cleaner handling of bi-implication for THF output of first-order type encodings
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 datatype scope = Global | Local | Assum | Chained
121 datatype status = General | Induction | Intro | Inductive | Elim | Simp | Def
122 type stature = scope * status
126 Higher_Order of thf_flavor
127 datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
128 datatype strictness = Strict | Non_Strict
129 datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars
130 datatype type_level =
132 Noninf_Nonmono_Types of strictness * granularity |
133 Fin_Nonmono_Types of granularity |
138 Native of order * polymorphism * type_level |
139 Guards of polymorphism * type_level |
140 Tags of polymorphism * type_level
142 fun is_type_enc_native (Native _) = true
143 | is_type_enc_native _ = false
144 fun is_type_enc_higher_order (Native (Higher_Order _, _, _)) = true
145 | is_type_enc_higher_order _ = false
147 fun polymorphism_of_type_enc (Native (_, poly, _)) = poly
148 | polymorphism_of_type_enc (Guards (poly, _)) = poly
149 | polymorphism_of_type_enc (Tags (poly, _)) = poly
151 fun level_of_type_enc (Native (_, _, level)) = level
152 | level_of_type_enc (Guards (_, level)) = level
153 | level_of_type_enc (Tags (_, level)) = level
155 fun granularity_of_type_level (Noninf_Nonmono_Types (_, grain)) = grain
156 | granularity_of_type_level (Fin_Nonmono_Types grain) = grain
157 | granularity_of_type_level _ = All_Vars
159 fun is_type_level_quasi_sound All_Types = true
160 | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
161 | is_type_level_quasi_sound _ = false
162 val is_type_enc_quasi_sound = is_type_level_quasi_sound o level_of_type_enc
164 fun is_type_level_fairly_sound (Fin_Nonmono_Types _) = true
165 | is_type_level_fairly_sound level = is_type_level_quasi_sound level
166 val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
168 fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
169 | is_type_level_monotonicity_based (Fin_Nonmono_Types _) = true
170 | is_type_level_monotonicity_based _ = false
172 val no_lamsN = "no_lams" (* used internally; undocumented *)
173 val hide_lamsN = "hide_lams"
174 val liftingN = "lifting"
176 val combs_and_liftingN = "combs_and_lifting"
177 val combs_or_liftingN = "combs_or_lifting"
178 val keep_lamsN = "keep_lams"
179 val lam_liftingN = "lam_lifting" (* legacy *)
181 (* It's still unclear whether all TFF1 implementations will support type
182 signatures such as "!>[A : $tType] : $o", with ghost type variables. *)
183 val avoid_first_order_ghost_type_vars = false
185 val bound_var_prefix = "B_"
186 val all_bound_var_prefix = "A_"
187 val exist_bound_var_prefix = "E_"
188 val schematic_var_prefix = "V_"
189 val fixed_var_prefix = "v_"
190 val tvar_prefix = "T_"
191 val tfree_prefix = "t_"
192 val const_prefix = "c_"
193 val type_const_prefix = "tc_"
194 val native_type_prefix = "n_"
195 val class_prefix = "cl_"
197 (* Freshness almost guaranteed! *)
198 val atp_prefix = "ATP" ^ Long_Name.separator
199 val atp_weak_prefix = "ATP:"
201 val lam_lifted_prefix = atp_weak_prefix ^ "Lam"
202 val lam_lifted_mono_prefix = lam_lifted_prefix ^ "m"
203 val lam_lifted_poly_prefix = lam_lifted_prefix ^ "p"
205 val skolem_const_prefix = atp_prefix ^ "Sko"
206 val old_skolem_const_prefix = skolem_const_prefix ^ "o"
207 val new_skolem_const_prefix = skolem_const_prefix ^ "n"
209 val combinator_prefix = "COMB"
211 val type_decl_prefix = "ty_"
212 val sym_decl_prefix = "sy_"
213 val guards_sym_formula_prefix = "gsy_"
214 val tags_sym_formula_prefix = "tsy_"
215 val uncurried_alias_eq_prefix = "unc_"
216 val fact_prefix = "fact_"
217 val conjecture_prefix = "conj_"
218 val helper_prefix = "help_"
219 val class_rel_clause_prefix = "clar_"
220 val arity_clause_prefix = "arity_"
221 val tfree_clause_prefix = "tfree_"
223 val lam_fact_prefix = "ATP.lambda_"
224 val typed_helper_suffix = "_T"
225 val untyped_helper_suffix = "_U"
227 val predicator_name = "pp"
228 val app_op_name = "aa"
229 val type_guard_name = "gg"
230 val type_tag_name = "tt"
232 val prefixed_predicator_name = const_prefix ^ predicator_name
233 val prefixed_app_op_name = const_prefix ^ app_op_name
234 val prefixed_type_tag_name = const_prefix ^ type_tag_name
236 (*Escaping of special characters.
237 Alphanumeric characters are left unchanged.
238 The character _ goes to __
239 Characters in the range ASCII space to / go to _A to _P, respectively.
240 Other characters go to _nnn where nnn is the decimal ASCII code.*)
241 val upper_a_minus_space = Char.ord #"A" - Char.ord #" "
243 fun stringN_of_int 0 _ = ""
244 | stringN_of_int k n =
245 stringN_of_int (k - 1) (n div 10) ^ string_of_int (n mod 10)
247 fun ascii_of_char c =
248 if Char.isAlphaNum c then
250 else if c = #"_" then
252 else if #" " <= c andalso c <= #"/" then
253 "_" ^ String.str (Char.chr (Char.ord c + upper_a_minus_space))
255 (* fixed width, in case more digits follow *)
256 "_" ^ stringN_of_int 3 (Char.ord c)
258 val ascii_of = String.translate ascii_of_char
260 (** Remove ASCII armoring from names in proof files **)
262 (* We don't raise error exceptions because this code can run inside a worker
263 thread. Also, the errors are impossible. *)
266 fun un rcs [] = String.implode(rev rcs)
267 | un rcs [#"_"] = un (#"_" :: rcs) [] (* ERROR *)
268 (* Three types of _ escapes: __, _A to _P, _nnn *)
269 | un rcs (#"_" :: #"_" :: cs) = un (#"_" :: rcs) cs
270 | un rcs (#"_" :: c :: cs) =
271 if #"A" <= c andalso c<= #"P" then
272 (* translation of #" " to #"/" *)
273 un (Char.chr (Char.ord c - upper_a_minus_space) :: rcs) cs
275 let val digits = List.take (c :: cs, 3) handle General.Subscript => [] in
276 case Int.fromString (String.implode digits) of
277 SOME n => un (Char.chr n :: rcs) (List.drop (cs, 2))
278 | NONE => un (c :: #"_" :: rcs) cs (* ERROR *)
280 | un rcs (c :: cs) = un (c :: rcs) cs
281 in un [] o String.explode end
283 (* If string s has the prefix s1, return the result of deleting it,
285 fun unprefix_and_unascii s1 s =
286 if String.isPrefix s1 s then
287 SOME (unascii_of (String.extract (s, size s1, NONE)))
292 [("c_False", (@{const_name False}, (@{thm fFalse_def},
293 ("fFalse", @{const_name ATP.fFalse})))),
294 ("c_True", (@{const_name True}, (@{thm fTrue_def},
295 ("fTrue", @{const_name ATP.fTrue})))),
296 ("c_Not", (@{const_name Not}, (@{thm fNot_def},
297 ("fNot", @{const_name ATP.fNot})))),
298 ("c_conj", (@{const_name conj}, (@{thm fconj_def},
299 ("fconj", @{const_name ATP.fconj})))),
300 ("c_disj", (@{const_name disj}, (@{thm fdisj_def},
301 ("fdisj", @{const_name ATP.fdisj})))),
302 ("c_implies", (@{const_name implies}, (@{thm fimplies_def},
303 ("fimplies", @{const_name ATP.fimplies})))),
304 ("equal", (@{const_name HOL.eq}, (@{thm fequal_def},
305 ("fequal", @{const_name ATP.fequal})))),
306 ("c_All", (@{const_name All}, (@{thm fAll_def},
307 ("fAll", @{const_name ATP.fAll})))),
308 ("c_Ex", (@{const_name Ex}, (@{thm fEx_def},
309 ("fEx", @{const_name ATP.fEx}))))]
311 val proxify_const = AList.lookup (op =) proxy_table #> Option.map (snd o snd)
313 (* Readable names for the more common symbolic functions. Do not mess with the
314 table unless you know what you are doing. *)
315 val const_trans_table =
316 [(@{type_name Product_Type.prod}, "prod"),
317 (@{type_name Sum_Type.sum}, "sum"),
318 (@{const_name False}, "False"),
319 (@{const_name True}, "True"),
320 (@{const_name Not}, "Not"),
321 (@{const_name conj}, "conj"),
322 (@{const_name disj}, "disj"),
323 (@{const_name implies}, "implies"),
324 (@{const_name HOL.eq}, "equal"),
325 (@{const_name All}, "All"),
326 (@{const_name Ex}, "Ex"),
327 (@{const_name If}, "If"),
328 (@{const_name Set.member}, "member"),
329 (@{const_name Meson.COMBI}, combinator_prefix ^ "I"),
330 (@{const_name Meson.COMBK}, combinator_prefix ^ "K"),
331 (@{const_name Meson.COMBB}, combinator_prefix ^ "B"),
332 (@{const_name Meson.COMBC}, combinator_prefix ^ "C"),
333 (@{const_name Meson.COMBS}, combinator_prefix ^ "S")]
335 |> fold (Symtab.update o swap o snd o snd o snd) proxy_table
337 (* Invert the table of translations between Isabelle and ATPs. *)
338 val const_trans_table_inv =
339 const_trans_table |> Symtab.dest |> map swap |> Symtab.make
340 val const_trans_table_unprox =
342 |> fold (fn (_, (isa, (_, (_, atp)))) => Symtab.update (atp, isa)) proxy_table
344 val invert_const = perhaps (Symtab.lookup const_trans_table_inv)
345 val unproxify_const = perhaps (Symtab.lookup const_trans_table_unprox)
348 case Symtab.lookup const_trans_table c of
352 fun ascii_of_indexname (v, 0) = ascii_of v
353 | ascii_of_indexname (v, i) = ascii_of v ^ "_" ^ string_of_int i
355 fun make_bound_var x = bound_var_prefix ^ ascii_of x
356 fun make_all_bound_var x = all_bound_var_prefix ^ ascii_of x
357 fun make_exist_bound_var x = exist_bound_var_prefix ^ ascii_of x
358 fun make_schematic_var v = schematic_var_prefix ^ ascii_of_indexname v
359 fun make_fixed_var x = fixed_var_prefix ^ ascii_of x
361 fun make_schematic_type_var (x, i) =
362 tvar_prefix ^ (ascii_of_indexname (unprefix "'" x, i))
363 fun make_fixed_type_var x = tfree_prefix ^ (ascii_of (unprefix "'" x))
365 (* "HOL.eq" and choice are mapped to the ATP's equivalents *)
367 val choice_const = (fst o dest_Const o HOLogic.choice_const) Term.dummyT
368 fun default c = const_prefix ^ lookup_const c
370 fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
371 | make_fixed_const (SOME (Native (Higher_Order THF_With_Choice, _, _))) c =
372 if c = choice_const then tptp_choice else default c
373 | make_fixed_const _ c = default c
376 fun make_fixed_type_const c = type_const_prefix ^ lookup_const c
378 fun make_type_class clas = class_prefix ^ ascii_of clas
380 fun new_skolem_var_name_from_const s =
381 let val ss = s |> space_explode Long_Name.separator in
382 nth ss (length ss - 2)
385 (* These are either simplified away by "Meson.presimplify" (most of the time) or
386 handled specially via "fFalse", "fTrue", ..., "fequal". *)
387 val atp_irrelevant_consts =
388 [@{const_name False}, @{const_name True}, @{const_name Not},
389 @{const_name conj}, @{const_name disj}, @{const_name implies},
390 @{const_name HOL.eq}, @{const_name If}, @{const_name Let}]
392 val atp_monomorph_bad_consts =
393 atp_irrelevant_consts @
394 (* These are ignored anyway by the relevance filter (unless they appear in
395 higher-order places) but not by the monomorphizer. *)
396 [@{const_name all}, @{const_name "==>"}, @{const_name "=="},
397 @{const_name Trueprop}, @{const_name All}, @{const_name Ex},
398 @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}]
400 fun add_schematic_const (x as (_, T)) =
401 Monomorph.typ_has_tvars T ? Symtab.insert_list (op =) x
402 val add_schematic_consts_of =
403 Term.fold_aterms (fn Const (x as (s, _)) =>
404 not (member (op =) atp_monomorph_bad_consts s)
405 ? add_schematic_const x
407 fun atp_schematic_consts_of t = add_schematic_consts_of t Symtab.empty
409 (** Definitions and functions for FOL clauses and formulas for TPTP **)
411 (** Isabelle arities **)
413 type arity_atom = name * name * name list
415 val type_class = the_single @{sort type}
419 prem_atoms : arity_atom list,
420 concl_atom : arity_atom}
422 fun add_prem_atom tvar =
423 fold (fn s => s <> type_class ? cons (`make_type_class s, `I tvar, []))
425 (* Arity of type constructor "tcon :: (arg1, ..., argN) res" *)
426 fun make_axiom_arity_clause (tcons, name, (cls, args)) =
428 val tvars = map (prefix tvar_prefix o string_of_int) (1 upto length args)
429 val tvars_srts = ListPair.zip (tvars, args)
432 prem_atoms = [] |> fold (uncurry add_prem_atom) tvars_srts,
433 concl_atom = (`make_type_class cls, `make_fixed_type_const tcons,
437 fun arity_clause _ _ (_, []) = []
438 | arity_clause seen n (tcons, ("HOL.type", _) :: ars) = (* ignore *)
439 arity_clause seen n (tcons, ars)
440 | arity_clause seen n (tcons, (ar as (class, _)) :: ars) =
441 if member (op =) seen class then
442 (* multiple arities for the same (tycon, class) pair *)
443 make_axiom_arity_clause (tcons,
444 lookup_const tcons ^ "___" ^ ascii_of class ^ "_" ^ string_of_int n,
446 arity_clause seen (n + 1) (tcons, ars)
448 make_axiom_arity_clause (tcons, lookup_const tcons ^ "___" ^
449 ascii_of class, ar) ::
450 arity_clause (class :: seen) n (tcons, ars)
452 fun multi_arity_clause [] = []
453 | multi_arity_clause ((tcons, ars) :: tc_arlists) =
454 arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
456 (* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
457 theory thy provided its arguments have the corresponding sorts. *)
458 fun type_class_pairs thy tycons classes =
460 val alg = Sign.classes_of thy
461 fun domain_sorts tycon = Sorts.mg_domain alg tycon o single
462 fun add_class tycon class =
463 cons (class, domain_sorts tycon class)
464 handle Sorts.CLASS_ERROR _ => I
465 fun try_classes tycon = (tycon, fold (add_class tycon) classes [])
466 in map try_classes tycons end
468 (*Proving one (tycon, class) membership may require proving others, so iterate.*)
469 fun iter_type_class_pairs _ _ [] = ([], [])
470 | iter_type_class_pairs thy tycons classes =
472 fun maybe_insert_class s =
473 (s <> type_class andalso not (member (op =) classes s))
475 val cpairs = type_class_pairs thy tycons classes
477 [] |> fold (fold (fold (fold maybe_insert_class) o snd) o snd) cpairs
478 val (classes', cpairs') = iter_type_class_pairs thy tycons newclasses
479 in (classes' @ classes, union (op =) cpairs' cpairs) end
481 fun make_arity_clauses thy tycons =
482 iter_type_class_pairs thy tycons ##> multi_arity_clause
485 (** Isabelle class relations **)
487 type class_rel_clause =
492 (* Generate all pairs (sub, super) such that sub is a proper subclass of super
494 fun class_pairs _ [] _ = []
495 | class_pairs thy subs supers =
497 val class_less = Sorts.class_less (Sign.classes_of thy)
498 fun add_super sub super = class_less (sub, super) ? cons (sub, super)
499 fun add_supers sub = fold (add_super sub) supers
500 in fold add_supers subs [] end
502 fun make_class_rel_clause (sub, super) =
503 {name = sub ^ "_" ^ super, subclass = `make_type_class sub,
504 superclass = `make_type_class super}
506 fun make_class_rel_clauses thy subs supers =
507 map make_class_rel_clause (class_pairs thy subs supers)
509 (* intermediate terms *)
511 IConst of name * typ * typ list |
513 IApp of iterm * iterm |
514 IAbs of (name * typ) * iterm
516 fun ityp_of (IConst (_, T, _)) = T
517 | ityp_of (IVar (_, T)) = T
518 | ityp_of (IApp (t1, _)) = snd (dest_funT (ityp_of t1))
519 | ityp_of (IAbs ((_, T), tm)) = T --> ityp_of tm
521 (*gets the head of a combinator application, along with the list of arguments*)
522 fun strip_iterm_comb u =
524 fun stripc (IApp (t, u), ts) = stripc (t, u :: ts)
526 in stripc (u, []) end
528 fun atomic_types_of T = fold_atyps (insert (op =)) T []
530 val tvar_a_str = "'a"
531 val tvar_a = TVar ((tvar_a_str, 0), HOLogic.typeS)
532 val tvar_a_name = (make_schematic_type_var (tvar_a_str, 0), tvar_a_str)
533 val itself_name = `make_fixed_type_const @{type_name itself}
534 val TYPE_name = `(make_fixed_const NONE) @{const_name TYPE}
535 val tvar_a_atype = AType (tvar_a_name, [])
536 val a_itself_atype = AType (itself_name, [tvar_a_atype])
538 fun new_skolem_const_name s num_T_args =
539 [new_skolem_const_prefix, s, string_of_int num_T_args]
542 fun robust_const_type thy s =
543 if s = app_op_name then
544 Logic.varifyT_global @{typ "('a => 'b) => 'a => 'b"}
545 else if String.isPrefix lam_lifted_prefix s then
546 Logic.varifyT_global @{typ "'a => 'b"}
548 (* Old Skolems throw a "TYPE" exception here, which will be caught. *)
549 s |> Sign.the_const_type thy
551 val robust_const_ary =
553 fun ary (Type (@{type_name fun}, [_, T])) = 1 + ary T
555 in ary oo robust_const_type end
557 (* This function only makes sense if "T" is as general as possible. *)
558 fun robust_const_typargs thy (s, T) =
559 if s = app_op_name then
560 let val (T1, T2) = T |> domain_type |> dest_funT in [T1, T2] end
561 else if String.isPrefix old_skolem_const_prefix s then
562 [] |> Term.add_tvarsT T |> rev |> map TVar
563 else if String.isPrefix lam_lifted_prefix s then
564 if String.isPrefix lam_lifted_poly_prefix s then
565 let val (T1, T2) = T |> dest_funT in [T1, T2] end
569 (s, T) |> Sign.const_typargs thy
571 (* Converts an Isabelle/HOL term (with combinators) into an intermediate term.
572 Also accumulates sort infomation. *)
573 fun iterm_from_term thy type_enc bs (P $ Q) =
575 val (P', P_atomics_Ts) = iterm_from_term thy type_enc bs P
576 val (Q', Q_atomics_Ts) = iterm_from_term thy type_enc bs Q
577 in (IApp (P', Q'), union (op =) P_atomics_Ts Q_atomics_Ts) end
578 | iterm_from_term thy type_enc _ (Const (c, T)) =
579 (IConst (`(make_fixed_const (SOME type_enc)) c, T,
580 robust_const_typargs thy (c, T)),
582 | iterm_from_term _ _ _ (Free (s, T)) =
583 (IConst (`make_fixed_var s, T, []), atomic_types_of T)
584 | iterm_from_term _ type_enc _ (Var (v as (s, _), T)) =
585 (if String.isPrefix Meson_Clausify.new_skolem_var_prefix s then
587 val Ts = T |> strip_type |> swap |> op ::
588 val s' = new_skolem_const_name s (length Ts)
589 in IConst (`(make_fixed_const (SOME type_enc)) s', T, Ts) end
591 IVar ((make_schematic_var v, s), T), atomic_types_of T)
592 | iterm_from_term _ _ bs (Bound j) =
593 nth bs j |> (fn (_, (name, T)) => (IConst (name, T, []), atomic_types_of T))
594 | iterm_from_term thy type_enc bs (Abs (s, T, t)) =
596 fun vary s = s |> AList.defined (op =) bs s ? vary o Symbol.bump_string
598 val name = `make_bound_var s
599 val (tm, atomic_Ts) =
600 iterm_from_term thy type_enc ((s, (name, T)) :: bs) t
601 in (IAbs ((name, T), tm), union (op =) atomic_Ts (atomic_types_of T)) end
603 (* "_query", "_bang", and "_at" are for the ASCII-challenged Metis and
605 val queries = ["?", "_query"]
606 val bangs = ["!", "_bang"]
607 val ats = ["@", "_at"]
609 fun try_unsuffixes ss s =
610 fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
612 fun try_nonmono constr suffixes fallback s =
613 case try_unsuffixes suffixes s of
615 (case try_unsuffixes suffixes s of
616 SOME s => (constr Positively_Naked_Vars, s)
618 case try_unsuffixes ats s of
619 SOME s => (constr Ghost_Type_Arg_Vars, s)
620 | NONE => (constr All_Vars, s))
623 fun type_enc_from_string strictness s =
624 (case try (unprefix "poly_") s of
625 SOME s => (SOME Polymorphic, s)
627 case try (unprefix "raw_mono_") s of
628 SOME s => (SOME Raw_Monomorphic, s)
630 case try (unprefix "mono_") s of
631 SOME s => (SOME Mangled_Monomorphic, s)
634 |> try_nonmono Fin_Nonmono_Types bangs
635 |> try_nonmono (curry Noninf_Nonmono_Types strictness) queries)
636 |> (fn (poly, (level, core)) =>
637 case (core, (poly, level)) of
638 ("native", (SOME poly, _)) =>
639 (case (poly, level) of
640 (Polymorphic, All_Types) =>
641 Native (First_Order, Polymorphic, All_Types)
642 | (Mangled_Monomorphic, _) =>
643 if granularity_of_type_level level = All_Vars then
644 Native (First_Order, Mangled_Monomorphic, level)
647 | _ => raise Same.SAME)
648 | ("native_higher", (SOME poly, _)) =>
649 (case (poly, level) of
650 (Polymorphic, All_Types) =>
651 Native (Higher_Order THF_With_Choice, Polymorphic, All_Types)
652 | (_, Noninf_Nonmono_Types _) => raise Same.SAME
653 | (Mangled_Monomorphic, _) =>
654 if granularity_of_type_level level = All_Vars then
655 Native (Higher_Order THF_With_Choice, Mangled_Monomorphic,
659 | _ => raise Same.SAME)
660 | ("guards", (SOME poly, _)) =>
661 if poly = Mangled_Monomorphic andalso
662 granularity_of_type_level level = Ghost_Type_Arg_Vars then
666 | ("tags", (SOME poly, _)) =>
667 if granularity_of_type_level level = Ghost_Type_Arg_Vars then
671 | ("args", (SOME poly, All_Types (* naja *))) =>
672 Guards (poly, Const_Arg_Types)
673 | ("erased", (NONE, All_Types (* naja *))) =>
674 Guards (Polymorphic, No_Types)
675 | _ => raise Same.SAME)
676 handle Same.SAME => error ("Unknown type encoding: " ^ quote s ^ ".")
678 fun adjust_order THF_Without_Choice (Higher_Order _) =
679 Higher_Order THF_Without_Choice
680 | adjust_order _ type_enc = type_enc
682 fun adjust_type_enc (THF (TPTP_Polymorphic, _, flavor))
683 (Native (order, poly, level)) =
684 Native (adjust_order flavor order, poly, level)
685 | adjust_type_enc (THF (TPTP_Monomorphic, _, flavor))
686 (Native (order, _, level)) =
687 Native (adjust_order flavor order, Mangled_Monomorphic, level)
688 | adjust_type_enc (TFF (TPTP_Monomorphic, _)) (Native (_, _, level)) =
689 Native (First_Order, Mangled_Monomorphic, level)
690 | adjust_type_enc (DFG DFG_Sorted) (Native (_, _, level)) =
691 Native (First_Order, Mangled_Monomorphic, level)
692 | adjust_type_enc (TFF _) (Native (_, poly, level)) =
693 Native (First_Order, poly, level)
694 | adjust_type_enc format (Native (_, poly, level)) =
695 adjust_type_enc format (Guards (poly, level))
696 | adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
697 (if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
698 | adjust_type_enc _ type_enc = type_enc
702 @{const Not} $ t1 => is_fol_term t1
703 | Const (@{const_name All}, _) $ Abs (_, _, t') => is_fol_term t'
704 | Const (@{const_name All}, _) $ t1 => is_fol_term t1
705 | Const (@{const_name Ex}, _) $ Abs (_, _, t') => is_fol_term t'
706 | Const (@{const_name Ex}, _) $ t1 => is_fol_term t1
707 | @{const HOL.conj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
708 | @{const HOL.disj} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
709 | @{const HOL.implies} $ t1 $ t2 => is_fol_term t1 andalso is_fol_term t2
710 | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
711 is_fol_term t1 andalso is_fol_term t2
712 | _ => not (exists_subterm (fn Abs _ => true | _ => false) t)
714 fun simple_translate_lambdas do_lambdas ctxt t =
715 if is_fol_term t then
721 @{const Not} $ t1 => @{const Not} $ trans Ts t1
722 | (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
723 t0 $ Abs (s, T, trans (T :: Ts) t')
724 | (t0 as Const (@{const_name All}, _)) $ t1 =>
725 trans Ts (t0 $ eta_expand Ts t1 1)
726 | (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
727 t0 $ Abs (s, T, trans (T :: Ts) t')
728 | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
729 trans Ts (t0 $ eta_expand Ts t1 1)
730 | (t0 as @{const HOL.conj}) $ t1 $ t2 =>
731 t0 $ trans Ts t1 $ trans Ts t2
732 | (t0 as @{const HOL.disj}) $ t1 $ t2 =>
733 t0 $ trans Ts t1 $ trans Ts t2
734 | (t0 as @{const HOL.implies}) $ t1 $ t2 =>
735 t0 $ trans Ts t1 $ trans Ts t2
736 | (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
738 t0 $ trans Ts t1 $ trans Ts t2
740 if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
741 else t |> Envir.eta_contract |> do_lambdas ctxt Ts
742 val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
743 in t |> trans [] |> singleton (Variable.export_terms ctxt' ctxt) end
745 fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
746 do_cheaply_conceal_lambdas Ts t1
747 $ do_cheaply_conceal_lambdas Ts t2
748 | do_cheaply_conceal_lambdas Ts (Abs (_, T, t)) =
749 Const (lam_lifted_poly_prefix ^ serial_string (),
750 T --> fastype_of1 (T :: Ts, t))
751 | do_cheaply_conceal_lambdas _ t = t
753 fun concealed_bound_name j = atp_weak_prefix ^ string_of_int j
754 fun conceal_bounds Ts t =
755 subst_bounds (map (Free o apfst concealed_bound_name)
756 (0 upto length Ts - 1 ~~ Ts), t)
757 fun reveal_bounds Ts =
758 subst_atomic (map (fn (j, T) => (Free (concealed_bound_name j, T), Bound j))
759 (0 upto length Ts - 1 ~~ Ts))
761 fun do_introduce_combinators ctxt Ts t =
762 let val thy = Proof_Context.theory_of ctxt in
763 t |> conceal_bounds Ts
765 |> Meson_Clausify.introduce_combinators_in_cterm
766 |> prop_of |> Logic.dest_equals |> snd
769 (* A type variable of sort "{}" will make abstraction fail. *)
770 handle THM _ => t |> do_cheaply_conceal_lambdas Ts
771 val introduce_combinators = simple_translate_lambdas do_introduce_combinators
773 fun constify_lifted (t $ u) = constify_lifted t $ constify_lifted u
774 | constify_lifted (Abs (s, T, t)) = Abs (s, T, constify_lifted t)
775 | constify_lifted (Free (x as (s, _))) =
776 (if String.isPrefix lam_lifted_prefix s then Const else Free) x
777 | constify_lifted t = t
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 (* Requires bound variables not to clash with any schematic variables (as
796 should be the case right after lambda-lifting). *)
797 |>> map (open_form (unprefix close_form_prefix))
798 ||> map (open_form I)
800 fun lift_lams ctxt = lift_lams_part_2 ctxt oo lift_lams_part_1 ctxt
802 fun intentionalize_def (Const (@{const_name All}, _) $ Abs (_, _, t)) =
804 | intentionalize_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
806 fun lam T t = Abs (Name.uu, T, t)
807 val (head, args) = strip_comb t ||> rev
808 val head_T = fastype_of head
810 val arg_Ts = head_T |> binder_types |> take n |> rev
811 val u = u |> subst_atomic (args ~~ map Bound (0 upto n - 1))
812 in HOLogic.eq_const head_T $ head $ fold lam arg_Ts u end
813 | intentionalize_def t = t
815 type translated_formula =
819 iformula : (name, typ, iterm) formula,
820 atomic_types : typ list}
822 fun update_iformula f ({name, stature, kind, iformula, atomic_types}
823 : translated_formula) =
824 {name = name, stature = stature, kind = kind, iformula = f iformula,
825 atomic_types = atomic_types} : translated_formula
827 fun fact_lift f ({iformula, ...} : translated_formula) = f iformula
829 fun insert_type thy get_T x xs =
830 let val T = get_T x in
831 if exists (type_instance thy T o get_T) xs then xs
832 else x :: filter_out (type_generalization thy T o get_T) xs
835 (* The Booleans indicate whether all type arguments should be kept. *)
836 datatype type_arg_policy =
837 Explicit_Type_Args of bool (* infer_from_term_args *) |
841 fun type_arg_policy monom_constrs type_enc s =
842 let val poly = polymorphism_of_type_enc type_enc in
843 if s = type_tag_name then
844 if poly = Mangled_Monomorphic then Mangled_Type_Args
845 else Explicit_Type_Args false
846 else case type_enc of
847 Native (_, Polymorphic, _) => Explicit_Type_Args false
848 | Tags (_, All_Types) => No_Type_Args
850 let val level = level_of_type_enc type_enc in
851 if level = No_Types orelse s = @{const_name HOL.eq} orelse
852 (s = app_op_name andalso level = Const_Arg_Types) then
854 else if poly = Mangled_Monomorphic then
856 else if member (op =) monom_constrs s andalso
857 granularity_of_type_level level = Positively_Naked_Vars then
861 (level = All_Types orelse
862 granularity_of_type_level level = Ghost_Type_Arg_Vars)
866 (* Make atoms for sorted type variables. *)
867 fun generic_add_sorts_on_type (_, []) = I
868 | generic_add_sorts_on_type ((x, i), s :: ss) =
869 generic_add_sorts_on_type ((x, i), ss)
870 #> (if s = the_single @{sort HOL.type} then
873 insert (op =) (`make_type_class s, `make_fixed_type_var x)
875 insert (op =) (`make_type_class s,
876 (make_schematic_type_var (x, i), x)))
877 fun add_sorts_on_tfree (TFree (s, S)) = generic_add_sorts_on_type ((s, ~1), S)
878 | add_sorts_on_tfree _ = I
879 fun add_sorts_on_tvar (TVar z) = generic_add_sorts_on_type z
880 | add_sorts_on_tvar _ = I
882 fun type_class_formula type_enc class arg =
883 AAtom (ATerm (class, arg ::
885 Native (First_Order, Polymorphic, _) =>
886 if avoid_first_order_ghost_type_vars then [ATerm (TYPE_name, [arg])]
889 fun formulas_for_types type_enc add_sorts_on_typ Ts =
890 [] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
891 |> map (fn (class, name) =>
892 type_class_formula type_enc class (ATerm (name, [])))
894 fun mk_aconns c phis =
895 let val (phis', phi') = split_last phis in
896 fold_rev (mk_aconn c) phis' phi'
898 fun mk_ahorn [] phi = phi
899 | mk_ahorn phis psi = AConn (AImplies, [mk_aconns AAnd phis, psi])
900 fun mk_aquant _ [] phi = phi
901 | mk_aquant q xs (phi as AQuant (q', xs', phi')) =
902 if q = q' then AQuant (q, xs @ xs', phi') else AQuant (q, xs, phi)
903 | mk_aquant q xs phi = AQuant (q, xs, phi)
905 fun close_universally add_term_vars phi =
907 fun add_formula_vars bounds (AQuant (_, xs, phi)) =
908 add_formula_vars (map fst xs @ bounds) phi
909 | add_formula_vars bounds (AConn (_, phis)) =
910 fold (add_formula_vars bounds) phis
911 | add_formula_vars bounds (AAtom tm) = add_term_vars bounds tm
912 in mk_aquant AForall (add_formula_vars [] phi []) phi end
914 fun add_term_vars bounds (ATerm (name as (s, _), tms)) =
915 (if is_tptp_variable s andalso
916 not (String.isPrefix tvar_prefix s) andalso
917 not (member (op =) bounds name) then
918 insert (op =) (name, NONE)
921 #> fold (add_term_vars bounds) tms
922 | add_term_vars bounds (AAbs ((name, _), tm)) =
923 add_term_vars (name :: bounds) tm
924 fun close_formula_universally phi = close_universally add_term_vars phi
926 fun add_iterm_vars bounds (IApp (tm1, tm2)) =
927 fold (add_iterm_vars bounds) [tm1, tm2]
928 | add_iterm_vars _ (IConst _) = I
929 | add_iterm_vars bounds (IVar (name, T)) =
930 not (member (op =) bounds name) ? insert (op =) (name, SOME T)
931 | add_iterm_vars bounds (IAbs (_, tm)) = add_iterm_vars bounds tm
932 fun close_iformula_universally phi = close_universally add_iterm_vars phi
934 val fused_infinite_type_name = "ATP.fused_inf" (* shouldn't clash *)
935 val fused_infinite_type = Type (fused_infinite_type_name, [])
937 fun tvar_name (x as (s, _)) = (make_schematic_type_var x, s)
939 fun ho_term_from_typ type_enc =
941 fun term (Type (s, Ts)) =
942 ATerm (case (is_type_enc_higher_order type_enc, s) of
943 (true, @{type_name bool}) => `I tptp_bool_type
944 | (true, @{type_name fun}) => `I tptp_fun_type
945 | _ => if s = fused_infinite_type_name andalso
946 is_type_enc_native type_enc then
947 `I tptp_individual_type
949 `make_fixed_type_const s,
951 | term (TFree (s, _)) = ATerm (`make_fixed_type_var s, [])
952 | term (TVar (x, _)) = ATerm (tvar_name x, [])
955 fun ho_term_for_type_arg type_enc T =
956 if T = dummyT then NONE else SOME (ho_term_from_typ type_enc T)
958 (* This shouldn't clash with anything else. *)
959 val uncurried_alias_sep = "\000"
960 val mangled_type_sep = "\001"
962 val ascii_of_uncurried_alias_sep = ascii_of uncurried_alias_sep
964 fun generic_mangled_type_name f (ATerm (name, [])) = f name
965 | generic_mangled_type_name f (ATerm (name, tys)) =
966 f name ^ "(" ^ space_implode "," (map (generic_mangled_type_name f) tys)
968 | generic_mangled_type_name _ _ = raise Fail "unexpected type abstraction"
970 fun mangled_type type_enc =
971 generic_mangled_type_name fst o ho_term_from_typ type_enc
973 fun make_native_type s =
974 if s = tptp_bool_type orelse s = tptp_fun_type orelse
975 s = tptp_individual_type then
978 native_type_prefix ^ ascii_of s
980 fun ho_type_from_ho_term type_enc pred_sym ary =
982 fun to_mangled_atype ty =
983 AType ((make_native_type (generic_mangled_type_name fst ty),
984 generic_mangled_type_name snd ty), [])
985 fun to_poly_atype (ATerm (name, tys)) = AType (name, map to_poly_atype tys)
986 | to_poly_atype _ = raise Fail "unexpected type abstraction"
988 if polymorphism_of_type_enc type_enc = Polymorphic then to_poly_atype
989 else to_mangled_atype
990 fun to_afun f1 f2 tys = AFun (f1 (hd tys), f2 (nth tys 1))
991 fun to_fo 0 ty = if pred_sym then bool_atype else to_atype ty
992 | to_fo ary (ATerm (_, tys)) = to_afun to_atype (to_fo (ary - 1)) tys
993 | to_fo _ _ = raise Fail "unexpected type abstraction"
994 fun to_ho (ty as ATerm ((s, _), tys)) =
995 if s = tptp_fun_type then to_afun to_ho to_ho tys else to_atype ty
996 | to_ho _ = raise Fail "unexpected type abstraction"
997 in if is_type_enc_higher_order type_enc then to_ho else to_fo ary end
999 fun ho_type_from_typ type_enc pred_sym ary =
1000 ho_type_from_ho_term type_enc pred_sym ary
1001 o ho_term_from_typ type_enc
1003 fun aliased_uncurried ary (s, s') =
1004 (s ^ ascii_of_uncurried_alias_sep ^ string_of_int ary, s' ^ string_of_int ary)
1005 fun unaliased_uncurried (s, s') =
1006 case space_explode uncurried_alias_sep s of
1008 | [s1, s2] => (s1, unsuffix s2 s')
1009 | _ => raise Fail "ill-formed explicit application alias"
1011 fun raw_mangled_const_name type_name ty_args (s, s') =
1013 fun type_suffix f g =
1014 fold_rev (curry (op ^) o g o prefix mangled_type_sep o type_name f)
1016 in (s ^ type_suffix fst ascii_of, s' ^ type_suffix snd I) end
1017 fun mangled_const_name type_enc =
1018 map_filter (ho_term_for_type_arg type_enc)
1019 #> raw_mangled_const_name generic_mangled_type_name
1021 val parse_mangled_ident =
1022 Scan.many1 (not o member (op =) ["(", ")", ","]) >> implode
1024 fun parse_mangled_type x =
1025 (parse_mangled_ident
1026 -- Scan.optional ($$ "(" |-- Scan.optional parse_mangled_types [] --| $$ ")")
1028 and parse_mangled_types x =
1029 (parse_mangled_type ::: Scan.repeat ($$ "," |-- parse_mangled_type)) x
1031 fun unmangled_type s =
1032 s |> suffix ")" |> raw_explode
1033 |> Scan.finite Symbol.stopper
1034 (Scan.error (!! (fn _ => raise Fail ("unrecognized mangled type " ^
1035 quote s)) parse_mangled_type))
1038 fun unmangled_const_name s =
1039 (s, s) |> unaliased_uncurried |> fst |> space_explode mangled_type_sep
1040 fun unmangled_const s =
1041 let val ss = unmangled_const_name s in
1042 (hd ss, map unmangled_type (tl ss))
1045 fun introduce_proxies_in_iterm type_enc =
1047 fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
1048 | tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
1050 (* Eta-expand "!!" and "??", to work around LEO-II 1.2.8 parser
1051 limitation. This works in conjuction with special code in
1052 "ATP_Problem" that uses the syntactic sugar "!" and "?" whenever
1054 IAbs ((`I "P", p_T),
1055 IApp (IConst (`I ho_quant, T, []),
1056 IAbs ((`I "X", x_T),
1057 IApp (IConst (`I "P", p_T, []),
1058 IConst (`I "X", x_T, [])))))
1059 | tweak_ho_quant _ _ _ = raise Fail "unexpected type for quantifier"
1060 fun intro top_level args (IApp (tm1, tm2)) =
1061 IApp (intro top_level (tm2 :: args) tm1, intro false [] tm2)
1062 | intro top_level args (IConst (name as (s, _), T, T_args)) =
1063 (case proxify_const s of
1065 if top_level orelse is_type_enc_higher_order type_enc then
1066 case (top_level, s) of
1067 (_, "c_False") => IConst (`I tptp_false, T, [])
1068 | (_, "c_True") => IConst (`I tptp_true, T, [])
1069 | (false, "c_Not") => IConst (`I tptp_not, T, [])
1070 | (false, "c_conj") => IConst (`I tptp_and, T, [])
1071 | (false, "c_disj") => IConst (`I tptp_or, T, [])
1072 | (false, "c_implies") => IConst (`I tptp_implies, T, [])
1073 | (false, "c_All") => tweak_ho_quant tptp_ho_forall T args
1074 | (false, "c_Ex") => tweak_ho_quant tptp_ho_exists T args
1076 if is_tptp_equal s andalso length args = 2 then
1077 IConst (`I tptp_equal, T, [])
1079 (* Eta-expand partially applied THF equality, because the
1080 LEO-II and Satallax parsers complain about not being able to
1081 infer the type of "=". *)
1082 let val i_T = domain_type T in
1083 IAbs ((`I "X", i_T),
1084 IAbs ((`I "Y", i_T),
1085 IApp (IApp (IConst (`I tptp_equal, T, []),
1086 IConst (`I "X", i_T, [])),
1087 IConst (`I "Y", i_T, []))))
1089 | _ => IConst (name, T, [])
1091 IConst (proxy_base |>> prefix const_prefix, T, T_args)
1092 | NONE => if s = tptp_choice then tweak_ho_quant tptp_choice T args
1093 else IConst (name, T, T_args))
1094 | intro _ _ (IAbs (bound, tm)) = IAbs (bound, intro false [] tm)
1096 in intro true [] end
1098 fun mangle_type_args_in_const type_enc (name as (s, _)) T_args =
1099 case unprefix_and_unascii const_prefix s of
1100 NONE => (name, T_args)
1102 case type_arg_policy [] type_enc (invert_const s'') of
1103 Mangled_Type_Args => (mangled_const_name type_enc T_args name, [])
1104 | _ => (name, T_args)
1105 fun mangle_type_args_in_iterm type_enc =
1106 if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
1108 fun mangle (IApp (tm1, tm2)) = IApp (mangle tm1, mangle tm2)
1109 | mangle (tm as IConst (_, _, [])) = tm
1110 | mangle (IConst (name, T, T_args)) =
1111 mangle_type_args_in_const type_enc name T_args
1112 |> (fn (name, T_args) => IConst (name, T, T_args))
1113 | mangle (IAbs (bound, tm)) = IAbs (bound, mangle tm)
1119 fun chop_fun 0 T = ([], T)
1120 | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
1121 chop_fun (n - 1) ran_T |>> cons dom_T
1122 | chop_fun _ T = ([], T)
1124 fun filter_const_type_args _ _ _ [] = []
1125 | filter_const_type_args thy s ary T_args =
1127 val U = robust_const_type thy s
1128 val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun ary |> fst) []
1129 val U_args = (s, U) |> robust_const_typargs thy
1132 |> map (fn (U, T) =>
1133 if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
1135 handle TYPE _ => T_args
1137 fun filter_type_args_in_const _ _ _ _ _ [] = []
1138 | filter_type_args_in_const thy monom_constrs type_enc ary s T_args =
1139 case unprefix_and_unascii const_prefix s of
1141 if level_of_type_enc type_enc = No_Types orelse s = tptp_choice then []
1145 val s'' = invert_const s''
1146 fun filter_T_args false = T_args
1147 | filter_T_args true = filter_const_type_args thy s'' ary T_args
1149 case type_arg_policy monom_constrs type_enc s'' of
1150 Explicit_Type_Args infer_from_term_args =>
1151 filter_T_args infer_from_term_args
1152 | No_Type_Args => []
1153 | Mangled_Type_Args => raise Fail "unexpected (un)mangled symbol"
1155 fun filter_type_args_in_iterm thy monom_constrs type_enc =
1157 fun filt ary (IApp (tm1, tm2)) = IApp (filt (ary + 1) tm1, filt 0 tm2)
1158 | filt ary (IConst (name as (s, _), T, T_args)) =
1159 filter_type_args_in_const thy monom_constrs type_enc ary s T_args
1160 |> (fn T_args => IConst (name, T, T_args))
1161 | filt _ (IAbs (bound, tm)) = IAbs (bound, filt 0 tm)
1165 fun iformula_from_prop ctxt type_enc iff_for_eq =
1167 val thy = Proof_Context.theory_of ctxt
1168 fun do_term bs t atomic_Ts =
1169 iterm_from_term thy type_enc bs (Envir.eta_contract t)
1170 |>> (introduce_proxies_in_iterm type_enc
1171 #> mangle_type_args_in_iterm type_enc #> AAtom)
1172 ||> union (op =) atomic_Ts
1173 fun do_quant bs q pos s T t' =
1175 val s = singleton (Name.variant_list (map fst bs)) s
1176 val universal = Option.map (q = AExists ? not) pos
1178 s |> `(case universal of
1179 SOME true => make_all_bound_var
1180 | SOME false => make_exist_bound_var
1181 | NONE => make_bound_var)
1183 do_formula ((s, (name, T)) :: bs) pos t'
1184 #>> mk_aquant q [(name, SOME T)]
1185 ##> union (op =) (atomic_types_of T)
1187 and do_conn bs c pos1 t1 pos2 t2 =
1188 do_formula bs pos1 t1 ##>> do_formula bs pos2 t2 #>> uncurry (mk_aconn c)
1189 and do_formula bs pos t =
1191 @{const Trueprop} $ t1 => do_formula bs pos t1
1192 | @{const Not} $ t1 => do_formula bs (Option.map not pos) t1 #>> mk_anot
1193 | Const (@{const_name All}, _) $ Abs (s, T, t') =>
1194 do_quant bs AForall pos s T t'
1195 | (t0 as Const (@{const_name All}, _)) $ t1 =>
1196 do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
1197 | Const (@{const_name Ex}, _) $ Abs (s, T, t') =>
1198 do_quant bs AExists pos s T t'
1199 | (t0 as Const (@{const_name Ex}, _)) $ t1 =>
1200 do_formula bs pos (t0 $ eta_expand (map (snd o snd) bs) t1 1)
1201 | @{const HOL.conj} $ t1 $ t2 => do_conn bs AAnd pos t1 pos t2
1202 | @{const HOL.disj} $ t1 $ t2 => do_conn bs AOr pos t1 pos t2
1203 | @{const HOL.implies} $ t1 $ t2 =>
1204 do_conn bs AImplies (Option.map not pos) t1 pos t2
1205 | Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])) $ t1 $ t2 =>
1206 if iff_for_eq then do_conn bs AIff NONE t1 NONE t2 else do_term bs t
1208 in do_formula [] end
1210 fun presimplify_term thy t =
1211 if exists_Const (member (op =) Meson.presimplified_consts o fst) t then
1212 t |> Skip_Proof.make_thm thy
1213 |> Meson.presimplify
1218 fun preprocess_abstractions_in_terms trans_lams facts =
1220 val (facts, lambda_ts) =
1221 facts |> map (snd o snd) |> trans_lams
1222 |>> map2 (fn (name, (kind, _)) => fn t => (name, (kind, t))) facts
1224 map2 (fn t => fn j =>
1225 ((lam_fact_prefix ^ Int.toString j, (Global, Def)), (Axiom, t)))
1226 lambda_ts (1 upto length lambda_ts)
1227 in (facts, lam_facts) end
1229 (* Metis's use of "resolve_tac" freezes the schematic variables. We simulate the
1230 same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
1233 fun freeze (t $ u) = freeze t $ freeze u
1234 | freeze (Abs (s, T, t)) = Abs (s, T, freeze t)
1235 | freeze (Var ((s, i), T)) =
1236 Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
1238 in t |> exists_subterm is_Var t ? freeze end
1240 fun presimp_prop ctxt type_enc t =
1242 val thy = Proof_Context.theory_of ctxt
1243 val t = t |> Envir.beta_eta_contract
1244 |> transform_elim_prop
1245 |> Object_Logic.atomize_term thy
1246 val need_trueprop = (fastype_of t = @{typ bool})
1248 t |> need_trueprop ? HOLogic.mk_Trueprop
1249 |> not (is_type_enc_higher_order type_enc) ? extensionalize_term ctxt
1250 |> presimplify_term thy
1251 |> HOLogic.dest_Trueprop
1253 handle TERM _ => @{const True}
1255 (* Satallax prefers "=" to "<=>" (for definitions) and Metis (CNF) requires "="
1256 for obscure technical reasons. *)
1257 fun should_use_iff_for_eq CNF _ = false
1258 | should_use_iff_for_eq (THF _) format = not (is_type_enc_higher_order format)
1259 | should_use_iff_for_eq _ _ = true
1261 fun make_formula ctxt format type_enc iff_for_eq name stature kind t =
1263 val iff_for_eq = iff_for_eq andalso should_use_iff_for_eq format type_enc
1264 val (iformula, atomic_Ts) =
1265 iformula_from_prop ctxt type_enc iff_for_eq (SOME (kind <> Conjecture)) t
1267 |>> close_iformula_universally
1269 {name = name, stature = stature, kind = kind, iformula = iformula,
1270 atomic_types = atomic_Ts}
1273 fun make_fact ctxt format type_enc iff_for_eq ((name, stature), t) =
1274 case t |> make_formula ctxt format type_enc iff_for_eq name stature Axiom of
1275 formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
1276 if s = tptp_true then NONE else SOME formula
1277 | formula => SOME formula
1279 fun s_not_prop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
1280 | s_not_prop _ = @{prop True} (* "t" is too meta for "metis" *)
1282 | s_not_prop (@{const "==>"} $ t $ @{prop False}) = t
1283 | s_not_prop t = @{const "==>"} $ t $ @{prop False}
1286 fun make_conjecture ctxt format type_enc =
1287 map (fn ((name, stature), (kind, t)) =>
1288 t |> kind = Conjecture ? s_not
1289 |> make_formula ctxt format type_enc true name stature kind)
1291 (** Finite and infinite type inference **)
1293 fun tvar_footprint thy s ary =
1294 (case unprefix_and_unascii const_prefix s of
1296 s |> invert_const |> robust_const_type thy |> chop_fun ary |> fst
1297 |> map (fn T => Term.add_tvarsT T [] |> map fst)
1301 fun ghost_type_args thy s ary =
1302 if is_tptp_equal s then
1306 val footprint = tvar_footprint thy s ary
1307 val eq = (s = @{const_name HOL.eq})
1308 fun ghosts _ [] = []
1309 | ghosts seen ((i, tvars) :: args) =
1310 ghosts (union (op =) seen tvars) args
1311 |> (eq orelse exists (fn tvar => not (member (op =) seen tvar)) tvars)
1314 if forall null footprint then
1317 0 upto length footprint - 1 ~~ footprint
1318 |> sort (rev_order o list_ord Term_Ord.indexname_ord o pairself snd)
1322 type monotonicity_info =
1323 {maybe_finite_Ts : typ list,
1324 surely_finite_Ts : typ list,
1325 maybe_infinite_Ts : typ list,
1326 surely_infinite_Ts : typ list,
1327 maybe_nonmono_Ts : typ list}
1329 (* These types witness that the type classes they belong to allow infinite
1330 models and hence that any types with these type classes is monotonic. *)
1331 val known_infinite_types =
1332 [@{typ nat}, HOLogic.intT, HOLogic.realT, @{typ "nat => bool"}]
1334 fun is_type_kind_of_surely_infinite ctxt strictness cached_Ts T =
1335 strictness <> Strict andalso is_type_surely_infinite ctxt true cached_Ts T
1337 (* Finite types such as "unit", "bool", "bool * bool", and "bool => bool" are
1338 dangerous because their "exhaust" properties can easily lead to unsound ATP
1339 proofs. On the other hand, all HOL infinite types can be given the same
1340 models in first-order logic (via Löwenheim-Skolem). *)
1342 fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
1343 | should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
1344 maybe_nonmono_Ts, ...}
1345 (Noninf_Nonmono_Types (strictness, grain)) T =
1346 let val thy = Proof_Context.theory_of ctxt in
1347 grain = Ghost_Type_Arg_Vars orelse
1348 (exists (type_intersect thy T) maybe_nonmono_Ts andalso
1349 not (exists (type_instance thy T) surely_infinite_Ts orelse
1350 (not (member (type_equiv thy) maybe_finite_Ts T) andalso
1351 is_type_kind_of_surely_infinite ctxt strictness surely_infinite_Ts
1354 | should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
1355 maybe_nonmono_Ts, ...}
1356 (Fin_Nonmono_Types grain) T =
1357 let val thy = Proof_Context.theory_of ctxt in
1358 grain = Ghost_Type_Arg_Vars orelse
1359 (exists (type_intersect thy T) maybe_nonmono_Ts andalso
1360 (exists (type_generalization thy T) surely_finite_Ts orelse
1361 (not (member (type_equiv thy) maybe_infinite_Ts T) andalso
1362 is_type_surely_finite ctxt T)))
1364 | should_encode_type _ _ _ _ = false
1366 fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =
1367 should_guard_var () andalso should_encode_type ctxt mono level T
1368 | should_guard_type _ _ _ _ _ = false
1370 fun is_maybe_universal_var (IConst ((s, _), _, _)) =
1371 String.isPrefix bound_var_prefix s orelse
1372 String.isPrefix all_bound_var_prefix s
1373 | is_maybe_universal_var (IVar _) = true
1374 | is_maybe_universal_var _ = false
1377 Top_Level of bool option |
1378 Eq_Arg of bool option |
1381 fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
1382 | should_tag_with_type ctxt mono (Tags (_, level)) site u T =
1383 if granularity_of_type_level level = All_Vars then
1384 should_encode_type ctxt mono level T
1386 (case (site, is_maybe_universal_var u) of
1387 (Eq_Arg _, true) => should_encode_type ctxt mono level T
1389 | should_tag_with_type _ _ _ _ _ _ = false
1391 fun fused_type ctxt mono level =
1393 val should_encode = should_encode_type ctxt mono level
1394 fun fuse 0 T = if should_encode T then T else fused_infinite_type
1395 | fuse ary (Type (@{type_name fun}, [T1, T2])) =
1396 fuse 0 T1 --> fuse (ary - 1) T2
1397 | fuse _ _ = raise Fail "expected function type"
1400 (** predicators and application operators **)
1403 {pred_sym : bool, min_ary : int, max_ary : int, types : typ list,
1406 fun default_sym_tab_entries type_enc =
1407 (make_fixed_const NONE @{const_name undefined},
1408 {pred_sym = false, min_ary = 0, max_ary = 0, types = [],
1409 in_conj = false}) ::
1410 ([tptp_false, tptp_true]
1411 |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = [],
1412 in_conj = false})) @
1413 ([tptp_equal, tptp_old_equal]
1414 |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = [],
1416 |> not (is_type_enc_higher_order type_enc)
1417 ? cons (prefixed_predicator_name,
1418 {pred_sym = true, min_ary = 1, max_ary = 1, types = [],
1421 datatype app_op_level =
1424 Sufficient_App_Op_And_Predicator |
1425 Full_App_Op_And_Predicator
1427 fun sym_table_for_facts ctxt type_enc app_op_level conjs facts =
1429 val thy = Proof_Context.theory_of ctxt
1430 fun consider_var_ary const_T var_T max_ary =
1433 if ary = max_ary orelse type_instance thy var_T T orelse
1434 type_instance thy T var_T then
1437 iter (ary + 1) (range_type T)
1438 in iter 0 const_T end
1439 fun add_universal_var T (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1440 if (app_op_level = Sufficient_App_Op andalso can dest_funT T) orelse
1441 (app_op_level = Sufficient_App_Op_And_Predicator andalso
1442 (can dest_funT T orelse T = @{typ bool})) then
1446 (app_op_level = Sufficient_App_Op_And_Predicator andalso
1447 body_type T = @{typ bool})
1448 fun repair_min_ary {pred_sym, min_ary, max_ary, types, in_conj} =
1449 {pred_sym = pred_sym andalso not bool_vars',
1450 min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,
1451 max_ary = max_ary, types = types, in_conj = in_conj}
1453 fun_var_Ts |> can dest_funT T ? insert_type thy I T
1455 if bool_vars' = bool_vars andalso
1456 pointer_eq (fun_var_Ts', fun_var_Ts) then
1459 ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_ary) sym_tab)
1463 fun add_iterm_syms conj_fact top_level tm
1464 (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
1465 let val (head, args) = strip_iterm_comb tm in
1467 IConst ((s, _), T, _) =>
1468 if String.isPrefix bound_var_prefix s orelse
1469 String.isPrefix all_bound_var_prefix s then
1470 add_universal_var T accum
1471 else if String.isPrefix exist_bound_var_prefix s then
1474 let val ary = length args in
1475 ((bool_vars, fun_var_Ts),
1476 case Symtab.lookup sym_tab s of
1477 SOME {pred_sym, min_ary, max_ary, types, in_conj} =>
1480 pred_sym andalso top_level andalso not bool_vars
1481 val types' = types |> insert_type thy I T
1482 val in_conj = in_conj orelse conj_fact
1484 if (app_op_level = Sufficient_App_Op orelse
1485 app_op_level = Sufficient_App_Op_And_Predicator)
1486 andalso not (pointer_eq (types', types)) then
1487 fold (consider_var_ary T) fun_var_Ts min_ary
1491 Symtab.update (s, {pred_sym = pred_sym,
1492 min_ary = Int.min (ary, min_ary),
1493 max_ary = Int.max (ary, max_ary),
1494 types = types', in_conj = in_conj})
1499 val pred_sym = top_level andalso not bool_vars
1501 case unprefix_and_unascii const_prefix s of
1503 (if String.isSubstring uncurried_alias_sep s then
1505 else case try (robust_const_ary thy
1507 o unmangled_const_name) s of
1508 SOME ary0 => Int.min (ary0, ary)
1512 case app_op_level of
1514 | Full_App_Op_And_Predicator => 0
1515 | _ => fold (consider_var_ary T) fun_var_Ts ary
1517 Symtab.update_new (s,
1518 {pred_sym = pred_sym, min_ary = min_ary,
1519 max_ary = ary, types = [T], in_conj = conj_fact})
1523 | IVar (_, T) => add_universal_var T accum
1524 | IAbs ((_, T), tm) =>
1525 accum |> add_universal_var T |> add_iterm_syms conj_fact false tm
1527 |> fold (add_iterm_syms conj_fact false) args
1529 fun add_fact_syms conj_fact =
1530 K (add_iterm_syms conj_fact true) |> formula_fold NONE |> fact_lift
1532 ((false, []), Symtab.empty)
1533 |> fold (add_fact_syms true) conjs
1534 |> fold (add_fact_syms false) facts
1536 |> fold Symtab.update (default_sym_tab_entries type_enc)
1539 fun min_ary_of sym_tab s =
1540 case Symtab.lookup sym_tab s of
1541 SOME ({min_ary, ...} : sym_info) => min_ary
1543 case unprefix_and_unascii const_prefix s of
1545 let val s = s |> unmangled_const_name |> hd |> invert_const in
1546 if s = predicator_name then 1
1547 else if s = app_op_name then 2
1548 else if s = type_guard_name then 1
1553 (* True if the constant ever appears outside of the top-level position in
1554 literals, or if it appears with different arities (e.g., because of different
1555 type instantiations). If false, the constant always receives all of its
1556 arguments and is used as a predicate. *)
1557 fun is_pred_sym sym_tab s =
1558 case Symtab.lookup sym_tab s of
1559 SOME ({pred_sym, min_ary, max_ary, ...} : sym_info) =>
1560 pred_sym andalso min_ary = max_ary
1563 val app_op = `(make_fixed_const NONE) app_op_name
1564 val predicator_combconst =
1565 IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
1567 fun list_app head args = fold (curry (IApp o swap)) args head
1568 fun predicator tm = IApp (predicator_combconst, tm)
1570 fun mk_app_op type_enc head arg =
1572 val head_T = ityp_of head
1573 val (arg_T, res_T) = dest_funT head_T
1575 IConst (app_op, head_T --> head_T, [arg_T, res_T])
1576 |> mangle_type_args_in_iterm type_enc
1577 in list_app app [head, arg] end
1579 fun firstorderize_fact thy monom_constrs type_enc sym_tab uncurried_aliases =
1581 fun do_app arg head = mk_app_op type_enc head arg
1582 fun list_app_ops head args = fold do_app args head
1583 fun introduce_app_ops tm =
1584 let val (head, args) = tm |> strip_iterm_comb ||> map introduce_app_ops in
1586 IConst (name as (s, _), T, T_args) =>
1587 if uncurried_aliases andalso String.isPrefix const_prefix s then
1589 val ary = length args
1591 name |> ary > min_ary_of sym_tab s ? aliased_uncurried ary
1592 in list_app (IConst (name, T, T_args)) args end
1594 args |> chop (min_ary_of sym_tab s)
1595 |>> list_app head |-> list_app_ops
1596 | _ => list_app_ops head args
1598 fun introduce_predicators tm =
1599 case strip_iterm_comb tm of
1600 (IConst ((s, _), _, _), _) =>
1601 if is_pred_sym sym_tab s then tm else predicator tm
1602 | _ => predicator tm
1604 not (is_type_enc_higher_order type_enc)
1605 ? (introduce_app_ops #> introduce_predicators)
1606 #> filter_type_args_in_iterm thy monom_constrs type_enc
1607 in update_iformula (formula_map do_iterm) end
1609 (** Helper facts **)
1611 val not_ffalse = @{lemma "~ fFalse" by (unfold fFalse_def) fast}
1612 val ftrue = @{lemma "fTrue" by (unfold fTrue_def) fast}
1614 (* The Boolean indicates that a fairly sound type encoding is needed. *)
1616 [(("COMBI", false), @{thms Meson.COMBI_def}),
1617 (("COMBK", false), @{thms Meson.COMBK_def}),
1618 (("COMBB", false), @{thms Meson.COMBB_def}),
1619 (("COMBC", false), @{thms Meson.COMBC_def}),
1620 (("COMBS", false), @{thms Meson.COMBS_def}),
1621 ((predicator_name, false), [not_ffalse, ftrue]),
1622 (("fFalse", false), [not_ffalse]),
1623 (("fFalse", true), @{thms True_or_False}),
1624 (("fTrue", false), [ftrue]),
1625 (("fTrue", true), @{thms True_or_False}),
1627 @{thms fNot_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1628 fNot_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1630 @{lemma "~ P | ~ Q | fconj P Q" "~ fconj P Q | P" "~ fconj P Q | Q"
1631 by (unfold fconj_def) fast+}),
1633 @{lemma "~ P | fdisj P Q" "~ Q | fdisj P Q" "~ fdisj P Q | P | Q"
1634 by (unfold fdisj_def) fast+}),
1635 (("fimplies", false),
1636 @{lemma "P | fimplies P Q" "~ Q | fimplies P Q" "~ fimplies P Q | ~ P | Q"
1637 by (unfold fimplies_def) fast+}),
1639 (* This is a lie: Higher-order equality doesn't need a sound type encoding.
1640 However, this is done so for backward compatibility: Including the
1641 equality helpers by default in Metis breaks a few existing proofs. *)
1642 @{thms fequal_def [THEN Meson.iff_to_disjD, THEN conjunct1]
1643 fequal_def [THEN Meson.iff_to_disjD, THEN conjunct2]}),
1644 (* Partial characterization of "fAll" and "fEx". A complete characterization
1645 would require the axiom of choice for replay with Metis. *)
1646 (("fAll", false), [@{lemma "~ fAll P | P x" by (auto simp: fAll_def)}]),
1647 (("fEx", false), [@{lemma "~ P x | fEx P" by (auto simp: fEx_def)}]),
1648 (("If", true), @{thms if_True if_False True_or_False})]
1649 |> map (apsnd (map zero_var_indexes))
1651 fun atype_of_type_vars (Native (_, Polymorphic, _)) = SOME atype_of_types
1652 | atype_of_type_vars _ = NONE
1654 fun bound_tvars type_enc sorts Ts =
1655 (sorts ? mk_ahorn (formulas_for_types type_enc add_sorts_on_tvar Ts))
1656 #> mk_aquant AForall
1657 (map_filter (fn TVar (x as (s, _), _) =>
1658 SOME ((make_schematic_type_var x, s),
1659 atype_of_type_vars type_enc)
1662 fun eq_formula type_enc atomic_Ts bounds pred_sym tm1 tm2 =
1663 (if pred_sym then AConn (AIff, [AAtom tm1, AAtom tm2])
1664 else AAtom (ATerm (`I tptp_equal, [tm1, tm2])))
1665 |> mk_aquant AForall bounds
1666 |> close_formula_universally
1667 |> bound_tvars type_enc true atomic_Ts
1669 val helper_rank = default_rank
1670 val min_rank = 9 * helper_rank div 10
1671 val max_rank = 4 * min_rank
1673 fun rank_of_fact_num n j = min_rank + (max_rank - min_rank) * j div n
1675 val type_tag = `(make_fixed_const NONE) type_tag_name
1677 fun should_specialize_helper type_enc t =
1678 polymorphism_of_type_enc type_enc <> Polymorphic andalso
1679 level_of_type_enc type_enc <> No_Types andalso
1680 not (null (Term.hidden_polymorphism t))
1682 fun add_helper_facts_for_sym ctxt format type_enc (s, {types, ...} : sym_info) =
1683 case unprefix_and_unascii const_prefix s of
1686 val thy = Proof_Context.theory_of ctxt
1687 val unmangled_s = mangled_s |> unmangled_const_name |> hd
1688 fun dub needs_fairly_sound j k =
1689 unmangled_s ^ "_" ^ string_of_int j ^ "_" ^ string_of_int k ^
1690 (if mangled_s = unmangled_s then "" else "_" ^ ascii_of mangled_s) ^
1691 (if needs_fairly_sound then typed_helper_suffix
1692 else untyped_helper_suffix)
1693 fun dub_and_inst needs_fairly_sound (th, j) =
1694 let val t = prop_of th in
1695 if should_specialize_helper type_enc t then
1696 map (fn T => specialize_type thy (invert_const unmangled_s, T) t)
1702 |> map (fn (k, t) => ((dub needs_fairly_sound j k, (Global, Def)), t))
1703 val make_facts = map_filter (make_fact ctxt format type_enc false)
1704 val fairly_sound = is_type_enc_fairly_sound type_enc
1706 fold (fn ((helper_s, needs_fairly_sound), ths) =>
1707 if helper_s <> unmangled_s orelse
1708 (needs_fairly_sound andalso not fairly_sound) then
1711 ths ~~ (1 upto length ths)
1712 |> maps (dub_and_inst needs_fairly_sound)
1714 |> union (op = o pairself #iformula))
1718 fun helper_facts_for_sym_table ctxt format type_enc sym_tab =
1719 Symtab.fold_rev (add_helper_facts_for_sym ctxt format type_enc) sym_tab []
1721 (***************************************************************)
1722 (* Type Classes Present in the Axiom or Conjecture Clauses *)
1723 (***************************************************************)
1725 fun set_insert (x, s) = Symtab.update (x, ()) s
1727 fun add_classes (sorts, cset) = List.foldl set_insert cset (flat sorts)
1729 (* Remove this trivial type class (FIXME: similar code elsewhere) *)
1730 fun delete_type cset = Symtab.delete_safe (the_single @{sort HOL.type}) cset
1732 fun classes_of_terms get_Ts =
1733 map (map snd o get_Ts)
1734 #> List.foldl add_classes Symtab.empty
1735 #> delete_type #> Symtab.keys
1737 val tfree_classes_of_terms = classes_of_terms Misc_Legacy.term_tfrees
1738 val tvar_classes_of_terms = classes_of_terms Misc_Legacy.term_tvars
1740 fun fold_type_constrs f (Type (s, Ts)) x =
1741 fold (fold_type_constrs f) Ts (f (s, x))
1742 | fold_type_constrs _ _ x = x
1744 (* Type constructors used to instantiate overloaded constants are the only ones
1746 fun add_type_constrs_in_term thy =
1748 fun add (Const (@{const_name Meson.skolem}, _) $ _) = I
1749 | add (t $ u) = add t #> add u
1751 x |> robust_const_typargs thy |> fold (fold_type_constrs set_insert)
1752 | add (Abs (_, _, u)) = add u
1756 fun type_constrs_of_terms thy ts =
1757 Symtab.keys (fold (add_type_constrs_in_term thy) ts Symtab.empty)
1759 fun extract_lambda_def (Const (@{const_name HOL.eq}, _) $ t $ u) =
1760 let val (head, args) = strip_comb t in
1761 (head |> dest_Const |> fst,
1762 fold_rev (fn t as Var ((s, _), T) =>
1763 (fn u => Abs (s, T, abstract_over (t, u)))
1764 | _ => raise Fail "expected Var") args u)
1766 | extract_lambda_def _ = raise Fail "malformed lifted lambda"
1768 fun trans_lams_from_string ctxt type_enc lam_trans =
1769 if lam_trans = no_lamsN then
1771 else if lam_trans = hide_lamsN then
1772 lift_lams ctxt type_enc ##> K []
1773 else if lam_trans = liftingN orelse lam_trans = lam_liftingN then
1774 lift_lams ctxt type_enc
1775 else if lam_trans = combsN then
1776 map (introduce_combinators ctxt) #> rpair []
1777 else if lam_trans = combs_and_liftingN then
1778 lift_lams_part_1 ctxt type_enc
1779 ##> maps (fn t => [t, introduce_combinators ctxt (intentionalize_def t)])
1780 #> lift_lams_part_2 ctxt
1781 else if lam_trans = combs_or_liftingN then
1782 lift_lams_part_1 ctxt type_enc
1783 ##> map (fn t => case head_of (strip_qnt_body @{const_name All} t) of
1784 @{term "op =::bool => bool => bool"} => t
1785 | _ => introduce_combinators ctxt (intentionalize_def t))
1786 #> lift_lams_part_2 ctxt
1787 else if lam_trans = keep_lamsN then
1788 map (Envir.eta_contract) #> rpair []
1790 error ("Unknown lambda translation scheme: " ^ quote lam_trans ^ ".")
1792 fun translate_formulas ctxt prem_kind format type_enc lam_trans presimp hyp_ts
1795 val thy = Proof_Context.theory_of ctxt
1796 val trans_lams = trans_lams_from_string ctxt type_enc lam_trans
1797 val fact_ts = facts |> map snd
1798 (* Remove existing facts from the conjecture, as this can dramatically
1799 boost an ATP's performance (for some reason). *)
1802 |> map (fn t => if member (op aconv) fact_ts t then @{prop True} else t)
1803 val facts = facts |> map (apsnd (pair Axiom))
1805 map (pair prem_kind) hyp_ts @ [(Conjecture, s_not_prop concl_t)]
1806 |> map (apsnd freeze_term)
1807 |> map2 (pair o rpair (Local, General) o string_of_int)
1808 (0 upto length hyp_ts)
1809 val ((conjs, facts), lam_facts) =
1811 |> presimp ? pairself (map (apsnd (apsnd (presimp_prop ctxt type_enc))))
1812 |> (if lam_trans = no_lamsN then
1816 #> preprocess_abstractions_in_terms trans_lams
1817 #>> chop (length conjs))
1818 val conjs = conjs |> make_conjecture ctxt format type_enc
1819 val (fact_names, facts) =
1821 |> map_filter (fn (name, (_, t)) =>
1822 make_fact ctxt format type_enc true (name, t)
1823 |> Option.map (pair name))
1825 val lifted = lam_facts |> map (extract_lambda_def o snd o snd)
1827 lam_facts |> map_filter (make_fact ctxt format type_enc true o apsnd snd)
1828 val all_ts = concl_t :: hyp_ts @ fact_ts
1829 val subs = tfree_classes_of_terms all_ts
1830 val supers = tvar_classes_of_terms all_ts
1831 val tycons = type_constrs_of_terms thy all_ts
1832 val (supers, arity_clauses) =
1833 if level_of_type_enc type_enc = No_Types then ([], [])
1834 else make_arity_clauses thy tycons supers
1835 val class_rel_clauses = make_class_rel_clauses thy subs supers
1837 (fact_names |> map single, union (op =) subs supers, conjs,
1838 facts @ lam_facts, class_rel_clauses, arity_clauses, lifted)
1841 val type_guard = `(make_fixed_const NONE) type_guard_name
1843 fun type_guard_iterm type_enc T tm =
1844 IApp (IConst (type_guard, T --> @{typ bool}, [T])
1845 |> mangle_type_args_in_iterm type_enc, tm)
1847 fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
1848 | is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
1849 accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
1850 | is_var_positively_naked_in_term _ _ _ _ = true
1852 fun is_var_ghost_type_arg_in_term thy polym_constrs name pos tm accum =
1853 is_var_positively_naked_in_term name pos tm accum orelse
1855 val var = ATerm (name, [])
1856 fun is_nasty_in_term (ATerm (_, [])) = false
1857 | is_nasty_in_term (ATerm ((s, _), tms)) =
1859 val ary = length tms
1860 val polym_constr = member (op =) polym_constrs s
1861 val ghosts = ghost_type_args thy s ary
1863 exists (fn (j, tm) =>
1864 if polym_constr then
1865 member (op =) ghosts j andalso
1866 (tm = var orelse is_nasty_in_term tm)
1868 tm = var andalso member (op =) ghosts j)
1869 (0 upto ary - 1 ~~ tms)
1870 orelse (not polym_constr andalso exists is_nasty_in_term tms)
1872 | is_nasty_in_term _ = true
1873 in is_nasty_in_term tm end
1875 fun should_guard_var_in_formula thy polym_constrs level pos phi (SOME true)
1877 (case granularity_of_type_level level of
1879 | Positively_Naked_Vars =>
1880 formula_fold pos (is_var_positively_naked_in_term name) phi false
1881 | Ghost_Type_Arg_Vars =>
1882 formula_fold pos (is_var_ghost_type_arg_in_term thy polym_constrs name) phi
1884 | should_guard_var_in_formula _ _ _ _ _ _ _ = true
1886 fun always_guard_var_in_formula _ _ _ _ _ _ _ = true
1888 fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
1889 | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =
1890 granularity_of_type_level level <> All_Vars andalso
1891 should_encode_type ctxt mono level T
1892 | should_generate_tag_bound_decl _ _ _ _ _ = false
1894 fun mk_aterm type_enc name T_args args =
1895 ATerm (name, map_filter (ho_term_for_type_arg type_enc) T_args @ args)
1897 fun do_bound_type ctxt mono type_enc =
1899 Native (_, _, level) =>
1900 fused_type ctxt mono level 0 #> ho_type_from_typ type_enc false 0 #> SOME
1903 fun tag_with_type ctxt mono type_enc pos T tm =
1904 IConst (type_tag, T --> T, [T])
1905 |> mangle_type_args_in_iterm type_enc
1906 |> ho_term_from_iterm ctxt mono type_enc pos
1907 |> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
1908 | _ => raise Fail "unexpected lambda-abstraction")
1909 and ho_term_from_iterm ctxt mono type_enc pos =
1911 fun beta_red bs (IApp (IAbs ((name, _), tm), tm')) =
1912 beta_red ((name, beta_red bs tm') :: bs) tm
1913 | beta_red bs (IApp tmp) = IApp (pairself (beta_red bs) tmp)
1914 | beta_red bs (tm as IConst (name, _, _)) =
1915 (case AList.lookup (op =) bs name of
1918 | beta_red bs (IAbs ((name, T), tm)) =
1919 IAbs ((name, T), beta_red (AList.delete (op =) name bs) tm)
1920 | beta_red _ tm = tm
1923 val (head, args) = strip_iterm_comb u
1926 Top_Level pos => pos
1931 IConst (name as (s, _), _, T_args) =>
1933 val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
1934 in map (term arg_site) args |> mk_aterm type_enc name T_args end
1936 map (term Elsewhere) args |> mk_aterm type_enc name []
1937 | IAbs ((name, T), tm) =>
1938 if is_type_enc_higher_order type_enc then
1939 AAbs ((name, ho_type_from_typ type_enc true 0 T),
1942 raise Fail "unexpected lambda-abstraction"
1943 | IApp _ => raise Fail "impossible \"IApp\""
1946 if should_tag_with_type ctxt mono type_enc site u T then
1947 tag_with_type ctxt mono type_enc pos T t
1951 in term (Top_Level pos) o beta_red [] end
1952 and formula_from_iformula ctxt polym_constrs mono type_enc should_guard_var =
1954 val thy = Proof_Context.theory_of ctxt
1955 val level = level_of_type_enc type_enc
1956 val do_term = ho_term_from_iterm ctxt mono type_enc
1957 fun do_out_of_bound_type pos phi universal (name, T) =
1958 if should_guard_type ctxt mono type_enc
1959 (fn () => should_guard_var thy polym_constrs level pos phi
1960 universal name) T then
1962 |> type_guard_iterm type_enc T
1963 |> do_term pos |> AAtom |> SOME
1964 else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
1966 val var = ATerm (name, [])
1967 val tagged_var = tag_with_type ctxt mono type_enc pos T var
1968 in SOME (AAtom (ATerm (`I tptp_equal, [tagged_var, var]))) end
1971 fun do_formula pos (AQuant (q, xs, phi)) =
1973 val phi = phi |> do_formula pos
1974 val universal = Option.map (q = AExists ? not) pos
1975 val do_bound_type = do_bound_type ctxt mono type_enc
1977 AQuant (q, xs |> map (apsnd (fn NONE => NONE
1978 | SOME T => do_bound_type T)),
1979 (if q = AForall then mk_ahorn else fold_rev (mk_aconn AAnd))
1981 (fn (_, NONE) => NONE
1983 do_out_of_bound_type pos phi universal (s, T))
1987 | do_formula pos (AConn conn) = aconn_map pos do_formula conn
1988 | do_formula pos (AAtom tm) = AAtom (do_term pos tm)
1991 (* Each fact is given a unique fact number to avoid name clashes (e.g., because
1992 of monomorphization). The TPTP explicitly forbids name clashes, and some of
1993 the remote provers might care. *)
1994 fun formula_line_for_fact ctxt polym_constrs prefix encode freshen pos
1995 mono type_enc rank (j, {name, stature, kind, iformula, atomic_types}) =
1996 (prefix ^ (if freshen then string_of_int j ^ "_" else "") ^ encode name, kind,
1998 |> formula_from_iformula ctxt polym_constrs mono type_enc
1999 should_guard_var_in_formula (if pos then SOME true else NONE)
2000 |> close_formula_universally
2001 |> bound_tvars type_enc true atomic_types,
2003 let val rank = rank j in
2005 Intro => isabelle_info introN rank
2006 | Inductive => isabelle_info inductiveN rank
2007 | Elim => isabelle_info elimN rank
2008 | Simp => isabelle_info simpN rank
2009 | Def => isabelle_info defN rank
2010 | _ => isabelle_info "" rank
2014 fun formula_line_for_class_rel_clause type_enc
2015 ({name, subclass, superclass, ...} : class_rel_clause) =
2016 let val ty_arg = ATerm (tvar_a_name, []) in
2017 Formula (class_rel_clause_prefix ^ ascii_of name, Axiom,
2019 [type_class_formula type_enc subclass ty_arg,
2020 type_class_formula type_enc superclass ty_arg])
2021 |> mk_aquant AForall
2022 [(tvar_a_name, atype_of_type_vars type_enc)],
2023 NONE, isabelle_info inductiveN helper_rank)
2026 fun formula_from_arity_atom type_enc (class, t, args) =
2027 ATerm (t, map (fn arg => ATerm (arg, [])) args)
2028 |> type_class_formula type_enc class
2030 fun formula_line_for_arity_clause type_enc
2031 ({name, prem_atoms, concl_atom} : arity_clause) =
2032 Formula (arity_clause_prefix ^ name, Axiom,
2033 mk_ahorn (map (formula_from_arity_atom type_enc) prem_atoms)
2034 (formula_from_arity_atom type_enc concl_atom)
2035 |> mk_aquant AForall
2036 (map (rpair (atype_of_type_vars type_enc)) (#3 concl_atom)),
2037 NONE, isabelle_info inductiveN helper_rank)
2039 fun formula_line_for_conjecture ctxt polym_constrs mono type_enc
2040 ({name, kind, iformula, atomic_types, ...} : translated_formula) =
2041 Formula (conjecture_prefix ^ name, kind,
2043 |> formula_from_iformula ctxt polym_constrs mono type_enc
2044 should_guard_var_in_formula (SOME false)
2045 |> close_formula_universally
2046 |> bound_tvars type_enc true atomic_types, NONE, [])
2048 fun type_enc_needs_free_types (Native (_, Polymorphic, _)) = true
2049 | type_enc_needs_free_types (Native _) = false
2050 | type_enc_needs_free_types _ = true
2052 fun formula_line_for_free_type j phi =
2053 Formula (tfree_clause_prefix ^ string_of_int j, Hypothesis, phi, NONE, [])
2054 fun formula_lines_for_free_types type_enc (facts : translated_formula list) =
2055 if type_enc_needs_free_types type_enc then
2058 fold (union (op =)) (map #atomic_types facts) []
2059 |> formulas_for_types type_enc add_sorts_on_tfree
2060 in map2 formula_line_for_free_type (0 upto length phis - 1) phis end
2064 (** Symbol declarations **)
2066 fun decl_line_for_class order s =
2067 let val name as (s, _) = `make_type_class s in
2068 Decl (sym_decl_prefix ^ s, name,
2069 if order = First_Order then
2070 ATyAbs ([tvar_a_name],
2071 if avoid_first_order_ghost_type_vars then
2072 AFun (a_itself_atype, bool_atype)
2076 AFun (atype_of_types, bool_atype))
2079 fun decl_lines_for_classes type_enc classes =
2081 Native (order, Polymorphic, _) => map (decl_line_for_class order) classes
2084 fun sym_decl_table_for_facts thy type_enc sym_tab (conjs, facts, extra_tms) =
2086 fun add_iterm_syms tm =
2087 let val (head, args) = strip_iterm_comb tm in
2089 IConst ((s, s'), T, T_args) =>
2091 val (pred_sym, in_conj) =
2092 case Symtab.lookup sym_tab s of
2093 SOME ({pred_sym, in_conj, ...} : sym_info) =>
2095 | NONE => (false, false)
2098 Guards _ => not pred_sym
2099 | _ => true) andalso
2100 is_tptp_user_symbol s
2103 Symtab.map_default (s, [])
2104 (insert_type thy #3 (s', T_args, T, pred_sym, length args,
2109 | IAbs (_, tm) => add_iterm_syms tm
2111 #> fold add_iterm_syms args
2113 val add_fact_syms = K add_iterm_syms |> formula_fold NONE |> fact_lift
2114 fun add_formula_var_types (AQuant (_, xs, phi)) =
2115 fold (fn (_, SOME T) => insert_type thy I T | _ => I) xs
2116 #> add_formula_var_types phi
2117 | add_formula_var_types (AConn (_, phis)) =
2118 fold add_formula_var_types phis
2119 | add_formula_var_types _ = I
2121 if polymorphism_of_type_enc type_enc = Polymorphic then [tvar_a]
2122 else fold (fact_lift add_formula_var_types) (conjs @ facts) []
2123 fun add_undefined_const T =
2126 `(make_fixed_const NONE) @{const_name undefined}
2127 |> (case type_arg_policy [] type_enc @{const_name undefined} of
2128 Mangled_Type_Args => mangled_const_name type_enc [T]
2131 Symtab.map_default (s, [])
2132 (insert_type thy #3 (s', [T], T, false, 0, false))
2134 fun add_TYPE_const () =
2135 let val (s, s') = TYPE_name in
2136 Symtab.map_default (s, [])
2138 (s', [tvar_a], @{typ "'a itself"}, false, 0, false))
2142 |> is_type_enc_fairly_sound type_enc
2143 ? (fold (fold add_fact_syms) [conjs, facts]
2144 #> fold add_iterm_syms extra_tms
2145 #> (case type_enc of
2146 Native (First_Order, Polymorphic, _) =>
2147 if avoid_first_order_ghost_type_vars then add_TYPE_const ()
2150 | _ => fold add_undefined_const (var_types ())))
2153 (* We add "bool" in case the helper "True_or_False" is included later. *)
2154 fun default_mono level =
2155 {maybe_finite_Ts = [@{typ bool}],
2156 surely_finite_Ts = [@{typ bool}],
2157 maybe_infinite_Ts = known_infinite_types,
2158 surely_infinite_Ts =
2160 Noninf_Nonmono_Types (Strict, _) => []
2161 | _ => known_infinite_types,
2162 maybe_nonmono_Ts = [@{typ bool}]}
2164 (* This inference is described in section 2.3 of Claessen et al.'s "Sorting it
2165 out with monotonicity" paper presented at CADE 2011. *)
2166 fun add_iterm_mononotonicity_info _ _ (SOME false) _ mono = mono
2167 | add_iterm_mononotonicity_info ctxt level _
2168 (IApp (IApp (IConst ((s, _), Type (_, [T, _]), _), tm1), tm2))
2169 (mono as {maybe_finite_Ts, surely_finite_Ts, maybe_infinite_Ts,
2170 surely_infinite_Ts, maybe_nonmono_Ts}) =
2171 let val thy = Proof_Context.theory_of ctxt in
2172 if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
2174 Noninf_Nonmono_Types (strictness, _) =>
2175 if exists (type_instance thy T) surely_infinite_Ts orelse
2176 member (type_equiv thy) maybe_finite_Ts T then
2178 else if is_type_kind_of_surely_infinite ctxt strictness
2179 surely_infinite_Ts T then
2180 {maybe_finite_Ts = maybe_finite_Ts,
2181 surely_finite_Ts = surely_finite_Ts,
2182 maybe_infinite_Ts = maybe_infinite_Ts,
2183 surely_infinite_Ts = surely_infinite_Ts |> insert_type thy I T,
2184 maybe_nonmono_Ts = maybe_nonmono_Ts}
2186 {maybe_finite_Ts = maybe_finite_Ts |> insert (type_equiv thy) T,
2187 surely_finite_Ts = surely_finite_Ts,
2188 maybe_infinite_Ts = maybe_infinite_Ts,
2189 surely_infinite_Ts = surely_infinite_Ts,
2190 maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}
2191 | Fin_Nonmono_Types _ =>
2192 if exists (type_instance thy T) surely_finite_Ts orelse
2193 member (type_equiv thy) maybe_infinite_Ts T then
2195 else if is_type_surely_finite ctxt T then
2196 {maybe_finite_Ts = maybe_finite_Ts,
2197 surely_finite_Ts = surely_finite_Ts |> insert_type thy I T,
2198 maybe_infinite_Ts = maybe_infinite_Ts,
2199 surely_infinite_Ts = surely_infinite_Ts,
2200 maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type thy I T}
2202 {maybe_finite_Ts = maybe_finite_Ts,
2203 surely_finite_Ts = surely_finite_Ts,
2204 maybe_infinite_Ts = maybe_infinite_Ts |> insert (type_equiv thy) T,
2205 surely_infinite_Ts = surely_infinite_Ts,
2206 maybe_nonmono_Ts = maybe_nonmono_Ts}
2211 | add_iterm_mononotonicity_info _ _ _ _ mono = mono
2212 fun add_fact_mononotonicity_info ctxt level
2213 ({kind, iformula, ...} : translated_formula) =
2214 formula_fold (SOME (kind <> Conjecture))
2215 (add_iterm_mononotonicity_info ctxt level) iformula
2216 fun mononotonicity_info_for_facts ctxt type_enc facts =
2217 let val level = level_of_type_enc type_enc in
2219 |> is_type_level_monotonicity_based level
2220 ? fold (add_fact_mononotonicity_info ctxt level) facts
2223 fun add_iformula_monotonic_types ctxt mono type_enc =
2225 val thy = Proof_Context.theory_of ctxt
2226 val level = level_of_type_enc type_enc
2227 val should_encode = should_encode_type ctxt mono level
2228 fun add_type T = not (should_encode T) ? insert_type thy I T
2229 fun add_args (IApp (tm1, tm2)) = add_args tm1 #> add_term tm2
2231 and add_term tm = add_type (ityp_of tm) #> add_args tm
2232 in formula_fold NONE (K add_term) end
2233 fun add_fact_monotonic_types ctxt mono type_enc =
2234 add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
2235 fun monotonic_types_for_facts ctxt mono type_enc facts =
2236 let val level = level_of_type_enc type_enc in
2237 [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
2238 is_type_level_monotonicity_based level andalso
2239 granularity_of_type_level level <> Ghost_Type_Arg_Vars)
2240 ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
2243 fun formula_line_for_guards_mono_type ctxt mono type_enc T =
2244 Formula (guards_sym_formula_prefix ^
2245 ascii_of (mangled_type type_enc T),
2247 IConst (`make_bound_var "X", T, [])
2248 |> type_guard_iterm type_enc T
2250 |> formula_from_iformula ctxt [] mono type_enc
2251 always_guard_var_in_formula (SOME true)
2252 |> close_formula_universally
2253 |> bound_tvars type_enc true (atomic_types_of T),
2254 NONE, isabelle_info inductiveN helper_rank)
2256 fun formula_line_for_tags_mono_type ctxt mono type_enc T =
2257 let val x_var = ATerm (`make_bound_var "X", []) in
2258 Formula (tags_sym_formula_prefix ^
2259 ascii_of (mangled_type type_enc T),
2261 eq_formula type_enc (atomic_types_of T) [] false
2262 (tag_with_type ctxt mono type_enc NONE T x_var) x_var,
2263 NONE, isabelle_info defN helper_rank)
2266 fun problem_lines_for_mono_types ctxt mono type_enc Ts =
2269 | Guards _ => map (formula_line_for_guards_mono_type ctxt mono type_enc) Ts
2270 | Tags _ => map (formula_line_for_tags_mono_type ctxt mono type_enc) Ts
2272 fun decl_line_for_sym ctxt mono type_enc s
2273 (s', T_args, T, pred_sym, ary, _) =
2275 val thy = Proof_Context.theory_of ctxt
2279 else case unprefix_and_unascii const_prefix s of
2282 val s' = s' |> invert_const
2283 val T = s' |> robust_const_type thy
2284 in (T, robust_const_typargs thy (s', T)) end
2285 | NONE => raise Fail "unexpected type arguments"
2287 Decl (sym_decl_prefix ^ s, (s, s'),
2288 T |> fused_type ctxt mono (level_of_type_enc type_enc) ary
2289 |> ho_type_from_typ type_enc pred_sym ary
2290 |> not (null T_args)
2291 ? curry ATyAbs (map (tvar_name o fst o dest_TVar) T_args))
2294 fun honor_conj_sym_kind in_conj conj_sym_kind =
2295 if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
2298 fun formula_line_for_guards_sym_decl ctxt conj_sym_kind mono type_enc n s j
2299 (s', T_args, T, _, ary, in_conj) =
2301 val thy = Proof_Context.theory_of ctxt
2302 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2303 val (arg_Ts, res_T) = chop_fun ary T
2304 val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
2306 bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
2308 if exists (curry (op =) dummyT) T_args then
2309 case level_of_type_enc type_enc of
2310 All_Types => map SOME arg_Ts
2312 if granularity_of_type_level level = Ghost_Type_Arg_Vars then
2313 let val ghosts = ghost_type_args thy s ary in
2314 map2 (fn j => if member (op =) ghosts j then SOME else K NONE)
2315 (0 upto ary - 1) arg_Ts
2322 Formula (guards_sym_formula_prefix ^ s ^
2323 (if n > 1 then "_" ^ string_of_int j else ""), kind,
2324 IConst ((s, s'), T, T_args)
2325 |> fold (curry (IApp o swap)) bounds
2326 |> type_guard_iterm type_enc res_T
2327 |> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
2328 |> formula_from_iformula ctxt [] mono type_enc
2329 always_guard_var_in_formula (SOME true)
2330 |> close_formula_universally
2331 |> bound_tvars type_enc (n > 1) (atomic_types_of T)
2333 NONE, isabelle_info inductiveN helper_rank)
2336 fun formula_lines_for_tags_sym_decl ctxt conj_sym_kind mono type_enc n s
2337 (j, (s', T_args, T, pred_sym, ary, in_conj)) =
2339 val thy = Proof_Context.theory_of ctxt
2340 val level = level_of_type_enc type_enc
2341 val grain = granularity_of_type_level level
2343 tags_sym_formula_prefix ^ s ^
2344 (if n > 1 then "_" ^ string_of_int j else "")
2345 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2346 val (arg_Ts, res_T) = chop_fun ary T
2347 val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
2348 val bounds = bound_names |> map (fn name => ATerm (name, []))
2349 val cst = mk_aterm type_enc (s, s') T_args
2350 val eq = maybe_negate oo eq_formula type_enc (atomic_types_of T) [] pred_sym
2351 val should_encode = should_encode_type ctxt mono level
2352 val tag_with = tag_with_type ctxt mono type_enc NONE
2353 val add_formula_for_res =
2354 if should_encode res_T then
2357 if grain = Ghost_Type_Arg_Vars then
2358 let val ghosts = ghost_type_args thy s ary in
2359 map2 (fn (j, arg_T) => member (op =) ghosts j ? tag_with arg_T)
2360 (0 upto ary - 1 ~~ arg_Ts) bounds
2365 cons (Formula (ident_base ^ "_res", kind,
2366 eq (tag_with res_T (cst bounds)) (cst tagged_bounds),
2367 NONE, isabelle_info defN helper_rank))
2371 in [] |> not pred_sym ? add_formula_for_res end
2373 fun result_type_of_decl (_, _, T, _, ary, _) = chop_fun ary T |> snd
2375 fun rationalize_decls thy (decls as decl :: (decls' as _ :: _)) =
2377 val T = result_type_of_decl decl
2378 |> map_type_tvar (fn (z, _) => TVar (z, HOLogic.typeS))
2380 if forall (type_generalization thy T o result_type_of_decl) decls' then
2385 | rationalize_decls _ decls = decls
2387 fun problem_lines_for_sym_decls ctxt conj_sym_kind mono type_enc (s, decls) =
2389 Native _ => [decl_line_for_sym ctxt mono type_enc s (hd decls)]
2390 | Guards (_, level) =>
2392 val thy = Proof_Context.theory_of ctxt
2393 val decls = decls |> rationalize_decls thy
2394 val n = length decls
2396 decls |> filter (should_encode_type ctxt mono level
2397 o result_type_of_decl)
2399 (0 upto length decls - 1, decls)
2400 |-> map2 (formula_line_for_guards_sym_decl ctxt conj_sym_kind mono
2403 | Tags (_, level) =>
2404 if granularity_of_type_level level = All_Vars then
2407 let val n = length decls in
2408 (0 upto n - 1 ~~ decls)
2409 |> maps (formula_lines_for_tags_sym_decl ctxt conj_sym_kind mono
2413 fun problem_lines_for_sym_decl_table ctxt conj_sym_kind mono type_enc mono_Ts
2416 val syms = sym_decl_tab |> Symtab.dest |> sort_wrt fst
2417 val mono_lines = problem_lines_for_mono_types ctxt mono type_enc mono_Ts
2419 fold_rev (append o problem_lines_for_sym_decls ctxt conj_sym_kind mono
2422 in mono_lines @ decl_lines end
2424 fun pair_append (xs1, xs2) (ys1, ys2) = (xs1 @ ys1, xs2 @ ys2)
2426 fun do_uncurried_alias_lines_for_sym ctxt monom_constrs conj_sym_kind mono
2427 type_enc sym_tab0 sym_tab base_s0 types in_conj =
2431 val thy = Proof_Context.theory_of ctxt
2432 val (kind, maybe_negate) = honor_conj_sym_kind in_conj conj_sym_kind
2433 val base_name = base_s0 |> `(make_fixed_const (SOME type_enc))
2434 val T = case types of [T] => T | _ => robust_const_type thy base_s0
2435 val T_args = robust_const_typargs thy (base_s0, T)
2436 val (base_name as (base_s, _), T_args) =
2437 mangle_type_args_in_const type_enc base_name T_args
2438 val base_ary = min_ary_of sym_tab0 base_s
2439 fun do_const name = IConst (name, T, T_args)
2440 val filter_ty_args =
2441 filter_type_args_in_iterm thy monom_constrs type_enc
2442 val ho_term_of = ho_term_from_iterm ctxt mono type_enc (SOME true)
2443 val name1 as (s1, _) =
2444 base_name |> ary - 1 > base_ary ? aliased_uncurried (ary - 1)
2445 val name2 as (s2, _) = base_name |> aliased_uncurried ary
2446 val (arg_Ts, _) = chop_fun ary T
2448 1 upto ary |> map (`I o make_bound_var o string_of_int)
2449 val bounds = bound_names ~~ arg_Ts
2450 val (first_bounds, last_bound) =
2451 bounds |> map (fn (name, T) => IConst (name, T, [])) |> split_last
2453 mk_app_op type_enc (list_app (do_const name1) first_bounds) last_bound
2456 list_app (do_const name2) (first_bounds @ [last_bound])
2458 val do_bound_type = do_bound_type ctxt mono type_enc
2460 eq_formula type_enc (atomic_types_of T)
2461 (map (apsnd do_bound_type) bounds) false
2462 (ho_term_of tm1) (ho_term_of tm2)
2465 [Formula (uncurried_alias_eq_prefix ^ s2, kind, eq |> maybe_negate,
2466 NONE, isabelle_info defN helper_rank)])
2467 |> (if ary - 1 = base_ary orelse Symtab.defined sym_tab s1 then I
2468 else pair_append (do_alias (ary - 1)))
2471 fun uncurried_alias_lines_for_sym ctxt monom_constrs conj_sym_kind mono type_enc
2472 sym_tab0 sym_tab (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 conj_sym_kind mono
2480 type_enc sym_tab0 sym_tab base_s0 types in_conj min_ary
2485 fun uncurried_alias_lines_for_sym_table ctxt monom_constrs conj_sym_kind mono
2486 type_enc uncurried_aliases sym_tab0 sym_tab =
2488 |> uncurried_aliases
2491 o uncurried_alias_lines_for_sym ctxt monom_constrs conj_sym_kind
2492 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 prem_kind format 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 type_enc))
2618 fun firstorderize in_helper =
2619 firstorderize_fact thy monom_constrs 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 conj_sym_kind mono
2631 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 type_enc sym_tab
2635 |> problem_lines_for_sym_decl_table ctxt conj_sym_kind mono type_enc
2637 val num_facts = length facts
2639 map (formula_line_for_fact ctxt polym_constrs 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 helper_prefix I false
2646 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 mono type_enc)
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)