1 (* Title: HOL/Tools/ATP/atp_problem_generate.ML
2 Author: Fabian Immler, TU Muenchen
4 Author: Jasmin Blanchette, TU Muenchen
6 Translation of HOL to FOL for Metis and Sledgehammer.
9 signature ATP_PROBLEM_GENERATE =
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 atp_format = ATP_Problem.atp_format
15 type formula_kind = ATP_Problem.formula_kind
16 type 'a problem = 'a ATP_Problem.problem
18 datatype scope = Global | Local | Assum | Chained
20 General | Induction | Intro | Inductive | Elim | Simp | Def
21 type stature = scope * status
23 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
24 datatype strictness = Strict | Non_Strict
25 datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars
28 Noninf_Nonmono_Types of strictness * granularity |
29 Fin_Nonmono_Types of granularity |
35 val hide_lamsN : string
38 val combs_and_liftingN : string
39 val combs_or_liftingN : string
40 val lam_liftingN : string
41 val keep_lamsN : string
42 val schematic_var_prefix : string
43 val fixed_var_prefix : string
44 val tvar_prefix : string
45 val tfree_prefix : string
46 val const_prefix : string
47 val type_const_prefix : string
48 val class_prefix : string
49 val lam_lifted_prefix : string
50 val lam_lifted_mono_prefix : string
51 val lam_lifted_poly_prefix : string
52 val skolem_const_prefix : string
53 val old_skolem_const_prefix : string
54 val new_skolem_const_prefix : string
55 val combinator_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 lam_fact_prefix : string
67 val typed_helper_suffix : string
68 val untyped_helper_suffix : 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 native_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 unprefix_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 is_type_enc_higher_order : type_enc -> bool
88 val polymorphism_of_type_enc : type_enc -> polymorphism
89 val level_of_type_enc : type_enc -> type_level
90 val is_type_enc_quasi_sound : type_enc -> bool
91 val is_type_enc_fairly_sound : type_enc -> bool
92 val type_enc_from_string : strictness -> string -> type_enc
93 val adjust_type_enc : atp_format -> type_enc -> type_enc
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 list
98 val helper_table : ((string * bool) * thm list) list
99 val trans_lams_from_string :
100 Proof.context -> type_enc -> string -> term list -> term list * term list
102 val prepare_atp_problem :
103 Proof.context -> atp_format -> formula_kind -> formula_kind -> type_enc
104 -> bool -> string -> bool -> bool -> bool -> term list -> term
105 -> ((string * stature) * term) list
106 -> string problem * string Symtab.table * (string * stature) list vector
107 * (string * term) list * int Symtab.table
108 val atp_problem_selection_weights : string problem -> (string * real) list
109 val atp_problem_term_order_info : string problem -> (string * int) list
112 structure ATP_Problem_Generate : ATP_PROBLEM_GENERATE =
118 type name = string * string
120 val no_lamsN = "no_lams" (* used internally; undocumented *)
121 val hide_lamsN = "hide_lams"
122 val liftingN = "lifting"
124 val combs_and_liftingN = "combs_and_lifting"
125 val combs_or_liftingN = "combs_or_lifting"
126 val keep_lamsN = "keep_lams"
127 val lam_liftingN = "lam_lifting" (* legacy *)
129 (* It's still unclear whether all TFF1 implementations will support type
130 signatures such as "!>[A : $tType] : $o", with ghost type variables. *)
131 val avoid_first_order_ghost_type_vars = false
133 val bound_var_prefix = "B_"
134 val all_bound_var_prefix = "A_"
135 val exist_bound_var_prefix = "E_"
136 val schematic_var_prefix = "V_"
137 val fixed_var_prefix = "v_"
138 val tvar_prefix = "T_"
139 val tfree_prefix = "t_"
140 val const_prefix = "c_"
141 val type_const_prefix = "tc_"
142 val native_type_prefix = "n_"
143 val class_prefix = "cl_"
145 (* Freshness almost guaranteed! *)
146 val atp_prefix = "ATP" ^ Long_Name.separator
147 val atp_weak_prefix = "ATP:"
149 val lam_lifted_prefix = atp_weak_prefix ^ "Lam"
150 val lam_lifted_mono_prefix = lam_lifted_prefix ^ "m"
151 val lam_lifted_poly_prefix = lam_lifted_prefix ^ "p"
153 val skolem_const_prefix = atp_prefix ^ "Sko"
154 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
155 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
157 val combinator_prefix = "COMB"
159 val type_decl_prefix = "ty_"
160 val sym_decl_prefix = "sy_"
161 val guards_sym_formula_prefix = "gsy_"
162 val tags_sym_formula_prefix = "tsy_"
163 val uncurried_alias_eq_prefix = "unc_"
164 val fact_prefix = "fact_"
165 val conjecture_prefix = "conj_"
166 val helper_prefix = "help_"
167 val class_rel_clause_prefix = "clar_"
168 val arity_clause_prefix = "arity_"
169 val tfree_clause_prefix = "tfree_"
171 val lam_fact_prefix = "ATP.lambda_"
172 val typed_helper_suffix = "_T"
173 val untyped_helper_suffix = "_U"
175 val predicator_name = "pp"
176 val app_op_name = "aa"
177 val type_guard_name = "gg"
178 val type_tag_name = "tt"
180 val prefixed_predicator_name = const_prefix ^ predicator_name
181 val prefixed_app_op_name = const_prefix ^ app_op_name
182 val prefixed_type_tag_name = const_prefix ^ type_tag_name
184 (*Escaping of special characters.
185 Alphanumeric characters are left unchanged.
186 The character _ goes to __
187 Characters in the range ASCII space to / go to _A to _P, respectively.
188 Other characters go to _nnn where nnn is the decimal ASCII code.*)
189 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
191 fun stringN_of_int 0 _ = ""
192 | stringN_of_int k n =
193 stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
195 fun ascii_of_char c =
196 if Char.isAlphaNum c then
198 else if c = #"_" then
200 else if #" " <= c andalso c <= #"/" then
201 "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
203 (* fixed width, in case more digits follow *)
204 "_" ^ stringN_of_int 3 (Char.ord c)
206 val ascii_of = String.translate ascii_of_char
208 (** Remove ASCII armoring from names in proof files **)
210 (* We don't raise error exceptions because this code can run inside a worker
211 thread. Also, the errors are impossible. *)
214 fun un rcs [] = String.implode(rev rcs)
215 | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
216 (* Three types of _ escapes: __, _A to _P, _nnn *)
217 | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
218 | un rcs (#"_" :: c :: cs) =
219 if #"A" <= c andalso c<= #"P" then
220 (* translation of #" " to #"/" *)
221 un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
223 let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
224 case Int.fromString (String.implode digits) of
225 SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
226 | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
228 | un rcs (c :: cs) = un (c :: rcs) cs
229 in un [] o String.explode end
231 (* If string s has the prefix s1, return the result of deleting it,
233 fun unprefix_and_unascii s1 s =
234 if String.isPrefix s1 s then
235 SOME (unascii_of (String.extract (s, size s1, NONE)))
240 [("c_False", (@{const_name False}, (@{thm fFalse_def},
241 ("fFalse", @{const_name ATP.fFalse})))),
242 ("c_True", (@{const_name True}, (@{thm fTrue_def},
243 ("fTrue", @{const_name ATP.fTrue})))),
244 ("c_Not", (@{const_name Not}, (@{thm fNot_def},
245 ("fNot", @{const_name ATP.fNot})))),
246 ("c_conj", (@{const_name conj}, (@{thm fconj_def},
247 ("fconj", @{const_name ATP.fconj})))),
248 ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
249 ("fdisj", @{const_name ATP.fdisj})))),
250 ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
251 ("fimplies", @{const_name ATP.fimplies})))),
252 ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
253 ("fequal", @{const_name ATP.fequal})))),
254 ("c_All", (@{const_name All}, (@{thm fAll_def},
255 ("fAll", @{const_name ATP.fAll})))),
256 ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
257 ("fEx", @{const_name ATP.fEx}))))]
259 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
261 (* Readable names for the more common symbolic functions. Do not mess with the
262 table unless you know what you are doing. *)
263 val const_trans_table =
264 [(@{type_name Product_Type.prod}, "prod"),
265 (@{type_name Sum_Type.sum}, "sum"),
266 (@{const_name False}, "False"),
267 (@{const_name True}, "True"),
268 (@{const_name Not}, "Not"),
269 (@{const_name conj}, "conj"),
270 (@{const_name disj}, "disj"),
271 (@{const_name implies}, "implies"),
272 (@{const_name HOL.eq}, "equal"),
273 (@{const_name All}, "All"),
274 (@{const_name Ex}, "Ex"),
275 (@{const_name If}, "If"),
276 (@{const_name Set.member}, "member"),
277 (@{const_name Meson.COMBI}, combinator_prefix ^ "I"),
278 (@{const_name Meson.COMBK}, combinator_prefix ^ "K"),
279 (@{const_name Meson.COMBB}, combinator_prefix ^ "B"),
280 (@{const_name Meson.COMBC}, combinator_prefix ^ "C"),
281 (@{const_name Meson.COMBS}, combinator_prefix ^ "S")]
283 |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
285 (* Invert the table of translations between Isabelle and ATPs. *)
286 val const_trans_table_inv =
287 const_trans_table |> Symtab.dest |> map swap |> Symtab.make
288 val const_trans_table_unprox =
290 |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
292 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
293 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
296 case Symtab.lookup const_trans_table c of
300 fun ascii_of_indexname (v, 0) = ascii_of v
301 | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
303 fun make_bound_var x = bound_var_prefix ^ ascii_of x
304 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
305 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
306 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
307 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
309 fun make_schematic_type_var (x, i) =
310 tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
311 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
313 (* "HOL.eq" and choice are mapped to the ATP's equivalents *)
315 val choice_const = (fst o dest_Const o HOLogic.choice_const) Term.dummyT
316 fun default c = const_prefix ^ lookup_const c
318 fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
319 | make_fixed_const (SOME (THF (_, _, THF_With_Choice))) c =
320 if c = choice_const then tptp_choice else default c
321 | make_fixed_const _ c = default c
324 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
326 fun make_type_class clas = class_prefix ^ ascii_of clas
328 fun new_skolem_var_name_from_const s =
329 let val ss = s |> space_explode Long_Name.separator in
330 nth ss (length ss - 2)
333 (* These are either simplified away by "Meson.presimplify" (most of the time) or
334 handled specially via "fFalse", "fTrue", ..., "fequal". *)
335 val atp_irrelevant_consts =
336 [@{const_name False}, @{const_name True}, @{const_name Not},
337 @{const_name conj}, @{const_name disj}, @{const_name implies},
338 @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
340 val atp_monomorph_bad_consts =
341 atp_irrelevant_consts @
342 (* These are ignored anyway by the relevance filter (unless they appear in
343 higher-order places) but not by the monomorphizer. *)
344 [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
345 @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
346 @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
348 fun add_schematic_const (x as (_, T)) =
349 Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
350 val add_schematic_consts_of =
351 Term.fold_aterms (fn Const (x as (s, _)) =>
352 not (member (op =) atp_monomorph_bad_consts s)
353 ? add_schematic_const x
355 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
357 (** Definitions and functions for FOL clauses and formulas for TPTP **)
359 (** Isabelle arities **)
361 type arity_atom = name * name * name list
363 val type_class = the_single @{sort type}
367 prem_atoms : arity_atom list,
368 concl_atom : arity_atom}
370 fun add_prem_atom tvar =
371 fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar, []))
373 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
374 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
376 val tvars = map (prefix tvar_prefix o string_of_int) (1 upto length args)
377 val tvars_srts = ListPair.zip (tvars, args)
380 prem_atoms = [] |> fold (uncurry add_prem_atom) tvars_srts,
381 concl_atom = (`make_type_class cls, `make_fixed_type_const tcons,
385 fun arity_clause _ _ (_, []) = []
386 | arity_clause seen n (tcons, ("HOL.type", _) :: ars) = (* ignore *)
387 arity_clause seen n (tcons, ars)
388 | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
389 if member (op =) seen class then
390 (* multiple arities for the same (tycon, class) pair *)
391 make_axiom_arity_clause (tcons,
392 lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
394 arity_clause seen (n + 1) (tcons, ars)
396 make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
397 ascii_of class, ar) ::
398 arity_clause (class :: seen) n (tcons, ars)
400 fun multi_arity_clause [] = []
401 | multi_arity_clause ((tcons, ars) :: tc_arlists) =
402 arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
404 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
405 theory thy provided its arguments have the corresponding sorts. *)
406 fun type_class_pairs thy tycons classes =
408 val alg = Sign.classes_of thy
409 fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
410 fun add_class tycon class =
411 cons (class, domain_sorts tycon class)
412 handle Sorts.CLASS_ERROR _ => I
413 fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
414 in map try_classes tycons end
416 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
417 fun iter_type_class_pairs _ _ [] = ([], [])
418 | iter_type_class_pairs thy tycons classes =
420 fun maybe_insert_class s =
421 (s <> type_class andalso not (member (op =) classes s))
423 val cpairs = type_class_pairs thy tycons classes
425 [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
426 val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
427 in (classes' @ classes, union (op =) cpairs' cpairs) end
429 fun make_arity_clauses thy tycons =
430 iter_type_class_pairs thy tycons ##> multi_arity_clause
433 (** Isabelle class relations **)
435 type class_rel_clause =
440 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
442 fun class_pairs _ [] _ = []
443 | class_pairs thy subs supers =
445 val class_less = Sorts.class_less (Sign.classes_of thy)
446 fun add_super sub super = class_less (sub, super) ? cons (sub, super)
447 fun add_supers sub = fold (add_super sub) supers
448 in fold add_supers subs [] end
450 fun make_class_rel_clause (sub, super) =
451 {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
452 superclass = `make_type_class super}
454 fun make_class_rel_clauses thy subs supers =
455 map make_class_rel_clause (class_pairs thy subs supers)
457 (* intermediate terms *)
459 IConst of name * typ * typ list |
461 IApp of iterm * iterm |
462 IAbs of (name * typ) * iterm
464 fun ityp_of (IConst (_, T, _)) = T
465 | ityp_of (IVar (_, T)) = T
466 | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
467 | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
469 (*gets the head of a combinator application, along with the list of arguments*)
470 fun strip_iterm_comb u =
472 fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
474 in stripc (u, []) end
476 fun atomic_types_of T = fold_atyps (insert (op =)) T []
478 val tvar_a_str = "'a"
479 val tvar_a = TVar ((tvar_a_str, 0), HOLogic.typeS)
480 val tvar_a_name = (make_schematic_type_var (tvar_a_str, 0), tvar_a_str)
481 val itself_name = `make_fixed_type_const @{type_name itself}
482 val TYPE_name = `(make_fixed_const NONE) @{const_name TYPE}
483 val tvar_a_atype = AType (tvar_a_name, [])
484 val a_itself_atype = AType (itself_name, [tvar_a_atype])
486 fun new_skolem_const_name s num_T_args =
487 [new_skolem_const_prefix, s, string_of_int num_T_args]
490 fun robust_const_type thy s =
491 if s = app_op_name then
492 Logic.varifyT_global @{typ "('a => 'b) => 'a => 'b"}
493 else if String.isPrefix lam_lifted_prefix s then
494 Logic.varifyT_global @{typ "'a => 'b"}
496 (* Old Skolems throw a "TYPE" exception here, which will be caught. *)
497 s |> Sign.the_const_type thy
499 val robust_const_ary =
501 fun ary (Type (@{type_name fun}, [_, T])) = 1 + ary T
503 in ary oo robust_const_type end
505 (* This function only makes sense if "T" is as general as possible. *)
506 fun robust_const_typargs thy (s, T) =
507 if s = app_op_name then
508 let val (T1, T2) = T |> domain_type |> dest_funT in [T1, T2] end
509 else if String.isPrefix old_skolem_const_prefix s then
510 [] |> Term.add_tvarsT T |> rev |> map TVar
511 else if String.isPrefix lam_lifted_prefix s then
512 if String.isPrefix lam_lifted_poly_prefix s then
513 let val (T1, T2) = T |> dest_funT in [T1, T2] end
517 (s, T) |> Sign.const_typargs thy
519 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
520 Also accumulates sort infomation. *)
521 fun iterm_from_term thy format bs (P $ Q) =
523 val (P', P_atomics_Ts) = iterm_from_term thy format bs P
524 val (Q', Q_atomics_Ts) = iterm_from_term thy format bs Q
525 in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
526 | iterm_from_term thy format _ (Const (c, T)) =
527 (IConst (`(make_fixed_const (SOME format)) c, T,
528 robust_const_typargs thy (c, T)),
530 | iterm_from_term _ _ _ (Free (s, T)) =
531 (IConst (`make_fixed_var s, T, []), atomic_types_of T)
532 | iterm_from_term _ format _ (Var (v as (s, _), T)) =
533 (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
535 val Ts = T |> strip_type |> swap |> op ::
536 val s' = new_skolem_const_name s (length Ts)
537 in IConst (`(make_fixed_const (SOME format)) s', T, Ts) end
539 IVar ((make_schematic_var v, s), T), atomic_types_of T)
540 | iterm_from_term _ _ bs (Bound j) =
541 nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atomic_types_of T))
542 | iterm_from_term thy format bs (Abs (s, T, t)) =
544 fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
546 val name = `make_bound_var s
547 val (tm, atomic_Ts) = iterm_from_term thy format ((s, (name, T)) :: bs) t
548 in (IAbs ((name, T), tm), union (op =) atomic_Ts (atomic_types_of T)) end
550 datatype scope = Global | Local | Assum | Chained
551 datatype status = General | Induction | Intro | Inductive | Elim | Simp | Def
552 type stature = scope * status
554 datatype order = First_Order | Higher_Order
555 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
556 datatype strictness = Strict | Non_Strict
557 datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars
558 datatype type_level =
560 Noninf_Nonmono_Types of strictness * granularity |
561 Fin_Nonmono_Types of granularity |
566 Simple_Types of order * polymorphism * type_level |
567 Guards of polymorphism * type_level |
568 Tags of polymorphism * type_level
570 fun is_type_enc_higher_order (Simple_Types (Higher_Order, _, _)) = true
571 | is_type_enc_higher_order _ = false
573 fun polymorphism_of_type_enc (Simple_Types (_, poly, _)) = poly
574 | polymorphism_of_type_enc (Guards (poly, _)) = poly
575 | polymorphism_of_type_enc (Tags (poly, _)) = poly
577 fun level_of_type_enc (Simple_Types (_, _, level)) = level
578 | level_of_type_enc (Guards (_, level)) = level
579 | level_of_type_enc (Tags (_, level)) = level
581 fun granularity_of_type_level (Noninf_Nonmono_Types (_, grain)) = grain
582 | granularity_of_type_level (Fin_Nonmono_Types grain) = grain
583 | granularity_of_type_level _ = All_Vars
585 fun is_type_level_quasi_sound All_Types = true
586 | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
587 | is_type_level_quasi_sound _ = false
588 val is_type_enc_quasi_sound = is_type_level_quasi_sound o level_of_type_enc
590 fun is_type_level_fairly_sound (Fin_Nonmono_Types _) = true
591 | is_type_level_fairly_sound level = is_type_level_quasi_sound level
592 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
594 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
595 | is_type_level_monotonicity_based (Fin_Nonmono_Types _) = true
596 | is_type_level_monotonicity_based _ = false
598 (* "_query", "_bang", and "_at" are for the ASCII-challenged Metis and
600 val queries = ["?", "_query"]
601 val bangs = ["!", "_bang"]
602 val ats = ["@", "_at"]
604 fun try_unsuffixes ss s =
605 fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
607 fun try_nonmono constr suffixes fallback s =
608 case try_unsuffixes suffixes s of
610 (case try_unsuffixes suffixes s of
611 SOME s => (constr Positively_Naked_Vars, s)
613 case try_unsuffixes ats s of
614 SOME s => (constr Ghost_Type_Arg_Vars, s)
615 | NONE => (constr All_Vars, s))
618 fun type_enc_from_string strictness s =
619 (case try (unprefix "poly_") s of
620 SOME s => (SOME Polymorphic, s)
622 case try (unprefix "raw_mono_") s of
623 SOME s => (SOME Raw_Monomorphic, s)
625 case try (unprefix "mono_") s of
626 SOME s => (SOME Mangled_Monomorphic, s)
629 |> try_nonmono Fin_Nonmono_Types bangs
630 |> try_nonmono (curry Noninf_Nonmono_Types strictness) queries)
631 |> (fn (poly, (level, core)) =>
632 case (core, (poly, level)) of
633 ("native", (SOME poly, _)) =>
634 (case (poly, level) of
635 (Polymorphic, All_Types) =>
636 Simple_Types (First_Order, Polymorphic, All_Types)
637 | (Mangled_Monomorphic, _) =>
638 if granularity_of_type_level level = All_Vars then
639 Simple_Types (First_Order, Mangled_Monomorphic, level)
642 | _ => raise Same.SAME)
643 | ("native_higher", (SOME poly, _)) =>
644 (case (poly, level) of
645 (Polymorphic, All_Types) =>
646 Simple_Types (Higher_Order, Polymorphic, All_Types)
647 | (_, Noninf_Nonmono_Types _) => raise Same.SAME
648 | (Mangled_Monomorphic, _) =>
649 if granularity_of_type_level level = All_Vars then
650 Simple_Types (Higher_Order, Mangled_Monomorphic, level)
653 | _ => raise Same.SAME)
654 | ("guards", (SOME poly, _)) =>
655 if poly = Mangled_Monomorphic andalso
656 granularity_of_type_level level = Ghost_Type_Arg_Vars then
660 | ("tags", (SOME poly, _)) =>
661 if granularity_of_type_level level = Ghost_Type_Arg_Vars then
665 | ("args", (SOME poly, All_Types (* naja *))) =>
666 Guards (poly, Const_Arg_Types)
667 | ("erased", (NONE, All_Types (* naja *))) =>
668 Guards (Polymorphic, No_Types)
669 | _ => raise Same.SAME)
670 handle Same.SAME => error ("Unknown type encoding: " ^ quote s ^ ".")
672 fun adjust_type_enc (THF (TPTP_Monomorphic, _, _))
673 (Simple_Types (order, _, level)) =
674 Simple_Types (order, Mangled_Monomorphic, level)
675 | adjust_type_enc (THF _) type_enc = type_enc
676 | adjust_type_enc (TFF (TPTP_Monomorphic, _)) (Simple_Types (_, _, level)) =
677 Simple_Types (First_Order, Mangled_Monomorphic, level)
678 | adjust_type_enc (DFG DFG_Sorted) (Simple_Types (_, _, level)) =
679 Simple_Types (First_Order, Mangled_Monomorphic, level)
680 | adjust_type_enc (TFF _) (Simple_Types (_, poly, level)) =
681 Simple_Types (First_Order, poly, level)
682 | adjust_type_enc format (Simple_Types (_, poly, level)) =
683 adjust_type_enc format (Guards (poly, level))
684 | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
685 (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
686 | adjust_type_enc _ type_enc = type_enc
690 @{const Not} $ t1 => is_fol_term t1
691 | Const (@{const_name All}, _) $ Abs (_, _, t') => is_fol_term t'
692 | Const (@{const_name All}, _) $ t1 => is_fol_term t1
693 | Const (@{const_name Ex}, _) $ Abs (_, _, t') => is_fol_term t'
694 | Const (@{const_name Ex}, _) $ t1 => is_fol_term t1
695 | @{const HOL.conj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
696 | @{const HOL.disj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
697 | @{const HOL.implies} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
698 | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
699 is_fol_term t1 andalso is_fol_term t2
700 | _ => not (exists_subterm (fn Abs _ => true | _ => false) t)
702 fun simple_translate_lambdas do_lambdas ctxt t =
703 if is_fol_term t then
709 @{const Not} $ t1 => @{const Not} $ trans Ts t1
710 | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
711 t0 $ Abs (s, T, trans (T :: Ts) t')
712 | (t0 as Const (@{const_name All}, _)) $ t1 =>
713 trans Ts (t0 $ eta_expand Ts t1 1)
714 | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
715 t0 $ Abs (s, T, trans (T :: Ts) t')
716 | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
717 trans Ts (t0 $ eta_expand Ts t1 1)
718 | (t0 as @{const HOL.conj}) $ t1 $ t2 =>
719 t0 $ trans Ts t1 $ trans Ts t2
720 | (t0 as @{const HOL.disj}) $ t1 $ t2 =>
721 t0 $ trans Ts t1 $ trans Ts t2
722 | (t0 as @{const HOL.implies}) $ t1 $ t2 =>
723 t0 $ trans Ts t1 $ trans Ts t2
724 | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
726 t0 $ trans Ts t1 $ trans Ts t2
728 if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
729 else t |> Envir.eta_contract |> do_lambdas ctxt Ts
730 val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
731 in t |> trans [] |> singleton (Variable.export_terms ctxt' ctxt) end
733 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
734 do_cheaply_conceal_lambdas Ts t1
735 $ do_cheaply_conceal_lambdas Ts t2
736 | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
737 Const (lam_lifted_poly_prefix ^ serial_string (),
738 T --> fastype_of1 (T :: Ts, t))
739 | do_cheaply_conceal_lambdas _ t = t
741 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
742 fun conceal_bounds Ts t =
743 subst_bounds (map (Free o apfst concealed_bound_name)
744 (0 upto length Ts - 1 ~~ Ts), t)
745 fun reveal_bounds Ts =
746 subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
747 (0 upto length Ts - 1 ~~ Ts))
749 fun do_introduce_combinators ctxt Ts t =
750 let val thy = Proof_Context.theory_of ctxt in
751 t |> conceal_bounds Ts
753 |> Meson_Clausify.introduce_combinators_in_cterm
754 |> prop_of |> Logic.dest_equals |> snd
757 (* A type variable of sort "{}" will make abstraction fail. *)
758 handle THM _ => t |> do_cheaply_conceal_lambdas Ts
759 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
761 fun constify_lifted (t $ u) = constify_lifted t $ constify_lifted u
762 | constify_lifted (Abs (s, T, t)) = Abs (s, T, constify_lifted t)
763 | constify_lifted (Free (x as (s, _))) =
764 (if String.isPrefix lam_lifted_prefix s then Const else Free) x
765 | constify_lifted t = t
767 (* Requires bound variables not to clash with any schematic variables (as should
768 be the case right after lambda-lifting). *)
769 fun open_form unprefix (t as Const (@{const_name All}, _) $ Abs (s, T, t')) =
770 (case try unprefix s of
773 val names = Name.make_context (map fst (Term.add_var_names t' []))
774 val (s, _) = Name.variant s names
775 in open_form unprefix (subst_bound (Var ((s, 0), T), t')) end
779 fun lift_lams_part_1 ctxt type_enc =
780 map close_form #> rpair ctxt
781 #-> Lambda_Lifting.lift_lambdas
782 (SOME ((if polymorphism_of_type_enc type_enc = Polymorphic then
783 lam_lifted_poly_prefix
785 lam_lifted_mono_prefix) ^ "_a"))
786 Lambda_Lifting.is_quantifier
789 fun lift_lams_part_2 ctxt (facts, lifted) =
791 (* Lambda-lifting sometimes leaves some lambdas around; we need some way to get rid
793 |> pairself (map (introduce_combinators ctxt))
794 |> pairself (map constify_lifted)
795 |>> map (open_form (unprefix close_form_prefix))
796 ||> map (open_form I)
798 fun lift_lams ctxt = lift_lams_part_2 ctxt oo lift_lams_part_1 ctxt
800 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
802 | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
804 fun lam T t = Abs (Name.uu, T, t)
805 val (head, args) = strip_comb t ||> rev
806 val head_T = fastype_of head
808 val arg_Ts = head_T |> binder_types |> take n |> rev
809 val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
810 in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
811 | intentionalize_def t = t
813 type translated_formula =
817 iformula : (name, typ, iterm) formula,
818 atomic_types : typ list}
820 fun update_iformula f ({name, stature, kind, iformula, atomic_types}
821 : translated_formula) =
822 {name = name, stature = stature, kind = kind, iformula = f iformula,
823 atomic_types = atomic_types} : translated_formula
825 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
827 fun insert_type thy get_T x xs =
828 let val T = get_T x in
829 if exists (type_instance thy T o get_T) xs then xs
830 else x :: filter_out (type_generalization thy T o get_T) xs
833 (* The Booleans indicate whether all type arguments should be kept. *)
834 datatype type_arg_policy =
835 Explicit_Type_Args of bool (* infer_from_term_args *) |
839 fun type_arg_policy monom_constrs type_enc s =
840 let val poly = polymorphism_of_type_enc type_enc in
841 if s = type_tag_name then
842 if poly = Mangled_Monomorphic then Mangled_Type_Args
843 else Explicit_Type_Args false
844 else case type_enc of
845 Simple_Types (_, Polymorphic, _) => Explicit_Type_Args false
846 | Tags (_, All_Types) => No_Type_Args
848 let val level = level_of_type_enc type_enc in
849 if level = No_Types orelse s = @{const_name HOL.eq} orelse
850 (s = app_op_name andalso level = Const_Arg_Types) then
852 else if poly = Mangled_Monomorphic then
854 else if member (op =) monom_constrs s andalso
855 granularity_of_type_level level = Positively_Naked_Vars then
859 (level = All_Types orelse
860 granularity_of_type_level level = Ghost_Type_Arg_Vars)
864 (* Make atoms for sorted type variables. *)
865 fun generic_add_sorts_on_type (_, []) = I
866 | generic_add_sorts_on_type ((x, i), s :: ss) =
867 generic_add_sorts_on_type ((x, i), ss)
868 #> (if s = the_single @{sort HOL.type} then
871 insert (op =) (`make_type_class s, `make_fixed_type_var x)
873 insert (op =) (`make_type_class s,
874 (make_schematic_type_var (x, i), x)))
875 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
876 | add_sorts_on_tfree _ = I
877 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
878 | add_sorts_on_tvar _ = I
880 fun type_class_formula type_enc class arg =
881 AAtom (ATerm (class, arg ::
883 Simple_Types (First_Order, Polymorphic, _) =>
884 if avoid_first_order_ghost_type_vars then [ATerm (TYPE_name, [arg])]
887 fun formulas_for_types type_enc add_sorts_on_typ Ts =
888 [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
889 |> map (fn (class, name) =>
890 type_class_formula type_enc class (ATerm (name, [])))
892 fun mk_aconns c phis =
893 let val (phis', phi') = split_last phis in
894 fold_rev (mk_aconn c) phis' phi'
896 fun mk_ahorn [] phi = phi
897 | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
898 fun mk_aquant _ [] phi = phi
899 | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
900 if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
901 | mk_aquant q xs phi = AQuant (q, xs, phi)
903 fun close_universally add_term_vars phi =
905 fun add_formula_vars bounds (AQuant (_, xs, phi)) =
906 add_formula_vars (map fst xs @ bounds) phi
907 | add_formula_vars bounds (AConn (_, phis)) =
908 fold (add_formula_vars bounds) phis
909 | add_formula_vars bounds (AAtom tm) = add_term_vars bounds tm
910 in mk_aquant AForall (add_formula_vars [] phi []) phi end
912 fun add_term_vars bounds (ATerm (name as (s, _), tms)) =
913 (if is_tptp_variable s andalso
914 not (String.isPrefix tvar_prefix s) andalso
915 not (member (op =) bounds name) then
916 insert (op =) (name, NONE)
919 #> fold (add_term_vars bounds) tms
920 | add_term_vars bounds (AAbs ((name, _), tm)) =
921 add_term_vars (name :: bounds) tm
922 fun close_formula_universally phi = close_universally add_term_vars phi
924 fun add_iterm_vars bounds (IApp (tm1, tm2)) =
925 fold (add_iterm_vars bounds) [tm1, tm2]
926 | add_iterm_vars _ (IConst _) = I
927 | add_iterm_vars bounds (IVar (name, T)) =
928 not (member (op =) bounds name) ? insert (op =) (name, SOME T)
929 | add_iterm_vars bounds (IAbs (_, tm)) = add_iterm_vars bounds tm
930 fun close_iformula_universally phi = close_universally add_iterm_vars phi
932 val fused_infinite_type_name = "ATP.fused_inf" (* shouldn't clash *)
933 val fused_infinite_type = Type (fused_infinite_type_name, [])
935 fun tvar_name (x as (s, _)) = (make_schematic_type_var x, s)
937 fun ho_term_from_typ format type_enc =
939 fun term (Type (s, Ts)) =
940 ATerm (case (is_type_enc_higher_order type_enc, s) of
941 (true, @{type_name bool}) => `I tptp_bool_type
942 | (true, @{type_name fun}) => `I tptp_fun_type
943 | _ => if s = fused_infinite_type_name andalso
944 is_format_typed format then
945 `I tptp_individual_type
947 `make_fixed_type_const s,
949 | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
950 | term (TVar (x, _)) = ATerm (tvar_name x, [])
953 fun ho_term_for_type_arg format type_enc T =
954 if T = dummyT then NONE else SOME (ho_term_from_typ format type_enc T)
956 (* This shouldn't clash with anything else. *)
957 val uncurried_alias_sep = "\000"
958 val mangled_type_sep = "\001"
960 val ascii_of_uncurried_alias_sep = ascii_of uncurried_alias_sep
962 fun generic_mangled_type_name f (ATerm (name, [])) = f name
963 | generic_mangled_type_name f (ATerm (name, tys)) =
964 f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
966 | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
968 fun mangled_type format type_enc =
969 generic_mangled_type_name fst o ho_term_from_typ format type_enc
971 fun make_native_type s =
972 if s = tptp_bool_type orelse s = tptp_fun_type orelse
973 s = tptp_individual_type then
976 native_type_prefix ^ ascii_of s
978 fun ho_type_from_ho_term type_enc pred_sym ary =
980 fun to_mangled_atype ty =
981 AType ((make_native_type (generic_mangled_type_name fst ty),
982 generic_mangled_type_name snd ty), [])
983 fun to_poly_atype (ATerm (name, tys)) = AType (name, map to_poly_atype tys)
984 | to_poly_atype _ = raise Fail "unexpected type abstraction"
986 if polymorphism_of_type_enc type_enc = Polymorphic then to_poly_atype
987 else to_mangled_atype
988 fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
989 fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
990 | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
991 | to_fo _ _ = raise Fail "unexpected type abstraction"
992 fun to_ho (ty as ATerm ((s, _), tys)) =
993 if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
994 | to_ho _ = raise Fail "unexpected type abstraction"
995 in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
997 fun ho_type_from_typ format type_enc pred_sym ary =
998 ho_type_from_ho_term type_enc pred_sym ary
999 o ho_term_from_typ format type_enc
1001 fun aliased_uncurried ary (s, s') =
1002 (s ^ ascii_of_uncurried_alias_sep ^ string_of_int ary, s' ^ string_of_int ary)
1003 fun unaliased_uncurried (s, s') =
1004 case space_explode uncurried_alias_sep s of
1006 | [s1, s2] => (s1, unsuffix s2 s')
1007 | _ => raise Fail "ill-formed explicit application alias"
1009 fun raw_mangled_const_name type_name ty_args (s, s') =
1011 fun type_suffix f g =
1012 fold_rev (curry (op ^) o g o prefix mangled_type_sep o type_name f)
1014 in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
1015 fun mangled_const_name format type_enc =
1016 map_filter (ho_term_for_type_arg format type_enc)
1017 #> raw_mangled_const_name generic_mangled_type_name
1019 val parse_mangled_ident =
1020 Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
1022 fun parse_mangled_type x =
1023 (parse_mangled_ident
1024 -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
1026 and parse_mangled_types x =
1027 (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
1029 fun unmangled_type s =
1030 s |> suffix ")" |> raw_explode
1031 |> Scan.finite Symbol.stopper
1032 (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
1033 quote s)) parse_mangled_type))
1036 fun unmangled_const_name s =
1037 (s, s) |> unaliased_uncurried |> fst |> space_explode mangled_type_sep
1038 fun unmangled_const s =
1039 let val ss = unmangled_const_name s in
1040 (hd ss, map unmangled_type (tl ss))
1043 fun introduce_proxies_in_iterm type_enc =
1045 fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
1046 | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
1048 (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
1049 limitation. This works in conjuction with special code in
1050 "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
1052 IAbs ((`I "P", p_T),
1053 IApp (IConst (`I ho_quant, T, []),
1054 IAbs ((`I "X", x_T),
1055 IApp (IConst (`I "P", p_T, []),
1056 IConst (`I "X", x_T, [])))))
1057 | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
1058 fun intro top_level args (IApp (tm1, tm2)) =
1059 IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
1060 | intro top_level args (IConst (name as (s, _), T, T_args)) =
1061 (case proxify_const s of
1063 if top_level orelse is_type_enc_higher_order type_enc then
1064 case (top_level, s) of
1065 (_, "c_False") => IConst (`I tptp_false, T, [])
1066 | (_, "c_True") => IConst (`I tptp_true, T, [])
1067 | (false, "c_Not") => IConst (`I tptp_not, T, [])
1068 | (false, "c_conj") => IConst (`I tptp_and, T, [])
1069 | (false, "c_disj") => IConst (`I tptp_or, T, [])
1070 | (false, "c_implies") => IConst (`I tptp_implies, T, [])
1071 | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
1072 | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
1074 if is_tptp_equal s andalso length args = 2 then
1075 IConst (`I tptp_equal, T, [])
1077 (* Use a proxy even for partially applied THF0 equality,
1078 because the LEO-II and Satallax parsers complain about not
1079 being able to infer the type of "=". *)
1080 IConst (proxy_base |>> prefix const_prefix, T, T_args)
1081 | _ => IConst (name, T, [])
1083 IConst (proxy_base |>> prefix const_prefix, T, T_args)
1084 | NONE => if s = tptp_choice then tweak_ho_quant tptp_choice T args
1085 else IConst (name, T, T_args))
1086 | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
1088 in intro true [] end
1090 fun mangle_type_args_in_const format type_enc (name as (s, _)) T_args =
1091 case unprefix_and_unascii const_prefix s of
1092 NONE => (name, T_args)
1094 case type_arg_policy [] type_enc (invert_const s'') of
1095 Mangled_Type_Args => (mangled_const_name format type_enc T_args name, [])
1096 | _ => (name, T_args)
1097 fun mangle_type_args_in_iterm format type_enc =
1098 if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
1100 fun mangle (IApp (tm1, tm2)) = IApp (mangle tm1, mangle tm2)
1101 | mangle (tm as IConst (_, _, [])) = tm
1102 | mangle (IConst (name, T, T_args)) =
1103 mangle_type_args_in_const format type_enc name T_args
1104 |> (fn (name, T_args) => IConst (name, T, T_args))
1105 | mangle (IAbs (bound, tm)) = IAbs (bound, mangle tm)
1111 fun chop_fun 0 T = ([], T)
1112 | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
1113 chop_fun (n - 1) ran_T |>> cons dom_T
1114 | chop_fun _ T = ([], T)
1116 fun filter_const_type_args _ _ _ [] = []
1117 | filter_const_type_args thy s ary T_args =
1119 val U = robust_const_type thy s
1120 val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun ary |> fst) []
1121 val U_args = (s, U) |> robust_const_typargs thy
1124 |> map (fn (U, T) =>
1125 if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
1127 handle TYPE _ => T_args
1129 fun filter_type_args_in_const _ _ _ _ _ [] = []
1130 | filter_type_args_in_const thy monom_constrs type_enc ary s T_args =
1131 case unprefix_and_unascii const_prefix s of
1133 if level_of_type_enc type_enc = No_Types orelse s = tptp_choice then []
1137 val s'' = invert_const s''
1138 fun filter_T_args false = T_args
1139 | filter_T_args true = filter_const_type_args thy s'' ary T_args
1141 case type_arg_policy monom_constrs type_enc s'' of
1142 Explicit_Type_Args infer_from_term_args =>
1143 filter_T_args infer_from_term_args
1144 | No_Type_Args => []
1145 | Mangled_Type_Args => raise Fail "unexpected (un)mangled symbol"
1147 fun filter_type_args_in_iterm thy monom_constrs type_enc =
1149 fun filt ary (IApp (tm1, tm2)) = IApp (filt (ary + 1) tm1, filt 0 tm2)
1150 | filt ary (IConst (name as (s, _), T, T_args)) =
1151 filter_type_args_in_const thy monom_constrs type_enc ary s T_args
1152 |> (fn T_args => IConst (name, T, T_args))
1153 | filt _ (IAbs (bound, tm)) = IAbs (bound, filt 0 tm)
1157 fun iformula_from_prop ctxt format type_enc eq_as_iff =
1159 val thy = Proof_Context.theory_of ctxt
1160 fun do_term bs t atomic_Ts =
1161 iterm_from_term thy format bs (Envir.eta_contract t)
1162 |>> (introduce_proxies_in_iterm type_enc
1163 #> mangle_type_args_in_iterm format type_enc #> AAtom)
1164 ||> union (op =) atomic_Ts
1165 fun do_quant bs q pos s T t' =
1167 val s = singleton (Name.variant_list (map fst bs)) s
1168 val universal = Option.map (q = AExists ? not) pos
1170 s |> `(case universal of
1171 SOME true => make_all_bound_var
1172 | SOME false => make_exist_bound_var
1173 | NONE => make_bound_var)
1175 do_formula ((s, (name, T)) :: bs) pos t'
1176 #>> mk_aquant q [(name, SOME T)]
1177 ##> union (op =) (atomic_types_of T)
1179 and do_conn bs c pos1 t1 pos2 t2 =
1180 do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
1181 and do_formula bs pos t =
1183 @{const Trueprop} $ t1 => do_formula bs pos t1
1184 | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
1185 | Const (@{const_name All}, _) $ Abs (s, T, t') =>
1186 do_quant bs AForall pos s T t'
1187 | (t0 as Const (@{const_name All}, _)) $ t1 =>
1188 do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
1189 | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
1190 do_quant bs AExists pos s T t'
1191 | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
1192 do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
1193 | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
1194 | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
1195 | @{const HOL.implies} $ t1 $ t2 =>
1196 do_conn bs AImplies (Option.map not pos) t1 pos t2
1197 | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
1198 if eq_as_iff then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
1200 in do_formula [] end
1202 fun presimplify_term thy t =
1203 if exists_Const (member (op =) Meson.presimplified_consts o fst) t then
1204 t |> Skip_Proof.make_thm thy
1205 |> Meson.presimplify
1210 fun preprocess_abstractions_in_terms trans_lams facts =
1212 val (facts, lambda_ts) =
1213 facts |> map (snd o snd) |> trans_lams
1214 |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
1216 map2 (fn t => fn j =>
1217 ((lam_fact_prefix ^ Int.toString j, (Global, Def)), (Axiom, t)))
1218 lambda_ts (1 upto length lambda_ts)
1219 in (facts, lam_facts) end
1221 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
1222 same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
1225 fun freeze (t $ u) = freeze t $ freeze u
1226 | freeze (Abs (s, T, t)) = Abs (s, T, freeze t)
1227 | freeze (Var ((s, i), T)) =
1228 Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
1230 in t |> exists_subterm is_Var t ? freeze end
1232 fun presimp_prop ctxt t =
1234 val thy = Proof_Context.theory_of ctxt
1235 val t = t |> Envir.beta_eta_contract
1236 |> transform_elim_prop
1237 |> Object_Logic.atomize_term thy
1238 val need_trueprop = (fastype_of t = @{typ bool})
1240 t |> need_trueprop ? HOLogic.mk_Trueprop
1241 |> extensionalize_term ctxt
1242 |> presimplify_term thy
1243 |> HOLogic.dest_Trueprop
1245 handle TERM _ => @{const True}
1247 fun make_formula ctxt format type_enc eq_as_iff name stature kind t =
1249 val (iformula, atomic_Ts) =
1250 iformula_from_prop ctxt format type_enc eq_as_iff
1251 (SOME (kind <> Conjecture)) t []
1252 |>> close_iformula_universally
1254 {name = name, stature = stature, kind = kind, iformula = iformula,
1255 atomic_types = atomic_Ts}
1258 fun make_fact ctxt format type_enc eq_as_iff ((name, stature), t) =
1259 case t |> make_formula ctxt format type_enc (eq_as_iff andalso format <> CNF)
1260 name stature Axiom of
1261 formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
1262 if s = tptp_true then NONE else SOME formula
1263 | formula => SOME formula
1265 fun s_not_prop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
1266 | s_not_prop _ = @{prop True} (* "t" is too meta for "metis" *)
1268 | s_not_prop (@{const "==>"} $ t $ @{prop False}) = t
1269 | s_not_prop t = @{const "==>"} $ t $ @{prop False}
1272 fun make_conjecture ctxt format type_enc =
1273 map (fn ((name, stature), (kind, t)) =>
1274 t |> kind = Conjecture ? s_not
1275 |> make_formula ctxt format type_enc (format <> CNF) name stature
1278 (** Finite and infinite type inference **)
1280 fun tvar_footprint thy s ary =
1281 (case unprefix_and_unascii const_prefix s of
1283 s |> invert_const |> robust_const_type thy |> chop_fun ary |> fst
1284 |> map (fn T => Term.add_tvarsT T [] |> map fst)
1288 fun ghost_type_args thy s ary =
1289 if is_tptp_equal s then
1293 val footprint = tvar_footprint thy s ary
1294 val eq = (s = @{const_name HOL.eq})
1295 fun ghosts _ [] = []
1296 | ghosts seen ((i, tvars) :: args) =
1297 ghosts (union (op =) seen tvars) args
1298 |> (eq orelse exists (fn tvar => not (member (op =) seen tvar)) tvars)
1301 if forall null footprint then
1304 0 upto length footprint - 1 ~~ footprint
1305 |> sort (rev_order o list_ord Term_Ord.indexname_ord o pairself snd)
1309 type monotonicity_info =
1310 {maybe_finite_Ts : typ list,
1311 surely_finite_Ts : typ list,
1312 maybe_infinite_Ts : typ list,
1313 surely_infinite_Ts : typ list,
1314 maybe_nonmono_Ts : typ list}
1316 (* These types witness that the type classes they belong to allow infinite
1317 models and hence that any types with these type classes is monotonic. *)
1318 val known_infinite_types =
1319 [@{typ nat}, HOLogic.intT, HOLogic.realT, @{typ "nat => bool"}]
1321 fun is_type_kind_of_surely_infinite ctxt strictness cached_Ts T =
1322 strictness <> Strict andalso is_type_surely_infinite ctxt true cached_Ts T
1324 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
1325 dangerous because their "exhaust" properties can easily lead to unsound ATP
1326 proofs. On the other hand, all HOL infinite types can be given the same
1327 models in first-order logic (via Löwenheim-Skolem). *)
1329 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
1330 | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
1331 maybe_nonmono_Ts, ...}
1332 (Noninf_Nonmono_Types (strictness, grain)) T =
1333 let val thy = Proof_Context.theory_of ctxt in
1334 grain = Ghost_Type_Arg_Vars orelse
1335 (exists (type_intersect thy T) maybe_nonmono_Ts andalso
1336 not (exists (type_instance thy T) surely_infinite_Ts orelse
1337 (not (member (type_equiv thy) maybe_finite_Ts T) andalso
1338 is_type_kind_of_surely_infinite ctxt strictness surely_infinite_Ts
1341 | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
1342 maybe_nonmono_Ts, ...}
1343 (Fin_Nonmono_Types grain) T =
1344 let val thy = Proof_Context.theory_of ctxt in
1345 grain = Ghost_Type_Arg_Vars orelse
1346 (exists (type_intersect thy T) maybe_nonmono_Ts andalso
1347 (exists (type_generalization thy T) surely_finite_Ts orelse
1348 (not (member (type_equiv thy) maybe_infinite_Ts T) andalso
1349 is_type_surely_finite ctxt T)))
1351 | should_encode_type _ _ _ _ = false
1353 fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =
1354 should_guard_var () andalso should_encode_type ctxt mono level T
1355 | should_guard_type _ _ _ _ _ = false
1357 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
1358 String.isPrefix bound_var_prefix s orelse
1359 String.isPrefix all_bound_var_prefix s
1360 | is_maybe_universal_var (IVar _) = true
1361 | is_maybe_universal_var _ = false
1364 Top_Level of bool option |
1365 Eq_Arg of bool option |
1368 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
1369 | should_tag_with_type ctxt mono (Tags (_, level)) site u T =
1370 if granularity_of_type_level level = All_Vars then
1371 should_encode_type ctxt mono level T
1373 (case (site, is_maybe_universal_var u) of
1374 (Eq_Arg _, true) => should_encode_type ctxt mono level T
1376 | should_tag_with_type _ _ _ _ _ _ = false
1378 fun fused_type ctxt mono level =
1380 val should_encode = should_encode_type ctxt mono level
1381 fun fuse 0 T = if should_encode T then T else fused_infinite_type
1382 | fuse ary (Type (@{type_name fun}, [T1, T2])) =
1383 fuse 0 T1 --> fuse (ary - 1) T2
1384 | fuse _ _ = raise Fail "expected function type"
1387 (** predicators and application operators **)
1390 {pred_sym : bool, min_ary : int, max_ary : int, types : typ list,
1393 fun default_sym_tab_entries type_enc =
1394 (make_fixed_const NONE @{const_name undefined},
1395 {pred_sym = false, min_ary = 0, max_ary = 0, types = [],
1396 in_conj = false}) ::
1397 ([tptp_false, tptp_true]
1398 |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = [],
1399 in_conj = false})) @
1400 ([tptp_equal, tptp_old_equal]
1401 |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = [],
1403 |> not (is_type_enc_higher_order type_enc)
1404 ? cons (prefixed_predicator_name,
1405 {pred_sym = true, min_ary = 1, max_ary = 1, types = [],
1408 datatype app_op_level =
1411 Sufficient_App_Op_And_Predicator |
1412 Full_App_Op_And_Predicator
1414 fun sym_table_for_facts ctxt type_enc app_op_level conjs facts =
1416 val thy = Proof_Context.theory_of ctxt
1417 fun consider_var_ary const_T var_T max_ary =
1420 if ary = max_ary orelse type_instance thy var_T T orelse
1421 type_instance thy T var_T then
1424 iter (ary + 1) (range_type T)
1425 in iter 0 const_T end
1426 fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1427 if (app_op_level = Sufficient_App_Op andalso can dest_funT T) orelse
1428 (app_op_level = Sufficient_App_Op_And_Predicator andalso
1429 (can dest_funT T orelse T = @{typ bool})) then
1433 (app_op_level = Sufficient_App_Op_And_Predicator andalso
1434 body_type T = @{typ bool})
1435 fun repair_min_ary {pred_sym, min_ary, max_ary, types, in_conj} =
1436 {pred_sym = pred_sym andalso not bool_vars',
1437 min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,
1438 max_ary = max_ary, types = types, in_conj = in_conj}
1440 fun_var_Ts |> can dest_funT T ? insert_type thy I T
1442 if bool_vars' = bool_vars andalso
1443 pointer_eq (fun_var_Ts', fun_var_Ts) then
1446 ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_ary) sym_tab)
1450 fun add_iterm_syms conj_fact top_level tm
1451 (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1452 let val (head, args) = strip_iterm_comb tm in
1454 IConst ((s, _), T, _) =>
1455 if String.isPrefix bound_var_prefix s orelse
1456 String.isPrefix all_bound_var_prefix s then
1457 add_universal_var T accum
1458 else if String.isPrefix exist_bound_var_prefix s then
1461 let val ary = length args in
1462 ((bool_vars, fun_var_Ts),
1463 case Symtab.lookup sym_tab s of
1464 SOME {pred_sym, min_ary, max_ary, types, in_conj} =>
1467 pred_sym andalso top_level andalso not bool_vars
1468 val types' = types |> insert_type thy I T
1469 val in_conj = in_conj orelse conj_fact
1471 if (app_op_level = Sufficient_App_Op orelse
1472 app_op_level = Sufficient_App_Op_And_Predicator)
1473 andalso not (pointer_eq (types', types)) then
1474 fold (consider_var_ary T) fun_var_Ts min_ary
1478 Symtab.update (s, {pred_sym = pred_sym,
1479 min_ary = Int.min (ary, min_ary),
1480 max_ary = Int.max (ary, max_ary),
1481 types = types', in_conj = in_conj})
1486 val pred_sym = top_level andalso not bool_vars
1488 case unprefix_and_unascii const_prefix s of
1490 (if String.isSubstring uncurried_alias_sep s then
1492 else case try (robust_const_ary thy
1494 o unmangled_const_name) s of
1495 SOME ary0 => Int.min (ary0, ary)
1499 case app_op_level of
1501 | Full_App_Op_And_Predicator => 0
1502 | _ => fold (consider_var_ary T) fun_var_Ts ary
1504 Symtab.update_new (s,
1505 {pred_sym = pred_sym, min_ary = min_ary,
1506 max_ary = ary, types = [T], in_conj = conj_fact})
1510 | IVar (_, T) => add_universal_var T accum
1511 | IAbs ((_, T), tm) =>
1512 accum |> add_universal_var T |> add_iterm_syms conj_fact false tm
1514 |> fold (add_iterm_syms conj_fact false) args
1516 fun add_fact_syms conj_fact =
1517 K (add_iterm_syms conj_fact true) |> formula_fold NONE |> fact_lift
1519 ((false, []), Symtab.empty)
1520 |> fold (add_fact_syms true) conjs
1521 |> fold (add_fact_syms false) facts
1523 |> fold Symtab.update (default_sym_tab_entries type_enc)
1526 fun min_ary_of sym_tab s =
1527 case Symtab.lookup sym_tab s of
1528 SOME ({min_ary, ...} : sym_info) => min_ary
1530 case unprefix_and_unascii const_prefix s of
1532 let val s = s |> unmangled_const_name |> hd |> invert_const in
1533 if s = predicator_name then 1
1534 else if s = app_op_name then 2
1535 else if s = type_guard_name then 1
1540 (* True if the constant ever appears outside of the top-level position in
1541 literals, or if it appears with different arities (e.g., because of different
1542 type instantiations). If false, the constant always receives all of its
1543 arguments and is used as a predicate. *)
1544 fun is_pred_sym sym_tab s =
1545 case Symtab.lookup sym_tab s of
1546 SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
1547 pred_sym andalso min_ary = max_ary
1550 val app_op = `(make_fixed_const NONE) app_op_name
1551 val predicator_combconst =
1552 IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
1554 fun list_app head args = fold (curry (IApp o swap)) args head
1555 fun predicator tm = IApp (predicator_combconst, tm)
1557 fun mk_app_op format type_enc head arg =
1559 val head_T = ityp_of head
1560 val (arg_T, res_T) = dest_funT head_T
1562 IConst (app_op, head_T --> head_T, [arg_T, res_T])
1563 |> mangle_type_args_in_iterm format type_enc
1564 in list_app app [head, arg] end
1566 fun firstorderize_fact thy monom_constrs format type_enc sym_tab
1569 fun do_app arg head = mk_app_op format type_enc head arg
1570 fun list_app_ops head args = fold do_app args head
1571 fun introduce_app_ops tm =
1572 let val (head, args) = tm |> strip_iterm_comb ||> map introduce_app_ops in
1574 IConst (name as (s, _), T, T_args) =>
1575 if uncurried_aliases andalso String.isPrefix const_prefix s then
1577 val ary = length args
1579 name |> ary > min_ary_of sym_tab s ? aliased_uncurried ary
1580 in list_app (IConst (name, T, T_args)) args end
1582 args |> chop (min_ary_of sym_tab s)
1583 |>> list_app head |-> list_app_ops
1584 | _ => list_app_ops head args
1586 fun introduce_predicators tm =
1587 case strip_iterm_comb tm of
1588 (IConst ((s, _), _, _), _) =>
1589 if is_pred_sym sym_tab s then tm else predicator tm
1590 | _ => predicator tm
1592 not (is_type_enc_higher_order type_enc)
1593 ? (introduce_app_ops #> introduce_predicators)
1594 #> filter_type_args_in_iterm thy monom_constrs type_enc
1595 in update_iformula (formula_map do_iterm) end
1597 (** Helper facts **)
1599 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
1600 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
1602 (* The Boolean indicates that a fairly sound type encoding is needed. *)
1604 [(("COMBI", false), @{thms Meson.COMBI_def}),
1605 (("COMBK", false), @{thms Meson.COMBK_def}),
1606 (("COMBB", false), @{thms Meson.COMBB_def}),
1607 (("COMBC", false), @{thms Meson.COMBC_def}),
1608 (("COMBS", false), @{thms Meson.COMBS_def}),
1609 ((predicator_name, false), [not_ffalse, ftrue]),
1610 (("fFalse", false), [not_ffalse]),
1611 (("fFalse", true), @{thms True_or_False}),
1612 (("fTrue", false), [ftrue]),
1613 (("fTrue", true), @{thms True_or_False}),
1615 @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1616 fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1618 @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
1619 by (unfold fconj_def) fast+}),
1621 @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
1622 by (unfold fdisj_def) fast+}),
1623 (("fimplies", false),
1624 @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
1625 by (unfold fimplies_def) fast+}),
1627 (* This is a lie: Higher-order equality doesn't need a sound type encoding.
1628 However, this is done so for backward compatibility: Including the
1629 equality helpers by default in Metis breaks a few existing proofs. *)
1630 @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1631 fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1632 (* Partial characterization of "fAll" and "fEx". A complete characterization
1633 would require the axiom of choice for replay with Metis. *)
1634 (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
1635 (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
1636 (("If", true), @{thms if_True if_False True_or_False})]
1637 |> map (apsnd (map zero_var_indexes))
1639 fun atype_of_type_vars (Simple_Types (_, Polymorphic, _)) = SOME atype_of_types
1640 | atype_of_type_vars _ = NONE
1642 fun bound_tvars type_enc sorts Ts =
1643 (sorts ? mk_ahorn (formulas_for_types type_enc add_sorts_on_tvar Ts))
1644 #> mk_aquant AForall
1645 (map_filter (fn TVar (x as (s, _), _) =>
1646 SOME ((make_schematic_type_var x, s),
1647 atype_of_type_vars type_enc)
1650 fun eq_formula type_enc atomic_Ts bounds pred_sym tm1 tm2 =
1651 (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
1652 else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
1653 |> mk_aquant AForall bounds
1654 |> close_formula_universally
1655 |> bound_tvars type_enc true atomic_Ts
1657 val helper_rank = default_rank
1658 val min_rank = 9 * helper_rank div 10
1659 val max_rank = 4 * min_rank
1661 fun rank_of_fact_num n j = min_rank + (max_rank - min_rank) * j div n
1663 val type_tag = `(make_fixed_const NONE) type_tag_name
1665 fun should_specialize_helper type_enc t =
1666 polymorphism_of_type_enc type_enc <> Polymorphic andalso
1667 level_of_type_enc type_enc <> No_Types andalso
1668 not (null (Term.hidden_polymorphism t))
1670 fun helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
1671 case unprefix_and_unascii const_prefix s of
1674 val thy = Proof_Context.theory_of ctxt
1675 val unmangled_s = mangled_s |> unmangled_const_name |> hd
1676 fun dub needs_fairly_sound j k =
1677 unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
1678 (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
1679 (if needs_fairly_sound then typed_helper_suffix
1680 else untyped_helper_suffix)
1681 fun dub_and_inst needs_fairly_sound (th, j) =
1682 let val t = prop_of th in
1683 if should_specialize_helper type_enc t then
1684 map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
1690 |> map (fn (k, t) => ((dub needs_fairly_sound j k, (Global, Def)), t))
1691 val make_facts = map_filter (make_fact ctxt format type_enc false)
1692 val fairly_sound = is_type_enc_fairly_sound type_enc
1695 |> maps (fn ((helper_s, needs_fairly_sound), ths) =>
1696 if helper_s <> unmangled_s orelse
1697 (needs_fairly_sound andalso not fairly_sound) then
1700 ths ~~ (1 upto length ths)
1701 |> maps (dub_and_inst needs_fairly_sound)
1705 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
1706 Symtab.fold_rev (append o helper_facts_for_sym ctxt format type_enc) sym_tab
1709 (***************************************************************)
1710 (* Type Classes Present in the Axiom or Conjecture Clauses *)
1711 (***************************************************************)
1713 fun set_insert (x, s) = Symtab.update (x, ()) s
1715 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
1717 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
1718 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
1720 fun classes_of_terms get_Ts =
1721 map (map snd o get_Ts)
1722 #> List.foldl add_classes Symtab.empty
1723 #> delete_type #> Symtab.keys
1725 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
1726 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
1728 fun fold_type_constrs f (Type (s, Ts)) x =
1729 fold (fold_type_constrs f) Ts (f (s, x))
1730 | fold_type_constrs _ _ x = x
1732 (* Type constructors used to instantiate overloaded constants are the only ones
1734 fun add_type_constrs_in_term thy =
1736 fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
1737 | add (t $ u) = add t #> add u
1739 x |> robust_const_typargs thy |> fold (fold_type_constrs set_insert)
1740 | add (Abs (_, _, u)) = add u
1744 fun type_constrs_of_terms thy ts =
1745 Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
1747 fun extract_lambda_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
1748 let val (head, args) = strip_comb t in
1749 (head |> dest_Const |> fst,
1750 fold_rev (fn t as Var ((s, _), T) =>
1751 (fn u => Abs (s, T, abstract_over (t, u)))
1752 | _ => raise Fail "expected Var") args u)
1754 | extract_lambda_def _ = raise Fail "malformed lifted lambda"
1756 fun trans_lams_from_string ctxt type_enc lam_trans =
1757 if lam_trans = no_lamsN then
1759 else if lam_trans = hide_lamsN then
1760 lift_lams ctxt type_enc ##> K []
1761 else if lam_trans = liftingN orelse lam_trans = lam_liftingN then
1762 lift_lams ctxt type_enc
1763 else if lam_trans = combsN then
1764 map (introduce_combinators ctxt) #> rpair []
1765 else if lam_trans = combs_and_liftingN then
1766 lift_lams_part_1 ctxt type_enc
1767 ##> maps (fn t => [t, introduce_combinators ctxt (intentionalize_def t)])
1768 #> lift_lams_part_2 ctxt
1769 else if lam_trans = combs_or_liftingN then
1770 lift_lams_part_1 ctxt type_enc
1771 ##> map (fn t => case head_of (strip_qnt_body @{const_name All} t) of
1772 @{term "op =::bool => bool => bool"} => t
1773 | _ => introduce_combinators ctxt (intentionalize_def t))
1774 #> lift_lams_part_2 ctxt
1775 else if lam_trans = keep_lamsN then
1776 map (Envir.eta_contract) #> rpair []
1778 error ("Unknown lambda translation scheme: " ^ quote lam_trans ^ ".")
1780 fun translate_formulas ctxt format prem_kind type_enc lam_trans presimp hyp_ts
1783 val thy = Proof_Context.theory_of ctxt
1784 val trans_lams = trans_lams_from_string ctxt type_enc lam_trans
1785 val fact_ts = facts |> map snd
1786 (* Remove existing facts from the conjecture, as this can dramatically
1787 boost an ATP's performance (for some reason). *)
1790 |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
1791 val facts = facts |> map (apsnd (pair Axiom))
1793 map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_prop concl_t)]
1794 |> map (apsnd freeze_term)
1795 |> map2 (pair o rpair (Local, General) o string_of_int)
1796 (0 upto length hyp_ts)
1797 val ((conjs, facts), lam_facts) =
1799 |> presimp ? pairself (map (apsnd (apsnd (presimp_prop ctxt))))
1800 |> (if lam_trans = no_lamsN then
1804 #> preprocess_abstractions_in_terms trans_lams
1805 #>> chop (length conjs))
1806 val conjs = conjs |> make_conjecture ctxt format type_enc
1807 val (fact_names, facts) =
1809 |> map_filter (fn (name, (_, t)) =>
1810 make_fact ctxt format type_enc true (name, t)
1811 |> Option.map (pair name))
1813 val lifted = lam_facts |> map (extract_lambda_def o snd o snd)
1815 lam_facts |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
1816 val all_ts = concl_t :: hyp_ts @ fact_ts
1817 val subs = tfree_classes_of_terms all_ts
1818 val supers = tvar_classes_of_terms all_ts
1819 val tycons = type_constrs_of_terms thy all_ts
1820 val (supers, arity_clauses) =
1821 if level_of_type_enc type_enc = No_Types then ([], [])
1822 else make_arity_clauses thy tycons supers
1823 val class_rel_clauses = make_class_rel_clauses thy subs supers
1825 (fact_names |> map single, union (op =) subs supers, conjs,
1826 facts @ lam_facts, class_rel_clauses, arity_clauses, lifted)
1829 val type_guard = `(make_fixed_const NONE) type_guard_name
1831 fun type_guard_iterm format type_enc T tm =
1832 IApp (IConst (type_guard, T --> @{typ bool}, [T])
1833 |> mangle_type_args_in_iterm format type_enc, tm)
1835 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
1836 | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
1837 accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
1838 | is_var_positively_naked_in_term _ _ _ _ = true
1840 fun is_var_ghost_type_arg_in_term thy polym_constrs name pos tm accum =
1841 is_var_positively_naked_in_term name pos tm accum orelse
1843 val var = ATerm (name, [])
1844 fun is_nasty_in_term (ATerm (_, [])) = false
1845 | is_nasty_in_term (ATerm ((s, _), tms)) =
1847 val ary = length tms
1848 val polym_constr = member (op =) polym_constrs s
1849 val ghosts = ghost_type_args thy s ary
1851 exists (fn (j, tm) =>
1852 if polym_constr then
1853 member (op =) ghosts j andalso
1854 (tm = var orelse is_nasty_in_term tm)
1856 tm = var andalso member (op =) ghosts j)
1857 (0 upto ary - 1 ~~ tms)
1858 orelse (not polym_constr andalso exists is_nasty_in_term tms)
1860 | is_nasty_in_term _ = true
1861 in is_nasty_in_term tm end
1863 fun should_guard_var_in_formula thy polym_constrs level pos phi (SOME true)
1865 (case granularity_of_type_level level of
1867 | Positively_Naked_Vars =>
1868 formula_fold pos (is_var_positively_naked_in_term name) phi false
1869 | Ghost_Type_Arg_Vars =>
1870 formula_fold pos (is_var_ghost_type_arg_in_term thy polym_constrs name) phi
1872 | should_guard_var_in_formula _ _ _ _ _ _ _ = true
1874 fun always_guard_var_in_formula _ _ _ _ _ _ _ = true
1876 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
1877 | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =
1878 granularity_of_type_level level <> All_Vars andalso
1879 should_encode_type ctxt mono level T
1880 | should_generate_tag_bound_decl _ _ _ _ _ = false
1882 fun mk_aterm format type_enc name T_args args =
1883 ATerm (name, map_filter (ho_term_for_type_arg format type_enc) T_args @ args)
1885 fun do_bound_type ctxt format mono type_enc =
1887 Simple_Types (_, _, level) =>
1888 fused_type ctxt mono level 0
1889 #> ho_type_from_typ format type_enc false 0 #> SOME
1892 fun tag_with_type ctxt format mono type_enc pos T tm =
1893 IConst (type_tag, T --> T, [T])
1894 |> mangle_type_args_in_iterm format type_enc
1895 |> ho_term_from_iterm ctxt format mono type_enc pos
1896 |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
1897 | _ => raise Fail "unexpected lambda-abstraction")
1898 and ho_term_from_iterm ctxt format mono type_enc pos =
1900 fun beta_red bs (IApp (IAbs ((name, _), tm), tm')) =
1901 beta_red ((name, beta_red bs tm') :: bs) tm
1902 | beta_red bs (IApp tmp) = IApp (pairself (beta_red bs) tmp)
1903 | beta_red bs (tm as IConst (name, _, _)) =
1904 (case AList.lookup (op =) bs name of
1907 | beta_red bs (IAbs ((name, T), tm)) =
1908 IAbs ((name, T), beta_red (AList.delete (op =) name bs) tm)
1909 | beta_red _ tm = tm
1912 val (head, args) = strip_iterm_comb u
1915 Top_Level pos => pos
1920 IConst (name as (s, _), _, T_args) =>
1922 val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
1924 map (term arg_site) args |> mk_aterm format type_enc name T_args
1927 map (term Elsewhere) args |> mk_aterm format type_enc name []
1928 | IAbs ((name, T), tm) =>
1929 if is_type_enc_higher_order type_enc then
1930 AAbs ((name, ho_type_from_typ format type_enc true 0 T),
1933 raise Fail "unexpected lambda-abstraction"
1934 | IApp _ => raise Fail "impossible \"IApp\""
1937 if should_tag_with_type ctxt mono type_enc site u T then
1938 tag_with_type ctxt format mono type_enc pos T t
1942 in term (Top_Level pos) o beta_red [] end
1943 and formula_from_iformula ctxt polym_constrs format mono type_enc
1946 val thy = Proof_Context.theory_of ctxt
1947 val level = level_of_type_enc type_enc
1948 val do_term = ho_term_from_iterm ctxt format mono type_enc
1949 fun do_out_of_bound_type pos phi universal (name, T) =
1950 if should_guard_type ctxt mono type_enc
1951 (fn () => should_guard_var thy polym_constrs level pos phi
1952 universal name) T then
1954 |> type_guard_iterm format type_enc T
1955 |> do_term pos |> AAtom |> SOME
1956 else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
1958 val var = ATerm (name, [])
1959 val tagged_var = tag_with_type ctxt format mono type_enc pos T var
1960 in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
1963 fun do_formula pos (AQuant (q, xs, phi)) =
1965 val phi = phi |> do_formula pos
1966 val universal = Option.map (q = AExists ? not) pos
1967 val do_bound_type = do_bound_type ctxt format mono type_enc
1969 AQuant (q, xs |> map (apsnd (fn NONE => NONE
1970 | SOME T => do_bound_type T)),
1971 (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
1973 (fn (_, NONE) => NONE
1975 do_out_of_bound_type pos phi universal (s, T))
1979 | do_formula pos (AConn conn) = aconn_map pos do_formula conn
1980 | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
1983 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
1984 of monomorphization). The TPTP explicitly forbids name clashes, and some of
1985 the remote provers might care. *)
1986 fun formula_line_for_fact ctxt polym_constrs format prefix encode freshen pos
1987 mono type_enc rank (j, {name, stature, kind, iformula, atomic_types}) =
1988 (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
1990 |> formula_from_iformula ctxt polym_constrs format mono type_enc
1991 should_guard_var_in_formula (if pos then SOME true else NONE)
1992 |> close_formula_universally
1993 |> bound_tvars type_enc true atomic_types,
1995 let val rank = rank j in
1997 Intro => isabelle_info introN rank
1998 | Inductive => isabelle_info inductiveN rank
1999 | Elim => isabelle_info elimN rank
2000 | Simp => isabelle_info simpN rank
2001 | Def => isabelle_info defN rank
2002 | _ => isabelle_info "" rank
2006 fun formula_line_for_class_rel_clause type_enc
2007 ({name, subclass, superclass, ...} : class_rel_clause) =
2008 let val ty_arg = ATerm (tvar_a_name, []) in
2009 Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
2011 [type_class_formula type_enc subclass ty_arg,
2012 type_class_formula type_enc superclass ty_arg])
2013 |> mk_aquant AForall
2014 [(tvar_a_name, atype_of_type_vars type_enc)],
2015 NONE, isabelle_info inductiveN helper_rank)
2018 fun formula_from_arity_atom type_enc (class, t, args) =
2019 ATerm (t, map (fn arg => ATerm (arg, [])) args)
2020 |> type_class_formula type_enc class
2022 fun formula_line_for_arity_clause type_enc
2023 ({name, prem_atoms, concl_atom} : arity_clause) =
2024 Formula (arity_clause_prefix ^ name, Axiom,
2025 mk_ahorn (map (formula_from_arity_atom type_enc) prem_atoms)
2026 (formula_from_arity_atom type_enc concl_atom)
2027 |> mk_aquant AForall
2028 (map (rpair (atype_of_type_vars type_enc)) (#3 concl_atom)),
2029 NONE, isabelle_info inductiveN helper_rank)
2031 fun formula_line_for_conjecture ctxt polym_constrs format mono type_enc
2032 ({name, kind, iformula, atomic_types, ...} : translated_formula) =
2033 Formula (conjecture_prefix ^ name, kind,
2035 |> formula_from_iformula ctxt polym_constrs format mono type_enc
2036 should_guard_var_in_formula (SOME false)
2037 |> close_formula_universally
2038 |> bound_tvars type_enc true atomic_types, NONE, [])
2040 fun type_enc_needs_free_types (Simple_Types (_, Polymorphic, _)) = true
2041 | type_enc_needs_free_types (Simple_Types _) = false
2042 | type_enc_needs_free_types _ = true
2044 fun formula_line_for_free_type j phi =
2045 Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis, phi, NONE, [])
2046 fun formula_lines_for_free_types type_enc (facts : translated_formula list) =
2047 if type_enc_needs_free_types type_enc then
2050 fold (union (op =)) (map #atomic_types facts) []
2051 |> formulas_for_types type_enc add_sorts_on_tfree
2052 in map2 formula_line_for_free_type (0 upto length phis - 1) phis end
2056 (** Symbol declarations **)
2058 fun decl_line_for_class order s =
2059 let val name as (s, _) = `make_type_class s in
2060 Decl (sym_decl_prefix ^ s, name,
2061 if order = First_Order then
2062 ATyAbs ([tvar_a_name],
2063 if avoid_first_order_ghost_type_vars then
2064 AFun (a_itself_atype, bool_atype)
2068 AFun (atype_of_types, bool_atype))
2071 fun decl_lines_for_classes type_enc classes =
2073 Simple_Types (order, Polymorphic, _) =>
2074 map (decl_line_for_class order) classes
2077 fun sym_decl_table_for_facts thy format type_enc sym_tab
2078 (conjs, facts, extra_tms) =
2080 fun add_iterm_syms tm =
2081 let val (head, args) = strip_iterm_comb tm in
2083 IConst ((s, s'), T, T_args) =>
2085 val (pred_sym, in_conj) =
2086 case Symtab.lookup sym_tab s of
2087 SOME ({pred_sym, in_conj, ...} : sym_info) =>
2089 | NONE => (false, false)
2092 Guards _ => not pred_sym
2093 | _ => true) andalso
2094 is_tptp_user_symbol s
2097 Symtab.map_default (s, [])
2098 (insert_type thy #3 (s', T_args, T, pred_sym, length args,
2103 | IAbs (_, tm) => add_iterm_syms tm
2105 #> fold add_iterm_syms args
2107 val add_fact_syms = K add_iterm_syms |> formula_fold NONE |> fact_lift
2108 fun add_formula_var_types (AQuant (_, xs, phi)) =
2109 fold (fn (_, SOME T) => insert_type thy I T | _ => I) xs
2110 #> add_formula_var_types phi
2111 | add_formula_var_types (AConn (_, phis)) =
2112 fold add_formula_var_types phis
2113 | add_formula_var_types _ = I
2115 if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
2116 else fold (fact_lift add_formula_var_types) (conjs @ facts) []
2117 fun add_undefined_const T =
2120 `(make_fixed_const NONE) @{const_name undefined}
2121 |> (case type_arg_policy [] type_enc @{const_name undefined} of
2122 Mangled_Type_Args => mangled_const_name format type_enc [T]
2125 Symtab.map_default (s, [])
2126 (insert_type thy #3 (s', [T], T, false, 0, false))
2128 fun add_TYPE_const () =
2129 let val (s, s') = TYPE_name in
2130 Symtab.map_default (s, [])
2132 (s', [tvar_a], @{typ "'a itself"}, false, 0, false))
2136 |> is_type_enc_fairly_sound type_enc
2137 ? (fold (fold add_fact_syms) [conjs, facts]
2138 #> fold add_iterm_syms extra_tms
2139 #> (case type_enc of
2140 Simple_Types (First_Order, Polymorphic, _) =>
2141 if avoid_first_order_ghost_type_vars then add_TYPE_const ()
2143 | Simple_Types _ => I
2144 | _ => fold add_undefined_const (var_types ())))
2147 (* We add "bool" in case the helper "True_or_False" is included later. *)
2148 fun default_mono level =
2149 {maybe_finite_Ts = [@{typ bool}],
2150 surely_finite_Ts = [@{typ bool}],
2151 maybe_infinite_Ts = known_infinite_types,
2152 surely_infinite_Ts =
2154 Noninf_Nonmono_Types (Strict, _) => []
2155 | _ => known_infinite_types,
2156 maybe_nonmono_Ts = [@{typ bool}]}
2158 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
2159 out with monotonicity" paper presented at CADE 2011. *)
2160 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
2161 | add_iterm_mononotonicity_info ctxt level _
2162 (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
2163 (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
2164 surely_infinite_Ts, maybe_nonmono_Ts}) =
2165 let val thy = Proof_Context.theory_of ctxt in
2166 if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
2168 Noninf_Nonmono_Types (strictness, _) =>
2169 if exists (type_instance thy T) surely_infinite_Ts orelse
2170 member (type_equiv thy) maybe_finite_Ts T then
2172 else if is_type_kind_of_surely_infinite ctxt strictness
2173 surely_infinite_Ts T then
2174 {maybe_finite_Ts = maybe_finite_Ts,
2175 surely_finite_Ts = surely_finite_Ts,
2176 maybe_infinite_Ts = maybe_infinite_Ts,
2177 surely_infinite_Ts = surely_infinite_Ts |> insert_type thy I T,
2178 maybe_nonmono_Ts = maybe_nonmono_Ts}
2180 {maybe_finite_Ts = maybe_finite_Ts |> insert (type_equiv thy) T,
2181 surely_finite_Ts = surely_finite_Ts,
2182 maybe_infinite_Ts = maybe_infinite_Ts,
2183 surely_infinite_Ts = surely_infinite_Ts,
2184 maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}
2185 | Fin_Nonmono_Types _ =>
2186 if exists (type_instance thy T) surely_finite_Ts orelse
2187 member (type_equiv thy) maybe_infinite_Ts T then
2189 else if is_type_surely_finite ctxt T then
2190 {maybe_finite_Ts = maybe_finite_Ts,
2191 surely_finite_Ts = surely_finite_Ts |> insert_type thy I T,
2192 maybe_infinite_Ts = maybe_infinite_Ts,
2193 surely_infinite_Ts = surely_infinite_Ts,
2194 maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}
2196 {maybe_finite_Ts = maybe_finite_Ts,
2197 surely_finite_Ts = surely_finite_Ts,
2198 maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_equiv thy) T,
2199 surely_infinite_Ts = surely_infinite_Ts,
2200 maybe_nonmono_Ts = maybe_nonmono_Ts}
2205 | add_iterm_mononotonicity_info _ _ _ _ mono = mono
2206 fun add_fact_mononotonicity_info ctxt level
2207 ({kind, iformula, ...} : translated_formula) =
2208 formula_fold (SOME (kind <> Conjecture))
2209 (add_iterm_mononotonicity_info ctxt level) iformula
2210 fun mononotonicity_info_for_facts ctxt type_enc facts =
2211 let val level = level_of_type_enc type_enc in
2213 |> is_type_level_monotonicity_based level
2214 ? fold (add_fact_mononotonicity_info ctxt level) facts
2217 fun add_iformula_monotonic_types ctxt mono type_enc =
2219 val thy = Proof_Context.theory_of ctxt
2220 val level = level_of_type_enc type_enc
2221 val should_encode = should_encode_type ctxt mono level
2222 fun add_type T = not (should_encode T) ? insert_type thy I T
2223 fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2
2225 and add_term tm = add_type (ityp_of tm) #> add_args tm
2226 in formula_fold NONE (K add_term) end
2227 fun add_fact_monotonic_types ctxt mono type_enc =
2228 add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
2229 fun monotonic_types_for_facts ctxt mono type_enc facts =
2230 let val level = level_of_type_enc type_enc in
2231 [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
2232 is_type_level_monotonicity_based level andalso
2233 granularity_of_type_level level <> Ghost_Type_Arg_Vars)
2234 ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
2237 fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
2238 Formula (guards_sym_formula_prefix ^
2239 ascii_of (mangled_type format type_enc T),
2241 IConst (`make_bound_var "X", T, [])
2242 |> type_guard_iterm format type_enc T
2244 |> formula_from_iformula ctxt [] format mono type_enc
2245 always_guard_var_in_formula (SOME true)
2246 |> close_formula_universally
2247 |> bound_tvars type_enc true (atomic_types_of T),
2248 NONE, isabelle_info inductiveN helper_rank)
2250 fun formula_line_for_tags_mono_type ctxt format mono type_enc T =
2251 let val x_var = ATerm (`make_bound_var "X", []) in
2252 Formula (tags_sym_formula_prefix ^
2253 ascii_of (mangled_type format type_enc T),
2255 eq_formula type_enc (atomic_types_of T) [] false
2256 (tag_with_type ctxt format mono type_enc NONE T x_var) x_var,
2257 NONE, isabelle_info defN helper_rank)
2260 fun problem_lines_for_mono_types ctxt format mono type_enc Ts =
2262 Simple_Types _ => []
2264 map (formula_line_for_guards_mono_type ctxt format mono type_enc) Ts
2265 | Tags _ => map (formula_line_for_tags_mono_type ctxt format mono type_enc) Ts
2267 fun decl_line_for_sym ctxt format mono type_enc s
2268 (s', T_args, T, pred_sym, ary, _) =
2270 val thy = Proof_Context.theory_of ctxt
2274 else case unprefix_and_unascii const_prefix s of
2277 val s' = s' |> invert_const
2278 val T = s' |> robust_const_type thy
2279 in (T, robust_const_typargs thy (s', T)) end
2280 | NONE => raise Fail "unexpected type arguments"
2282 Decl (sym_decl_prefix ^ s, (s, s'),
2283 T |> fused_type ctxt mono (level_of_type_enc type_enc) ary
2284 |> ho_type_from_typ format type_enc pred_sym ary
2285 |> not (null T_args)
2286 ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))
2289 fun honor_conj_sym_kind in_conj conj_sym_kind =
2290 if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
2293 fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
2294 j (s', T_args, T, _, ary, in_conj) =
2296 val thy = Proof_Context.theory_of ctxt
2297 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2298 val (arg_Ts, res_T) = chop_fun ary T
2299 val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
2301 bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
2303 if exists (curry (op =) dummyT) T_args then
2304 case level_of_type_enc type_enc of
2305 All_Types => map SOME arg_Ts
2307 if granularity_of_type_level level = Ghost_Type_Arg_Vars then
2308 let val ghosts = ghost_type_args thy s ary in
2309 map2 (fn j => if member (op =) ghosts j then SOME else K NONE)
2310 (0 upto ary - 1) arg_Ts
2317 Formula (guards_sym_formula_prefix ^ s ^
2318 (if n > 1 then "_" ^ string_of_int j else ""), kind,
2319 IConst ((s, s'), T, T_args)
2320 |> fold (curry (IApp o swap)) bounds
2321 |> type_guard_iterm format type_enc res_T
2322 |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
2323 |> formula_from_iformula ctxt [] format mono type_enc
2324 always_guard_var_in_formula (SOME true)
2325 |> close_formula_universally
2326 |> bound_tvars type_enc (n > 1) (atomic_types_of T)
2328 NONE, isabelle_info inductiveN helper_rank)
2331 fun formula_lines_for_tags_sym_decl ctxt format conj_sym_kind mono type_enc n s
2332 (j, (s', T_args, T, pred_sym, ary, in_conj)) =
2334 val thy = Proof_Context.theory_of ctxt
2335 val level = level_of_type_enc type_enc
2336 val grain = granularity_of_type_level level
2338 tags_sym_formula_prefix ^ s ^
2339 (if n > 1 then "_" ^ string_of_int j else "")
2340 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2341 val (arg_Ts, res_T) = chop_fun ary T
2342 val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
2343 val bounds = bound_names |> map (fn name => ATerm (name, []))
2344 val cst = mk_aterm format type_enc (s, s') T_args
2345 val eq = maybe_negate oo eq_formula type_enc (atomic_types_of T) [] pred_sym
2346 val should_encode = should_encode_type ctxt mono level
2347 val tag_with = tag_with_type ctxt format mono type_enc NONE
2348 val add_formula_for_res =
2349 if should_encode res_T then
2352 if grain = Ghost_Type_Arg_Vars then
2353 let val ghosts = ghost_type_args thy s ary in
2354 map2 (fn (j, arg_T) => member (op =) ghosts j ? tag_with arg_T)
2355 (0 upto ary - 1 ~~ arg_Ts) bounds
2360 cons (Formula (ident_base ^ "_res", kind,
2361 eq (tag_with res_T (cst bounds)) (cst tagged_bounds),
2362 NONE, isabelle_info defN helper_rank))
2366 in [] |> not pred_sym ? add_formula_for_res end
2368 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
2370 fun rationalize_decls thy (decls as decl :: (decls' as _ :: _)) =
2372 val T = result_type_of_decl decl
2373 |> map_type_tvar (fn (z, _) => TVar (z, HOLogic.typeS))
2375 if forall (type_generalization thy T o result_type_of_decl) decls' then
2380 | rationalize_decls _ decls = decls
2382 fun problem_lines_for_sym_decls ctxt format conj_sym_kind mono type_enc
2385 Simple_Types _ => [decl_line_for_sym ctxt format mono type_enc s (hd decls)]
2386 | Guards (_, level) =>
2388 val thy = Proof_Context.theory_of ctxt
2389 val decls = decls |> rationalize_decls thy
2390 val n = length decls
2392 decls |> filter (should_encode_type ctxt mono level
2393 o result_type_of_decl)
2395 (0 upto length decls - 1, decls)
2396 |-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
2399 | Tags (_, level) =>
2400 if granularity_of_type_level level = All_Vars then
2403 let val n = length decls in
2404 (0 upto n - 1 ~~ decls)
2405 |> maps (formula_lines_for_tags_sym_decl ctxt format conj_sym_kind mono
2409 fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
2410 mono_Ts sym_decl_tab =
2412 val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
2414 problem_lines_for_mono_types ctxt format mono type_enc mono_Ts
2416 fold_rev (append o problem_lines_for_sym_decls ctxt format conj_sym_kind
2419 in mono_lines @ decl_lines end
2421 fun pair_append (xs1, xs2) (ys1, ys2) = (xs1 @ ys1, xs2 @ ys2)
2423 fun do_uncurried_alias_lines_for_sym ctxt monom_constrs format conj_sym_kind
2424 mono type_enc sym_tab0 sym_tab base_s0 types in_conj =
2428 val thy = Proof_Context.theory_of ctxt
2429 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2430 val base_name = base_s0 |> `(make_fixed_const (SOME format))
2431 val T = case types of [T] => T | _ => robust_const_type thy base_s0
2432 val T_args = robust_const_typargs thy (base_s0, T)
2433 val (base_name as (base_s, _), T_args) =
2434 mangle_type_args_in_const format type_enc base_name T_args
2435 val base_ary = min_ary_of sym_tab0 base_s
2436 fun do_const name = IConst (name, T, T_args)
2437 val filter_ty_args =
2438 filter_type_args_in_iterm thy monom_constrs type_enc
2440 ho_term_from_iterm ctxt format mono type_enc (SOME true)
2441 val name1 as (s1, _) =
2442 base_name |> ary - 1 > base_ary ? aliased_uncurried (ary - 1)
2443 val name2 as (s2, _) = base_name |> aliased_uncurried ary
2444 val (arg_Ts, _) = chop_fun ary T
2446 1 upto ary |> map (`I o make_bound_var o string_of_int)
2447 val bounds = bound_names ~~ arg_Ts
2448 val (first_bounds, last_bound) =
2449 bounds |> map (fn (name, T) => IConst (name, T, [])) |> split_last
2451 mk_app_op format type_enc (list_app (do_const name1) first_bounds)
2455 list_app (do_const name2) (first_bounds @ [last_bound])
2457 val do_bound_type = do_bound_type ctxt format mono type_enc
2459 eq_formula type_enc (atomic_types_of T)
2460 (map (apsnd do_bound_type) bounds) false
2461 (ho_term_of tm1) (ho_term_of tm2)
2464 [Formula (uncurried_alias_eq_prefix ^ s2, kind, eq |> maybe_negate,
2465 NONE, isabelle_info defN helper_rank)])
2466 |> (if ary - 1 = base_ary orelse Symtab.defined sym_tab s1 then I
2467 else pair_append (do_alias (ary - 1)))
2470 fun uncurried_alias_lines_for_sym ctxt monom_constrs format conj_sym_kind mono
2471 type_enc sym_tab0 sym_tab
2472 (s, {min_ary, types, in_conj, ...} : sym_info) =
2473 case unprefix_and_unascii const_prefix s of
2475 if String.isSubstring uncurried_alias_sep mangled_s then
2477 val base_s0 = mangled_s |> unmangled_const_name |> hd |> invert_const
2479 do_uncurried_alias_lines_for_sym ctxt monom_constrs format conj_sym_kind
2480 mono type_enc sym_tab0 sym_tab base_s0 types in_conj min_ary
2485 fun uncurried_alias_lines_for_sym_table ctxt monom_constrs format conj_sym_kind
2486 mono type_enc uncurried_aliases sym_tab0 sym_tab =
2488 |> uncurried_aliases
2491 o uncurried_alias_lines_for_sym ctxt monom_constrs format
2492 conj_sym_kind mono type_enc sym_tab0 sym_tab) sym_tab
2494 val implicit_declsN = "Should-be-implicit typings"
2495 val explicit_declsN = "Explicit typings"
2496 val uncurried_alias_eqsN = "Uncurried aliases"
2497 val factsN = "Relevant facts"
2498 val class_relsN = "Class relationships"
2499 val aritiesN = "Arities"
2500 val helpersN = "Helper facts"
2501 val conjsN = "Conjectures"
2502 val free_typesN = "Type variables"
2504 (* TFF allows implicit declarations of types, function symbols, and predicate
2505 symbols (with "$i" as the type of individuals), but some provers (e.g.,
2506 SNARK) require explicit declarations. The situation is similar for THF. *)
2508 fun default_type type_enc pred_sym s =
2513 if String.isPrefix type_const_prefix s orelse
2514 String.isPrefix tfree_prefix s then
2518 | _ => individual_atype
2519 fun typ 0 = if pred_sym then bool_atype else ind
2520 | typ ary = AFun (ind, typ (ary - 1))
2523 fun nary_type_constr_type n =
2524 funpow n (curry AFun atype_of_types) atype_of_types
2526 fun undeclared_syms_in_problem type_enc problem =
2528 fun do_sym (name as (s, _)) ty =
2529 if is_tptp_user_symbol s then
2530 Symtab.default (s, (name, ty))
2533 fun do_type (AType (name, tys)) =
2534 do_sym name (fn () => nary_type_constr_type (length tys))
2536 | do_type (AFun (ty1, ty2)) = do_type ty1 #> do_type ty2
2537 | do_type (ATyAbs (_, ty)) = do_type ty
2538 fun do_term pred_sym (ATerm (name as (s, _), tms)) =
2539 do_sym name (fn _ => default_type type_enc pred_sym s (length tms))
2540 #> fold (do_term false) tms
2541 | do_term _ (AAbs ((_, ty), tm)) = do_type ty #> do_term false tm
2542 fun do_formula (AQuant (_, xs, phi)) =
2543 fold do_type (map_filter snd xs) #> do_formula phi
2544 | do_formula (AConn (_, phis)) = fold do_formula phis
2545 | do_formula (AAtom tm) = do_term true tm
2546 fun do_problem_line (Decl (_, _, ty)) = do_type ty
2547 | do_problem_line (Formula (_, _, phi, _, _)) = do_formula phi
2550 |> fold (fn (s, _) => Symtab.default (s, (("", ""), K tvar_a_atype)))
2551 (declared_syms_in_problem problem)
2552 |> fold (fold do_problem_line o snd) problem
2555 fun declare_undeclared_syms_in_atp_problem type_enc problem =
2558 Symtab.fold (fn (_, (("", ""), _)) => I (* already declared *)
2560 cons (Decl (type_decl_prefix ^ s, sym, ty ())))
2561 (undeclared_syms_in_problem type_enc problem) []
2562 in (implicit_declsN, decls) :: problem end
2564 fun exists_subdtype P =
2566 fun ex U = P U orelse
2567 (case U of Datatype.DtType (_, Us) => exists ex Us | _ => false)
2570 fun is_poly_constr (_, Us) =
2571 exists (exists_subdtype (fn Datatype.DtTFree _ => true | _ => false)) Us
2573 fun all_constrs_of_polymorphic_datatypes thy =
2577 #> (fn cs => exists is_poly_constr cs ? append cs))
2578 (Datatype.get_all thy) []
2579 |> List.partition is_poly_constr
2580 |> pairself (map fst)
2582 val app_op_and_predicator_threshold = 50
2584 fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
2585 lam_trans uncurried_aliases readable_names preproc
2586 hyp_ts concl_t facts =
2588 val thy = Proof_Context.theory_of ctxt
2589 val type_enc = type_enc |> adjust_type_enc format
2590 (* Forcing explicit applications is expensive for polymorphic encodings,
2591 because it takes only one existential variable ranging over "'a => 'b" to
2592 ruin everything. Hence we do it only if there are few facts (which is
2593 normally the case for "metis" and the minimizer). *)
2595 if length facts > app_op_and_predicator_threshold then
2596 if polymorphism_of_type_enc type_enc = Polymorphic then
2601 Sufficient_App_Op_And_Predicator
2603 if lam_trans = keep_lamsN andalso
2604 not (is_type_enc_higher_order type_enc) then
2605 error ("Lambda translation scheme incompatible with first-order \
2609 val (fact_names, classes, conjs, facts, class_rel_clauses, arity_clauses,
2611 translate_formulas ctxt format prem_kind type_enc lam_trans preproc hyp_ts
2613 val sym_tab0 = sym_table_for_facts ctxt type_enc app_op_level conjs facts
2614 val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
2615 val (polym_constrs, monom_constrs) =
2616 all_constrs_of_polymorphic_datatypes thy
2617 |>> map (make_fixed_const (SOME format))
2618 fun firstorderize in_helper =
2619 firstorderize_fact thy monom_constrs format type_enc sym_tab0
2620 (uncurried_aliases andalso not in_helper)
2621 val (conjs, facts) = (conjs, facts) |> pairself (map (firstorderize false))
2622 val sym_tab = sym_table_for_facts ctxt type_enc Min_App_Op conjs facts
2624 sym_tab |> helper_facts_for_sym_table ctxt format type_enc
2625 |> map (firstorderize true)
2627 helpers @ conjs @ facts |> monotonic_types_for_facts ctxt mono type_enc
2628 val class_decl_lines = decl_lines_for_classes type_enc classes
2629 val (uncurried_alias_eq_tms, uncurried_alias_eq_lines) =
2630 uncurried_alias_lines_for_sym_table ctxt monom_constrs format
2631 conj_sym_kind mono type_enc uncurried_aliases sym_tab0 sym_tab
2632 val sym_decl_lines =
2633 (conjs, helpers @ facts, uncurried_alias_eq_tms)
2634 |> sym_decl_table_for_facts thy format type_enc sym_tab
2635 |> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
2637 val num_facts = length facts
2639 map (formula_line_for_fact ctxt polym_constrs format fact_prefix
2640 ascii_of (not exporter) (not exporter) mono type_enc
2641 (rank_of_fact_num num_facts))
2642 (0 upto num_facts - 1 ~~ facts)
2644 0 upto length helpers - 1 ~~ helpers
2645 |> map (formula_line_for_fact ctxt polym_constrs format helper_prefix I
2646 false true mono type_enc (K default_rank))
2647 (* Reordering these might confuse the proof reconstruction code or the SPASS
2650 [(explicit_declsN, class_decl_lines @ sym_decl_lines),
2651 (uncurried_alias_eqsN, uncurried_alias_eq_lines),
2652 (factsN, fact_lines),
2654 map (formula_line_for_class_rel_clause type_enc) class_rel_clauses),
2655 (aritiesN, map (formula_line_for_arity_clause type_enc) arity_clauses),
2656 (helpersN, helper_lines),
2657 (free_typesN, formula_lines_for_free_types type_enc (facts @ conjs)),
2659 map (formula_line_for_conjecture ctxt polym_constrs format mono
2664 CNF => ensure_cnf_problem
2665 | CNF_UEQ => filter_cnf_ueq_problem
2667 | TFF (_, TPTP_Implicit) => I
2668 | THF (_, TPTP_Implicit, _) => I
2669 | _ => declare_undeclared_syms_in_atp_problem type_enc)
2670 val (problem, pool) = problem |> nice_atp_problem readable_names format
2671 fun add_sym_ary (s, {min_ary, ...} : sym_info) =
2672 min_ary > 0 ? Symtab.insert (op =) (s, min_ary)
2675 case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
2676 fact_names |> Vector.fromList,
2678 Symtab.empty |> Symtab.fold add_sym_ary sym_tab)
2682 val conj_weight = 0.0
2683 val hyp_weight = 0.1
2684 val fact_min_weight = 0.2
2685 val fact_max_weight = 1.0
2686 val type_info_default_weight = 0.8
2688 (* Weights are from 0.0 (most important) to 1.0 (least important). *)
2689 fun atp_problem_selection_weights problem =
2691 fun add_term_weights weight (ATerm (s, tms)) =
2692 is_tptp_user_symbol s ? Symtab.default (s, weight)
2693 #> fold (add_term_weights weight) tms
2694 | add_term_weights weight (AAbs (_, tm)) = add_term_weights weight tm
2695 fun add_line_weights weight (Formula (_, _, phi, _, _)) =
2696 formula_fold NONE (K (add_term_weights weight)) phi
2697 | add_line_weights _ _ = I
2698 fun add_conjectures_weights [] = I
2699 | add_conjectures_weights conjs =
2700 let val (hyps, conj) = split_last conjs in
2701 add_line_weights conj_weight conj
2702 #> fold (add_line_weights hyp_weight) hyps
2704 fun add_facts_weights facts =
2706 val num_facts = length facts
2708 fact_min_weight + (fact_max_weight - fact_min_weight) * Real.fromInt j
2709 / Real.fromInt num_facts
2711 map weight_of (0 upto num_facts - 1) ~~ facts
2712 |> fold (uncurry add_line_weights)
2714 val get = these o AList.lookup (op =) problem
2717 |> add_conjectures_weights (get free_typesN @ get conjsN)
2718 |> add_facts_weights (get factsN)
2719 |> fold (fold (add_line_weights type_info_default_weight) o get)
2720 [explicit_declsN, class_relsN, aritiesN]
2722 |> sort (prod_ord Real.compare string_ord o pairself swap)
2725 (* Ugly hack: may make innocent victims (collateral damage) *)
2726 fun may_be_app s args = String.isPrefix app_op_name s andalso length args = 2
2727 fun may_be_predicator s =
2728 member (op =) [predicator_name, prefixed_predicator_name] s
2730 fun strip_predicator (tm as ATerm (s, [tm'])) =
2731 if may_be_predicator s then tm' else tm
2732 | strip_predicator tm = tm
2734 fun make_head_roll (ATerm (s, tms)) =
2735 if may_be_app s tms then make_head_roll (hd tms) ||> append (tl tms)
2737 | make_head_roll _ = ("", [])
2739 fun strip_up_to_predicator (AQuant (_, _, phi)) = strip_up_to_predicator phi
2740 | strip_up_to_predicator (AConn (_, phis)) = maps strip_up_to_predicator phis
2741 | strip_up_to_predicator (AAtom tm) = [strip_predicator tm]
2743 fun strip_ahorn_etc (AQuant (_, _, phi)) = strip_ahorn_etc phi
2744 | strip_ahorn_etc (AConn (AImplies, [phi1, phi2])) =
2745 strip_ahorn_etc phi2 |>> append (strip_up_to_predicator phi1)
2746 | strip_ahorn_etc phi = ([], hd (strip_up_to_predicator phi))
2748 fun strip_iff_etc (AQuant (_, _, phi)) = strip_iff_etc phi
2749 | strip_iff_etc (AConn (AIff, [phi1, phi2])) =
2750 pairself strip_up_to_predicator (phi1, phi2)
2751 | strip_iff_etc _ = ([], [])
2753 val max_term_order_weight = 2147483647
2755 fun atp_problem_term_order_info problem =
2758 Graph.default_node (s, ())
2759 #> Graph.default_node (s', ())
2760 #> Graph.add_edge_acyclic (s, s')
2761 fun add_term_deps head (ATerm (s, args)) =
2762 if is_tptp_user_symbol head then
2763 (if is_tptp_user_symbol s then perhaps (try (add_edge s head)) else I)
2764 #> fold (add_term_deps head) args
2767 | add_term_deps head (AAbs (_, tm)) = add_term_deps head tm
2768 fun add_intro_deps pred (Formula (_, role, phi, _, info)) =
2769 if pred (role, info) then
2770 let val (hyps, concl) = strip_ahorn_etc phi in
2771 case make_head_roll concl of
2772 (head, args as _ :: _) => fold (add_term_deps head) (hyps @ args)
2777 | add_intro_deps _ _ = I
2778 fun add_atom_eq_deps (SOME true) (ATerm (s, [lhs as _, rhs])) =
2779 if is_tptp_equal s then
2780 case make_head_roll lhs of
2781 (head, args as _ :: _) => fold (add_term_deps head) (rhs :: args)
2785 | add_atom_eq_deps _ _ = I
2786 fun add_eq_deps pred (Formula (_, role, phi, _, info)) =
2787 if pred (role, info) then
2788 case strip_iff_etc phi of
2790 (case make_head_roll lhs of
2791 (head, args as _ :: _) => fold (add_term_deps head) (rhs @ args)
2793 | _ => formula_fold (SOME (role <> Conjecture)) add_atom_eq_deps phi
2796 | add_eq_deps _ _ = I
2797 fun has_status status (_, info) =
2798 extract_isabelle_status info = SOME status
2799 fun is_conj (role, _) = (role = Conjecture orelse role = Hypothesis)
2802 |> fold (fold (add_eq_deps (has_status defN)) o snd) problem
2803 |> fold (fold (add_eq_deps (has_status simpN orf is_conj)) o snd) problem
2804 |> fold (fold (add_intro_deps (has_status inductiveN)) o snd) problem
2805 |> fold (fold (add_intro_deps (has_status introN)) o snd) problem
2806 fun next_weight w = if w + w <= max_term_order_weight then w + w else w + 1
2807 fun add_weights _ [] = I
2808 | add_weights weight syms =
2809 fold (AList.update (op =) o rpair weight) syms
2810 #> add_weights (next_weight weight)
2811 (fold (union (op =) o Graph.immediate_succs graph) syms [])
2813 (* Sorting is not just for aesthetics: It specifies the precedence order
2814 for the term ordering (KBO or LPO), from smaller to larger values. *)
2815 [] |> add_weights 1 (Graph.minimals graph) |> sort (int_ord o pairself snd)