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 (*Generation of unique names -- maxidx cannot be relied upon to increase!
163 Cannot rely on "variant", since variables might coincide when literals
164 are joined to make a clause...
165 19 chooses "U" as the first variable name*)
166 val name_ref = ref 19;
168 (*Replaces universally quantified variables by FREE variables -- because
169 assumptions may not contain scheme variables. Later, call "generalize". *)
171 let val names = add_term_names (prop_of th, [])
172 val newname = (name_ref := !name_ref + 1;
173 variant names (radixstring(26, "A", !name_ref)))
174 val spec' = read_instantiate [("x", newname)] spec
177 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
178 and then normalized via function nf. The normal form is given to resolve_tac,
179 presumably to instantiate a Boolean variable.*)
180 fun resop nf [prem] = resolve_tac (nf prem) 1;
183 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
185 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
186 Strips universal quantifiers and breaks up conjunctions.
187 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
188 fun cnf skoths (th,ths) =
189 let fun cnf_aux (th,ths) =
190 if has_meta_conn (prop_of th) then ths (*meta-level: ignore*)
191 else if not (has_consts ["All","Ex","op &"] (prop_of th))
192 then th::ths (*no work to do, terminate*)
193 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
194 Const ("op &", _) => (*conjunction*)
195 cnf_aux (th RS conjunct1,
196 cnf_aux (th RS conjunct2, ths))
197 | Const ("All", _) => (*universal quantifier*)
198 cnf_aux (freeze_spec th, ths)
200 (*existential quantifier: Insert Skolem functions*)
201 cnf_aux (tryres (th,skoths), ths)
202 | Const ("op |", _) => (*disjunction*)
204 (METAHYPS (resop cnf_nil) 1) THEN
205 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
206 in Seq.list_of (tac (th RS disj_forward)) @ ths end
207 | _ => (*no work to do*) th::ths
208 and cnf_nil th = cnf_aux (th,[])
210 name_ref := 19; (*It's safe to reset this in a top-level call to cnf*)
214 (*Convert all suitable free variables to schematic variables,
215 but don't discharge assumptions.*)
216 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
218 fun make_cnf skoths th =
219 filter (not o is_taut)
220 (map (refl_clause o generalize) (cnf skoths (th, [])));
223 (**** Removal of duplicate literals ****)
225 (*Forward proof, passing extra assumptions as theorems to the tactic*)
226 fun forward_res2 nf hyps st =
229 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
232 | NONE => raise THM("forward_res2", 0, [st]);
234 (*Remove duplicates in P|Q by assuming ~P in Q
235 rls (initially []) accumulates assumptions of the form P==>False*)
236 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
237 handle THM _ => tryres(th,rls)
238 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
239 [disj_FalseD1, disj_FalseD2, asm_rl])
242 (*Remove duplicate literals, if there are any*)
244 if null(findrep(literals(prop_of th))) then th
245 else nodups_aux [] th;
248 (**** Generation of contrapositives ****)
250 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
251 fun assoc_right th = assoc_right (th RS disj_assoc)
254 (*Must check for negative literal first!*)
255 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
257 (*For ordinary resolution. *)
258 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
260 (*Create a goal or support clause, conclusing False*)
261 fun make_goal th = (*Must check for negative literal first!*)
262 make_goal (tryres(th, clause_rules))
263 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
265 (*Sort clauses by number of literals*)
266 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
268 fun sort_clauses ths = sort (make_ord fewerlits) ths;
270 (*True if the given type contains bool anywhere*)
271 fun has_bool (Type("bool",_)) = true
272 | has_bool (Type(_, Ts)) = exists has_bool Ts
273 | has_bool _ = false;
275 (*Is the string the name of a connective? It doesn't matter if this list is
276 incomplete, since when actually called, the only connectives likely to
277 remain are & | Not.*)
278 fun is_conn c = c mem_string
279 ["Trueprop", "HOL.tag", "op &", "op |", "op -->", "op =", "Not",
280 "All", "Ex", "Ball", "Bex"];
282 (*True if the term contains a function where the type of any argument contains
284 val has_bool_arg_const =
286 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
288 (*Raises an exception if any Vars in the theorem mention type bool; they
289 could cause make_horn to loop! Also rejects functions whose arguments are
290 Booleans or other functions.*)
292 not (exists (has_bool o fastype_of) (term_vars t) orelse
293 not (Term.is_first_order ["all","All","Ex"] t) orelse
294 has_bool_arg_const t orelse
297 (*FIXME: replace this by the boolean-valued version above!*)
298 fun check_is_fol_term t =
299 if is_fol_term t then t else raise TERM("check_is_fol_term",[t]);
301 fun check_is_fol th = (check_is_fol_term (prop_of th); th);
304 (*Create a meta-level Horn clause*)
305 fun make_horn crules th = make_horn crules (tryres(th,crules))
308 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
309 is a HOL disjunction.*)
310 fun add_contras crules (th,hcs) =
311 let fun rots (0,th) = hcs
312 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
313 rots(k-1, assoc_right (th RS disj_comm))
314 in case nliterals(prop_of th) of
316 | n => rots(n, assoc_right th)
319 (*Use "theorem naming" to label the clauses*)
320 fun name_thms label =
321 let fun name1 (th, (k,ths)) =
322 (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
324 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
326 (*Is the given disjunction an all-negative support clause?*)
327 fun is_negative th = forall (not o #1) (literals (prop_of th));
329 val neg_clauses = List.filter is_negative;
332 (***** MESON PROOF PROCEDURE *****)
334 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
335 As) = rhyps(phi, A::As)
336 | rhyps (_, As) = As;
338 (** Detecting repeated assumptions in a subgoal **)
340 (*The stringtree detects repeated assumptions.*)
341 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
343 (*detects repetitions in a list of terms*)
344 fun has_reps [] = false
345 | has_reps [_] = false
346 | has_reps [t,u] = (t aconv u)
347 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
348 handle INSERT => true;
350 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
351 fun TRYING_eq_assume_tac 0 st = Seq.single st
352 | TRYING_eq_assume_tac i st =
353 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
354 handle THM _ => TRYING_eq_assume_tac (i-1) st;
356 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
358 (*Loop checking: FAIL if trying to prove the same thing twice
359 -- if *ANY* subgoal has repeated literals*)
361 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
362 then Seq.empty else Seq.single st;
365 (* net_resolve_tac actually made it slower... *)
366 fun prolog_step_tac horns i =
367 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
368 TRYALL_eq_assume_tac;
370 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
371 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
373 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
376 (*Negation Normal Form*)
377 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
378 not_impD, not_iffD, not_allD, not_exD, not_notD];
380 fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
382 forward_res make_nnf1
383 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
386 (*The simplification removes defined quantifiers and occurrences of True and False,
387 as well as tags applied to True and False. nnf_ss also includes the one-point simprocs,
388 which are needed to avoid the various one-point theorems from generating junk clauses.*)
389 val tag_True = thm "tag_True";
390 val tag_False = thm "tag_False";
391 val nnf_simps = [Ex1_def,Ball_def,Bex_def,tag_True,tag_False]
394 HOL_basic_ss addsimps
395 (nnf_simps @ [if_True, if_False, if_cancel, if_eq_cancel, cases_simp] @
396 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms)
397 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
399 fun make_nnf th = th |> simplify nnf_ss
402 (*Pull existential quantifiers to front. This accomplishes Skolemization for
403 clauses that arise from a subgoal.*)
405 if not (has_consts ["Ex"] (prop_of th)) then th
406 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
407 disj_exD, disj_exD1, disj_exD2])))
409 skolemize (forward_res skolemize
410 (tryres (th, [conj_forward, disj_forward, all_forward])))
411 handle THM _ => forward_res skolemize (th RS ex_forward);
414 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
415 The resulting clauses are HOL disjunctions.*)
416 fun make_clauses ths =
417 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
420 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
423 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
425 (*Could simply use nprems_of, which would count remaining subgoals -- no
426 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
428 fun best_prolog_tac sizef horns =
429 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
431 fun depth_prolog_tac horns =
432 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
434 (*Return all negative clauses, as possible goal clauses*)
435 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
437 fun skolemize_prems_tac prems =
438 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
441 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
442 fun expand_defs_tac st =
443 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
444 in LocalDefs.def_export false defs st end;
446 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
447 fun MESON cltac i st =
449 (EVERY [rtac ccontr 1,
451 EVERY1 [skolemize_prems_tac negs,
452 METAHYPS (cltac o make_clauses)]) 1,
453 expand_defs_tac]) i st
454 handle TERM _ => no_tac st; (*probably from check_is_fol*)
456 (** Best-first search versions **)
458 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
459 fun best_meson_tac sizef =
461 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
462 (has_fewer_prems 1, sizef)
463 (prolog_step_tac (make_horns cls) 1));
465 (*First, breaks the goal into independent units*)
466 val safe_best_meson_tac =
467 SELECT_GOAL (TRY Safe_tac THEN
468 TRYALL (best_meson_tac size_of_subgoals));
470 (** Depth-first search version **)
472 val depth_meson_tac =
473 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
474 depth_prolog_tac (make_horns cls)]);
477 (** Iterative deepening version **)
479 (*This version does only one inference per call;
480 having only one eq_assume_tac speeds it up!*)
481 fun prolog_step_tac' horns =
482 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
483 take_prefix Thm.no_prems horns
484 val nrtac = net_resolve_tac horns
485 in fn i => eq_assume_tac i ORELSE
486 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
487 ((assume_tac i APPEND nrtac i) THEN check_tac)
490 fun iter_deepen_prolog_tac horns =
491 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
493 fun iter_deepen_meson_tac ths =
495 case (gocls (cls@ths)) of
496 [] => no_tac (*no goal clauses*)
498 (THEN_ITER_DEEPEN (resolve_tac goes 1)
500 (prolog_step_tac' (make_horns (cls@ths)))));
502 fun meson_claset_tac ths cs =
503 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
505 val meson_tac = CLASET' (meson_claset_tac []);
508 (**** Code to support ordinary resolution, rather than Model Elimination ****)
510 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
511 with no contrapositives, for ordinary resolution.*)
513 (*Rules to convert the head literal into a negated assumption. If the head
514 literal is already negated, then using notEfalse instead of notEfalse'
515 prevents a double negation.*)
516 val notEfalse = read_instantiate [("R","False")] notE;
517 val notEfalse' = rotate_prems 1 notEfalse;
519 fun negated_asm_of_head th =
520 th RS notEfalse handle THM _ => th RS notEfalse';
522 (*Converting one clause*)
523 fun make_meta_clause th =
524 negated_asm_of_head (make_horn resolution_clause_rules (check_is_fol th));
526 fun make_meta_clauses ths =
528 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
530 (*Permute a rule's premises to move the i-th premise to the last position.*)
532 let val n = nprems_of th
533 in if 1 <= i andalso i <= n
534 then Thm.permute_prems (i-1) 1 th
535 else raise THM("select_literal", i, [th])
538 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
540 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
542 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
543 fun select_literal i cl = negate_head (make_last i cl);
546 (*Top-level Skolemization. Allows part of the conversion to clauses to be
547 expressed as a tactic (or Isar method). Each assumption of the selected
548 goal is converted to NNF and then its existential quantifiers are pulled
549 to the front. Finally, all existential quantifiers are eliminated,
550 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
551 might generate many subgoals.*)
553 fun skolemize_tac i st =
554 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
557 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
558 THEN REPEAT (etac exE 1))),
559 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
561 handle Subscript => Seq.empty;
563 (*Top-level conversion to meta-level clauses. Each clause has
564 leading !!-bound universal variables, to express generality. To get
565 disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
566 val make_clauses_tac =
569 let val ts = Logic.strip_assums_hyp prop
574 (map forall_intr_vars
575 (make_meta_clauses (make_clauses hyps))) 1)),
576 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
580 (*** setup the special skoklemization methods ***)
582 (*No CHANGED_PROP here, since these always appear in the preamble*)
584 val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
586 val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
588 val skolemize_setup =
590 [("skolemize", Method.no_args skolemize_meth,
591 "Skolemization into existential quantifiers"),
592 ("make_clauses", Method.no_args make_clauses_meth,
593 "Conversion to !!-quantified meta-level clauses")];
597 structure BasicMeson: BASIC_MESON = Meson;