1 (* Title: HOL/Tools/meson.ML
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1992 University of Cambridge
6 The MESON resolution proof procedure for HOL.
8 When making clauses, avoids using the rewriter -- instead uses RS recursively
10 NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E. ELIMINATES NEED FOR
11 FUNCTION nodups -- if done to goal clauses too!
14 signature BASIC_MESON =
16 val size_of_subgoals : thm -> int
17 val make_cnf : thm list -> thm -> thm list
18 val make_nnf : thm -> thm
19 val make_nnf1 : thm -> thm
20 val skolemize : thm -> thm
21 val make_clauses : thm list -> thm list
22 val make_horns : thm list -> thm list
23 val best_prolog_tac : (thm -> int) -> thm list -> tactic
24 val depth_prolog_tac : thm list -> tactic
25 val gocls : thm list -> thm list
26 val skolemize_prems_tac : thm list -> int -> tactic
27 val MESON : (thm list -> tactic) -> int -> tactic
28 val best_meson_tac : (thm -> int) -> int -> tactic
29 val safe_best_meson_tac : int -> tactic
30 val depth_meson_tac : int -> tactic
31 val prolog_step_tac' : thm list -> int -> tactic
32 val iter_deepen_prolog_tac : thm list -> tactic
33 val iter_deepen_meson_tac : thm list -> int -> tactic
34 val meson_tac : int -> tactic
35 val negate_head : thm -> thm
36 val select_literal : int -> thm -> thm
37 val skolemize_tac : int -> tactic
38 val make_clauses_tac : int -> tactic
39 val check_is_fol_term : term -> term
46 val not_conjD = thm "meson_not_conjD";
47 val not_disjD = thm "meson_not_disjD";
48 val not_notD = thm "meson_not_notD";
49 val not_allD = thm "meson_not_allD";
50 val not_exD = thm "meson_not_exD";
51 val imp_to_disjD = thm "meson_imp_to_disjD";
52 val not_impD = thm "meson_not_impD";
53 val iff_to_disjD = thm "meson_iff_to_disjD";
54 val not_iffD = thm "meson_not_iffD";
55 val conj_exD1 = thm "meson_conj_exD1";
56 val conj_exD2 = thm "meson_conj_exD2";
57 val disj_exD = thm "meson_disj_exD";
58 val disj_exD1 = thm "meson_disj_exD1";
59 val disj_exD2 = thm "meson_disj_exD2";
60 val disj_assoc = thm "meson_disj_assoc";
61 val disj_comm = thm "meson_disj_comm";
62 val disj_FalseD1 = thm "meson_disj_FalseD1";
63 val disj_FalseD2 = thm "meson_disj_FalseD2";
65 val depth_limit = ref 2000;
67 (**** Operators for forward proof ****)
69 (*Like RS, but raises Option if there are no unifiers and allows multiple unifiers.*)
70 fun resolve1 (tha,thb) = Seq.hd (biresolution false [(false,tha)] 1 thb);
72 (*raises exception if no rules apply -- unlike RL*)
73 fun tryres (th, rls) =
74 let fun tryall [] = raise THM("tryres", 0, th::rls)
75 | tryall (rl::rls) = (resolve1(th,rl) handle Option => tryall rls)
78 (*Permits forward proof from rules that discharge assumptions*)
79 fun forward_res nf st =
80 case Seq.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
82 | NONE => raise THM("forward_res", 0, [st]);
85 (*Are any of the constants in "bs" present in the term?*)
87 let fun has (Const(a,_)) = a mem bs
88 | has (Const ("Hilbert_Choice.Eps",_) $ _) = false
89 (*ignore constants within @-terms*)
90 | has (f$u) = has f orelse has u
91 | has (Abs(_,_,t)) = has t
96 (**** Clause handling ****)
98 fun literals (Const("Trueprop",_) $ P) = literals P
99 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
100 | literals (Const("Not",_) $ P) = [(false,P)]
101 | literals P = [(true,P)];
103 (*number of literals in a term*)
104 val nliterals = length o literals;
107 (*** Tautology Checking ***)
109 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
110 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
111 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
112 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
114 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
116 (*Literals like X=X are tautologous*)
117 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
118 | taut_poslit (Const("True",_)) = true
119 | taut_poslit _ = false;
122 let val (poslits,neglits) = signed_lits th
123 in exists taut_poslit poslits
125 exists (fn t => mem_term (t, neglits)) (HOLogic.false_const :: poslits)
129 (*** To remove trivial negated equality literals from clauses ***)
131 (*They are typically functional reflexivity axioms and are the converses of
132 injectivity equivalences*)
134 val not_refl_disj_D = thm"meson_not_refl_disj_D";
136 fun refl_clause_aux 0 th = th
137 | refl_clause_aux n th =
138 case HOLogic.dest_Trueprop (concl_of th) of
139 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
140 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
141 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
142 if is_Var t orelse is_Var u then (*Var inequation: delete or ignore*)
143 (refl_clause_aux (n-1) (th RS not_refl_disj_D) (*delete*)
144 handle THM _ => refl_clause_aux (n-1) (th RS disj_comm)) (*ignore*)
145 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
146 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
147 | _ => (*not a disjunction*) th;
149 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
150 notequal_lits_count P + notequal_lits_count Q
151 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
152 | notequal_lits_count _ = 0;
154 (*Simplify a clause by applying reflexivity to its negated equality literals*)
156 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
157 in zero_var_indexes (refl_clause_aux neqs th) end;
160 (*** The basic CNF transformation ***)
162 (*Replaces universally quantified variables by FREE variables -- because
163 assumptions may not contain scheme variables. Later, call "generalize". *)
165 let val newname = gensym "A_"
166 val spec' = read_instantiate [("x", newname)] spec
169 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
170 and then normalized via function nf. The normal form is given to resolve_tac,
171 presumably to instantiate a Boolean variable.*)
172 fun resop nf [prem] = resolve_tac (nf prem) 1;
175 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
177 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
178 Strips universal quantifiers and breaks up conjunctions.
179 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
180 fun cnf skoths (th,ths) =
181 let fun cnf_aux (th,ths) =
182 if has_meta_conn (prop_of th) then ths (*meta-level: ignore*)
183 else if not (has_consts ["All","Ex","op &"] (prop_of th))
184 then th::ths (*no work to do, terminate*)
185 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
186 Const ("op &", _) => (*conjunction*)
187 cnf_aux (th RS conjunct1,
188 cnf_aux (th RS conjunct2, ths))
189 | Const ("All", _) => (*universal quantifier*)
190 cnf_aux (freeze_spec th, ths)
192 (*existential quantifier: Insert Skolem functions*)
193 cnf_aux (tryres (th,skoths), ths)
194 | Const ("op |", _) => (*disjunction*)
196 (METAHYPS (resop cnf_nil) 1) THEN
197 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
198 in Seq.list_of (tac (th RS disj_forward)) @ ths end
199 | _ => (*no work to do*) th::ths
200 and cnf_nil th = cnf_aux (th,[])
205 (*Convert all suitable free variables to schematic variables,
206 but don't discharge assumptions.*)
207 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
209 fun make_cnf skoths th =
210 filter (not o is_taut)
211 (map (refl_clause o generalize) (cnf skoths (th, [])));
214 (**** Removal of duplicate literals ****)
216 (*Forward proof, passing extra assumptions as theorems to the tactic*)
217 fun forward_res2 nf hyps st =
220 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
223 | NONE => raise THM("forward_res2", 0, [st]);
225 (*Remove duplicates in P|Q by assuming ~P in Q
226 rls (initially []) accumulates assumptions of the form P==>False*)
227 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
228 handle THM _ => tryres(th,rls)
229 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
230 [disj_FalseD1, disj_FalseD2, asm_rl])
233 (*Remove duplicate literals, if there are any*)
235 if null(findrep(literals(prop_of th))) then th
236 else nodups_aux [] th;
239 (**** Generation of contrapositives ****)
241 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
242 fun assoc_right th = assoc_right (th RS disj_assoc)
245 (*Must check for negative literal first!*)
246 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
248 (*For ordinary resolution. *)
249 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
251 (*Create a goal or support clause, conclusing False*)
252 fun make_goal th = (*Must check for negative literal first!*)
253 make_goal (tryres(th, clause_rules))
254 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
256 (*Sort clauses by number of literals*)
257 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
259 fun sort_clauses ths = sort (make_ord fewerlits) ths;
261 (*True if the given type contains bool anywhere*)
262 fun has_bool (Type("bool",_)) = true
263 | has_bool (Type(_, Ts)) = exists has_bool Ts
264 | has_bool _ = false;
266 (*Is the string the name of a connective? It doesn't matter if this list is
267 incomplete, since when actually called, the only connectives likely to
268 remain are & | Not.*)
269 fun is_conn c = c mem_string
270 ["Trueprop", "HOL.tag", "op &", "op |", "op -->", "op =", "Not",
271 "All", "Ex", "Ball", "Bex"];
273 (*True if the term contains a function where the type of any argument contains
275 val has_bool_arg_const =
277 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
279 (*Raises an exception if any Vars in the theorem mention type bool; they
280 could cause make_horn to loop! Also rejects functions whose arguments are
281 Booleans or other functions.*)
283 not (exists (has_bool o fastype_of) (term_vars t) orelse
284 not (Term.is_first_order ["all","All","Ex"] t) orelse
285 has_bool_arg_const t orelse
288 (*FIXME: replace this by the boolean-valued version above!*)
289 fun check_is_fol_term t =
290 if is_fol_term t then t else raise TERM("check_is_fol_term",[t]);
292 fun check_is_fol th = (check_is_fol_term (prop_of th); th);
295 (*Create a meta-level Horn clause*)
296 fun make_horn crules th = make_horn crules (tryres(th,crules))
299 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
300 is a HOL disjunction.*)
301 fun add_contras crules (th,hcs) =
302 let fun rots (0,th) = hcs
303 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
304 rots(k-1, assoc_right (th RS disj_comm))
305 in case nliterals(prop_of th) of
307 | n => rots(n, assoc_right th)
310 (*Use "theorem naming" to label the clauses*)
311 fun name_thms label =
312 let fun name1 (th, (k,ths)) =
313 (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
315 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
317 (*Is the given disjunction an all-negative support clause?*)
318 fun is_negative th = forall (not o #1) (literals (prop_of th));
320 val neg_clauses = List.filter is_negative;
323 (***** MESON PROOF PROCEDURE *****)
325 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
326 As) = rhyps(phi, A::As)
327 | rhyps (_, As) = As;
329 (** Detecting repeated assumptions in a subgoal **)
331 (*The stringtree detects repeated assumptions.*)
332 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
334 (*detects repetitions in a list of terms*)
335 fun has_reps [] = false
336 | has_reps [_] = false
337 | has_reps [t,u] = (t aconv u)
338 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
339 handle INSERT => true;
341 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
342 fun TRYING_eq_assume_tac 0 st = Seq.single st
343 | TRYING_eq_assume_tac i st =
344 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
345 handle THM _ => TRYING_eq_assume_tac (i-1) st;
347 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
349 (*Loop checking: FAIL if trying to prove the same thing twice
350 -- if *ANY* subgoal has repeated literals*)
352 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
353 then Seq.empty else Seq.single st;
356 (* net_resolve_tac actually made it slower... *)
357 fun prolog_step_tac horns i =
358 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
359 TRYALL_eq_assume_tac;
361 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
362 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
364 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
367 (*Negation Normal Form*)
368 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
369 not_impD, not_iffD, not_allD, not_exD, not_notD];
371 fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
373 forward_res make_nnf1
374 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
377 (*The simplification removes defined quantifiers and occurrences of True and False,
378 as well as tags applied to True and False. nnf_ss also includes the one-point simprocs,
379 which are needed to avoid the various one-point theorems from generating junk clauses.*)
380 val tag_True = thm "tag_True";
381 val tag_False = thm "tag_False";
382 val nnf_simps = [Ex1_def,Ball_def,Bex_def,tag_True,tag_False]
385 HOL_basic_ss addsimps
386 (nnf_simps @ [if_True, if_False, if_cancel, if_eq_cancel, cases_simp] @
387 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms)
388 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
390 fun make_nnf th = th |> simplify nnf_ss
393 (*Pull existential quantifiers to front. This accomplishes Skolemization for
394 clauses that arise from a subgoal.*)
396 if not (has_consts ["Ex"] (prop_of th)) then th
397 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
398 disj_exD, disj_exD1, disj_exD2])))
400 skolemize (forward_res skolemize
401 (tryres (th, [conj_forward, disj_forward, all_forward])))
402 handle THM _ => forward_res skolemize (th RS ex_forward);
405 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
406 The resulting clauses are HOL disjunctions.*)
407 fun make_clauses ths =
408 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
411 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
414 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
416 (*Could simply use nprems_of, which would count remaining subgoals -- no
417 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
419 fun best_prolog_tac sizef horns =
420 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
422 fun depth_prolog_tac horns =
423 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
425 (*Return all negative clauses, as possible goal clauses*)
426 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
428 fun skolemize_prems_tac prems =
429 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
432 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
433 fun expand_defs_tac st =
434 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
435 in LocalDefs.def_export false defs st end;
437 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
438 fun MESON cltac i st =
440 (EVERY [rtac ccontr 1,
442 EVERY1 [skolemize_prems_tac negs,
443 METAHYPS (cltac o make_clauses)]) 1,
444 expand_defs_tac]) i st
445 handle TERM _ => no_tac st; (*probably from check_is_fol*)
447 (** Best-first search versions **)
449 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
450 fun best_meson_tac sizef =
452 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
453 (has_fewer_prems 1, sizef)
454 (prolog_step_tac (make_horns cls) 1));
456 (*First, breaks the goal into independent units*)
457 val safe_best_meson_tac =
458 SELECT_GOAL (TRY Safe_tac THEN
459 TRYALL (best_meson_tac size_of_subgoals));
461 (** Depth-first search version **)
463 val depth_meson_tac =
464 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
465 depth_prolog_tac (make_horns cls)]);
468 (** Iterative deepening version **)
470 (*This version does only one inference per call;
471 having only one eq_assume_tac speeds it up!*)
472 fun prolog_step_tac' horns =
473 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
474 take_prefix Thm.no_prems horns
475 val nrtac = net_resolve_tac horns
476 in fn i => eq_assume_tac i ORELSE
477 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
478 ((assume_tac i APPEND nrtac i) THEN check_tac)
481 fun iter_deepen_prolog_tac horns =
482 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
484 fun iter_deepen_meson_tac ths =
486 case (gocls (cls@ths)) of
487 [] => no_tac (*no goal clauses*)
489 (THEN_ITER_DEEPEN (resolve_tac goes 1)
491 (prolog_step_tac' (make_horns (cls@ths)))));
493 fun meson_claset_tac ths cs =
494 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
496 val meson_tac = CLASET' (meson_claset_tac []);
499 (**** Code to support ordinary resolution, rather than Model Elimination ****)
501 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
502 with no contrapositives, for ordinary resolution.*)
504 (*Rules to convert the head literal into a negated assumption. If the head
505 literal is already negated, then using notEfalse instead of notEfalse'
506 prevents a double negation.*)
507 val notEfalse = read_instantiate [("R","False")] notE;
508 val notEfalse' = rotate_prems 1 notEfalse;
510 fun negated_asm_of_head th =
511 th RS notEfalse handle THM _ => th RS notEfalse';
513 (*Converting one clause*)
514 fun make_meta_clause th =
515 negated_asm_of_head (make_horn resolution_clause_rules (check_is_fol th));
517 fun make_meta_clauses ths =
519 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
521 (*Permute a rule's premises to move the i-th premise to the last position.*)
523 let val n = nprems_of th
524 in if 1 <= i andalso i <= n
525 then Thm.permute_prems (i-1) 1 th
526 else raise THM("select_literal", i, [th])
529 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
531 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
533 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
534 fun select_literal i cl = negate_head (make_last i cl);
537 (*Top-level Skolemization. Allows part of the conversion to clauses to be
538 expressed as a tactic (or Isar method). Each assumption of the selected
539 goal is converted to NNF and then its existential quantifiers are pulled
540 to the front. Finally, all existential quantifiers are eliminated,
541 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
542 might generate many subgoals.*)
544 fun skolemize_tac i st =
545 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
548 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
549 THEN REPEAT (etac exE 1))),
550 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
552 handle Subscript => Seq.empty;
554 (*Top-level conversion to meta-level clauses. Each clause has
555 leading !!-bound universal variables, to express generality. To get
556 disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
557 val make_clauses_tac =
560 let val ts = Logic.strip_assums_hyp prop
565 (map forall_intr_vars
566 (make_meta_clauses (make_clauses hyps))) 1)),
567 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
571 (*** setup the special skoklemization methods ***)
573 (*No CHANGED_PROP here, since these always appear in the preamble*)
575 val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
577 val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
579 val skolemize_setup =
581 [("skolemize", Method.no_args skolemize_meth,
582 "Skolemization into existential quantifiers"),
583 ("make_clauses", Method.no_args make_clauses_meth,
584 "Conversion to !!-quantified meta-level clauses")];
588 structure BasicMeson: BASIC_MESON = Meson;