MESON tactical takes an additional argument: the clausification function.
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 fun match pat = Pattern.first_order_match thy (pat,tmA) (tyenv0,tenv0)
81 val Const("==>",_) $ tmB $ _ = prop_of thB
82 val (tyenv,tenv) = match tmB
83 val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
84 in thA RS (cterm_instantiate ct_pairs thB) end
85 handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
87 (*raises exception if no rules apply -- unlike RL*)
88 fun tryres (th, rls) =
89 let fun tryall [] = raise THM("tryres", 0, th::rls)
90 | tryall (rl::rls) = (th RS rl handle THM _ => tryall rls)
93 (*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
94 e.g. from conj_forward, should have the form
95 "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
96 and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
97 fun forward_res nf st =
98 let fun forward_tacf [prem] = rtac (nf prem) 1
99 | forward_tacf prems =
100 error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
103 cat_lines (map string_of_thm prems))
105 case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
107 | NONE => raise THM("forward_res", 0, [st])
110 (*Are any of the logical connectives in "bs" present in the term?*)
112 let fun has (Const(a,_)) = false
113 | has (Const("Trueprop",_) $ p) = has p
114 | has (Const("Not",_) $ p) = has p
115 | has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
116 | has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
117 | has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
118 | has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
123 (**** Clause handling ****)
125 fun literals (Const("Trueprop",_) $ P) = literals P
126 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
127 | literals (Const("Not",_) $ P) = [(false,P)]
128 | literals P = [(true,P)];
130 (*number of literals in a term*)
131 val nliterals = length o literals;
134 (*** Tautology Checking ***)
136 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
137 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
138 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
139 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
141 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
143 (*Literals like X=X are tautologous*)
144 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
145 | taut_poslit (Const("True",_)) = true
146 | taut_poslit _ = false;
149 let val (poslits,neglits) = signed_lits th
150 in exists taut_poslit poslits
152 exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
154 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
157 (*** To remove trivial negated equality literals from clauses ***)
159 (*They are typically functional reflexivity axioms and are the converses of
160 injectivity equivalences*)
162 val not_refl_disj_D = thm"meson_not_refl_disj_D";
164 (*Is either term a Var that does not properly occur in the other term?*)
165 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
166 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
167 | eliminable _ = false;
169 fun refl_clause_aux 0 th = th
170 | refl_clause_aux n th =
171 case HOLogic.dest_Trueprop (concl_of th) of
172 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
173 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
174 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
176 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
177 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
178 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
179 | _ => (*not a disjunction*) th;
181 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
182 notequal_lits_count P + notequal_lits_count Q
183 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
184 | notequal_lits_count _ = 0;
186 (*Simplify a clause by applying reflexivity to its negated equality literals*)
188 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
189 in zero_var_indexes (refl_clause_aux neqs th) end
190 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
193 (*** The basic CNF transformation ***)
195 val max_clauses = ref 40;
197 fun sum x y = if x < !max_clauses andalso y < !max_clauses then x+y else !max_clauses;
198 fun prod x y = if x < !max_clauses andalso y < !max_clauses then x*y else !max_clauses;
200 (*Estimate the number of clauses in order to detect infeasible theorems*)
201 fun signed_nclauses b (Const("Trueprop",_) $ t) = signed_nclauses b t
202 | signed_nclauses b (Const("Not",_) $ t) = signed_nclauses (not b) t
203 | signed_nclauses b (Const("op &",_) $ t $ u) =
204 if b then sum (signed_nclauses b t) (signed_nclauses b u)
205 else prod (signed_nclauses b t) (signed_nclauses b u)
206 | signed_nclauses b (Const("op |",_) $ t $ u) =
207 if b then prod (signed_nclauses b t) (signed_nclauses b u)
208 else sum (signed_nclauses b t) (signed_nclauses b u)
209 | signed_nclauses b (Const("op -->",_) $ t $ u) =
210 if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)
211 else sum (signed_nclauses (not b) t) (signed_nclauses b u)
212 | signed_nclauses b (Const("op =", Type ("fun", [T, _])) $ t $ u) =
213 if T = HOLogic.boolT then (*Boolean equality is if-and-only-if*)
214 if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))
215 (prod (signed_nclauses (not b) u) (signed_nclauses b t))
216 else sum (prod (signed_nclauses b t) (signed_nclauses b u))
217 (prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))
219 | signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
220 | signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
221 | signed_nclauses _ _ = 1; (* literal *)
223 val nclauses = signed_nclauses true;
225 fun too_many_clauses t = nclauses t >= !max_clauses;
227 (*Replaces universally quantified variables by FREE variables -- because
228 assumptions may not contain scheme variables. Later, we call "generalize". *)
230 let val newname = gensym "mes_"
231 val spec' = read_instantiate [("x", newname)] spec
234 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
235 and then normalized via function nf. The normal form is given to resolve_tac,
236 instantiate a Boolean variable created by resolution with disj_forward. Since
237 (nf prem) returns a LIST of theorems, we can backtrack to get all combinations.*)
238 fun resop nf [prem] = resolve_tac (nf prem) 1;
240 (*Any need to extend this list with
241 "HOL.type_class","Code_Generator.eq_class","ProtoPure.term"?*)
243 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
245 fun apply_skolem_ths (th, rls) =
246 let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
247 | tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
250 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
251 Strips universal quantifiers and breaks up conjunctions.
252 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
253 fun cnf skoths (th,ths) =
254 let fun cnf_aux (th,ths) =
255 if not (can HOLogic.dest_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
256 else if not (has_conns ["All","Ex","op &"] (prop_of th))
257 then th::ths (*no work to do, terminate*)
258 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
259 Const ("op &", _) => (*conjunction*)
260 cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
261 | Const ("All", _) => (*universal quantifier*)
262 cnf_aux (freeze_spec th, ths)
264 (*existential quantifier: Insert Skolem functions*)
265 cnf_aux (apply_skolem_ths (th,skoths), ths)
266 | Const ("op |", _) =>
267 (*Disjunction of P, Q: Create new goal of proving ?P | ?Q and solve it using
268 all combinations of converting P, Q to CNF.*)
270 (METAHYPS (resop cnf_nil) 1) THEN
271 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
272 in Seq.list_of (tac (th RS disj_forward)) @ ths end
273 | _ => (*no work to do*) th::ths
274 and cnf_nil th = cnf_aux (th,[])
276 if too_many_clauses (concl_of th)
277 then (Output.debug (fn () => ("cnf is ignoring: " ^ string_of_thm th)); ths)
278 else cnf_aux (th,ths)
281 fun all_names (Const ("all", _) $ Abs(x,_,P)) = x :: all_names P
284 fun new_names n [] = []
285 | new_names n (x::xs) =
286 if String.isPrefix "mes_" x then (x, radixstring(26,"A",n)) :: new_names (n+1) xs
289 (*The gensym names are ugly, so don't let the user see them. When forall_elim_vars
290 is called, it will ensure that no name clauses ensue.*)
292 let val old_names = all_names (prop_of th)
293 in Drule.rename_bvars (new_names 0 old_names) th end;
295 (*Convert all suitable free variables to schematic variables,
296 but don't discharge assumptions.*)
297 fun generalize th = Thm.varifyT (forall_elim_vars 0 (nice_names (forall_intr_frees th)));
299 fun make_cnf skoths th = cnf skoths (th, []);
301 (*Generalization, removal of redundant equalities, removal of tautologies.*)
302 fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
305 (**** Removal of duplicate literals ****)
307 (*Forward proof, passing extra assumptions as theorems to the tactic*)
308 fun forward_res2 nf hyps st =
311 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
314 | NONE => raise THM("forward_res2", 0, [st]);
316 (*Remove duplicates in P|Q by assuming ~P in Q
317 rls (initially []) accumulates assumptions of the form P==>False*)
318 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
319 handle THM _ => tryres(th,rls)
320 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
321 [disj_FalseD1, disj_FalseD2, asm_rl])
324 (*Remove duplicate literals, if there are any*)
326 if has_duplicates (op =) (literals (prop_of th))
327 then nodups_aux [] th
331 (**** Generation of contrapositives ****)
333 fun is_left (Const ("Trueprop", _) $
334 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _)) = true
337 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
339 if is_left (prop_of th) then assoc_right (th RS disj_assoc)
342 (*Must check for negative literal first!*)
343 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
345 (*For ordinary resolution. *)
346 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
348 (*Create a goal or support clause, conclusing False*)
349 fun make_goal th = (*Must check for negative literal first!*)
350 make_goal (tryres(th, clause_rules))
351 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
353 (*Sort clauses by number of literals*)
354 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
356 fun sort_clauses ths = sort (make_ord fewerlits) ths;
358 (*True if the given type contains bool anywhere*)
359 fun has_bool (Type("bool",_)) = true
360 | has_bool (Type(_, Ts)) = exists has_bool Ts
361 | has_bool _ = false;
363 (*Is the string the name of a connective? Really only | and Not can remain,
364 since this code expects to be called on a clause form.*)
365 val is_conn = member (op =)
366 ["Trueprop", "op &", "op |", "op -->", "Not",
367 "All", "Ex", "Ball", "Bex"];
369 (*True if the term contains a function--not a logical connective--where the type
370 of any argument contains bool.*)
371 val has_bool_arg_const =
373 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
375 (*A higher-order instance of a first-order constant? Example is the definition of
376 HOL.one, 1, at a function type in theory SetsAndFunctions.*)
377 fun higher_inst_const thy (c,T) =
378 case binder_types T of
379 [] => false (*not a function type, OK*)
380 | Ts => length (binder_types (Sign.the_const_type thy c)) <> length Ts;
382 (*Raises an exception if any Vars in the theorem mention type bool.
383 Also rejects functions whose arguments are Booleans or other functions.*)
384 fun is_fol_term thy t =
385 Term.is_first_order ["all","All","Ex"] t andalso
386 not (exists (has_bool o fastype_of) (term_vars t) orelse
387 has_bool_arg_const t orelse
388 exists_Const (higher_inst_const thy) t orelse
391 fun rigid t = not (is_Var (head_of t));
393 fun ok4horn (Const ("Trueprop",_) $ (Const ("op |", _) $ t $ _)) = rigid t
394 | ok4horn (Const ("Trueprop",_) $ t) = rigid t
397 (*Create a meta-level Horn clause*)
398 fun make_horn crules th =
399 if ok4horn (concl_of th)
400 then make_horn crules (tryres(th,crules)) handle THM _ => th
403 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
404 is a HOL disjunction.*)
405 fun add_contras crules (th,hcs) =
406 let fun rots (0,th) = hcs
407 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
408 rots(k-1, assoc_right (th RS disj_comm))
409 in case nliterals(prop_of th) of
411 | n => rots(n, assoc_right th)
414 (*Use "theorem naming" to label the clauses*)
415 fun name_thms label =
416 let fun name1 (th, (k,ths)) =
417 (k-1, PureThy.put_name_hint (label ^ string_of_int k) th :: ths)
418 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
420 (*Is the given disjunction an all-negative support clause?*)
421 fun is_negative th = forall (not o #1) (literals (prop_of th));
423 val neg_clauses = List.filter is_negative;
426 (***** MESON PROOF PROCEDURE *****)
428 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
429 As) = rhyps(phi, A::As)
430 | rhyps (_, As) = As;
432 (** Detecting repeated assumptions in a subgoal **)
434 (*The stringtree detects repeated assumptions.*)
435 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
437 (*detects repetitions in a list of terms*)
438 fun has_reps [] = false
439 | has_reps [_] = false
440 | has_reps [t,u] = (t aconv u)
441 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
442 handle Net.INSERT => true;
444 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
445 fun TRYING_eq_assume_tac 0 st = Seq.single st
446 | TRYING_eq_assume_tac i st =
447 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
448 handle THM _ => TRYING_eq_assume_tac (i-1) st;
450 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
452 (*Loop checking: FAIL if trying to prove the same thing twice
453 -- if *ANY* subgoal has repeated literals*)
455 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
456 then Seq.empty else Seq.single st;
459 (* net_resolve_tac actually made it slower... *)
460 fun prolog_step_tac horns i =
461 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
462 TRYALL_eq_assume_tac;
464 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
465 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
467 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
470 (*Negation Normal Form*)
471 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
472 not_impD, not_iffD, not_allD, not_exD, not_notD];
474 fun ok4nnf (Const ("Trueprop",_) $ (Const ("Not", _) $ t)) = rigid t
475 | ok4nnf (Const ("Trueprop",_) $ t) = rigid t
479 if ok4nnf (concl_of th)
480 then make_nnf1 (tryres(th, nnf_rls))
482 forward_res make_nnf1
483 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
487 (*The simplification removes defined quantifiers and occurrences of True and False.
488 nnf_ss also includes the one-point simprocs,
489 which are needed to avoid the various one-point theorems from generating junk clauses.*)
491 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
492 if_False, if_cancel, if_eq_cancel, cases_simp];
493 val nnf_extra_simps =
494 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
497 HOL_basic_ss addsimps nnf_extra_simps
498 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
500 fun make_nnf th = case prems_of th of
501 [] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
504 | _ => raise THM ("make_nnf: premises in argument", 0, [th]);
506 (*Pull existential quantifiers to front. This accomplishes Skolemization for
507 clauses that arise from a subgoal.*)
509 if not (has_conns ["Ex"] (prop_of th)) then th
510 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
511 disj_exD, disj_exD1, disj_exD2])))
513 skolemize (forward_res skolemize
514 (tryres (th, [conj_forward, disj_forward, all_forward])))
515 handle THM _ => forward_res skolemize (th RS ex_forward);
518 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
519 The resulting clauses are HOL disjunctions.*)
520 fun make_clauses ths =
521 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
523 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
526 (distinct Thm.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
528 (*Could simply use nprems_of, which would count remaining subgoals -- no
529 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
531 fun best_prolog_tac sizef horns =
532 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
534 fun depth_prolog_tac horns =
535 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
537 (*Return all negative clauses, as possible goal clauses*)
538 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
540 fun skolemize_prems_tac prems =
541 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
544 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
545 fun expand_defs_tac st =
546 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
547 in PRIMITIVE (LocalDefs.expand defs) st end;
549 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions.
550 Function mkcl converts theorems to clauses.*)
551 fun MESON mkcl cltac i st =
553 (EVERY [rtac ccontr 1,
555 EVERY1 [skolemize_prems_tac negs,
556 METAHYPS (cltac o mkcl)]) 1,
557 expand_defs_tac]) i st
558 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
560 (** Best-first search versions **)
562 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
563 fun best_meson_tac sizef =
566 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
567 (has_fewer_prems 1, sizef)
568 (prolog_step_tac (make_horns cls) 1));
570 (*First, breaks the goal into independent units*)
571 val safe_best_meson_tac =
572 SELECT_GOAL (TRY Safe_tac THEN
573 TRYALL (best_meson_tac size_of_subgoals));
575 (** Depth-first search version **)
577 val depth_meson_tac =
579 (fn cls => EVERY [resolve_tac (gocls cls) 1, depth_prolog_tac (make_horns cls)]);
582 (** Iterative deepening version **)
584 (*This version does only one inference per call;
585 having only one eq_assume_tac speeds it up!*)
586 fun prolog_step_tac' horns =
587 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
588 take_prefix Thm.no_prems horns
589 val nrtac = net_resolve_tac horns
590 in fn i => eq_assume_tac i ORELSE
591 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
592 ((assume_tac i APPEND nrtac i) THEN check_tac)
595 fun iter_deepen_prolog_tac horns =
596 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
598 fun iter_deepen_meson_tac ths = MESON make_clauses
600 case (gocls (cls@ths)) of
601 [] => no_tac (*no goal clauses*)
603 let val horns = make_horns (cls@ths)
605 Output.debug (fn () => ("meson method called:\n" ^
606 space_implode "\n" (map string_of_thm (cls@ths)) ^
608 space_implode "\n" (map string_of_thm horns)))
609 in THEN_ITER_DEEPEN (resolve_tac goes 1) (has_fewer_prems 1) (prolog_step_tac' horns)
613 fun meson_claset_tac ths cs =
614 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
616 val meson_tac = CLASET' (meson_claset_tac []);
619 (**** Code to support ordinary resolution, rather than Model Elimination ****)
621 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
622 with no contrapositives, for ordinary resolution.*)
624 (*Rules to convert the head literal into a negated assumption. If the head
625 literal is already negated, then using notEfalse instead of notEfalse'
626 prevents a double negation.*)
627 val notEfalse = read_instantiate [("R","False")] notE;
628 val notEfalse' = rotate_prems 1 notEfalse;
630 fun negated_asm_of_head th =
631 th RS notEfalse handle THM _ => th RS notEfalse';
633 (*Converting one clause*)
634 fun make_meta_clause th =
635 negated_asm_of_head (make_horn resolution_clause_rules th);
637 fun make_meta_clauses ths =
639 (distinct Thm.eq_thm_prop (map make_meta_clause ths));
641 (*Permute a rule's premises to move the i-th premise to the last position.*)
643 let val n = nprems_of th
644 in if 1 <= i andalso i <= n
645 then Thm.permute_prems (i-1) 1 th
646 else raise THM("select_literal", i, [th])
649 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
651 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
653 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
654 fun select_literal i cl = negate_head (make_last i cl);
657 (*Top-level Skolemization. Allows part of the conversion to clauses to be
658 expressed as a tactic (or Isar method). Each assumption of the selected
659 goal is converted to NNF and then its existential quantifiers are pulled
660 to the front. Finally, all existential quantifiers are eliminated,
661 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
662 might generate many subgoals.*)
664 fun skolemize_tac i st =
665 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
668 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
669 THEN REPEAT (etac exE 1))),
670 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
672 handle Subscript => Seq.empty;
676 structure BasicMeson: BASIC_MESON = Meson;