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 finish_cnf : thm list -> thm list
19 val make_nnf : thm -> thm
20 val make_nnf1 : thm -> thm
21 val skolemize : thm -> thm
22 val make_clauses : thm list -> thm list
23 val make_horns : thm list -> thm list
24 val best_prolog_tac : (thm -> int) -> thm list -> tactic
25 val depth_prolog_tac : thm list -> tactic
26 val gocls : thm list -> thm list
27 val skolemize_prems_tac : thm list -> int -> tactic
28 val MESON : (thm list -> thm list) -> (thm list -> tactic) -> int -> tactic
29 val best_meson_tac : (thm -> int) -> int -> tactic
30 val safe_best_meson_tac : int -> tactic
31 val depth_meson_tac : int -> tactic
32 val prolog_step_tac' : thm list -> int -> tactic
33 val iter_deepen_prolog_tac : thm list -> tactic
34 val iter_deepen_meson_tac : thm list -> int -> tactic
35 val meson_tac : int -> tactic
36 val negate_head : thm -> thm
37 val select_literal : int -> thm -> thm
38 val skolemize_tac : int -> tactic
45 val not_conjD = thm "meson_not_conjD";
46 val not_disjD = thm "meson_not_disjD";
47 val not_notD = thm "meson_not_notD";
48 val not_allD = thm "meson_not_allD";
49 val not_exD = thm "meson_not_exD";
50 val imp_to_disjD = thm "meson_imp_to_disjD";
51 val not_impD = thm "meson_not_impD";
52 val iff_to_disjD = thm "meson_iff_to_disjD";
53 val not_iffD = thm "meson_not_iffD";
54 val conj_exD1 = thm "meson_conj_exD1";
55 val conj_exD2 = thm "meson_conj_exD2";
56 val disj_exD = thm "meson_disj_exD";
57 val disj_exD1 = thm "meson_disj_exD1";
58 val disj_exD2 = thm "meson_disj_exD2";
59 val disj_assoc = thm "meson_disj_assoc";
60 val disj_comm = thm "meson_disj_comm";
61 val disj_FalseD1 = thm "meson_disj_FalseD1";
62 val disj_FalseD2 = thm "meson_disj_FalseD2";
64 val depth_limit = ref 2000;
66 (**** Operators for forward proof ****)
69 (** First-order Resolution **)
71 fun typ_pair_of (ix, (sort,ty)) = (TVar (ix,sort), ty);
72 fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);
74 val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
76 (*FIXME: currently does not "rename variables apart"*)
77 fun first_order_resolve thA thB =
78 let val thy = theory_of_thm thA
79 val tmA = concl_of thA
80 val Const("==>",_) $ tmB $ _ = prop_of thB
81 val (tyenv,tenv) = Pattern.first_order_match thy (tmB,tmA) (tyenv0,tenv0)
82 val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
83 in thA RS (cterm_instantiate ct_pairs thB) end
84 handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
86 fun flexflex_first_order th =
87 case (tpairs_of th) of
90 let val thy = theory_of_thm th
92 foldl (uncurry (Pattern.first_order_match thy)) (tyenv0,tenv0) pairs
93 val t_pairs = map term_pair_of (Vartab.dest tenv)
94 val th' = Thm.instantiate ([], map (pairself (cterm_of thy)) t_pairs) th
98 (*raises exception if no rules apply -- unlike RL*)
99 fun tryres (th, rls) =
100 let fun tryall [] = raise THM("tryres", 0, th::rls)
101 | tryall (rl::rls) = (th RS rl handle THM _ => tryall rls)
104 (*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
105 e.g. from conj_forward, should have the form
106 "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
107 and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
108 fun forward_res nf st =
109 let fun forward_tacf [prem] = rtac (nf prem) 1
110 | forward_tacf prems =
111 error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
114 cat_lines (map string_of_thm prems))
116 case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
118 | NONE => raise THM("forward_res", 0, [st])
121 (*Are any of the logical connectives in "bs" present in the term?*)
123 let fun has (Const(a,_)) = false
124 | has (Const("Trueprop",_) $ p) = has p
125 | has (Const("Not",_) $ p) = has p
126 | has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
127 | has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
128 | has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
129 | has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
134 (**** Clause handling ****)
136 fun literals (Const("Trueprop",_) $ P) = literals P
137 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
138 | literals (Const("Not",_) $ P) = [(false,P)]
139 | literals P = [(true,P)];
141 (*number of literals in a term*)
142 val nliterals = length o literals;
145 (*** Tautology Checking ***)
147 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
148 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
149 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
150 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
152 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
154 (*Literals like X=X are tautologous*)
155 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
156 | taut_poslit (Const("True",_)) = true
157 | taut_poslit _ = false;
160 let val (poslits,neglits) = signed_lits th
161 in exists taut_poslit poslits
163 exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
165 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
168 (*** To remove trivial negated equality literals from clauses ***)
170 (*They are typically functional reflexivity axioms and are the converses of
171 injectivity equivalences*)
173 val not_refl_disj_D = thm"meson_not_refl_disj_D";
175 (*Is either term a Var that does not properly occur in the other term?*)
176 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
177 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
178 | eliminable _ = false;
180 fun refl_clause_aux 0 th = th
181 | refl_clause_aux n th =
182 case HOLogic.dest_Trueprop (concl_of th) of
183 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
184 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
185 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
187 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
188 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
189 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
190 | _ => (*not a disjunction*) th;
192 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
193 notequal_lits_count P + notequal_lits_count Q
194 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
195 | notequal_lits_count _ = 0;
197 (*Simplify a clause by applying reflexivity to its negated equality literals*)
199 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
200 in zero_var_indexes (refl_clause_aux neqs th) end
201 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
204 (*** The basic CNF transformation ***)
206 val max_clauses = ref 40;
208 fun sum x y = if x < !max_clauses andalso y < !max_clauses then x+y else !max_clauses;
209 fun prod x y = if x < !max_clauses andalso y < !max_clauses then x*y else !max_clauses;
211 (*Estimate the number of clauses in order to detect infeasible theorems*)
212 fun signed_nclauses b (Const("Trueprop",_) $ t) = signed_nclauses b t
213 | signed_nclauses b (Const("Not",_) $ t) = signed_nclauses (not b) t
214 | signed_nclauses b (Const("op &",_) $ t $ u) =
215 if b then sum (signed_nclauses b t) (signed_nclauses b u)
216 else prod (signed_nclauses b t) (signed_nclauses b u)
217 | signed_nclauses b (Const("op |",_) $ t $ u) =
218 if b then prod (signed_nclauses b t) (signed_nclauses b u)
219 else sum (signed_nclauses b t) (signed_nclauses b u)
220 | signed_nclauses b (Const("op -->",_) $ t $ u) =
221 if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)
222 else sum (signed_nclauses (not b) t) (signed_nclauses b u)
223 | signed_nclauses b (Const("op =", Type ("fun", [T, _])) $ t $ u) =
224 if T = HOLogic.boolT then (*Boolean equality is if-and-only-if*)
225 if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))
226 (prod (signed_nclauses (not b) u) (signed_nclauses b t))
227 else sum (prod (signed_nclauses b t) (signed_nclauses b u))
228 (prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))
230 | signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
231 | signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
232 | signed_nclauses _ _ = 1; (* literal *)
234 val nclauses = signed_nclauses true;
236 fun too_many_clauses t = nclauses t >= !max_clauses;
238 (*Replaces universally quantified variables by FREE variables -- because
239 assumptions may not contain scheme variables. Later, we call "generalize". *)
241 let val newname = gensym "mes_"
242 val spec' = read_instantiate [("x", newname)] spec
245 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
246 and then normalized via function nf. The normal form is given to resolve_tac,
247 instantiate a Boolean variable created by resolution with disj_forward. Since
248 (nf prem) returns a LIST of theorems, we can backtrack to get all combinations.*)
249 fun resop nf [prem] = resolve_tac (nf prem) 1;
251 (*Any need to extend this list with
252 "HOL.type_class","HOL.eq_class","ProtoPure.term"?*)
254 exists_Const (member (op =) ["==", "==>", "all", "prop"] o #1);
256 fun apply_skolem_ths (th, rls) =
257 let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
258 | tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
261 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
262 Strips universal quantifiers and breaks up conjunctions.
263 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
264 fun cnf skoths (th,ths) =
265 let fun cnf_aux (th,ths) =
266 if not (can HOLogic.dest_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
267 else if not (has_conns ["All","Ex","op &"] (prop_of th))
268 then th::ths (*no work to do, terminate*)
269 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
270 Const ("op &", _) => (*conjunction*)
271 cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
272 | Const ("All", _) => (*universal quantifier*)
273 cnf_aux (freeze_spec th, ths)
275 (*existential quantifier: Insert Skolem functions*)
276 cnf_aux (apply_skolem_ths (th,skoths), ths)
277 | Const ("op |", _) =>
278 (*Disjunction of P, Q: Create new goal of proving ?P | ?Q and solve it using
279 all combinations of converting P, Q to CNF.*)
281 (METAHYPS (resop cnf_nil) 1) THEN
282 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
283 in Seq.list_of (tac (th RS disj_forward)) @ ths end
284 | _ => (*no work to do*) th::ths
285 and cnf_nil th = cnf_aux (th,[])
287 if too_many_clauses (concl_of th)
288 then (Output.debug (fn () => ("cnf is ignoring: " ^ string_of_thm th)); ths)
289 else cnf_aux (th,ths)
292 fun all_names (Const ("all", _) $ Abs(x,_,P)) = x :: all_names P
295 fun new_names n [] = []
296 | new_names n (x::xs) =
297 if String.isPrefix "mes_" x then (x, radixstring(26,"A",n)) :: new_names (n+1) xs
300 (*The gensym names are ugly, so don't let the user see them. When forall_elim_vars
301 is called, it will ensure that no name clauses ensue.*)
303 let val old_names = all_names (prop_of th)
304 in Drule.rename_bvars (new_names 0 old_names) th end;
306 (*Convert all suitable free variables to schematic variables,
307 but don't discharge assumptions.*)
308 fun generalize th = Thm.varifyT (forall_elim_vars 0 (nice_names (forall_intr_frees th)));
310 fun make_cnf skoths th = cnf skoths (th, []);
312 (*Generalization, removal of redundant equalities, removal of tautologies.*)
313 fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
316 (**** Removal of duplicate literals ****)
318 (*Forward proof, passing extra assumptions as theorems to the tactic*)
319 fun forward_res2 nf hyps st =
322 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
325 | NONE => raise THM("forward_res2", 0, [st]);
327 (*Remove duplicates in P|Q by assuming ~P in Q
328 rls (initially []) accumulates assumptions of the form P==>False*)
329 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
330 handle THM _ => tryres(th,rls)
331 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
332 [disj_FalseD1, disj_FalseD2, asm_rl])
335 (*Remove duplicate literals, if there are any*)
337 if has_duplicates (op =) (literals (prop_of th))
338 then nodups_aux [] th
342 (**** Generation of contrapositives ****)
344 fun is_left (Const ("Trueprop", _) $
345 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _)) = true
348 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
350 if is_left (prop_of th) then assoc_right (th RS disj_assoc)
353 (*Must check for negative literal first!*)
354 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
356 (*For ordinary resolution. *)
357 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
359 (*Create a goal or support clause, conclusing False*)
360 fun make_goal th = (*Must check for negative literal first!*)
361 make_goal (tryres(th, clause_rules))
362 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
364 (*Sort clauses by number of literals*)
365 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
367 fun sort_clauses ths = sort (make_ord fewerlits) ths;
369 (*True if the given type contains bool anywhere*)
370 fun has_bool (Type("bool",_)) = true
371 | has_bool (Type(_, Ts)) = exists has_bool Ts
372 | has_bool _ = false;
374 (*Is the string the name of a connective? Really only | and Not can remain,
375 since this code expects to be called on a clause form.*)
376 val is_conn = member (op =)
377 ["Trueprop", "op &", "op |", "op -->", "Not",
378 "All", "Ex", "Ball", "Bex"];
380 (*True if the term contains a function--not a logical connective--where the type
381 of any argument contains bool.*)
382 val has_bool_arg_const =
384 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
386 (*A higher-order instance of a first-order constant? Example is the definition of
387 HOL.one, 1, at a function type in theory SetsAndFunctions.*)
388 fun higher_inst_const thy (c,T) =
389 case binder_types T of
390 [] => false (*not a function type, OK*)
391 | Ts => length (binder_types (Sign.the_const_type thy c)) <> length Ts;
393 (*Raises an exception if any Vars in the theorem mention type bool.
394 Also rejects functions whose arguments are Booleans or other functions.*)
395 fun is_fol_term thy t =
396 Term.is_first_order ["all","All","Ex"] t andalso
397 not (exists (has_bool o fastype_of) (term_vars t) orelse
398 has_bool_arg_const t orelse
399 exists_Const (higher_inst_const thy) t orelse
402 fun rigid t = not (is_Var (head_of t));
404 fun ok4horn (Const ("Trueprop",_) $ (Const ("op |", _) $ t $ _)) = rigid t
405 | ok4horn (Const ("Trueprop",_) $ t) = rigid t
408 (*Create a meta-level Horn clause*)
409 fun make_horn crules th =
410 if ok4horn (concl_of th)
411 then make_horn crules (tryres(th,crules)) handle THM _ => th
414 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
415 is a HOL disjunction.*)
416 fun add_contras crules (th,hcs) =
417 let fun rots (0,th) = hcs
418 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
419 rots(k-1, assoc_right (th RS disj_comm))
420 in case nliterals(prop_of th) of
422 | n => rots(n, assoc_right th)
425 (*Use "theorem naming" to label the clauses*)
426 fun name_thms label =
427 let fun name1 (th, (k,ths)) =
428 (k-1, PureThy.put_name_hint (label ^ string_of_int k) th :: ths)
429 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
431 (*Is the given disjunction an all-negative support clause?*)
432 fun is_negative th = forall (not o #1) (literals (prop_of th));
434 val neg_clauses = List.filter is_negative;
437 (***** MESON PROOF PROCEDURE *****)
439 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
440 As) = rhyps(phi, A::As)
441 | rhyps (_, As) = As;
443 (** Detecting repeated assumptions in a subgoal **)
445 (*The stringtree detects repeated assumptions.*)
446 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
448 (*detects repetitions in a list of terms*)
449 fun has_reps [] = false
450 | has_reps [_] = false
451 | has_reps [t,u] = (t aconv u)
452 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
453 handle Net.INSERT => true;
455 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
456 fun TRYING_eq_assume_tac 0 st = Seq.single st
457 | TRYING_eq_assume_tac i st =
458 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
459 handle THM _ => TRYING_eq_assume_tac (i-1) st;
461 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
463 (*Loop checking: FAIL if trying to prove the same thing twice
464 -- if *ANY* subgoal has repeated literals*)
466 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
467 then Seq.empty else Seq.single st;
470 (* net_resolve_tac actually made it slower... *)
471 fun prolog_step_tac horns i =
472 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
473 TRYALL_eq_assume_tac;
475 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
476 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
478 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
481 (*Negation Normal Form*)
482 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
483 not_impD, not_iffD, not_allD, not_exD, not_notD];
485 fun ok4nnf (Const ("Trueprop",_) $ (Const ("Not", _) $ t)) = rigid t
486 | ok4nnf (Const ("Trueprop",_) $ t) = rigid t
490 if ok4nnf (concl_of th)
491 then make_nnf1 (tryres(th, nnf_rls))
493 forward_res make_nnf1
494 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
498 (*The simplification removes defined quantifiers and occurrences of True and False.
499 nnf_ss also includes the one-point simprocs,
500 which are needed to avoid the various one-point theorems from generating junk clauses.*)
502 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
503 if_False, if_cancel, if_eq_cancel, cases_simp];
504 val nnf_extra_simps =
505 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
508 HOL_basic_ss addsimps nnf_extra_simps
509 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
511 fun make_nnf th = case prems_of th of
512 [] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
515 | _ => raise THM ("make_nnf: premises in argument", 0, [th]);
517 (*Pull existential quantifiers to front. This accomplishes Skolemization for
518 clauses that arise from a subgoal.*)
520 if not (has_conns ["Ex"] (prop_of th)) then th
521 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
522 disj_exD, disj_exD1, disj_exD2])))
524 skolemize (forward_res skolemize
525 (tryres (th, [conj_forward, disj_forward, all_forward])))
526 handle THM _ => forward_res skolemize (th RS ex_forward);
529 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
530 The resulting clauses are HOL disjunctions.*)
531 fun make_clauses ths =
532 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
534 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
537 (distinct Thm.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
539 (*Could simply use nprems_of, which would count remaining subgoals -- no
540 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
542 fun best_prolog_tac sizef horns =
543 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
545 fun depth_prolog_tac horns =
546 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
548 (*Return all negative clauses, as possible goal clauses*)
549 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
551 fun skolemize_prems_tac prems =
552 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
555 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions.
556 Function mkcl converts theorems to clauses.*)
557 fun MESON mkcl cltac i st =
559 (EVERY [ObjectLogic.atomize_prems_tac 1,
562 EVERY1 [skolemize_prems_tac negs,
563 METAHYPS (cltac o mkcl)]) 1]) i st
564 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
566 (** Best-first search versions **)
568 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
569 fun best_meson_tac sizef =
572 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
573 (has_fewer_prems 1, sizef)
574 (prolog_step_tac (make_horns cls) 1));
576 (*First, breaks the goal into independent units*)
577 val safe_best_meson_tac =
578 SELECT_GOAL (TRY Safe_tac THEN
579 TRYALL (best_meson_tac size_of_subgoals));
581 (** Depth-first search version **)
583 val depth_meson_tac =
585 (fn cls => EVERY [resolve_tac (gocls cls) 1, depth_prolog_tac (make_horns cls)]);
588 (** Iterative deepening version **)
590 (*This version does only one inference per call;
591 having only one eq_assume_tac speeds it up!*)
592 fun prolog_step_tac' horns =
593 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
594 take_prefix Thm.no_prems horns
595 val nrtac = net_resolve_tac horns
596 in fn i => eq_assume_tac i ORELSE
597 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
598 ((assume_tac i APPEND nrtac i) THEN check_tac)
601 fun iter_deepen_prolog_tac horns =
602 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
604 fun iter_deepen_meson_tac ths = MESON make_clauses
606 case (gocls (cls@ths)) of
607 [] => no_tac (*no goal clauses*)
609 let val horns = make_horns (cls@ths)
611 Output.debug (fn () => ("meson method called:\n" ^
612 space_implode "\n" (map string_of_thm (cls@ths)) ^
614 space_implode "\n" (map string_of_thm horns)))
615 in THEN_ITER_DEEPEN (resolve_tac goes 1) (has_fewer_prems 1) (prolog_step_tac' horns)
619 fun meson_claset_tac ths cs =
620 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
622 val meson_tac = CLASET' (meson_claset_tac []);
625 (**** Code to support ordinary resolution, rather than Model Elimination ****)
627 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
628 with no contrapositives, for ordinary resolution.*)
630 (*Rules to convert the head literal into a negated assumption. If the head
631 literal is already negated, then using notEfalse instead of notEfalse'
632 prevents a double negation.*)
633 val notEfalse = read_instantiate [("R","False")] notE;
634 val notEfalse' = rotate_prems 1 notEfalse;
636 fun negated_asm_of_head th =
637 th RS notEfalse handle THM _ => th RS notEfalse';
639 (*Converting one clause*)
640 val make_meta_clause =
641 zero_var_indexes o negated_asm_of_head o make_horn resolution_clause_rules;
643 fun make_meta_clauses ths =
645 (distinct Thm.eq_thm_prop (map make_meta_clause ths));
647 (*Permute a rule's premises to move the i-th premise to the last position.*)
649 let val n = nprems_of th
650 in if 1 <= i andalso i <= n
651 then Thm.permute_prems (i-1) 1 th
652 else raise THM("select_literal", i, [th])
655 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
657 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
659 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
660 fun select_literal i cl = negate_head (make_last i cl);
663 (*Top-level Skolemization. Allows part of the conversion to clauses to be
664 expressed as a tactic (or Isar method). Each assumption of the selected
665 goal is converted to NNF and then its existential quantifiers are pulled
666 to the front. Finally, all existential quantifiers are eliminated,
667 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
668 might generate many subgoals.*)
670 fun skolemize_tac i st =
671 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
674 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
675 THEN REPEAT (etac exE 1))),
676 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
678 handle Subscript => Seq.empty;
682 structure BasicMeson: BASIC_MESON = Meson;