotonic. *)
  1092 val known_infinite_types =
  1093   [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]
  1094 
  1095 fun is_type_kind_of_surely_infinite ctxt soundness cached_Ts T =
  1096   soundness <> Sound andalso
  1097   is_type_surely_infinite ctxt (soundness <> Unsound) cached_Ts T
  1098 
  1099 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
  1100    dangerous because their "exhaust" properties can easily lead to unsound ATP
  1101    proofs. On the other hand, all HOL infinite types can be given the same
  1102    models in first-order logic (via Löwenheim-Skolem). *)
  1103 
  1104 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  1105   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
  1106                              maybe_nonmono_Ts, ...}
  1107                        (Noninf_Nonmono_Types soundness) T =
  1108     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1109     not (exists (type_instance ctxt T) surely_infinite_Ts orelse
  1110          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
  1111           is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts T))
  1112   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
  1113                              maybe_nonmono_Ts, ...}
  1114                        Fin_Nonmono_Types T =
  1115     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1116     (exists (type_generalization ctxt T) surely_finite_Ts orelse
  1117      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
  1118       is_type_surely_finite ctxt T))
  1119   | should_encode_type _ _ _ _ = false
  1120 
  1121 fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var
  1122                       T =
  1123     (uniformity = Uniform orelse should_guard_var ()) andalso
  1124     should_encode_type ctxt mono level T
  1125   | should_guard_type _ _ _ _ _ = false
  1126 
  1127 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
  1128     String.isPrefix bound_var_prefix s orelse
  1129     String.isPrefix all_bound_var_prefix s
  1130   | is_maybe_universal_var (IVar _) = true
  1131   | is_maybe_universal_var _ = false
  1132 
  1133 datatype tag_site =
  1134   Top_Level of bool option |
  1135   Eq_Arg of bool option |
  1136   Elsewhere
  1137 
  1138 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  1139   | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =
  1140     (case uniformity of
  1141        Uniform => should_encode_type ctxt mono level T
  1142      | Nonuniform =>
  1143        case (site, is_maybe_universal_var u) of
  1144          (Eq_Arg _, true) => should_encode_type ctxt mono level T
  1145        | _ => false)
  1146   | should_tag_with_type _ _ _ _ _ _ = false
  1147 
  1148 fun homogenized_type ctxt mono level =
  1149   let
  1150     val should_encode = should_encode_type ctxt mono level
  1151     fun homo 0 T = if should_encode T then T else homo_infinite_type
  1152       | homo ary (Type (@{type_name fun}, [T1, T2])) =
  1153         homo 0 T1 --> homo (ary - 1) T2
  1154       | homo _ _ = raise Fail "expected function type"
  1155   in homo end
  1156 
  1157 (** predicators and application operators **)
  1158 
  1159 type sym_info =
  1160   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
  1161 
  1162 fun add_iterm_syms_to_table ctxt explicit_apply =
  1163   let
  1164     fun consider_var_arity const_T var_T max_ary =
  1165       let
  1166         fun iter ary T =
  1167           if ary = max_ary orelse type_instance ctxt var_T T orelse
  1168              type_instance ctxt T var_T then
  1169             ary
  1170           else
  1171             iter (ary + 1) (range_type T)
  1172       in iter 0 const_T end
  1173     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1174       if explicit_apply = NONE andalso
  1175          (can dest_funT T orelse T = @{typ bool}) then
  1176         let
  1177           val bool_vars' = bool_vars orelse body_type T = @{typ bool}
  1178           fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
  1179             {pred_sym = pred_sym andalso not bool_vars',
  1180              min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
  1181              max_ary = max_ary, types = types}
  1182           val fun_var_Ts' =
  1183             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
  1184         in
  1185           if bool_vars' = bool_vars andalso
  1186              pointer_eq (fun_var_Ts', fun_var_Ts) then
  1187             accum
  1188           else
  1189             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
  1190         end
  1191       else
  1192         accum
  1193     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1194       let val (head, args) = strip_iterm_comb tm in
  1195         (case head of
  1196            IConst ((s, _), T, _) =>
  1197            if String.isPrefix bound_var_prefix s orelse
  1198               String.isPrefix all_bound_var_prefix s then
  1199              add_universal_var T accum
  1200            else if String.isPrefix exist_bound_var_prefix s then
  1201              accum
  1202            else
  1203              let val ary = length args in
  1204                ((bool_vars, fun_var_Ts),
  1205                 case Symtab.lookup sym_tab s of
  1206                   SOME {pred_sym, min_ary, max_ary, types} =>
  1207                   let
  1208                     val pred_sym =
  1209                       pred_sym andalso top_level andalso not bool_vars
  1210                     val types' = types |> insert_type ctxt I T
  1211                     val min_ary =
  1212                       if is_some explicit_apply orelse
  1213                          pointer_eq (types', types) then
  1214                         min_ary
  1215                       else
  1216                         fold (consider_var_arity T) fun_var_Ts min_ary
  1217                   in
  1218                     Symtab.update (s, {pred_sym = pred_sym,
  1219                                        min_ary = Int.min (ary, min_ary),
  1220                                        max_ary = Int.max (ary, max_ary),
  1221                                        types = types'})
  1222                                   sym_tab
  1223                   end
  1224                 | NONE =>
  1225                   let
  1226                     val pred_sym = top_level andalso not bool_vars
  1227                     val min_ary =
  1228                       case explicit_apply of
  1229                         SOME true => 0
  1230                       | SOME false => ary
  1231                       | NONE => fold (consider_var_arity T) fun_var_Ts ary
  1232                   in
  1233                     Symtab.update_new (s, {pred_sym = pred_sym,
  1234                                            min_ary = min_ary, max_ary = ary,
  1235                                            types = [T]})
  1236                                       sym_tab
  1237                   end)
  1238              end
  1239          | IVar (_, T) => add_universal_var T accum
  1240          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
  1241          | _ => accum)
  1242         |> fold (add false) args
  1243       end
  1244   in add true end
  1245 fun add_fact_syms_to_table ctxt explicit_apply =
  1246   K (add_iterm_syms_to_table ctxt explicit_apply)
  1247   |> formula_fold NONE |> fact_lift
  1248 
  1249 val tvar_a = TVar (("'a", 0), HOLogic.typeS)
  1250 
  1251 val default_sym_tab_entries : (string * sym_info) list =
  1252   (prefixed_predicator_name,
  1253    {pred_sym = true, min_ary = 1, max_ary = 1, types = []})
  1254        (* FIXME: needed? *) ::
  1255   (make_fixed_const NONE @{const_name undefined},
  1256    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  1257   ([tptp_false, tptp_true]
  1258    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  1259   ([tptp_equal, tptp_old_equal]
  1260    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
  1261 
  1262 fun sym_table_for_facts ctxt explicit_apply facts =
  1263   ((false, []), Symtab.empty)
  1264   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  1265   |> fold Symtab.update default_sym_tab_entries
  1266 
  1267 fun min_arity_of sym_tab s =
  1268   case Symtab.lookup sym_tab s of
  1269     SOME ({min_ary, ...} : sym_info) => min_ary
  1270   | NONE =>
  1271     case strip_prefix_and_unascii const_prefix s of
  1272       SOME s =>
  1273       let val s = s |> unmangled_const_name |> invert_const in
  1274         if s = predicator_name then 1
  1275         else if s = app_op_name then 2
  1276         else if s = type_guard_name then 1
  1277         else 0
  1278       end
  1279     | NONE => 0
  1280 
  1281 (* True if the constant ever appears outside of the top-level position in
  1282    literals, or if it appears with different arities (e.g., because of different
  1283    type instantiations). If false, the constant always receives all of its
  1284    arguments and is used as a predicate. *)
  1285 fun is_pred_sym sym_tab s =
  1286   case Symtab.lookup sym_tab s of
  1287     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
  1288     pred_sym andalso min_ary = max_ary
  1289   | NONE => false
  1290 
  1291 val predicator_combconst =
  1292   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
  1293 fun predicator tm = IApp (predicator_combconst, tm)
  1294 
  1295 fun introduce_predicators_in_iterm sym_tab tm =
  1296   case strip_iterm_comb tm of
  1297     (IConst ((s, _), _, _), _) =>
  1298     if is_pred_sym sym_tab s then tm else predicator tm
  1299   | _ => predicator tm
  1300 
  1301 fun list_app head args = fold (curry (IApp o swap)) args head
  1302 
  1303 val app_op = `(make_fixed_const NONE) app_op_name
  1304 
  1305 fun explicit_app arg head =
  1306   let
  1307     val head_T = ityp_of head
  1308     val (arg_T, res_T) = dest_funT head_T
  1309     val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
  1310   in list_app explicit_app [head, arg] end
  1311 fun list_explicit_app head args = fold explicit_app args head
  1312 
  1313 fun introduce_explicit_apps_in_iterm sym_tab =
  1314   let
  1315     fun aux tm =
  1316       case strip_iterm_comb tm of
  1317         (head as IConst ((s, _), _, _), args) =>
  1318         args |> map aux
  1319              |> chop (min_arity_of sym_tab s)
  1320              |>> list_app head
  1321              |-> list_explicit_app
  1322       | (head, args) => list_explicit_app head (map aux args)
  1323   in aux end
  1324 
  1325 fun chop_fun 0 T = ([], T)
  1326   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
  1327     chop_fun (n - 1) ran_T |>> cons dom_T
  1328   | chop_fun _ _ = raise Fail "unexpected non-function"
  1329 
  1330 fun filter_type_args _ _ _ [] = []
  1331   | filter_type_args thy s arity T_args =
  1332     let
  1333       (* will throw "TYPE" for pseudo-constants *)
  1334       val U = if s = app_op_name then
  1335                 @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global
  1336               else
  1337                 s |> Sign.the_const_type thy
  1338     in
  1339       case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of
  1340         [] => []
  1341       | res_U_vars =>
  1342         let val U_args = (s, U) |> Sign.const_typargs thy in
  1343           U_args ~~ T_args
  1344           |> map (fn (U, T) =>
  1345                      if member (op =) res_U_vars (dest_TVar U) then T
  1346                      else dummyT)
  1347         end
  1348     end
  1349     handle TYPE _ => T_args
  1350 
  1351 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
  1352   let
  1353     val thy = Proof_Context.theory_of ctxt
  1354     fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
  1355       | aux arity (IConst (name as (s, _), T, T_args)) =
  1356         (case strip_prefix_and_unascii const_prefix s of
  1357            NONE =>
  1358            (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice
  1359                   then [] else T_args)
  1360          | SOME s'' =>
  1361            let
  1362              val s'' = invert_const s''
  1363              fun filtered_T_args false = T_args
  1364                | filtered_T_args true = filter_type_args thy s'' arity T_args
  1365            in
  1366              case type_arg_policy type_enc s'' of
  1367                Explicit_Type_Args drop_args =>
  1368                (name, filtered_T_args drop_args)
  1369              | Mangled_Type_Args drop_args =>
  1370                (mangled_const_name format type_enc (filtered_T_args drop_args)
  1371                                    name, [])
  1372              | No_Type_Args => (name, [])
  1373            end)
  1374         |> (fn (name, T_args) => IConst (name, T, T_args))
  1375       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
  1376       | aux _ tm = tm
  1377   in aux 0 end
  1378 
  1379 fun repair_iterm ctxt format type_enc sym_tab =
  1380   not (is_type_enc_higher_order type_enc)
  1381   ? (introduce_explicit_apps_in_iterm sym_tab
  1382      #> introduce_predicators_in_iterm sym_tab)
  1383   #> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1384 fun repair_fact ctxt format type_enc sym_tab =
  1385   update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))
  1386 
  1387 (** Helper facts **)
  1388 
  1389 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
  1390 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
  1391 
  1392 (* The Boolean indicates that a fairly sound type encoding is needed. *)
  1393 val helper_table =
  1394   [(("COMBI", false), @{thms Meson.COMBI_def}),
  1395    (("COMBK", false), @{thms Meson.COMBK_def}),
  1396    (("COMBB", false), @{thms Meson.COMBB_def}),
  1397    (("COMBC", false), @{thms Meson.COMBC_def}),
  1398    (("COMBS", false), @{thms Meson.COMBS_def}),
  1399    ((predicator_name, false), [not_ffalse, ftrue]),
  1400    (("fFalse", false), [not_ffalse]),
  1401    (("fFalse", true), @{thms True_or_False}),
  1402    (("fTrue", false), [ftrue]),
  1403    (("fTrue", true), @{thms True_or_False}),
  1404    (("fNot", false),
  1405     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1406            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1407    (("fconj", false),
  1408     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
  1409         by (unfold fconj_def) fast+}),
  1410    (("fdisj", false),
  1411     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
  1412         by (unfold fdisj_def) fast+}),
  1413    (("fimplies", false),
  1414     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
  1415         by (unfold fimplies_def) fast+}),
  1416    (("fequal", true),
  1417     (* This is a lie: Higher-order equality doesn't need a sound type encoding.
  1418        However, this is done so for backward compatibility: Including the
  1419        equality helpers by default in Metis breaks a few existing proofs. *)
  1420     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1421            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1422    (* Partial characterization of "fAll" and "fEx". A complete characterization
  1423       would require the axiom of choice for replay with Metis. *)
  1424    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
  1425    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
  1426    (("If", true), @{thms if_True if_False True_or_False})]
  1427   |> map (apsnd (map zero_var_indexes))
  1428 
  1429 fun fo_literal_from_type_literal (TyLitVar (class, name)) =
  1430     (true, ATerm (class, [ATerm (name, [])]))
  1431   | fo_literal_from_type_literal (TyLitFree (class, name)) =
  1432     (true, ATerm (class, [ATerm (name, [])]))
  1433 
  1434 fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
  1435 
  1436 fun bound_tvars type_enc Ts =
  1437   mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
  1438                 (type_literals_for_types type_enc add_sorts_on_tvar Ts))
  1439 
  1440 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  1441   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
  1442    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  1443   |> bound_tvars type_enc atomic_Ts
  1444   |> close_formula_universally
  1445 
  1446 val type_tag = `(make_fixed_const NONE) type_tag_name
  1447 
  1448 fun type_tag_idempotence_fact type_enc =
  1449   let
  1450     fun var s = ATerm (`I s, [])
  1451     fun tag tm = ATerm (type_tag, [var "A", tm])
  1452     val tagged_var = tag (var "X")
  1453   in
  1454     Formula (type_tag_idempotence_helper_name, Axiom,
  1455              eq_formula type_enc [] false (tag tagged_var) tagged_var,
  1456              isabelle_info simpN, NONE)
  1457   end
  1458 
  1459 fun should_specialize_helper type_enc t =
  1460   polymorphism_of_type_enc type_enc <> Polymorphic andalso
  1461   level_of_type_enc type_enc <> No_Types andalso
  1462   not (null (Term.hidden_polymorphism t))
  1463 
  1464 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  1465   case strip_prefix_and_unascii const_prefix s of
  1466     SOME mangled_s =>
  1467     let
  1468       val thy = Proof_Context.theory_of ctxt
  1469       val unmangled_s = mangled_s |> unmangled_const_name
  1470       fun dub needs_fairly_sound j k =
  1471         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
  1472          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
  1473          (if needs_fairly_sound then typed_helper_suffix
  1474           else untyped_helper_suffix),
  1475          Helper)
  1476       fun dub_and_inst needs_fairly_sound (th, j) =
  1477         let val t = prop_of th in
  1478           if should_specialize_helper type_enc t then
  1479             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
  1480                 types
  1481           else
  1482             [t]
  1483         end
  1484         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
  1485       val make_facts = map_filter (make_fact ctxt format type_enc false)
  1486       val fairly_sound = is_type_enc_fairly_sound type_enc
  1487     in
  1488       helper_table
  1489       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
  1490                   if helper_s <> unmangled_s orelse
  1491                      (needs_fairly_sound andalso not fairly_sound) then
  1492                     []
  1493                   else
  1494                     ths ~~ (1 upto length ths)
  1495                     |> maps (dub_and_inst needs_fairly_sound)
  1496                     |> make_facts)
  1497     end
  1498   | NONE => []
  1499 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  1500   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
  1501                   []
  1502 
  1503 (***************************************************************)
  1504 (* Type Classes Present in the Axiom or Conjecture Clauses     *)
  1505 (***************************************************************)
  1506 
  1507 fun set_insert (x, s) = Symtab.update (x, ()) s
  1508 
  1509 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
  1510 
  1511 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
  1512 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
  1513 
  1514 fun classes_of_terms get_Ts =
  1515   map (map snd o get_Ts)
  1516   #> List.foldl add_classes Symtab.empty
  1517   #> delete_type #> Symtab.keys
  1518 
  1519 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
  1520 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
  1521 
  1522 fun fold_type_constrs f (Type (s, Ts)) x =
  1523     fold (fold_type_constrs f) Ts (f (s, x))
  1524   | fold_type_constrs _ _ x = x
  1525 
  1526 (* Type constructors used to instantiate overloaded constants are the only ones
  1527    needed. *)
  1528 fun add_type_constrs_in_term thy =
  1529   let
  1530     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
  1531       | add (t $ u) = add t #> add u
  1532       | add (Const (x as (s, _))) =
  1533         if String.isPrefix skolem_const_prefix s then I
  1534         else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)
  1535       | add (Free (s, T)) =
  1536         if String.isPrefix polymorphic_free_prefix s then
  1537           T |> fold_type_constrs set_insert
  1538         else
  1539           I
  1540       | add (Abs (_, _, u)) = add u
  1541       | add _ = I
  1542   in add end
  1543 
  1544 fun type_constrs_of_terms thy ts =
  1545   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
  1546 
  1547 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  1548                        hyp_ts concl_t facts =
  1549   let
  1550     val thy = Proof_Context.theory_of ctxt
  1551     val presimp_consts = Meson.presimplified_consts ctxt
  1552     val fact_ts = facts |> map snd
  1553     (* Remove existing facts from the conjecture, as this can dramatically
  1554        boost an ATP's performance (for some reason). *)
  1555     val hyp_ts =
  1556       hyp_ts
  1557       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
  1558     val facts = facts |> map (apsnd (pair Axiom))
  1559     val conjs =
  1560       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]
  1561       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
  1562     val ((conjs, facts), lambdas) =
  1563       if preproc then
  1564         conjs @ facts
  1565         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
  1566         |> preprocess_abstractions_in_terms trans_lambdas
  1567         |>> chop (length conjs)
  1568         |>> apfst (map (apsnd (apsnd freeze_term)))
  1569       else
  1570         ((conjs, facts), [])
  1571     val conjs = conjs |> make_conjecture thy format type_enc
  1572     val (fact_names, facts) =
  1573       facts
  1574       |> map_filter (fn (name, (_, t)) =>
  1575                         make_fact ctxt format type_enc true (name, t)
  1576                         |> Option.map (pair name))
  1577       |> ListPair.unzip
  1578     val lambdas =
  1579       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
  1580     val all_ts = concl_t :: hyp_ts @ fact_ts
  1581     val subs = tfree_classes_of_terms all_ts
  1582     val supers = tvar_classes_of_terms all_ts
  1583     val tycons = type_constrs_of_terms thy all_ts
  1584     val (supers, arity_clauses) =
  1585       if level_of_type_enc type_enc = No_Types then ([], [])
  1586       else make_arity_clauses thy tycons supers
  1587     val class_rel_clauses = make_class_rel_clauses thy subs supers
  1588   in
  1589     (fact_names |> map single,
  1590      (conjs, facts @ lambdas, class_rel_clauses, arity_clauses))
  1591   end
  1592 
  1593 val type_guard = `(make_fixed_const NONE) type_guard_name
  1594 
  1595 fun type_guard_iterm ctxt format type_enc T tm =
  1596   IApp (IConst (type_guard, T --> @{typ bool}, [T])
  1597         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
  1598 
  1599 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  1600   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
  1601     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  1602   | is_var_positively_naked_in_term _ _ _ _ = true
  1603 
  1604 fun should_guard_var_in_formula pos phi (SOME true) name =
  1605     formula_fold pos (is_var_positively_naked_in_term name) phi false
  1606   | should_guard_var_in_formula _ _ _ _ = true
  1607 
  1608 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  1609   | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =
  1610     should_encode_type ctxt mono level T
  1611   | should_generate_tag_bound_decl _ _ _ _ _ = false
  1612 
  1613 fun mk_aterm format type_enc name T_args args =
  1614   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
  1615 
  1616 fun tag_with_type ctxt format mono type_enc pos T tm =
  1617   IConst (type_tag, T --> T, [T])
  1618   |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1619   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  1620   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
  1621        | _ => raise Fail "unexpected lambda-abstraction")
  1622 and ho_term_from_iterm ctxt format mono type_enc =
  1623   let
  1624     fun aux site u =
  1625       let
  1626         val (head, args) = strip_iterm_comb u
  1627         val pos =
  1628           case site of
  1629             Top_Level pos => pos
  1630           | Eq_Arg pos => pos
  1631           | Elsewhere => NONE
  1632         val t =
  1633           case head of
  1634             IConst (name as (s, _), _, T_args) =>
  1635             let
  1636               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
  1637             in
  1638               mk_aterm format type_enc name T_args (map (aux arg_site) args)
  1639             end
  1640           | IVar (name, _) =>
  1641             mk_aterm format type_enc name [] (map (aux Elsewhere) args)
  1642           | IAbs ((name, T), tm) =>
  1643             AAbs ((name, ho_type_from_typ format type_enc true 0 T),
  1644                   aux Elsewhere tm)
  1645           | IApp _ => raise Fail "impossible \"IApp\""
  1646         val T = ityp_of u
  1647       in
  1648         t |> (if should_tag_with_type ctxt mono type_enc site u T then
  1649                 tag_with_type ctxt format mono type_enc pos T
  1650               else
  1651                 I)
  1652       end
  1653   in aux end
  1654 and formula_from_iformula ctxt format mono type_enc should_guard_var =
  1655   let
  1656     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
  1657     val do_bound_type =
  1658       case type_enc of
  1659         Simple_Types (_, level) =>
  1660         homogenized_type ctxt mono level 0
  1661         #> ho_type_from_typ format type_enc false 0 #> SOME
  1662       | _ => K NONE
  1663     fun do_out_of_bound_type pos phi universal (name, T) =
  1664       if should_guard_type ctxt mono type_enc
  1665              (fn () => should_guard_var pos phi universal name) T then
  1666         IVar (name, T)
  1667         |> type_guard_iterm ctxt format type_enc T
  1668         |> do_term pos |> AAtom |> SOME
  1669       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
  1670         let
  1671           val var = ATerm (name, [])
  1672           val tagged_var = var |> tag_with_type ctxt format mono type_enc pos T
  1673         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
  1674       else
  1675         NONE
  1676     fun do_formula pos (AQuant (q, xs, phi)) =
  1677         let
  1678           val phi = phi |> do_formula pos
  1679           val universal = Option.map (q = AExists ? not) pos
  1680         in
  1681           AQuant (q, xs |> map (apsnd (fn NONE => NONE
  1682                                         | SOME T => do_bound_type T)),
  1683                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
  1684                       (map_filter
  1685                            (fn (_, NONE) => NONE
  1686                              | (s, SOME T) =>
  1687                                do_out_of_bound_type pos phi universal (s, T))
  1688                            xs)
  1689                       phi)
  1690         end
  1691       | do_formula pos (AConn conn) = aconn_map pos do_formula conn
  1692       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  1693   in do_formula end
  1694 
  1695 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
  1696    of monomorphization). The TPTP explicitly forbids name clashes, and some of
  1697    the remote provers might care. *)
  1698 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
  1699                           (j, {name, locality, kind, iformula, atomic_types}) =
  1700   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
  1701    iformula
  1702    |> close_iformula_universally
  1703    |> formula_from_iformula ctxt format mono type_enc
  1704                             should_guard_var_in_formula
  1705                             (if pos then SOME true else NONE)
  1706    |> bound_tvars type_enc atomic_types
  1707    |> close_formula_universally,
  1708    NONE,
  1709    case locality of
  1710      Intro => isabelle_info introN
  1711    | Elim => isabelle_info elimN
  1712    | Simp => isabelle_info simpN
  1713    | _ => NONE)
  1714   |> Formula
  1715 
  1716 fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}
  1717                                        : class_rel_clause) =
  1718   let val ty_arg = ATerm (`I "T", []) in
  1719     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
  1720              AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
  1721                                AAtom (ATerm (superclass, [ty_arg]))])
  1722              |> close_formula_universally, isabelle_info introN, NONE)
  1723   end
  1724 
  1725 fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
  1726     (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
  1727   | fo_literal_from_arity_literal (TVarLit (c, sort)) =
  1728     (false, ATerm (c, [ATerm (sort, [])]))
  1729 
  1730 fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}
  1731                                    : arity_clause) =
  1732   Formula (arity_clause_prefix ^ name, Axiom,
  1733            mk_ahorn (map (formula_from_fo_literal o apfst not
  1734                           o fo_literal_from_arity_literal) prem_lits)
  1735                     (formula_from_fo_literal
  1736                          (fo_literal_from_arity_literal concl_lits))
  1737            |> close_formula_universally, isabelle_info introN, NONE)
  1738 
  1739 fun formula_line_for_conjecture ctxt format mono type_enc
  1740         ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  1741   Formula (conjecture_prefix ^ name, kind,
  1742            formula_from_iformula ctxt format mono type_enc
  1743                should_guard_var_in_formula (SOME false)
  1744                (close_iformula_universally iformula)
  1745            |> bound_tvars type_enc atomic_types
  1746            |> close_formula_universally, NONE, NONE)
  1747 
  1748 fun free_type_literals type_enc ({atomic_types, ...} : translated_formula) =
  1749   atomic_types |> type_literals_for_types type_enc add_sorts_on_tfree
  1750                |> map fo_literal_from_type_literal
  1751 
  1752 fun formula_line_for_free_type j lit =
  1753   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,
  1754            formula_from_fo_literal lit, NONE, NONE)
  1755 fun formula_lines_for_free_types type_enc facts =
  1756   let
  1757     val litss = map (free_type_literals type_enc) facts
  1758     val lits = fold (union (op =)) litss []
  1759   in map2 formula_line_for_free_type (0 upto length lits - 1) lits end
  1760 
  1761 (** Symbol declarations **)
  1762 
  1763 fun should_declare_sym type_enc pred_sym s =
  1764   (case type_enc of
  1765      Guards _ => not pred_sym
  1766    | _ => true) andalso
  1767   is_tptp_user_symbol s
  1768 
  1769 fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  1770                              (conjs, facts) =
  1771   let
  1772     fun add_iterm_syms in_conj tm =
  1773       let val (head, args) = strip_iterm_comb tm in
  1774         (case head of
  1775            IConst ((s, s'), T, T_args) =>
  1776            let val pred_sym = is_pred_sym repaired_sym_tab s in
  1777              if should_declare_sym type_enc pred_sym s then
  1778                Symtab.map_default (s, [])
  1779                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
  1780                                          in_conj))
  1781              else
  1782                I
  1783            end
  1784          | IAbs (_, tm) => add_iterm_syms in_conj tm
  1785          | _ => I)
  1786         #> fold (add_iterm_syms in_conj) args
  1787       end
  1788     fun add_fact_syms in_conj =
  1789       K (add_iterm_syms in_conj) |> formula_fold NONE |> fact_lift
  1790     fun add_formula_var_types (AQuant (_, xs, phi)) =
  1791         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
  1792         #> add_formula_var_types phi
  1793       | add_formula_var_types (AConn (_, phis)) =
  1794         fold add_formula_var_types phis
  1795       | add_formula_var_types _ = I
  1796     fun var_types () =
  1797       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
  1798       else fold (fact_lift add_formula_var_types) (conjs @ facts) []
  1799     fun add_undefined_const T =
  1800       let
  1801         val (s, s') =
  1802           `(make_fixed_const (SOME format)) @{const_name undefined}
  1803           |> (case type_arg_policy type_enc @{const_name undefined} of
  1804                 Mangled_Type_Args _ => mangled_const_name format type_enc [T]
  1805               | _ => I)
  1806       in
  1807         Symtab.map_default (s, [])
  1808                            (insert_type ctxt #3 (s', [T], T, false, 0, false))
  1809       end
  1810   in
  1811     Symtab.empty
  1812     |> is_type_enc_fairly_sound type_enc
  1813        ? (fold (add_fact_syms true) conjs
  1814           #> fold (add_fact_syms false) facts
  1815           #> (case type_enc of
  1816                 Simple_Types _ => I
  1817               | _ => fold add_undefined_const (var_types ())))
  1818   end
  1819 
  1820 (* We add "bool" in case the helper "True_or_False" is included later. *)
  1821 val default_mono =
  1822   {maybe_finite_Ts = [@{typ bool}],
  1823    surely_finite_Ts = [@{typ bool}],
  1824    maybe_infinite_Ts = known_infinite_types,
  1825    surely_infinite_Ts = known_infinite_types,
  1826    maybe_nonmono_Ts = [@{typ bool}]}
  1827 
  1828 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
  1829    out with monotonicity" paper presented at CADE 2011. *)
  1830 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  1831   | add_iterm_mononotonicity_info ctxt level _
  1832         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
  1833         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
  1834                   surely_infinite_Ts, maybe_nonmono_Ts}) =
  1835     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
  1836       case level of
  1837         Noninf_Nonmono_Types soundness =>
  1838         if exists (type_instance ctxt T) surely_infinite_Ts orelse
  1839            member (type_aconv ctxt) maybe_finite_Ts T then
  1840           mono
  1841         else if is_type_kind_of_surely_infinite ctxt soundness
  1842                                                 surely_infinite_Ts T then
  1843           {maybe_finite_Ts = maybe_finite_Ts,
  1844            surely_finite_Ts = surely_finite_Ts,
  1845            maybe_infinite_Ts = maybe_infinite_Ts,
  1846            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
  1847            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1848         else
  1849           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
  1850            surely_finite_Ts = surely_finite_Ts,
  1851            maybe_infinite_Ts = maybe_infinite_Ts,
  1852            surely_infinite_Ts = surely_infinite_Ts,
  1853            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1854       | Fin_Nonmono_Types =>
  1855         if exists (type_instance ctxt T) surely_finite_Ts orelse
  1856            member (type_aconv ctxt) maybe_infinite_Ts T then
  1857           mono
  1858         else if is_type_surely_finite ctxt T then
  1859           {maybe_finite_Ts = maybe_finite_Ts,
  1860            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
  1861            maybe_infinite_Ts = maybe_infinite_Ts,
  1862            surely_infinite_Ts = surely_infinite_Ts,
  1863            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1864         else
  1865           {maybe_finite_Ts = maybe_finite_Ts,
  1866            surely_finite_Ts = surely_finite_Ts,
  1867            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
  1868            surely_infinite_Ts = surely_infinite_Ts,
  1869            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1870       | _ => mono
  1871     else
  1872       mono
  1873   | add_iterm_mononotonicity_info _ _ _ _ mono = mono
  1874 fun add_fact_mononotonicity_info ctxt level
  1875         ({kind, iformula, ...} : translated_formula) =
  1876   formula_fold (SOME (kind <> Conjecture))
  1877                (add_iterm_mononotonicity_info ctxt level) iformula
  1878 fun mononotonicity_info_for_facts ctxt type_enc facts =
  1879   let val level = level_of_type_enc type_enc in
  1880     default_mono
  1881     |> is_type_level_monotonicity_based level
  1882        ? fold (add_fact_mononotonicity_info ctxt level) facts
  1883   end
  1884 
  1885 fun add_iformula_monotonic_types ctxt mono type_enc =
  1886   let
  1887     val level = level_of_type_enc type_enc
  1888     val should_encode = should_encode_type ctxt mono level
  1889     fun add_type T = not (should_encode T) ? insert_type ctxt I T
  1890     fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2
  1891       | add_args _ = I
  1892     and add_term tm = add_type (ityp_of tm) #> add_args tm
  1893   in formula_fold NONE (K add_term) end
  1894 fun add_fact_monotonic_types ctxt mono type_enc =
  1895   add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
  1896 fun monotonic_types_for_facts ctxt mono type_enc facts =
  1897   [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
  1898          is_type_level_monotonicity_based (level_of_type_enc type_enc))
  1899         ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
  1900 
  1901 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  1902   Formula (guards_sym_formula_prefix ^
  1903            ascii_of (mangled_type format type_enc T),
  1904            Axiom,
  1905            IConst (`make_bound_var "X", T, [])
  1906            |> type_guard_iterm ctxt format type_enc T
  1907            |> AAtom
  1908            |> formula_from_iformula ctxt format mono type_enc
  1909                                     (K (K (K (K true)))) (SOME true)
  1910            |> bound_tvars type_enc (atyps_of T)
  1911            |> close_formula_universally,
  1912            isabelle_info introN, NONE)
  1913 
  1914 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  1915   let val x_var = ATerm (`make_bound_var "X", []) in
  1916     Formula (tags_sym_formula_prefix ^
  1917              ascii_of (mangled_type format type_enc T),
  1918              Axiom,
  1919              eq_formula type_enc (atyps_of T) false
  1920                         (tag_with_type ctxt format mono type_enc NONE T x_var)
  1921                         x_var,
  1922              isabelle_info simpN, NONE)
  1923   end
  1924 
  1925 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  1926   case type_enc of
  1927     Simple_Types _ => []
  1928   | Guards _ =>
  1929     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  1930   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
  1931 
  1932 fun decl_line_for_sym ctxt format mono type_enc s
  1933                       (s', T_args, T, pred_sym, ary, _) =
  1934   let
  1935     val (T_arg_Ts, level) =
  1936       case type_enc of
  1937         Simple_Types (_, level) => ([], level)
  1938       | _ => (replicate (length T_args) homo_infinite_type, No_Types)
  1939   in
  1940     Decl (sym_decl_prefix ^ s, (s, s'),
  1941           (T_arg_Ts ---> (T |> homogenized_type ctxt mono level ary))
  1942           |> ho_type_from_typ format type_enc pred_sym (length T_arg_Ts + ary))
  1943   end
  1944 
  1945 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
  1946                                      j (s', T_args, T, _, ary, in_conj) =
  1947   let
  1948     val (kind, maybe_negate) =
  1949       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1950       else (Axiom, I)
  1951     val (arg_Ts, res_T) = chop_fun ary T
  1952     val num_args = length arg_Ts
  1953     val bound_names =
  1954       1 upto num_args |> map (`I o make_bound_var o string_of_int)
  1955     val bounds =
  1956       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
  1957     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
  1958     fun should_keep_arg_type T =
  1959       sym_needs_arg_types andalso
  1960       should_guard_type ctxt mono type_enc (K true) T
  1961     val bound_Ts =
  1962       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  1963   in
  1964     Formula (guards_sym_formula_prefix ^ s ^
  1965              (if n > 1 then "_" ^ string_of_int j else ""), kind,
  1966              IConst ((s, s'), T, T_args)
  1967              |> fold (curry (IApp o swap)) bounds
  1968              |> type_guard_iterm ctxt format type_enc res_T
  1969              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
  1970              |> formula_from_iformula ctxt format mono type_enc
  1971                                       (K (K (K (K true)))) (SOME true)
  1972              |> n > 1 ? bound_tvars type_enc (atyps_of T)
  1973              |> close_formula_universally
  1974              |> maybe_negate,
  1975              isabelle_info introN, NONE)
  1976   end
  1977 
  1978 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
  1979         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  1980   let
  1981     val ident_base =
  1982       tags_sym_formula_prefix ^ s ^
  1983       (if n > 1 then "_" ^ string_of_int j else "")
  1984     val (kind, maybe_negate) =
  1985       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1986       else (Axiom, I)
  1987     val (arg_Ts, res_T) = chop_fun ary T
  1988     val bound_names =
  1989       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
  1990     val bounds = bound_names |> map (fn name => ATerm (name, []))
  1991     val cst = mk_aterm format type_enc (s, s') T_args
  1992     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
  1993     val should_encode =
  1994       should_encode_type ctxt mono (level_of_type_enc type_enc)
  1995     val tag_with = tag_with_type ctxt format mono type_enc NONE
  1996     val add_formula_for_res =
  1997       if should_encode res_T then
  1998         cons (Formula (ident_base ^ "_res", kind,
  1999                        eq (tag_with res_T (cst bounds)) (cst bounds),
  2000                        isabelle_info simpN, NONE))
  2001       else
  2002         I
  2003     fun add_formula_for_arg k =
  2004       let val arg_T = nth arg_Ts k in
  2005         if should_encode arg_T then
  2006           case chop k bounds of
  2007             (bounds1, bound :: bounds2) =>
  2008             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
  2009                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
  2010                               (cst bounds),
  2011                            isabelle_info simpN, NONE))
  2012           | _ => raise Fail "expected nonempty tail"
  2013         else
  2014           I
  2015       end
  2016   in
  2017     [] |> not pred_sym ? add_formula_for_res
  2018        |> Config.get ctxt type_tag_arguments
  2019           ? fold add_formula_for_arg (ary - 1 downto 0)
  2020   end
  2021 
  2022 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
  2023 
  2024 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
  2025                                 (s, decls) =
  2026   case type_enc of
  2027     Simple_Types _ =>
  2028     decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  2029   | Guards (_, level, _) =>
  2030     let
  2031       val decls =
  2032         case decls of
  2033           decl :: (decls' as _ :: _) =>
  2034           let val T = result_type_of_decl decl in
  2035             if forall (type_generalization ctxt T o result_type_of_decl)
  2036                       decls' then
  2037               [decl]
  2038             else
  2039               decls
  2040           end
  2041         | _ => decls
  2042       val n = length decls
  2043       val decls =
  2044         decls |> filter (should_encode_type ctxt mono level
  2045                          o result_type_of_decl)
  2046     in
  2047       (0 upto length decls - 1, decls)
  2048       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
  2049                                                  type_enc n s)
  2050     end
  2051   | Tags (_, _, uniformity) =>
  2052     (case uniformity of
  2053        Uniform => []
  2054      | Nonuniform =>
  2055        let val n = length decls in
  2056          (0 upto n - 1 ~~ decls)
  2057          |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
  2058                       conj_sym_kind mono type_enc n s)
  2059        end)
  2060 
  2061 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
  2062                                      mono_Ts sym_decl_tab =
  2063   let
  2064     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
  2065     val mono_lines =
  2066       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
  2067     val decl_lines =
  2068       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
  2069                                                      mono type_enc)
  2070                syms []
  2071   in mono_lines @ decl_lines end
  2072 
  2073 fun needs_type_tag_idempotence ctxt (Tags (poly, level, uniformity)) =
  2074     Config.get ctxt type_tag_idempotence andalso
  2075     poly <> Mangled_Monomorphic andalso
  2076     ((level = All_Types andalso uniformity = Nonuniform) orelse
  2077      is_type_level_monotonicity_based level)
  2078   | needs_type_tag_idempotence _ _ = false
  2079 
  2080 fun offset_of_heading_in_problem _ [] j = j
  2081   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
  2082     if heading = needle then j
  2083     else offset_of_heading_in_problem needle problem (j + length lines)
  2084 
  2085 val implicit_declsN = "Should-be-implicit typings"
  2086 val explicit_declsN = "Explicit typings"
  2087 val factsN = "Relevant facts"
  2088 val class_relsN = "Class relationships"
  2089 val aritiesN = "Arities"
  2090 val helpersN = "Helper facts"
  2091 val conjsN = "Conjectures"
  2092 val free_typesN = "Type variables"
  2093 
  2094 val explicit_apply = NONE (* for experiments *)
  2095 
  2096 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
  2097         lambda_trans readable_names preproc hyp_ts concl_t facts =
  2098   let
  2099     val type_enc = type_enc |> adjust_type_enc format
  2100     val lambda_trans =
  2101       if lambda_trans = smartN then
  2102         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
  2103       else if lambda_trans = lambdasN andalso
  2104               not (is_type_enc_higher_order type_enc) then
  2105         error ("Lambda translation method incompatible with first-order \
  2106                \encoding.")
  2107       else
  2108         lambda_trans
  2109     val trans_lambdas =
  2110       if lambda_trans = no_lambdasN then
  2111         rpair []
  2112       else if lambda_trans = concealedN then
  2113         lift_lambdas ctxt type_enc ##> K []
  2114       else if lambda_trans = liftingN then
  2115         lift_lambdas ctxt type_enc
  2116       else if lambda_trans = combinatorsN then
  2117         map (introduce_combinators ctxt) #> rpair []
  2118       else if lambda_trans = hybridN then
  2119         lift_lambdas ctxt type_enc
  2120         ##> maps (fn t => [t, introduce_combinators ctxt
  2121                                   (intentionalize_def t)])
  2122       else if lambda_trans = lambdasN then
  2123         map (Envir.eta_contract) #> rpair []
  2124       else
  2125         error ("Unknown lambda translation method: " ^
  2126                quote lambda_trans ^ ".")
  2127     val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
  2128       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  2129                          hyp_ts concl_t facts
  2130     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
  2131     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
  2132     val repair = repair_fact ctxt format type_enc sym_tab
  2133     val (conjs, facts) = (conjs, facts) |> pairself (map repair)
  2134     val repaired_sym_tab =
  2135       conjs @ facts |> sym_table_for_facts ctxt (SOME false)
  2136     val helpers =
  2137       repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
  2138                        |> map repair
  2139     val mono_Ts =
  2140       helpers @ conjs @ facts
  2141       |> monotonic_types_for_facts ctxt mono type_enc
  2142     val sym_decl_lines =
  2143       (conjs, helpers @ facts)
  2144       |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  2145       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
  2146                                                type_enc mono_Ts
  2147     val helper_lines =
  2148       0 upto length helpers - 1 ~~ helpers
  2149       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
  2150                                     type_enc)
  2151       |> (if needs_type_tag_idempotence ctxt type_enc then
  2152             cons (type_tag_idempotence_fact type_enc)
  2153           else
  2154             I)
  2155     (* Reordering these might confuse the proof reconstruction code or the SPASS
  2156        FLOTTER hack. *)
  2157     val problem =
  2158       [(explicit_declsN, sym_decl_lines),
  2159        (factsN,
  2160         map (formula_line_for_fact ctxt format fact_prefix ascii_of
  2161                                    (not exporter) (not exporter) mono
  2162                                    type_enc)
  2163             (0 upto length facts - 1 ~~ facts)),
  2164        (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
  2165        (aritiesN, map formula_line_for_arity_clause arity_clauses),
  2166        (helpersN, helper_lines),
  2167        (conjsN,
  2168         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
  2169        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
  2170     val problem =
  2171       problem
  2172       |> (case format of
  2173             CNF => ensure_cnf_problem
  2174           | CNF_UEQ => filter_cnf_ueq_problem
  2175           | _ => I)
  2176       |> (if is_format_typed format andalso format <> TFF Implicit then
  2177             declare_undeclared_syms_in_atp_problem type_decl_prefix
  2178                                                    implicit_declsN
  2179           else
  2180             I)
  2181     val (problem, pool) = problem |> nice_atp_problem readable_names
  2182     val helpers_offset = offset_of_heading_in_problem helpersN problem 0
  2183     val typed_helpers =
  2184       map_filter (fn (j, {name, ...}) =>
  2185                      if String.isSuffix typed_helper_suffix name then SOME j
  2186                      else NONE)
  2187                  ((helpers_offset + 1 upto helpers_offset + length helpers)
  2188                   ~~ helpers)
  2189     fun add_sym_arity (s, {min_ary, ...} : sym_info) =
  2190       if min_ary > 0 then
  2191         case strip_prefix_and_unascii const_prefix s of
  2192           SOME s => Symtab.insert (op =) (s, min_ary)
  2193         | NONE => I
  2194       else
  2195         I
  2196   in
  2197     (problem,
  2198      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
  2199      offset_of_heading_in_problem conjsN problem 0,
  2200      offset_of_heading_in_problem factsN problem 0,
  2201      fact_names |> Vector.fromList,
  2202      typed_helpers,
  2203      Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
  2204   end
  2205 
  2206 (* FUDGE *)
  2207 val conj_weight = 0.0
  2208 val hyp_weight = 0.1
  2209 val fact_min_weight = 0.2
  2210 val fact_max_weight = 1.0
  2211 val type_info_default_weight = 0.8
  2212 
  2213 fun add_term_weights weight (ATerm (s, tms)) =
  2214     is_tptp_user_symbol s ? Symtab.default (s, weight)
  2215     #> fold (add_term_weights weight) tms
  2216   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
  2217 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
  2218     formula_fold NONE (K (add_term_weights weight)) phi
  2219   | add_problem_line_weights _ _ = I
  2220 
  2221 fun add_conjectures_weights [] = I
  2222   | add_conjectures_weights conjs =
  2223     let val (hyps, conj) = split_last conjs in
  2224       add_problem_line_weights conj_weight conj
  2225       #> fold (add_problem_line_weights hyp_weight) hyps
  2226     end
  2227 
  2228 fun add_facts_weights facts =
  2229   let
  2230     val num_facts = length facts
  2231     fun weight_of j =
  2232       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
  2233                         / Real.fromInt num_facts
  2234   in
  2235     map weight_of (0 upto num_facts - 1) ~~ facts
  2236     |> fold (uncurry add_problem_line_weights)
  2237   end
  2238 
  2239 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
  2240 fun atp_problem_weights problem =
  2241   let val get = these o AList.lookup (op =) problem in
  2242     Symtab.empty
  2243     |> add_conjectures_weights (get free_typesN @ get conjsN)
  2244     |> add_facts_weights (get factsN)
  2245     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
  2246             [explicit_declsN, class_relsN, aritiesN]
  2247     |> Symtab.dest
  2248     |> sort (prod_ord Real.compare string_ord o pairself swap)
  2249   end
  2250 
  2251 end;