src/HOL/Tools/ATP/atp_translate.ML
author blanchet
Thu Aug 25 19:02:47 2011 +0200 (2011-08-25)
changeset 44496 c1884789ff80
parent 44495 4c2242c8a96c
child 44499 8870232a87ad
permissions -rw-r--r--
added config options to control two aspects of the translation, for evaluation purposes
     1 (*  Title:      HOL/Tools/ATP/atp_translate.ML
     2     Author:     Fabian Immler, TU Muenchen
     3     Author:     Makarius
     4     Author:     Jasmin Blanchette, TU Muenchen
     5 
     6 Translation of HOL to FOL for Metis and Sledgehammer.
     7 *)
     8 
     9 signature ATP_TRANSLATE =
    10 sig
    11   type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
    12   type connective = ATP_Problem.connective
    13   type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
    14   type format = ATP_Problem.format
    15   type formula_kind = ATP_Problem.formula_kind
    16   type 'a problem = 'a ATP_Problem.problem
    17 
    18   datatype locality =
    19     General | Helper | Extensionality | Intro | Elim | Simp | Local | Assum |
    20     Chained
    21 
    22   datatype order = First_Order | Higher_Order
    23   datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
    24   datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
    25   datatype type_level =
    26     All_Types |
    27     Noninf_Nonmono_Types of soundness |
    28     Fin_Nonmono_Types |
    29     Const_Arg_Types |
    30     No_Types
    31   datatype type_uniformity = Uniform | Nonuniform
    32 
    33   datatype type_enc =
    34     Simple_Types of order * type_level |
    35     Guards of polymorphism * type_level * type_uniformity |
    36     Tags of polymorphism * type_level * type_uniformity
    37 
    38   val type_tag_idempotence : bool Config.T
    39   val type_tag_arguments : bool Config.T
    40   val no_lambdasN : string
    41   val concealedN : string
    42   val liftingN : string
    43   val combinatorsN : string
    44   val hybridN : string
    45   val lambdasN : string
    46   val smartN : string
    47   val schematic_var_prefix : string
    48   val fixed_var_prefix : string
    49   val tvar_prefix : string
    50   val tfree_prefix : string
    51   val const_prefix : string
    52   val type_const_prefix : string
    53   val class_prefix : string
    54   val polymorphic_free_prefix : string
    55   val skolem_const_prefix : string
    56   val old_skolem_const_prefix : string
    57   val new_skolem_const_prefix : string
    58   val type_decl_prefix : string
    59   val sym_decl_prefix : string
    60   val guards_sym_formula_prefix : string
    61   val tags_sym_formula_prefix : string
    62   val fact_prefix : string
    63   val conjecture_prefix : string
    64   val helper_prefix : string
    65   val class_rel_clause_prefix : string
    66   val arity_clause_prefix : string
    67   val tfree_clause_prefix : string
    68   val typed_helper_suffix : string
    69   val untyped_helper_suffix : string
    70   val type_tag_idempotence_helper_name : string
    71   val predicator_name : string
    72   val app_op_name : string
    73   val type_guard_name : string
    74   val type_tag_name : string
    75   val simple_type_prefix : string
    76   val prefixed_predicator_name : string
    77   val prefixed_app_op_name : string
    78   val prefixed_type_tag_name : string
    79   val ascii_of : string -> string
    80   val unascii_of : string -> string
    81   val strip_prefix_and_unascii : string -> string -> string option
    82   val proxy_table : (string * (string * (thm * (string * string)))) list
    83   val proxify_const : string -> (string * string) option
    84   val invert_const : string -> string
    85   val unproxify_const : string -> string
    86   val new_skolem_var_name_from_const : string -> string
    87   val atp_irrelevant_consts : string list
    88   val atp_schematic_consts_of : term -> typ list Symtab.table
    89   val type_enc_from_string : soundness -> string -> type_enc
    90   val is_type_enc_higher_order : type_enc -> bool
    91   val polymorphism_of_type_enc : type_enc -> polymorphism
    92   val level_of_type_enc : type_enc -> type_level
    93   val is_type_enc_quasi_sound : type_enc -> bool
    94   val is_type_enc_fairly_sound : type_enc -> bool
    95   val adjust_type_enc : format -> type_enc -> type_enc
    96   val mk_aconns :
    97     connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
    98   val unmangled_const : string -> string * (string, 'b) ho_term list
    99   val unmangled_const_name : string -> string
   100   val helper_table : ((string * bool) * thm list) list
   101   val factsN : string
   102   val prepare_atp_problem :
   103     Proof.context -> format -> formula_kind -> formula_kind -> type_enc
   104     -> bool -> string -> bool -> bool -> term list -> term
   105     -> ((string * locality) * term) list
   106     -> string problem * string Symtab.table * int * int
   107        * (string * locality) list vector * int list * int Symtab.table
   108   val atp_problem_weights : string problem -> (string * real) list
   109 end;
   110 
   111 structure ATP_Translate : ATP_TRANSLATE =
   112 struct
   113 
   114 open ATP_Util
   115 open ATP_Problem
   116 
   117 type name = string * string
   118 
   119 val type_tag_idempotence =
   120   Attrib.setup_config_bool @{binding atp_type_tag_idempotence} (K true)
   121 val type_tag_arguments =
   122   Attrib.setup_config_bool @{binding atp_type_tag_arguments} (K true)
   123 
   124 val no_lambdasN = "no_lambdas"
   125 val concealedN = "concealed"
   126 val liftingN = "lifting"
   127 val combinatorsN = "combinators"
   128 val hybridN = "hybrid"
   129 val lambdasN = "lambdas"
   130 val smartN = "smart"
   131 
   132 val generate_info = false (* experimental *)
   133 
   134 fun isabelle_info s =
   135   if generate_info then SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
   136   else NONE
   137 
   138 val introN = "intro"
   139 val elimN = "elim"
   140 val simpN = "simp"
   141 
   142 val bound_var_prefix = "B_"
   143 val all_bound_var_prefix = "BA_"
   144 val exist_bound_var_prefix = "BE_"
   145 val schematic_var_prefix = "V_"
   146 val fixed_var_prefix = "v_"
   147 val tvar_prefix = "T_"
   148 val tfree_prefix = "t_"
   149 val const_prefix = "c_"
   150 val type_const_prefix = "tc_"
   151 val simple_type_prefix = "s_"
   152 val class_prefix = "cl_"
   153 
   154 val polymorphic_free_prefix = "poly_free"
   155 
   156 val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"
   157 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
   158 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
   159 
   160 val type_decl_prefix = "ty_"
   161 val sym_decl_prefix = "sy_"
   162 val guards_sym_formula_prefix = "gsy_"
   163 val tags_sym_formula_prefix = "tsy_"
   164 val fact_prefix = "fact_"
   165 val conjecture_prefix = "conj_"
   166 val helper_prefix = "help_"
   167 val class_rel_clause_prefix = "clar_"
   168 val arity_clause_prefix = "arity_"
   169 val tfree_clause_prefix = "tfree_"
   170 
   171 val lambda_fact_prefix = "ATP.lambda_"
   172 val typed_helper_suffix = "_T"
   173 val untyped_helper_suffix = "_U"
   174 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"
   175 
   176 val predicator_name = "pp"
   177 val app_op_name = "aa"
   178 val type_guard_name = "gg"
   179 val type_tag_name = "tt"
   180 
   181 val prefixed_predicator_name = const_prefix ^ predicator_name
   182 val prefixed_app_op_name = const_prefix ^ app_op_name
   183 val prefixed_type_tag_name = const_prefix ^ type_tag_name
   184 
   185 (* Freshness almost guaranteed! *)
   186 val atp_weak_prefix = "ATP:"
   187 
   188 (*Escaping of special characters.
   189   Alphanumeric characters are left unchanged.
   190   The character _ goes to __
   191   Characters in the range ASCII space to / go to _A to _P, respectively.
   192   Other characters go to _nnn where nnn is the decimal ASCII code.*)
   193 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
   194 
   195 fun stringN_of_int 0 _ = ""
   196   | stringN_of_int k n =
   197     stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
   198 
   199 fun ascii_of_char c =
   200   if Char.isAlphaNum c then
   201     String.str c
   202   else if c = #"_" then
   203     "__"
   204   else if #" " <= c andalso c <= #"/" then
   205     "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
   206   else
   207     (* fixed width, in case more digits follow *)
   208     "_" ^ stringN_of_int 3 (Char.ord c)
   209 
   210 val ascii_of = String.translate ascii_of_char
   211 
   212 (** Remove ASCII armoring from names in proof files **)
   213 
   214 (* We don't raise error exceptions because this code can run inside a worker
   215    thread. Also, the errors are impossible. *)
   216 val unascii_of =
   217   let
   218     fun un rcs [] = String.implode(rev rcs)
   219       | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
   220         (* Three types of _ escapes: __, _A to _P, _nnn *)
   221       | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
   222       | un rcs (#"_" :: c :: cs) =
   223         if #"A" <= c andalso c<= #"P" then
   224           (* translation of #" " to #"/" *)
   225           un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
   226         else
   227           let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
   228             case Int.fromString (String.implode digits) of
   229               SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
   230             | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
   231           end
   232       | un rcs (c :: cs) = un (c :: rcs) cs
   233   in un [] o String.explode end
   234 
   235 (* If string s has the prefix s1, return the result of deleting it,
   236    un-ASCII'd. *)
   237 fun strip_prefix_and_unascii s1 s =
   238   if String.isPrefix s1 s then
   239     SOME (unascii_of (String.extract (s, size s1, NONE)))
   240   else
   241     NONE
   242 
   243 val proxy_table =
   244   [("c_False", (@{const_name False}, (@{thm fFalse_def},
   245        ("fFalse", @{const_name ATP.fFalse})))),
   246    ("c_True", (@{const_name True}, (@{thm fTrue_def},
   247        ("fTrue", @{const_name ATP.fTrue})))),
   248    ("c_Not", (@{const_name Not}, (@{thm fNot_def},
   249        ("fNot", @{const_name ATP.fNot})))),
   250    ("c_conj", (@{const_name conj}, (@{thm fconj_def},
   251        ("fconj", @{const_name ATP.fconj})))),
   252    ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
   253        ("fdisj", @{const_name ATP.fdisj})))),
   254    ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
   255        ("fimplies", @{const_name ATP.fimplies})))),
   256    ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
   257        ("fequal", @{const_name ATP.fequal})))),
   258    ("c_All", (@{const_name All}, (@{thm fAll_def},
   259        ("fAll", @{const_name ATP.fAll})))),
   260    ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
   261        ("fEx", @{const_name ATP.fEx}))))]
   262 
   263 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
   264 
   265 (* Readable names for the more common symbolic functions. Do not mess with the
   266    table unless you know what you are doing. *)
   267 val const_trans_table =
   268   [(@{type_name Product_Type.prod}, "prod"),
   269    (@{type_name Sum_Type.sum}, "sum"),
   270    (@{const_name False}, "False"),
   271    (@{const_name True}, "True"),
   272    (@{const_name Not}, "Not"),
   273    (@{const_name conj}, "conj"),
   274    (@{const_name disj}, "disj"),
   275    (@{const_name implies}, "implies"),
   276    (@{const_name HOL.eq}, "equal"),
   277    (@{const_name All}, "All"),
   278    (@{const_name Ex}, "Ex"),
   279    (@{const_name If}, "If"),
   280    (@{const_name Set.member}, "member"),
   281    (@{const_name Meson.COMBI}, "COMBI"),
   282    (@{const_name Meson.COMBK}, "COMBK"),
   283    (@{const_name Meson.COMBB}, "COMBB"),
   284    (@{const_name Meson.COMBC}, "COMBC"),
   285    (@{const_name Meson.COMBS}, "COMBS")]
   286   |> Symtab.make
   287   |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
   288 
   289 (* Invert the table of translations between Isabelle and ATPs. *)
   290 val const_trans_table_inv =
   291   const_trans_table |> Symtab.dest |> map swap |> Symtab.make
   292 val const_trans_table_unprox =
   293   Symtab.empty
   294   |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
   295 
   296 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
   297 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
   298 
   299 fun lookup_const c =
   300   case Symtab.lookup const_trans_table c of
   301     SOME c' => c'
   302   | NONE => ascii_of c
   303 
   304 fun ascii_of_indexname (v, 0) = ascii_of v
   305   | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
   306 
   307 fun make_bound_var x = bound_var_prefix ^ ascii_of x
   308 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
   309 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
   310 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
   311 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
   312 
   313 fun make_schematic_type_var (x, i) =
   314       tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
   315 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
   316 
   317 (* "HOL.eq" is mapped to the ATP's equality. *)
   318 fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
   319   | make_fixed_const (SOME (THF With_Choice)) "Hilbert_Choice.Eps"(*FIXME*) =
   320       tptp_choice
   321   | make_fixed_const _ c = const_prefix ^ lookup_const c
   322 
   323 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
   324 
   325 fun make_type_class clas = class_prefix ^ ascii_of clas
   326 
   327 fun new_skolem_var_name_from_const s =
   328   let val ss = s |> space_explode Long_Name.separator in
   329     nth ss (length ss - 2)
   330   end
   331 
   332 (* These are either simplified away by "Meson.presimplify" (most of the time) or
   333    handled specially via "fFalse", "fTrue", ..., "fequal". *)
   334 val atp_irrelevant_consts =
   335   [@{const_name False}, @{const_name True}, @{const_name Not},
   336    @{const_name conj}, @{const_name disj}, @{const_name implies},
   337    @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
   338 
   339 val atp_monomorph_bad_consts =
   340   atp_irrelevant_consts @
   341   (* These are ignored anyway by the relevance filter (unless they appear in
   342      higher-order places) but not by the monomorphizer. *)
   343   [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
   344    @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
   345    @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
   346 
   347 fun add_schematic_const (x as (_, T)) =
   348   Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
   349 val add_schematic_consts_of =
   350   Term.fold_aterms (fn Const (x as (s, _)) =>
   351                        not (member (op =) atp_monomorph_bad_consts s)
   352                        ? add_schematic_const x
   353                       | _ => I)
   354 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
   355 
   356 (** Definitions and functions for FOL clauses and formulas for TPTP **)
   357 
   358 (* The first component is the type class; the second is a "TVar" or "TFree". *)
   359 datatype type_literal =
   360   TyLitVar of name * name |
   361   TyLitFree of name * name
   362 
   363 
   364 (** Isabelle arities **)
   365 
   366 datatype arity_literal =
   367   TConsLit of name * name * name list |
   368   TVarLit of name * name
   369 
   370 fun gen_TVars 0 = []
   371   | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)
   372 
   373 val type_class = the_single @{sort type}
   374 
   375 fun add_packed_sort tvar =
   376   fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))
   377 
   378 type arity_clause =
   379   {name : string,
   380    prem_lits : arity_literal list,
   381    concl_lits : arity_literal}
   382 
   383 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
   384 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
   385   let
   386     val tvars = gen_TVars (length args)
   387     val tvars_srts = ListPair.zip (tvars, args)
   388   in
   389     {name = name,
   390      prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,
   391      concl_lits = TConsLit (`make_type_class cls,
   392                             `make_fixed_type_const tcons,
   393                             tvars ~~ tvars)}
   394   end
   395 
   396 fun arity_clause _ _ (_, []) = []
   397   | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)
   398     arity_clause seen n (tcons, ars)
   399   | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
   400     if member (op =) seen class then
   401       (* multiple arities for the same (tycon, class) pair *)
   402       make_axiom_arity_clause (tcons,
   403           lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
   404           ar) ::
   405       arity_clause seen (n + 1) (tcons, ars)
   406     else
   407       make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
   408                                ascii_of class, ar) ::
   409       arity_clause (class :: seen) n (tcons, ars)
   410 
   411 fun multi_arity_clause [] = []
   412   | multi_arity_clause ((tcons, ars) :: tc_arlists) =
   413       arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
   414 
   415 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
   416    theory thy provided its arguments have the corresponding sorts. *)
   417 fun type_class_pairs thy tycons classes =
   418   let
   419     val alg = Sign.classes_of thy
   420     fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
   421     fun add_class tycon class =
   422       cons (class, domain_sorts tycon class)
   423       handle Sorts.CLASS_ERROR _ => I
   424     fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
   425   in map try_classes tycons end
   426 
   427 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
   428 fun iter_type_class_pairs _ _ [] = ([], [])
   429   | iter_type_class_pairs thy tycons classes =
   430       let
   431         fun maybe_insert_class s =
   432           (s <> type_class andalso not (member (op =) classes s))
   433           ? insert (op =) s
   434         val cpairs = type_class_pairs thy tycons classes
   435         val newclasses =
   436           [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
   437         val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
   438       in (classes' @ classes, union (op =) cpairs' cpairs) end
   439 
   440 fun make_arity_clauses thy tycons =
   441   iter_type_class_pairs thy tycons ##> multi_arity_clause
   442 
   443 
   444 (** Isabelle class relations **)
   445 
   446 type class_rel_clause =
   447   {name : string,
   448    subclass : name,
   449    superclass : name}
   450 
   451 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
   452    in theory "thy". *)
   453 fun class_pairs _ [] _ = []
   454   | class_pairs thy subs supers =
   455       let
   456         val class_less = Sorts.class_less (Sign.classes_of thy)
   457         fun add_super sub super = class_less (sub, super) ? cons (sub, super)
   458         fun add_supers sub = fold (add_super sub) supers
   459       in fold add_supers subs [] end
   460 
   461 fun make_class_rel_clause (sub, super) =
   462   {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
   463    superclass = `make_type_class super}
   464 
   465 fun make_class_rel_clauses thy subs supers =
   466   map make_class_rel_clause (class_pairs thy subs supers)
   467 
   468 (* intermediate terms *)
   469 datatype iterm =
   470   IConst of name * typ * typ list |
   471   IVar of name * typ |
   472   IApp of iterm * iterm |
   473   IAbs of (name * typ) * iterm
   474 
   475 fun ityp_of (IConst (_, T, _)) = T
   476   | ityp_of (IVar (_, T)) = T
   477   | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
   478   | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
   479 
   480 (*gets the head of a combinator application, along with the list of arguments*)
   481 fun strip_iterm_comb u =
   482   let
   483     fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
   484       | stripc x = x
   485   in stripc (u, []) end
   486 
   487 fun atyps_of T = fold_atyps (insert (op =)) T []
   488 
   489 fun new_skolem_const_name s num_T_args =
   490   [new_skolem_const_prefix, s, string_of_int num_T_args]
   491   |> space_implode Long_Name.separator
   492 
   493 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
   494    Also accumulates sort infomation. *)
   495 fun iterm_from_term thy format bs (P $ Q) =
   496     let
   497       val (P', P_atomics_Ts) = iterm_from_term thy format bs P
   498       val (Q', Q_atomics_Ts) = iterm_from_term thy format bs Q
   499     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
   500   | iterm_from_term thy format _ (Const (c, T)) =
   501     (IConst (`(make_fixed_const (SOME format)) c, T,
   502              if String.isPrefix old_skolem_const_prefix c then
   503                [] |> Term.add_tvarsT T |> map TVar
   504              else
   505                (c, T) |> Sign.const_typargs thy),
   506      atyps_of T)
   507   | iterm_from_term _ _ _ (Free (s, T)) =
   508     (IConst (`make_fixed_var s, T,
   509              if String.isPrefix polymorphic_free_prefix s then [T] else []),
   510      atyps_of T)
   511   | iterm_from_term _ format _ (Var (v as (s, _), T)) =
   512     (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
   513        let
   514          val Ts = T |> strip_type |> swap |> op ::
   515          val s' = new_skolem_const_name s (length Ts)
   516        in IConst (`(make_fixed_const (SOME format)) s', T, Ts) end
   517      else
   518        IVar ((make_schematic_var v, s), T), atyps_of T)
   519   | iterm_from_term _ _ bs (Bound j) =
   520     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))
   521   | iterm_from_term thy format bs (Abs (s, T, t)) =
   522     let
   523       fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
   524       val s = vary s
   525       val name = `make_bound_var s
   526       val (tm, atomic_Ts) = iterm_from_term thy format ((s, (name, T)) :: bs) t
   527     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end
   528 
   529 datatype locality =
   530   General | Helper | Extensionality | Intro | Elim | Simp | Local | Assum |
   531   Chained
   532 
   533 datatype order = First_Order | Higher_Order
   534 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
   535 datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
   536 datatype type_level =
   537   All_Types |
   538   Noninf_Nonmono_Types of soundness |
   539   Fin_Nonmono_Types |
   540   Const_Arg_Types |
   541   No_Types
   542 datatype type_uniformity = Uniform | Nonuniform
   543 
   544 datatype type_enc =
   545   Simple_Types of order * type_level |
   546   Guards of polymorphism * type_level * type_uniformity |
   547   Tags of polymorphism * type_level * type_uniformity
   548 
   549 fun try_unsuffixes ss s =
   550   fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
   551 
   552 fun type_enc_from_string soundness s =
   553   (case try (unprefix "poly_") s of
   554      SOME s => (SOME Polymorphic, s)
   555    | NONE =>
   556      case try (unprefix "raw_mono_") s of
   557        SOME s => (SOME Raw_Monomorphic, s)
   558      | NONE =>
   559        case try (unprefix "mono_") s of
   560          SOME s => (SOME Mangled_Monomorphic, s)
   561        | NONE => (NONE, s))
   562   ||> (fn s =>
   563           (* "_query" and "_bang" are for the ASCII-challenged Metis and
   564              Mirabelle. *)
   565           case try_unsuffixes ["?", "_query"] s of
   566             SOME s => (Noninf_Nonmono_Types soundness, s)
   567           | NONE =>
   568             case try_unsuffixes ["!", "_bang"] s of
   569               SOME s => (Fin_Nonmono_Types, s)
   570             | NONE => (All_Types, s))
   571   ||> apsnd (fn s =>
   572                 case try (unsuffix "_uniform") s of
   573                   SOME s => (Uniform, s)
   574                 | NONE => (Nonuniform, s))
   575   |> (fn (poly, (level, (uniformity, core))) =>
   576          case (core, (poly, level, uniformity)) of
   577            ("simple", (NONE, _, Nonuniform)) =>
   578            Simple_Types (First_Order, level)
   579          | ("simple_higher", (NONE, _, Nonuniform)) =>
   580            (case level of
   581               Noninf_Nonmono_Types _ => raise Same.SAME
   582             | _ => Simple_Types (Higher_Order, level))
   583          | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)
   584          | ("tags", (SOME Polymorphic, _, _)) =>
   585            Tags (Polymorphic, level, uniformity)
   586          | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)
   587          | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>
   588            Guards (poly, Const_Arg_Types, Nonuniform)
   589          | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>
   590            Guards (Polymorphic, No_Types, Nonuniform)
   591          | _ => raise Same.SAME)
   592   handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
   593 
   594 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _)) = true
   595   | is_type_enc_higher_order _ = false
   596 
   597 fun polymorphism_of_type_enc (Simple_Types _) = Mangled_Monomorphic
   598   | polymorphism_of_type_enc (Guards (poly, _, _)) = poly
   599   | polymorphism_of_type_enc (Tags (poly, _, _)) = poly
   600 
   601 fun level_of_type_enc (Simple_Types (_, level)) = level
   602   | level_of_type_enc (Guards (_, level, _)) = level
   603   | level_of_type_enc (Tags (_, level, _)) = level
   604 
   605 fun uniformity_of_type_enc (Simple_Types _) = Uniform
   606   | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity
   607   | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity
   608 
   609 fun is_type_level_quasi_sound All_Types = true
   610   | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
   611   | is_type_level_quasi_sound _ = false
   612 val is_type_enc_quasi_sound =
   613   is_type_level_quasi_sound o level_of_type_enc
   614 
   615 fun is_type_level_fairly_sound level =
   616   is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types
   617 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
   618 
   619 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
   620   | is_type_level_monotonicity_based Fin_Nonmono_Types = true
   621   | is_type_level_monotonicity_based _ = false
   622 
   623 fun adjust_type_enc (THF _) type_enc = type_enc
   624   | adjust_type_enc TFF (Simple_Types (_, level)) =
   625     Simple_Types (First_Order, level)
   626   | adjust_type_enc format (Simple_Types (_, level)) =
   627     adjust_type_enc format (Guards (Mangled_Monomorphic, level, Uniform))
   628   | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
   629     (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
   630   | adjust_type_enc _ type_enc = type_enc
   631 
   632 fun lift_lambdas ctxt type_enc =
   633   map (close_form o Envir.eta_contract) #> rpair ctxt
   634   #-> Lambda_Lifting.lift_lambdas
   635           (if polymorphism_of_type_enc type_enc = Polymorphic then
   636              SOME polymorphic_free_prefix
   637            else
   638              NONE)
   639           Lambda_Lifting.is_quantifier
   640   #> fst
   641 
   642 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
   643     intentionalize_def t
   644   | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
   645     let
   646       fun lam T t = Abs (Name.uu, T, t)
   647       val (head, args) = strip_comb t ||> rev
   648       val head_T = fastype_of head
   649       val n = length args
   650       val arg_Ts = head_T |> binder_types |> take n |> rev
   651       val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
   652     in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
   653   | intentionalize_def t = t
   654 
   655 type translated_formula =
   656   {name : string,
   657    locality : locality,
   658    kind : formula_kind,
   659    iformula : (name, typ, iterm) formula,
   660    atomic_types : typ list}
   661 
   662 fun update_iformula f ({name, locality, kind, iformula, atomic_types}
   663                        : translated_formula) =
   664   {name = name, locality = locality, kind = kind, iformula = f iformula,
   665    atomic_types = atomic_types} : translated_formula
   666 
   667 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
   668 
   669 fun insert_type ctxt get_T x xs =
   670   let val T = get_T x in
   671     if exists (type_instance ctxt T o get_T) xs then xs
   672     else x :: filter_out (type_generalization ctxt T o get_T) xs
   673   end
   674 
   675 (* The Booleans indicate whether all type arguments should be kept. *)
   676 datatype type_arg_policy =
   677   Explicit_Type_Args of bool |
   678   Mangled_Type_Args of bool |
   679   No_Type_Args
   680 
   681 fun should_drop_arg_type_args (Simple_Types _) =
   682     false (* since TFF doesn't support overloading *)
   683   | should_drop_arg_type_args type_enc =
   684     level_of_type_enc type_enc = All_Types andalso
   685     uniformity_of_type_enc type_enc = Uniform
   686 
   687 fun type_arg_policy type_enc s =
   688   if s = type_tag_name then
   689     (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   690        Mangled_Type_Args
   691      else
   692        Explicit_Type_Args) false
   693   else case type_enc of
   694     Tags (_, All_Types, Uniform) => No_Type_Args
   695   | _ =>
   696     let val level = level_of_type_enc type_enc in
   697       if level = No_Types orelse s = @{const_name HOL.eq} orelse
   698          (s = app_op_name andalso level = Const_Arg_Types) then
   699         No_Type_Args
   700       else
   701         should_drop_arg_type_args type_enc
   702         |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   703               Mangled_Type_Args
   704             else
   705               Explicit_Type_Args)
   706     end
   707 
   708 (* Make literals for sorted type variables. *)
   709 fun generic_add_sorts_on_type (_, []) = I
   710   | generic_add_sorts_on_type ((x, i), s :: ss) =
   711     generic_add_sorts_on_type ((x, i), ss)
   712     #> (if s = the_single @{sort HOL.type} then
   713           I
   714         else if i = ~1 then
   715           insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))
   716         else
   717           insert (op =) (TyLitVar (`make_type_class s,
   718                                    (make_schematic_type_var (x, i), x))))
   719 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
   720   | add_sorts_on_tfree _ = I
   721 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
   722   | add_sorts_on_tvar _ = I
   723 
   724 fun type_literals_for_types type_enc add_sorts_on_typ Ts =
   725   [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
   726 
   727 fun mk_aconns c phis =
   728   let val (phis', phi') = split_last phis in
   729     fold_rev (mk_aconn c) phis' phi'
   730   end
   731 fun mk_ahorn [] phi = phi
   732   | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
   733 fun mk_aquant _ [] phi = phi
   734   | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
   735     if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
   736   | mk_aquant q xs phi = AQuant (q, xs, phi)
   737 
   738 fun close_universally atom_vars phi =
   739   let
   740     fun formula_vars bounds (AQuant (_, xs, phi)) =
   741         formula_vars (map fst xs @ bounds) phi
   742       | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
   743       | formula_vars bounds (AAtom tm) =
   744         union (op =) (atom_vars tm []
   745                       |> filter_out (member (op =) bounds o fst))
   746   in mk_aquant AForall (formula_vars [] phi []) phi end
   747 
   748 fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]
   749   | iterm_vars (IConst _) = I
   750   | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)
   751   | iterm_vars (IAbs (_, tm)) = iterm_vars tm
   752 fun close_iformula_universally phi = close_universally iterm_vars phi
   753 
   754 fun term_vars bounds (ATerm (name as (s, _), tms)) =
   755     (is_tptp_variable s andalso not (member (op =) bounds name))
   756     ? insert (op =) (name, NONE) #> fold (term_vars bounds) tms
   757   | term_vars bounds (AAbs ((name, _), tm)) = term_vars (name :: bounds) tm
   758 fun close_formula_universally phi = close_universally (term_vars []) phi
   759 
   760 val homo_infinite_type_name = @{type_name ind} (* any infinite type *)
   761 val homo_infinite_type = Type (homo_infinite_type_name, [])
   762 
   763 fun ho_term_from_typ format type_enc =
   764   let
   765     fun term (Type (s, Ts)) =
   766       ATerm (case (is_type_enc_higher_order type_enc, s) of
   767                (true, @{type_name bool}) => `I tptp_bool_type
   768              | (true, @{type_name fun}) => `I tptp_fun_type
   769              | _ => if s = homo_infinite_type_name andalso
   770                        is_format_typed format then
   771                       `I tptp_individual_type
   772                     else
   773                       `make_fixed_type_const s,
   774              map term Ts)
   775     | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
   776     | term (TVar ((x as (s, _)), _)) =
   777       ATerm ((make_schematic_type_var x, s), [])
   778   in term end
   779 
   780 fun ho_term_for_type_arg format type_enc T =
   781   if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)
   782 
   783 (* This shouldn't clash with anything else. *)
   784 val mangled_type_sep = "\000"
   785 
   786 fun generic_mangled_type_name f (ATerm (name, [])) = f name
   787   | generic_mangled_type_name f (ATerm (name, tys)) =
   788     f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
   789     ^ ")"
   790   | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
   791 
   792 fun mangled_type format type_enc =
   793   generic_mangled_type_name fst o ho_term_from_typ format type_enc
   794 
   795 val bool_atype = AType (`I tptp_bool_type)
   796 
   797 fun make_simple_type s =
   798   if s = tptp_bool_type orelse s = tptp_fun_type orelse
   799      s = tptp_individual_type then
   800     s
   801   else
   802     simple_type_prefix ^ ascii_of s
   803 
   804 fun ho_type_from_ho_term type_enc pred_sym ary =
   805   let
   806     fun to_atype ty =
   807       AType ((make_simple_type (generic_mangled_type_name fst ty),
   808               generic_mangled_type_name snd ty))
   809     fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
   810     fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
   811       | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
   812       | to_fo _ _ = raise Fail "unexpected type abstraction"
   813     fun to_ho (ty as ATerm ((s, _), tys)) =
   814         if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
   815       | to_ho _ = raise Fail "unexpected type abstraction"
   816   in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
   817 
   818 fun ho_type_from_typ format type_enc pred_sym ary =
   819   ho_type_from_ho_term type_enc pred_sym ary
   820   o ho_term_from_typ format type_enc
   821 
   822 fun mangled_const_name format type_enc T_args (s, s') =
   823   let
   824     val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)
   825     fun type_suffix f g =
   826       fold_rev (curry (op ^) o g o prefix mangled_type_sep
   827                 o generic_mangled_type_name f) ty_args ""
   828   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
   829 
   830 val parse_mangled_ident =
   831   Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
   832 
   833 fun parse_mangled_type x =
   834   (parse_mangled_ident
   835    -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
   836                     [] >> ATerm) x
   837 and parse_mangled_types x =
   838   (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
   839 
   840 fun unmangled_type s =
   841   s |> suffix ")" |> raw_explode
   842     |> Scan.finite Symbol.stopper
   843            (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
   844                                                 quote s)) parse_mangled_type))
   845     |> fst
   846 
   847 val unmangled_const_name = space_explode mangled_type_sep #> hd
   848 fun unmangled_const s =
   849   let val ss = space_explode mangled_type_sep s in
   850     (hd ss, map unmangled_type (tl ss))
   851   end
   852 
   853 fun introduce_proxies type_enc =
   854   let
   855     fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
   856       | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
   857                        _ =
   858         (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
   859            limitation. This works in conjuction with special code in
   860            "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
   861            possible. *)
   862         IAbs ((`I "P", p_T),
   863               IApp (IConst (`I ho_quant, T, []),
   864                     IAbs ((`I "X", x_T),
   865                           IApp (IConst (`I "P", p_T, []),
   866                                 IConst (`I "X", x_T, [])))))
   867       | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
   868     fun intro top_level args (IApp (tm1, tm2)) =
   869         IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
   870       | intro top_level args (IConst (name as (s, _), T, T_args)) =
   871         (case proxify_const s of
   872            SOME proxy_base =>
   873            if top_level orelse is_type_enc_higher_order type_enc then
   874              case (top_level, s) of
   875                (_, "c_False") => IConst (`I tptp_false, T, [])
   876              | (_, "c_True") => IConst (`I tptp_true, T, [])
   877              | (false, "c_Not") => IConst (`I tptp_not, T, [])
   878              | (false, "c_conj") => IConst (`I tptp_and, T, [])
   879              | (false, "c_disj") => IConst (`I tptp_or, T, [])
   880              | (false, "c_implies") => IConst (`I tptp_implies, T, [])
   881              | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
   882              | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
   883              | (false, s) =>
   884                if is_tptp_equal s andalso length args = 2 then
   885                  IConst (`I tptp_equal, T, [])
   886                else
   887                  (* Use a proxy even for partially applied THF equality, because
   888                     the LEO-II and Satallax parsers complain about not being
   889                     able to infer the type of "=". *)
   890                  IConst (proxy_base |>> prefix const_prefix, T, T_args)
   891              | _ => IConst (name, T, [])
   892            else
   893              IConst (proxy_base |>> prefix const_prefix, T, T_args)
   894           | NONE => if s = tptp_choice then
   895                       (*this could be made neater by adding c_Eps as a proxy,
   896                         but we'd need to be able to "see" Hilbert_Choice.Eps
   897                         at this level in order to define fEps*)
   898                       tweak_ho_quant tptp_choice T args
   899                     else IConst (name, T, T_args))
   900       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
   901       | intro _ _ tm = tm
   902   in intro true [] end
   903 
   904 fun iformula_from_prop thy format type_enc eq_as_iff =
   905   let
   906     fun do_term bs t atomic_types =
   907       iterm_from_term thy format bs (Envir.eta_contract t)
   908       |>> (introduce_proxies type_enc #> AAtom)
   909       ||> union (op =) atomic_types
   910     fun do_quant bs q pos s T t' =
   911       let
   912         val s = singleton (Name.variant_list (map fst bs)) s
   913         val universal = Option.map (q = AExists ? not) pos
   914         val name =
   915           s |> `(case universal of
   916                    SOME true => make_all_bound_var
   917                  | SOME false => make_exist_bound_var
   918                  | NONE => make_bound_var)
   919       in
   920         do_formula ((s, (name, T)) :: bs) pos t'
   921         #>> mk_aquant q [(name, SOME T)]
   922       end
   923     and do_conn bs c pos1 t1 pos2 t2 =
   924       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
   925     and do_formula bs pos t =
   926       case t of
   927         @{const Trueprop} $ t1 => do_formula bs pos t1
   928       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
   929       | Const (@{const_name All}, _) $ Abs (s, T, t') =>
   930         do_quant bs AForall pos s T t'
   931       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
   932         do_quant bs AExists pos s T t'
   933       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
   934       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
   935       | @{const HOL.implies} $ t1 $ t2 =>
   936         do_conn bs AImplies (Option.map not pos) t1 pos t2
   937       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
   938         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
   939       | _ => do_term bs t
   940   in do_formula [] end
   941 
   942 fun presimplify_term _ [] t = t
   943   | presimplify_term ctxt presimp_consts t =
   944     t |> exists_Const (member (op =) presimp_consts o fst) t
   945          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
   946             #> Meson.presimplify ctxt
   947             #> prop_of)
   948 
   949 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
   950 fun conceal_bounds Ts t =
   951   subst_bounds (map (Free o apfst concealed_bound_name)
   952                     (0 upto length Ts - 1 ~~ Ts), t)
   953 fun reveal_bounds Ts =
   954   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
   955                     (0 upto length Ts - 1 ~~ Ts))
   956 
   957 fun is_fun_equality (@{const_name HOL.eq},
   958                      Type (_, [Type (@{type_name fun}, _), _])) = true
   959   | is_fun_equality _ = false
   960 
   961 fun extensionalize_term ctxt t =
   962   if exists_Const is_fun_equality t then
   963     let val thy = Proof_Context.theory_of ctxt in
   964       t |> cterm_of thy |> Meson.extensionalize_conv ctxt
   965         |> prop_of |> Logic.dest_equals |> snd
   966     end
   967   else
   968     t
   969 
   970 fun simple_translate_lambdas do_lambdas ctxt t =
   971   let val thy = Proof_Context.theory_of ctxt in
   972     if Meson.is_fol_term thy t then
   973       t
   974     else
   975       let
   976         fun aux Ts t =
   977           case t of
   978             @{const Not} $ t1 => @{const Not} $ aux Ts t1
   979           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
   980             t0 $ Abs (s, T, aux (T :: Ts) t')
   981           | (t0 as Const (@{const_name All}, _)) $ t1 =>
   982             aux Ts (t0 $ eta_expand Ts t1 1)
   983           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
   984             t0 $ Abs (s, T, aux (T :: Ts) t')
   985           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
   986             aux Ts (t0 $ eta_expand Ts t1 1)
   987           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   988           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   989           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   990           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
   991               $ t1 $ t2 =>
   992             t0 $ aux Ts t1 $ aux Ts t2
   993           | _ =>
   994             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
   995             else t |> Envir.eta_contract |> do_lambdas ctxt Ts
   996         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
   997       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
   998   end
   999 
  1000 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
  1001     do_cheaply_conceal_lambdas Ts t1
  1002     $ do_cheaply_conceal_lambdas Ts t2
  1003   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
  1004     Free (polymorphic_free_prefix ^ serial_string (),
  1005           T --> fastype_of1 (T :: Ts, t))
  1006   | do_cheaply_conceal_lambdas _ t = t
  1007 
  1008 fun do_introduce_combinators ctxt Ts t =
  1009   let val thy = Proof_Context.theory_of ctxt in
  1010     t |> conceal_bounds Ts
  1011       |> cterm_of thy
  1012       |> Meson_Clausify.introduce_combinators_in_cterm
  1013       |> prop_of |> Logic.dest_equals |> snd
  1014       |> reveal_bounds Ts
  1015   end
  1016   (* A type variable of sort "{}" will make abstraction fail. *)
  1017   handle THM _ => t |> do_cheaply_conceal_lambdas Ts
  1018 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
  1019 
  1020 fun preprocess_abstractions_in_terms trans_lambdas facts =
  1021   let
  1022     val (facts, lambda_ts) =
  1023       facts |> map (snd o snd) |> trans_lambdas 
  1024             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
  1025     val lambda_facts =
  1026       map2 (fn t => fn j =>
  1027                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
  1028            lambda_ts (1 upto length lambda_ts)
  1029   in (facts, lambda_facts) end
  1030 
  1031 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
  1032    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
  1033 fun freeze_term t =
  1034   let
  1035     fun aux (t $ u) = aux t $ aux u
  1036       | aux (Abs (s, T, t)) = Abs (s, T, aux t)
  1037       | aux (Var ((s, i), T)) =
  1038         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
  1039       | aux t = t
  1040   in t |> exists_subterm is_Var t ? aux end
  1041 
  1042 fun presimp_prop ctxt presimp_consts t =
  1043   let
  1044     val thy = Proof_Context.theory_of ctxt
  1045     val t = t |> Envir.beta_eta_contract
  1046               |> transform_elim_prop
  1047               |> Object_Logic.atomize_term thy
  1048     val need_trueprop = (fastype_of t = @{typ bool})
  1049   in
  1050     t |> need_trueprop ? HOLogic.mk_Trueprop
  1051       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
  1052       |> extensionalize_term ctxt
  1053       |> presimplify_term ctxt presimp_consts
  1054       |> perhaps (try (HOLogic.dest_Trueprop))
  1055   end
  1056 
  1057 (* making fact and conjecture formulas *)
  1058 fun make_formula thy format type_enc eq_as_iff name loc kind t =
  1059   let
  1060     val (iformula, atomic_types) =
  1061       iformula_from_prop thy format type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
  1062   in
  1063     {name = name, locality = loc, kind = kind, iformula = iformula,
  1064      atomic_types = atomic_types}
  1065   end
  1066 
  1067 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  1068   let val thy = Proof_Context.theory_of ctxt in
  1069     case t |> make_formula thy format type_enc (eq_as_iff andalso format <> CNF)
  1070                            name loc Axiom of
  1071       formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
  1072       if s = tptp_true then NONE else SOME formula
  1073     | formula => SOME formula
  1074   end
  1075 
  1076 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
  1077   | s_not_trueprop t = s_not t
  1078 
  1079 fun make_conjecture thy format type_enc =
  1080   map (fn ((name, loc), (kind, t)) =>
  1081           t |> kind = Conjecture ? s_not_trueprop
  1082             |> make_formula thy format type_enc (format <> CNF) name loc kind)
  1083 
  1084 (** Finite and infinite type inference **)
  1085 
  1086 type monotonicity_info =
  1087   {maybe_finite_Ts : typ list,
  1088    surely_finite_Ts : typ list,
  1089    maybe_infinite_Ts : typ list,
  1090    surely_infinite_Ts : typ list,
  1091    maybe_nonmono_Ts : typ list}
  1092 
  1093 (* These types witness that the type classes they belong to allow infinite
  1094    models and hence that any types with these type classes is monotonic. *)
  1095 val known_infinite_types =
  1096   [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]
  1097 
  1098 fun is_type_surely_infinite' ctxt soundness cached_Ts T =
  1099   (* Unlike virtually any other polymorphic fact whose type variables can be
  1100      instantiated by a known infinite type, extensionality actually encodes a
  1101      cardinality constraints. *)
  1102   soundness <> Sound andalso
  1103   is_type_surely_infinite ctxt (soundness = Unsound) cached_Ts T
  1104 
  1105 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
  1106    dangerous because their "exhaust" properties can easily lead to unsound ATP
  1107    proofs. On the other hand, all HOL infinite types can be given the same
  1108    models in first-order logic (via Löwenheim-Skolem). *)
  1109 
  1110 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  1111   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
  1112                              maybe_nonmono_Ts, ...}
  1113                        (Noninf_Nonmono_Types soundness) T =
  1114     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1115     not (exists (type_instance ctxt T) surely_infinite_Ts orelse
  1116          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
  1117           is_type_surely_infinite' ctxt soundness surely_infinite_Ts T))
  1118   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
  1119                              maybe_nonmono_Ts, ...}
  1120                        Fin_Nonmono_Types T =
  1121     exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
  1122     (exists (type_generalization ctxt T) surely_finite_Ts orelse
  1123      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
  1124       is_type_surely_finite ctxt T))
  1125   | should_encode_type _ _ _ _ = false
  1126 
  1127 fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var
  1128                       T =
  1129     (uniformity = Uniform orelse should_guard_var ()) andalso
  1130     should_encode_type ctxt mono level T
  1131   | should_guard_type _ _ _ _ _ = false
  1132 
  1133 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
  1134     String.isPrefix bound_var_prefix s orelse
  1135     String.isPrefix all_bound_var_prefix s
  1136   | is_maybe_universal_var (IVar _) = true
  1137   | is_maybe_universal_var _ = false
  1138 
  1139 datatype tag_site =
  1140   Top_Level of bool option |
  1141   Eq_Arg of bool option |
  1142   Elsewhere
  1143 
  1144 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  1145   | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =
  1146     (case uniformity of
  1147        Uniform => should_encode_type ctxt mono level T
  1148      | Nonuniform =>
  1149        case (site, is_maybe_universal_var u) of
  1150          (Eq_Arg _, true) => should_encode_type ctxt mono level T
  1151        | _ => false)
  1152   | should_tag_with_type _ _ _ _ _ _ = false
  1153 
  1154 fun homogenized_type ctxt mono level =
  1155   let
  1156     val should_encode = should_encode_type ctxt mono level
  1157     fun homo 0 T = if should_encode T then T else homo_infinite_type
  1158       | homo ary (Type (@{type_name fun}, [T1, T2])) =
  1159         homo 0 T1 --> homo (ary - 1) T2
  1160       | homo _ _ = raise Fail "expected function type"
  1161   in homo end
  1162 
  1163 (** predicators and application operators **)
  1164 
  1165 type sym_info =
  1166   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
  1167 
  1168 fun add_iterm_syms_to_table ctxt explicit_apply =
  1169   let
  1170     fun consider_var_arity const_T var_T max_ary =
  1171       let
  1172         fun iter ary T =
  1173           if ary = max_ary orelse type_instance ctxt var_T T orelse
  1174              type_instance ctxt T var_T then
  1175             ary
  1176           else
  1177             iter (ary + 1) (range_type T)
  1178       in iter 0 const_T end
  1179     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1180       if explicit_apply = NONE andalso
  1181          (can dest_funT T orelse T = @{typ bool}) then
  1182         let
  1183           val bool_vars' = bool_vars orelse body_type T = @{typ bool}
  1184           fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
  1185             {pred_sym = pred_sym andalso not bool_vars',
  1186              min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
  1187              max_ary = max_ary, types = types}
  1188           val fun_var_Ts' =
  1189             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
  1190         in
  1191           if bool_vars' = bool_vars andalso
  1192              pointer_eq (fun_var_Ts', fun_var_Ts) then
  1193             accum
  1194           else
  1195             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
  1196         end
  1197       else
  1198         accum
  1199     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1200       let val (head, args) = strip_iterm_comb tm in
  1201         (case head of
  1202            IConst ((s, _), T, _) =>
  1203            if String.isPrefix bound_var_prefix s orelse
  1204               String.isPrefix all_bound_var_prefix s then
  1205              add_universal_var T accum
  1206            else if String.isPrefix exist_bound_var_prefix s then
  1207              accum
  1208            else
  1209              let val ary = length args in
  1210                ((bool_vars, fun_var_Ts),
  1211                 case Symtab.lookup sym_tab s of
  1212                   SOME {pred_sym, min_ary, max_ary, types} =>
  1213                   let
  1214                     val pred_sym =
  1215                       pred_sym andalso top_level andalso not bool_vars
  1216                     val types' = types |> insert_type ctxt I T
  1217                     val min_ary =
  1218                       if is_some explicit_apply orelse
  1219                          pointer_eq (types', types) then
  1220                         min_ary
  1221                       else
  1222                         fold (consider_var_arity T) fun_var_Ts min_ary
  1223                   in
  1224                     Symtab.update (s, {pred_sym = pred_sym,
  1225                                        min_ary = Int.min (ary, min_ary),
  1226                                        max_ary = Int.max (ary, max_ary),
  1227                                        types = types'})
  1228                                   sym_tab
  1229                   end
  1230                 | NONE =>
  1231                   let
  1232                     val pred_sym = top_level andalso not bool_vars
  1233                     val min_ary =
  1234                       case explicit_apply of
  1235                         SOME true => 0
  1236                       | SOME false => ary
  1237                       | NONE => fold (consider_var_arity T) fun_var_Ts ary
  1238                   in
  1239                     Symtab.update_new (s, {pred_sym = pred_sym,
  1240                                            min_ary = min_ary, max_ary = ary,
  1241                                            types = [T]})
  1242                                       sym_tab
  1243                   end)
  1244              end
  1245          | IVar (_, T) => add_universal_var T accum
  1246          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
  1247          | _ => accum)
  1248         |> fold (add false) args
  1249       end
  1250   in add true end
  1251 fun add_fact_syms_to_table ctxt explicit_apply =
  1252   formula_fold NONE (K (add_iterm_syms_to_table ctxt explicit_apply))
  1253   |> fact_lift
  1254 
  1255 val tvar_a = TVar (("'a", 0), HOLogic.typeS)
  1256 
  1257 val default_sym_tab_entries : (string * sym_info) list =
  1258   (prefixed_predicator_name,
  1259    {pred_sym = true, min_ary = 1, max_ary = 1, types = []}) ::
  1260   (make_fixed_const NONE @{const_name undefined},
  1261    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  1262   ([tptp_false, tptp_true]
  1263    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  1264   ([tptp_equal, tptp_old_equal]
  1265    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
  1266 
  1267 fun sym_table_for_facts ctxt explicit_apply facts =
  1268   ((false, []), Symtab.empty)
  1269   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  1270   |> fold Symtab.update default_sym_tab_entries
  1271 
  1272 fun min_arity_of sym_tab s =
  1273   case Symtab.lookup sym_tab s of
  1274     SOME ({min_ary, ...} : sym_info) => min_ary
  1275   | NONE =>
  1276     case strip_prefix_and_unascii const_prefix s of
  1277       SOME s =>
  1278       let val s = s |> unmangled_const_name |> invert_const in
  1279         if s = predicator_name then 1
  1280         else if s = app_op_name then 2
  1281         else if s = type_guard_name then 1
  1282         else 0
  1283       end
  1284     | NONE => 0
  1285 
  1286 (* True if the constant ever appears outside of the top-level position in
  1287    literals, or if it appears with different arities (e.g., because of different
  1288    type instantiations). If false, the constant always receives all of its
  1289    arguments and is used as a predicate. *)
  1290 fun is_pred_sym sym_tab s =
  1291   case Symtab.lookup sym_tab s of
  1292     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
  1293     pred_sym andalso min_ary = max_ary
  1294   | NONE => false
  1295 
  1296 val predicator_combconst =
  1297   IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
  1298 fun predicator tm = IApp (predicator_combconst, tm)
  1299 
  1300 fun introduce_predicators_in_iterm sym_tab tm =
  1301   case strip_iterm_comb tm of
  1302     (IConst ((s, _), _, _), _) =>
  1303     if is_pred_sym sym_tab s then tm else predicator tm
  1304   | _ => predicator tm
  1305 
  1306 fun list_app head args = fold (curry (IApp o swap)) args head
  1307 
  1308 val app_op = `(make_fixed_const NONE) app_op_name
  1309 
  1310 fun explicit_app arg head =
  1311   let
  1312     val head_T = ityp_of head
  1313     val (arg_T, res_T) = dest_funT head_T
  1314     val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
  1315   in list_app explicit_app [head, arg] end
  1316 fun list_explicit_app head args = fold explicit_app args head
  1317 
  1318 fun introduce_explicit_apps_in_iterm sym_tab =
  1319   let
  1320     fun aux tm =
  1321       case strip_iterm_comb tm of
  1322         (head as IConst ((s, _), _, _), args) =>
  1323         args |> map aux
  1324              |> chop (min_arity_of sym_tab s)
  1325              |>> list_app head
  1326              |-> list_explicit_app
  1327       | (head, args) => list_explicit_app head (map aux args)
  1328   in aux end
  1329 
  1330 fun chop_fun 0 T = ([], T)
  1331   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
  1332     chop_fun (n - 1) ran_T |>> cons dom_T
  1333   | chop_fun _ _ = raise Fail "unexpected non-function"
  1334 
  1335 fun filter_type_args _ _ _ [] = []
  1336   | filter_type_args thy s arity T_args =
  1337     let
  1338       (* will throw "TYPE" for pseudo-constants *)
  1339       val U = if s = app_op_name then
  1340                 @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global
  1341               else
  1342                 s |> Sign.the_const_type thy
  1343     in
  1344       case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of
  1345         [] => []
  1346       | res_U_vars =>
  1347         let val U_args = (s, U) |> Sign.const_typargs thy in
  1348           U_args ~~ T_args
  1349           |> map (fn (U, T) =>
  1350                      if member (op =) res_U_vars (dest_TVar U) then T
  1351                      else dummyT)
  1352         end
  1353     end
  1354     handle TYPE _ => T_args
  1355 
  1356 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
  1357   let
  1358     val thy = Proof_Context.theory_of ctxt
  1359     fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
  1360       | aux arity (IConst (name as (s, _), T, T_args)) =
  1361         (case strip_prefix_and_unascii const_prefix s of
  1362            NONE =>
  1363            (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice
  1364                   then [] else T_args)
  1365          | SOME s'' =>
  1366            let
  1367              val s'' = invert_const s''
  1368              fun filtered_T_args false = T_args
  1369                | filtered_T_args true = filter_type_args thy s'' arity T_args
  1370            in
  1371              case type_arg_policy type_enc s'' of
  1372                Explicit_Type_Args drop_args =>
  1373                (name, filtered_T_args drop_args)
  1374              | Mangled_Type_Args drop_args =>
  1375                (mangled_const_name format type_enc (filtered_T_args drop_args)
  1376                                    name, [])
  1377              | No_Type_Args => (name, [])
  1378            end)
  1379         |> (fn (name, T_args) => IConst (name, T, T_args))
  1380       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
  1381       | aux _ tm = tm
  1382   in aux 0 end
  1383 
  1384 fun repair_iterm ctxt format type_enc sym_tab =
  1385   not (is_type_enc_higher_order type_enc)
  1386   ? (introduce_explicit_apps_in_iterm sym_tab
  1387      #> introduce_predicators_in_iterm sym_tab)
  1388   #> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1389 fun repair_fact ctxt format type_enc sym_tab =
  1390   update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))
  1391 
  1392 (** Helper facts **)
  1393 
  1394 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
  1395 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
  1396 
  1397 (* The Boolean indicates that a fairly sound type encoding is needed. *)
  1398 val helper_table =
  1399   [(("COMBI", false), @{thms Meson.COMBI_def}),
  1400    (("COMBK", false), @{thms Meson.COMBK_def}),
  1401    (("COMBB", false), @{thms Meson.COMBB_def}),
  1402    (("COMBC", false), @{thms Meson.COMBC_def}),
  1403    (("COMBS", false), @{thms Meson.COMBS_def}),
  1404    ((predicator_name, false), [not_ffalse, ftrue]),
  1405    (("fFalse", false), [not_ffalse]),
  1406    (("fFalse", true), @{thms True_or_False}),
  1407    (("fTrue", false), [ftrue]),
  1408    (("fTrue", true), @{thms True_or_False}),
  1409    (("fNot", false),
  1410     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1411            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1412    (("fconj", false),
  1413     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
  1414         by (unfold fconj_def) fast+}),
  1415    (("fdisj", false),
  1416     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
  1417         by (unfold fdisj_def) fast+}),
  1418    (("fimplies", false),
  1419     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
  1420         by (unfold fimplies_def) fast+}),
  1421    (("fequal", true),
  1422     (* This is a lie: Higher-order equality doesn't need a sound type encoding.
  1423        However, this is done so for backward compatibility: Including the
  1424        equality helpers by default in Metis breaks a few existing proofs. *)
  1425     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1426            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1427    (* Partial characterization of "fAll" and "fEx". A complete characterization
  1428       would require the axiom of choice for replay with Metis. *)
  1429    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
  1430    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
  1431    (("If", true), @{thms if_True if_False True_or_False})]
  1432   |> map (apsnd (map zero_var_indexes))
  1433 
  1434 fun fo_literal_from_type_literal (TyLitVar (class, name)) =
  1435     (true, ATerm (class, [ATerm (name, [])]))
  1436   | fo_literal_from_type_literal (TyLitFree (class, name)) =
  1437     (true, ATerm (class, [ATerm (name, [])]))
  1438 
  1439 fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
  1440 
  1441 fun bound_tvars type_enc Ts =
  1442   mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
  1443                 (type_literals_for_types type_enc add_sorts_on_tvar Ts))
  1444 
  1445 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  1446   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
  1447    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  1448   |> bound_tvars type_enc atomic_Ts
  1449   |> close_formula_universally
  1450 
  1451 val type_tag = `(make_fixed_const NONE) type_tag_name
  1452 
  1453 fun type_tag_idempotence_fact type_enc =
  1454   let
  1455     fun var s = ATerm (`I s, [])
  1456     fun tag tm = ATerm (type_tag, [var "A", tm])
  1457     val tagged_var = tag (var "X")
  1458   in
  1459     Formula (type_tag_idempotence_helper_name, Axiom,
  1460              eq_formula type_enc [] false (tag tagged_var) tagged_var,
  1461              isabelle_info simpN, NONE)
  1462   end
  1463 
  1464 fun should_specialize_helper type_enc t =
  1465   polymorphism_of_type_enc type_enc <> Polymorphic andalso
  1466   level_of_type_enc type_enc <> No_Types andalso
  1467   not (null (Term.hidden_polymorphism t))
  1468 
  1469 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  1470   case strip_prefix_and_unascii const_prefix s of
  1471     SOME mangled_s =>
  1472     let
  1473       val thy = Proof_Context.theory_of ctxt
  1474       val unmangled_s = mangled_s |> unmangled_const_name
  1475       fun dub needs_fairly_sound j k =
  1476         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
  1477          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
  1478          (if needs_fairly_sound then typed_helper_suffix
  1479           else untyped_helper_suffix),
  1480          Helper)
  1481       fun dub_and_inst needs_fairly_sound (th, j) =
  1482         let val t = prop_of th in
  1483           if should_specialize_helper type_enc t then
  1484             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
  1485                 types
  1486           else
  1487             [t]
  1488         end
  1489         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
  1490       val make_facts = map_filter (make_fact ctxt format type_enc false)
  1491       val fairly_sound = is_type_enc_fairly_sound type_enc
  1492     in
  1493       helper_table
  1494       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
  1495                   if helper_s <> unmangled_s orelse
  1496                      (needs_fairly_sound andalso not fairly_sound) then
  1497                     []
  1498                   else
  1499                     ths ~~ (1 upto length ths)
  1500                     |> maps (dub_and_inst needs_fairly_sound)
  1501                     |> make_facts)
  1502     end
  1503   | NONE => []
  1504 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  1505   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
  1506                   []
  1507 
  1508 (***************************************************************)
  1509 (* Type Classes Present in the Axiom or Conjecture Clauses     *)
  1510 (***************************************************************)
  1511 
  1512 fun set_insert (x, s) = Symtab.update (x, ()) s
  1513 
  1514 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
  1515 
  1516 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
  1517 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
  1518 
  1519 fun classes_of_terms get_Ts =
  1520   map (map snd o get_Ts)
  1521   #> List.foldl add_classes Symtab.empty
  1522   #> delete_type #> Symtab.keys
  1523 
  1524 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
  1525 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
  1526 
  1527 fun fold_type_constrs f (Type (s, Ts)) x =
  1528     fold (fold_type_constrs f) Ts (f (s, x))
  1529   | fold_type_constrs _ _ x = x
  1530 
  1531 (* Type constructors used to instantiate overloaded constants are the only ones
  1532    needed. *)
  1533 fun add_type_constrs_in_term thy =
  1534   let
  1535     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
  1536       | add (t $ u) = add t #> add u
  1537       | add (Const (x as (s, _))) =
  1538         if String.isPrefix skolem_const_prefix s then I
  1539         else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)
  1540       | add (Free (s, T)) =
  1541         if String.isPrefix polymorphic_free_prefix s then
  1542           T |> fold_type_constrs set_insert
  1543         else
  1544           I
  1545       | add (Abs (_, _, u)) = add u
  1546       | add _ = I
  1547   in add end
  1548 
  1549 fun type_constrs_of_terms thy ts =
  1550   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
  1551 
  1552 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  1553                        hyp_ts concl_t facts =
  1554   let
  1555     val thy = Proof_Context.theory_of ctxt
  1556     val presimp_consts = Meson.presimplified_consts ctxt
  1557     val fact_ts = facts |> map snd
  1558     (* Remove existing facts from the conjecture, as this can dramatically
  1559        boost an ATP's performance (for some reason). *)
  1560     val hyp_ts =
  1561       hyp_ts
  1562       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
  1563     val facts = facts |> map (apsnd (pair Axiom))
  1564     val conjs =
  1565       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]
  1566       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
  1567     val ((conjs, facts), lambdas) =
  1568       if preproc then
  1569         conjs @ facts
  1570         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
  1571         |> preprocess_abstractions_in_terms trans_lambdas
  1572         |>> chop (length conjs)
  1573         |>> apfst (map (apsnd (apsnd freeze_term)))
  1574       else
  1575         ((conjs, facts), [])
  1576     val conjs = conjs |> make_conjecture thy format type_enc
  1577     val (fact_names, facts) =
  1578       facts
  1579       |> map_filter (fn (name, (_, t)) =>
  1580                         make_fact ctxt format type_enc true (name, t)
  1581                         |> Option.map (pair name))
  1582       |> ListPair.unzip
  1583     val lambdas =
  1584       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
  1585     val all_ts = concl_t :: hyp_ts @ fact_ts
  1586     val subs = tfree_classes_of_terms all_ts
  1587     val supers = tvar_classes_of_terms all_ts
  1588     val tycons = type_constrs_of_terms thy all_ts
  1589     val (supers, arity_clauses) =
  1590       if level_of_type_enc type_enc = No_Types then ([], [])
  1591       else make_arity_clauses thy tycons supers
  1592     val class_rel_clauses = make_class_rel_clauses thy subs supers
  1593   in
  1594     (fact_names |> map single,
  1595      (conjs, facts @ lambdas, class_rel_clauses, arity_clauses))
  1596   end
  1597 
  1598 val type_guard = `(make_fixed_const NONE) type_guard_name
  1599 
  1600 fun type_guard_iterm ctxt format type_enc T tm =
  1601   IApp (IConst (type_guard, T --> @{typ bool}, [T])
  1602         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
  1603 
  1604 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  1605   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
  1606     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  1607   | is_var_positively_naked_in_term _ _ _ _ = true
  1608 
  1609 fun should_guard_var_in_formula pos phi (SOME true) name =
  1610     formula_fold pos (is_var_positively_naked_in_term name) phi false
  1611   | should_guard_var_in_formula _ _ _ _ = true
  1612 
  1613 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  1614   | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =
  1615     should_encode_type ctxt mono level T
  1616   | should_generate_tag_bound_decl _ _ _ _ _ = false
  1617 
  1618 fun mk_aterm format type_enc name T_args args =
  1619   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
  1620 
  1621 fun tag_with_type ctxt format mono type_enc pos T tm =
  1622   IConst (type_tag, T --> T, [T])
  1623   |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1624   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  1625   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
  1626        | _ => raise Fail "unexpected lambda-abstraction")
  1627 and ho_term_from_iterm ctxt format mono type_enc =
  1628   let
  1629     fun aux site u =
  1630       let
  1631         val (head, args) = strip_iterm_comb u
  1632         val pos =
  1633           case site of
  1634             Top_Level pos => pos
  1635           | Eq_Arg pos => pos
  1636           | Elsewhere => NONE
  1637         val t =
  1638           case head of
  1639             IConst (name as (s, _), _, T_args) =>
  1640             let
  1641               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
  1642             in
  1643               mk_aterm format type_enc name T_args (map (aux arg_site) args)
  1644             end
  1645           | IVar (name, _) =>
  1646             mk_aterm format type_enc name [] (map (aux Elsewhere) args)
  1647           | IAbs ((name, T), tm) =>
  1648             AAbs ((name, ho_type_from_typ format type_enc true 0 T),
  1649                   aux Elsewhere tm)
  1650           | IApp _ => raise Fail "impossible \"IApp\""
  1651         val T = ityp_of u
  1652       in
  1653         t |> (if should_tag_with_type ctxt mono type_enc site u T then
  1654                 tag_with_type ctxt format mono type_enc pos T
  1655               else
  1656                 I)
  1657       end
  1658   in aux end
  1659 and formula_from_iformula ctxt format mono type_enc should_guard_var =
  1660   let
  1661     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
  1662     val do_bound_type =
  1663       case type_enc of
  1664         Simple_Types (_, level) =>
  1665         homogenized_type ctxt mono level 0
  1666         #> ho_type_from_typ format type_enc false 0 #> SOME
  1667       | _ => K NONE
  1668     fun do_out_of_bound_type pos phi universal (name, T) =
  1669       if should_guard_type ctxt mono type_enc
  1670              (fn () => should_guard_var pos phi universal name) T then
  1671         IVar (name, T)
  1672         |> type_guard_iterm ctxt format type_enc T
  1673         |> do_term pos |> AAtom |> SOME
  1674       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
  1675         let
  1676           val var = ATerm (name, [])
  1677           val tagged_var =
  1678             ATerm (type_tag, [ho_term_from_typ format type_enc T, var])
  1679         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
  1680       else
  1681         NONE
  1682     fun do_formula pos (AQuant (q, xs, phi)) =
  1683         let
  1684           val phi = phi |> do_formula pos
  1685           val universal = Option.map (q = AExists ? not) pos
  1686         in
  1687           AQuant (q, xs |> map (apsnd (fn NONE => NONE
  1688                                         | SOME T => do_bound_type T)),
  1689                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
  1690                       (map_filter
  1691                            (fn (_, NONE) => NONE
  1692                              | (s, SOME T) =>
  1693                                do_out_of_bound_type pos phi universal (s, T))
  1694                            xs)
  1695                       phi)
  1696         end
  1697       | do_formula pos (AConn conn) = aconn_map pos do_formula conn
  1698       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  1699   in do_formula end
  1700 
  1701 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
  1702    of monomorphization). The TPTP explicitly forbids name clashes, and some of
  1703    the remote provers might care. *)
  1704 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
  1705                           (j, {name, locality, kind, iformula, atomic_types}) =
  1706   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
  1707    iformula
  1708    |> close_iformula_universally
  1709    |> formula_from_iformula ctxt format mono type_enc
  1710                             should_guard_var_in_formula
  1711                             (if pos then SOME true else NONE)
  1712    |> bound_tvars type_enc atomic_types
  1713    |> close_formula_universally,
  1714    NONE,
  1715    case locality of
  1716      Intro => isabelle_info introN
  1717    | Elim => isabelle_info elimN
  1718    | Simp => isabelle_info simpN
  1719    | _ => NONE)
  1720   |> Formula
  1721 
  1722 fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}
  1723                                        : class_rel_clause) =
  1724   let val ty_arg = ATerm (`I "T", []) in
  1725     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
  1726              AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
  1727                                AAtom (ATerm (superclass, [ty_arg]))])
  1728              |> close_formula_universally, isabelle_info introN, NONE)
  1729   end
  1730 
  1731 fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
  1732     (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
  1733   | fo_literal_from_arity_literal (TVarLit (c, sort)) =
  1734     (false, ATerm (c, [ATerm (sort, [])]))
  1735 
  1736 fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}
  1737                                    : arity_clause) =
  1738   Formula (arity_clause_prefix ^ name, Axiom,
  1739            mk_ahorn (map (formula_from_fo_literal o apfst not
  1740                           o fo_literal_from_arity_literal) prem_lits)
  1741                     (formula_from_fo_literal
  1742                          (fo_literal_from_arity_literal concl_lits))
  1743            |> close_formula_universally, isabelle_info introN, NONE)
  1744 
  1745 fun formula_line_for_conjecture ctxt format mono type_enc
  1746         ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  1747   Formula (conjecture_prefix ^ name, kind,
  1748            formula_from_iformula ctxt format mono type_enc
  1749                should_guard_var_in_formula (SOME false)
  1750                (close_iformula_universally iformula)
  1751            |> bound_tvars type_enc atomic_types
  1752            |> close_formula_universally, NONE, NONE)
  1753 
  1754 fun free_type_literals type_enc ({atomic_types, ...} : translated_formula) =
  1755   atomic_types |> type_literals_for_types type_enc add_sorts_on_tfree
  1756                |> map fo_literal_from_type_literal
  1757 
  1758 fun formula_line_for_free_type j lit =
  1759   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,
  1760            formula_from_fo_literal lit, NONE, NONE)
  1761 fun formula_lines_for_free_types type_enc facts =
  1762   let
  1763     val litss = map (free_type_literals type_enc) facts
  1764     val lits = fold (union (op =)) litss []
  1765   in map2 formula_line_for_free_type (0 upto length lits - 1) lits end
  1766 
  1767 (** Symbol declarations **)
  1768 
  1769 fun should_declare_sym type_enc pred_sym s =
  1770   (case type_enc of
  1771      Guards _ => not pred_sym
  1772    | _ => true) andalso
  1773   is_tptp_user_symbol s
  1774 
  1775 fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  1776                              (conjs, facts) =
  1777   let
  1778     fun add_iterm_syms in_conj tm =
  1779       let val (head, args) = strip_iterm_comb tm in
  1780         (case head of
  1781            IConst ((s, s'), T, T_args) =>
  1782            let val pred_sym = is_pred_sym repaired_sym_tab s in
  1783              if should_declare_sym type_enc pred_sym s then
  1784                Symtab.map_default (s, [])
  1785                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
  1786                                          in_conj))
  1787              else
  1788                I
  1789            end
  1790          | IAbs (_, tm) => add_iterm_syms in_conj tm
  1791          | _ => I)
  1792         #> fold (add_iterm_syms in_conj) args
  1793       end
  1794     fun add_fact_syms in_conj =
  1795       fact_lift (formula_fold NONE (K (add_iterm_syms in_conj)))
  1796     fun add_formula_var_types (AQuant (_, xs, phi)) =
  1797         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
  1798         #> add_formula_var_types phi
  1799       | add_formula_var_types (AConn (_, phis)) =
  1800         fold add_formula_var_types phis
  1801       | add_formula_var_types _ = I
  1802     fun var_types () =
  1803       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
  1804       else fold (fact_lift add_formula_var_types) (conjs @ facts) []
  1805     fun add_undefined_const T =
  1806       let
  1807         val (s, s') =
  1808           `(make_fixed_const (SOME format)) @{const_name undefined}
  1809           |> (case type_arg_policy type_enc @{const_name undefined} of
  1810                 Mangled_Type_Args _ => mangled_const_name format type_enc [T]
  1811               | _ => I)
  1812       in
  1813         Symtab.map_default (s, [])
  1814                            (insert_type ctxt #3 (s', [T], T, false, 0, false))
  1815       end
  1816   in
  1817     Symtab.empty
  1818     |> is_type_enc_fairly_sound type_enc
  1819        ? (fold (add_fact_syms true) conjs
  1820           #> fold (add_fact_syms false) facts
  1821           #> (case type_enc of
  1822                 Simple_Types _ => I
  1823               | _ => fold add_undefined_const (var_types ())))
  1824   end
  1825 
  1826 (* We add "bool" in case the helper "True_or_False" is included later. *)
  1827 val default_mono =
  1828   {maybe_finite_Ts = [@{typ bool}],
  1829    surely_finite_Ts = [@{typ bool}],
  1830    maybe_infinite_Ts = known_infinite_types,
  1831    surely_infinite_Ts = known_infinite_types,
  1832    maybe_nonmono_Ts = [@{typ bool}]}
  1833 
  1834 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
  1835    out with monotonicity" paper presented at CADE 2011. *)
  1836 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  1837   | add_iterm_mononotonicity_info ctxt level _
  1838         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
  1839         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
  1840                   surely_infinite_Ts, maybe_nonmono_Ts}) =
  1841     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
  1842       case level of
  1843         Noninf_Nonmono_Types soundness =>
  1844         if exists (type_instance ctxt T) surely_infinite_Ts orelse
  1845            member (type_aconv ctxt) maybe_finite_Ts T then
  1846           mono
  1847         else if is_type_surely_infinite' ctxt soundness surely_infinite_Ts
  1848                                          T then
  1849           {maybe_finite_Ts = maybe_finite_Ts,
  1850            surely_finite_Ts = surely_finite_Ts,
  1851            maybe_infinite_Ts = maybe_infinite_Ts,
  1852            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
  1853            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1854         else
  1855           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
  1856            surely_finite_Ts = surely_finite_Ts,
  1857            maybe_infinite_Ts = maybe_infinite_Ts,
  1858            surely_infinite_Ts = surely_infinite_Ts,
  1859            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1860       | Fin_Nonmono_Types =>
  1861         if exists (type_instance ctxt T) surely_finite_Ts orelse
  1862            member (type_aconv ctxt) maybe_infinite_Ts T then
  1863           mono
  1864         else if is_type_surely_finite ctxt T then
  1865           {maybe_finite_Ts = maybe_finite_Ts,
  1866            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
  1867            maybe_infinite_Ts = maybe_infinite_Ts,
  1868            surely_infinite_Ts = surely_infinite_Ts,
  1869            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1870         else
  1871           {maybe_finite_Ts = maybe_finite_Ts,
  1872            surely_finite_Ts = surely_finite_Ts,
  1873            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
  1874            surely_infinite_Ts = surely_infinite_Ts,
  1875            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1876       | _ => mono
  1877     else
  1878       mono
  1879   | add_iterm_mononotonicity_info _ _ _ _ mono = mono
  1880 fun add_fact_mononotonicity_info ctxt level
  1881         ({kind, iformula, ...} : translated_formula) =
  1882   formula_fold (SOME (kind <> Conjecture))
  1883                (add_iterm_mononotonicity_info ctxt level) iformula
  1884 fun mononotonicity_info_for_facts ctxt type_enc facts =
  1885   let val level = level_of_type_enc type_enc in
  1886     default_mono
  1887     |> is_type_level_monotonicity_based level
  1888        ? fold (add_fact_mononotonicity_info ctxt level) facts
  1889   end
  1890 
  1891 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  1892   Formula (guards_sym_formula_prefix ^
  1893            ascii_of (mangled_type format type_enc T),
  1894            Axiom,
  1895            IConst (`make_bound_var "X", T, [])
  1896            |> type_guard_iterm ctxt format type_enc T
  1897            |> AAtom
  1898            |> formula_from_iformula ctxt format mono type_enc
  1899                                     (K (K (K (K true)))) (SOME true)
  1900            |> bound_tvars type_enc (atyps_of T)
  1901            |> close_formula_universally,
  1902            isabelle_info introN, NONE)
  1903 
  1904 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  1905   let val x_var = ATerm (`make_bound_var "X", []) in
  1906     Formula (tags_sym_formula_prefix ^
  1907              ascii_of (mangled_type format type_enc T),
  1908              Axiom,
  1909              eq_formula type_enc (atyps_of T) false
  1910                         (tag_with_type ctxt format mono type_enc NONE T x_var)
  1911                         x_var,
  1912              isabelle_info simpN, NONE)
  1913   end
  1914 
  1915 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  1916   case type_enc of
  1917     Simple_Types _ => []
  1918   | Guards _ =>
  1919     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  1920   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
  1921 
  1922 fun decl_line_for_sym ctxt format mono type_enc s
  1923                       (s', T_args, T, pred_sym, ary, _) =
  1924   let
  1925     val (T_arg_Ts, level) =
  1926       case type_enc of
  1927         Simple_Types (_, level) => ([], level)
  1928       | _ => (replicate (length T_args) homo_infinite_type, No_Types)
  1929   in
  1930     Decl (sym_decl_prefix ^ s, (s, s'),
  1931           (T_arg_Ts ---> (T |> homogenized_type ctxt mono level ary))
  1932           |> ho_type_from_typ format type_enc pred_sym (length T_arg_Ts + ary))
  1933   end
  1934 
  1935 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
  1936                                      j (s', T_args, T, _, ary, in_conj) =
  1937   let
  1938     val (kind, maybe_negate) =
  1939       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1940       else (Axiom, I)
  1941     val (arg_Ts, res_T) = chop_fun ary T
  1942     val num_args = length arg_Ts
  1943     val bound_names =
  1944       1 upto num_args |> map (`I o make_bound_var o string_of_int)
  1945     val bounds =
  1946       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
  1947     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
  1948     fun should_keep_arg_type T =
  1949       sym_needs_arg_types andalso
  1950       should_guard_type ctxt mono type_enc (K true) T
  1951     val bound_Ts =
  1952       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  1953   in
  1954     Formula (guards_sym_formula_prefix ^ s ^
  1955              (if n > 1 then "_" ^ string_of_int j else ""), kind,
  1956              IConst ((s, s'), T, T_args)
  1957              |> fold (curry (IApp o swap)) bounds
  1958              |> type_guard_iterm ctxt format type_enc res_T
  1959              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
  1960              |> formula_from_iformula ctxt format mono type_enc
  1961                                       (K (K (K (K true)))) (SOME true)
  1962              |> n > 1 ? bound_tvars type_enc (atyps_of T)
  1963              |> close_formula_universally
  1964              |> maybe_negate,
  1965              isabelle_info introN, NONE)
  1966   end
  1967 
  1968 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
  1969         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  1970   let
  1971     val ident_base =
  1972       tags_sym_formula_prefix ^ s ^
  1973       (if n > 1 then "_" ^ string_of_int j else "")
  1974     val (kind, maybe_negate) =
  1975       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1976       else (Axiom, I)
  1977     val (arg_Ts, res_T) = chop_fun ary T
  1978     val bound_names =
  1979       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
  1980     val bounds = bound_names |> map (fn name => ATerm (name, []))
  1981     val cst = mk_aterm format type_enc (s, s') T_args
  1982     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
  1983     val should_encode =
  1984       should_encode_type ctxt mono (level_of_type_enc type_enc)
  1985     val tag_with = tag_with_type ctxt format mono type_enc NONE
  1986     val add_formula_for_res =
  1987       if should_encode res_T then
  1988         cons (Formula (ident_base ^ "_res", kind,
  1989                        eq (tag_with res_T (cst bounds)) (cst bounds),
  1990                        isabelle_info simpN, NONE))
  1991       else
  1992         I
  1993     fun add_formula_for_arg k =
  1994       let val arg_T = nth arg_Ts k in
  1995         if should_encode arg_T then
  1996           case chop k bounds of
  1997             (bounds1, bound :: bounds2) =>
  1998             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
  1999                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
  2000                               (cst bounds),
  2001                            isabelle_info simpN, NONE))
  2002           | _ => raise Fail "expected nonempty tail"
  2003         else
  2004           I
  2005       end
  2006   in
  2007     [] |> not pred_sym ? add_formula_for_res
  2008        |> Config.get ctxt type_tag_arguments
  2009           ? fold add_formula_for_arg (ary - 1 downto 0)
  2010   end
  2011 
  2012 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
  2013 
  2014 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
  2015                                 (s, decls) =
  2016   case type_enc of
  2017     Simple_Types _ =>
  2018     decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  2019   | Guards (_, level, _) =>
  2020     let
  2021       val decls =
  2022         case decls of
  2023           decl :: (decls' as _ :: _) =>
  2024           let val T = result_type_of_decl decl in
  2025             if forall (type_generalization ctxt T o result_type_of_decl)
  2026                       decls' then
  2027               [decl]
  2028             else
  2029               decls
  2030           end
  2031         | _ => decls
  2032       val n = length decls
  2033       val decls =
  2034         decls |> filter (should_encode_type ctxt mono level
  2035                          o result_type_of_decl)
  2036     in
  2037       (0 upto length decls - 1, decls)
  2038       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
  2039                                                  type_enc n s)
  2040     end
  2041   | Tags (_, _, uniformity) =>
  2042     (case uniformity of
  2043        Uniform => []
  2044      | Nonuniform =>
  2045        let val n = length decls in
  2046          (0 upto n - 1 ~~ decls)
  2047          |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
  2048                       conj_sym_kind mono type_enc n s)
  2049        end)
  2050 
  2051 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
  2052                                      sym_decl_tab =
  2053   let
  2054     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
  2055     val mono_Ts =
  2056       if polymorphism_of_type_enc type_enc = Polymorphic then
  2057         syms |> maps (map result_type_of_decl o snd)
  2058              |> filter_out (should_encode_type ctxt mono
  2059                                 (level_of_type_enc type_enc))
  2060              |> rpair [] |-> fold (insert_type ctxt I)
  2061       else
  2062         []
  2063     val mono_lines =
  2064       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
  2065     val decl_lines =
  2066       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
  2067                                                      mono type_enc)
  2068                syms []
  2069   in mono_lines @ decl_lines end
  2070 
  2071 fun needs_type_tag_idempotence ctxt (Tags (poly, level, uniformity)) =
  2072     Config.get ctxt type_tag_idempotence andalso
  2073     poly <> Mangled_Monomorphic andalso
  2074     ((level = All_Types andalso uniformity = Nonuniform) orelse
  2075      is_type_level_monotonicity_based level)
  2076   | needs_type_tag_idempotence _ _ = false
  2077 
  2078 fun offset_of_heading_in_problem _ [] j = j
  2079   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
  2080     if heading = needle then j
  2081     else offset_of_heading_in_problem needle problem (j + length lines)
  2082 
  2083 val implicit_declsN = "Should-be-implicit typings"
  2084 val explicit_declsN = "Explicit typings"
  2085 val factsN = "Relevant facts"
  2086 val class_relsN = "Class relationships"
  2087 val aritiesN = "Arities"
  2088 val helpersN = "Helper facts"
  2089 val conjsN = "Conjectures"
  2090 val free_typesN = "Type variables"
  2091 
  2092 val explicit_apply = NONE (* for experiments *)
  2093 
  2094 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
  2095         lambda_trans readable_names preproc hyp_ts concl_t facts =
  2096   let
  2097     val type_enc = type_enc |> adjust_type_enc format
  2098     val lambda_trans =
  2099       if lambda_trans = smartN then
  2100         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
  2101       else if lambda_trans = lambdasN andalso
  2102               not (is_type_enc_higher_order type_enc) then
  2103         error ("Lambda translation method incompatible with first-order \
  2104                \encoding.")
  2105       else
  2106         lambda_trans
  2107     val trans_lambdas =
  2108       if lambda_trans = no_lambdasN then
  2109         rpair []
  2110       else if lambda_trans = concealedN then
  2111         lift_lambdas ctxt type_enc ##> K []
  2112       else if lambda_trans = liftingN then
  2113         lift_lambdas ctxt type_enc
  2114       else if lambda_trans = combinatorsN then
  2115         map (introduce_combinators ctxt) #> rpair []
  2116       else if lambda_trans = hybridN then
  2117         lift_lambdas ctxt type_enc
  2118         ##> maps (fn t => [t, introduce_combinators ctxt
  2119                                   (intentionalize_def t)])
  2120       else if lambda_trans = lambdasN then
  2121         map (Envir.eta_contract) #> rpair []
  2122       else
  2123         error ("Unknown lambda translation method: " ^
  2124                quote lambda_trans ^ ".")
  2125     val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
  2126       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  2127                          hyp_ts concl_t facts
  2128     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
  2129     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
  2130     val repair = repair_fact ctxt format type_enc sym_tab
  2131     val (conjs, facts) = (conjs, facts) |> pairself (map repair)
  2132     val repaired_sym_tab =
  2133       conjs @ facts |> sym_table_for_facts ctxt (SOME false)
  2134     val helpers =
  2135       repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
  2136                        |> map repair
  2137     val sym_decl_lines =
  2138       (conjs, helpers @ facts)
  2139       |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  2140       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
  2141                                                type_enc
  2142     val helper_lines =
  2143       0 upto length helpers - 1 ~~ helpers
  2144       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
  2145                                     type_enc)
  2146       |> (if needs_type_tag_idempotence ctxt type_enc then
  2147             cons (type_tag_idempotence_fact type_enc)
  2148           else
  2149             I)
  2150     (* Reordering these might confuse the proof reconstruction code or the SPASS
  2151        FLOTTER hack. *)
  2152     val problem =
  2153       [(explicit_declsN, sym_decl_lines),
  2154        (factsN,
  2155         map (formula_line_for_fact ctxt format fact_prefix ascii_of
  2156                                    (not exporter) (not exporter) mono
  2157                                    type_enc)
  2158             (0 upto length facts - 1 ~~ facts)),
  2159        (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
  2160        (aritiesN, map formula_line_for_arity_clause arity_clauses),
  2161        (helpersN, helper_lines),
  2162        (conjsN,
  2163         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
  2164        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
  2165     val problem =
  2166       problem
  2167       |> (case format of
  2168             CNF => ensure_cnf_problem
  2169           | CNF_UEQ => filter_cnf_ueq_problem
  2170           | _ => I)
  2171       |> (if is_format_typed format then
  2172             declare_undeclared_syms_in_atp_problem type_decl_prefix
  2173                                                    implicit_declsN
  2174           else
  2175             I)
  2176     val (problem, pool) = problem |> nice_atp_problem readable_names
  2177     val helpers_offset = offset_of_heading_in_problem helpersN problem 0
  2178     val typed_helpers =
  2179       map_filter (fn (j, {name, ...}) =>
  2180                      if String.isSuffix typed_helper_suffix name then SOME j
  2181                      else NONE)
  2182                  ((helpers_offset + 1 upto helpers_offset + length helpers)
  2183                   ~~ helpers)
  2184     fun add_sym_arity (s, {min_ary, ...} : sym_info) =
  2185       if min_ary > 0 then
  2186         case strip_prefix_and_unascii const_prefix s of
  2187           SOME s => Symtab.insert (op =) (s, min_ary)
  2188         | NONE => I
  2189       else
  2190         I
  2191   in
  2192     (problem,
  2193      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
  2194      offset_of_heading_in_problem conjsN problem 0,
  2195      offset_of_heading_in_problem factsN problem 0,
  2196      fact_names |> Vector.fromList,
  2197      typed_helpers,
  2198      Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
  2199   end
  2200 
  2201 (* FUDGE *)
  2202 val conj_weight = 0.0
  2203 val hyp_weight = 0.1
  2204 val fact_min_weight = 0.2
  2205 val fact_max_weight = 1.0
  2206 val type_info_default_weight = 0.8
  2207 
  2208 fun add_term_weights weight (ATerm (s, tms)) =
  2209     is_tptp_user_symbol s ? Symtab.default (s, weight)
  2210     #> fold (add_term_weights weight) tms
  2211   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
  2212 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
  2213     formula_fold NONE (K (add_term_weights weight)) phi
  2214   | add_problem_line_weights _ _ = I
  2215 
  2216 fun add_conjectures_weights [] = I
  2217   | add_conjectures_weights conjs =
  2218     let val (hyps, conj) = split_last conjs in
  2219       add_problem_line_weights conj_weight conj
  2220       #> fold (add_problem_line_weights hyp_weight) hyps
  2221     end
  2222 
  2223 fun add_facts_weights facts =
  2224   let
  2225     val num_facts = length facts
  2226     fun weight_of j =
  2227       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
  2228                         / Real.fromInt num_facts
  2229   in
  2230     map weight_of (0 upto num_facts - 1) ~~ facts
  2231     |> fold (uncurry add_problem_line_weights)
  2232   end
  2233 
  2234 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
  2235 fun atp_problem_weights problem =
  2236   let val get = these o AList.lookup (op =) problem in
  2237     Symtab.empty
  2238     |> add_conjectures_weights (get free_typesN @ get conjsN)
  2239     |> add_facts_weights (get factsN)
  2240     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
  2241             [explicit_declsN, class_relsN, aritiesN]
  2242     |> Symtab.dest
  2243     |> sort (prod_ord Real.compare string_ord o pairself swap)
  2244   end
  2245 
  2246 end;