src/HOL/Tools/ATP/atp_translate.ML
author blanchet
Wed Sep 07 09:10:41 2011 +0200 (2011-09-07)
changeset 44771 0e5d4388bbac
parent 44770 3b1b4d805441
child 44772 60ac7b56296a
permissions -rw-r--r--
make mangling sound w.r.t. type arguments
     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 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 iformula_from_prop thy format type_enc eq_as_iff =
   959   let
   960     fun do_term bs t atomic_types =
   961       iterm_from_term thy format bs (Envir.eta_contract t)
   962       |>> (introduce_proxies type_enc #> AAtom)
   963       ||> union (op =) atomic_types
   964     fun do_quant bs q pos s T t' =
   965       let
   966         val s = singleton (Name.variant_list (map fst bs)) s
   967         val universal = Option.map (q = AExists ? not) pos
   968         val name =
   969           s |> `(case universal of
   970                    SOME true => make_all_bound_var
   971                  | SOME false => make_exist_bound_var
   972                  | NONE => make_bound_var)
   973       in
   974         do_formula ((s, (name, T)) :: bs) pos t'
   975         #>> mk_aquant q [(name, SOME T)]
   976       end
   977     and do_conn bs c pos1 t1 pos2 t2 =
   978       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
   979     and do_formula bs pos t =
   980       case t of
   981         @{const Trueprop} $ t1 => do_formula bs pos t1
   982       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
   983       | Const (@{const_name All}, _) $ Abs (s, T, t') =>
   984         do_quant bs AForall pos s T t'
   985       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
   986         do_quant bs AExists pos s T t'
   987       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
   988       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
   989       | @{const HOL.implies} $ t1 $ t2 =>
   990         do_conn bs AImplies (Option.map not pos) t1 pos t2
   991       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
   992         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
   993       | _ => do_term bs t
   994   in do_formula [] end
   995 
   996 fun presimplify_term _ [] t = t
   997   | presimplify_term ctxt presimp_consts t =
   998     t |> exists_Const (member (op =) presimp_consts o fst) t
   999          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
  1000             #> Meson.presimplify ctxt
  1001             #> prop_of)
  1002 
  1003 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
  1004 fun conceal_bounds Ts t =
  1005   subst_bounds (map (Free o apfst concealed_bound_name)
  1006                     (0 upto length Ts - 1 ~~ Ts), t)
  1007 fun reveal_bounds Ts =
  1008   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
  1009                     (0 upto length Ts - 1 ~~ Ts))
  1010 
  1011 fun is_fun_equality (@{const_name HOL.eq},
  1012                      Type (_, [Type (@{type_name fun}, _), _])) = true
  1013   | is_fun_equality _ = false
  1014 
  1015 fun extensionalize_term ctxt t =
  1016   if exists_Const is_fun_equality t then
  1017     let val thy = Proof_Context.theory_of ctxt in
  1018       t |> cterm_of thy |> Meson.extensionalize_conv ctxt
  1019         |> prop_of |> Logic.dest_equals |> snd
  1020     end
  1021   else
  1022     t
  1023 
  1024 fun simple_translate_lambdas do_lambdas ctxt t =
  1025   let val thy = Proof_Context.theory_of ctxt in
  1026     if Meson.is_fol_term thy t then
  1027       t
  1028     else
  1029       let
  1030         fun aux Ts t =
  1031           case t of
  1032             @{const Not} $ t1 => @{const Not} $ aux Ts t1
  1033           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
  1034             t0 $ Abs (s, T, aux (T :: Ts) t')
  1035           | (t0 as Const (@{const_name All}, _)) $ t1 =>
  1036             aux Ts (t0 $ eta_expand Ts t1 1)
  1037           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
  1038             t0 $ Abs (s, T, aux (T :: Ts) t')
  1039           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
  1040             aux Ts (t0 $ eta_expand Ts t1 1)
  1041           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1042           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1043           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
  1044           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
  1045               $ t1 $ t2 =>
  1046             t0 $ aux Ts t1 $ aux Ts t2
  1047           | _ =>
  1048             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
  1049             else t |> Envir.eta_contract |> do_lambdas ctxt Ts
  1050         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
  1051       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
  1052   end
  1053 
  1054 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
  1055     do_cheaply_conceal_lambdas Ts t1
  1056     $ do_cheaply_conceal_lambdas Ts t2
  1057   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
  1058     Free (polymorphic_free_prefix ^ serial_string (),
  1059           T --> fastype_of1 (T :: Ts, t))
  1060   | do_cheaply_conceal_lambdas _ t = t
  1061 
  1062 fun do_introduce_combinators ctxt Ts t =
  1063   let val thy = Proof_Context.theory_of ctxt in
  1064     t |> conceal_bounds Ts
  1065       |> cterm_of thy
  1066       |> Meson_Clausify.introduce_combinators_in_cterm
  1067       |> prop_of |> Logic.dest_equals |> snd
  1068       |> reveal_bounds Ts
  1069   end
  1070   (* A type variable of sort "{}" will make abstraction fail. *)
  1071   handle THM _ => t |> do_cheaply_conceal_lambdas Ts
  1072 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
  1073 
  1074 fun preprocess_abstractions_in_terms trans_lambdas facts =
  1075   let
  1076     val (facts, lambda_ts) =
  1077       facts |> map (snd o snd) |> trans_lambdas
  1078             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
  1079     val lambda_facts =
  1080       map2 (fn t => fn j =>
  1081                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
  1082            lambda_ts (1 upto length lambda_ts)
  1083   in (facts, lambda_facts) end
  1084 
  1085 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
  1086    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
  1087 fun freeze_term t =
  1088   let
  1089     fun aux (t $ u) = aux t $ aux u
  1090       | aux (Abs (s, T, t)) = Abs (s, T, aux t)
  1091       | aux (Var ((s, i), T)) =
  1092         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
  1093       | aux t = t
  1094   in t |> exists_subterm is_Var t ? aux end
  1095 
  1096 fun presimp_prop ctxt presimp_consts t =
  1097   let
  1098     val thy = Proof_Context.theory_of ctxt
  1099     val t = t |> Envir.beta_eta_contract
  1100               |> transform_elim_prop
  1101               |> Object_Logic.atomize_term thy
  1102     val need_trueprop = (fastype_of t = @{typ bool})
  1103   in
  1104     t |> need_trueprop ? HOLogic.mk_Trueprop
  1105       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
  1106       |> extensionalize_term ctxt
  1107       |> presimplify_term ctxt presimp_consts
  1108       |> perhaps (try (HOLogic.dest_Trueprop))
  1109   end
  1110 
  1111 (* making fact and conjecture formulas *)
  1112 fun make_formula thy format type_enc eq_as_iff name loc kind t =
  1113   let
  1114     val (iformula, atomic_types) =
  1115       iformula_from_prop thy format type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
  1116   in
  1117     {name = name, locality = loc, kind = kind, iformula = iformula,
  1118      atomic_types = atomic_types}
  1119   end
  1120 
  1121 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  1122   let val thy = Proof_Context.theory_of ctxt in
  1123     case t |> make_formula thy format type_enc (eq_as_iff andalso format <> CNF)
  1124                            name loc Axiom of
  1125       formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
  1126       if s = tptp_true then NONE else SOME formula
  1127     | formula => SOME formula
  1128   end
  1129 
  1130 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
  1131   | s_not_trueprop t = s_not t
  1132 
  1133 fun make_conjecture thy format type_enc =
  1134   map (fn ((name, loc), (kind, t)) =>
  1135           t |> kind = Conjecture ? s_not_trueprop
  1136             |> make_formula thy format type_enc (format <> CNF) name loc kind)
  1137 
  1138 (** Finite and infinite type inference **)
  1139 
  1140 type monotonicity_info =
  1141   {maybe_finite_Ts : typ list,
  1142    surely_finite_Ts : typ list,
  1143    maybe_infinite_Ts : typ list,
  1144    surely_infinite_Ts : typ list,
  1145    maybe_nonmono_Ts : typ list}
  1146 
  1147 (* These types witness that the type classes they belong to allow infinite
  1148    models and hence that any types with these type classes is monotonic. *)
  1149 val known_infinite_types =
  1150   [@{typ nat}, HOLogic.intT, HOLogic.realT, @{typ "nat => bool"}]
  1151 
  1152 fun is_type_kind_of_surely_infinite ctxt soundness cached_Ts T =
  1153   soundness <> Sound andalso is_type_surely_infinite ctxt true cached_Ts T
  1154 
  1155 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
  1156    dangerous because their "exhaust" properties can easily lead to unsound ATP
  1157    proofs. On the other hand, all HOL infinite types can be given the same
  1158    models in first-order logic (via Löwenheim-Skolem). *)
  1159 
  1160 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  1161   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
  1162                              maybe_nonmono_Ts, ...}
  1163                        (Noninf_Nonmono_Types (soundness, _)) T =
  1164     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1165     not (exists (type_instance ctxt T) surely_infinite_Ts orelse
  1166          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
  1167           is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts T))
  1168   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
  1169                              maybe_nonmono_Ts, ...}
  1170                        (Fin_Nonmono_Types _) T =
  1171     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1172     (exists (type_generalization ctxt T) surely_finite_Ts orelse
  1173      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
  1174       is_type_surely_finite ctxt T))
  1175   | should_encode_type _ _ _ _ = false
  1176 
  1177 fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =
  1178     (is_level_uniform level orelse should_guard_var ()) andalso
  1179     should_encode_type ctxt mono level T
  1180   | should_guard_type _ _ _ _ _ = false
  1181 
  1182 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
  1183     String.isPrefix bound_var_prefix s orelse
  1184     String.isPrefix all_bound_var_prefix s
  1185   | is_maybe_universal_var (IVar _) = true
  1186   | is_maybe_universal_var _ = false
  1187 
  1188 datatype tag_site =
  1189   Top_Level of bool option |
  1190   Eq_Arg of bool option |
  1191   Elsewhere
  1192 
  1193 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  1194   | should_tag_with_type ctxt mono (Tags (_, level)) site u T =
  1195     (if is_level_uniform level then
  1196        should_encode_type ctxt mono level T
  1197      else case (site, is_maybe_universal_var u) of
  1198        (Eq_Arg _, true) => should_encode_type ctxt mono level T
  1199      | _ => false)
  1200   | should_tag_with_type _ _ _ _ _ _ = false
  1201 
  1202 fun fused_type ctxt mono level =
  1203   let
  1204     val should_encode = should_encode_type ctxt mono level
  1205     fun fuse 0 T = if should_encode T then T else fused_infinite_type
  1206       | fuse ary (Type (@{type_name fun}, [T1, T2])) =
  1207         fuse 0 T1 --> fuse (ary - 1) T2
  1208       | fuse _ _ = raise Fail "expected function type"
  1209   in fuse end
  1210 
  1211 (** predicators and application operators **)
  1212 
  1213 type sym_info =
  1214   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
  1215 
  1216 fun add_iterm_syms_to_table ctxt explicit_apply =
  1217   let
  1218     fun consider_var_arity const_T var_T max_ary =
  1219       let
  1220         fun iter ary T =
  1221           if ary = max_ary orelse type_instance ctxt var_T T orelse
  1222              type_instance ctxt T var_T then
  1223             ary
  1224           else
  1225             iter (ary + 1) (range_type T)
  1226       in iter 0 const_T end
  1227     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1228       if explicit_apply = NONE andalso
  1229          (can dest_funT T orelse T = @{typ bool}) then
  1230         let
  1231           val bool_vars' = bool_vars orelse body_type T = @{typ bool}
  1232           fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
  1233             {pred_sym = pred_sym andalso not bool_vars',
  1234              min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
  1235              max_ary = max_ary, types = types}
  1236           val fun_var_Ts' =
  1237             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
  1238         in
  1239           if bool_vars' = bool_vars andalso
  1240              pointer_eq (fun_var_Ts', fun_var_Ts) then
  1241             accum
  1242           else
  1243             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
  1244         end
  1245       else
  1246         accum
  1247     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1248       let val (head, args) = strip_iterm_comb tm in
  1249         (case head of
  1250            IConst ((s, _), T, _) =>
  1251            if String.isPrefix bound_var_prefix s orelse
  1252               String.isPrefix all_bound_var_prefix s then
  1253              add_universal_var T accum
  1254            else if String.isPrefix exist_bound_var_prefix s then
  1255              accum
  1256            else
  1257              let val ary = length args in
  1258                ((bool_vars, fun_var_Ts),
  1259                 case Symtab.lookup sym_tab s of
  1260                   SOME {pred_sym, min_ary, max_ary, types} =>
  1261                   let
  1262                     val pred_sym =
  1263                       pred_sym andalso top_level andalso not bool_vars
  1264                     val types' = types |> insert_type ctxt I T
  1265                     val min_ary =
  1266                       if is_some explicit_apply orelse
  1267                          pointer_eq (types', types) then
  1268                         min_ary
  1269                       else
  1270                         fold (consider_var_arity T) fun_var_Ts min_ary
  1271                   in
  1272                     Symtab.update (s, {pred_sym = pred_sym,
  1273                                        min_ary = Int.min (ary, min_ary),
  1274                                        max_ary = Int.max (ary, max_ary),
  1275                                        types = types'})
  1276                                   sym_tab
  1277                   end
  1278                 | NONE =>
  1279                   let
  1280                     val pred_sym = top_level andalso not bool_vars
  1281                     val min_ary =
  1282                       case explicit_apply of
  1283                         SOME true => 0
  1284                       | SOME false => ary
  1285                       | NONE => fold (consider_var_arity T) fun_var_Ts ary
  1286                   in
  1287                     Symtab.update_new (s, {pred_sym = pred_sym,
  1288                                            min_ary = min_ary, max_ary = ary,
  1289                                            types = [T]})
  1290                                       sym_tab
  1291                   end)
  1292              end
  1293          | IVar (_, T) => add_universal_var T accum
  1294          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
  1295          | _ => accum)
  1296         |> fold (add false) args
  1297       end
  1298   in add true end
  1299 fun add_fact_syms_to_table ctxt explicit_apply =
  1300   K (add_iterm_syms_to_table ctxt explicit_apply)
  1301   |> formula_fold NONE |> fact_lift
  1302 
  1303 val default_sym_tab_entries : (string * sym_info) list =
  1304   (prefixed_predicator_name,
  1305    {pred_sym = true, min_ary = 1, max_ary = 1, types = []})
  1306        (* FIXME: needed? *) ::
  1307   (make_fixed_const NONE @{const_name undefined},
  1308    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  1309   ([tptp_false, tptp_true]
  1310    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  1311   ([tptp_equal, tptp_old_equal]
  1312    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
  1313 
  1314 fun sym_table_for_facts ctxt explicit_apply facts =
  1315   ((false, []), Symtab.empty)
  1316   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  1317   |> fold Symtab.update default_sym_tab_entries
  1318 
  1319 fun min_arity_of sym_tab s =
  1320   case Symtab.lookup sym_tab s of
  1321     SOME ({min_ary, ...} : sym_info) => min_ary
  1322   | NONE =>
  1323     case strip_prefix_and_unascii const_prefix s of
  1324       SOME s =>
  1325       let val s = s |> unmangled_const_name |> invert_const in
  1326         if s = predicator_name then 1
  1327         else if s = app_op_name then 2
  1328         else if s = type_guard_name then 1
  1329         else 0
  1330       end
  1331     | NONE => 0
  1332 
  1333 (* True if the constant ever appears outside of the top-level position in
  1334    literals, or if it appears with different arities (e.g., because of different
  1335    type instantiations). If false, the constant always receives all of its
  1336    arguments and is used as a predicate. *)
  1337 fun is_pred_sym sym_tab s =
  1338   case Symtab.lookup sym_tab s of
  1339     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
  1340     pred_sym andalso min_ary = max_ary
  1341   | NONE => false
  1342 
  1343 val predicator_combconst =
  1344   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
  1345 fun predicator tm = IApp (predicator_combconst, tm)
  1346 
  1347 fun introduce_predicators_in_iterm sym_tab tm =
  1348   case strip_iterm_comb tm of
  1349     (IConst ((s, _), _, _), _) =>
  1350     if is_pred_sym sym_tab s then tm else predicator tm
  1351   | _ => predicator tm
  1352 
  1353 fun list_app head args = fold (curry (IApp o swap)) args head
  1354 
  1355 val app_op = `(make_fixed_const NONE) app_op_name
  1356 
  1357 fun explicit_app arg head =
  1358   let
  1359     val head_T = ityp_of head
  1360     val (arg_T, res_T) = dest_funT head_T
  1361     val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
  1362   in list_app explicit_app [head, arg] end
  1363 fun list_explicit_app head args = fold explicit_app args head
  1364 
  1365 fun introduce_explicit_apps_in_iterm sym_tab =
  1366   let
  1367     fun aux tm =
  1368       case strip_iterm_comb tm of
  1369         (head as IConst ((s, _), _, _), args) =>
  1370         args |> map aux
  1371              |> chop (min_arity_of sym_tab s)
  1372              |>> list_app head
  1373              |-> list_explicit_app
  1374       | (head, args) => list_explicit_app head (map aux args)
  1375   in aux end
  1376 
  1377 fun chop_fun 0 T = ([], T)
  1378   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
  1379     chop_fun (n - 1) ran_T |>> cons dom_T
  1380   | chop_fun _ T = ([], T)
  1381 
  1382 fun filter_type_args _ _ _ [] = []
  1383   | filter_type_args thy s arity T_args =
  1384     let
  1385       val U = robust_const_type thy s
  1386       val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun arity |> fst) []
  1387       val U_args = (s, U) |> robust_const_typargs thy
  1388     in
  1389       U_args ~~ T_args
  1390       |> map (fn (U, T) =>
  1391                  if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
  1392     end
  1393     handle TYPE _ => T_args
  1394 
  1395 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
  1396   let
  1397     val thy = Proof_Context.theory_of ctxt
  1398     fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
  1399       | aux arity (IConst (name as (s, _), T, T_args)) =
  1400         (case strip_prefix_and_unascii const_prefix s of
  1401            NONE =>
  1402            (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice
  1403                   then [] else T_args)
  1404          | SOME s'' =>
  1405            let
  1406              val s'' = invert_const s''
  1407              fun filter_T_args false = T_args
  1408                | filter_T_args true = filter_type_args thy s'' arity T_args
  1409            in
  1410              case type_arg_policy type_enc s'' of
  1411                Explicit_Type_Args drop_args => (name, filter_T_args drop_args)
  1412              | Mangled_Type_Args =>
  1413                (mangled_const_name format type_enc T_args name, [])
  1414              | No_Type_Args => (name, [])
  1415            end)
  1416         |> (fn (name, T_args) => IConst (name, T, T_args))
  1417       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
  1418       | aux _ tm = tm
  1419   in aux 0 end
  1420 
  1421 fun repair_iterm ctxt format type_enc sym_tab =
  1422   not (is_type_enc_higher_order type_enc)
  1423   ? (introduce_explicit_apps_in_iterm sym_tab
  1424      #> introduce_predicators_in_iterm sym_tab)
  1425   #> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1426 fun repair_fact ctxt format type_enc sym_tab =
  1427   update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))
  1428 
  1429 (** Helper facts **)
  1430 
  1431 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
  1432 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
  1433 
  1434 (* The Boolean indicates that a fairly sound type encoding is needed. *)
  1435 val helper_table =
  1436   [(("COMBI", false), @{thms Meson.COMBI_def}),
  1437    (("COMBK", false), @{thms Meson.COMBK_def}),
  1438    (("COMBB", false), @{thms Meson.COMBB_def}),
  1439    (("COMBC", false), @{thms Meson.COMBC_def}),
  1440    (("COMBS", false), @{thms Meson.COMBS_def}),
  1441    ((predicator_name, false), [not_ffalse, ftrue]),
  1442    (("fFalse", false), [not_ffalse]),
  1443    (("fFalse", true), @{thms True_or_False}),
  1444    (("fTrue", false), [ftrue]),
  1445    (("fTrue", true), @{thms True_or_False}),
  1446    (("fNot", false),
  1447     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1448            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1449    (("fconj", false),
  1450     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
  1451         by (unfold fconj_def) fast+}),
  1452    (("fdisj", false),
  1453     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
  1454         by (unfold fdisj_def) fast+}),
  1455    (("fimplies", false),
  1456     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
  1457         by (unfold fimplies_def) fast+}),
  1458    (("fequal", true),
  1459     (* This is a lie: Higher-order equality doesn't need a sound type encoding.
  1460        However, this is done so for backward compatibility: Including the
  1461        equality helpers by default in Metis breaks a few existing proofs. *)
  1462     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1463            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1464    (* Partial characterization of "fAll" and "fEx". A complete characterization
  1465       would require the axiom of choice for replay with Metis. *)
  1466    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
  1467    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
  1468    (("If", true), @{thms if_True if_False True_or_False})]
  1469   |> map (apsnd (map zero_var_indexes))
  1470 
  1471 fun bound_tvars type_enc =
  1472   mk_ahorn o formulas_for_types type_enc add_sorts_on_tvar
  1473 
  1474 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  1475   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
  1476    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  1477   |> bound_tvars type_enc atomic_Ts
  1478   |> close_formula_universally type_enc
  1479 
  1480 val type_tag = `(make_fixed_const NONE) type_tag_name
  1481 
  1482 fun type_tag_idempotence_fact type_enc =
  1483   let
  1484     fun var s = ATerm (`I s, [])
  1485     fun tag tm = ATerm (type_tag, [var "A", tm])
  1486     val tagged_var = tag (var "X")
  1487   in
  1488     Formula (type_tag_idempotence_helper_name, Axiom,
  1489              eq_formula type_enc [] false (tag tagged_var) tagged_var,
  1490              isabelle_info simpN, NONE)
  1491   end
  1492 
  1493 fun should_specialize_helper type_enc t =
  1494   polymorphism_of_type_enc type_enc <> Polymorphic andalso
  1495   level_of_type_enc type_enc <> No_Types andalso
  1496   not (null (Term.hidden_polymorphism t))
  1497 
  1498 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  1499   case strip_prefix_and_unascii const_prefix s of
  1500     SOME mangled_s =>
  1501     let
  1502       val thy = Proof_Context.theory_of ctxt
  1503       val unmangled_s = mangled_s |> unmangled_const_name
  1504       fun dub needs_fairly_sound j k =
  1505         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
  1506          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
  1507          (if needs_fairly_sound then typed_helper_suffix
  1508           else untyped_helper_suffix),
  1509          Helper)
  1510       fun dub_and_inst needs_fairly_sound (th, j) =
  1511         let val t = prop_of th in
  1512           if should_specialize_helper type_enc t then
  1513             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
  1514                 types
  1515           else
  1516             [t]
  1517         end
  1518         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
  1519       val make_facts = map_filter (make_fact ctxt format type_enc false)
  1520       val fairly_sound = is_type_enc_fairly_sound type_enc
  1521     in
  1522       helper_table
  1523       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
  1524                   if helper_s <> unmangled_s orelse
  1525                      (needs_fairly_sound andalso not fairly_sound) then
  1526                     []
  1527                   else
  1528                     ths ~~ (1 upto length ths)
  1529                     |> maps (dub_and_inst needs_fairly_sound)
  1530                     |> make_facts)
  1531     end
  1532   | NONE => []
  1533 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  1534   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
  1535                   []
  1536 
  1537 (***************************************************************)
  1538 (* Type Classes Present in the Axiom or Conjecture Clauses     *)
  1539 (***************************************************************)
  1540 
  1541 fun set_insert (x, s) = Symtab.update (x, ()) s
  1542 
  1543 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
  1544 
  1545 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
  1546 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
  1547 
  1548 fun classes_of_terms get_Ts =
  1549   map (map snd o get_Ts)
  1550   #> List.foldl add_classes Symtab.empty
  1551   #> delete_type #> Symtab.keys
  1552 
  1553 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
  1554 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
  1555 
  1556 fun fold_type_constrs f (Type (s, Ts)) x =
  1557     fold (fold_type_constrs f) Ts (f (s, x))
  1558   | fold_type_constrs _ _ x = x
  1559 
  1560 (* Type constructors used to instantiate overloaded constants are the only ones
  1561    needed. *)
  1562 fun add_type_constrs_in_term thy =
  1563   let
  1564     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
  1565       | add (t $ u) = add t #> add u
  1566       | add (Const x) =
  1567         x |> robust_const_typargs thy |> fold (fold_type_constrs set_insert)
  1568       | add (Free (s, T)) =
  1569         if String.isPrefix polymorphic_free_prefix s then
  1570           T |> fold_type_constrs set_insert
  1571         else
  1572           I
  1573       | add (Abs (_, _, u)) = add u
  1574       | add _ = I
  1575   in add end
  1576 
  1577 fun type_constrs_of_terms thy ts =
  1578   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
  1579 
  1580 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  1581                        hyp_ts concl_t facts =
  1582   let
  1583     val thy = Proof_Context.theory_of ctxt
  1584     val presimp_consts = Meson.presimplified_consts ctxt
  1585     val fact_ts = facts |> map snd
  1586     (* Remove existing facts from the conjecture, as this can dramatically
  1587        boost an ATP's performance (for some reason). *)
  1588     val hyp_ts =
  1589       hyp_ts
  1590       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
  1591     val facts = facts |> map (apsnd (pair Axiom))
  1592     val conjs =
  1593       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]
  1594       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
  1595     val ((conjs, facts), lambdas) =
  1596       if preproc then
  1597         conjs @ facts
  1598         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
  1599         |> preprocess_abstractions_in_terms trans_lambdas
  1600         |>> chop (length conjs)
  1601         |>> apfst (map (apsnd (apsnd freeze_term)))
  1602       else
  1603         ((conjs, facts), [])
  1604     val conjs = conjs |> make_conjecture thy format type_enc
  1605     val (fact_names, facts) =
  1606       facts
  1607       |> map_filter (fn (name, (_, t)) =>
  1608                         make_fact ctxt format type_enc true (name, t)
  1609                         |> Option.map (pair name))
  1610       |> ListPair.unzip
  1611     val lambdas =
  1612       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
  1613     val all_ts = concl_t :: hyp_ts @ fact_ts
  1614     val subs = tfree_classes_of_terms all_ts
  1615     val supers = tvar_classes_of_terms all_ts
  1616     val tycons = type_constrs_of_terms thy all_ts
  1617     val (supers, arity_clauses) =
  1618       if level_of_type_enc type_enc = No_Types then ([], [])
  1619       else make_arity_clauses thy tycons supers
  1620     val class_rel_clauses = make_class_rel_clauses thy subs supers
  1621   in
  1622     (fact_names |> map single, union (op =) subs supers, conjs, facts @ lambdas,
  1623      class_rel_clauses, arity_clauses)
  1624   end
  1625 
  1626 val type_guard = `(make_fixed_const NONE) type_guard_name
  1627 
  1628 fun type_guard_iterm ctxt format type_enc T tm =
  1629   IApp (IConst (type_guard, T --> @{typ bool}, [T])
  1630         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
  1631 
  1632 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  1633   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
  1634     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  1635   | is_var_positively_naked_in_term _ _ _ _ = true
  1636 
  1637 fun should_guard_var_in_formula pos phi (SOME true) name =
  1638     formula_fold pos (is_var_positively_naked_in_term name) phi false
  1639   | should_guard_var_in_formula _ _ _ _ = true
  1640 
  1641 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  1642   | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =
  1643     not (is_level_uniform level) andalso should_encode_type ctxt mono level T
  1644   | should_generate_tag_bound_decl _ _ _ _ _ = false
  1645 
  1646 fun mk_aterm format type_enc name T_args args =
  1647   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
  1648 
  1649 fun tag_with_type ctxt format mono type_enc pos T tm =
  1650   IConst (type_tag, T --> T, [T])
  1651   |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1652   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  1653   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
  1654        | _ => raise Fail "unexpected lambda-abstraction")
  1655 and ho_term_from_iterm ctxt format mono type_enc =
  1656   let
  1657     fun aux site u =
  1658       let
  1659         val (head, args) = strip_iterm_comb u
  1660         val pos =
  1661           case site of
  1662             Top_Level pos => pos
  1663           | Eq_Arg pos => pos
  1664           | Elsewhere => NONE
  1665         val t =
  1666           case head of
  1667             IConst (name as (s, _), _, T_args) =>
  1668             let
  1669               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
  1670             in
  1671               mk_aterm format type_enc name T_args (map (aux arg_site) args)
  1672             end
  1673           | IVar (name, _) =>
  1674             mk_aterm format type_enc name [] (map (aux Elsewhere) args)
  1675           | IAbs ((name, T), tm) =>
  1676             AAbs ((name, ho_type_from_typ format type_enc true 0 T),
  1677                   aux Elsewhere tm)
  1678           | IApp _ => raise Fail "impossible \"IApp\""
  1679         val T = ityp_of u
  1680       in
  1681         t |> (if should_tag_with_type ctxt mono type_enc site u T then
  1682                 tag_with_type ctxt format mono type_enc pos T
  1683               else
  1684                 I)
  1685       end
  1686   in aux end
  1687 and formula_from_iformula ctxt format mono type_enc should_guard_var =
  1688   let
  1689     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
  1690     val do_bound_type =
  1691       case type_enc of
  1692         Simple_Types (_, _, level) => fused_type ctxt mono level 0
  1693         #> ho_type_from_typ format type_enc false 0 #> SOME
  1694       | _ => K NONE
  1695     fun do_out_of_bound_type pos phi universal (name, T) =
  1696       if should_guard_type ctxt mono type_enc
  1697              (fn () => should_guard_var pos phi universal name) T then
  1698         IVar (name, T)
  1699         |> type_guard_iterm ctxt format type_enc T
  1700         |> do_term pos |> AAtom |> SOME
  1701       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
  1702         let
  1703           val var = ATerm (name, [])
  1704           val tagged_var = var |> tag_with_type ctxt format mono type_enc pos T
  1705         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
  1706       else
  1707         NONE
  1708     fun do_formula pos (AQuant (q, xs, phi)) =
  1709         let
  1710           val phi = phi |> do_formula pos
  1711           val universal = Option.map (q = AExists ? not) pos
  1712         in
  1713           AQuant (q, xs |> map (apsnd (fn NONE => NONE
  1714                                         | SOME T => do_bound_type T)),
  1715                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
  1716                       (map_filter
  1717                            (fn (_, NONE) => NONE
  1718                              | (s, SOME T) =>
  1719                                do_out_of_bound_type pos phi universal (s, T))
  1720                            xs)
  1721                       phi)
  1722         end
  1723       | do_formula pos (AConn conn) = aconn_map pos do_formula conn
  1724       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  1725   in do_formula end
  1726 
  1727 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
  1728    of monomorphization). The TPTP explicitly forbids name clashes, and some of
  1729    the remote provers might care. *)
  1730 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
  1731                           (j, {name, locality, kind, iformula, atomic_types}) =
  1732   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
  1733    iformula
  1734    |> close_iformula_universally
  1735    |> formula_from_iformula ctxt format mono type_enc
  1736                             should_guard_var_in_formula
  1737                             (if pos then SOME true else NONE)
  1738    |> bound_tvars type_enc atomic_types
  1739    |> close_formula_universally type_enc,
  1740    NONE,
  1741    case locality of
  1742      Intro => isabelle_info introN
  1743    | Elim => isabelle_info elimN
  1744    | Simp => isabelle_info simpN
  1745    | _ => NONE)
  1746   |> Formula
  1747 
  1748 fun formula_line_for_class_rel_clause type_enc
  1749         ({name, subclass, superclass, ...} : class_rel_clause) =
  1750   let val ty_arg = ATerm (tvar_a_name, []) in
  1751     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
  1752              AConn (AImplies,
  1753                     [type_class_formula type_enc subclass ty_arg,
  1754                      type_class_formula type_enc superclass ty_arg])
  1755              |> close_formula_universally type_enc, isabelle_info introN, NONE)
  1756   end
  1757 
  1758 fun formula_from_arity_atom type_enc (class, t, args) =
  1759   ATerm (t, map (fn arg => ATerm (arg, [])) args)
  1760   |> type_class_formula type_enc class
  1761 
  1762 fun formula_line_for_arity_clause type_enc
  1763         ({name, prem_atoms, concl_atom, ...} : arity_clause) =
  1764   Formula (arity_clause_prefix ^ name, Axiom,
  1765            mk_ahorn (map (formula_from_arity_atom type_enc) prem_atoms)
  1766                     (formula_from_arity_atom type_enc concl_atom)
  1767            |> close_formula_universally type_enc, isabelle_info introN, NONE)
  1768 
  1769 fun formula_line_for_conjecture ctxt format mono type_enc
  1770         ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  1771   Formula (conjecture_prefix ^ name, kind,
  1772            formula_from_iformula ctxt format mono type_enc
  1773                should_guard_var_in_formula (SOME false)
  1774                (close_iformula_universally iformula)
  1775            |> bound_tvars type_enc atomic_types
  1776            |> close_formula_universally type_enc, NONE, NONE)
  1777 
  1778 fun formula_line_for_free_type j phi =
  1779   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis, phi, NONE, NONE)
  1780 fun formula_lines_for_free_types type_enc (facts : translated_formula list) =
  1781   let
  1782     val phis =
  1783       fold (union (op =)) (map #atomic_types facts) []
  1784       |> formulas_for_types type_enc add_sorts_on_tfree
  1785   in map2 formula_line_for_free_type (0 upto length phis - 1) phis end
  1786 
  1787 (** Symbol declarations **)
  1788 
  1789 fun decl_line_for_class order s =
  1790   let val name as (s, _) = `make_type_class s in
  1791     Decl (sym_decl_prefix ^ s, name,
  1792           if order = First_Order andalso avoid_first_order_dummy_type_vars then
  1793             ATyAbs ([tvar_a_name], AFun (a_itself_atype, bool_atype))
  1794           else
  1795             AFun (atype_of_types, bool_atype))
  1796   end
  1797 
  1798 fun decl_lines_for_classes type_enc classes =
  1799   case type_enc of
  1800     Simple_Types (order, Polymorphic, _) =>
  1801     map (decl_line_for_class order) classes
  1802   | _ => []
  1803 
  1804 fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  1805                              (conjs, facts) =
  1806   let
  1807     fun add_iterm_syms in_conj tm =
  1808       let val (head, args) = strip_iterm_comb tm in
  1809         (case head of
  1810            IConst ((s, s'), T, T_args) =>
  1811            let
  1812              val pred_sym = is_pred_sym repaired_sym_tab s
  1813              val decl_sym =
  1814                (case type_enc of
  1815                   Guards _ => not pred_sym
  1816                 | _ => true) andalso
  1817                is_tptp_user_symbol s
  1818            in
  1819              if decl_sym then
  1820                Symtab.map_default (s, [])
  1821                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
  1822                                          in_conj))
  1823              else
  1824                I
  1825            end
  1826          | IAbs (_, tm) => add_iterm_syms in_conj tm
  1827          | _ => I)
  1828         #> fold (add_iterm_syms in_conj) args
  1829       end
  1830     fun add_fact_syms in_conj =
  1831       K (add_iterm_syms in_conj) |> formula_fold NONE |> fact_lift
  1832     fun add_formula_var_types (AQuant (_, xs, phi)) =
  1833         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
  1834         #> add_formula_var_types phi
  1835       | add_formula_var_types (AConn (_, phis)) =
  1836         fold add_formula_var_types phis
  1837       | add_formula_var_types _ = I
  1838     fun var_types () =
  1839       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
  1840       else fold (fact_lift add_formula_var_types) (conjs @ facts) []
  1841     fun add_undefined_const T =
  1842       let
  1843         val (s, s') =
  1844           `(make_fixed_const NONE) @{const_name undefined}
  1845           |> (case type_arg_policy type_enc @{const_name undefined} of
  1846                 Mangled_Type_Args => mangled_const_name format type_enc [T]
  1847               | _ => I)
  1848       in
  1849         Symtab.map_default (s, [])
  1850                            (insert_type ctxt #3 (s', [T], T, false, 0, false))
  1851       end
  1852     fun add_TYPE_const () =
  1853       let val (s, s') = TYPE_name in
  1854         Symtab.map_default (s, [])
  1855             (insert_type ctxt #3
  1856                          (s', [tvar_a], @{typ "'a itself"}, false, 0, false))
  1857       end
  1858   in
  1859     Symtab.empty
  1860     |> is_type_enc_fairly_sound type_enc
  1861        ? (fold (add_fact_syms true) conjs
  1862           #> fold (add_fact_syms false) facts
  1863           #> (case type_enc of
  1864                 Simple_Types (_, poly, _) =>
  1865                 if poly = Polymorphic then add_TYPE_const () else I
  1866               | _ => fold add_undefined_const (var_types ())))
  1867   end
  1868 
  1869 (* We add "bool" in case the helper "True_or_False" is included later. *)
  1870 fun default_mono level =
  1871   {maybe_finite_Ts = [@{typ bool}],
  1872    surely_finite_Ts = [@{typ bool}],
  1873    maybe_infinite_Ts = known_infinite_types,
  1874    surely_infinite_Ts =
  1875      case level of
  1876        Noninf_Nonmono_Types (Sound, _) => []
  1877      | _ => known_infinite_types,
  1878    maybe_nonmono_Ts = [@{typ bool}]}
  1879 
  1880 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
  1881    out with monotonicity" paper presented at CADE 2011. *)
  1882 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  1883   | add_iterm_mononotonicity_info ctxt level _
  1884         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
  1885         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
  1886                   surely_infinite_Ts, maybe_nonmono_Ts}) =
  1887     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
  1888       case level of
  1889         Noninf_Nonmono_Types (soundness, _) =>
  1890         if exists (type_instance ctxt T) surely_infinite_Ts orelse
  1891            member (type_aconv ctxt) maybe_finite_Ts T then
  1892           mono
  1893         else if is_type_kind_of_surely_infinite ctxt soundness
  1894                                                 surely_infinite_Ts T then
  1895           {maybe_finite_Ts = maybe_finite_Ts,
  1896            surely_finite_Ts = surely_finite_Ts,
  1897            maybe_infinite_Ts = maybe_infinite_Ts,
  1898            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
  1899            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1900         else
  1901           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
  1902            surely_finite_Ts = surely_finite_Ts,
  1903            maybe_infinite_Ts = maybe_infinite_Ts,
  1904            surely_infinite_Ts = surely_infinite_Ts,
  1905            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1906       | Fin_Nonmono_Types _ =>
  1907         if exists (type_instance ctxt T) surely_finite_Ts orelse
  1908            member (type_aconv ctxt) maybe_infinite_Ts T then
  1909           mono
  1910         else if is_type_surely_finite ctxt T then
  1911           {maybe_finite_Ts = maybe_finite_Ts,
  1912            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
  1913            maybe_infinite_Ts = maybe_infinite_Ts,
  1914            surely_infinite_Ts = surely_infinite_Ts,
  1915            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1916         else
  1917           {maybe_finite_Ts = maybe_finite_Ts,
  1918            surely_finite_Ts = surely_finite_Ts,
  1919            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
  1920            surely_infinite_Ts = surely_infinite_Ts,
  1921            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1922       | _ => mono
  1923     else
  1924       mono
  1925   | add_iterm_mononotonicity_info _ _ _ _ mono = mono
  1926 fun add_fact_mononotonicity_info ctxt level
  1927         ({kind, iformula, ...} : translated_formula) =
  1928   formula_fold (SOME (kind <> Conjecture))
  1929                (add_iterm_mononotonicity_info ctxt level) iformula
  1930 fun mononotonicity_info_for_facts ctxt type_enc facts =
  1931   let val level = level_of_type_enc type_enc in
  1932     default_mono level
  1933     |> is_type_level_monotonicity_based level
  1934        ? fold (add_fact_mononotonicity_info ctxt level) facts
  1935   end
  1936 
  1937 fun add_iformula_monotonic_types ctxt mono type_enc =
  1938   let
  1939     val level = level_of_type_enc type_enc
  1940     val should_encode = should_encode_type ctxt mono level
  1941     fun add_type T = not (should_encode T) ? insert_type ctxt I T
  1942     fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2
  1943       | add_args _ = I
  1944     and add_term tm = add_type (ityp_of tm) #> add_args tm
  1945   in formula_fold NONE (K add_term) end
  1946 fun add_fact_monotonic_types ctxt mono type_enc =
  1947   add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
  1948 fun monotonic_types_for_facts ctxt mono type_enc facts =
  1949   [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
  1950          is_type_level_monotonicity_based (level_of_type_enc type_enc))
  1951         ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
  1952 
  1953 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  1954   Formula (guards_sym_formula_prefix ^
  1955            ascii_of (mangled_type format type_enc T),
  1956            Axiom,
  1957            IConst (`make_bound_var "X", T, [])
  1958            |> type_guard_iterm ctxt format type_enc T
  1959            |> AAtom
  1960            |> formula_from_iformula ctxt format mono type_enc
  1961                                     (K (K (K (K true)))) (SOME true)
  1962            |> bound_tvars type_enc (atyps_of T)
  1963            |> close_formula_universally type_enc,
  1964            isabelle_info introN, NONE)
  1965 
  1966 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  1967   let val x_var = ATerm (`make_bound_var "X", []) in
  1968     Formula (tags_sym_formula_prefix ^
  1969              ascii_of (mangled_type format type_enc T),
  1970              Axiom,
  1971              eq_formula type_enc (atyps_of T) false
  1972                         (tag_with_type ctxt format mono type_enc NONE T x_var)
  1973                         x_var,
  1974              isabelle_info simpN, NONE)
  1975   end
  1976 
  1977 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  1978   case type_enc of
  1979     Simple_Types _ => []
  1980   | Guards _ =>
  1981     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  1982   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
  1983 
  1984 fun decl_line_for_sym ctxt format mono type_enc s
  1985                       (s', T_args, T, pred_sym, ary, _) =
  1986   let
  1987     val thy = Proof_Context.theory_of ctxt
  1988     val (T, T_args) =
  1989       if null T_args then
  1990         (T, [])
  1991       else case strip_prefix_and_unascii const_prefix s of
  1992         SOME s' =>
  1993         let
  1994           val s' = s' |> invert_const
  1995           val T = s' |> robust_const_type thy
  1996         in (T, robust_const_typargs thy (s', T)) end
  1997       | NONE =>
  1998         case strip_prefix_and_unascii fixed_var_prefix s of
  1999           SOME s' =>
  2000           if String.isPrefix polymorphic_free_prefix s' then (tvar_a, [tvar_a])
  2001           else raise Fail "unexpected type arguments to free variable"
  2002         | NONE => raise Fail "unexpected type arguments"
  2003   in
  2004     Decl (sym_decl_prefix ^ s, (s, s'),
  2005           T |> fused_type ctxt mono (level_of_type_enc type_enc) ary
  2006             |> ho_type_from_typ format type_enc pred_sym ary
  2007             |> not (null T_args)
  2008                ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))
  2009   end
  2010 
  2011 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
  2012                                      j (s', T_args, T, _, ary, in_conj) =
  2013   let
  2014     val (kind, maybe_negate) =
  2015       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  2016       else (Axiom, I)
  2017     val (arg_Ts, res_T) = chop_fun ary T
  2018     val num_args = length arg_Ts
  2019     val bound_names =
  2020       1 upto num_args |> map (`I o make_bound_var o string_of_int)
  2021     val bounds =
  2022       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
  2023     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
  2024     fun should_keep_arg_type T =
  2025       sym_needs_arg_types andalso
  2026       should_guard_type ctxt mono type_enc (K true) T
  2027     val bound_Ts =
  2028       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  2029   in
  2030     Formula (guards_sym_formula_prefix ^ s ^
  2031              (if n > 1 then "_" ^ string_of_int j else ""), kind,
  2032              IConst ((s, s'), T, T_args)
  2033              |> fold (curry (IApp o swap)) bounds
  2034              |> type_guard_iterm ctxt format type_enc res_T
  2035              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
  2036              |> formula_from_iformula ctxt format mono type_enc
  2037                                       (K (K (K (K true)))) (SOME true)
  2038              |> n > 1 ? bound_tvars type_enc (atyps_of T)
  2039              |> close_formula_universally type_enc
  2040              |> maybe_negate,
  2041              isabelle_info introN, NONE)
  2042   end
  2043 
  2044 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
  2045         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  2046   let
  2047     val ident_base =
  2048       tags_sym_formula_prefix ^ s ^
  2049       (if n > 1 then "_" ^ string_of_int j else "")
  2050     val (kind, maybe_negate) =
  2051       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  2052       else (Axiom, I)
  2053     val (arg_Ts, res_T) = chop_fun ary T
  2054     val bound_names =
  2055       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
  2056     val bounds = bound_names |> map (fn name => ATerm (name, []))
  2057     val cst = mk_aterm format type_enc (s, s') T_args
  2058     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
  2059     val should_encode =
  2060       should_encode_type ctxt mono (level_of_type_enc type_enc)
  2061     val tag_with = tag_with_type ctxt format mono type_enc NONE
  2062     val add_formula_for_res =
  2063       if should_encode res_T then
  2064         cons (Formula (ident_base ^ "_res", kind,
  2065                        eq (tag_with res_T (cst bounds)) (cst bounds),
  2066                        isabelle_info simpN, NONE))
  2067       else
  2068         I
  2069     fun add_formula_for_arg k =
  2070       let val arg_T = nth arg_Ts k in
  2071         if should_encode arg_T then
  2072           case chop k bounds of
  2073             (bounds1, bound :: bounds2) =>
  2074             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
  2075                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
  2076                               (cst bounds),
  2077                            isabelle_info simpN, NONE))
  2078           | _ => raise Fail "expected nonempty tail"
  2079         else
  2080           I
  2081       end
  2082   in
  2083     [] |> not pred_sym ? add_formula_for_res
  2084        |> Config.get ctxt type_tag_arguments
  2085           ? fold add_formula_for_arg (ary - 1 downto 0)
  2086   end
  2087 
  2088 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
  2089 
  2090 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
  2091                                 (s, decls) =
  2092   case type_enc of
  2093     Simple_Types _ =>
  2094     decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  2095   | Guards (_, level) =>
  2096     let
  2097       val decls =
  2098         case decls of
  2099           decl :: (decls' as _ :: _) =>
  2100           let val T = result_type_of_decl decl in
  2101             if forall (type_generalization ctxt T o result_type_of_decl)
  2102                       decls' then
  2103               [decl]
  2104             else
  2105               decls
  2106           end
  2107         | _ => decls
  2108       val n = length decls
  2109       val decls =
  2110         decls |> filter (should_encode_type ctxt mono level
  2111                          o result_type_of_decl)
  2112     in
  2113       (0 upto length decls - 1, decls)
  2114       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
  2115                                                  type_enc n s)
  2116     end
  2117   | Tags (_, level) =>
  2118     if is_level_uniform level then
  2119       []
  2120     else
  2121       let val n = length decls in
  2122         (0 upto n - 1 ~~ decls)
  2123         |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
  2124                      conj_sym_kind mono type_enc n s)
  2125       end
  2126 
  2127 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
  2128                                      mono_Ts sym_decl_tab =
  2129   let
  2130     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
  2131     val mono_lines =
  2132       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
  2133     val decl_lines =
  2134       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
  2135                                                      mono type_enc)
  2136                syms []
  2137   in mono_lines @ decl_lines end
  2138 
  2139 fun needs_type_tag_idempotence ctxt (Tags (poly, level)) =
  2140     Config.get ctxt type_tag_idempotence andalso
  2141     is_type_level_monotonicity_based level andalso
  2142     poly <> Mangled_Monomorphic
  2143   | needs_type_tag_idempotence _ _ = false
  2144 
  2145 fun offset_of_heading_in_problem _ [] j = j
  2146   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
  2147     if heading = needle then j
  2148     else offset_of_heading_in_problem needle problem (j + length lines)
  2149 
  2150 val implicit_declsN = "Should-be-implicit typings"
  2151 val explicit_declsN = "Explicit typings"
  2152 val factsN = "Relevant facts"
  2153 val class_relsN = "Class relationships"
  2154 val aritiesN = "Arities"
  2155 val helpersN = "Helper facts"
  2156 val conjsN = "Conjectures"
  2157 val free_typesN = "Type variables"
  2158 
  2159 val explicit_apply = NONE (* for experiments *)
  2160 
  2161 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
  2162         lambda_trans readable_names preproc hyp_ts concl_t facts =
  2163   let
  2164     val type_enc = type_enc |> adjust_type_enc format
  2165     val lambda_trans =
  2166       if lambda_trans = smartN then
  2167         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
  2168       else if lambda_trans = lambdasN andalso
  2169               not (is_type_enc_higher_order type_enc) then
  2170         error ("Lambda translation method incompatible with first-order \
  2171                \encoding.")
  2172       else
  2173         lambda_trans
  2174     val trans_lambdas =
  2175       if lambda_trans = no_lambdasN then
  2176         rpair []
  2177       else if lambda_trans = concealedN then
  2178         lift_lambdas ctxt type_enc ##> K []
  2179       else if lambda_trans = liftingN then
  2180         lift_lambdas ctxt type_enc
  2181       else if lambda_trans = combinatorsN then
  2182         map (introduce_combinators ctxt) #> rpair []
  2183       else if lambda_trans = hybridN then
  2184         lift_lambdas ctxt type_enc
  2185         ##> maps (fn t => [t, introduce_combinators ctxt
  2186                                   (intentionalize_def t)])
  2187       else if lambda_trans = lambdasN then
  2188         map (Envir.eta_contract) #> rpair []
  2189       else
  2190         error ("Unknown lambda translation method: " ^
  2191                quote lambda_trans ^ ".")
  2192     val (fact_names, classes, conjs, facts, class_rel_clauses, arity_clauses) =
  2193       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  2194                          hyp_ts concl_t facts
  2195     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
  2196     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
  2197     val repair = repair_fact ctxt format type_enc sym_tab
  2198     val (conjs, facts) = (conjs, facts) |> pairself (map repair)
  2199     val repaired_sym_tab =
  2200       conjs @ facts |> sym_table_for_facts ctxt (SOME false)
  2201     val helpers =
  2202       repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
  2203                        |> map repair
  2204     val mono_Ts =
  2205       helpers @ conjs @ facts
  2206       |> monotonic_types_for_facts ctxt mono type_enc
  2207     val class_decl_lines = decl_lines_for_classes type_enc classes
  2208     val sym_decl_lines =
  2209       (conjs, helpers @ facts)
  2210       |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  2211       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
  2212                                                type_enc mono_Ts
  2213     val helper_lines =
  2214       0 upto length helpers - 1 ~~ helpers
  2215       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
  2216                                     type_enc)
  2217       |> (if needs_type_tag_idempotence ctxt type_enc then
  2218             cons (type_tag_idempotence_fact type_enc)
  2219           else
  2220             I)
  2221     (* Reordering these might confuse the proof reconstruction code or the SPASS
  2222        FLOTTER hack. *)
  2223     val problem =
  2224       [(explicit_declsN, class_decl_lines @ sym_decl_lines),
  2225        (factsN,
  2226         map (formula_line_for_fact ctxt format fact_prefix ascii_of
  2227                                    (not exporter) (not exporter) mono type_enc)
  2228             (0 upto length facts - 1 ~~ facts)),
  2229        (class_relsN,
  2230         map (formula_line_for_class_rel_clause type_enc) class_rel_clauses),
  2231        (aritiesN, map (formula_line_for_arity_clause type_enc) arity_clauses),
  2232        (helpersN, helper_lines),
  2233        (conjsN,
  2234         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
  2235        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
  2236     val problem =
  2237       problem
  2238       |> (case format of
  2239             CNF => ensure_cnf_problem
  2240           | CNF_UEQ => filter_cnf_ueq_problem
  2241           | FOF => I
  2242           | TFF (_, TPTP_Implicit) => I
  2243           | THF (_, TPTP_Implicit, _) => I
  2244           | _ => declare_undeclared_syms_in_atp_problem type_decl_prefix
  2245                                                         implicit_declsN)
  2246     val (problem, pool) = problem |> nice_atp_problem readable_names
  2247     val helpers_offset = offset_of_heading_in_problem helpersN problem 0
  2248     val typed_helpers =
  2249       map_filter (fn (j, {name, ...}) =>
  2250                      if String.isSuffix typed_helper_suffix name then SOME j
  2251                      else NONE)
  2252                  ((helpers_offset + 1 upto helpers_offset + length helpers)
  2253                   ~~ helpers)
  2254     fun add_sym_arity (s, {min_ary, ...} : sym_info) =
  2255       if min_ary > 0 then
  2256         case strip_prefix_and_unascii const_prefix s of
  2257           SOME s => Symtab.insert (op =) (s, min_ary)
  2258         | NONE => I
  2259       else
  2260         I
  2261   in
  2262     (problem,
  2263      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
  2264      offset_of_heading_in_problem conjsN problem 0,
  2265      offset_of_heading_in_problem factsN problem 0,
  2266      fact_names |> Vector.fromList,
  2267      typed_helpers,
  2268      Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
  2269   end
  2270 
  2271 (* FUDGE *)
  2272 val conj_weight = 0.0
  2273 val hyp_weight = 0.1
  2274 val fact_min_weight = 0.2
  2275 val fact_max_weight = 1.0
  2276 val type_info_default_weight = 0.8
  2277 
  2278 fun add_term_weights weight (ATerm (s, tms)) =
  2279     is_tptp_user_symbol s ? Symtab.default (s, weight)
  2280     #> fold (add_term_weights weight) tms
  2281   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
  2282 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
  2283     formula_fold NONE (K (add_term_weights weight)) phi
  2284   | add_problem_line_weights _ _ = I
  2285 
  2286 fun add_conjectures_weights [] = I
  2287   | add_conjectures_weights conjs =
  2288     let val (hyps, conj) = split_last conjs in
  2289       add_problem_line_weights conj_weight conj
  2290       #> fold (add_problem_line_weights hyp_weight) hyps
  2291     end
  2292 
  2293 fun add_facts_weights facts =
  2294   let
  2295     val num_facts = length facts
  2296     fun weight_of j =
  2297       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
  2298                         / Real.fromInt num_facts
  2299   in
  2300     map weight_of (0 upto num_facts - 1) ~~ facts
  2301     |> fold (uncurry add_problem_line_weights)
  2302   end
  2303 
  2304 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
  2305 fun atp_problem_weights problem =
  2306   let val get = these o AList.lookup (op =) problem in
  2307     Symtab.empty
  2308     |> add_conjectures_weights (get free_typesN @ get conjsN)
  2309     |> add_facts_weights (get factsN)
  2310     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
  2311             [explicit_declsN, class_relsN, aritiesN]
  2312     |> Symtab.dest
  2313     |> sort (prod_ord Real.compare string_ord o pairself swap)
  2314   end
  2315 
  2316 end;