src/HOL/Tools/ATP/atp_translate.ML
author blanchet
Wed Sep 07 09:10:41 2011 +0200 (2011-09-07)
changeset 44773 e701dabbfe37
parent 44772 60ac7b56296a
child 44774 72785558a6ff
permissions -rw-r--r--
perform mangling before computing symbol arity, to avoid needless "hAPP"s and "hBOOL"s
     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 tptp_format = ATP_Problem.tptp_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 = Sound_Modulo_Infiniteness | Sound
    25   datatype uniformity = Uniform | Nonuniform
    26   datatype type_level =
    27     All_Types |
    28     Noninf_Nonmono_Types of soundness * uniformity |
    29     Fin_Nonmono_Types of uniformity |
    30     Const_Arg_Types |
    31     No_Types
    32 
    33   datatype type_enc =
    34     Simple_Types of order * polymorphism * type_level |
    35     Guards of polymorphism * type_level |
    36     Tags of polymorphism * type_level
    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 is_type_enc_higher_order : type_enc -> bool
    90   val polymorphism_of_type_enc : type_enc -> polymorphism
    91   val level_of_type_enc : type_enc -> type_level
    92   val is_type_enc_quasi_sound : type_enc -> bool
    93   val is_type_enc_fairly_sound : type_enc -> bool
    94   val type_enc_from_string : soundness -> string -> type_enc
    95   val adjust_type_enc : tptp_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 -> tptp_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 (* TFF1 is still in development, and it is still unclear whether symbols will be
   143    allowed to have types like "$tType > $o" (i.e., "![A : $tType] : $o"), with
   144    "dummy" type variables. *)
   145 val avoid_first_order_dummy_type_vars = true
   146 
   147 val bound_var_prefix = "B_"
   148 val all_bound_var_prefix = "BA_"
   149 val exist_bound_var_prefix = "BE_"
   150 val schematic_var_prefix = "V_"
   151 val fixed_var_prefix = "v_"
   152 val tvar_prefix = "T_"
   153 val tfree_prefix = "t_"
   154 val const_prefix = "c_"
   155 val type_const_prefix = "tc_"
   156 val simple_type_prefix = "s_"
   157 val class_prefix = "cl_"
   158 
   159 val polymorphic_free_prefix = "poly_free"
   160 
   161 val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"
   162 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
   163 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
   164 
   165 val type_decl_prefix = "ty_"
   166 val sym_decl_prefix = "sy_"
   167 val guards_sym_formula_prefix = "gsy_"
   168 val tags_sym_formula_prefix = "tsy_"
   169 val fact_prefix = "fact_"
   170 val conjecture_prefix = "conj_"
   171 val helper_prefix = "help_"
   172 val class_rel_clause_prefix = "clar_"
   173 val arity_clause_prefix = "arity_"
   174 val tfree_clause_prefix = "tfree_"
   175 
   176 val lambda_fact_prefix = "ATP.lambda_"
   177 val typed_helper_suffix = "_T"
   178 val untyped_helper_suffix = "_U"
   179 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"
   180 
   181 val predicator_name = "pp"
   182 val app_op_name = "aa"
   183 val type_guard_name = "gg"
   184 val type_tag_name = "tt"
   185 
   186 val prefixed_predicator_name = const_prefix ^ predicator_name
   187 val prefixed_app_op_name = const_prefix ^ app_op_name
   188 val prefixed_type_tag_name = const_prefix ^ type_tag_name
   189 
   190 (* Freshness almost guaranteed! *)
   191 val atp_weak_prefix = "ATP:"
   192 
   193 (*Escaping of special characters.
   194   Alphanumeric characters are left unchanged.
   195   The character _ goes to __
   196   Characters in the range ASCII space to / go to _A to _P, respectively.
   197   Other characters go to _nnn where nnn is the decimal ASCII code.*)
   198 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
   199 
   200 fun stringN_of_int 0 _ = ""
   201   | stringN_of_int k n =
   202     stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
   203 
   204 fun ascii_of_char c =
   205   if Char.isAlphaNum c then
   206     String.str c
   207   else if c = #"_" then
   208     "__"
   209   else if #" " <= c andalso c <= #"/" then
   210     "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
   211   else
   212     (* fixed width, in case more digits follow *)
   213     "_" ^ stringN_of_int 3 (Char.ord c)
   214 
   215 val ascii_of = String.translate ascii_of_char
   216 
   217 (** Remove ASCII armoring from names in proof files **)
   218 
   219 (* We don't raise error exceptions because this code can run inside a worker
   220    thread. Also, the errors are impossible. *)
   221 val unascii_of =
   222   let
   223     fun un rcs [] = String.implode(rev rcs)
   224       | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
   225         (* Three types of _ escapes: __, _A to _P, _nnn *)
   226       | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
   227       | un rcs (#"_" :: c :: cs) =
   228         if #"A" <= c andalso c<= #"P" then
   229           (* translation of #" " to #"/" *)
   230           un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
   231         else
   232           let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
   233             case Int.fromString (String.implode digits) of
   234               SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
   235             | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
   236           end
   237       | un rcs (c :: cs) = un (c :: rcs) cs
   238   in un [] o String.explode end
   239 
   240 (* If string s has the prefix s1, return the result of deleting it,
   241    un-ASCII'd. *)
   242 fun strip_prefix_and_unascii s1 s =
   243   if String.isPrefix s1 s then
   244     SOME (unascii_of (String.extract (s, size s1, NONE)))
   245   else
   246     NONE
   247 
   248 val proxy_table =
   249   [("c_False", (@{const_name False}, (@{thm fFalse_def},
   250        ("fFalse", @{const_name ATP.fFalse})))),
   251    ("c_True", (@{const_name True}, (@{thm fTrue_def},
   252        ("fTrue", @{const_name ATP.fTrue})))),
   253    ("c_Not", (@{const_name Not}, (@{thm fNot_def},
   254        ("fNot", @{const_name ATP.fNot})))),
   255    ("c_conj", (@{const_name conj}, (@{thm fconj_def},
   256        ("fconj", @{const_name ATP.fconj})))),
   257    ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
   258        ("fdisj", @{const_name ATP.fdisj})))),
   259    ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
   260        ("fimplies", @{const_name ATP.fimplies})))),
   261    ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
   262        ("fequal", @{const_name ATP.fequal})))),
   263    ("c_All", (@{const_name All}, (@{thm fAll_def},
   264        ("fAll", @{const_name ATP.fAll})))),
   265    ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
   266        ("fEx", @{const_name ATP.fEx}))))]
   267 
   268 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
   269 
   270 (* Readable names for the more common symbolic functions. Do not mess with the
   271    table unless you know what you are doing. *)
   272 val const_trans_table =
   273   [(@{type_name Product_Type.prod}, "prod"),
   274    (@{type_name Sum_Type.sum}, "sum"),
   275    (@{const_name False}, "False"),
   276    (@{const_name True}, "True"),
   277    (@{const_name Not}, "Not"),
   278    (@{const_name conj}, "conj"),
   279    (@{const_name disj}, "disj"),
   280    (@{const_name implies}, "implies"),
   281    (@{const_name HOL.eq}, "equal"),
   282    (@{const_name All}, "All"),
   283    (@{const_name Ex}, "Ex"),
   284    (@{const_name If}, "If"),
   285    (@{const_name Set.member}, "member"),
   286    (@{const_name Meson.COMBI}, "COMBI"),
   287    (@{const_name Meson.COMBK}, "COMBK"),
   288    (@{const_name Meson.COMBB}, "COMBB"),
   289    (@{const_name Meson.COMBC}, "COMBC"),
   290    (@{const_name Meson.COMBS}, "COMBS")]
   291   |> Symtab.make
   292   |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
   293 
   294 (* Invert the table of translations between Isabelle and ATPs. *)
   295 val const_trans_table_inv =
   296   const_trans_table |> Symtab.dest |> map swap |> Symtab.make
   297 val const_trans_table_unprox =
   298   Symtab.empty
   299   |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
   300 
   301 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
   302 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
   303 
   304 fun lookup_const c =
   305   case Symtab.lookup const_trans_table c of
   306     SOME c' => c'
   307   | NONE => ascii_of c
   308 
   309 fun ascii_of_indexname (v, 0) = ascii_of v
   310   | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
   311 
   312 fun make_bound_var x = bound_var_prefix ^ ascii_of x
   313 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
   314 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
   315 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
   316 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
   317 
   318 fun make_schematic_type_var (x, i) =
   319   tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
   320 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
   321 
   322 (* "HOL.eq" and Choice are mapped to the ATP's equivalents *)
   323 local
   324   val choice_const = (fst o dest_Const o HOLogic.choice_const) Term.dummyT
   325   fun default c = const_prefix ^ lookup_const c
   326 in
   327   fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
   328     | make_fixed_const (SOME (THF (_, _, THF_With_Choice))) c =
   329       if c = choice_const then tptp_choice else default c
   330     | make_fixed_const _ c = default c
   331 end
   332 
   333 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
   334 
   335 fun make_type_class clas = class_prefix ^ ascii_of clas
   336 
   337 fun new_skolem_var_name_from_const s =
   338   let val ss = s |> space_explode Long_Name.separator in
   339     nth ss (length ss - 2)
   340   end
   341 
   342 (* These are either simplified away by "Meson.presimplify" (most of the time) or
   343    handled specially via "fFalse", "fTrue", ..., "fequal". *)
   344 val atp_irrelevant_consts =
   345   [@{const_name False}, @{const_name True}, @{const_name Not},
   346    @{const_name conj}, @{const_name disj}, @{const_name implies},
   347    @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
   348 
   349 val atp_monomorph_bad_consts =
   350   atp_irrelevant_consts @
   351   (* These are ignored anyway by the relevance filter (unless they appear in
   352      higher-order places) but not by the monomorphizer. *)
   353   [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
   354    @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
   355    @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
   356 
   357 fun add_schematic_const (x as (_, T)) =
   358   Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
   359 val add_schematic_consts_of =
   360   Term.fold_aterms (fn Const (x as (s, _)) =>
   361                        not (member (op =) atp_monomorph_bad_consts s)
   362                        ? add_schematic_const x
   363                       | _ => I)
   364 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
   365 
   366 (** Definitions and functions for FOL clauses and formulas for TPTP **)
   367 
   368 (** Isabelle arities **)
   369 
   370 type arity_atom = name * name * name list
   371 
   372 val type_class = the_single @{sort type}
   373 
   374 type arity_clause =
   375   {name : string,
   376    prem_atoms : arity_atom list,
   377    concl_atom : arity_atom}
   378 
   379 fun add_prem_atom tvar =
   380   fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar, []))
   381 
   382 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
   383 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
   384   let
   385     val tvars = map (prefix tvar_prefix o string_of_int) (1 upto length args)
   386     val tvars_srts = ListPair.zip (tvars, args)
   387   in
   388     {name = name,
   389      prem_atoms = [] |> fold (uncurry add_prem_atom) tvars_srts,
   390      concl_atom = (`make_type_class cls, `make_fixed_type_const tcons,
   391                    tvars ~~ tvars)}
   392   end
   393 
   394 fun arity_clause _ _ (_, []) = []
   395   | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)
   396     arity_clause seen n (tcons, ars)
   397   | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
   398     if member (op =) seen class then
   399       (* multiple arities for the same (tycon, class) pair *)
   400       make_axiom_arity_clause (tcons,
   401           lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
   402           ar) ::
   403       arity_clause seen (n + 1) (tcons, ars)
   404     else
   405       make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
   406                                ascii_of class, ar) ::
   407       arity_clause (class :: seen) n (tcons, ars)
   408 
   409 fun multi_arity_clause [] = []
   410   | multi_arity_clause ((tcons, ars) :: tc_arlists) =
   411     arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
   412 
   413 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
   414    theory thy provided its arguments have the corresponding sorts. *)
   415 fun type_class_pairs thy tycons classes =
   416   let
   417     val alg = Sign.classes_of thy
   418     fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
   419     fun add_class tycon class =
   420       cons (class, domain_sorts tycon class)
   421       handle Sorts.CLASS_ERROR _ => I
   422     fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
   423   in map try_classes tycons end
   424 
   425 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
   426 fun iter_type_class_pairs _ _ [] = ([], [])
   427   | iter_type_class_pairs thy tycons classes =
   428       let
   429         fun maybe_insert_class s =
   430           (s <> type_class andalso not (member (op =) classes s))
   431           ? insert (op =) s
   432         val cpairs = type_class_pairs thy tycons classes
   433         val newclasses =
   434           [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
   435         val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
   436       in (classes' @ classes, union (op =) cpairs' cpairs) end
   437 
   438 fun make_arity_clauses thy tycons =
   439   iter_type_class_pairs thy tycons ##> multi_arity_clause
   440 
   441 
   442 (** Isabelle class relations **)
   443 
   444 type class_rel_clause =
   445   {name : string,
   446    subclass : name,
   447    superclass : name}
   448 
   449 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
   450    in theory "thy". *)
   451 fun class_pairs _ [] _ = []
   452   | class_pairs thy subs supers =
   453       let
   454         val class_less = Sorts.class_less (Sign.classes_of thy)
   455         fun add_super sub super = class_less (sub, super) ? cons (sub, super)
   456         fun add_supers sub = fold (add_super sub) supers
   457       in fold add_supers subs [] end
   458 
   459 fun make_class_rel_clause (sub, super) =
   460   {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
   461    superclass = `make_type_class super}
   462 
   463 fun make_class_rel_clauses thy subs supers =
   464   map make_class_rel_clause (class_pairs thy subs supers)
   465 
   466 (* intermediate terms *)
   467 datatype iterm =
   468   IConst of name * typ * typ list |
   469   IVar of name * typ |
   470   IApp of iterm * iterm |
   471   IAbs of (name * typ) * iterm
   472 
   473 fun ityp_of (IConst (_, T, _)) = T
   474   | ityp_of (IVar (_, T)) = T
   475   | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
   476   | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
   477 
   478 (*gets the head of a combinator application, along with the list of arguments*)
   479 fun strip_iterm_comb u =
   480   let
   481     fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
   482       | stripc x = x
   483   in stripc (u, []) end
   484 
   485 fun atyps_of T = fold_atyps (insert (op =)) T []
   486 
   487 fun new_skolem_const_name s num_T_args =
   488   [new_skolem_const_prefix, s, string_of_int num_T_args]
   489   |> space_implode Long_Name.separator
   490 
   491 fun robust_const_type thy s =
   492   if s = app_op_name then Logic.varifyT_global @{typ "('a => 'b) => 'a => 'b"}
   493   else s |> Sign.the_const_type thy
   494 
   495 (* This function only makes sense if "T" is as general as possible. *)
   496 fun robust_const_typargs thy (s, T) =
   497   (s, T) |> Sign.const_typargs thy
   498   handle TYPE _ => [] |> Term.add_tvarsT T |> rev |> map TVar
   499 
   500 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
   501    Also accumulates sort infomation. *)
   502 fun iterm_from_term thy format bs (P $ Q) =
   503     let
   504       val (P', P_atomics_Ts) = iterm_from_term thy format bs P
   505       val (Q', Q_atomics_Ts) = iterm_from_term thy format bs Q
   506     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
   507   | iterm_from_term thy format _ (Const (c, T)) =
   508     (IConst (`(make_fixed_const (SOME format)) c, T,
   509              robust_const_typargs thy (c, T)),
   510      atyps_of T)
   511   | iterm_from_term _ _ _ (Free (s, T)) =
   512     (IConst (`make_fixed_var s, T,
   513              if String.isPrefix polymorphic_free_prefix s then [T] else []),
   514      atyps_of T)
   515   | iterm_from_term _ format _ (Var (v as (s, _), T)) =
   516     (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
   517        let
   518          val Ts = T |> strip_type |> swap |> op ::
   519          val s' = new_skolem_const_name s (length Ts)
   520        in IConst (`(make_fixed_const (SOME format)) s', T, Ts) end
   521      else
   522        IVar ((make_schematic_var v, s), T), atyps_of T)
   523   | iterm_from_term _ _ bs (Bound j) =
   524     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))
   525   | iterm_from_term thy format bs (Abs (s, T, t)) =
   526     let
   527       fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
   528       val s = vary s
   529       val name = `make_bound_var s
   530       val (tm, atomic_Ts) = iterm_from_term thy format ((s, (name, T)) :: bs) t
   531     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end
   532 
   533 datatype locality =
   534   General | Helper | Induction | Extensionality | Intro | Elim | Simp |
   535   Local | Assum | Chained
   536 
   537 datatype order = First_Order | Higher_Order
   538 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
   539 datatype soundness = Sound_Modulo_Infiniteness | Sound
   540 datatype uniformity = Uniform | Nonuniform
   541 datatype type_level =
   542   All_Types |
   543   Noninf_Nonmono_Types of soundness * uniformity |
   544   Fin_Nonmono_Types of uniformity |
   545   Const_Arg_Types |
   546   No_Types
   547 
   548 datatype type_enc =
   549   Simple_Types of order * polymorphism * type_level |
   550   Guards of polymorphism * type_level |
   551   Tags of polymorphism * type_level
   552 
   553 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _, _)) = true
   554   | is_type_enc_higher_order _ = false
   555 
   556 fun polymorphism_of_type_enc (Simple_Types (_, poly, _)) = poly
   557   | polymorphism_of_type_enc (Guards (poly, _)) = poly
   558   | polymorphism_of_type_enc (Tags (poly, _)) = poly
   559 
   560 fun level_of_type_enc (Simple_Types (_, _, level)) = level
   561   | level_of_type_enc (Guards (_, level)) = level
   562   | level_of_type_enc (Tags (_, level)) = level
   563 
   564 fun is_level_uniform (Noninf_Nonmono_Types (_, Nonuniform)) = false
   565   | is_level_uniform (Fin_Nonmono_Types Nonuniform) = false
   566   | is_level_uniform _ = true
   567 
   568 fun is_type_level_quasi_sound All_Types = true
   569   | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
   570   | is_type_level_quasi_sound _ = false
   571 val is_type_enc_quasi_sound = is_type_level_quasi_sound o level_of_type_enc
   572 
   573 fun is_type_level_fairly_sound (Fin_Nonmono_Types _) = true
   574   | is_type_level_fairly_sound level = is_type_level_quasi_sound level
   575 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
   576 
   577 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
   578   | is_type_level_monotonicity_based (Fin_Nonmono_Types _) = true
   579   | is_type_level_monotonicity_based _ = false
   580 
   581 fun try_unsuffixes ss s =
   582   fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
   583 
   584 val queries = ["?", "_query"]
   585 val bangs = ["!", "_bang"]
   586 
   587 fun type_enc_from_string soundness s =
   588   (case try (unprefix "poly_") s of
   589      SOME s => (SOME Polymorphic, s)
   590    | NONE =>
   591      case try (unprefix "raw_mono_") s of
   592        SOME s => (SOME Raw_Monomorphic, s)
   593      | NONE =>
   594        case try (unprefix "mono_") s of
   595          SOME s => (SOME Mangled_Monomorphic, s)
   596        | NONE => (NONE, s))
   597   ||> (fn s =>
   598           (* "_query" and "_bang" are for the ASCII-challenged Metis and
   599              Mirabelle. *)
   600           case try_unsuffixes queries s of
   601             SOME s =>
   602             (case try_unsuffixes queries s of
   603                SOME s => (Noninf_Nonmono_Types (soundness, Nonuniform), s)
   604              | NONE => (Noninf_Nonmono_Types (soundness, Uniform), s))
   605           | NONE =>
   606             case try_unsuffixes bangs s of
   607               SOME s =>
   608               (case try_unsuffixes bangs s of
   609                  SOME s => (Fin_Nonmono_Types Nonuniform, s)
   610                | NONE => (Fin_Nonmono_Types Uniform, s))
   611             | NONE => (All_Types, s))
   612   |> (fn (poly, (level, core)) =>
   613          case (core, (poly, level)) of
   614            ("simple", (SOME poly, _)) =>
   615            (case (poly, level) of
   616               (Polymorphic, All_Types) =>
   617               Simple_Types (First_Order, Polymorphic, All_Types)
   618             | (Mangled_Monomorphic, _) =>
   619               if is_level_uniform level then
   620                 Simple_Types (First_Order, Mangled_Monomorphic, level)
   621               else
   622                 raise Same.SAME
   623             | _ => raise Same.SAME)
   624          | ("simple_higher", (SOME poly, _)) =>
   625            (case (poly, level) of
   626               (Polymorphic, All_Types) =>
   627               Simple_Types (Higher_Order, Polymorphic, All_Types)
   628             | (_, Noninf_Nonmono_Types _) => raise Same.SAME
   629             | (Mangled_Monomorphic, _) =>
   630               if is_level_uniform level then
   631                 Simple_Types (Higher_Order, Mangled_Monomorphic, level)
   632               else
   633                 raise Same.SAME
   634             | _ => raise Same.SAME)
   635          | ("guards", (SOME poly, _)) => Guards (poly, level)
   636          | ("tags", (SOME Polymorphic, _)) => Tags (Polymorphic, level)
   637          | ("tags", (SOME poly, _)) => Tags (poly, level)
   638          | ("args", (SOME poly, All_Types (* naja *))) =>
   639            Guards (poly, Const_Arg_Types)
   640          | ("erased", (NONE, All_Types (* naja *))) =>
   641            Guards (Polymorphic, No_Types)
   642          | _ => raise Same.SAME)
   643   handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
   644 
   645 fun adjust_type_enc (THF (TPTP_Monomorphic, _, _))
   646                     (Simple_Types (order, _, level)) =
   647     Simple_Types (order, Mangled_Monomorphic, level)
   648   | adjust_type_enc (THF _) type_enc = type_enc
   649   | adjust_type_enc (TFF (TPTP_Monomorphic, _)) (Simple_Types (_, _, level)) =
   650     Simple_Types (First_Order, Mangled_Monomorphic, level)
   651   | adjust_type_enc (TFF _) (Simple_Types (_, poly, level)) =
   652     Simple_Types (First_Order, poly, level)
   653   | adjust_type_enc format (Simple_Types (_, poly, level)) =
   654     adjust_type_enc format (Guards (poly, level))
   655   | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
   656     (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
   657   | adjust_type_enc _ type_enc = type_enc
   658 
   659 fun lift_lambdas ctxt type_enc =
   660   map (close_form o Envir.eta_contract) #> rpair ctxt
   661   #-> Lambda_Lifting.lift_lambdas
   662           (if polymorphism_of_type_enc type_enc = Polymorphic then
   663              SOME polymorphic_free_prefix
   664            else
   665              NONE)
   666           Lambda_Lifting.is_quantifier
   667   #> fst
   668 
   669 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
   670     intentionalize_def t
   671   | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
   672     let
   673       fun lam T t = Abs (Name.uu, T, t)
   674       val (head, args) = strip_comb t ||> rev
   675       val head_T = fastype_of head
   676       val n = length args
   677       val arg_Ts = head_T |> binder_types |> take n |> rev
   678       val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
   679     in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
   680   | intentionalize_def t = t
   681 
   682 type translated_formula =
   683   {name : string,
   684    locality : locality,
   685    kind : formula_kind,
   686    iformula : (name, typ, iterm) formula,
   687    atomic_types : typ list}
   688 
   689 fun update_iformula f ({name, locality, kind, iformula, atomic_types}
   690                        : translated_formula) =
   691   {name = name, locality = locality, kind = kind, iformula = f iformula,
   692    atomic_types = atomic_types} : translated_formula
   693 
   694 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
   695 
   696 fun insert_type ctxt get_T x xs =
   697   let val T = get_T x in
   698     if exists (type_instance ctxt T o get_T) xs then xs
   699     else x :: filter_out (type_generalization ctxt T o get_T) xs
   700   end
   701 
   702 (* The Booleans indicate whether all type arguments should be kept. *)
   703 datatype type_arg_policy =
   704   Explicit_Type_Args of bool |
   705   Mangled_Type_Args |
   706   No_Type_Args
   707 
   708 fun should_drop_arg_type_args (Simple_Types _) = false
   709   | should_drop_arg_type_args type_enc =
   710     level_of_type_enc type_enc = All_Types
   711 
   712 fun type_arg_policy type_enc s =
   713   if s = type_tag_name then
   714     if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   715       Mangled_Type_Args
   716     else
   717       Explicit_Type_Args false
   718   else case type_enc of
   719     Tags (_, All_Types) => No_Type_Args
   720   | _ =>
   721     let val level = level_of_type_enc type_enc in
   722       if level = No_Types orelse s = @{const_name HOL.eq} orelse
   723          (s = app_op_name andalso level = Const_Arg_Types) then
   724         No_Type_Args
   725       else if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   726         Mangled_Type_Args
   727       else
   728         Explicit_Type_Args (should_drop_arg_type_args type_enc)
   729     end
   730 
   731 (* Make atoms for sorted type variables. *)
   732 fun generic_add_sorts_on_type (_, []) = I
   733   | generic_add_sorts_on_type ((x, i), s :: ss) =
   734     generic_add_sorts_on_type ((x, i), ss)
   735     #> (if s = the_single @{sort HOL.type} then
   736           I
   737         else if i = ~1 then
   738           insert (op =) (`make_type_class s, `make_fixed_type_var x)
   739         else
   740           insert (op =) (`make_type_class s,
   741                          (make_schematic_type_var (x, i), x)))
   742 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
   743   | add_sorts_on_tfree _ = I
   744 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
   745   | add_sorts_on_tvar _ = I
   746 
   747 val tvar_a_str = "'a"
   748 val tvar_a = TVar ((tvar_a_str, 0), HOLogic.typeS)
   749 val tvar_a_name = (make_schematic_type_var (tvar_a_str, 0), tvar_a_str)
   750 val itself_name = `make_fixed_type_const @{type_name itself}
   751 val TYPE_name = `(make_fixed_const NONE) @{const_name TYPE}
   752 val tvar_a_atype = AType (tvar_a_name, [])
   753 val a_itself_atype = AType (itself_name, [tvar_a_atype])
   754 
   755 fun type_class_formula type_enc class arg =
   756   AAtom (ATerm (class, arg ::
   757       (case type_enc of
   758          Simple_Types (First_Order, Polymorphic, _) =>
   759          if avoid_first_order_dummy_type_vars then [ATerm (TYPE_name, [arg])]
   760          else []
   761        | _ => [])))
   762 fun formulas_for_types type_enc add_sorts_on_typ Ts =
   763   [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
   764      |> map (fn (class, name) =>
   765                 type_class_formula type_enc class (ATerm (name, [])))
   766 
   767 fun mk_aconns c phis =
   768   let val (phis', phi') = split_last phis in
   769     fold_rev (mk_aconn c) phis' phi'
   770   end
   771 fun mk_ahorn [] phi = phi
   772   | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
   773 fun mk_aquant _ [] phi = phi
   774   | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
   775     if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
   776   | mk_aquant q xs phi = AQuant (q, xs, phi)
   777 
   778 fun close_universally atom_vars phi =
   779   let
   780     fun formula_vars bounds (AQuant (_, xs, phi)) =
   781         formula_vars (map fst xs @ bounds) phi
   782       | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
   783       | formula_vars bounds (AAtom tm) =
   784         union (op =) (atom_vars tm []
   785                       |> filter_out (member (op =) bounds o fst))
   786   in mk_aquant AForall (formula_vars [] phi []) phi end
   787 
   788 fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]
   789   | iterm_vars (IConst _) = I
   790   | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)
   791   | iterm_vars (IAbs (_, tm)) = iterm_vars tm
   792 fun close_iformula_universally phi = close_universally iterm_vars phi
   793 
   794 fun term_vars type_enc =
   795   let
   796     fun vars bounds (ATerm (name as (s, _), tms)) =
   797         (if is_tptp_variable s andalso not (member (op =) bounds name) then
   798            (case type_enc of
   799               Simple_Types (_, Polymorphic, _) =>
   800               if String.isPrefix tvar_prefix s then SOME atype_of_types
   801               else NONE
   802             | _ => NONE)
   803            |> pair name |> insert (op =)
   804          else
   805            I)
   806         #> fold (vars bounds) tms
   807       | vars bounds (AAbs ((name, _), tm)) = vars (name :: bounds) tm
   808   in vars end
   809 fun close_formula_universally type_enc =
   810   close_universally (term_vars type_enc [])
   811 
   812 val fused_infinite_type_name = @{type_name ind} (* any infinite type *)
   813 val fused_infinite_type = Type (fused_infinite_type_name, [])
   814 
   815 fun tvar_name (x as (s, _)) = (make_schematic_type_var x, s)
   816 
   817 fun ho_term_from_typ format type_enc =
   818   let
   819     fun term (Type (s, Ts)) =
   820       ATerm (case (is_type_enc_higher_order type_enc, s) of
   821                (true, @{type_name bool}) => `I tptp_bool_type
   822              | (true, @{type_name fun}) => `I tptp_fun_type
   823              | _ => if s = fused_infinite_type_name andalso
   824                        is_format_typed format then
   825                       `I tptp_individual_type
   826                     else
   827                       `make_fixed_type_const s,
   828              map term Ts)
   829     | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
   830     | term (TVar (x, _)) = ATerm (tvar_name x, [])
   831   in term end
   832 
   833 fun ho_term_for_type_arg format type_enc T =
   834   if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)
   835 
   836 (* This shouldn't clash with anything else. *)
   837 val mangled_type_sep = "\000"
   838 
   839 fun generic_mangled_type_name f (ATerm (name, [])) = f name
   840   | generic_mangled_type_name f (ATerm (name, tys)) =
   841     f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
   842     ^ ")"
   843   | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
   844 
   845 fun mangled_type format type_enc =
   846   generic_mangled_type_name fst o ho_term_from_typ format type_enc
   847 
   848 fun make_simple_type s =
   849   if s = tptp_bool_type orelse s = tptp_fun_type orelse
   850      s = tptp_individual_type then
   851     s
   852   else
   853     simple_type_prefix ^ ascii_of s
   854 
   855 fun ho_type_from_ho_term type_enc pred_sym ary =
   856   let
   857     fun to_mangled_atype ty =
   858       AType ((make_simple_type (generic_mangled_type_name fst ty),
   859               generic_mangled_type_name snd ty), [])
   860     fun to_poly_atype (ATerm (name, tys)) = AType (name, map to_poly_atype tys)
   861       | to_poly_atype _ = raise Fail "unexpected type abstraction"
   862     val to_atype =
   863       if polymorphism_of_type_enc type_enc = Polymorphic then to_poly_atype
   864       else to_mangled_atype
   865     fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
   866     fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
   867       | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
   868       | to_fo _ _ = raise Fail "unexpected type abstraction"
   869     fun to_ho (ty as ATerm ((s, _), tys)) =
   870         if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
   871       | to_ho _ = raise Fail "unexpected type abstraction"
   872   in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
   873 
   874 fun ho_type_from_typ format type_enc pred_sym ary =
   875   ho_type_from_ho_term type_enc pred_sym ary
   876   o ho_term_from_typ format type_enc
   877 
   878 fun mangled_const_name format type_enc T_args (s, s') =
   879   let
   880     val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)
   881     fun type_suffix f g =
   882       fold_rev (curry (op ^) o g o prefix mangled_type_sep
   883                 o generic_mangled_type_name f) ty_args ""
   884   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
   885 
   886 val parse_mangled_ident =
   887   Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
   888 
   889 fun parse_mangled_type x =
   890   (parse_mangled_ident
   891    -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
   892                     [] >> ATerm) x
   893 and parse_mangled_types x =
   894   (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
   895 
   896 fun unmangled_type s =
   897   s |> suffix ")" |> raw_explode
   898     |> Scan.finite Symbol.stopper
   899            (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
   900                                                 quote s)) parse_mangled_type))
   901     |> fst
   902 
   903 val unmangled_const_name = space_explode mangled_type_sep #> hd
   904 fun unmangled_const s =
   905   let val ss = space_explode mangled_type_sep s in
   906     (hd ss, map unmangled_type (tl ss))
   907   end
   908 
   909 fun introduce_proxies_in_iterm type_enc =
   910   let
   911     fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
   912       | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
   913                        _ =
   914         (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
   915            limitation. This works in conjuction with special code in
   916            "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
   917            possible. *)
   918         IAbs ((`I "P", p_T),
   919               IApp (IConst (`I ho_quant, T, []),
   920                     IAbs ((`I "X", x_T),
   921                           IApp (IConst (`I "P", p_T, []),
   922                                 IConst (`I "X", x_T, [])))))
   923       | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
   924     fun intro top_level args (IApp (tm1, tm2)) =
   925         IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
   926       | intro top_level args (IConst (name as (s, _), T, T_args)) =
   927         (case proxify_const s of
   928            SOME proxy_base =>
   929            if top_level orelse is_type_enc_higher_order type_enc then
   930              case (top_level, s) of
   931                (_, "c_False") => IConst (`I tptp_false, T, [])
   932              | (_, "c_True") => IConst (`I tptp_true, T, [])
   933              | (false, "c_Not") => IConst (`I tptp_not, T, [])
   934              | (false, "c_conj") => IConst (`I tptp_and, T, [])
   935              | (false, "c_disj") => IConst (`I tptp_or, T, [])
   936              | (false, "c_implies") => IConst (`I tptp_implies, T, [])
   937              | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
   938              | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
   939              | (false, s) =>
   940                if is_tptp_equal s andalso length args = 2 then
   941                  IConst (`I tptp_equal, T, [])
   942                else
   943                  (* Use a proxy even for partially applied THF0 equality,
   944                     because the LEO-II and Satallax parsers complain about not
   945                     being able to infer the type of "=". *)
   946                  IConst (proxy_base |>> prefix const_prefix, T, T_args)
   947              | _ => IConst (name, T, [])
   948            else
   949              IConst (proxy_base |>> prefix const_prefix, T, T_args)
   950           | NONE => if s = tptp_choice then
   951                       tweak_ho_quant tptp_choice T args
   952                     else
   953                       IConst (name, T, T_args))
   954       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
   955       | intro _ _ tm = tm
   956   in intro true [] end
   957 
   958 fun chop_fun 0 T = ([], T)
   959   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
   960     chop_fun (n - 1) ran_T |>> cons dom_T
   961   | chop_fun _ T = ([], T)
   962 
   963 fun filter_type_args _ _ _ [] = []
   964   | filter_type_args thy s ary T_args =
   965     let
   966       val U = robust_const_type thy s
   967       val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun ary |> fst) []
   968       val U_args = (s, U) |> robust_const_typargs thy
   969     in
   970       U_args ~~ T_args
   971       |> map (fn (U, T) =>
   972                  if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
   973     end
   974     handle TYPE _ => T_args
   975 
   976 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
   977   let
   978     val thy = Proof_Context.theory_of ctxt
   979     fun aux ary (IApp (tm1, tm2)) = IApp (aux (ary + 1) tm1, aux 0 tm2)
   980       | aux ary (IConst (name as (s, _), T, T_args)) =
   981         (case strip_prefix_and_unascii const_prefix s of
   982            NONE =>
   983            (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice
   984                   then [] else T_args)
   985          | SOME s'' =>
   986            let
   987              val s'' = invert_const s''
   988              fun filter_T_args false = T_args
   989                | filter_T_args true = filter_type_args thy s'' ary T_args
   990            in
   991              case type_arg_policy type_enc s'' of
   992                Explicit_Type_Args drop_args => (name, filter_T_args drop_args)
   993              | Mangled_Type_Args =>
   994                (mangled_const_name format type_enc T_args name, [])
   995              | No_Type_Args => (name, [])
   996            end)
   997         |> (fn (name, T_args) => IConst (name, T, T_args))
   998       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
   999       | aux _ tm = tm
  1000   in aux 0 end
  1001 
  1002 fun iformula_from_prop ctxt format type_enc eq_as_iff =
  1003   let
  1004     val thy = Proof_Context.theory_of ctxt
  1005     fun do_term bs t atomic_types =
  1006       iterm_from_term thy format bs (Envir.eta_contract t)
  1007       |>> (introduce_proxies_in_iterm type_enc
  1008            #> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1009            #> AAtom)
  1010       ||> union (op =) atomic_types
  1011     fun do_quant bs q pos s T t' =
  1012       let
  1013         val s = singleton (Name.variant_list (map fst bs)) s
  1014         val universal = Option.map (q = AExists ? not) pos
  1015         val name =
  1016           s |> `(case universal of
  1017                    SOME true => make_all_bound_var
  1018                  | SOME false => make_exist_bound_var
  1019                  | NONE => make_bound_var)
  1020       in
  1021         do_formula ((s, (name, T)) :: bs) pos t'
  1022         #>> mk_aquant q [(name, SOME T)]
  1023       end
  1024     and do_conn bs c pos1 t1 pos2 t2 =
  1025       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
  1026     and do_formula bs pos t =
  1027       case t of
  1028         @{const Trueprop} $ t1 => do_formula bs pos t1
  1029       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
  1030       | Const (@{const_name All}, _) $ Abs (s, T, t') =>
  1031         do_quant bs AForall pos s T t'
  1032       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
  1033         do_quant bs AExists pos s T t'
  1034       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
  1035       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
  1036       | @{const HOL.implies} $ t1 $ t2 =>
  1037         do_conn bs AImplies (Option.map not pos) t1 pos t2
  1038       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
  1039         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
  1040       | _ => do_term bs t
  1041   in do_formula [] end
  1042 
  1043 fun presimplify_term _ [] t = t
  1044   | presimplify_term ctxt presimp_consts t =
  1045     t |> exists_Const (member (op =) presimp_consts o fst) t
  1046          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
  1047             #> Meson.presimplify ctxt
  1048             #> prop_of)
  1049 
  1050 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
  1051 fun conceal_bounds Ts t =
  1052   subst_bounds (map (Free o apfst concealed_bound_name)
  1053                     (0 upto length Ts - 1 ~~ Ts), t)
  1054 fun reveal_bounds Ts =
  1055   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
  1056                     (0 upto length Ts - 1 ~~ Ts))
  1057 
  1058 fun is_fun_equality (@{const_name HOL.eq},
  1059                      Type (_, [Type (@{type_name fun}, _), _])) = true
  1060   | is_fun_equality _ = false
  1061 
  1062 fun extensionalize_term ctxt t =
  1063   if exists_Const is_fun_equality t then
  1064     let val thy = Proof_Context.theory_of ctxt in
  1065       t |> cterm_of thy |> Meson.extensionalize_conv ctxt
  1066         |> prop_of |> Logic.dest_equals |> snd
  1067     end
  1068   else
  1069     t
  1070 
  1071 fun simple_translate_lambdas do_lambdas ctxt t =
  1072   let val thy = Proof_Context.theory_of ctxt in
  1073     if Meson.is_fol_term thy t then
  1074       t
  1075     else
  1076       let
  1077         fun aux Ts t =
  1078           case t of
  1079             @{const Not} $ t1 => @{const Not} $ aux Ts t1
  1080           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
  1081             t0 $ Abs (s, T, aux (T :: Ts) t')
  1082           | (t0 as Const (@{const_name All}, _)) $ t1 =>
  1083             aux Ts (t0 $ eta_expand Ts t1 1)
  1084           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
  1085             t0 $ Abs (s, T, aux (T :: Ts) t')
  1086           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
  1087             aux Ts (t0 $ eta_expand Ts t1 1)
  1088           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1089           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1090           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1091           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
  1092               $ t1 $ t2 =>
  1093             t0 $ aux Ts t1 $ aux Ts t2
  1094           | _ =>
  1095             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
  1096             else t |> Envir.eta_contract |> do_lambdas ctxt Ts
  1097         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
  1098       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
  1099   end
  1100 
  1101 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
  1102     do_cheaply_conceal_lambdas Ts t1
  1103     $ do_cheaply_conceal_lambdas Ts t2
  1104   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
  1105     Free (polymorphic_free_prefix ^ serial_string (),
  1106           T --> fastype_of1 (T :: Ts, t))
  1107   | do_cheaply_conceal_lambdas _ t = t
  1108 
  1109 fun do_introduce_combinators ctxt Ts t =
  1110   let val thy = Proof_Context.theory_of ctxt in
  1111     t |> conceal_bounds Ts
  1112       |> cterm_of thy
  1113       |> Meson_Clausify.introduce_combinators_in_cterm
  1114       |> prop_of |> Logic.dest_equals |> snd
  1115       |> reveal_bounds Ts
  1116   end
  1117   (* A type variable of sort "{}" will make abstraction fail. *)
  1118   handle THM _ => t |> do_cheaply_conceal_lambdas Ts
  1119 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
  1120 
  1121 fun preprocess_abstractions_in_terms trans_lambdas facts =
  1122   let
  1123     val (facts, lambda_ts) =
  1124       facts |> map (snd o snd) |> trans_lambdas
  1125             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
  1126     val lambda_facts =
  1127       map2 (fn t => fn j =>
  1128                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
  1129            lambda_ts (1 upto length lambda_ts)
  1130   in (facts, lambda_facts) end
  1131 
  1132 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
  1133    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
  1134 fun freeze_term t =
  1135   let
  1136     fun aux (t $ u) = aux t $ aux u
  1137       | aux (Abs (s, T, t)) = Abs (s, T, aux t)
  1138       | aux (Var ((s, i), T)) =
  1139         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
  1140       | aux t = t
  1141   in t |> exists_subterm is_Var t ? aux end
  1142 
  1143 fun presimp_prop ctxt presimp_consts t =
  1144   let
  1145     val thy = Proof_Context.theory_of ctxt
  1146     val t = t |> Envir.beta_eta_contract
  1147               |> transform_elim_prop
  1148               |> Object_Logic.atomize_term thy
  1149     val need_trueprop = (fastype_of t = @{typ bool})
  1150   in
  1151     t |> need_trueprop ? HOLogic.mk_Trueprop
  1152       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
  1153       |> extensionalize_term ctxt
  1154       |> presimplify_term ctxt presimp_consts
  1155       |> perhaps (try (HOLogic.dest_Trueprop))
  1156   end
  1157 
  1158 (* making fact and conjecture formulas *)
  1159 fun make_formula ctxt format type_enc eq_as_iff name loc kind t =
  1160   let
  1161     val (iformula, atomic_types) =
  1162       iformula_from_prop ctxt format type_enc eq_as_iff
  1163                          (SOME (kind <> Conjecture)) t []
  1164   in
  1165     {name = name, locality = loc, kind = kind, iformula = iformula,
  1166      atomic_types = atomic_types}
  1167   end
  1168 
  1169 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  1170   case t |> make_formula ctxt format type_enc (eq_as_iff andalso format <> CNF)
  1171                          name loc Axiom of
  1172     formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
  1173     if s = tptp_true then NONE else SOME formula
  1174   | formula => SOME formula
  1175 
  1176 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
  1177   | s_not_trueprop t = s_not t
  1178 
  1179 fun make_conjecture ctxt format type_enc =
  1180   map (fn ((name, loc), (kind, t)) =>
  1181           t |> kind = Conjecture ? s_not_trueprop
  1182             |> make_formula ctxt format type_enc (format <> CNF) name loc kind)
  1183 
  1184 (** Finite and infinite type inference **)
  1185 
  1186 type monotonicity_info =
  1187   {maybe_finite_Ts : typ list,
  1188    surely_finite_Ts : typ list,
  1189    maybe_infinite_Ts : typ list,
  1190    surely_infinite_Ts : typ list,
  1191    maybe_nonmono_Ts : typ list}
  1192 
  1193 (* These types witness that the type classes they belong to allow infinite
  1194    models and hence that any types with these type classes is monotonic. *)
  1195 val known_infinite_types =
  1196   [@{typ nat}, HOLogic.intT, HOLogic.realT, @{typ "nat => bool"}]
  1197 
  1198 fun is_type_kind_of_surely_infinite ctxt soundness cached_Ts T =
  1199   soundness <> Sound andalso is_type_surely_infinite ctxt true cached_Ts T
  1200 
  1201 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
  1202    dangerous because their "exhaust" properties can easily lead to unsound ATP
  1203    proofs. On the other hand, all HOL infinite types can be given the same
  1204    models in first-order logic (via Löwenheim-Skolem). *)
  1205 
  1206 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  1207   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
  1208                              maybe_nonmono_Ts, ...}
  1209                        (Noninf_Nonmono_Types (soundness, _)) T =
  1210     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1211     not (exists (type_instance ctxt T) surely_infinite_Ts orelse
  1212          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
  1213           is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts T))
  1214   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
  1215                              maybe_nonmono_Ts, ...}
  1216                        (Fin_Nonmono_Types _) T =
  1217     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1218     (exists (type_generalization ctxt T) surely_finite_Ts orelse
  1219      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
  1220       is_type_surely_finite ctxt T))
  1221   | should_encode_type _ _ _ _ = false
  1222 
  1223 fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =
  1224     (is_level_uniform level orelse should_guard_var ()) andalso
  1225     should_encode_type ctxt mono level T
  1226   | should_guard_type _ _ _ _ _ = false
  1227 
  1228 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
  1229     String.isPrefix bound_var_prefix s orelse
  1230     String.isPrefix all_bound_var_prefix s
  1231   | is_maybe_universal_var (IVar _) = true
  1232   | is_maybe_universal_var _ = false
  1233 
  1234 datatype tag_site =
  1235   Top_Level of bool option |
  1236   Eq_Arg of bool option |
  1237   Elsewhere
  1238 
  1239 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  1240   | should_tag_with_type ctxt mono (Tags (_, level)) site u T =
  1241     (if is_level_uniform level then
  1242        should_encode_type ctxt mono level T
  1243      else case (site, is_maybe_universal_var u) of
  1244        (Eq_Arg _, true) => should_encode_type ctxt mono level T
  1245      | _ => false)
  1246   | should_tag_with_type _ _ _ _ _ _ = false
  1247 
  1248 fun fused_type ctxt mono level =
  1249   let
  1250     val should_encode = should_encode_type ctxt mono level
  1251     fun fuse 0 T = if should_encode T then T else fused_infinite_type
  1252       | fuse ary (Type (@{type_name fun}, [T1, T2])) =
  1253         fuse 0 T1 --> fuse (ary - 1) T2
  1254       | fuse _ _ = raise Fail "expected function type"
  1255   in fuse end
  1256 
  1257 (** predicators and application operators **)
  1258 
  1259 type sym_info =
  1260   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
  1261 
  1262 fun add_iterm_syms_to_table ctxt explicit_apply =
  1263   let
  1264     fun consider_var_ary const_T var_T max_ary =
  1265       let
  1266         fun iter ary T =
  1267           if ary = max_ary orelse type_instance ctxt var_T T orelse
  1268              type_instance ctxt T var_T then
  1269             ary
  1270           else
  1271             iter (ary + 1) (range_type T)
  1272       in iter 0 const_T end
  1273     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1274       if explicit_apply = NONE andalso
  1275          (can dest_funT T orelse T = @{typ bool}) then
  1276         let
  1277           val bool_vars' = bool_vars orelse body_type T = @{typ bool}
  1278           fun repair_min_ary {pred_sym, min_ary, max_ary, types} =
  1279             {pred_sym = pred_sym andalso not bool_vars',
  1280              min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,
  1281              max_ary = max_ary, types = types}
  1282           val fun_var_Ts' =
  1283             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
  1284         in
  1285           if bool_vars' = bool_vars andalso
  1286              pointer_eq (fun_var_Ts', fun_var_Ts) then
  1287             accum
  1288           else
  1289             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_ary) sym_tab)
  1290         end
  1291       else
  1292         accum
  1293     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1294       let val (head, args) = strip_iterm_comb tm in
  1295         (case head of
  1296            IConst ((s, _), T, _) =>
  1297            if String.isPrefix bound_var_prefix s orelse
  1298               String.isPrefix all_bound_var_prefix s then
  1299              add_universal_var T accum
  1300            else if String.isPrefix exist_bound_var_prefix s then
  1301              accum
  1302            else
  1303              let val ary = length args in
  1304                ((bool_vars, fun_var_Ts),
  1305                 case Symtab.lookup sym_tab s of
  1306                   SOME {pred_sym, min_ary, max_ary, types} =>
  1307                   let
  1308                     val pred_sym =
  1309                       pred_sym andalso top_level andalso not bool_vars
  1310                     val types' = types |> insert_type ctxt I T
  1311                     val min_ary =
  1312                       if is_some explicit_apply orelse
  1313                          pointer_eq (types', types) then
  1314                         min_ary
  1315                       else
  1316                         fold (consider_var_ary T) fun_var_Ts min_ary
  1317                   in
  1318                     Symtab.update (s, {pred_sym = pred_sym,
  1319                                        min_ary = Int.min (ary, min_ary),
  1320                                        max_ary = Int.max (ary, max_ary),
  1321                                        types = types'})
  1322                                   sym_tab
  1323                   end
  1324                 | NONE =>
  1325                   let
  1326                     val pred_sym = top_level andalso not bool_vars
  1327                     val min_ary =
  1328                       case explicit_apply of
  1329                         SOME true => 0
  1330                       | SOME false => ary
  1331                       | NONE => fold (consider_var_ary T) fun_var_Ts ary
  1332                   in
  1333                     Symtab.update_new (s, {pred_sym = pred_sym,
  1334                                            min_ary = min_ary, max_ary = ary,
  1335                                            types = [T]})
  1336                                       sym_tab
  1337                   end)
  1338              end
  1339          | IVar (_, T) => add_universal_var T accum
  1340          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
  1341          | _ => accum)
  1342         |> fold (add false) args
  1343       end
  1344   in add true end
  1345 fun add_fact_syms_to_table ctxt explicit_apply =
  1346   K (add_iterm_syms_to_table ctxt explicit_apply)
  1347   |> formula_fold NONE |> fact_lift
  1348 
  1349 val default_sym_tab_entries : (string * sym_info) list =
  1350   (prefixed_predicator_name,
  1351    {pred_sym = true, min_ary = 1, max_ary = 1, types = []})
  1352        (* FIXME: needed? *) ::
  1353   (make_fixed_const NONE @{const_name undefined},
  1354    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  1355   ([tptp_false, tptp_true]
  1356    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  1357   ([tptp_equal, tptp_old_equal]
  1358    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
  1359 
  1360 fun sym_table_for_facts ctxt explicit_apply facts =
  1361   ((false, []), Symtab.empty)
  1362   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  1363   |> fold Symtab.update default_sym_tab_entries
  1364 
  1365 fun min_ary_of sym_tab s =
  1366   case Symtab.lookup sym_tab s of
  1367     SOME ({min_ary, ...} : sym_info) => min_ary
  1368   | NONE =>
  1369     case strip_prefix_and_unascii const_prefix s of
  1370       SOME s =>
  1371       let val s = s |> unmangled_const_name |> invert_const in
  1372         if s = predicator_name then 1
  1373         else if s = app_op_name then 2
  1374         else if s = type_guard_name then 1
  1375         else 0
  1376       end
  1377     | NONE => 0
  1378 
  1379 (* True if the constant ever appears outside of the top-level position in
  1380    literals, or if it appears with different arities (e.g., because of different
  1381    type instantiations). If false, the constant always receives all of its
  1382    arguments and is used as a predicate. *)
  1383 fun is_pred_sym sym_tab s =
  1384   case Symtab.lookup sym_tab s of
  1385     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
  1386     pred_sym andalso min_ary = max_ary
  1387   | NONE => false
  1388 
  1389 val app_op = `(make_fixed_const NONE) app_op_name
  1390 val predicator_combconst =
  1391   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
  1392 
  1393 fun list_app head args = fold (curry (IApp o swap)) args head
  1394 fun predicator tm = IApp (predicator_combconst, tm)
  1395 
  1396 fun firstorderize_fact ctxt format type_enc sym_tab =
  1397   let
  1398     fun do_app arg head =
  1399       let
  1400         val head_T = ityp_of head
  1401         val (arg_T, res_T) = dest_funT head_T
  1402         val app =
  1403           IConst (app_op, head_T --> head_T, [arg_T, res_T])
  1404           |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1405       in list_app app [head, arg] end
  1406     fun list_app_ops head args = fold do_app args head
  1407     fun introduce_app_ops tm =
  1408       case strip_iterm_comb tm of
  1409         (head as IConst ((s, _), _, _), args) =>
  1410         args |> map introduce_app_ops
  1411              |> chop (min_ary_of sym_tab s)
  1412              |>> list_app head
  1413              |-> list_app_ops
  1414       | (head, args) => list_app_ops head (map introduce_app_ops args)
  1415     fun introduce_predicators tm =
  1416       case strip_iterm_comb tm of
  1417         (IConst ((s, _), _, _), _) =>
  1418         if is_pred_sym sym_tab s then tm else predicator tm
  1419       | _ => predicator tm
  1420     val do_iterm =
  1421       not (is_type_enc_higher_order type_enc)
  1422       ? (introduce_app_ops #> introduce_predicators)
  1423   in update_iformula (formula_map do_iterm) end
  1424 
  1425 (** Helper facts **)
  1426 
  1427 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
  1428 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
  1429 
  1430 (* The Boolean indicates that a fairly sound type encoding is needed. *)
  1431 val helper_table =
  1432   [(("COMBI", false), @{thms Meson.COMBI_def}),
  1433    (("COMBK", false), @{thms Meson.COMBK_def}),
  1434    (("COMBB", false), @{thms Meson.COMBB_def}),
  1435    (("COMBC", false), @{thms Meson.COMBC_def}),
  1436    (("COMBS", false), @{thms Meson.COMBS_def}),
  1437    ((predicator_name, false), [not_ffalse, ftrue]),
  1438    (("fFalse", false), [not_ffalse]),
  1439    (("fFalse", true), @{thms True_or_False}),
  1440    (("fTrue", false), [ftrue]),
  1441    (("fTrue", true), @{thms True_or_False}),
  1442    (("fNot", false),
  1443     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1444            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1445    (("fconj", false),
  1446     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
  1447         by (unfold fconj_def) fast+}),
  1448    (("fdisj", false),
  1449     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
  1450         by (unfold fdisj_def) fast+}),
  1451    (("fimplies", false),
  1452     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
  1453         by (unfold fimplies_def) fast+}),
  1454    (("fequal", true),
  1455     (* This is a lie: Higher-order equality doesn't need a sound type encoding.
  1456        However, this is done so for backward compatibility: Including the
  1457        equality helpers by default in Metis breaks a few existing proofs. *)
  1458     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1459            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1460    (* Partial characterization of "fAll" and "fEx". A complete characterization
  1461       would require the axiom of choice for replay with Metis. *)
  1462    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
  1463    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
  1464    (("If", true), @{thms if_True if_False True_or_False})]
  1465   |> map (apsnd (map zero_var_indexes))
  1466 
  1467 fun bound_tvars type_enc =
  1468   mk_ahorn o formulas_for_types type_enc add_sorts_on_tvar
  1469 
  1470 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  1471   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
  1472    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  1473   |> bound_tvars type_enc atomic_Ts
  1474   |> close_formula_universally type_enc
  1475 
  1476 val type_tag = `(make_fixed_const NONE) type_tag_name
  1477 
  1478 fun type_tag_idempotence_fact type_enc =
  1479   let
  1480     fun var s = ATerm (`I s, [])
  1481     fun tag tm = ATerm (type_tag, [var "A", tm])
  1482     val tagged_var = tag (var "X")
  1483   in
  1484     Formula (type_tag_idempotence_helper_name, Axiom,
  1485              eq_formula type_enc [] false (tag tagged_var) tagged_var,
  1486              isabelle_info simpN, NONE)
  1487   end
  1488 
  1489 fun should_specialize_helper type_enc t =
  1490   polymorphism_of_type_enc type_enc <> Polymorphic andalso
  1491   level_of_type_enc type_enc <> No_Types andalso
  1492   not (null (Term.hidden_polymorphism t))
  1493 
  1494 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  1495   case strip_prefix_and_unascii const_prefix s of
  1496     SOME mangled_s =>
  1497     let
  1498       val thy = Proof_Context.theory_of ctxt
  1499       val unmangled_s = mangled_s |> unmangled_const_name
  1500       fun dub needs_fairly_sound j k =
  1501         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
  1502          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
  1503          (if needs_fairly_sound then typed_helper_suffix
  1504           else untyped_helper_suffix),
  1505          Helper)
  1506       fun dub_and_inst needs_fairly_sound (th, j) =
  1507         let val t = prop_of th in
  1508           if should_specialize_helper type_enc t then
  1509             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
  1510                 types
  1511           else
  1512             [t]
  1513         end
  1514         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
  1515       val make_facts = map_filter (make_fact ctxt format type_enc false)
  1516       val fairly_sound = is_type_enc_fairly_sound type_enc
  1517     in
  1518       helper_table
  1519       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
  1520                   if helper_s <> unmangled_s orelse
  1521                      (needs_fairly_sound andalso not fairly_sound) then
  1522                     []
  1523                   else
  1524                     ths ~~ (1 upto length ths)
  1525                     |> maps (dub_and_inst needs_fairly_sound)
  1526                     |> make_facts)
  1527     end
  1528   | NONE => []
  1529 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  1530   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
  1531                   []
  1532 
  1533 (***************************************************************)
  1534 (* Type Classes Present in the Axiom or Conjecture Clauses     *)
  1535 (***************************************************************)
  1536 
  1537 fun set_insert (x, s) = Symtab.update (x, ()) s
  1538 
  1539 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
  1540 
  1541 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
  1542 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
  1543 
  1544 fun classes_of_terms get_Ts =
  1545   map (map snd o get_Ts)
  1546   #> List.foldl add_classes Symtab.empty
  1547   #> delete_type #> Symtab.keys
  1548 
  1549 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
  1550 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
  1551 
  1552 fun fold_type_constrs f (Type (s, Ts)) x =
  1553     fold (fold_type_constrs f) Ts (f (s, x))
  1554   | fold_type_constrs _ _ x = x
  1555 
  1556 (* Type constructors used to instantiate overloaded constants are the only ones
  1557    needed. *)
  1558 fun add_type_constrs_in_term thy =
  1559   let
  1560     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
  1561       | add (t $ u) = add t #> add u
  1562       | add (Const x) =
  1563         x |> robust_const_typargs thy |> fold (fold_type_constrs set_insert)
  1564       | add (Free (s, T)) =
  1565         if String.isPrefix polymorphic_free_prefix s then
  1566           T |> fold_type_constrs set_insert
  1567         else
  1568           I
  1569       | add (Abs (_, _, u)) = add u
  1570       | add _ = I
  1571   in add end
  1572 
  1573 fun type_constrs_of_terms thy ts =
  1574   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
  1575 
  1576 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  1577                        hyp_ts concl_t facts =
  1578   let
  1579     val thy = Proof_Context.theory_of ctxt
  1580     val presimp_consts = Meson.presimplified_consts ctxt
  1581     val fact_ts = facts |> map snd
  1582     (* Remove existing facts from the conjecture, as this can dramatically
  1583        boost an ATP's performance (for some reason). *)
  1584     val hyp_ts =
  1585       hyp_ts
  1586       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
  1587     val facts = facts |> map (apsnd (pair Axiom))
  1588     val conjs =
  1589       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]
  1590       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
  1591     val ((conjs, facts), lambdas) =
  1592       if preproc then
  1593         conjs @ facts
  1594         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
  1595         |> preprocess_abstractions_in_terms trans_lambdas
  1596         |>> chop (length conjs)
  1597         |>> apfst (map (apsnd (apsnd freeze_term)))
  1598       else
  1599         ((conjs, facts), [])
  1600     val conjs = conjs |> make_conjecture ctxt format type_enc
  1601     val (fact_names, facts) =
  1602       facts
  1603       |> map_filter (fn (name, (_, t)) =>
  1604                         make_fact ctxt format type_enc true (name, t)
  1605                         |> Option.map (pair name))
  1606       |> ListPair.unzip
  1607     val lambdas =
  1608       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
  1609     val all_ts = concl_t :: hyp_ts @ fact_ts
  1610     val subs = tfree_classes_of_terms all_ts
  1611     val supers = tvar_classes_of_terms all_ts
  1612     val tycons = type_constrs_of_terms thy all_ts
  1613     val (supers, arity_clauses) =
  1614       if level_of_type_enc type_enc = No_Types then ([], [])
  1615       else make_arity_clauses thy tycons supers
  1616     val class_rel_clauses = make_class_rel_clauses thy subs supers
  1617   in
  1618     (fact_names |> map single, union (op =) subs supers, conjs, facts @ lambdas,
  1619      class_rel_clauses, arity_clauses)
  1620   end
  1621 
  1622 val type_guard = `(make_fixed_const NONE) type_guard_name
  1623 
  1624 fun type_guard_iterm ctxt format type_enc T tm =
  1625   IApp (IConst (type_guard, T --> @{typ bool}, [T])
  1626         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
  1627 
  1628 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  1629   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
  1630     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  1631   | is_var_positively_naked_in_term _ _ _ _ = true
  1632 
  1633 fun should_guard_var_in_formula pos phi (SOME true) name =
  1634     formula_fold pos (is_var_positively_naked_in_term name) phi false
  1635   | should_guard_var_in_formula _ _ _ _ = true
  1636 
  1637 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  1638   | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =
  1639     not (is_level_uniform level) andalso should_encode_type ctxt mono level T
  1640   | should_generate_tag_bound_decl _ _ _ _ _ = false
  1641 
  1642 fun mk_aterm format type_enc name T_args args =
  1643   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
  1644 
  1645 fun tag_with_type ctxt format mono type_enc pos T tm =
  1646   IConst (type_tag, T --> T, [T])
  1647   |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1648   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  1649   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
  1650        | _ => raise Fail "unexpected lambda-abstraction")
  1651 and ho_term_from_iterm ctxt format mono type_enc =
  1652   let
  1653     fun aux site u =
  1654       let
  1655         val (head, args) = strip_iterm_comb u
  1656         val pos =
  1657           case site of
  1658             Top_Level pos => pos
  1659           | Eq_Arg pos => pos
  1660           | Elsewhere => NONE
  1661         val t =
  1662           case head of
  1663             IConst (name as (s, _), _, T_args) =>
  1664             let
  1665               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
  1666             in
  1667               mk_aterm format type_enc name T_args (map (aux arg_site) args)
  1668             end
  1669           | IVar (name, _) =>
  1670             mk_aterm format type_enc name [] (map (aux Elsewhere) args)
  1671           | IAbs ((name, T), tm) =>
  1672             AAbs ((name, ho_type_from_typ format type_enc true 0 T),
  1673                   aux Elsewhere tm)
  1674           | IApp _ => raise Fail "impossible \"IApp\""
  1675         val T = ityp_of u
  1676       in
  1677         t |> (if should_tag_with_type ctxt mono type_enc site u T then
  1678                 tag_with_type ctxt format mono type_enc pos T
  1679               else
  1680                 I)
  1681       end
  1682   in aux end
  1683 and formula_from_iformula ctxt format mono type_enc should_guard_var =
  1684   let
  1685     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
  1686     val do_bound_type =
  1687       case type_enc of
  1688         Simple_Types (_, _, level) => fused_type ctxt mono level 0
  1689         #> ho_type_from_typ format type_enc false 0 #> SOME
  1690       | _ => K NONE
  1691     fun do_out_of_bound_type pos phi universal (name, T) =
  1692       if should_guard_type ctxt mono type_enc
  1693              (fn () => should_guard_var pos phi universal name) T then
  1694         IVar (name, T)
  1695         |> type_guard_iterm ctxt format type_enc T
  1696         |> do_term pos |> AAtom |> SOME
  1697       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
  1698         let
  1699           val var = ATerm (name, [])
  1700           val tagged_var = var |> tag_with_type ctxt format mono type_enc pos T
  1701         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
  1702       else
  1703         NONE
  1704     fun do_formula pos (AQuant (q, xs, phi)) =
  1705         let
  1706           val phi = phi |> do_formula pos
  1707           val universal = Option.map (q = AExists ? not) pos
  1708         in
  1709           AQuant (q, xs |> map (apsnd (fn NONE => NONE
  1710                                         | SOME T => do_bound_type T)),
  1711                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
  1712                       (map_filter
  1713                            (fn (_, NONE) => NONE
  1714                              | (s, SOME T) =>
  1715                                do_out_of_bound_type pos phi universal (s, T))
  1716                            xs)
  1717                       phi)
  1718         end
  1719       | do_formula pos (AConn conn) = aconn_map pos do_formula conn
  1720       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  1721   in do_formula end
  1722 
  1723 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
  1724    of monomorphization). The TPTP explicitly forbids name clashes, and some of
  1725    the remote provers might care. *)
  1726 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
  1727                           (j, {name, locality, kind, iformula, atomic_types}) =
  1728   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
  1729    iformula
  1730    |> close_iformula_universally
  1731    |> formula_from_iformula ctxt format mono type_enc
  1732                             should_guard_var_in_formula
  1733                             (if pos then SOME true else NONE)
  1734    |> bound_tvars type_enc atomic_types
  1735    |> close_formula_universally type_enc,
  1736    NONE,
  1737    case locality of
  1738      Intro => isabelle_info introN
  1739    | Elim => isabelle_info elimN
  1740    | Simp => isabelle_info simpN
  1741    | _ => NONE)
  1742   |> Formula
  1743 
  1744 fun formula_line_for_class_rel_clause type_enc
  1745         ({name, subclass, superclass, ...} : class_rel_clause) =
  1746   let val ty_arg = ATerm (tvar_a_name, []) in
  1747     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
  1748              AConn (AImplies,
  1749                     [type_class_formula type_enc subclass ty_arg,
  1750                      type_class_formula type_enc superclass ty_arg])
  1751              |> close_formula_universally type_enc, isabelle_info introN, NONE)
  1752   end
  1753 
  1754 fun formula_from_arity_atom type_enc (class, t, args) =
  1755   ATerm (t, map (fn arg => ATerm (arg, [])) args)
  1756   |> type_class_formula type_enc class
  1757 
  1758 fun formula_line_for_arity_clause type_enc
  1759         ({name, prem_atoms, concl_atom, ...} : arity_clause) =
  1760   Formula (arity_clause_prefix ^ name, Axiom,
  1761            mk_ahorn (map (formula_from_arity_atom type_enc) prem_atoms)
  1762                     (formula_from_arity_atom type_enc concl_atom)
  1763            |> close_formula_universally type_enc, isabelle_info introN, NONE)
  1764 
  1765 fun formula_line_for_conjecture ctxt format mono type_enc
  1766         ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  1767   Formula (conjecture_prefix ^ name, kind,
  1768            formula_from_iformula ctxt format mono type_enc
  1769                should_guard_var_in_formula (SOME false)
  1770                (close_iformula_universally iformula)
  1771            |> bound_tvars type_enc atomic_types
  1772            |> close_formula_universally type_enc, NONE, NONE)
  1773 
  1774 fun formula_line_for_free_type j phi =
  1775   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis, phi, NONE, NONE)
  1776 fun formula_lines_for_free_types type_enc (facts : translated_formula list) =
  1777   let
  1778     val phis =
  1779       fold (union (op =)) (map #atomic_types facts) []
  1780       |> formulas_for_types type_enc add_sorts_on_tfree
  1781   in map2 formula_line_for_free_type (0 upto length phis - 1) phis end
  1782 
  1783 (** Symbol declarations **)
  1784 
  1785 fun decl_line_for_class order s =
  1786   let val name as (s, _) = `make_type_class s in
  1787     Decl (sym_decl_prefix ^ s, name,
  1788           if order = First_Order andalso avoid_first_order_dummy_type_vars then
  1789             ATyAbs ([tvar_a_name], AFun (a_itself_atype, bool_atype))
  1790           else
  1791             AFun (atype_of_types, bool_atype))
  1792   end
  1793 
  1794 fun decl_lines_for_classes type_enc classes =
  1795   case type_enc of
  1796     Simple_Types (order, Polymorphic, _) =>
  1797     map (decl_line_for_class order) classes
  1798   | _ => []
  1799 
  1800 fun sym_decl_table_for_facts ctxt format type_enc sym_tab (conjs, facts) =
  1801   let
  1802     fun add_iterm_syms in_conj tm =
  1803       let val (head, args) = strip_iterm_comb tm in
  1804         (case head of
  1805            IConst ((s, s'), T, T_args) =>
  1806            let
  1807              val pred_sym = is_pred_sym sym_tab s
  1808              val decl_sym =
  1809                (case type_enc of
  1810                   Guards _ => not pred_sym
  1811                 | _ => true) andalso
  1812                is_tptp_user_symbol s
  1813            in
  1814              if decl_sym then
  1815                Symtab.map_default (s, [])
  1816                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
  1817                                          in_conj))
  1818              else
  1819                I
  1820            end
  1821          | IAbs (_, tm) => add_iterm_syms in_conj tm
  1822          | _ => I)
  1823         #> fold (add_iterm_syms in_conj) args
  1824       end
  1825     fun add_fact_syms in_conj =
  1826       K (add_iterm_syms in_conj) |> formula_fold NONE |> fact_lift
  1827     fun add_formula_var_types (AQuant (_, xs, phi)) =
  1828         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
  1829         #> add_formula_var_types phi
  1830       | add_formula_var_types (AConn (_, phis)) =
  1831         fold add_formula_var_types phis
  1832       | add_formula_var_types _ = I
  1833     fun var_types () =
  1834       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
  1835       else fold (fact_lift add_formula_var_types) (conjs @ facts) []
  1836     fun add_undefined_const T =
  1837       let
  1838         val (s, s') =
  1839           `(make_fixed_const NONE) @{const_name undefined}
  1840           |> (case type_arg_policy type_enc @{const_name undefined} of
  1841                 Mangled_Type_Args => mangled_const_name format type_enc [T]
  1842               | _ => I)
  1843       in
  1844         Symtab.map_default (s, [])
  1845                            (insert_type ctxt #3 (s', [T], T, false, 0, false))
  1846       end
  1847     fun add_TYPE_const () =
  1848       let val (s, s') = TYPE_name in
  1849         Symtab.map_default (s, [])
  1850             (insert_type ctxt #3
  1851                          (s', [tvar_a], @{typ "'a itself"}, false, 0, false))
  1852       end
  1853   in
  1854     Symtab.empty
  1855     |> is_type_enc_fairly_sound type_enc
  1856        ? (fold (add_fact_syms true) conjs
  1857           #> fold (add_fact_syms false) facts
  1858           #> (case type_enc of
  1859                 Simple_Types (_, poly, _) =>
  1860                 if poly = Polymorphic then add_TYPE_const () else I
  1861               | _ => fold add_undefined_const (var_types ())))
  1862   end
  1863 
  1864 (* We add "bool" in case the helper "True_or_False" is included later. *)
  1865 fun default_mono level =
  1866   {maybe_finite_Ts = [@{typ bool}],
  1867    surely_finite_Ts = [@{typ bool}],
  1868    maybe_infinite_Ts = known_infinite_types,
  1869    surely_infinite_Ts =
  1870      case level of
  1871        Noninf_Nonmono_Types (Sound, _) => []
  1872      | _ => known_infinite_types,
  1873    maybe_nonmono_Ts = [@{typ bool}]}
  1874 
  1875 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
  1876    out with monotonicity" paper presented at CADE 2011. *)
  1877 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  1878   | add_iterm_mononotonicity_info ctxt level _
  1879         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
  1880         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
  1881                   surely_infinite_Ts, maybe_nonmono_Ts}) =
  1882     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
  1883       case level of
  1884         Noninf_Nonmono_Types (soundness, _) =>
  1885         if exists (type_instance ctxt T) surely_infinite_Ts orelse
  1886            member (type_aconv ctxt) maybe_finite_Ts T then
  1887           mono
  1888         else if is_type_kind_of_surely_infinite ctxt soundness
  1889                                                 surely_infinite_Ts T then
  1890           {maybe_finite_Ts = maybe_finite_Ts,
  1891            surely_finite_Ts = surely_finite_Ts,
  1892            maybe_infinite_Ts = maybe_infinite_Ts,
  1893            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
  1894            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1895         else
  1896           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
  1897            surely_finite_Ts = surely_finite_Ts,
  1898            maybe_infinite_Ts = maybe_infinite_Ts,
  1899            surely_infinite_Ts = surely_infinite_Ts,
  1900            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1901       | Fin_Nonmono_Types _ =>
  1902         if exists (type_instance ctxt T) surely_finite_Ts orelse
  1903            member (type_aconv ctxt) maybe_infinite_Ts T then
  1904           mono
  1905         else if is_type_surely_finite ctxt T then
  1906           {maybe_finite_Ts = maybe_finite_Ts,
  1907            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
  1908            maybe_infinite_Ts = maybe_infinite_Ts,
  1909            surely_infinite_Ts = surely_infinite_Ts,
  1910            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1911         else
  1912           {maybe_finite_Ts = maybe_finite_Ts,
  1913            surely_finite_Ts = surely_finite_Ts,
  1914            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
  1915            surely_infinite_Ts = surely_infinite_Ts,
  1916            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1917       | _ => mono
  1918     else
  1919       mono
  1920   | add_iterm_mononotonicity_info _ _ _ _ mono = mono
  1921 fun add_fact_mononotonicity_info ctxt level
  1922         ({kind, iformula, ...} : translated_formula) =
  1923   formula_fold (SOME (kind <> Conjecture))
  1924                (add_iterm_mononotonicity_info ctxt level) iformula
  1925 fun mononotonicity_info_for_facts ctxt type_enc facts =
  1926   let val level = level_of_type_enc type_enc in
  1927     default_mono level
  1928     |> is_type_level_monotonicity_based level
  1929        ? fold (add_fact_mononotonicity_info ctxt level) facts
  1930   end
  1931 
  1932 fun add_iformula_monotonic_types ctxt mono type_enc =
  1933   let
  1934     val level = level_of_type_enc type_enc
  1935     val should_encode = should_encode_type ctxt mono level
  1936     fun add_type T = not (should_encode T) ? insert_type ctxt I T
  1937     fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2
  1938       | add_args _ = I
  1939     and add_term tm = add_type (ityp_of tm) #> add_args tm
  1940   in formula_fold NONE (K add_term) end
  1941 fun add_fact_monotonic_types ctxt mono type_enc =
  1942   add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
  1943 fun monotonic_types_for_facts ctxt mono type_enc facts =
  1944   [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
  1945          is_type_level_monotonicity_based (level_of_type_enc type_enc))
  1946         ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
  1947 
  1948 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  1949   Formula (guards_sym_formula_prefix ^
  1950            ascii_of (mangled_type format type_enc T),
  1951            Axiom,
  1952            IConst (`make_bound_var "X", T, [])
  1953            |> type_guard_iterm ctxt format type_enc T
  1954            |> AAtom
  1955            |> formula_from_iformula ctxt format mono type_enc
  1956                                     (K (K (K (K true)))) (SOME true)
  1957            |> bound_tvars type_enc (atyps_of T)
  1958            |> close_formula_universally type_enc,
  1959            isabelle_info introN, NONE)
  1960 
  1961 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  1962   let val x_var = ATerm (`make_bound_var "X", []) in
  1963     Formula (tags_sym_formula_prefix ^
  1964              ascii_of (mangled_type format type_enc T),
  1965              Axiom,
  1966              eq_formula type_enc (atyps_of T) false
  1967                         (tag_with_type ctxt format mono type_enc NONE T x_var)
  1968                         x_var,
  1969              isabelle_info simpN, NONE)
  1970   end
  1971 
  1972 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  1973   case type_enc of
  1974     Simple_Types _ => []
  1975   | Guards _ =>
  1976     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  1977   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
  1978 
  1979 fun decl_line_for_sym ctxt format mono type_enc s
  1980                       (s', T_args, T, pred_sym, ary, _) =
  1981   let
  1982     val thy = Proof_Context.theory_of ctxt
  1983     val (T, T_args) =
  1984       if null T_args then
  1985         (T, [])
  1986       else case strip_prefix_and_unascii const_prefix s of
  1987         SOME s' =>
  1988         let
  1989           val s' = s' |> invert_const
  1990           val T = s' |> robust_const_type thy
  1991         in (T, robust_const_typargs thy (s', T)) end
  1992       | NONE =>
  1993         case strip_prefix_and_unascii fixed_var_prefix s of
  1994           SOME s' =>
  1995           if String.isPrefix polymorphic_free_prefix s' then (tvar_a, [tvar_a])
  1996           else raise Fail "unexpected type arguments to free variable"
  1997         | NONE => raise Fail "unexpected type arguments"
  1998   in
  1999     Decl (sym_decl_prefix ^ s, (s, s'),
  2000           T |> fused_type ctxt mono (level_of_type_enc type_enc) ary
  2001             |> ho_type_from_typ format type_enc pred_sym ary
  2002             |> not (null T_args)
  2003                ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))
  2004   end
  2005 
  2006 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
  2007                                      j (s', T_args, T, _, ary, in_conj) =
  2008   let
  2009     val (kind, maybe_negate) =
  2010       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  2011       else (Axiom, I)
  2012     val (arg_Ts, res_T) = chop_fun ary T
  2013     val num_args = length arg_Ts
  2014     val bound_names =
  2015       1 upto num_args |> map (`I o make_bound_var o string_of_int)
  2016     val bounds =
  2017       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
  2018     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
  2019     fun should_keep_arg_type T =
  2020       sym_needs_arg_types andalso
  2021       should_guard_type ctxt mono type_enc (K true) T
  2022     val bound_Ts =
  2023       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  2024   in
  2025     Formula (guards_sym_formula_prefix ^ s ^
  2026              (if n > 1 then "_" ^ string_of_int j else ""), kind,
  2027              IConst ((s, s'), T, T_args)
  2028              |> fold (curry (IApp o swap)) bounds
  2029              |> type_guard_iterm ctxt format type_enc res_T
  2030              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
  2031              |> formula_from_iformula ctxt format mono type_enc
  2032                                       (K (K (K (K true)))) (SOME true)
  2033              |> n > 1 ? bound_tvars type_enc (atyps_of T)
  2034              |> close_formula_universally type_enc
  2035              |> maybe_negate,
  2036              isabelle_info introN, NONE)
  2037   end
  2038 
  2039 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
  2040         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  2041   let
  2042     val ident_base =
  2043       tags_sym_formula_prefix ^ s ^
  2044       (if n > 1 then "_" ^ string_of_int j else "")
  2045     val (kind, maybe_negate) =
  2046       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  2047       else (Axiom, I)
  2048     val (arg_Ts, res_T) = chop_fun ary T
  2049     val bound_names =
  2050       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
  2051     val bounds = bound_names |> map (fn name => ATerm (name, []))
  2052     val cst = mk_aterm format type_enc (s, s') T_args
  2053     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
  2054     val should_encode =
  2055       should_encode_type ctxt mono (level_of_type_enc type_enc)
  2056     val tag_with = tag_with_type ctxt format mono type_enc NONE
  2057     val add_formula_for_res =
  2058       if should_encode res_T then
  2059         cons (Formula (ident_base ^ "_res", kind,
  2060                        eq (tag_with res_T (cst bounds)) (cst bounds),
  2061                        isabelle_info simpN, NONE))
  2062       else
  2063         I
  2064     fun add_formula_for_arg k =
  2065       let val arg_T = nth arg_Ts k in
  2066         if should_encode arg_T then
  2067           case chop k bounds of
  2068             (bounds1, bound :: bounds2) =>
  2069             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
  2070                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
  2071                               (cst bounds),
  2072                            isabelle_info simpN, NONE))
  2073           | _ => raise Fail "expected nonempty tail"
  2074         else
  2075           I
  2076       end
  2077   in
  2078     [] |> not pred_sym ? add_formula_for_res
  2079        |> Config.get ctxt type_tag_arguments
  2080           ? fold add_formula_for_arg (ary - 1 downto 0)
  2081   end
  2082 
  2083 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
  2084 
  2085 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
  2086                                 (s, decls) =
  2087   case type_enc of
  2088     Simple_Types _ =>
  2089     decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  2090   | Guards (_, level) =>
  2091     let
  2092       val decls =
  2093         case decls of
  2094           decl :: (decls' as _ :: _) =>
  2095           let val T = result_type_of_decl decl in
  2096             if forall (type_generalization ctxt T o result_type_of_decl)
  2097                       decls' then
  2098               [decl]
  2099             else
  2100               decls
  2101           end
  2102         | _ => decls
  2103       val n = length decls
  2104       val decls =
  2105         decls |> filter (should_encode_type ctxt mono level
  2106                          o result_type_of_decl)
  2107     in
  2108       (0 upto length decls - 1, decls)
  2109       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
  2110                                                  type_enc n s)
  2111     end
  2112   | Tags (_, level) =>
  2113     if is_level_uniform level then
  2114       []
  2115     else
  2116       let val n = length decls in
  2117         (0 upto n - 1 ~~ decls)
  2118         |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
  2119                      conj_sym_kind mono type_enc n s)
  2120       end
  2121 
  2122 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
  2123                                      mono_Ts sym_decl_tab =
  2124   let
  2125     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
  2126     val mono_lines =
  2127       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
  2128     val decl_lines =
  2129       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
  2130                                                      mono type_enc)
  2131                syms []
  2132   in mono_lines @ decl_lines end
  2133 
  2134 fun needs_type_tag_idempotence ctxt (Tags (poly, level)) =
  2135     Config.get ctxt type_tag_idempotence andalso
  2136     is_type_level_monotonicity_based level andalso
  2137     poly <> Mangled_Monomorphic
  2138   | needs_type_tag_idempotence _ _ = false
  2139 
  2140 fun offset_of_heading_in_problem _ [] j = j
  2141   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
  2142     if heading = needle then j
  2143     else offset_of_heading_in_problem needle problem (j + length lines)
  2144 
  2145 val implicit_declsN = "Should-be-implicit typings"
  2146 val explicit_declsN = "Explicit typings"
  2147 val factsN = "Relevant facts"
  2148 val class_relsN = "Class relationships"
  2149 val aritiesN = "Arities"
  2150 val helpersN = "Helper facts"
  2151 val conjsN = "Conjectures"
  2152 val free_typesN = "Type variables"
  2153 
  2154 val explicit_apply = NONE (* for experiments *)
  2155 
  2156 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
  2157         lambda_trans readable_names preproc hyp_ts concl_t facts =
  2158   let
  2159     val type_enc = type_enc |> adjust_type_enc format
  2160     val lambda_trans =
  2161       if lambda_trans = smartN then
  2162         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
  2163       else if lambda_trans = lambdasN andalso
  2164               not (is_type_enc_higher_order type_enc) then
  2165         error ("Lambda translation method incompatible with first-order \
  2166                \encoding.")
  2167       else
  2168         lambda_trans
  2169     val trans_lambdas =
  2170       if lambda_trans = no_lambdasN then
  2171         rpair []
  2172       else if lambda_trans = concealedN then
  2173         lift_lambdas ctxt type_enc ##> K []
  2174       else if lambda_trans = liftingN then
  2175         lift_lambdas ctxt type_enc
  2176       else if lambda_trans = combinatorsN then
  2177         map (introduce_combinators ctxt) #> rpair []
  2178       else if lambda_trans = hybridN then
  2179         lift_lambdas ctxt type_enc
  2180         ##> maps (fn t => [t, introduce_combinators ctxt
  2181                                   (intentionalize_def t)])
  2182       else if lambda_trans = lambdasN then
  2183         map (Envir.eta_contract) #> rpair []
  2184       else
  2185         error ("Unknown lambda translation method: " ^
  2186                quote lambda_trans ^ ".")
  2187     val (fact_names, classes, conjs, facts, class_rel_clauses, arity_clauses) =
  2188       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  2189                          hyp_ts concl_t facts
  2190     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
  2191     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
  2192     val firstorderize = firstorderize_fact ctxt format type_enc sym_tab
  2193     val (conjs, facts) = (conjs, facts) |> pairself (map firstorderize)
  2194     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt (SOME false)
  2195     val helpers =
  2196       sym_tab |> helper_facts_for_sym_table ctxt format type_enc
  2197               |> map firstorderize
  2198     val mono_Ts =
  2199       helpers @ conjs @ facts
  2200       |> monotonic_types_for_facts ctxt mono type_enc
  2201     val class_decl_lines = decl_lines_for_classes type_enc classes
  2202     val sym_decl_lines =
  2203       (conjs, helpers @ facts)
  2204       |> sym_decl_table_for_facts ctxt format type_enc sym_tab
  2205       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
  2206                                                type_enc mono_Ts
  2207     val helper_lines =
  2208       0 upto length helpers - 1 ~~ helpers
  2209       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
  2210                                     type_enc)
  2211       |> (if needs_type_tag_idempotence ctxt type_enc then
  2212             cons (type_tag_idempotence_fact type_enc)
  2213           else
  2214             I)
  2215     (* Reordering these might confuse the proof reconstruction code or the SPASS
  2216        FLOTTER hack. *)
  2217     val problem =
  2218       [(explicit_declsN, class_decl_lines @ sym_decl_lines),
  2219        (factsN,
  2220         map (formula_line_for_fact ctxt format fact_prefix ascii_of
  2221                                    (not exporter) (not exporter) mono type_enc)
  2222             (0 upto length facts - 1 ~~ facts)),
  2223        (class_relsN,
  2224         map (formula_line_for_class_rel_clause type_enc) class_rel_clauses),
  2225        (aritiesN, map (formula_line_for_arity_clause type_enc) arity_clauses),
  2226        (helpersN, helper_lines),
  2227        (conjsN,
  2228         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
  2229        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
  2230     val problem =
  2231       problem
  2232       |> (case format of
  2233             CNF => ensure_cnf_problem
  2234           | CNF_UEQ => filter_cnf_ueq_problem
  2235           | FOF => I
  2236           | TFF (_, TPTP_Implicit) => I
  2237           | THF (_, TPTP_Implicit, _) => I
  2238           | _ => declare_undeclared_syms_in_atp_problem type_decl_prefix
  2239                                                         implicit_declsN)
  2240     val (problem, pool) = problem |> nice_atp_problem readable_names
  2241     val helpers_offset = offset_of_heading_in_problem helpersN problem 0
  2242     val typed_helpers =
  2243       map_filter (fn (j, {name, ...}) =>
  2244                      if String.isSuffix typed_helper_suffix name then SOME j
  2245                      else NONE)
  2246                  ((helpers_offset + 1 upto helpers_offset + length helpers)
  2247                   ~~ helpers)
  2248     fun add_sym_ary (s, {min_ary, ...} : sym_info) =
  2249       min_ary > 0 ? Symtab.insert (op =) (s, min_ary)
  2250   in
  2251     (problem,
  2252      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
  2253      offset_of_heading_in_problem conjsN problem 0,
  2254      offset_of_heading_in_problem factsN problem 0,
  2255      fact_names |> Vector.fromList,
  2256      typed_helpers,
  2257      Symtab.empty |> Symtab.fold add_sym_ary sym_tab)
  2258   end
  2259 
  2260 (* FUDGE *)
  2261 val conj_weight = 0.0
  2262 val hyp_weight = 0.1
  2263 val fact_min_weight = 0.2
  2264 val fact_max_weight = 1.0
  2265 val type_info_default_weight = 0.8
  2266 
  2267 fun add_term_weights weight (ATerm (s, tms)) =
  2268     is_tptp_user_symbol s ? Symtab.default (s, weight)
  2269     #> fold (add_term_weights weight) tms
  2270   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
  2271 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
  2272     formula_fold NONE (K (add_term_weights weight)) phi
  2273   | add_problem_line_weights _ _ = I
  2274 
  2275 fun add_conjectures_weights [] = I
  2276   | add_conjectures_weights conjs =
  2277     let val (hyps, conj) = split_last conjs in
  2278       add_problem_line_weights conj_weight conj
  2279       #> fold (add_problem_line_weights hyp_weight) hyps
  2280     end
  2281 
  2282 fun add_facts_weights facts =
  2283   let
  2284     val num_facts = length facts
  2285     fun weight_of j =
  2286       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
  2287                         / Real.fromInt num_facts
  2288   in
  2289     map weight_of (0 upto num_facts - 1) ~~ facts
  2290     |> fold (uncurry add_problem_line_weights)
  2291   end
  2292 
  2293 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
  2294 fun atp_problem_weights problem =
  2295   let val get = these o AList.lookup (op =) problem in
  2296     Symtab.empty
  2297     |> add_conjectures_weights (get free_typesN @ get conjsN)
  2298     |> add_facts_weights (get factsN)
  2299     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
  2300             [explicit_declsN, class_relsN, aritiesN]
  2301     |> Symtab.dest
  2302     |> sort (prod_ord Real.compare string_ord o pairself swap)
  2303   end
  2304 
  2305 end;