1 (* Title: HOL/Tools/ATP/atp_translate.ML
2 Author: Fabian Immler, TU Muenchen
4 Author: Jasmin Blanchette, TU Muenchen
6 Translation of HOL to FOL for Sledgehammer.
9 signature ATP_TRANSLATE =
11 type 'a fo_term = 'a ATP_Problem.fo_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
18 type name = string * string
20 datatype type_literal =
21 TyLitVar of name * name |
22 TyLitFree of name * name
24 datatype arity_literal =
25 TConsLit of name * name * name list |
26 TVarLit of name * name
30 prem_lits: arity_literal list,
31 concl_lits: arity_literal}
33 type class_rel_clause =
39 CombConst of name * typ * typ list |
40 CombVar of name * typ |
41 CombApp of combterm * combterm
43 datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained
45 datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
47 All_Types | Noninf_Nonmono_Types | Fin_Nonmono_Types | Const_Arg_Types |
49 datatype type_heaviness = Heavyweight | Lightweight
52 Simple_Types of type_level |
53 Preds of polymorphism * type_level * type_heaviness |
54 Tags of polymorphism * type_level * type_heaviness
56 val bound_var_prefix : string
57 val schematic_var_prefix: string
58 val fixed_var_prefix: string
59 val tvar_prefix: string
60 val tfree_prefix: string
61 val const_prefix: string
62 val type_const_prefix: string
63 val class_prefix: string
64 val skolem_const_prefix : string
65 val old_skolem_const_prefix : string
66 val new_skolem_const_prefix : string
67 val type_decl_prefix : string
68 val sym_decl_prefix : string
69 val preds_sym_formula_prefix : string
70 val lightweight_tags_sym_formula_prefix : string
71 val fact_prefix : string
72 val conjecture_prefix : string
73 val helper_prefix : string
74 val class_rel_clause_prefix : string
75 val arity_clause_prefix : string
76 val tfree_clause_prefix : string
77 val typed_helper_suffix : string
78 val untyped_helper_suffix : string
79 val type_tag_idempotence_helper_name : string
80 val predicator_name : string
81 val app_op_name : string
82 val type_tag_name : string
83 val type_pred_name : string
84 val simple_type_prefix : string
85 val prefixed_predicator_name : string
86 val prefixed_app_op_name : string
87 val prefixed_type_tag_name : string
88 val ascii_of: string -> string
89 val unascii_of: string -> string
90 val strip_prefix_and_unascii : string -> string -> string option
91 val proxy_table : (string * (string * (thm * (string * string)))) list
92 val proxify_const : string -> (string * string) option
93 val invert_const: string -> string
94 val unproxify_const: string -> string
95 val make_bound_var : string -> string
96 val make_schematic_var : string * int -> string
97 val make_fixed_var : string -> string
98 val make_schematic_type_var : string * int -> string
99 val make_fixed_type_var : string -> string
100 val make_fixed_const : string -> string
101 val make_fixed_type_const : string -> string
102 val make_type_class : string -> string
103 val new_skolem_var_name_from_const : string -> string
104 val num_type_args : theory -> string -> int
105 val atp_irrelevant_consts : string list
106 val atp_schematic_consts_of : term -> typ list Symtab.table
107 val make_arity_clauses :
108 theory -> string list -> class list -> class list * arity_clause list
109 val make_class_rel_clauses :
110 theory -> class list -> class list -> class_rel_clause list
111 val combtyp_of : combterm -> typ
112 val strip_combterm_comb : combterm -> combterm * combterm list
113 val atyps_of : typ -> typ list
114 val combterm_from_term :
115 theory -> (string * typ) list -> term -> combterm * typ list
116 val is_locality_global : locality -> bool
117 val type_sys_from_string : string -> type_sys
118 val polymorphism_of_type_sys : type_sys -> polymorphism
119 val level_of_type_sys : type_sys -> type_level
120 val is_type_sys_virtually_sound : type_sys -> bool
121 val is_type_sys_fairly_sound : type_sys -> bool
122 val choose_format : format list -> type_sys -> format * type_sys
124 connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
125 val unmangled_const_name : string -> string
126 val unmangled_const : string -> string * string fo_term list
127 val helper_table : ((string * bool) * thm list) list
128 val should_specialize_helper : type_sys -> term -> bool
129 val tfree_classes_of_terms : term list -> string list
130 val tvar_classes_of_terms : term list -> string list
131 val type_constrs_of_terms : theory -> term list -> string list
132 val prepare_atp_problem :
133 Proof.context -> format -> formula_kind -> formula_kind -> type_sys
134 -> bool -> bool -> bool -> term list -> term
135 -> ((string * locality) * term) list
136 -> string problem * string Symtab.table * int * int
137 * (string * locality) list vector * int list * int Symtab.table
138 val atp_problem_weights : string problem -> (string * real) list
141 structure ATP_Translate : ATP_TRANSLATE =
147 type name = string * string
150 val generate_useful_info = false
152 fun useful_isabelle_info s =
153 if generate_useful_info then
154 SOME (ATerm ("[]", [ATerm ("isabelle_" ^ s, [])]))
158 val intro_info = useful_isabelle_info "intro"
159 val elim_info = useful_isabelle_info "elim"
160 val simp_info = useful_isabelle_info "simp"
162 val bound_var_prefix = "B_"
163 val schematic_var_prefix = "V_"
164 val fixed_var_prefix = "v_"
166 val tvar_prefix = "T_"
167 val tfree_prefix = "t_"
169 val const_prefix = "c_"
170 val type_const_prefix = "tc_"
171 val class_prefix = "cl_"
173 val skolem_const_prefix = "Sledgehammer" ^ Long_Name.separator ^ "Sko"
174 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
175 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
177 val type_decl_prefix = "ty_"
178 val sym_decl_prefix = "sy_"
179 val preds_sym_formula_prefix = "psy_"
180 val lightweight_tags_sym_formula_prefix = "tsy_"
181 val fact_prefix = "fact_"
182 val conjecture_prefix = "conj_"
183 val helper_prefix = "help_"
184 val class_rel_clause_prefix = "clar_"
185 val arity_clause_prefix = "arity_"
186 val tfree_clause_prefix = "tfree_"
188 val typed_helper_suffix = "_T"
189 val untyped_helper_suffix = "_U"
190 val type_tag_idempotence_helper_name = helper_prefix ^ "ti_idem"
192 val predicator_name = "hBOOL"
193 val app_op_name = "hAPP"
194 val type_tag_name = "ti"
195 val type_pred_name = "is"
196 val simple_type_prefix = "ty_"
198 val prefixed_predicator_name = const_prefix ^ predicator_name
199 val prefixed_app_op_name = const_prefix ^ app_op_name
200 val prefixed_type_tag_name = const_prefix ^ type_tag_name
202 (* Freshness almost guaranteed! *)
203 val sledgehammer_weak_prefix = "Sledgehammer:"
205 (*Escaping of special characters.
206 Alphanumeric characters are left unchanged.
207 The character _ goes to __
208 Characters in the range ASCII space to / go to _A to _P, respectively.
209 Other characters go to _nnn where nnn is the decimal ASCII code.*)
210 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
212 fun stringN_of_int 0 _ = ""
213 | stringN_of_int k n =
214 stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
216 fun ascii_of_char c =
217 if Char.isAlphaNum c then
219 else if c = #"_" then
221 else if #" " <= c andalso c <= #"/" then
222 "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
224 (* fixed width, in case more digits follow *)
225 "_" ^ stringN_of_int 3 (Char.ord c)
227 val ascii_of = String.translate ascii_of_char
229 (** Remove ASCII armoring from names in proof files **)
231 (* We don't raise error exceptions because this code can run inside a worker
232 thread. Also, the errors are impossible. *)
235 fun un rcs [] = String.implode(rev rcs)
236 | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
237 (* Three types of _ escapes: __, _A to _P, _nnn *)
238 | un rcs (#"_" :: #"_" :: cs) = un (#"_"::rcs) cs
239 | un rcs (#"_" :: c :: cs) =
240 if #"A" <= c andalso c<= #"P" then
241 (* translation of #" " to #"/" *)
242 un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
244 let val digits = List.take (c::cs, 3) handle General.Subscript => [] in
245 case Int.fromString (String.implode digits) of
246 SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
247 | NONE => un (c:: #"_"::rcs) cs (* ERROR *)
249 | un rcs (c :: cs) = un (c :: rcs) cs
250 in un [] o String.explode end
252 (* If string s has the prefix s1, return the result of deleting it,
254 fun strip_prefix_and_unascii s1 s =
255 if String.isPrefix s1 s then
256 SOME (unascii_of (String.extract (s, size s1, NONE)))
261 [("c_False", (@{const_name False}, (@{thm fFalse_def},
262 ("fFalse", @{const_name ATP.fFalse})))),
263 ("c_True", (@{const_name True}, (@{thm fTrue_def},
264 ("fTrue", @{const_name ATP.fTrue})))),
265 ("c_Not", (@{const_name Not}, (@{thm fNot_def},
266 ("fNot", @{const_name ATP.fNot})))),
267 ("c_conj", (@{const_name conj}, (@{thm fconj_def},
268 ("fconj", @{const_name ATP.fconj})))),
269 ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
270 ("fdisj", @{const_name ATP.fdisj})))),
271 ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
272 ("fimplies", @{const_name ATP.fimplies})))),
273 ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
274 ("fequal", @{const_name ATP.fequal}))))]
276 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
278 (* Readable names for the more common symbolic functions. Do not mess with the
279 table unless you know what you are doing. *)
280 val const_trans_table =
281 [(@{type_name Product_Type.prod}, "prod"),
282 (@{type_name Sum_Type.sum}, "sum"),
283 (@{const_name False}, "False"),
284 (@{const_name True}, "True"),
285 (@{const_name Not}, "Not"),
286 (@{const_name conj}, "conj"),
287 (@{const_name disj}, "disj"),
288 (@{const_name implies}, "implies"),
289 (@{const_name HOL.eq}, "equal"),
290 (@{const_name If}, "If"),
291 (@{const_name Set.member}, "member"),
292 (@{const_name Meson.COMBI}, "COMBI"),
293 (@{const_name Meson.COMBK}, "COMBK"),
294 (@{const_name Meson.COMBB}, "COMBB"),
295 (@{const_name Meson.COMBC}, "COMBC"),
296 (@{const_name Meson.COMBS}, "COMBS")]
298 |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
300 (* Invert the table of translations between Isabelle and ATPs. *)
301 val const_trans_table_inv =
302 const_trans_table |> Symtab.dest |> map swap |> Symtab.make
303 val const_trans_table_unprox =
305 |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
307 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
308 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
311 case Symtab.lookup const_trans_table c of
315 (*Remove the initial ' character from a type variable, if it is present*)
316 fun trim_type_var s =
317 if s <> "" andalso String.sub(s,0) = #"'" then String.extract(s,1,NONE)
318 else raise Fail ("trim_type: Malformed type variable encountered: " ^ s)
320 fun ascii_of_indexname (v,0) = ascii_of v
321 | ascii_of_indexname (v,i) = ascii_of v ^ "_" ^ string_of_int i
323 fun make_bound_var x = bound_var_prefix ^ ascii_of x
324 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
325 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
327 fun make_schematic_type_var (x,i) =
328 tvar_prefix ^ (ascii_of_indexname (trim_type_var x, i))
329 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (trim_type_var x))
331 (* HOL.eq MUST BE "equal" because it's built into ATPs. *)
332 fun make_fixed_const @{const_name HOL.eq} = "equal"
333 | make_fixed_const c = const_prefix ^ lookup_const c
335 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
337 fun make_type_class clas = class_prefix ^ ascii_of clas
339 fun new_skolem_var_name_from_const s =
340 let val ss = s |> space_explode Long_Name.separator in
341 nth ss (length ss - 2)
344 (* The number of type arguments of a constant, zero if it's monomorphic. For
345 (instances of) Skolem pseudoconstants, this information is encoded in the
347 fun num_type_args thy s =
348 if String.isPrefix skolem_const_prefix s then
349 s |> space_explode Long_Name.separator |> List.last |> Int.fromString |> the
351 (s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
353 (* These are either simplified away by "Meson.presimplify" (most of the time) or
354 handled specially via "fFalse", "fTrue", ..., "fequal". *)
355 val atp_irrelevant_consts =
356 [@{const_name False}, @{const_name True}, @{const_name Not},
357 @{const_name conj}, @{const_name disj}, @{const_name implies},
358 @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
360 val atp_monomorph_bad_consts =
361 atp_irrelevant_consts @
362 (* These are ignored anyway by the relevance filter (unless they appear in
363 higher-order places) but not by the monomorphizer. *)
364 [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
365 @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
366 @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
368 fun add_schematic_const (x as (_, T)) =
369 Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
370 val add_schematic_consts_of =
371 Term.fold_aterms (fn Const (x as (s, _)) =>
372 not (member (op =) atp_monomorph_bad_consts s)
373 ? add_schematic_const x
375 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
377 (** Definitions and functions for FOL clauses and formulas for TPTP **)
379 (* The first component is the type class; the second is a "TVar" or "TFree". *)
380 datatype type_literal =
381 TyLitVar of name * name |
382 TyLitFree of name * name
385 (** Isabelle arities **)
387 datatype arity_literal =
388 TConsLit of name * name * name list |
389 TVarLit of name * name
392 | gen_TVars n = ("T_" ^ string_of_int n) :: gen_TVars (n-1)
394 val type_class = the_single @{sort type}
396 fun add_packed_sort tvar =
397 fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar))
401 prem_lits: arity_literal list,
402 concl_lits: arity_literal}
404 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
405 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
407 val tvars = gen_TVars (length args)
408 val tvars_srts = ListPair.zip (tvars, args)
411 prem_lits = [] |> fold (uncurry add_packed_sort) tvars_srts |> map TVarLit,
412 concl_lits = TConsLit (`make_type_class cls,
413 `make_fixed_type_const tcons,
417 fun arity_clause _ _ (_, []) = []
418 | arity_clause seen n (tcons, ("HOL.type",_)::ars) = (*ignore*)
419 arity_clause seen n (tcons,ars)
420 | arity_clause seen n (tcons, (ar as (class,_)) :: ars) =
421 if member (op =) seen class then (*multiple arities for the same tycon, class pair*)
422 make_axiom_arity_clause (tcons, lookup_const tcons ^ "_" ^ class ^ "_" ^ string_of_int n, ar) ::
423 arity_clause seen (n+1) (tcons,ars)
425 make_axiom_arity_clause (tcons, lookup_const tcons ^ "_" ^ class, ar) ::
426 arity_clause (class::seen) n (tcons,ars)
428 fun multi_arity_clause [] = []
429 | multi_arity_clause ((tcons, ars) :: tc_arlists) =
430 arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
432 (*Generate all pairs (tycon,class,sorts) such that tycon belongs to class in theory thy
433 provided its arguments have the corresponding sorts.*)
434 fun type_class_pairs thy tycons classes =
436 val alg = Sign.classes_of thy
437 fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
438 fun add_class tycon class =
439 cons (class, domain_sorts tycon class)
440 handle Sorts.CLASS_ERROR _ => I
441 fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
442 in map try_classes tycons end
444 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
445 fun iter_type_class_pairs _ _ [] = ([], [])
446 | iter_type_class_pairs thy tycons classes =
448 fun maybe_insert_class s =
449 (s <> type_class andalso not (member (op =) classes s))
451 val cpairs = type_class_pairs thy tycons classes
453 [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
454 val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
455 in (classes' @ classes, union (op =) cpairs' cpairs) end
457 fun make_arity_clauses thy tycons =
458 iter_type_class_pairs thy tycons ##> multi_arity_clause
461 (** Isabelle class relations **)
463 type class_rel_clause =
468 (*Generate all pairs (sub,super) such that sub is a proper subclass of super in theory thy.*)
469 fun class_pairs _ [] _ = []
470 | class_pairs thy subs supers =
472 val class_less = Sorts.class_less (Sign.classes_of thy)
473 fun add_super sub super = class_less (sub, super) ? cons (sub, super)
474 fun add_supers sub = fold (add_super sub) supers
475 in fold add_supers subs [] end
477 fun make_class_rel_clause (sub,super) =
478 {name = sub ^ "_" ^ super,
479 subclass = `make_type_class sub,
480 superclass = `make_type_class super}
482 fun make_class_rel_clauses thy subs supers =
483 map make_class_rel_clause (class_pairs thy subs supers)
486 CombConst of name * typ * typ list |
487 CombVar of name * typ |
488 CombApp of combterm * combterm
490 fun combtyp_of (CombConst (_, T, _)) = T
491 | combtyp_of (CombVar (_, T)) = T
492 | combtyp_of (CombApp (t1, _)) = snd (dest_funT (combtyp_of t1))
494 (*gets the head of a combinator application, along with the list of arguments*)
495 fun strip_combterm_comb u =
496 let fun stripc (CombApp(t,u), ts) = stripc (t, u::ts)
500 fun atyps_of T = fold_atyps (insert (op =)) T []
502 fun new_skolem_const_name s num_T_args =
503 [new_skolem_const_prefix, s, string_of_int num_T_args]
504 |> space_implode Long_Name.separator
506 (* Converts a term (with combinators) into a combterm. Also accumulates sort
508 fun combterm_from_term thy bs (P $ Q) =
510 val (P', P_atomics_Ts) = combterm_from_term thy bs P
511 val (Q', Q_atomics_Ts) = combterm_from_term thy bs Q
512 in (CombApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
513 | combterm_from_term thy _ (Const (c, T)) =
516 (if String.isPrefix old_skolem_const_prefix c then
517 [] |> Term.add_tvarsT T |> map TVar
519 (c, T) |> Sign.const_typargs thy)
520 val c' = CombConst (`make_fixed_const c, T, tvar_list)
521 in (c', atyps_of T) end
522 | combterm_from_term _ _ (Free (v, T)) =
523 (CombConst (`make_fixed_var v, T, []), atyps_of T)
524 | combterm_from_term _ _ (Var (v as (s, _), T)) =
525 (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
527 val Ts = T |> strip_type |> swap |> op ::
528 val s' = new_skolem_const_name s (length Ts)
529 in CombConst (`make_fixed_const s', T, Ts) end
531 CombVar ((make_schematic_var v, s), T), atyps_of T)
532 | combterm_from_term _ bs (Bound j) =
534 |> (fn (s, T) => (CombConst (`make_bound_var s, T, []), atyps_of T))
535 | combterm_from_term _ _ (Abs _) = raise Fail "HOL clause: Abs"
537 datatype locality = General | Intro | Elim | Simp | Local | Assum | Chained
539 (* (quasi-)underapproximation of the truth *)
540 fun is_locality_global Local = false
541 | is_locality_global Assum = false
542 | is_locality_global Chained = false
543 | is_locality_global _ = true
545 datatype polymorphism = Polymorphic | Monomorphic | Mangled_Monomorphic
546 datatype type_level =
547 All_Types | Noninf_Nonmono_Types | Fin_Nonmono_Types | Const_Arg_Types |
549 datatype type_heaviness = Heavyweight | Lightweight
552 Simple_Types of type_level |
553 Preds of polymorphism * type_level * type_heaviness |
554 Tags of polymorphism * type_level * type_heaviness
556 fun try_unsuffixes ss s =
557 fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
559 fun type_sys_from_string s =
560 (case try (unprefix "poly_") s of
561 SOME s => (SOME Polymorphic, s)
563 case try (unprefix "mono_") s of
564 SOME s => (SOME Monomorphic, s)
566 case try (unprefix "mangled_") s of
567 SOME s => (SOME Mangled_Monomorphic, s)
570 (* "_query" and "_bang" are for the ASCII-challenged Mirabelle. *)
571 case try_unsuffixes ["?", "_query"] s of
572 SOME s => (Noninf_Nonmono_Types, s)
574 case try_unsuffixes ["!", "_bang"] s of
575 SOME s => (Fin_Nonmono_Types, s)
576 | NONE => (All_Types, s))
578 case try (unsuffix "_heavy") s of
579 SOME s => (Heavyweight, s)
580 | NONE => (Lightweight, s))
581 |> (fn (poly, (level, (heaviness, core))) =>
582 case (core, (poly, level, heaviness)) of
583 ("simple", (NONE, _, Lightweight)) => Simple_Types level
584 | ("preds", (SOME poly, _, _)) => Preds (poly, level, heaviness)
585 | ("tags", (SOME Polymorphic, _, _)) =>
586 Tags (Polymorphic, level, heaviness)
587 | ("tags", (SOME poly, _, _)) => Tags (poly, level, heaviness)
588 | ("args", (SOME poly, All_Types (* naja *), Lightweight)) =>
589 Preds (poly, Const_Arg_Types, Lightweight)
590 | ("erased", (NONE, All_Types (* naja *), Lightweight)) =>
591 Preds (Polymorphic, No_Types, Lightweight)
592 | _ => raise Same.SAME)
593 handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
595 fun polymorphism_of_type_sys (Simple_Types _) = Mangled_Monomorphic
596 | polymorphism_of_type_sys (Preds (poly, _, _)) = poly
597 | polymorphism_of_type_sys (Tags (poly, _, _)) = poly
599 fun level_of_type_sys (Simple_Types level) = level
600 | level_of_type_sys (Preds (_, level, _)) = level
601 | level_of_type_sys (Tags (_, level, _)) = level
603 fun heaviness_of_type_sys (Simple_Types _) = Heavyweight
604 | heaviness_of_type_sys (Preds (_, _, heaviness)) = heaviness
605 | heaviness_of_type_sys (Tags (_, _, heaviness)) = heaviness
607 fun is_type_level_virtually_sound level =
608 level = All_Types orelse level = Noninf_Nonmono_Types
609 val is_type_sys_virtually_sound =
610 is_type_level_virtually_sound o level_of_type_sys
612 fun is_type_level_fairly_sound level =
613 is_type_level_virtually_sound level orelse level = Fin_Nonmono_Types
614 val is_type_sys_fairly_sound = is_type_level_fairly_sound o level_of_type_sys
616 fun is_setting_higher_order THF (Simple_Types _) = true
617 | is_setting_higher_order _ _ = false
619 fun choose_format formats (Simple_Types level) =
620 if member (op =) formats THF then (THF, Simple_Types level)
621 else if member (op =) formats TFF then (TFF, Simple_Types level)
622 else choose_format formats (Preds (Mangled_Monomorphic, level, Heavyweight))
623 | choose_format formats type_sys =
626 (CNF_UEQ, case type_sys of
628 (if is_type_sys_fairly_sound type_sys then Tags else Preds)
631 | format => (format, type_sys))
633 type translated_formula =
637 combformula: (name, typ, combterm) formula,
638 atomic_types: typ list}
640 fun update_combformula f ({name, locality, kind, combformula, atomic_types}
641 : translated_formula) =
642 {name = name, locality = locality, kind = kind, combformula = f combformula,
643 atomic_types = atomic_types} : translated_formula
645 fun fact_lift f ({combformula, ...} : translated_formula) = f combformula
647 val type_instance = Sign.typ_instance o Proof_Context.theory_of
649 fun insert_type ctxt get_T x xs =
650 let val T = get_T x in
651 if exists (curry (type_instance ctxt) T o get_T) xs then xs
652 else x :: filter_out (curry (type_instance ctxt o swap) T o get_T) xs
655 (* The Booleans indicate whether all type arguments should be kept. *)
656 datatype type_arg_policy =
657 Explicit_Type_Args of bool |
658 Mangled_Type_Args of bool |
661 fun should_drop_arg_type_args (Simple_Types _) =
662 false (* since TFF doesn't support overloading *)
663 | should_drop_arg_type_args type_sys =
664 level_of_type_sys type_sys = All_Types andalso
665 heaviness_of_type_sys type_sys = Heavyweight
667 fun general_type_arg_policy (Tags (_, All_Types, Heavyweight)) = No_Type_Args
668 | general_type_arg_policy type_sys =
669 if level_of_type_sys type_sys = No_Types then
671 else if polymorphism_of_type_sys type_sys = Mangled_Monomorphic then
672 Mangled_Type_Args (should_drop_arg_type_args type_sys)
674 Explicit_Type_Args (should_drop_arg_type_args type_sys)
676 fun type_arg_policy type_sys s =
677 if s = @{const_name HOL.eq} orelse
678 (s = app_op_name andalso level_of_type_sys type_sys = Const_Arg_Types) then
680 else if s = type_tag_name then
681 Explicit_Type_Args false
683 general_type_arg_policy type_sys
685 (*Make literals for sorted type variables*)
686 fun generic_add_sorts_on_type (_, []) = I
687 | generic_add_sorts_on_type ((x, i), s :: ss) =
688 generic_add_sorts_on_type ((x, i), ss)
689 #> (if s = the_single @{sort HOL.type} then
692 insert (op =) (TyLitFree (`make_type_class s, `make_fixed_type_var x))
694 insert (op =) (TyLitVar (`make_type_class s,
695 (make_schematic_type_var (x, i), x))))
696 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
697 | add_sorts_on_tfree _ = I
698 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
699 | add_sorts_on_tvar _ = I
701 fun type_literals_for_types type_sys add_sorts_on_typ Ts =
702 [] |> level_of_type_sys type_sys <> No_Types ? fold add_sorts_on_typ Ts
704 fun mk_aconns c phis =
705 let val (phis', phi') = split_last phis in
706 fold_rev (mk_aconn c) phis' phi'
708 fun mk_ahorn [] phi = phi
709 | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
710 fun mk_aquant _ [] phi = phi
711 | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
712 if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
713 | mk_aquant q xs phi = AQuant (q, xs, phi)
715 fun close_universally atom_vars phi =
717 fun formula_vars bounds (AQuant (_, xs, phi)) =
718 formula_vars (map fst xs @ bounds) phi
719 | formula_vars bounds (AConn (_, phis)) = fold (formula_vars bounds) phis
720 | formula_vars bounds (AAtom tm) =
721 union (op =) (atom_vars tm []
722 |> filter_out (member (op =) bounds o fst))
723 in mk_aquant AForall (formula_vars [] phi []) phi end
725 fun combterm_vars (CombApp (tm1, tm2)) = fold combterm_vars [tm1, tm2]
726 | combterm_vars (CombConst _) = I
727 | combterm_vars (CombVar (name, T)) = insert (op =) (name, SOME T)
728 fun close_combformula_universally phi = close_universally combterm_vars phi
730 fun term_vars (ATerm (name as (s, _), tms)) =
731 is_tptp_variable s ? insert (op =) (name, NONE) #> fold term_vars tms
732 fun close_formula_universally phi = close_universally term_vars phi
734 val homo_infinite_type_name = @{type_name ind} (* any infinite type *)
735 val homo_infinite_type = Type (homo_infinite_type_name, [])
737 fun fo_term_from_typ format type_sys =
739 fun term (Type (s, Ts)) =
740 ATerm (case (is_setting_higher_order format type_sys, s) of
741 (true, @{type_name bool}) => `I tptp_bool_type
742 | (true, @{type_name fun}) => `I tptp_fun_type
743 | _ => if s = homo_infinite_type_name andalso
744 (format = TFF orelse format = THF) then
745 `I tptp_individual_type
747 `make_fixed_type_const s,
749 | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
750 | term (TVar ((x as (s, _)), _)) =
751 ATerm ((make_schematic_type_var x, s), [])
754 fun fo_term_for_type_arg format type_sys T =
755 if T = dummyT then NONE else SOME (fo_term_from_typ format type_sys T)
757 (* This shouldn't clash with anything else. *)
758 val mangled_type_sep = "\000"
760 fun generic_mangled_type_name f (ATerm (name, [])) = f name
761 | generic_mangled_type_name f (ATerm (name, tys)) =
762 f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
765 val bool_atype = AType (`I tptp_bool_type)
767 fun make_simple_type s =
768 if s = tptp_bool_type orelse s = tptp_fun_type orelse
769 s = tptp_individual_type then
772 simple_type_prefix ^ ascii_of s
774 fun ho_type_from_fo_term format type_sys pred_sym ary =
777 AType ((make_simple_type (generic_mangled_type_name fst ty),
778 generic_mangled_type_name snd ty))
779 fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
780 fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
781 | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
782 fun to_ho (ty as ATerm ((s, _), tys)) =
783 if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
784 in if is_setting_higher_order format type_sys then to_ho else to_fo ary end
786 fun mangled_type format type_sys pred_sym ary =
787 ho_type_from_fo_term format type_sys pred_sym ary
788 o fo_term_from_typ format type_sys
790 fun mangled_const_name format type_sys T_args (s, s') =
792 val ty_args = T_args |> map_filter (fo_term_for_type_arg format type_sys)
793 fun type_suffix f g =
794 fold_rev (curry (op ^) o g o prefix mangled_type_sep
795 o generic_mangled_type_name f) ty_args ""
796 in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
798 val parse_mangled_ident =
799 Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
801 fun parse_mangled_type x =
803 -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
805 and parse_mangled_types x =
806 (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
808 fun unmangled_type s =
809 s |> suffix ")" |> raw_explode
810 |> Scan.finite Symbol.stopper
811 (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
812 quote s)) parse_mangled_type))
815 val unmangled_const_name = space_explode mangled_type_sep #> hd
816 fun unmangled_const s =
817 let val ss = space_explode mangled_type_sep s in
818 (hd ss, map unmangled_type (tl ss))
821 fun introduce_proxies format type_sys =
823 fun intro top_level (CombApp (tm1, tm2)) =
824 CombApp (intro top_level tm1, intro false tm2)
825 | intro top_level (CombConst (name as (s, _), T, T_args)) =
826 (case proxify_const s of
828 if top_level orelse is_setting_higher_order format type_sys then
829 case (top_level, s) of
830 (_, "c_False") => (`I tptp_false, [])
831 | (_, "c_True") => (`I tptp_true, [])
832 | (false, "c_Not") => (`I tptp_not, [])
833 | (false, "c_conj") => (`I tptp_and, [])
834 | (false, "c_disj") => (`I tptp_or, [])
835 | (false, "c_implies") => (`I tptp_implies, [])
837 if is_tptp_equal s then (`I tptp_equal, [])
838 else (proxy_base |>> prefix const_prefix, T_args)
841 (proxy_base |>> prefix const_prefix, T_args)
842 | NONE => (name, T_args))
843 |> (fn (name, T_args) => CombConst (name, T, T_args))
847 fun combformula_from_prop thy format type_sys eq_as_iff =
849 fun do_term bs t atomic_types =
850 combterm_from_term thy bs (Envir.eta_contract t)
851 |>> (introduce_proxies format type_sys #> AAtom)
852 ||> union (op =) atomic_types
853 fun do_quant bs q s T t' =
854 let val s = singleton (Name.variant_list (map fst bs)) s in
855 do_formula ((s, T) :: bs) t'
856 #>> mk_aquant q [(`make_bound_var s, SOME T)]
858 and do_conn bs c t1 t2 =
859 do_formula bs t1 ##>> do_formula bs t2 #>> uncurry (mk_aconn c)
860 and do_formula bs t =
862 @{const Trueprop} $ t1 => do_formula bs t1
863 | @{const Not} $ t1 => do_formula bs t1 #>> mk_anot
864 | Const (@{const_name All}, _) $ Abs (s, T, t') =>
865 do_quant bs AForall s T t'
866 | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
867 do_quant bs AExists s T t'
868 | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd t1 t2
869 | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr t1 t2
870 | @{const HOL.implies} $ t1 $ t2 => do_conn bs AImplies t1 t2
871 | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
872 if eq_as_iff then do_conn bs AIff t1 t2 else do_term bs t
876 fun presimplify_term _ [] t = t
877 | presimplify_term ctxt presimp_consts t =
878 t |> exists_Const (member (op =) presimp_consts o fst) t
879 ? (Skip_Proof.make_thm (Proof_Context.theory_of ctxt)
880 #> Meson.presimplify ctxt
883 fun concealed_bound_name j = sledgehammer_weak_prefix ^ string_of_int j
884 fun conceal_bounds Ts t =
885 subst_bounds (map (Free o apfst concealed_bound_name)
886 (0 upto length Ts - 1 ~~ Ts), t)
887 fun reveal_bounds Ts =
888 subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
889 (0 upto length Ts - 1 ~~ Ts))
891 fun is_fun_equality (@{const_name HOL.eq},
892 Type (_, [Type (@{type_name fun}, _), _])) = true
893 | is_fun_equality _ = false
895 fun extensionalize_term ctxt t =
896 if exists_Const is_fun_equality t then
897 let val thy = Proof_Context.theory_of ctxt in
898 t |> cterm_of thy |> Meson.extensionalize_conv ctxt
899 |> prop_of |> Logic.dest_equals |> snd
904 fun introduce_combinators_in_term ctxt kind t =
905 let val thy = Proof_Context.theory_of ctxt in
906 if Meson.is_fol_term thy t then
912 @{const Not} $ t1 => @{const Not} $ aux Ts t1
913 | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
914 t0 $ Abs (s, T, aux (T :: Ts) t')
915 | (t0 as Const (@{const_name All}, _)) $ t1 =>
916 aux Ts (t0 $ eta_expand Ts t1 1)
917 | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
918 t0 $ Abs (s, T, aux (T :: Ts) t')
919 | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
920 aux Ts (t0 $ eta_expand Ts t1 1)
921 | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
922 | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
923 | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
924 | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
926 t0 $ aux Ts t1 $ aux Ts t2
927 | _ => if not (exists_subterm (fn Abs _ => true | _ => false) t) then
930 t |> conceal_bounds Ts
931 |> Envir.eta_contract
933 |> Meson_Clausify.introduce_combinators_in_cterm
934 |> prop_of |> Logic.dest_equals |> snd
936 val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
937 in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
939 (* A type variable of sort "{}" will make abstraction fail. *)
940 if kind = Conjecture then HOLogic.false_const
941 else HOLogic.true_const
944 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
945 same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
948 fun aux (t $ u) = aux t $ aux u
949 | aux (Abs (s, T, t)) = Abs (s, T, aux t)
950 | aux (Var ((s, i), T)) =
951 Free (sledgehammer_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
953 in t |> exists_subterm is_Var t ? aux end
955 fun preprocess_prop ctxt presimp_consts kind t =
957 val thy = Proof_Context.theory_of ctxt
958 val t = t |> Envir.beta_eta_contract
959 |> transform_elim_prop
960 |> Object_Logic.atomize_term thy
961 val need_trueprop = (fastype_of t = @{typ bool})
963 t |> need_trueprop ? HOLogic.mk_Trueprop
964 |> Raw_Simplifier.rewrite_term thy (Meson.unfold_set_const_simps ctxt) []
965 |> extensionalize_term ctxt
966 |> presimplify_term ctxt presimp_consts
967 |> perhaps (try (HOLogic.dest_Trueprop))
968 |> introduce_combinators_in_term ctxt kind
971 (* making fact and conjecture formulas *)
972 fun make_formula thy format type_sys eq_as_iff name loc kind t =
974 val (combformula, atomic_types) =
975 combformula_from_prop thy format type_sys eq_as_iff t []
977 {name = name, locality = loc, kind = kind, combformula = combformula,
978 atomic_types = atomic_types}
981 fun make_fact ctxt format type_sys eq_as_iff preproc presimp_consts
983 let val thy = Proof_Context.theory_of ctxt in
984 case t |> preproc ? preprocess_prop ctxt presimp_consts Axiom
985 |> make_formula thy format type_sys (eq_as_iff andalso format <> CNF)
987 formula as {combformula = AAtom (CombConst ((s, _), _, _)), ...} =>
988 if s = tptp_true then NONE else SOME formula
989 | formula => SOME formula
992 fun make_conjecture ctxt format prem_kind type_sys preproc presimp_consts ts =
994 val thy = Proof_Context.theory_of ctxt
995 val last = length ts - 1
997 map2 (fn j => fn t =>
999 val (kind, maybe_negate) =
1004 if prem_kind = Conjecture then update_combformula mk_anot
1008 (preprocess_prop ctxt presimp_consts kind #> freeze_term)
1009 |> make_formula thy format type_sys (format <> CNF)
1010 (string_of_int j) Local kind
1016 (** Finite and infinite type inference **)
1018 fun deep_freeze_atyp (TVar (_, S)) = TFree ("v", S)
1019 | deep_freeze_atyp T = T
1020 val deep_freeze_type = map_atyps deep_freeze_atyp
1022 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
1023 dangerous because their "exhaust" properties can easily lead to unsound ATP
1024 proofs. On the other hand, all HOL infinite types can be given the same
1025 models in first-order logic (via Löwenheim-Skolem). *)
1027 fun should_encode_type ctxt (nonmono_Ts as _ :: _) _ T =
1028 exists (curry (type_instance ctxt) (deep_freeze_type T)) nonmono_Ts
1029 | should_encode_type _ _ All_Types _ = true
1030 | should_encode_type ctxt _ Fin_Nonmono_Types T = is_type_surely_finite ctxt T
1031 | should_encode_type _ _ _ _ = false
1033 fun should_predicate_on_type ctxt nonmono_Ts (Preds (_, level, heaviness))
1034 should_predicate_on_var T =
1035 (heaviness = Heavyweight orelse should_predicate_on_var ()) andalso
1036 should_encode_type ctxt nonmono_Ts level T
1037 | should_predicate_on_type _ _ _ _ _ = false
1039 fun is_var_or_bound_var (CombConst ((s, _), _, _)) =
1040 String.isPrefix bound_var_prefix s
1041 | is_var_or_bound_var (CombVar _) = true
1042 | is_var_or_bound_var _ = false
1045 Top_Level of bool option |
1046 Eq_Arg of bool option |
1049 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
1050 | should_tag_with_type ctxt nonmono_Ts (Tags (poly, level, heaviness)) site
1053 Heavyweight => should_encode_type ctxt nonmono_Ts level T
1055 case (site, is_var_or_bound_var u) of
1056 (Eq_Arg pos, true) =>
1057 (* The first disjunct prevents a subtle soundness issue explained in
1058 Blanchette's Ph.D. thesis. See also
1059 "formula_lines_for_lightweight_tags_sym_decl". *)
1060 (pos <> SOME false andalso poly = Polymorphic andalso
1061 level <> All_Types andalso heaviness = Lightweight andalso
1062 exists (fn T' => type_instance ctxt (T', T)) nonmono_Ts) orelse
1063 should_encode_type ctxt nonmono_Ts level T
1065 | should_tag_with_type _ _ _ _ _ _ = false
1067 fun homogenized_type ctxt nonmono_Ts level =
1069 val should_encode = should_encode_type ctxt nonmono_Ts level
1070 fun homo 0 T = if should_encode T then T else homo_infinite_type
1071 | homo ary (Type (@{type_name fun}, [T1, T2])) =
1072 homo 0 T1 --> homo (ary - 1) T2
1073 | homo _ _ = raise Fail "expected function type"
1076 (** "hBOOL" and "hAPP" **)
1079 {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
1081 fun add_combterm_syms_to_table ctxt explicit_apply =
1083 fun consider_var_arity const_T var_T max_ary =
1086 if ary = max_ary orelse type_instance ctxt (var_T, T) orelse
1087 type_instance ctxt (T, var_T) then
1090 iter (ary + 1) (range_type T)
1091 in iter 0 const_T end
1092 fun add_var_or_bound_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1093 if explicit_apply = NONE andalso
1094 (can dest_funT T orelse T = @{typ bool}) then
1096 val bool_vars' = bool_vars orelse body_type T = @{typ bool}
1097 fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
1098 {pred_sym = pred_sym andalso not bool_vars',
1099 min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
1100 max_ary = max_ary, types = types}
1102 fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
1104 if bool_vars' = bool_vars andalso
1105 pointer_eq (fun_var_Ts', fun_var_Ts) then
1108 ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
1112 fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1113 let val (head, args) = strip_combterm_comb tm in
1115 CombConst ((s, _), T, _) =>
1116 if String.isPrefix bound_var_prefix s then
1117 add_var_or_bound_var T accum
1119 let val ary = length args in
1120 ((bool_vars, fun_var_Ts),
1121 case Symtab.lookup sym_tab s of
1122 SOME {pred_sym, min_ary, max_ary, types} =>
1125 pred_sym andalso top_level andalso not bool_vars
1126 val types' = types |> insert_type ctxt I T
1128 if is_some explicit_apply orelse
1129 pointer_eq (types', types) then
1132 fold (consider_var_arity T) fun_var_Ts min_ary
1134 Symtab.update (s, {pred_sym = pred_sym,
1135 min_ary = Int.min (ary, min_ary),
1136 max_ary = Int.max (ary, max_ary),
1142 val pred_sym = top_level andalso not bool_vars
1144 case explicit_apply of
1147 | NONE => fold (consider_var_arity T) fun_var_Ts ary
1149 Symtab.update_new (s, {pred_sym = pred_sym,
1150 min_ary = min_ary, max_ary = ary,
1155 | CombVar (_, T) => add_var_or_bound_var T accum
1157 |> fold (add false) args
1160 fun add_fact_syms_to_table ctxt explicit_apply =
1161 fact_lift (formula_fold NONE
1162 (K (add_combterm_syms_to_table ctxt explicit_apply)))
1164 val default_sym_tab_entries : (string * sym_info) list =
1165 (prefixed_predicator_name,
1166 {pred_sym = true, min_ary = 1, max_ary = 1, types = []}) ::
1167 ([tptp_false, tptp_true]
1168 |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
1169 ([tptp_equal, tptp_old_equal]
1170 |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
1172 fun sym_table_for_facts ctxt explicit_apply facts =
1173 ((false, []), Symtab.empty)
1174 |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
1175 |> fold Symtab.update default_sym_tab_entries
1177 fun min_arity_of sym_tab s =
1178 case Symtab.lookup sym_tab s of
1179 SOME ({min_ary, ...} : sym_info) => min_ary
1181 case strip_prefix_and_unascii const_prefix s of
1183 let val s = s |> unmangled_const_name |> invert_const in
1184 if s = predicator_name then 1
1185 else if s = app_op_name then 2
1186 else if s = type_pred_name then 1
1191 (* True if the constant ever appears outside of the top-level position in
1192 literals, or if it appears with different arities (e.g., because of different
1193 type instantiations). If false, the constant always receives all of its
1194 arguments and is used as a predicate. *)
1195 fun is_pred_sym sym_tab s =
1196 case Symtab.lookup sym_tab s of
1197 SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
1198 pred_sym andalso min_ary = max_ary
1201 val predicator_combconst =
1202 CombConst (`make_fixed_const predicator_name, @{typ "bool => bool"}, [])
1203 fun predicator tm = CombApp (predicator_combconst, tm)
1205 fun introduce_predicators_in_combterm sym_tab tm =
1206 case strip_combterm_comb tm of
1207 (CombConst ((s, _), _, _), _) =>
1208 if is_pred_sym sym_tab s then tm else predicator tm
1209 | _ => predicator tm
1211 fun list_app head args = fold (curry (CombApp o swap)) args head
1213 val app_op = `make_fixed_const app_op_name
1215 fun explicit_app arg head =
1217 val head_T = combtyp_of head
1218 val (arg_T, res_T) = dest_funT head_T
1220 CombConst (app_op, head_T --> head_T, [arg_T, res_T])
1221 in list_app explicit_app [head, arg] end
1222 fun list_explicit_app head args = fold explicit_app args head
1224 fun introduce_explicit_apps_in_combterm sym_tab =
1227 case strip_combterm_comb tm of
1228 (head as CombConst ((s, _), _, _), args) =>
1230 |> chop (min_arity_of sym_tab s)
1232 |-> list_explicit_app
1233 | (head, args) => list_explicit_app head (map aux args)
1236 fun chop_fun 0 T = ([], T)
1237 | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
1238 chop_fun (n - 1) ran_T |>> cons dom_T
1239 | chop_fun _ _ = raise Fail "unexpected non-function"
1241 fun filter_type_args _ _ _ [] = []
1242 | filter_type_args thy s arity T_args =
1244 (* will throw "TYPE" for pseudo-constants *)
1245 val U = if s = app_op_name then
1246 @{typ "('a => 'b) => 'a => 'b"} |> Logic.varifyT_global
1248 s |> Sign.the_const_type thy
1250 case Term.add_tvarsT (U |> chop_fun arity |> snd) [] of
1253 let val U_args = (s, U) |> Sign.const_typargs thy in
1255 |> map (fn (U, T) =>
1256 if member (op =) res_U_vars (dest_TVar U) then T
1260 handle TYPE _ => T_args
1262 fun enforce_type_arg_policy_in_combterm ctxt format type_sys =
1264 val thy = Proof_Context.theory_of ctxt
1265 fun aux arity (CombApp (tm1, tm2)) =
1266 CombApp (aux (arity + 1) tm1, aux 0 tm2)
1267 | aux arity (CombConst (name as (s, _), T, T_args)) =
1268 (case strip_prefix_and_unascii const_prefix s of
1269 NONE => (name, T_args)
1272 val s'' = invert_const s''
1273 fun filtered_T_args false = T_args
1274 | filtered_T_args true = filter_type_args thy s'' arity T_args
1276 case type_arg_policy type_sys s'' of
1277 Explicit_Type_Args drop_args =>
1278 (name, filtered_T_args drop_args)
1279 | Mangled_Type_Args drop_args =>
1280 (mangled_const_name format type_sys (filtered_T_args drop_args)
1282 | No_Type_Args => (name, [])
1284 |> (fn (name, T_args) => CombConst (name, T, T_args))
1288 fun repair_combterm ctxt format type_sys sym_tab =
1289 not (is_setting_higher_order format type_sys)
1290 ? (introduce_explicit_apps_in_combterm sym_tab
1291 #> introduce_predicators_in_combterm sym_tab)
1292 #> enforce_type_arg_policy_in_combterm ctxt format type_sys
1293 fun repair_fact ctxt format type_sys sym_tab =
1294 update_combformula (formula_map
1295 (repair_combterm ctxt format type_sys sym_tab))
1297 (** Helper facts **)
1299 (* The Boolean indicates that a fairly sound type encoding is needed. *)
1301 [(("COMBI", false), @{thms Meson.COMBI_def}),
1302 (("COMBK", false), @{thms Meson.COMBK_def}),
1303 (("COMBB", false), @{thms Meson.COMBB_def}),
1304 (("COMBC", false), @{thms Meson.COMBC_def}),
1305 (("COMBS", false), @{thms Meson.COMBS_def}),
1307 (* This is a lie: Higher-order equality doesn't need a sound type encoding.
1308 However, this is done so for backward compatibility: Including the
1309 equality helpers by default in Metis breaks a few existing proofs. *)
1310 @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1311 fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1312 (("fFalse", false), [@{lemma "~ fFalse" by (unfold fFalse_def) fast}]),
1313 (("fFalse", true), @{thms True_or_False}),
1314 (("fTrue", false), [@{lemma "fTrue" by (unfold fTrue_def) fast}]),
1315 (("fTrue", true), @{thms True_or_False}),
1317 @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1318 fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1320 @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
1321 by (unfold fconj_def) fast+}),
1323 @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
1324 by (unfold fdisj_def) fast+}),
1325 (("fimplies", false),
1326 @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
1327 by (unfold fimplies_def) fast+}),
1328 (("If", true), @{thms if_True if_False True_or_False})]
1329 |> map (apsnd (map zero_var_indexes))
1331 val type_tag = `make_fixed_const type_tag_name
1333 fun type_tag_idempotence_fact () =
1335 fun var s = ATerm (`I s, [])
1336 fun tag tm = ATerm (type_tag, [var "T", tm])
1337 val tagged_a = tag (var "A")
1339 Formula (type_tag_idempotence_helper_name, Axiom,
1340 AAtom (ATerm (`I tptp_equal, [tag tagged_a, tagged_a]))
1341 |> close_formula_universally, simp_info, NONE)
1344 fun should_specialize_helper type_sys t =
1345 case general_type_arg_policy type_sys of
1346 Mangled_Type_Args _ => not (null (Term.hidden_polymorphism t))
1349 fun helper_facts_for_sym ctxt format type_sys (s, {types, ...} : sym_info) =
1350 case strip_prefix_and_unascii const_prefix s of
1353 val thy = Proof_Context.theory_of ctxt
1354 val unmangled_s = mangled_s |> unmangled_const_name
1355 fun dub_and_inst needs_fairly_sound (th, j) =
1356 ((unmangled_s ^ "_" ^ string_of_int j ^
1357 (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
1358 (if needs_fairly_sound then typed_helper_suffix
1359 else untyped_helper_suffix),
1361 let val t = th |> prop_of in
1362 t |> should_specialize_helper type_sys t
1364 [T] => specialize_type thy (invert_const unmangled_s, T)
1368 map_filter (make_fact ctxt format type_sys false false [])
1369 val fairly_sound = is_type_sys_fairly_sound type_sys
1372 |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
1373 if helper_s <> unmangled_s orelse
1374 (needs_fairly_sound andalso not fairly_sound) then
1377 ths ~~ (1 upto length ths)
1378 |> map (dub_and_inst needs_fairly_sound)
1382 fun helper_facts_for_sym_table ctxt format type_sys sym_tab =
1383 Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_sys) sym_tab
1386 (***************************************************************)
1387 (* Type Classes Present in the Axiom or Conjecture Clauses *)
1388 (***************************************************************)
1390 fun set_insert (x, s) = Symtab.update (x, ()) s
1392 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
1394 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
1395 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
1397 fun classes_of_terms get_Ts =
1398 map (map snd o get_Ts)
1399 #> List.foldl add_classes Symtab.empty
1400 #> delete_type #> Symtab.keys
1402 val tfree_classes_of_terms = classes_of_terms OldTerm.term_tfrees
1403 val tvar_classes_of_terms = classes_of_terms OldTerm.term_tvars
1405 (*fold type constructors*)
1406 fun fold_type_constrs f (Type (a, Ts)) x =
1407 fold (fold_type_constrs f) Ts (f (a,x))
1408 | fold_type_constrs _ _ x = x
1410 (*Type constructors used to instantiate overloaded constants are the only ones needed.*)
1411 fun add_type_constrs_in_term thy =
1413 fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
1414 | add (t $ u) = add t #> add u
1415 | add (Const (x as (s, _))) =
1416 if String.isPrefix skolem_const_prefix s then I
1417 else x |> Sign.const_typargs thy |> fold (fold_type_constrs set_insert)
1418 | add (Abs (_, _, u)) = add u
1422 fun type_constrs_of_terms thy ts =
1423 Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
1425 fun translate_formulas ctxt format prem_kind type_sys preproc hyp_ts concl_t
1428 val thy = Proof_Context.theory_of ctxt
1429 val fact_ts = facts |> map snd
1430 val presimp_consts = Meson.presimplified_consts ctxt
1431 val make_fact = make_fact ctxt format type_sys true preproc presimp_consts
1432 val (facts, fact_names) =
1433 facts |> map (fn (name, t) => (name, t) |> make_fact |> rpair name)
1434 |> map_filter (try (apfst the))
1436 (* Remove existing facts from the conjecture, as this can dramatically
1437 boost an ATP's performance (for some reason). *)
1440 |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
1441 val goal_t = Logic.list_implies (hyp_ts, concl_t)
1442 val all_ts = goal_t :: fact_ts
1443 val subs = tfree_classes_of_terms all_ts
1444 val supers = tvar_classes_of_terms all_ts
1445 val tycons = type_constrs_of_terms thy all_ts
1448 |> make_conjecture ctxt format prem_kind type_sys preproc presimp_consts
1449 val (supers', arity_clauses) =
1450 if level_of_type_sys type_sys = No_Types then ([], [])
1451 else make_arity_clauses thy tycons supers
1452 val class_rel_clauses = make_class_rel_clauses thy subs supers'
1454 (fact_names |> map single, (conjs, facts, class_rel_clauses, arity_clauses))
1457 fun fo_literal_from_type_literal (TyLitVar (class, name)) =
1458 (true, ATerm (class, [ATerm (name, [])]))
1459 | fo_literal_from_type_literal (TyLitFree (class, name)) =
1460 (true, ATerm (class, [ATerm (name, [])]))
1462 fun formula_from_fo_literal (pos, t) = AAtom t |> not pos ? mk_anot
1464 val type_pred = `make_fixed_const type_pred_name
1466 fun type_pred_combterm ctxt format type_sys T tm =
1467 CombApp (CombConst (type_pred, T --> @{typ bool}, [T])
1468 |> enforce_type_arg_policy_in_combterm ctxt format type_sys, tm)
1470 fun var_occurs_positively_naked_in_term _ (SOME false) _ accum = accum
1471 | var_occurs_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
1472 accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
1473 fun is_var_nonmonotonic_in_formula _ _ (SOME false) _ = false
1474 | is_var_nonmonotonic_in_formula pos phi _ name =
1475 formula_fold pos (var_occurs_positively_naked_in_term name) phi false
1477 fun mk_const_aterm format type_sys x T_args args =
1478 ATerm (x, map_filter (fo_term_for_type_arg format type_sys) T_args @ args)
1480 fun tag_with_type ctxt format nonmono_Ts type_sys pos T tm =
1481 CombConst (type_tag, T --> T, [T])
1482 |> enforce_type_arg_policy_in_combterm ctxt format type_sys
1483 |> term_from_combterm ctxt format nonmono_Ts type_sys (Top_Level pos)
1484 |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm]))
1485 and term_from_combterm ctxt format nonmono_Ts type_sys =
1489 val (head, args) = strip_combterm_comb u
1490 val (x as (s, _), T_args) =
1492 CombConst (name, _, T_args) => (name, T_args)
1493 | CombVar (name, _) => (name, [])
1494 | CombApp _ => raise Fail "impossible \"CombApp\""
1495 val (pos, arg_site) =
1498 (pos, if is_tptp_equal s then Eq_Arg pos else Elsewhere)
1499 | Eq_Arg pos => (pos, Elsewhere)
1500 | Elsewhere => (NONE, Elsewhere)
1501 val t = mk_const_aterm format type_sys x T_args
1502 (map (aux arg_site) args)
1503 val T = combtyp_of u
1505 t |> (if should_tag_with_type ctxt nonmono_Ts type_sys site u T then
1506 tag_with_type ctxt format nonmono_Ts type_sys pos T
1511 and formula_from_combformula ctxt format nonmono_Ts type_sys
1512 should_predicate_on_var =
1515 term_from_combterm ctxt format nonmono_Ts type_sys (Top_Level pos)
1518 Simple_Types level =>
1519 homogenized_type ctxt nonmono_Ts level 0
1520 #> mangled_type format type_sys false 0 #> SOME
1522 fun do_out_of_bound_type pos phi universal (name, T) =
1523 if should_predicate_on_type ctxt nonmono_Ts type_sys
1524 (fn () => should_predicate_on_var pos phi universal name) T then
1526 |> type_pred_combterm ctxt format type_sys T
1527 |> do_term pos |> AAtom |> SOME
1530 fun do_formula pos (AQuant (q, xs, phi)) =
1532 val phi = phi |> do_formula pos
1533 val universal = Option.map (q = AExists ? not) pos
1535 AQuant (q, xs |> map (apsnd (fn NONE => NONE
1536 | SOME T => do_bound_type T)),
1537 (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
1539 (fn (_, NONE) => NONE
1541 do_out_of_bound_type pos phi universal (s, T))
1545 | do_formula pos (AConn conn) = aconn_map pos do_formula conn
1546 | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
1547 in do_formula o SOME end
1549 fun bound_tvars type_sys Ts =
1550 mk_ahorn (map (formula_from_fo_literal o fo_literal_from_type_literal)
1551 (type_literals_for_types type_sys add_sorts_on_tvar Ts))
1553 fun formula_for_fact ctxt format nonmono_Ts type_sys
1554 ({combformula, atomic_types, ...} : translated_formula) =
1556 |> close_combformula_universally
1557 |> formula_from_combformula ctxt format nonmono_Ts type_sys
1558 is_var_nonmonotonic_in_formula true
1559 |> bound_tvars type_sys atomic_types
1560 |> close_formula_universally
1562 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
1563 of monomorphization). The TPTP explicitly forbids name clashes, and some of
1564 the remote provers might care. *)
1565 fun formula_line_for_fact ctxt format prefix encode freshen nonmono_Ts type_sys
1566 (j, formula as {name, locality, kind, ...}) =
1567 Formula (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^
1569 kind, formula_for_fact ctxt format nonmono_Ts type_sys formula, NONE,
1576 fun formula_line_for_class_rel_clause ({name, subclass, superclass, ...}
1577 : class_rel_clause) =
1578 let val ty_arg = ATerm (`I "T", []) in
1579 Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
1580 AConn (AImplies, [AAtom (ATerm (subclass, [ty_arg])),
1581 AAtom (ATerm (superclass, [ty_arg]))])
1582 |> close_formula_universally, intro_info, NONE)
1585 fun fo_literal_from_arity_literal (TConsLit (c, t, args)) =
1586 (true, ATerm (c, [ATerm (t, map (fn arg => ATerm (arg, [])) args)]))
1587 | fo_literal_from_arity_literal (TVarLit (c, sort)) =
1588 (false, ATerm (c, [ATerm (sort, [])]))
1590 fun formula_line_for_arity_clause ({name, prem_lits, concl_lits, ...}
1592 Formula (arity_clause_prefix ^ ascii_of name, Axiom,
1593 mk_ahorn (map (formula_from_fo_literal o apfst not
1594 o fo_literal_from_arity_literal) prem_lits)
1595 (formula_from_fo_literal
1596 (fo_literal_from_arity_literal concl_lits))
1597 |> close_formula_universally, intro_info, NONE)
1599 fun formula_line_for_conjecture ctxt format nonmono_Ts type_sys
1600 ({name, kind, combformula, atomic_types, ...} : translated_formula) =
1601 Formula (conjecture_prefix ^ name, kind,
1602 formula_from_combformula ctxt format nonmono_Ts type_sys
1603 is_var_nonmonotonic_in_formula false
1604 (close_combformula_universally combformula)
1605 |> bound_tvars type_sys atomic_types
1606 |> close_formula_universally, NONE, NONE)
1608 fun free_type_literals type_sys ({atomic_types, ...} : translated_formula) =
1609 atomic_types |> type_literals_for_types type_sys add_sorts_on_tfree
1610 |> map fo_literal_from_type_literal
1612 fun formula_line_for_free_type j lit =
1613 Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis,
1614 formula_from_fo_literal lit, NONE, NONE)
1615 fun formula_lines_for_free_types type_sys facts =
1617 val litss = map (free_type_literals type_sys) facts
1618 val lits = fold (union (op =)) litss []
1619 in map2 formula_line_for_free_type (0 upto length lits - 1) lits end
1621 (** Symbol declarations **)
1623 fun should_declare_sym type_sys pred_sym s =
1624 is_tptp_user_symbol s andalso not (String.isPrefix bound_var_prefix s) andalso
1626 Simple_Types _ => true
1627 | Tags (_, _, Lightweight) => true
1628 | _ => not pred_sym)
1630 fun sym_decl_table_for_facts ctxt type_sys repaired_sym_tab (conjs, facts) =
1632 fun add_combterm in_conj tm =
1633 let val (head, args) = strip_combterm_comb tm in
1635 CombConst ((s, s'), T, T_args) =>
1636 let val pred_sym = is_pred_sym repaired_sym_tab s in
1637 if should_declare_sym type_sys pred_sym s then
1638 Symtab.map_default (s, [])
1639 (insert_type ctxt #3 (s', T_args, T, pred_sym, length args,
1645 #> fold (add_combterm in_conj) args
1647 fun add_fact in_conj =
1648 fact_lift (formula_fold NONE (K (add_combterm in_conj)))
1651 |> is_type_sys_fairly_sound type_sys
1652 ? (fold (add_fact true) conjs #> fold (add_fact false) facts)
1655 (* These types witness that the type classes they belong to allow infinite
1656 models and hence that any types with these type classes is monotonic. *)
1657 val known_infinite_types =
1658 [@{typ nat}, Type ("Int.int", []), @{typ "nat => bool"}]
1660 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
1661 out with monotonicity" paper presented at CADE 2011. *)
1662 fun add_combterm_nonmonotonic_types _ _ _ (SOME false) _ = I
1663 | add_combterm_nonmonotonic_types ctxt level locality _
1664 (CombApp (CombApp (CombConst ((s, _), Type (_, [T, _]), _), tm1),
1666 (is_tptp_equal s andalso exists is_var_or_bound_var [tm1, tm2] andalso
1668 Noninf_Nonmono_Types =>
1669 not (is_locality_global locality) orelse
1670 not (is_type_surely_infinite ctxt known_infinite_types T)
1671 | Fin_Nonmono_Types => is_type_surely_finite ctxt T
1672 | _ => true)) ? insert_type ctxt I (deep_freeze_type T)
1673 | add_combterm_nonmonotonic_types _ _ _ _ _ = I
1674 fun add_fact_nonmonotonic_types ctxt level ({kind, locality, combformula, ...}
1675 : translated_formula) =
1676 formula_fold (SOME (kind <> Conjecture))
1677 (add_combterm_nonmonotonic_types ctxt level locality) combformula
1678 fun nonmonotonic_types_for_facts ctxt type_sys facts =
1679 let val level = level_of_type_sys type_sys in
1680 if level = Noninf_Nonmono_Types orelse level = Fin_Nonmono_Types then
1681 [] |> fold (add_fact_nonmonotonic_types ctxt level) facts
1682 (* We must add "bool" in case the helper "True_or_False" is added
1683 later. In addition, several places in the code rely on the list of
1684 nonmonotonic types not being empty. *)
1685 |> insert_type ctxt I @{typ bool}
1690 fun decl_line_for_sym ctxt format nonmono_Ts type_sys s
1691 (s', T_args, T, pred_sym, ary, _) =
1693 val (T_arg_Ts, level) =
1695 Simple_Types level => ([], level)
1696 | _ => (replicate (length T_args) homo_infinite_type, No_Types)
1698 Decl (sym_decl_prefix ^ s, (s, s'),
1699 (T_arg_Ts ---> (T |> homogenized_type ctxt nonmono_Ts level ary))
1700 |> mangled_type format type_sys pred_sym (length T_arg_Ts + ary))
1703 fun formula_line_for_preds_sym_decl ctxt format conj_sym_kind nonmono_Ts
1704 poly_nonmono_Ts type_sys n s j (s', T_args, T, _, ary, in_conj) =
1706 val (kind, maybe_negate) =
1707 if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
1709 val (arg_Ts, res_T) = chop_fun ary T
1710 val num_args = length arg_Ts
1712 1 upto num_args |> map (`I o make_bound_var o string_of_int)
1714 bound_names ~~ arg_Ts |> map (fn (name, T) => CombConst (name, T, []))
1715 val sym_needs_arg_types = n > 1 orelse exists (curry (op =) dummyT) T_args
1716 fun should_keep_arg_type T =
1717 sym_needs_arg_types orelse
1718 not (should_predicate_on_type ctxt nonmono_Ts type_sys (K false) T)
1720 arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
1722 Formula (preds_sym_formula_prefix ^ s ^
1723 (if n > 1 then "_" ^ string_of_int j else ""), kind,
1724 CombConst ((s, s'), T, T_args)
1725 |> fold (curry (CombApp o swap)) bounds
1726 |> type_pred_combterm ctxt format type_sys res_T
1727 |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
1728 |> formula_from_combformula ctxt format poly_nonmono_Ts type_sys
1729 (K (K (K (K true)))) true
1730 |> n > 1 ? bound_tvars type_sys (atyps_of T)
1731 |> close_formula_universally
1736 fun formula_lines_for_lightweight_tags_sym_decl ctxt format conj_sym_kind
1737 poly_nonmono_Ts type_sys n s
1738 (j, (s', T_args, T, pred_sym, ary, in_conj)) =
1741 lightweight_tags_sym_formula_prefix ^ s ^
1742 (if n > 1 then "_" ^ string_of_int j else "")
1743 val (kind, maybe_negate) =
1744 if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
1746 val (arg_Ts, res_T) = chop_fun ary T
1748 1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
1749 val bounds = bound_names |> map (fn name => ATerm (name, []))
1750 val cst = mk_const_aterm format type_sys (s, s') T_args
1751 val atomic_Ts = atyps_of T
1753 (if pred_sym then AConn (AIff, map AAtom tms)
1754 else AAtom (ATerm (`I tptp_equal, tms)))
1755 |> bound_tvars type_sys atomic_Ts
1756 |> close_formula_universally
1758 (* See also "should_tag_with_type". *)
1759 fun should_encode T =
1760 should_encode_type ctxt poly_nonmono_Ts All_Types T orelse
1762 Tags (Polymorphic, level, Lightweight) =>
1763 level <> All_Types andalso Monomorph.typ_has_tvars T
1765 val tag_with = tag_with_type ctxt format poly_nonmono_Ts type_sys NONE
1766 val add_formula_for_res =
1767 if should_encode res_T then
1768 cons (Formula (ident_base ^ "_res", kind,
1769 eq [tag_with res_T (cst bounds), cst bounds],
1773 fun add_formula_for_arg k =
1774 let val arg_T = nth arg_Ts k in
1775 if should_encode arg_T then
1776 case chop k bounds of
1777 (bounds1, bound :: bounds2) =>
1778 cons (Formula (ident_base ^ "_arg" ^ string_of_int (k + 1), kind,
1779 eq [cst (bounds1 @ tag_with arg_T bound :: bounds2),
1782 | _ => raise Fail "expected nonempty tail"
1787 [] |> not pred_sym ? add_formula_for_res
1788 |> fold add_formula_for_arg (ary - 1 downto 0)
1791 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
1793 fun problem_lines_for_sym_decls ctxt format conj_sym_kind nonmono_Ts
1794 poly_nonmono_Ts type_sys (s, decls) =
1797 decls |> map (decl_line_for_sym ctxt format nonmono_Ts type_sys s)
1802 decl :: (decls' as _ :: _) =>
1803 let val T = result_type_of_decl decl in
1804 if forall (curry (type_instance ctxt o swap) T
1805 o result_type_of_decl) decls' then
1811 val n = length decls
1813 decls |> filter (should_predicate_on_type ctxt poly_nonmono_Ts type_sys
1815 o result_type_of_decl)
1817 (0 upto length decls - 1, decls)
1818 |-> map2 (formula_line_for_preds_sym_decl ctxt format conj_sym_kind
1819 nonmono_Ts poly_nonmono_Ts type_sys n s)
1821 | Tags (_, _, heaviness) =>
1825 let val n = length decls in
1826 (0 upto n - 1 ~~ decls)
1827 |> maps (formula_lines_for_lightweight_tags_sym_decl ctxt format
1828 conj_sym_kind poly_nonmono_Ts type_sys n s)
1831 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind nonmono_Ts
1832 poly_nonmono_Ts type_sys sym_decl_tab =
1837 |-> fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
1838 nonmono_Ts poly_nonmono_Ts type_sys)
1840 fun needs_type_tag_idempotence (Tags (poly, level, heaviness)) =
1841 poly <> Mangled_Monomorphic andalso
1842 ((level = All_Types andalso heaviness = Lightweight) orelse
1843 level = Noninf_Nonmono_Types orelse level = Fin_Nonmono_Types)
1844 | needs_type_tag_idempotence _ = false
1846 fun offset_of_heading_in_problem _ [] j = j
1847 | offset_of_heading_in_problem needle ((heading, lines) :: problem) j =
1848 if heading = needle then j
1849 else offset_of_heading_in_problem needle problem (j + length lines)
1851 val implicit_declsN = "Should-be-implicit typings"
1852 val explicit_declsN = "Explicit typings"
1853 val factsN = "Relevant facts"
1854 val class_relsN = "Class relationships"
1855 val aritiesN = "Arities"
1856 val helpersN = "Helper facts"
1857 val conjsN = "Conjectures"
1858 val free_typesN = "Type variables"
1860 val explicit_apply = NONE (* for experimental purposes *)
1862 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_sys
1863 freshen_facts readable_names preproc hyp_ts concl_t facts =
1865 val (format, type_sys) = choose_format [format] type_sys
1866 val (fact_names, (conjs, facts, class_rel_clauses, arity_clauses)) =
1867 translate_formulas ctxt format prem_kind type_sys preproc hyp_ts concl_t
1869 val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
1870 val nonmono_Ts = conjs @ facts |> nonmonotonic_types_for_facts ctxt type_sys
1871 val repair = repair_fact ctxt format type_sys sym_tab
1872 val (conjs, facts) = (conjs, facts) |> pairself (map repair)
1873 val repaired_sym_tab =
1874 conjs @ facts |> sym_table_for_facts ctxt (SOME false)
1876 repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_sys
1878 val poly_nonmono_Ts =
1879 if null nonmono_Ts orelse nonmono_Ts = [@{typ bool}] orelse
1880 polymorphism_of_type_sys type_sys <> Polymorphic then
1883 [TVar (("'a", 0), HOLogic.typeS)]
1884 val sym_decl_lines =
1885 (conjs, helpers @ facts)
1886 |> sym_decl_table_for_facts ctxt type_sys repaired_sym_tab
1887 |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind nonmono_Ts
1888 poly_nonmono_Ts type_sys
1890 0 upto length helpers - 1 ~~ helpers
1891 |> map (formula_line_for_fact ctxt format helper_prefix I false
1892 poly_nonmono_Ts type_sys)
1893 |> (if needs_type_tag_idempotence type_sys then
1894 cons (type_tag_idempotence_fact ())
1897 (* Reordering these might confuse the proof reconstruction code or the SPASS
1900 [(explicit_declsN, sym_decl_lines),
1902 map (formula_line_for_fact ctxt format fact_prefix ascii_of
1903 freshen_facts nonmono_Ts type_sys)
1904 (0 upto length facts - 1 ~~ facts)),
1905 (class_relsN, map formula_line_for_class_rel_clause class_rel_clauses),
1906 (aritiesN, map formula_line_for_arity_clause arity_clauses),
1907 (helpersN, helper_lines),
1909 map (formula_line_for_conjecture ctxt format nonmono_Ts type_sys)
1911 (free_typesN, formula_lines_for_free_types type_sys (facts @ conjs))]
1915 CNF => ensure_cnf_problem
1916 | CNF_UEQ => filter_cnf_ueq_problem
1918 |> (if is_format_typed format then
1919 declare_undeclared_syms_in_atp_problem type_decl_prefix
1923 val (problem, pool) = problem |> nice_atp_problem readable_names
1924 val helpers_offset = offset_of_heading_in_problem helpersN problem 0
1926 map_filter (fn (j, {name, ...}) =>
1927 if String.isSuffix typed_helper_suffix name then SOME j
1929 ((helpers_offset + 1 upto helpers_offset + length helpers)
1931 fun add_sym_arity (s, {min_ary, ...} : sym_info) =
1933 case strip_prefix_and_unascii const_prefix s of
1934 SOME s => Symtab.insert (op =) (s, min_ary)
1940 case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
1941 offset_of_heading_in_problem conjsN problem 0,
1942 offset_of_heading_in_problem factsN problem 0,
1943 fact_names |> Vector.fromList,
1945 Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
1949 val conj_weight = 0.0
1950 val hyp_weight = 0.1
1951 val fact_min_weight = 0.2
1952 val fact_max_weight = 1.0
1953 val type_info_default_weight = 0.8
1955 fun add_term_weights weight (ATerm (s, tms)) =
1956 is_tptp_user_symbol s ? Symtab.default (s, weight)
1957 #> fold (add_term_weights weight) tms
1958 fun add_problem_line_weights weight (Formula (_, _, phi, _, _)) =
1959 formula_fold NONE (K (add_term_weights weight)) phi
1960 | add_problem_line_weights _ _ = I
1962 fun add_conjectures_weights [] = I
1963 | add_conjectures_weights conjs =
1964 let val (hyps, conj) = split_last conjs in
1965 add_problem_line_weights conj_weight conj
1966 #> fold (add_problem_line_weights hyp_weight) hyps
1969 fun add_facts_weights facts =
1971 val num_facts = length facts
1973 fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
1974 / Real.fromInt num_facts
1976 map weight_of (0 upto num_facts - 1) ~~ facts
1977 |> fold (uncurry add_problem_line_weights)
1980 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
1981 fun atp_problem_weights problem =
1982 let val get = these o AList.lookup (op =) problem in
1984 |> add_conjectures_weights (get free_typesN @ get conjsN)
1985 |> add_facts_weights (get factsN)
1986 |> fold (fold (add_problem_line_weights type_info_default_weight) o get)
1987 [explicit_declsN, class_relsN, aritiesN]
1989 |> sort (prod_ord Real.compare string_ord o pairself swap)