src/HOL/Tools/ATP/atp_translate.ML
author blanchet
Wed Aug 24 15:25:39 2011 +0200 (2011-08-24)
changeset 44463 c471a2b48fa1
parent 44460 5d0754cf994a
child 44491 ba22ed224b20
permissions -rw-r--r--
make sure that all facts are passed to ATP from minimizer
     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 | 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 no_lambdasN : string
    39   val concealedN : string
    40   val liftingN : string
    41   val combinatorsN : string
    42   val hybridN : string
    43   val lambdasN : string
    44   val smartN : string
    45   val schematic_var_prefix : string
    46   val fixed_var_prefix : string
    47   val tvar_prefix : string
    48   val tfree_prefix : string
    49   val const_prefix : string
    50   val type_const_prefix : string
    51   val class_prefix : string
    52   val polymorphic_free_prefix : string
    53   val skolem_const_prefix : string
    54   val old_skolem_const_prefix : string
    55   val new_skolem_const_prefix : string
    56   val type_decl_prefix : string
    57   val sym_decl_prefix : string
    58   val guards_sym_formula_prefix : string
    59   val tags_sym_formula_prefix : string
    60   val fact_prefix : string
    61   val conjecture_prefix : string
    62   val helper_prefix : string
    63   val class_rel_clause_prefix : string
    64   val arity_clause_prefix : string
    65   val tfree_clause_prefix : string
    66   val typed_helper_suffix : string
    67   val untyped_helper_suffix : string
    68   val type_tag_idempotence_helper_name : string
    69   val predicator_name : string
    70   val app_op_name : string
    71   val type_guard_name : string
    72   val type_tag_name : string
    73   val simple_type_prefix : string
    74   val prefixed_predicator_name : string
    75   val prefixed_app_op_name : string
    76   val prefixed_type_tag_name : string
    77   val ascii_of : string -> string
    78   val unascii_of : string -> string
    79   val strip_prefix_and_unascii : string -> string -> string option
    80   val proxy_table : (string * (string * (thm * (string * string)))) list
    81   val proxify_const : string -> (string * string) option
    82   val invert_const : string -> string
    83   val unproxify_const : string -> string
    84   val new_skolem_var_name_from_const : string -> string
    85   val atp_irrelevant_consts : string list
    86   val atp_schematic_consts_of : term -> typ list Symtab.table
    87   val type_enc_from_string : soundness -> string -> type_enc
    88   val is_type_enc_higher_order : type_enc -> bool
    89   val polymorphism_of_type_enc : type_enc -> polymorphism
    90   val level_of_type_enc : type_enc -> type_level
    91   val is_type_enc_quasi_sound : type_enc -> bool
    92   val is_type_enc_fairly_sound : type_enc -> bool
    93   val adjust_type_enc : format -> type_enc -> type_enc
    94   val mk_aconns :
    95     connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
    96   val unmangled_const : string -> string * (string, 'b) ho_term list
    97   val unmangled_const_name : string -> string
    98   val helper_table : ((string * bool) * thm list) list
    99   val factsN : string
   100   val prepare_atp_problem :
   101     Proof.context -> format -> formula_kind -> formula_kind -> type_enc
   102     -> bool -> string -> bool -> bool -> term list -> term
   103     -> ((string * locality) * term) list
   104     -> string problem * string Symtab.table * int * int
   105        * (string * locality) list vector * int list * int Symtab.table
   106   val atp_problem_weights : string problem -> (string * real) list
   107 end;
   108 
   109 structure ATP_Translate : ATP_TRANSLATE =
   110 struct
   111 
   112 open ATP_Util
   113 open ATP_Problem
   114 
   115 type name = string * string
   116 
   117 val no_lambdasN = "no_lambdas"
   118 val concealedN = "concealed"
   119 val liftingN = "lifting"
   120 val combinatorsN = "combinators"
   121 val hybridN = "hybrid"
   122 val lambdasN = "lambdas"
   123 val smartN = "smart"
   124 
   125 val generate_info = false (* experimental *)
   126 
   127 fun isabelle_info s =
   128   if generate_info then SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
   129   else NONE
   130 
   131 val introN = "intro"
   132 val elimN = "elim"
   133 val simpN = "simp"
   134 
   135 val bound_var_prefix = "B_"
   136 val all_bound_var_prefix = "BA_"
   137 val exist_bound_var_prefix = "BE_"
   138 val schematic_var_prefix = "V_"
   139 val fixed_var_prefix = "v_"
   140 val tvar_prefix = "T_"
   141 val tfree_prefix = "t_"
   142 val const_prefix = "c_"
   143 val type_const_prefix = "tc_"
   144 val class_prefix = "cl_"
   145 
   146 val polymorphic_free_prefix = "poly_free"
   147 
   148 val skolem_const_prefix = "ATP" ^ Long_Name.separator ^ "Sko"
   149 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
   150 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
   151 
   152 val type_decl_prefix = "ty_"
   153 val sym_decl_prefix = "sy_"
   154 val guards_sym_formula_prefix = "gsy_"
   155 val tags_sym_formula_prefix = "tsy_"
   156 val fact_prefix = "fact_"
   157 val conjecture_prefix = "conj_"
   158 val helper_prefix = "help_"
   159 val class_rel_clause_prefix = "clar_"
   160 val arity_clause_prefix = "arity_"
   161 val tfree_clause_prefix = "tfree_"
   162 
   163 val lambda_fact_prefix = "ATP.lambda_"
   164 val typed_helper_suffix = "_T"
   165 val untyped_helper_suffix = "_U"
   166 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"
   167 
   168 val predicator_name = "p"
   169 val app_op_name = "a"
   170 val type_guard_name = "g"
   171 val type_tag_name = "t"
   172 val simple_type_prefix = "ty_"
   173 
   174 val prefixed_predicator_name = const_prefix ^ predicator_name
   175 val prefixed_app_op_name = const_prefix ^ app_op_name
   176 val prefixed_type_tag_name = const_prefix ^ type_tag_name
   177 
   178 (* Freshness almost guaranteed! *)
   179 val atp_weak_prefix = "ATP:"
   180 
   181 (*Escaping of special characters.
   182   Alphanumeric characters are left unchanged.
   183   The character _ goes to __
   184   Characters in the range ASCII space to / go to _A to _P, respectively.
   185   Other characters go to _nnn where nnn is the decimal ASCII code.*)
   186 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
   187 
   188 fun stringN_of_int 0 _ = ""
   189   | stringN_of_int k n =
   190     stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
   191 
   192 fun ascii_of_char c =
   193   if Char.isAlphaNum c then
   194     String.str c
   195   else if c = #"_" then
   196     "__"
   197   else if #" " <= c andalso c <= #"/" then
   198     "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
   199   else
   200     (* fixed width, in case more digits follow *)
   201     "_" ^ stringN_of_int 3 (Char.ord c)
   202 
   203 val ascii_of = String.translate ascii_of_char
   204 
   205 (** Remove ASCII armoring from names in proof files **)
   206 
   207 (* We don't raise error exceptions because this code can run inside a worker
   208    thread. Also, the errors are impossible. *)
   209 val unascii_of =
   210   let
   211     fun un rcs [] = String.implode(rev rcs)
   212       | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
   213         (* Three types of _ escapes: __, _A to _P, _nnn *)
   214       | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
   215       | un rcs (#"_" :: c :: cs) =
   216         if #"A" <= c andalso c<= #"P" then
   217           (* translation of #" " to #"/" *)
   218           un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
   219         else
   220           let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
   221             case Int.fromString (String.implode digits) of
   222               SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
   223             | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
   224           end
   225       | un rcs (c :: cs) = un (c :: rcs) cs
   226   in un [] o String.explode end
   227 
   228 (* If string s has the prefix s1, return the result of deleting it,
   229    un-ASCII'd. *)
   230 fun strip_prefix_and_unascii s1 s =
   231   if String.isPrefix s1 s then
   232     SOME (unascii_of (String.extract (s, size s1, NONE)))
   233   else
   234     NONE
   235 
   236 val proxy_table =
   237   [("c_False", (@{const_name False}, (@{thm fFalse_def},
   238        ("fFalse", @{const_name ATP.fFalse})))),
   239    ("c_True", (@{const_name True}, (@{thm fTrue_def},
   240        ("fTrue", @{const_name ATP.fTrue})))),
   241    ("c_Not", (@{const_name Not}, (@{thm fNot_def},
   242        ("fNot", @{const_name ATP.fNot})))),
   243    ("c_conj", (@{const_name conj}, (@{thm fconj_def},
   244        ("fconj", @{const_name ATP.fconj})))),
   245    ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
   246        ("fdisj", @{const_name ATP.fdisj})))),
   247    ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
   248        ("fimplies", @{const_name ATP.fimplies})))),
   249    ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
   250        ("fequal", @{const_name ATP.fequal})))),
   251    ("c_All", (@{const_name All}, (@{thm fAll_def},
   252        ("fAll", @{const_name ATP.fAll})))),
   253    ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
   254        ("fEx", @{const_name ATP.fEx}))))]
   255 
   256 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
   257 
   258 (* Readable names for the more common symbolic functions. Do not mess with the
   259    table unless you know what you are doing. *)
   260 val const_trans_table =
   261   [(@{type_name Product_Type.prod}, "prod"),
   262    (@{type_name Sum_Type.sum}, "sum"),
   263    (@{const_name False}, "False"),
   264    (@{const_name True}, "True"),
   265    (@{const_name Not}, "Not"),
   266    (@{const_name conj}, "conj"),
   267    (@{const_name disj}, "disj"),
   268    (@{const_name implies}, "implies"),
   269    (@{const_name HOL.eq}, "equal"),
   270    (@{const_name All}, "All"),
   271    (@{const_name Ex}, "Ex"),
   272    (@{const_name If}, "If"),
   273    (@{const_name Set.member}, "member"),
   274    (@{const_name Meson.COMBI}, "COMBI"),
   275    (@{const_name Meson.COMBK}, "COMBK"),
   276    (@{const_name Meson.COMBB}, "COMBB"),
   277    (@{const_name Meson.COMBC}, "COMBC"),
   278    (@{const_name Meson.COMBS}, "COMBS")]
   279   |> Symtab.make
   280   |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
   281 
   282 (* Invert the table of translations between Isabelle and ATPs. *)
   283 val const_trans_table_inv =
   284   const_trans_table |> Symtab.dest |> map swap |> Symtab.make
   285 val const_trans_table_unprox =
   286   Symtab.empty
   287   |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
   288 
   289 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
   290 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
   291 
   292 fun lookup_const c =
   293   case Symtab.lookup const_trans_table c of
   294     SOME c' => c'
   295   | NONE => ascii_of c
   296 
   297 fun ascii_of_indexname (v, 0) = ascii_of v
   298   | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
   299 
   300 fun make_bound_var x = bound_var_prefix ^ ascii_of x
   301 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
   302 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
   303 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
   304 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
   305 
   306 fun make_schematic_type_var (x, i) =
   307       tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
   308 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
   309 
   310 (* "HOL.eq" is mapped to the ATP's equality. *)
   311 fun make_fixed_const @{const_name HOL.eq} = tptp_old_equal
   312   | make_fixed_const c = const_prefix ^ lookup_const c
   313 
   314 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
   315 
   316 fun make_type_class clas = class_prefix ^ ascii_of clas
   317 
   318 fun new_skolem_var_name_from_const s =
   319   let val ss = s |> space_explode Long_Name.separator in
   320     nth ss (length ss - 2)
   321   end
   322 
   323 (* These are either simplified away by "Meson.presimplify" (most of the time) or
   324    handled specially via "fFalse", "fTrue", ..., "fequal". *)
   325 val atp_irrelevant_consts =
   326   [@{const_name False}, @{const_name True}, @{const_name Not},
   327    @{const_name conj}, @{const_name disj}, @{const_name implies},
   328    @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
   329 
   330 val atp_monomorph_bad_consts =
   331   atp_irrelevant_consts @
   332   (* These are ignored anyway by the relevance filter (unless they appear in
   333      higher-order places) but not by the monomorphizer. *)
   334   [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
   335    @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
   336    @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
   337 
   338 fun add_schematic_const (x as (_, T)) =
   339   Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
   340 val add_schematic_consts_of =
   341   Term.fold_aterms (fn Const (x as (s, _)) =>
   342                        not (member (op =) atp_monomorph_bad_consts s)
   343                        ? add_schematic_const x
   344                       | _ => I)
   345 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
   346 
   347 (** Definitions and functions for FOL clauses and formulas for TPTP **)
   348 
   349 (* The first component is the type class; the second is a "TVar" or "TFree". *)
   350 datatype type_literal =
   351   TyLitVar of name * name |
   352   TyLitFree of name * name
   353 
   354 
   355 (** Isabelle arities **)
   356 
   357 datatype arity_literal =
   358   TConsLit of name * name * name list |
   359   TVarLit of name * name
   360 
   361 fun gen_TVars 0 = []
   362   | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)
   363 
   364 val type_class = the_single @{sort type}
   365 
   366 fun add_packed_sort tvar =
   367   fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))
   368 
   369 type arity_clause =
   370   {name : string,
   371    prem_lits : arity_literal list,
   372    concl_lits : arity_literal}
   373 
   374 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
   375 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
   376   let
   377     val tvars = gen_TVars (length args)
   378     val tvars_srts = ListPair.zip (tvars, args)
   379   in
   380     {name = name,
   381      prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,
   382      concl_lits = TConsLit (`make_type_class cls,
   383                             `make_fixed_type_const tcons,
   384                             tvars ~~ tvars)}
   385   end
   386 
   387 fun arity_clause _ _ (_, []) = []
   388   | arity_clause seen n (tcons, ("HOL.type", _) :: ars) =  (* ignore *)
   389     arity_clause seen n (tcons, ars)
   390   | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
   391     if member (op =) seen class then
   392       (* multiple arities for the same (tycon, class) pair *)
   393       make_axiom_arity_clause (tcons,
   394           lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
   395           ar) ::
   396       arity_clause seen (n + 1) (tcons, ars)
   397     else
   398       make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
   399                                ascii_of class, ar) ::
   400       arity_clause (class :: seen) n (tcons, ars)
   401 
   402 fun multi_arity_clause [] = []
   403   | multi_arity_clause ((tcons, ars) :: tc_arlists) =
   404       arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
   405 
   406 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
   407    theory thy provided its arguments have the corresponding sorts. *)
   408 fun type_class_pairs thy tycons classes =
   409   let
   410     val alg = Sign.classes_of thy
   411     fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
   412     fun add_class tycon class =
   413       cons (class, domain_sorts tycon class)
   414       handle Sorts.CLASS_ERROR _ => I
   415     fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
   416   in map try_classes tycons end
   417 
   418 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
   419 fun iter_type_class_pairs _ _ [] = ([], [])
   420   | iter_type_class_pairs thy tycons classes =
   421       let
   422         fun maybe_insert_class s =
   423           (s <> type_class andalso not (member (op =) classes s))
   424           ? insert (op =) s
   425         val cpairs = type_class_pairs thy tycons classes
   426         val newclasses =
   427           [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
   428         val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
   429       in (classes' @ classes, union (op =) cpairs' cpairs) end
   430 
   431 fun make_arity_clauses thy tycons =
   432   iter_type_class_pairs thy tycons ##> multi_arity_clause
   433 
   434 
   435 (** Isabelle class relations **)
   436 
   437 type class_rel_clause =
   438   {name : string,
   439    subclass : name,
   440    superclass : name}
   441 
   442 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
   443    in theory "thy". *)
   444 fun class_pairs _ [] _ = []
   445   | class_pairs thy subs supers =
   446       let
   447         val class_less = Sorts.class_less (Sign.classes_of thy)
   448         fun add_super sub super = class_less (sub, super) ? cons (sub, super)
   449         fun add_supers sub = fold (add_super sub) supers
   450       in fold add_supers subs [] end
   451 
   452 fun make_class_rel_clause (sub, super) =
   453   {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
   454    superclass = `make_type_class super}
   455 
   456 fun make_class_rel_clauses thy subs supers =
   457   map make_class_rel_clause (class_pairs thy subs supers)
   458 
   459 (* intermediate terms *)
   460 datatype iterm =
   461   IConst of name * typ * typ list |
   462   IVar of name * typ |
   463   IApp of iterm * iterm |
   464   IAbs of (name * typ) * iterm
   465 
   466 fun ityp_of (IConst (_, T, _)) = T
   467   | ityp_of (IVar (_, T)) = T
   468   | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
   469   | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
   470 
   471 (*gets the head of a combinator application, along with the list of arguments*)
   472 fun strip_iterm_comb u =
   473   let
   474     fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
   475       | stripc x = x
   476   in stripc (u, []) end
   477 
   478 fun atyps_of T = fold_atyps (insert (op =)) T []
   479 
   480 fun new_skolem_const_name s num_T_args =
   481   [new_skolem_const_prefix, s, string_of_int num_T_args]
   482   |> space_implode Long_Name.separator
   483 
   484 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
   485    Also accumulates sort infomation. *)
   486 fun iterm_from_term thy bs (P $ Q) =
   487     let
   488       val (P', P_atomics_Ts) = iterm_from_term thy bs P
   489       val (Q', Q_atomics_Ts) = iterm_from_term thy bs Q
   490     in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
   491   | iterm_from_term thy _ (Const (c, T)) =
   492     (IConst (`make_fixed_const c, T,
   493              if String.isPrefix old_skolem_const_prefix c then
   494                [] |> Term.add_tvarsT T |> map TVar
   495              else
   496                (c, T) |> Sign.const_typargs thy),
   497      atyps_of T)
   498   | iterm_from_term _ _ (Free (s, T)) =
   499     (IConst (`make_fixed_var s, T,
   500              if String.isPrefix polymorphic_free_prefix s then [T] else []),
   501      atyps_of T)
   502   | iterm_from_term _ _ (Var (v as (s, _), T)) =
   503     (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
   504        let
   505          val Ts = T |> strip_type |> swap |> op ::
   506          val s' = new_skolem_const_name s (length Ts)
   507        in IConst (`make_fixed_const s', T, Ts) end
   508      else
   509        IVar ((make_schematic_var v, s), T), atyps_of T)
   510   | iterm_from_term _ bs (Bound j) =
   511     nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atyps_of T))
   512   | iterm_from_term thy bs (Abs (s, T, t)) =
   513     let
   514       fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
   515       val s = vary s
   516       val name = `make_bound_var s
   517       val (tm, atomic_Ts) = iterm_from_term thy ((s, (name, T)) :: bs) t
   518     in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end
   519 
   520 datatype locality =
   521   General | Helper | Extensionality | Intro | Elim | Simp | Local | Assum |
   522   Chained
   523 
   524 datatype order = First_Order | Higher_Order
   525 datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
   526 datatype soundness = Unsound | Sound_Modulo_Infiniteness | Sound
   527 datatype type_level =
   528   All_Types |
   529   Noninf_Nonmono_Types of soundness |
   530   Fin_Nonmono_Types |
   531   Const_Arg_Types |
   532   No_Types
   533 datatype type_uniformity = Uniform | Nonuniform
   534 
   535 datatype type_enc =
   536   Simple_Types of order * type_level |
   537   Guards of polymorphism * type_level * type_uniformity |
   538   Tags of polymorphism * type_level * type_uniformity
   539 
   540 fun try_unsuffixes ss s =
   541   fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
   542 
   543 fun type_enc_from_string soundness s =
   544   (case try (unprefix "poly_") s of
   545      SOME s => (SOME Polymorphic, s)
   546    | NONE =>
   547      case try (unprefix "mono_") s of
   548        SOME s => (SOME Monomorphic, s)
   549      | NONE =>
   550        case try (unprefix "mangled_") s of
   551          SOME s => (SOME Mangled_Monomorphic, s)
   552        | NONE => (NONE, s))
   553   ||> (fn s =>
   554           (* "_query" and "_bang" are for the ASCII-challenged Metis and
   555              Mirabelle. *)
   556           case try_unsuffixes ["?", "_query"] s of
   557             SOME s => (Noninf_Nonmono_Types soundness, s)
   558           | NONE =>
   559             case try_unsuffixes ["!", "_bang"] s of
   560               SOME s => (Fin_Nonmono_Types, s)
   561             | NONE => (All_Types, s))
   562   ||> apsnd (fn s =>
   563                 case try (unsuffix "_uniform") s of
   564                   SOME s => (Uniform, s)
   565                 | NONE => (Nonuniform, s))
   566   |> (fn (poly, (level, (uniformity, core))) =>
   567          case (core, (poly, level, uniformity)) of
   568            ("simple", (NONE, _, Nonuniform)) =>
   569            Simple_Types (First_Order, level)
   570          | ("simple_higher", (NONE, _, Nonuniform)) =>
   571            (case level of
   572               Noninf_Nonmono_Types _ => raise Same.SAME
   573             | _ => Simple_Types (Higher_Order, level))
   574          | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)
   575          | ("tags", (SOME Polymorphic, _, _)) =>
   576            Tags (Polymorphic, level, uniformity)
   577          | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)
   578          | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>
   579            Guards (poly, Const_Arg_Types, Nonuniform)
   580          | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>
   581            Guards (Polymorphic, No_Types, Nonuniform)
   582          | _ => raise Same.SAME)
   583   handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
   584 
   585 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _)) = true
   586   | is_type_enc_higher_order _ = false
   587 
   588 fun polymorphism_of_type_enc (Simple_Types _) = Mangled_Monomorphic
   589   | polymorphism_of_type_enc (Guards (poly, _, _)) = poly
   590   | polymorphism_of_type_enc (Tags (poly, _, _)) = poly
   591 
   592 fun level_of_type_enc (Simple_Types (_, level)) = level
   593   | level_of_type_enc (Guards (_, level, _)) = level
   594   | level_of_type_enc (Tags (_, level, _)) = level
   595 
   596 fun uniformity_of_type_enc (Simple_Types _) = Uniform
   597   | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity
   598   | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity
   599 
   600 fun is_type_level_quasi_sound All_Types = true
   601   | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
   602   | is_type_level_quasi_sound _ = false
   603 val is_type_enc_quasi_sound =
   604   is_type_level_quasi_sound o level_of_type_enc
   605 
   606 fun is_type_level_fairly_sound level =
   607   is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types
   608 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
   609 
   610 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
   611   | is_type_level_monotonicity_based Fin_Nonmono_Types = true
   612   | is_type_level_monotonicity_based _ = false
   613 
   614 fun adjust_type_enc (THF _) type_enc = type_enc
   615   | adjust_type_enc TFF (Simple_Types (_, level)) =
   616     Simple_Types (First_Order, level)
   617   | adjust_type_enc format (Simple_Types (_, level)) =
   618     adjust_type_enc format (Guards (Mangled_Monomorphic, level, Uniform))
   619   | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
   620     (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
   621   | adjust_type_enc _ type_enc = type_enc
   622 
   623 fun lift_lambdas ctxt type_enc =
   624   map (close_form o Envir.eta_contract) #> rpair ctxt
   625   #-> Lambda_Lifting.lift_lambdas
   626           (if polymorphism_of_type_enc type_enc = Polymorphic then
   627              SOME polymorphic_free_prefix
   628            else
   629              NONE)
   630           Lambda_Lifting.is_quantifier
   631   #> fst
   632 
   633 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
   634     intentionalize_def t
   635   | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
   636     let
   637       fun lam T t = Abs (Name.uu, T, t)
   638       val (head, args) = strip_comb t ||> rev
   639       val head_T = fastype_of head
   640       val n = length args
   641       val arg_Ts = head_T |> binder_types |> take n |> rev
   642       val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
   643     in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
   644   | intentionalize_def t = t
   645 
   646 type translated_formula =
   647   {name : string,
   648    locality : locality,
   649    kind : formula_kind,
   650    iformula : (name, typ, iterm) formula,
   651    atomic_types : typ list}
   652 
   653 fun update_iformula f ({name, locality, kind, iformula, atomic_types}
   654                        : translated_formula) =
   655   {name = name, locality = locality, kind = kind, iformula = f iformula,
   656    atomic_types = atomic_types} : translated_formula
   657 
   658 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
   659 
   660 fun insert_type ctxt get_T x xs =
   661   let val T = get_T x in
   662     if exists (type_instance ctxt T o get_T) xs then xs
   663     else x :: filter_out (type_generalization ctxt T o get_T) xs
   664   end
   665 
   666 (* The Booleans indicate whether all type arguments should be kept. *)
   667 datatype type_arg_policy =
   668   Explicit_Type_Args of bool |
   669   Mangled_Type_Args of bool |
   670   No_Type_Args
   671 
   672 fun should_drop_arg_type_args (Simple_Types _) =
   673     false (* since TFF doesn't support overloading *)
   674   | should_drop_arg_type_args type_enc =
   675     level_of_type_enc type_enc = All_Types andalso
   676     uniformity_of_type_enc type_enc = Uniform
   677 
   678 fun type_arg_policy type_enc s =
   679   if s = type_tag_name then
   680     (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   681        Mangled_Type_Args
   682      else
   683        Explicit_Type_Args) false
   684   else case type_enc of
   685     Tags (_, All_Types, Uniform) => No_Type_Args
   686   | _ =>
   687     let val level = level_of_type_enc type_enc in
   688       if level = No_Types orelse s = @{const_name HOL.eq} orelse
   689          (s = app_op_name andalso level = Const_Arg_Types) then
   690         No_Type_Args
   691       else
   692         should_drop_arg_type_args type_enc
   693         |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
   694               Mangled_Type_Args
   695             else
   696               Explicit_Type_Args)
   697     end
   698 
   699 (* Make literals for sorted type variables. *)
   700 fun generic_add_sorts_on_type (_, []) = I
   701   | generic_add_sorts_on_type ((x, i), s :: ss) =
   702     generic_add_sorts_on_type ((x, i), ss)
   703     #> (if s = the_single @{sort HOL.type} then
   704           I
   705         else if i = ~1 then
   706           insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))
   707         else
   708           insert (op =) (TyLitVar (`make_type_class s,
   709                                    (make_schematic_type_var (x, i), x))))
   710 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
   711   | add_sorts_on_tfree _ = I
   712 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
   713   | add_sorts_on_tvar _ = I
   714 
   715 fun type_literals_for_types type_enc add_sorts_on_typ Ts =
   716   [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
   717 
   718 fun mk_aconns c phis =
   719   let val (phis', phi') = split_last phis in
   720     fold_rev (mk_aconn c) phis' phi'
   721   end
   722 fun mk_ahorn [] phi = phi
   723   | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
   724 fun mk_aquant _ [] phi = phi
   725   | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
   726     if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
   727   | mk_aquant q xs phi = AQuant (q, xs, phi)
   728 
   729 fun close_universally atom_vars phi =
   730   let
   731     fun formula_vars bounds (AQuant (_, xs, phi)) =
   732         formula_vars (map fst xs @ bounds) phi
   733       | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
   734       | formula_vars bounds (AAtom tm) =
   735         union (op =) (atom_vars tm []
   736                       |> filter_out (member (op =) bounds o fst))
   737   in mk_aquant AForall (formula_vars [] phi []) phi end
   738 
   739 fun iterm_vars (IApp (tm1, tm2)) = fold iterm_vars [tm1, tm2]
   740   | iterm_vars (IConst _) = I
   741   | iterm_vars (IVar (name, T)) = insert (op =) (name, SOME T)
   742   | iterm_vars (IAbs (_, tm)) = iterm_vars tm
   743 fun close_iformula_universally phi = close_universally iterm_vars phi
   744 
   745 fun term_vars bounds (ATerm (name as (s, _), tms)) =
   746     (is_tptp_variable s andalso not (member (op =) bounds name))
   747     ? insert (op =) (name, NONE) #> fold (term_vars bounds) tms
   748   | term_vars bounds (AAbs ((name, _), tm)) = term_vars (name :: bounds) tm
   749 fun close_formula_universally phi = close_universally (term_vars []) phi
   750 
   751 val homo_infinite_type_name = @{type_name ind} (* any infinite type *)
   752 val homo_infinite_type = Type (homo_infinite_type_name, [])
   753 
   754 fun ho_term_from_typ format type_enc =
   755   let
   756     fun term (Type (s, Ts)) =
   757       ATerm (case (is_type_enc_higher_order type_enc, s) of
   758                (true, @{type_name bool}) => `I tptp_bool_type
   759              | (true, @{type_name fun}) => `I tptp_fun_type
   760              | _ => if s = homo_infinite_type_name andalso
   761                        is_format_typed format then
   762                       `I tptp_individual_type
   763                     else
   764                       `make_fixed_type_const s,
   765              map term Ts)
   766     | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
   767     | term (TVar ((x as (s, _)), _)) =
   768       ATerm ((make_schematic_type_var x, s), [])
   769   in term end
   770 
   771 fun ho_term_for_type_arg format type_enc T =
   772   if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)
   773 
   774 (* This shouldn't clash with anything else. *)
   775 val mangled_type_sep = "\000"
   776 
   777 fun generic_mangled_type_name f (ATerm (name, [])) = f name
   778   | generic_mangled_type_name f (ATerm (name, tys)) =
   779     f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
   780     ^ ")"
   781   | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
   782 
   783 fun mangled_type format type_enc =
   784   generic_mangled_type_name fst o ho_term_from_typ format type_enc
   785 
   786 val bool_atype = AType (`I tptp_bool_type)
   787 
   788 fun make_simple_type s =
   789   if s = tptp_bool_type orelse s = tptp_fun_type orelse
   790      s = tptp_individual_type then
   791     s
   792   else
   793     simple_type_prefix ^ ascii_of s
   794 
   795 fun ho_type_from_ho_term type_enc pred_sym ary =
   796   let
   797     fun to_atype ty =
   798       AType ((make_simple_type (generic_mangled_type_name fst ty),
   799               generic_mangled_type_name snd ty))
   800     fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
   801     fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
   802       | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
   803       | to_fo _ _ = raise Fail "unexpected type abstraction"
   804     fun to_ho (ty as ATerm ((s, _), tys)) =
   805         if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
   806       | to_ho _ = raise Fail "unexpected type abstraction"
   807   in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
   808 
   809 fun ho_type_from_typ format type_enc pred_sym ary =
   810   ho_type_from_ho_term type_enc pred_sym ary
   811   o ho_term_from_typ format type_enc
   812 
   813 fun mangled_const_name format type_enc T_args (s, s') =
   814   let
   815     val ty_args = T_args |> map_filter (ho_term_for_type_arg format type_enc)
   816     fun type_suffix f g =
   817       fold_rev (curry (op ^) o g o prefix mangled_type_sep
   818                 o generic_mangled_type_name f) ty_args ""
   819   in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
   820 
   821 val parse_mangled_ident =
   822   Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
   823 
   824 fun parse_mangled_type x =
   825   (parse_mangled_ident
   826    -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
   827                     [] >> ATerm) x
   828 and parse_mangled_types x =
   829   (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
   830 
   831 fun unmangled_type s =
   832   s |> suffix ")" |> raw_explode
   833     |> Scan.finite Symbol.stopper
   834            (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
   835                                                 quote s)) parse_mangled_type))
   836     |> fst
   837 
   838 val unmangled_const_name = space_explode mangled_type_sep #> hd
   839 fun unmangled_const s =
   840   let val ss = space_explode mangled_type_sep s in
   841     (hd ss, map unmangled_type (tl ss))
   842   end
   843 
   844 fun introduce_proxies type_enc =
   845   let
   846     fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
   847       | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
   848                        _ =
   849         (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
   850            limitation. This works in conjuction with special code in
   851            "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
   852            possible. *)
   853         IAbs ((`I "P", p_T),
   854               IApp (IConst (`I ho_quant, T, []),
   855                     IAbs ((`I "X", x_T),
   856                           IApp (IConst (`I "P", p_T, []),
   857                                 IConst (`I "X", x_T, [])))))
   858       | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
   859     fun intro top_level args (IApp (tm1, tm2)) =
   860         IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
   861       | intro top_level args (IConst (name as (s, _), T, T_args)) =
   862         (case proxify_const s of
   863            SOME proxy_base =>
   864            if top_level orelse is_type_enc_higher_order type_enc then
   865              case (top_level, s) of
   866                (_, "c_False") => IConst (`I tptp_false, T, [])
   867              | (_, "c_True") => IConst (`I tptp_true, T, [])
   868              | (false, "c_Not") => IConst (`I tptp_not, T, [])
   869              | (false, "c_conj") => IConst (`I tptp_and, T, [])
   870              | (false, "c_disj") => IConst (`I tptp_or, T, [])
   871              | (false, "c_implies") => IConst (`I tptp_implies, T, [])
   872              | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
   873              | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
   874              | (false, s) =>
   875                if is_tptp_equal s andalso length args = 2 then
   876                  IConst (`I tptp_equal, T, [])
   877                else
   878                  (* Use a proxy even for partially applied THF equality, because
   879                     the LEO-II and Satallax parsers complain about not being
   880                     able to infer the type of "=". *)
   881                  IConst (proxy_base |>> prefix const_prefix, T, T_args)
   882              | _ => IConst (name, T, [])
   883            else
   884              IConst (proxy_base |>> prefix const_prefix, T, T_args)
   885           | NONE => IConst (name, T, T_args))
   886       | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
   887       | intro _ _ tm = tm
   888   in intro true [] end
   889 
   890 fun iformula_from_prop thy type_enc eq_as_iff =
   891   let
   892     fun do_term bs t atomic_types =
   893       iterm_from_term thy bs (Envir.eta_contract t)
   894       |>> (introduce_proxies type_enc #> AAtom)
   895       ||> union (op =) atomic_types
   896     fun do_quant bs q pos s T t' =
   897       let
   898         val s = singleton (Name.variant_list (map fst bs)) s
   899         val universal = Option.map (q = AExists ? not) pos
   900         val name =
   901           s |> `(case universal of
   902                    SOME true => make_all_bound_var
   903                  | SOME false => make_exist_bound_var
   904                  | NONE => make_bound_var)
   905       in
   906         do_formula ((s, (name, T)) :: bs) pos t'
   907         #>> mk_aquant q [(name, SOME T)]
   908       end
   909     and do_conn bs c pos1 t1 pos2 t2 =
   910       do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
   911     and do_formula bs pos t =
   912       case t of
   913         @{const Trueprop} $ t1 => do_formula bs pos t1
   914       | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
   915       | Const (@{const_name All}, _) $ Abs (s, T, t') =>
   916         do_quant bs AForall pos s T t'
   917       | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
   918         do_quant bs AExists pos s T t'
   919       | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
   920       | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
   921       | @{const HOL.implies} $ t1 $ t2 =>
   922         do_conn bs AImplies (Option.map not pos) t1 pos t2
   923       | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
   924         if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
   925       | _ => do_term bs t
   926   in do_formula [] end
   927 
   928 fun presimplify_term _ [] t = t
   929   | presimplify_term ctxt presimp_consts t =
   930     t |> exists_Const (member (op =) presimp_consts o fst) t
   931          ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
   932             #> Meson.presimplify ctxt
   933             #> prop_of)
   934 
   935 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
   936 fun conceal_bounds Ts t =
   937   subst_bounds (map (Free o apfst concealed_bound_name)
   938                     (0 upto length Ts - 1 ~~ Ts), t)
   939 fun reveal_bounds Ts =
   940   subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
   941                     (0 upto length Ts - 1 ~~ Ts))
   942 
   943 fun is_fun_equality (@{const_name HOL.eq},
   944                      Type (_, [Type (@{type_name fun}, _), _])) = true
   945   | is_fun_equality _ = false
   946 
   947 fun extensionalize_term ctxt t =
   948   if exists_Const is_fun_equality t then
   949     let val thy = Proof_Context.theory_of ctxt in
   950       t |> cterm_of thy |> Meson.extensionalize_conv ctxt
   951         |> prop_of |> Logic.dest_equals |> snd
   952     end
   953   else
   954     t
   955 
   956 fun simple_translate_lambdas do_lambdas ctxt t =
   957   let val thy = Proof_Context.theory_of ctxt in
   958     if Meson.is_fol_term thy t then
   959       t
   960     else
   961       let
   962         fun aux Ts t =
   963           case t of
   964             @{const Not} $ t1 => @{const Not} $ aux Ts t1
   965           | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
   966             t0 $ Abs (s, T, aux (T :: Ts) t')
   967           | (t0 as Const (@{const_name All}, _)) $ t1 =>
   968             aux Ts (t0 $ eta_expand Ts t1 1)
   969           | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
   970             t0 $ Abs (s, T, aux (T :: Ts) t')
   971           | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
   972             aux Ts (t0 $ eta_expand Ts t1 1)
   973           | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   974           | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   975           | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
   976           | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
   977               $ t1 $ t2 =>
   978             t0 $ aux Ts t1 $ aux Ts t2
   979           | _ =>
   980             if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
   981             else t |> Envir.eta_contract |> do_lambdas ctxt Ts
   982         val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
   983       in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
   984   end
   985 
   986 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
   987     do_cheaply_conceal_lambdas Ts t1
   988     $ do_cheaply_conceal_lambdas Ts t2
   989   | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
   990     Free (polymorphic_free_prefix ^ serial_string (),
   991           T --> fastype_of1 (T :: Ts, t))
   992   | do_cheaply_conceal_lambdas _ t = t
   993 
   994 fun do_introduce_combinators ctxt Ts t =
   995   let val thy = Proof_Context.theory_of ctxt in
   996     t |> conceal_bounds Ts
   997       |> cterm_of thy
   998       |> Meson_Clausify.introduce_combinators_in_cterm
   999       |> prop_of |> Logic.dest_equals |> snd
  1000       |> reveal_bounds Ts
  1001   end
  1002   (* A type variable of sort "{}" will make abstraction fail. *)
  1003   handle THM _ => t |> do_cheaply_conceal_lambdas Ts
  1004 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
  1005 
  1006 fun preprocess_abstractions_in_terms trans_lambdas facts =
  1007   let
  1008     val (facts, lambda_ts) =
  1009       facts |> map (snd o snd) |> trans_lambdas 
  1010             |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
  1011     val lambda_facts =
  1012       map2 (fn t => fn j =>
  1013                ((lambda_fact_prefix ^ Int.toString j, Helper), (Axiom, t)))
  1014            lambda_ts (1 upto length lambda_ts)
  1015   in (facts, lambda_facts) end
  1016 
  1017 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
  1018    same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
  1019 fun freeze_term t =
  1020   let
  1021     fun aux (t $ u) = aux t $ aux u
  1022       | aux (Abs (s, T, t)) = Abs (s, T, aux t)
  1023       | aux (Var ((s, i), T)) =
  1024         Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
  1025       | aux t = t
  1026   in t |> exists_subterm is_Var t ? aux end
  1027 
  1028 fun presimp_prop ctxt presimp_consts t =
  1029   let
  1030     val thy = Proof_Context.theory_of ctxt
  1031     val t = t |> Envir.beta_eta_contract
  1032               |> transform_elim_prop
  1033               |> Object_Logic.atomize_term thy
  1034     val need_trueprop = (fastype_of t = @{typ bool})
  1035   in
  1036     t |> need_trueprop ? HOLogic.mk_Trueprop
  1037       |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
  1038       |> extensionalize_term ctxt
  1039       |> presimplify_term ctxt presimp_consts
  1040       |> perhaps (try (HOLogic.dest_Trueprop))
  1041   end
  1042 
  1043 (* making fact and conjecture formulas *)
  1044 fun make_formula thy type_enc eq_as_iff name loc kind t =
  1045   let
  1046     val (iformula, atomic_types) =
  1047       iformula_from_prop thy type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
  1048   in
  1049     {name = name, locality = loc, kind = kind, iformula = iformula,
  1050      atomic_types = atomic_types}
  1051   end
  1052 
  1053 fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
  1054   let val thy = Proof_Context.theory_of ctxt in
  1055     case t |> make_formula thy type_enc (eq_as_iff andalso format <> CNF) name
  1056                            loc Axiom of
  1057       formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
  1058       if s = tptp_true then NONE else SOME formula
  1059     | formula => SOME formula
  1060   end
  1061 
  1062 fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
  1063   | s_not_trueprop t = s_not t
  1064 
  1065 fun make_conjecture thy format type_enc =
  1066   map (fn ((name, loc), (kind, t)) =>
  1067           t |> kind = Conjecture ? s_not_trueprop
  1068             |> make_formula thy type_enc (format <> CNF) name loc kind)
  1069 
  1070 (** Finite and infinite type inference **)
  1071 
  1072 type monotonicity_info =
  1073   {maybe_finite_Ts : typ list,
  1074    surely_finite_Ts : typ list,
  1075    maybe_infinite_Ts : typ list,
  1076    surely_infinite_Ts : typ list,
  1077    maybe_nonmono_Ts : typ list}
  1078 
  1079 (* These types witness that the type classes they belong to allow infinite
  1080    models and hence that any types with these type classes is monotonic. *)
  1081 val known_infinite_types =
  1082   [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]
  1083 
  1084 fun is_type_surely_infinite' ctxt soundness cached_Ts T =
  1085   (* Unlike virtually any other polymorphic fact whose type variables can be
  1086      instantiated by a known infinite type, extensionality actually encodes a
  1087      cardinality constraints. *)
  1088   soundness <> Sound andalso
  1089   is_type_surely_infinite ctxt (soundness = Unsound) cached_Ts T
  1090 
  1091 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
  1092    dangerous because their "exhaust" properties can easily lead to unsound ATP
  1093    proofs. On the other hand, all HOL infinite types can be given the same
  1094    models in first-order logic (via Löwenheim-Skolem). *)
  1095 
  1096 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
  1097   | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
  1098                              maybe_nonmono_Ts, ...}
  1099                        (Noninf_Nonmono_Types soundness) T =
  1100     exists (type_instance ctxt T orf type_generalization ctxt T)
  1101            maybe_nonmono_Ts andalso
  1102     not (exists (type_instance ctxt T) surely_infinite_Ts orelse
  1103          (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
  1104           is_type_surely_infinite' ctxt soundness surely_infinite_Ts T))
  1105   | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
  1106                              maybe_nonmono_Ts, ...}
  1107                        Fin_Nonmono_Types T =
  1108     exists (type_instance ctxt T orf type_generalization ctxt T)
  1109            maybe_nonmono_Ts andalso
  1110     (exists (type_instance ctxt T) surely_finite_Ts orelse
  1111      (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
  1112       is_type_surely_finite ctxt T))
  1113   | should_encode_type _ _ _ _ = false
  1114 
  1115 fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var
  1116                       T =
  1117     (uniformity = Uniform orelse should_guard_var ()) andalso
  1118     should_encode_type ctxt mono level T
  1119   | should_guard_type _ _ _ _ _ = false
  1120 
  1121 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
  1122     String.isPrefix bound_var_prefix s orelse
  1123     String.isPrefix all_bound_var_prefix s
  1124   | is_maybe_universal_var (IVar _) = true
  1125   | is_maybe_universal_var _ = false
  1126 
  1127 datatype tag_site =
  1128   Top_Level of bool option |
  1129   Eq_Arg of bool option |
  1130   Elsewhere
  1131 
  1132 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
  1133   | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =
  1134     (case uniformity of
  1135        Uniform => should_encode_type ctxt mono level T
  1136      | Nonuniform =>
  1137        case (site, is_maybe_universal_var u) of
  1138          (Eq_Arg _, true) => should_encode_type ctxt mono level T
  1139        | _ => false)
  1140   | should_tag_with_type _ _ _ _ _ _ = false
  1141 
  1142 fun homogenized_type ctxt mono level =
  1143   let
  1144     val should_encode = should_encode_type ctxt mono level
  1145     fun homo 0 T = if should_encode T then T else homo_infinite_type
  1146       | homo ary (Type (@{type_name fun}, [T1, T2])) =
  1147         homo 0 T1 --> homo (ary - 1) T2
  1148       | homo _ _ = raise Fail "expected function type"
  1149   in homo end
  1150 
  1151 (** predicators and application operators **)
  1152 
  1153 type sym_info =
  1154   {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
  1155 
  1156 fun add_iterm_syms_to_table ctxt explicit_apply =
  1157   let
  1158     fun consider_var_arity const_T var_T max_ary =
  1159       let
  1160         fun iter ary T =
  1161           if ary = max_ary orelse type_instance ctxt var_T T orelse
  1162              type_instance ctxt T var_T then
  1163             ary
  1164           else
  1165             iter (ary + 1) (range_type T)
  1166       in iter 0 const_T end
  1167     fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1168       if explicit_apply = NONE andalso
  1169          (can dest_funT T orelse T = @{typ bool}) then
  1170         let
  1171           val bool_vars' = bool_vars orelse body_type T = @{typ bool}
  1172           fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
  1173             {pred_sym = pred_sym andalso not bool_vars',
  1174              min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
  1175              max_ary = max_ary, types = types}
  1176           val fun_var_Ts' =
  1177             fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
  1178         in
  1179           if bool_vars' = bool_vars andalso
  1180              pointer_eq (fun_var_Ts', fun_var_Ts) then
  1181             accum
  1182           else
  1183             ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
  1184         end
  1185       else
  1186         accum
  1187     fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
  1188       let val (head, args) = strip_iterm_comb tm in
  1189         (case head of
  1190            IConst ((s, _), T, _) =>
  1191            if String.isPrefix bound_var_prefix s orelse
  1192               String.isPrefix all_bound_var_prefix s then
  1193              add_universal_var T accum
  1194            else if String.isPrefix exist_bound_var_prefix s then
  1195              accum
  1196            else
  1197              let val ary = length args in
  1198                ((bool_vars, fun_var_Ts),
  1199                 case Symtab.lookup sym_tab s of
  1200                   SOME {pred_sym, min_ary, max_ary, types} =>
  1201                   let
  1202                     val pred_sym =
  1203                       pred_sym andalso top_level andalso not bool_vars
  1204                     val types' = types |> insert_type ctxt I T
  1205                     val min_ary =
  1206                       if is_some explicit_apply orelse
  1207                          pointer_eq (types', types) then
  1208                         min_ary
  1209                       else
  1210                         fold (consider_var_arity T) fun_var_Ts min_ary
  1211                   in
  1212                     Symtab.update (s, {pred_sym = pred_sym,
  1213                                        min_ary = Int.min (ary, min_ary),
  1214                                        max_ary = Int.max (ary, max_ary),
  1215                                        types = types'})
  1216                                   sym_tab
  1217                   end
  1218                 | NONE =>
  1219                   let
  1220                     val pred_sym = top_level andalso not bool_vars
  1221                     val min_ary =
  1222                       case explicit_apply of
  1223                         SOME true => 0
  1224                       | SOME false => ary
  1225                       | NONE => fold (consider_var_arity T) fun_var_Ts ary
  1226                   in
  1227                     Symtab.update_new (s, {pred_sym = pred_sym,
  1228                                            min_ary = min_ary, max_ary = ary,
  1229                                            types = [T]})
  1230                                       sym_tab
  1231                   end)
  1232              end
  1233          | IVar (_, T) => add_universal_var T accum
  1234          | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
  1235          | _ => accum)
  1236         |> fold (add false) args
  1237       end
  1238   in add true end
  1239 fun add_fact_syms_to_table ctxt explicit_apply =
  1240   formula_fold NONE (K (add_iterm_syms_to_table ctxt explicit_apply))
  1241   |> fact_lift
  1242 
  1243 val tvar_a = TVar (("'a", 0), HOLogic.typeS)
  1244 
  1245 val default_sym_tab_entries : (string * sym_info) list =
  1246   (prefixed_predicator_name,
  1247    {pred_sym = true, min_ary = 1, max_ary = 1, types = []}) ::
  1248   (make_fixed_const @{const_name undefined},
  1249    {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
  1250   ([tptp_false, tptp_true]
  1251    |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
  1252   ([tptp_equal, tptp_old_equal]
  1253    |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
  1254 
  1255 fun sym_table_for_facts ctxt explicit_apply facts =
  1256   ((false, []), Symtab.empty)
  1257   |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
  1258   |> fold Symtab.update default_sym_tab_entries
  1259 
  1260 fun min_arity_of sym_tab s =
  1261   case Symtab.lookup sym_tab s of
  1262     SOME ({min_ary, ...} : sym_info) => min_ary
  1263   | NONE =>
  1264     case strip_prefix_and_unascii const_prefix s of
  1265       SOME s =>
  1266       let val s = s |> unmangled_const_name |> invert_const in
  1267         if s = predicator_name then 1
  1268         else if s = app_op_name then 2
  1269         else if s = type_guard_name then 1
  1270         else 0
  1271       end
  1272     | NONE => 0
  1273 
  1274 (* True if the constant ever appears outside of the top-level position in
  1275    literals, or if it appears with different arities (e.g., because of different
  1276    type instantiations). If false, the constant always receives all of its
  1277    arguments and is used as a predicate. *)
  1278 fun is_pred_sym sym_tab s =
  1279   case Symtab.lookup sym_tab s of
  1280     SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
  1281     pred_sym andalso min_ary = max_ary
  1282   | NONE => false
  1283 
  1284 val predicator_combconst =
  1285   IConst (`make_fixed_const predicator_name, @{typ "bool => bool"}, [])
  1286 fun predicator tm = IApp (predicator_combconst, tm)
  1287 
  1288 fun introduce_predicators_in_iterm sym_tab tm =
  1289   case strip_iterm_comb tm of
  1290     (IConst ((s, _), _, _), _) =>
  1291     if is_pred_sym sym_tab s then tm else predicator tm
  1292   | _ => predicator tm
  1293 
  1294 fun list_app head args = fold (curry (IApp o swap)) args head
  1295 
  1296 val app_op = `make_fixed_const app_op_name
  1297 
  1298 fun explicit_app arg head =
  1299   let
  1300     val head_T = ityp_of head
  1301     val (arg_T, res_T) = dest_funT head_T
  1302     val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
  1303   in list_app explicit_app [head, arg] end
  1304 fun list_explicit_app head args = fold explicit_app args head
  1305 
  1306 fun introduce_explicit_apps_in_iterm sym_tab =
  1307   let
  1308     fun aux tm =
  1309       case strip_iterm_comb tm of
  1310         (head as IConst ((s, _), _, _), args) =>
  1311         args |> map aux
  1312              |> chop (min_arity_of sym_tab s)
  1313              |>> list_app head
  1314              |-> list_explicit_app
  1315       | (head, args) => list_explicit_app head (map aux args)
  1316   in aux end
  1317 
  1318 fun chop_fun 0 T = ([], T)
  1319   | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
  1320     chop_fun (n - 1) ran_T |>> cons dom_T
  1321   | chop_fun _ _ = raise Fail "unexpected non-function"
  1322 
  1323 fun filter_type_args _ _ _ [] = []
  1324   | filter_type_args thy s arity T_args =
  1325     let
  1326       (* will throw "TYPE" for pseudo-constants *)
  1327       val U = if s = app_op_name then
  1328                 @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global
  1329               else
  1330                 s |> Sign.the_const_type thy
  1331     in
  1332       case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of
  1333         [] => []
  1334       | res_U_vars =>
  1335         let val U_args = (s, U) |> Sign.const_typargs thy in
  1336           U_args ~~ T_args
  1337           |> map (fn (U, T) =>
  1338                      if member (op =) res_U_vars (dest_TVar U) then T
  1339                      else dummyT)
  1340         end
  1341     end
  1342     handle TYPE _ => T_args
  1343 
  1344 fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
  1345   let
  1346     val thy = Proof_Context.theory_of ctxt
  1347     fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
  1348       | aux arity (IConst (name as (s, _), T, T_args)) =
  1349         (case strip_prefix_and_unascii const_prefix s of
  1350            NONE =>
  1351            (name, if level_of_type_enc type_enc = No_Types then [] else T_args)
  1352          | SOME s'' =>
  1353            let
  1354              val s'' = invert_const s''
  1355              fun filtered_T_args false = T_args
  1356                | filtered_T_args true = filter_type_args thy s'' arity T_args
  1357            in
  1358              case type_arg_policy type_enc s'' of
  1359                Explicit_Type_Args drop_args =>
  1360                (name, filtered_T_args drop_args)
  1361              | Mangled_Type_Args drop_args =>
  1362                (mangled_const_name format type_enc (filtered_T_args drop_args)
  1363                                    name, [])
  1364              | No_Type_Args => (name, [])
  1365            end)
  1366         |> (fn (name, T_args) => IConst (name, T, T_args))
  1367       | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
  1368       | aux _ tm = tm
  1369   in aux 0 end
  1370 
  1371 fun repair_iterm ctxt format type_enc sym_tab =
  1372   not (is_type_enc_higher_order type_enc)
  1373   ? (introduce_explicit_apps_in_iterm sym_tab
  1374      #> introduce_predicators_in_iterm sym_tab)
  1375   #> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1376 fun repair_fact ctxt format type_enc sym_tab =
  1377   update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))
  1378 
  1379 (** Helper facts **)
  1380 
  1381 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
  1382 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
  1383 
  1384 (* The Boolean indicates that a fairly sound type encoding is needed. *)
  1385 val helper_table =
  1386   [(("COMBI", false), @{thms Meson.COMBI_def}),
  1387    (("COMBK", false), @{thms Meson.COMBK_def}),
  1388    (("COMBB", false), @{thms Meson.COMBB_def}),
  1389    (("COMBC", false), @{thms Meson.COMBC_def}),
  1390    (("COMBS", false), @{thms Meson.COMBS_def}),
  1391    ((predicator_name, false), [not_ffalse, ftrue]),
  1392    (("fFalse", false), [not_ffalse]),
  1393    (("fFalse", true), @{thms True_or_False}),
  1394    (("fTrue", false), [ftrue]),
  1395    (("fTrue", true), @{thms True_or_False}),
  1396    (("fNot", false),
  1397     @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1398            fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1399    (("fconj", false),
  1400     @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
  1401         by (unfold fconj_def) fast+}),
  1402    (("fdisj", false),
  1403     @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
  1404         by (unfold fdisj_def) fast+}),
  1405    (("fimplies", false),
  1406     @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
  1407         by (unfold fimplies_def) fast+}),
  1408    (("fequal", true),
  1409     (* This is a lie: Higher-order equality doesn't need a sound type encoding.
  1410        However, this is done so for backward compatibility: Including the
  1411        equality helpers by default in Metis breaks a few existing proofs. *)
  1412     @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
  1413            fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
  1414    (* Partial characterization of "fAll" and "fEx". A complete characterization
  1415       would require the axiom of choice for replay with Metis. *)
  1416    (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
  1417    (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
  1418    (("If", true), @{thms if_True if_False True_or_False})]
  1419   |> map (apsnd (map zero_var_indexes))
  1420 
  1421 fun fo_literal_from_type_literal (TyLitVar (class, name)) =
  1422     (true, ATerm (class, [ATerm (name, [])]))
  1423   | fo_literal_from_type_literal (TyLitFree (class, name)) =
  1424     (true, ATerm (class, [ATerm (name, [])]))
  1425 
  1426 fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
  1427 
  1428 fun bound_tvars type_enc Ts =
  1429   mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
  1430                 (type_literals_for_types type_enc add_sorts_on_tvar Ts))
  1431 
  1432 fun eq_formula type_enc atomic_Ts pred_sym tm1 tm2 =
  1433   (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
  1434    else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
  1435   |> bound_tvars type_enc atomic_Ts
  1436   |> close_formula_universally
  1437 
  1438 val type_tag = `make_fixed_const type_tag_name
  1439 
  1440 fun type_tag_idempotence_fact type_enc =
  1441   let
  1442     fun var s = ATerm (`I s, [])
  1443     fun tag tm = ATerm (type_tag, [var "A", tm])
  1444     val tagged_var = tag (var "X")
  1445   in
  1446     Formula (type_tag_idempotence_helper_name, Axiom,
  1447              eq_formula type_enc [] false (tag tagged_var) tagged_var,
  1448              isabelle_info simpN, NONE)
  1449   end
  1450 
  1451 fun should_specialize_helper type_enc t =
  1452   polymorphism_of_type_enc type_enc = Mangled_Monomorphic andalso
  1453   level_of_type_enc type_enc <> No_Types andalso
  1454   not (null (Term.hidden_polymorphism t))
  1455 
  1456 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
  1457   case strip_prefix_and_unascii const_prefix s of
  1458     SOME mangled_s =>
  1459     let
  1460       val thy = Proof_Context.theory_of ctxt
  1461       val unmangled_s = mangled_s |> unmangled_const_name
  1462       fun dub needs_fairly_sound j k =
  1463         (unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
  1464          (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
  1465          (if needs_fairly_sound then typed_helper_suffix
  1466           else untyped_helper_suffix),
  1467          Helper)
  1468       fun dub_and_inst needs_fairly_sound (th, j) =
  1469         let val t = prop_of th in
  1470           if should_specialize_helper type_enc t then
  1471             map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
  1472                 types
  1473           else
  1474             [t]
  1475         end
  1476         |> map (fn (k, t) => (dub needs_fairly_sound j k, t)) o tag_list 1
  1477       val make_facts = map_filter (make_fact ctxt format type_enc false)
  1478       val fairly_sound = is_type_enc_fairly_sound type_enc
  1479     in
  1480       helper_table
  1481       |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
  1482                   if helper_s <> unmangled_s orelse
  1483                      (needs_fairly_sound andalso not fairly_sound) then
  1484                     []
  1485                   else
  1486                     ths ~~ (1 upto length ths)
  1487                     |> maps (dub_and_inst needs_fairly_sound)
  1488                     |> make_facts)
  1489     end
  1490   | NONE => []
  1491 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
  1492   Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
  1493                   []
  1494 
  1495 (***************************************************************)
  1496 (* Type Classes Present in the Axiom or Conjecture Clauses     *)
  1497 (***************************************************************)
  1498 
  1499 fun set_insert (x, s) = Symtab.update (x, ()) s
  1500 
  1501 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
  1502 
  1503 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
  1504 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
  1505 
  1506 fun classes_of_terms get_Ts =
  1507   map (map snd o get_Ts)
  1508   #> List.foldl add_classes Symtab.empty
  1509   #> delete_type #> Symtab.keys
  1510 
  1511 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
  1512 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
  1513 
  1514 fun fold_type_constrs f (Type (s, Ts)) x =
  1515     fold (fold_type_constrs f) Ts (f (s, x))
  1516   | fold_type_constrs _ _ x = x
  1517 
  1518 (* Type constructors used to instantiate overloaded constants are the only ones
  1519    needed. *)
  1520 fun add_type_constrs_in_term thy =
  1521   let
  1522     fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
  1523       | add (t $ u) = add t #> add u
  1524       | add (Const (x as (s, _))) =
  1525         if String.isPrefix skolem_const_prefix s then I
  1526         else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)
  1527       | add (Free (s, T)) =
  1528         if String.isPrefix polymorphic_free_prefix s then
  1529           T |> fold_type_constrs set_insert
  1530         else
  1531           I
  1532       | add (Abs (_, _, u)) = add u
  1533       | add _ = I
  1534   in add end
  1535 
  1536 fun type_constrs_of_terms thy ts =
  1537   Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
  1538 
  1539 fun translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  1540                        hyp_ts concl_t facts =
  1541   let
  1542     val thy = Proof_Context.theory_of ctxt
  1543     val presimp_consts = Meson.presimplified_consts ctxt
  1544     val fact_ts = facts |> map snd
  1545     (* Remove existing facts from the conjecture, as this can dramatically
  1546        boost an ATP's performance (for some reason). *)
  1547     val hyp_ts =
  1548       hyp_ts
  1549       |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
  1550     val facts = facts |> map (apsnd (pair Axiom))
  1551     val conjs =
  1552       map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_trueprop concl_t)]
  1553       |> map2 (pair o rpair Local o string_of_int) (0 upto length hyp_ts)
  1554     val ((conjs, facts), lambdas) =
  1555       if preproc then
  1556         conjs @ facts
  1557         |> map (apsnd (apsnd (presimp_prop ctxt presimp_consts)))
  1558         |> preprocess_abstractions_in_terms trans_lambdas
  1559         |>> chop (length conjs)
  1560         |>> apfst (map (apsnd (apsnd freeze_term)))
  1561       else
  1562         ((conjs, facts), [])
  1563     val conjs = conjs |> make_conjecture thy format type_enc
  1564     val (fact_names, facts) =
  1565       facts
  1566       |> map_filter (fn (name, (_, t)) =>
  1567                         make_fact ctxt format type_enc true (name, t)
  1568                         |> Option.map (pair name))
  1569       |> ListPair.unzip
  1570     val lambdas =
  1571       lambdas |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
  1572     val all_ts = concl_t :: hyp_ts @ fact_ts
  1573     val subs = tfree_classes_of_terms all_ts
  1574     val supers = tvar_classes_of_terms all_ts
  1575     val tycons = type_constrs_of_terms thy all_ts
  1576     val (supers, arity_clauses) =
  1577       if level_of_type_enc type_enc = No_Types then ([], [])
  1578       else make_arity_clauses thy tycons supers
  1579     val class_rel_clauses = make_class_rel_clauses thy subs supers
  1580   in
  1581     (fact_names |> map single,
  1582      (conjs, facts @ lambdas, class_rel_clauses, arity_clauses))
  1583   end
  1584 
  1585 val type_guard = `make_fixed_const type_guard_name
  1586 
  1587 fun type_guard_iterm ctxt format type_enc T tm =
  1588   IApp (IConst (type_guard, T --> @{typ bool}, [T])
  1589         |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
  1590 
  1591 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
  1592   | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
  1593     accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
  1594   | is_var_positively_naked_in_term _ _ _ _ = true
  1595 
  1596 fun should_guard_var_in_formula pos phi (SOME true) name =
  1597     formula_fold pos (is_var_positively_naked_in_term name) phi false
  1598   | should_guard_var_in_formula _ _ _ _ = true
  1599 
  1600 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
  1601   | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =
  1602     should_encode_type ctxt mono level T
  1603   | should_generate_tag_bound_decl _ _ _ _ _ = false
  1604 
  1605 fun mk_aterm format type_enc name T_args args =
  1606   ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
  1607 
  1608 fun tag_with_type ctxt format mono type_enc pos T tm =
  1609   IConst (type_tag, T --> T, [T])
  1610   |> enforce_type_arg_policy_in_iterm ctxt format type_enc
  1611   |> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
  1612   |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
  1613        | _ => raise Fail "unexpected lambda-abstraction")
  1614 and ho_term_from_iterm ctxt format mono type_enc =
  1615   let
  1616     fun aux site u =
  1617       let
  1618         val (head, args) = strip_iterm_comb u
  1619         val pos =
  1620           case site of
  1621             Top_Level pos => pos
  1622           | Eq_Arg pos => pos
  1623           | Elsewhere => NONE
  1624         val t =
  1625           case head of
  1626             IConst (name as (s, _), _, T_args) =>
  1627             let
  1628               val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
  1629             in
  1630               mk_aterm format type_enc name T_args (map (aux arg_site) args)
  1631             end
  1632           | IVar (name, _) =>
  1633             mk_aterm format type_enc name [] (map (aux Elsewhere) args)
  1634           | IAbs ((name, T), tm) =>
  1635             AAbs ((name, ho_type_from_typ format type_enc true 0 T),
  1636                   aux Elsewhere tm)
  1637           | IApp _ => raise Fail "impossible \"IApp\""
  1638         val T = ityp_of u
  1639       in
  1640         t |> (if should_tag_with_type ctxt mono type_enc site u T then
  1641                 tag_with_type ctxt format mono type_enc pos T
  1642               else
  1643                 I)
  1644       end
  1645   in aux end
  1646 and formula_from_iformula ctxt format mono type_enc should_guard_var =
  1647   let
  1648     val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
  1649     val do_bound_type =
  1650       case type_enc of
  1651         Simple_Types (_, level) =>
  1652         homogenized_type ctxt mono level 0
  1653         #> ho_type_from_typ format type_enc false 0 #> SOME
  1654       | _ => K NONE
  1655     fun do_out_of_bound_type pos phi universal (name, T) =
  1656       if should_guard_type ctxt mono type_enc
  1657              (fn () => should_guard_var pos phi universal name) T then
  1658         IVar (name, T)
  1659         |> type_guard_iterm ctxt format type_enc T
  1660         |> do_term pos |> AAtom |> SOME
  1661       else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
  1662         let
  1663           val var = ATerm (name, [])
  1664           val tagged_var =
  1665             ATerm (type_tag, [ho_term_from_typ format type_enc T, var])
  1666         in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
  1667       else
  1668         NONE
  1669     fun do_formula pos (AQuant (q, xs, phi)) =
  1670         let
  1671           val phi = phi |> do_formula pos
  1672           val universal = Option.map (q = AExists ? not) pos
  1673         in
  1674           AQuant (q, xs |> map (apsnd (fn NONE => NONE
  1675                                         | SOME T => do_bound_type T)),
  1676                   (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
  1677                       (map_filter
  1678                            (fn (_, NONE) => NONE
  1679                              | (s, SOME T) =>
  1680                                do_out_of_bound_type pos phi universal (s, T))
  1681                            xs)
  1682                       phi)
  1683         end
  1684       | do_formula pos (AConn conn) = aconn_map pos do_formula conn
  1685       | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
  1686   in do_formula end
  1687 
  1688 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
  1689    of monomorphization). The TPTP explicitly forbids name clashes, and some of
  1690    the remote provers might care. *)
  1691 fun formula_line_for_fact ctxt format prefix encode freshen pos mono type_enc
  1692                           (j, {name, locality, kind, iformula, atomic_types}) =
  1693   (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
  1694    iformula
  1695    |> close_iformula_universally
  1696    |> formula_from_iformula ctxt format mono type_enc
  1697                             should_guard_var_in_formula
  1698                             (if pos then SOME true else NONE)
  1699    |> bound_tvars type_enc atomic_types
  1700    |> close_formula_universally,
  1701    NONE,
  1702    case locality of
  1703      Intro => isabelle_info introN
  1704    | Elim => isabelle_info elimN
  1705    | Simp => isabelle_info simpN
  1706    | _ => NONE)
  1707   |> Formula
  1708 
  1709 fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}
  1710                                        : class_rel_clause) =
  1711   let val ty_arg = ATerm (`I "T", []) in
  1712     Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
  1713              AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
  1714                                AAtom (ATerm (superclass, [ty_arg]))])
  1715              |> close_formula_universally, isabelle_info introN, NONE)
  1716   end
  1717 
  1718 fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
  1719     (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
  1720   | fo_literal_from_arity_literal (TVarLit (c, sort)) =
  1721     (false, ATerm (c, [ATerm (sort, [])]))
  1722 
  1723 fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}
  1724                                    : arity_clause) =
  1725   Formula (arity_clause_prefix ^ name, Axiom,
  1726            mk_ahorn (map (formula_from_fo_literal o apfst not
  1727                           o fo_literal_from_arity_literal) prem_lits)
  1728                     (formula_from_fo_literal
  1729                          (fo_literal_from_arity_literal concl_lits))
  1730            |> close_formula_universally, isabelle_info introN, NONE)
  1731 
  1732 fun formula_line_for_conjecture ctxt format mono type_enc
  1733         ({name, kind, iformula, atomic_types, ...} : translated_formula) =
  1734   Formula (conjecture_prefix ^ name, kind,
  1735            formula_from_iformula ctxt format mono type_enc
  1736                should_guard_var_in_formula (SOME false)
  1737                (close_iformula_universally iformula)
  1738            |> bound_tvars type_enc atomic_types
  1739            |> close_formula_universally, NONE, NONE)
  1740 
  1741 fun free_type_literals type_enc ({atomic_types, ...} : translated_formula) =
  1742   atomic_types |> type_literals_for_types type_enc add_sorts_on_tfree
  1743                |> map fo_literal_from_type_literal
  1744 
  1745 fun formula_line_for_free_type j lit =
  1746   Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,
  1747            formula_from_fo_literal lit, NONE, NONE)
  1748 fun formula_lines_for_free_types type_enc facts =
  1749   let
  1750     val litss = map (free_type_literals type_enc) facts
  1751     val lits = fold (union (op =)) litss []
  1752   in map2 formula_line_for_free_type (0 upto length lits - 1) lits end
  1753 
  1754 (** Symbol declarations **)
  1755 
  1756 fun should_declare_sym type_enc pred_sym s =
  1757   (case type_enc of
  1758      Guards _ => not pred_sym
  1759    | _ => true) andalso
  1760   is_tptp_user_symbol s
  1761 
  1762 fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  1763                              (conjs, facts) =
  1764   let
  1765     fun add_iterm_syms in_conj tm =
  1766       let val (head, args) = strip_iterm_comb tm in
  1767         (case head of
  1768            IConst ((s, s'), T, T_args) =>
  1769            let val pred_sym = is_pred_sym repaired_sym_tab s in
  1770              if should_declare_sym type_enc pred_sym s then
  1771                Symtab.map_default (s, [])
  1772                    (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
  1773                                          in_conj))
  1774              else
  1775                I
  1776            end
  1777          | IAbs (_, tm) => add_iterm_syms in_conj tm
  1778          | _ => I)
  1779         #> fold (add_iterm_syms in_conj) args
  1780       end
  1781     fun add_fact_syms in_conj =
  1782       fact_lift (formula_fold NONE (K (add_iterm_syms in_conj)))
  1783     fun add_formula_var_types (AQuant (_, xs, phi)) =
  1784         fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
  1785         #> add_formula_var_types phi
  1786       | add_formula_var_types (AConn (_, phis)) =
  1787         fold add_formula_var_types phis
  1788       | add_formula_var_types _ = I
  1789     fun var_types () =
  1790       if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
  1791       else fold (fact_lift add_formula_var_types) (conjs @ facts) []
  1792     fun add_undefined_const T =
  1793       let
  1794         val (s, s') =
  1795           `make_fixed_const @{const_name undefined}
  1796           |> (case type_arg_policy type_enc @{const_name undefined} of
  1797                 Mangled_Type_Args _ => mangled_const_name format type_enc [T]
  1798               | _ => I)
  1799       in
  1800         Symtab.map_default (s, [])
  1801                            (insert_type ctxt #3 (s', [T], T, false, 0, false))
  1802       end
  1803   in
  1804     Symtab.empty
  1805     |> is_type_enc_fairly_sound type_enc
  1806        ? (fold (add_fact_syms true) conjs
  1807           #> fold (add_fact_syms false) facts
  1808           #> (case type_enc of
  1809                 Simple_Types _ => I
  1810               | _ => fold add_undefined_const (var_types ())))
  1811   end
  1812 
  1813 (* We add "bool" in case the helper "True_or_False" is included later. *)
  1814 val default_mono =
  1815   {maybe_finite_Ts = [@{typ bool}],
  1816    surely_finite_Ts = [@{typ bool}],
  1817    maybe_infinite_Ts = known_infinite_types,
  1818    surely_infinite_Ts = known_infinite_types,
  1819    maybe_nonmono_Ts = [@{typ bool}]}
  1820 
  1821 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
  1822    out with monotonicity" paper presented at CADE 2011. *)
  1823 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
  1824   | add_iterm_mononotonicity_info ctxt level _
  1825         (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
  1826         (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
  1827                   surely_infinite_Ts, maybe_nonmono_Ts}) =
  1828     if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
  1829       case level of
  1830         Noninf_Nonmono_Types soundness =>
  1831         if exists (type_instance ctxt T) surely_infinite_Ts orelse
  1832            member (type_aconv ctxt) maybe_finite_Ts T then
  1833           mono
  1834         else if is_type_surely_infinite' ctxt soundness surely_infinite_Ts
  1835                                          T then
  1836           {maybe_finite_Ts = maybe_finite_Ts,
  1837            surely_finite_Ts = surely_finite_Ts,
  1838            maybe_infinite_Ts = maybe_infinite_Ts,
  1839            surely_infinite_Ts = surely_infinite_Ts |> insert_type ctxt I T,
  1840            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1841         else
  1842           {maybe_finite_Ts = maybe_finite_Ts |> insert (type_aconv ctxt) T,
  1843            surely_finite_Ts = surely_finite_Ts,
  1844            maybe_infinite_Ts = maybe_infinite_Ts,
  1845            surely_infinite_Ts = surely_infinite_Ts,
  1846            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1847       | Fin_Nonmono_Types =>
  1848         if exists (type_instance ctxt T) surely_finite_Ts orelse
  1849            member (type_aconv ctxt) maybe_infinite_Ts T then
  1850           mono
  1851         else if is_type_surely_finite ctxt T then
  1852           {maybe_finite_Ts = maybe_finite_Ts,
  1853            surely_finite_Ts = surely_finite_Ts |> insert_type ctxt I T,
  1854            maybe_infinite_Ts = maybe_infinite_Ts,
  1855            surely_infinite_Ts = surely_infinite_Ts,
  1856            maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
  1857         else
  1858           {maybe_finite_Ts = maybe_finite_Ts,
  1859            surely_finite_Ts = surely_finite_Ts,
  1860            maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_aconv ctxt) T,
  1861            surely_infinite_Ts = surely_infinite_Ts,
  1862            maybe_nonmono_Ts = maybe_nonmono_Ts}
  1863       | _ => mono
  1864     else
  1865       mono
  1866   | add_iterm_mononotonicity_info _ _ _ _ mono = mono
  1867 fun add_fact_mononotonicity_info ctxt level
  1868         ({kind, iformula, ...} : translated_formula) =
  1869   formula_fold (SOME (kind <> Conjecture))
  1870                (add_iterm_mononotonicity_info ctxt level) iformula
  1871 fun mononotonicity_info_for_facts ctxt type_enc facts =
  1872   let val level = level_of_type_enc type_enc in
  1873     default_mono
  1874     |> is_type_level_monotonicity_based level
  1875        ? fold (add_fact_mononotonicity_info ctxt level) facts
  1876   end
  1877 
  1878 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
  1879   Formula (guards_sym_formula_prefix ^
  1880            ascii_of (mangled_type format type_enc T),
  1881            Axiom,
  1882            IConst (`make_bound_var "X", T, [])
  1883            |> type_guard_iterm ctxt format type_enc T
  1884            |> AAtom
  1885            |> formula_from_iformula ctxt format mono type_enc
  1886                                     (K (K (K (K true)))) (SOME true)
  1887            |> bound_tvars type_enc (atyps_of T)
  1888            |> close_formula_universally,
  1889            isabelle_info introN, NONE)
  1890 
  1891 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
  1892   let val x_var = ATerm (`make_bound_var "X", []) in
  1893     Formula (tags_sym_formula_prefix ^
  1894              ascii_of (mangled_type format type_enc T),
  1895              Axiom,
  1896              eq_formula type_enc (atyps_of T) false
  1897                         (tag_with_type ctxt format mono type_enc NONE T x_var)
  1898                         x_var,
  1899              isabelle_info simpN, NONE)
  1900   end
  1901 
  1902 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
  1903   case type_enc of
  1904     Simple_Types _ => []
  1905   | Guards _ =>
  1906     map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
  1907   | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
  1908 
  1909 fun decl_line_for_sym ctxt format mono type_enc s
  1910                       (s', T_args, T, pred_sym, ary, _) =
  1911   let
  1912     val (T_arg_Ts, level) =
  1913       case type_enc of
  1914         Simple_Types (_, level) => ([], level)
  1915       | _ => (replicate (length T_args) homo_infinite_type, No_Types)
  1916   in
  1917     Decl (sym_decl_prefix ^ s, (s, s'),
  1918           (T_arg_Ts ---> (T |> homogenized_type ctxt mono level ary))
  1919           |> ho_type_from_typ format type_enc pred_sym (length T_arg_Ts + ary))
  1920   end
  1921 
  1922 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
  1923                                      j (s', T_args, T, _, ary, in_conj) =
  1924   let
  1925     val (kind, maybe_negate) =
  1926       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1927       else (Axiom, I)
  1928     val (arg_Ts, res_T) = chop_fun ary T
  1929     val num_args = length arg_Ts
  1930     val bound_names =
  1931       1 upto num_args |> map (`I o make_bound_var o string_of_int)
  1932     val bounds =
  1933       bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
  1934     val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
  1935     fun should_keep_arg_type T =
  1936       sym_needs_arg_types andalso
  1937       should_guard_type ctxt mono type_enc (K true) T
  1938     val bound_Ts =
  1939       arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
  1940   in
  1941     Formula (guards_sym_formula_prefix ^ s ^
  1942              (if n > 1 then "_" ^ string_of_int j else ""), kind,
  1943              IConst ((s, s'), T, T_args)
  1944              |> fold (curry (IApp o swap)) bounds
  1945              |> type_guard_iterm ctxt format type_enc res_T
  1946              |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
  1947              |> formula_from_iformula ctxt format mono type_enc
  1948                                       (K (K (K (K true)))) (SOME true)
  1949              |> n > 1 ? bound_tvars type_enc (atyps_of T)
  1950              |> close_formula_universally
  1951              |> maybe_negate,
  1952              isabelle_info introN, NONE)
  1953   end
  1954 
  1955 fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
  1956         type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
  1957   let
  1958     val ident_base =
  1959       tags_sym_formula_prefix ^ s ^
  1960       (if n > 1 then "_" ^ string_of_int j else "")
  1961     val (kind, maybe_negate) =
  1962       if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
  1963       else (Axiom, I)
  1964     val (arg_Ts, res_T) = chop_fun ary T
  1965     val bound_names =
  1966       1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
  1967     val bounds = bound_names |> map (fn name => ATerm (name, []))
  1968     val cst = mk_aterm format type_enc (s, s') T_args
  1969     val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
  1970     val should_encode =
  1971       should_encode_type ctxt mono (level_of_type_enc type_enc)
  1972     val tag_with = tag_with_type ctxt format mono type_enc NONE
  1973     val add_formula_for_res =
  1974       if should_encode res_T then
  1975         cons (Formula (ident_base ^ "_res", kind,
  1976                        eq (tag_with res_T (cst bounds)) (cst bounds),
  1977                        isabelle_info simpN, NONE))
  1978       else
  1979         I
  1980     fun add_formula_for_arg k =
  1981       let val arg_T = nth arg_Ts k in
  1982         if should_encode arg_T then
  1983           case chop k bounds of
  1984             (bounds1, bound :: bounds2) =>
  1985             cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
  1986                            eq (cst (bounds1 @ tag_with arg_T bound :: bounds2))
  1987                               (cst bounds),
  1988                            isabelle_info simpN, NONE))
  1989           | _ => raise Fail "expected nonempty tail"
  1990         else
  1991           I
  1992       end
  1993   in
  1994     [] |> not pred_sym ? add_formula_for_res
  1995        |> fold add_formula_for_arg (ary - 1 downto 0)
  1996   end
  1997 
  1998 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
  1999 
  2000 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
  2001                                 (s, decls) =
  2002   case type_enc of
  2003     Simple_Types _ =>
  2004     decls |> map (decl_line_for_sym ctxt format mono type_enc s)
  2005   | Guards (_, level, _) =>
  2006     let
  2007       val decls =
  2008         case decls of
  2009           decl :: (decls' as _ :: _) =>
  2010           let val T = result_type_of_decl decl in
  2011             if forall (type_generalization ctxt T o result_type_of_decl)
  2012                       decls' then
  2013               [decl]
  2014             else
  2015               decls
  2016           end
  2017         | _ => decls
  2018       val n = length decls
  2019       val decls =
  2020         decls |> filter (should_encode_type ctxt mono level
  2021                          o result_type_of_decl)
  2022     in
  2023       (0 upto length decls - 1, decls)
  2024       |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
  2025                                                  type_enc n s)
  2026     end
  2027   | Tags (_, _, uniformity) =>
  2028     (case uniformity of
  2029        Uniform => []
  2030      | Nonuniform =>
  2031        let val n = length decls in
  2032          (0 upto n - 1 ~~ decls)
  2033          |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
  2034                       conj_sym_kind mono type_enc n s)
  2035        end)
  2036 
  2037 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
  2038                                      sym_decl_tab =
  2039   let
  2040     val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
  2041     val mono_Ts =
  2042       if polymorphism_of_type_enc type_enc = Polymorphic then
  2043         syms |> maps (map result_type_of_decl o snd)
  2044              |> filter_out (should_encode_type ctxt mono
  2045                                 (level_of_type_enc type_enc))
  2046              |> rpair [] |-> fold (insert_type ctxt I)
  2047       else
  2048         []
  2049     val mono_lines =
  2050       problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
  2051     val decl_lines =
  2052       fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
  2053                                                      mono type_enc)
  2054                syms []
  2055   in mono_lines @ decl_lines end
  2056 
  2057 fun needs_type_tag_idempotence (Tags (poly, level, uniformity)) =
  2058     poly <> Mangled_Monomorphic andalso
  2059     ((level = All_Types andalso uniformity = Nonuniform) orelse
  2060      is_type_level_monotonicity_based level)
  2061   | needs_type_tag_idempotence _ = false
  2062 
  2063 fun offset_of_heading_in_problem _ [] j = j
  2064   | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
  2065     if heading = needle then j
  2066     else offset_of_heading_in_problem needle problem (j + length lines)
  2067 
  2068 val implicit_declsN = "Should-be-implicit typings"
  2069 val explicit_declsN = "Explicit typings"
  2070 val factsN = "Relevant facts"
  2071 val class_relsN = "Class relationships"
  2072 val aritiesN = "Arities"
  2073 val helpersN = "Helper facts"
  2074 val conjsN = "Conjectures"
  2075 val free_typesN = "Type variables"
  2076 
  2077 val explicit_apply = NONE (* for experiments *)
  2078 
  2079 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
  2080         lambda_trans readable_names preproc hyp_ts concl_t facts =
  2081   let
  2082     val type_enc = type_enc |> adjust_type_enc format
  2083     val lambda_trans =
  2084       if lambda_trans = smartN then
  2085         if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
  2086       else if lambda_trans = lambdasN andalso
  2087               not (is_type_enc_higher_order type_enc) then
  2088         error ("Lambda translation method incompatible with first-order \
  2089                \encoding.")
  2090       else
  2091         lambda_trans
  2092     val trans_lambdas =
  2093       if lambda_trans = no_lambdasN then
  2094         rpair []
  2095       else if lambda_trans = concealedN then
  2096         lift_lambdas ctxt type_enc ##> K []
  2097       else if lambda_trans = liftingN then
  2098         lift_lambdas ctxt type_enc
  2099       else if lambda_trans = combinatorsN then
  2100         map (introduce_combinators ctxt) #> rpair []
  2101       else if lambda_trans = hybridN then
  2102         lift_lambdas ctxt type_enc
  2103         ##> maps (fn t => [t, introduce_combinators ctxt
  2104                                   (intentionalize_def t)])
  2105       else if lambda_trans = lambdasN then
  2106         map (Envir.eta_contract) #> rpair []
  2107       else
  2108         error ("Unknown lambda translation method: " ^
  2109                quote lambda_trans ^ ".")
  2110     val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
  2111       translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
  2112                          hyp_ts concl_t facts
  2113     val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
  2114     val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
  2115     val repair = repair_fact ctxt format type_enc sym_tab
  2116     val (conjs, facts) = (conjs, facts) |> pairself (map repair)
  2117     val repaired_sym_tab =
  2118       conjs @ facts |> sym_table_for_facts ctxt (SOME false)
  2119     val helpers =
  2120       repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
  2121                        |> map repair
  2122     val sym_decl_lines =
  2123       (conjs, helpers @ facts)
  2124       |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
  2125       |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
  2126                                                type_enc
  2127     val helper_lines =
  2128       0 upto length helpers - 1 ~~ helpers
  2129       |> map (formula_line_for_fact ctxt format helper_prefix I false true mono
  2130                                     type_enc)
  2131       |> (if needs_type_tag_idempotence type_enc then
  2132             cons (type_tag_idempotence_fact type_enc)
  2133           else
  2134             I)
  2135     (* Reordering these might confuse the proof reconstruction code or the SPASS
  2136        FLOTTER hack. *)
  2137     val problem =
  2138       [(explicit_declsN, sym_decl_lines),
  2139        (factsN,
  2140         map (formula_line_for_fact ctxt format fact_prefix ascii_of
  2141                                    (not exporter) (not exporter) mono
  2142                                    type_enc)
  2143             (0 upto length facts - 1 ~~ facts)),
  2144        (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
  2145        (aritiesN, map formula_line_for_arity_clause arity_clauses),
  2146        (helpersN, helper_lines),
  2147        (conjsN,
  2148         map (formula_line_for_conjecture ctxt format mono type_enc) conjs),
  2149        (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs))]
  2150     val problem =
  2151       problem
  2152       |> (case format of
  2153             CNF => ensure_cnf_problem
  2154           | CNF_UEQ => filter_cnf_ueq_problem
  2155           | _ => I)
  2156       |> (if is_format_typed format then
  2157             declare_undeclared_syms_in_atp_problem type_decl_prefix
  2158                                                    implicit_declsN
  2159           else
  2160             I)
  2161     val (problem, pool) = problem |> nice_atp_problem readable_names
  2162     val helpers_offset = offset_of_heading_in_problem helpersN problem 0
  2163     val typed_helpers =
  2164       map_filter (fn (j, {name, ...}) =>
  2165                      if String.isSuffix typed_helper_suffix name then SOME j
  2166                      else NONE)
  2167                  ((helpers_offset + 1 upto helpers_offset + length helpers)
  2168                   ~~ helpers)
  2169     fun add_sym_arity (s, {min_ary, ...} : sym_info) =
  2170       if min_ary > 0 then
  2171         case strip_prefix_and_unascii const_prefix s of
  2172           SOME s => Symtab.insert (op =) (s, min_ary)
  2173         | NONE => I
  2174       else
  2175         I
  2176   in
  2177     (problem,
  2178      case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
  2179      offset_of_heading_in_problem conjsN problem 0,
  2180      offset_of_heading_in_problem factsN problem 0,
  2181      fact_names |> Vector.fromList,
  2182      typed_helpers,
  2183      Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
  2184   end
  2185 
  2186 (* FUDGE *)
  2187 val conj_weight = 0.0
  2188 val hyp_weight = 0.1
  2189 val fact_min_weight = 0.2
  2190 val fact_max_weight = 1.0
  2191 val type_info_default_weight = 0.8
  2192 
  2193 fun add_term_weights weight (ATerm (s, tms)) =
  2194     is_tptp_user_symbol s ? Symtab.default (s, weight)
  2195     #> fold (add_term_weights weight) tms
  2196   | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
  2197 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
  2198     formula_fold NONE (K (add_term_weights weight)) phi
  2199   | add_problem_line_weights _ _ = I
  2200 
  2201 fun add_conjectures_weights [] = I
  2202   | add_conjectures_weights conjs =
  2203     let val (hyps, conj) = split_last conjs in
  2204       add_problem_line_weights conj_weight conj
  2205       #> fold (add_problem_line_weights hyp_weight) hyps
  2206     end
  2207 
  2208 fun add_facts_weights facts =
  2209   let
  2210     val num_facts = length facts
  2211     fun weight_of j =
  2212       fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
  2213                         / Real.fromInt num_facts
  2214   in
  2215     map weight_of (0 upto num_facts - 1) ~~ facts
  2216     |> fold (uncurry add_problem_line_weights)
  2217   end
  2218 
  2219 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
  2220 fun atp_problem_weights problem =
  2221   let val get = these o AList.lookup (op =) problem in
  2222     Symtab.empty
  2223     |> add_conjectures_weights (get free_typesN @ get conjsN)
  2224     |> add_facts_weights (get factsN)
  2225     |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
  2226             [explicit_declsN, class_relsN, aritiesN]
  2227     |> Symtab.dest
  2228     |> sort (prod_ord Real.compare string_ord o pairself swap)
  2229   end
  2230 
  2231 end;