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.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,_)) = member (op =) bs a
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 (member (op aconv) neglits) (HOLogic.false_const :: poslits)
127 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
130 (*** To remove trivial negated equality literals from clauses ***)
132 (*They are typically functional reflexivity axioms and are the converses of
133 injectivity equivalences*)
135 val not_refl_disj_D = thm"meson_not_refl_disj_D";
137 (*Is either term a Var that does not properly occur in the other term?*)
138 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
139 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
140 | eliminable _ = false;
142 fun refl_clause_aux 0 th = th
143 | refl_clause_aux n th =
144 case HOLogic.dest_Trueprop (concl_of th) of
145 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
146 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
147 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
149 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
150 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
151 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
152 | _ => (*not a disjunction*) th;
154 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
155 notequal_lits_count P + notequal_lits_count Q
156 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
157 | notequal_lits_count _ = 0;
159 (*Simplify a clause by applying reflexivity to its negated equality literals*)
161 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
162 in zero_var_indexes (refl_clause_aux neqs th) end
163 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
166 (*** The basic CNF transformation ***)
168 (*Estimate the number of clauses in order to detect infeasible theorems*)
169 fun nclauses (Const("Trueprop",_) $ t) = nclauses t
170 | nclauses (Const("op &",_) $ t $ u) = nclauses t + nclauses u
171 | nclauses (Const("Ex", _) $ Abs (_,_,t)) = nclauses t
172 | nclauses (Const("All",_) $ Abs (_,_,t)) = nclauses t
173 | nclauses (Const("op |",_) $ t $ u) = nclauses t * nclauses u
174 | nclauses _ = 1; (* literal *)
176 (*Replaces universally quantified variables by FREE variables -- because
177 assumptions may not contain scheme variables. Later, call "generalize". *)
179 let val newname = gensym "A_"
180 val spec' = read_instantiate [("x", newname)] spec
183 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
184 and then normalized via function nf. The normal form is given to resolve_tac,
185 presumably to instantiate a Boolean variable.*)
186 fun resop nf [prem] = resolve_tac (nf prem) 1;
189 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
191 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
192 Strips universal quantifiers and breaks up conjunctions.
193 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
194 fun cnf skoths (th,ths) =
195 let fun cnf_aux (th,ths) =
196 if has_meta_conn (prop_of th) then ths (*meta-level: ignore*)
197 else if not (has_consts ["All","Ex","op &"] (prop_of th))
198 then th::ths (*no work to do, terminate*)
199 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
200 Const ("op &", _) => (*conjunction*)
201 cnf_aux (th RS conjunct1,
202 cnf_aux (th RS conjunct2, ths))
203 | Const ("All", _) => (*universal quantifier*)
204 cnf_aux (freeze_spec th, ths)
206 (*existential quantifier: Insert Skolem functions*)
207 cnf_aux (tryres (th,skoths), ths)
208 | Const ("op |", _) => (*disjunction*)
210 (METAHYPS (resop cnf_nil) 1) THEN
211 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
212 in Seq.list_of (tac (th RS disj_forward)) @ ths end
213 | _ => (*no work to do*) th::ths
214 and cnf_nil th = cnf_aux (th,[])
216 if nclauses (concl_of th) > 20
217 then (Output.debug ("cnf is ignoring: " ^ string_of_thm th); ths)
218 else cnf_aux (th,ths)
221 (*Convert all suitable free variables to schematic variables,
222 but don't discharge assumptions.*)
223 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
225 fun make_cnf skoths th =
226 filter (not o is_taut)
227 (map (refl_clause o generalize) (cnf skoths (th, [])));
230 (**** Removal of duplicate literals ****)
232 (*Forward proof, passing extra assumptions as theorems to the tactic*)
233 fun forward_res2 nf hyps st =
236 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
239 | NONE => raise THM("forward_res2", 0, [st]);
241 (*Remove duplicates in P|Q by assuming ~P in Q
242 rls (initially []) accumulates assumptions of the form P==>False*)
243 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
244 handle THM _ => tryres(th,rls)
245 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
246 [disj_FalseD1, disj_FalseD2, asm_rl])
249 (*Remove duplicate literals, if there are any*)
251 if null(findrep(literals(prop_of th))) then th
252 else nodups_aux [] th;
255 (**** Generation of contrapositives ****)
257 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
258 fun assoc_right th = assoc_right (th RS disj_assoc)
261 (*Must check for negative literal first!*)
262 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
264 (*For ordinary resolution. *)
265 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
267 (*Create a goal or support clause, conclusing False*)
268 fun make_goal th = (*Must check for negative literal first!*)
269 make_goal (tryres(th, clause_rules))
270 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
272 (*Sort clauses by number of literals*)
273 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
275 fun sort_clauses ths = sort (make_ord fewerlits) ths;
277 (*True if the given type contains bool anywhere*)
278 fun has_bool (Type("bool",_)) = true
279 | has_bool (Type(_, Ts)) = exists has_bool Ts
280 | has_bool _ = false;
282 (*Is the string the name of a connective? It doesn't matter if this list is
283 incomplete, since when actually called, the only connectives likely to
284 remain are & | Not.*)
285 val is_conn = member (op =)
286 ["Trueprop", "op &", "op |", "op -->", "op =", "Not",
287 "All", "Ex", "Ball", "Bex"];
289 (*True if the term contains a function where the type of any argument contains
291 val has_bool_arg_const =
293 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
295 (*Raises an exception if any Vars in the theorem mention type bool; they
296 could cause make_horn to loop! Also rejects functions whose arguments are
297 Booleans or other functions.*)
299 not (exists (has_bool o fastype_of) (term_vars t) orelse
300 not (Term.is_first_order ["all","All","Ex"] t) orelse
301 has_bool_arg_const t orelse
304 (*FIXME: replace this by the boolean-valued version above!*)
305 fun check_is_fol_term t =
306 if is_fol_term t then t else raise TERM("check_is_fol_term",[t]);
308 fun check_is_fol th = (check_is_fol_term (prop_of th); th);
311 (*Create a meta-level Horn clause*)
312 fun make_horn crules th = make_horn crules (tryres(th,crules))
315 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
316 is a HOL disjunction.*)
317 fun add_contras crules (th,hcs) =
318 let fun rots (0,th) = hcs
319 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
320 rots(k-1, assoc_right (th RS disj_comm))
321 in case nliterals(prop_of th) of
323 | n => rots(n, assoc_right th)
326 (*Use "theorem naming" to label the clauses*)
327 fun name_thms label =
328 let fun name1 (th, (k,ths)) =
329 (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
331 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
333 (*Is the given disjunction an all-negative support clause?*)
334 fun is_negative th = forall (not o #1) (literals (prop_of th));
336 val neg_clauses = List.filter is_negative;
339 (***** MESON PROOF PROCEDURE *****)
341 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
342 As) = rhyps(phi, A::As)
343 | rhyps (_, As) = As;
345 (** Detecting repeated assumptions in a subgoal **)
347 (*The stringtree detects repeated assumptions.*)
348 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
350 (*detects repetitions in a list of terms*)
351 fun has_reps [] = false
352 | has_reps [_] = false
353 | has_reps [t,u] = (t aconv u)
354 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
355 handle Net.INSERT => true;
357 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
358 fun TRYING_eq_assume_tac 0 st = Seq.single st
359 | TRYING_eq_assume_tac i st =
360 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
361 handle THM _ => TRYING_eq_assume_tac (i-1) st;
363 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
365 (*Loop checking: FAIL if trying to prove the same thing twice
366 -- if *ANY* subgoal has repeated literals*)
368 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
369 then Seq.empty else Seq.single st;
372 (* net_resolve_tac actually made it slower... *)
373 fun prolog_step_tac horns i =
374 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
375 TRYALL_eq_assume_tac;
377 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
378 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
380 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
383 (*Negation Normal Form*)
384 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
385 not_impD, not_iffD, not_allD, not_exD, not_notD];
387 fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
389 forward_res make_nnf1
390 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
393 (*The simplification removes defined quantifiers and occurrences of True and False.
394 nnf_ss also includes the one-point simprocs,
395 which are needed to avoid the various one-point theorems from generating junk clauses.*)
397 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
398 if_False, if_cancel, if_eq_cancel, cases_simp];
399 val nnf_extra_simps =
400 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
403 HOL_basic_ss addsimps nnf_extra_simps
404 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
406 fun make_nnf th = th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
407 |> simplify nnf_ss (*But this doesn't simplify premises...*)
410 (*Pull existential quantifiers to front. This accomplishes Skolemization for
411 clauses that arise from a subgoal.*)
413 if not (has_consts ["Ex"] (prop_of th)) then th
414 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
415 disj_exD, disj_exD1, disj_exD2])))
417 skolemize (forward_res skolemize
418 (tryres (th, [conj_forward, disj_forward, all_forward])))
419 handle THM _ => forward_res skolemize (th RS ex_forward);
422 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
423 The resulting clauses are HOL disjunctions.*)
424 fun make_clauses ths =
425 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
428 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
431 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
433 (*Could simply use nprems_of, which would count remaining subgoals -- no
434 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
436 fun best_prolog_tac sizef horns =
437 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
439 fun depth_prolog_tac horns =
440 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
442 (*Return all negative clauses, as possible goal clauses*)
443 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
445 fun skolemize_prems_tac prems =
446 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
449 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
450 fun expand_defs_tac st =
451 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
452 in LocalDefs.def_export false defs st end;
454 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
455 fun MESON cltac i st =
457 (EVERY [rtac ccontr 1,
459 EVERY1 [skolemize_prems_tac negs,
460 METAHYPS (cltac o make_clauses)]) 1,
461 expand_defs_tac]) i st
462 handle TERM _ => no_tac st; (*probably from check_is_fol*)
464 (** Best-first search versions **)
466 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
467 fun best_meson_tac sizef =
469 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
470 (has_fewer_prems 1, sizef)
471 (prolog_step_tac (make_horns cls) 1));
473 (*First, breaks the goal into independent units*)
474 val safe_best_meson_tac =
475 SELECT_GOAL (TRY Safe_tac THEN
476 TRYALL (best_meson_tac size_of_subgoals));
478 (** Depth-first search version **)
480 val depth_meson_tac =
481 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
482 depth_prolog_tac (make_horns cls)]);
485 (** Iterative deepening version **)
487 (*This version does only one inference per call;
488 having only one eq_assume_tac speeds it up!*)
489 fun prolog_step_tac' horns =
490 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
491 take_prefix Thm.no_prems horns
492 val nrtac = net_resolve_tac horns
493 in fn i => eq_assume_tac i ORELSE
494 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
495 ((assume_tac i APPEND nrtac i) THEN check_tac)
498 fun iter_deepen_prolog_tac horns =
499 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
501 fun iter_deepen_meson_tac ths =
503 case (gocls (cls@ths)) of
504 [] => no_tac (*no goal clauses*)
506 (THEN_ITER_DEEPEN (resolve_tac goes 1)
508 (prolog_step_tac' (make_horns (cls@ths)))));
510 fun meson_claset_tac ths cs =
511 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
513 val meson_tac = CLASET' (meson_claset_tac []);
516 (**** Code to support ordinary resolution, rather than Model Elimination ****)
518 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
519 with no contrapositives, for ordinary resolution.*)
521 (*Rules to convert the head literal into a negated assumption. If the head
522 literal is already negated, then using notEfalse instead of notEfalse'
523 prevents a double negation.*)
524 val notEfalse = read_instantiate [("R","False")] notE;
525 val notEfalse' = rotate_prems 1 notEfalse;
527 fun negated_asm_of_head th =
528 th RS notEfalse handle THM _ => th RS notEfalse';
530 (*Converting one clause*)
531 fun make_meta_clause th =
532 negated_asm_of_head (make_horn resolution_clause_rules (check_is_fol th));
534 fun make_meta_clauses ths =
536 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
538 (*Permute a rule's premises to move the i-th premise to the last position.*)
540 let val n = nprems_of th
541 in if 1 <= i andalso i <= n
542 then Thm.permute_prems (i-1) 1 th
543 else raise THM("select_literal", i, [th])
546 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
548 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
550 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
551 fun select_literal i cl = negate_head (make_last i cl);
554 (*Top-level Skolemization. Allows part of the conversion to clauses to be
555 expressed as a tactic (or Isar method). Each assumption of the selected
556 goal is converted to NNF and then its existential quantifiers are pulled
557 to the front. Finally, all existential quantifiers are eliminated,
558 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
559 might generate many subgoals.*)
561 fun skolemize_tac i st =
562 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
565 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
566 THEN REPEAT (etac exE 1))),
567 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
569 handle Subscript => Seq.empty;
571 (*Top-level conversion to meta-level clauses. Each clause has
572 leading !!-bound universal variables, to express generality. To get
573 disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
574 val make_clauses_tac =
577 let val ts = Logic.strip_assums_hyp prop
582 (map forall_intr_vars
583 (make_meta_clauses (make_clauses hyps))) 1)),
584 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
588 (*** setup the special skoklemization methods ***)
590 (*No CHANGED_PROP here, since these always appear in the preamble*)
592 val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
594 val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
596 val skolemize_setup =
598 [("skolemize", Method.no_args skolemize_meth,
599 "Skolemization into existential quantifiers"),
600 ("make_clauses", Method.no_args make_clauses_meth,
601 "Conversion to !!-quantified meta-level clauses")];
605 structure BasicMeson: BASIC_MESON = Meson;