Skolemization now catches exception THM, which may be raised if unification fails.
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
13 val term_pair_of: indexname * (typ * 'a) -> term * 'a
14 val first_order_resolve: thm -> thm -> thm
15 val flexflex_first_order: thm -> thm
16 val size_of_subgoals: thm -> int
17 val too_many_clauses: term -> bool
18 val make_cnf: thm list -> thm -> Proof.context -> thm list * Proof.context
19 val finish_cnf: thm list -> thm list
20 val make_nnf: thm -> thm
21 val make_nnf1: thm -> thm
22 val skolemize: thm -> thm
23 val is_fol_term: theory -> term -> bool
24 val make_clauses: thm list -> thm list
25 val make_horns: thm list -> thm list
26 val best_prolog_tac: (thm -> int) -> thm list -> tactic
27 val depth_prolog_tac: thm list -> tactic
28 val gocls: thm list -> thm list
29 val skolemize_prems_tac: thm list -> int -> tactic
30 val MESON: (thm list -> thm list) -> (thm list -> tactic) -> int -> tactic
31 val best_meson_tac: (thm -> int) -> int -> tactic
32 val safe_best_meson_tac: int -> tactic
33 val depth_meson_tac: int -> tactic
34 val prolog_step_tac': thm list -> int -> tactic
35 val iter_deepen_prolog_tac: thm list -> tactic
36 val iter_deepen_meson_tac: thm list -> int -> tactic
37 val make_meta_clause: thm -> thm
38 val make_meta_clauses: thm list -> thm list
39 val meson_claset_tac: thm list -> claset -> int -> tactic
40 val meson_tac: int -> tactic
41 val negate_head: thm -> thm
42 val select_literal: int -> thm -> thm
43 val skolemize_tac: int -> tactic
46 structure Meson: MESON =
49 val not_conjD = thm "meson_not_conjD";
50 val not_disjD = thm "meson_not_disjD";
51 val not_notD = thm "meson_not_notD";
52 val not_allD = thm "meson_not_allD";
53 val not_exD = thm "meson_not_exD";
54 val imp_to_disjD = thm "meson_imp_to_disjD";
55 val not_impD = thm "meson_not_impD";
56 val iff_to_disjD = thm "meson_iff_to_disjD";
57 val not_iffD = thm "meson_not_iffD";
58 val conj_exD1 = thm "meson_conj_exD1";
59 val conj_exD2 = thm "meson_conj_exD2";
60 val disj_exD = thm "meson_disj_exD";
61 val disj_exD1 = thm "meson_disj_exD1";
62 val disj_exD2 = thm "meson_disj_exD2";
63 val disj_assoc = thm "meson_disj_assoc";
64 val disj_comm = thm "meson_disj_comm";
65 val disj_FalseD1 = thm "meson_disj_FalseD1";
66 val disj_FalseD2 = thm "meson_disj_FalseD2";
69 (**** Operators for forward proof ****)
72 (** First-order Resolution **)
74 fun typ_pair_of (ix, (sort,ty)) = (TVar (ix,sort), ty);
75 fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);
77 val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
79 (*FIXME: currently does not "rename variables apart"*)
80 fun first_order_resolve thA thB =
81 let val thy = theory_of_thm thA
82 val tmA = concl_of thA
83 val Const("==>",_) $ tmB $ _ = prop_of thB
84 val (tyenv,tenv) = Pattern.first_order_match thy (tmB,tmA) (tyenv0,tenv0)
85 val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
86 in thA RS (cterm_instantiate ct_pairs thB) end
87 handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
89 fun flexflex_first_order th =
90 case (tpairs_of th) of
93 let val thy = theory_of_thm th
95 foldl (uncurry (Pattern.first_order_match thy)) (tyenv0,tenv0) pairs
96 val t_pairs = map term_pair_of (Vartab.dest tenv)
97 val th' = Thm.instantiate ([], map (pairself (cterm_of thy)) t_pairs) th
101 (*Forward proof while preserving bound variables names*)
102 fun rename_bvs_RS th rl =
103 let val th' = th RS rl
104 in Thm.rename_boundvars (concl_of th') (concl_of th) th' end;
106 (*raises exception if no rules apply*)
107 fun tryres (th, rls) =
108 let fun tryall [] = raise THM("tryres", 0, th::rls)
109 | tryall (rl::rls) = (rename_bvs_RS th rl handle THM _ => tryall rls)
112 (*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
113 e.g. from conj_forward, should have the form
114 "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
115 and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
116 fun forward_res nf st =
117 let fun forward_tacf [prem] = rtac (nf prem) 1
118 | forward_tacf prems =
119 error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
122 cat_lines (map string_of_thm prems))
124 case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
126 | NONE => raise THM("forward_res", 0, [st])
129 (*Are any of the logical connectives in "bs" present in the term?*)
131 let fun has (Const(a,_)) = false
132 | has (Const("Trueprop",_) $ p) = has p
133 | has (Const("Not",_) $ p) = has p
134 | has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
135 | has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
136 | has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
137 | has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
142 (**** Clause handling ****)
144 fun literals (Const("Trueprop",_) $ P) = literals P
145 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
146 | literals (Const("Not",_) $ P) = [(false,P)]
147 | literals P = [(true,P)];
149 (*number of literals in a term*)
150 val nliterals = length o literals;
153 (*** Tautology Checking ***)
155 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
156 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
157 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
158 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
160 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
162 (*Literals like X=X are tautologous*)
163 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
164 | taut_poslit (Const("True",_)) = true
165 | taut_poslit _ = false;
168 let val (poslits,neglits) = signed_lits th
169 in exists taut_poslit poslits
171 exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
173 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
176 (*** To remove trivial negated equality literals from clauses ***)
178 (*They are typically functional reflexivity axioms and are the converses of
179 injectivity equivalences*)
181 val not_refl_disj_D = thm"meson_not_refl_disj_D";
183 (*Is either term a Var that does not properly occur in the other term?*)
184 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
185 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
186 | eliminable _ = false;
188 fun refl_clause_aux 0 th = th
189 | refl_clause_aux n th =
190 case HOLogic.dest_Trueprop (concl_of th) of
191 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
192 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
193 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
195 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
196 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
197 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
198 | _ => (*not a disjunction*) th;
200 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
201 notequal_lits_count P + notequal_lits_count Q
202 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
203 | notequal_lits_count _ = 0;
205 (*Simplify a clause by applying reflexivity to its negated equality literals*)
207 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
208 in zero_var_indexes (refl_clause_aux neqs th) end
209 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
212 (*** Removal of duplicate literals ***)
214 (*Forward proof, passing extra assumptions as theorems to the tactic*)
215 fun forward_res2 nf hyps st =
218 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
221 | NONE => raise THM("forward_res2", 0, [st]);
223 (*Remove duplicates in P|Q by assuming ~P in Q
224 rls (initially []) accumulates assumptions of the form P==>False*)
225 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
226 handle THM _ => tryres(th,rls)
227 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
228 [disj_FalseD1, disj_FalseD2, asm_rl])
231 (*Remove duplicate literals, if there are any*)
233 if has_duplicates (op =) (literals (prop_of th))
234 then nodups_aux [] th
238 (*** The basic CNF transformation ***)
240 val max_clauses = 40;
242 fun sum x y = if x < max_clauses andalso y < max_clauses then x+y else max_clauses;
243 fun prod x y = if x < max_clauses andalso y < max_clauses then x*y else max_clauses;
245 (*Estimate the number of clauses in order to detect infeasible theorems*)
246 fun signed_nclauses b (Const("Trueprop",_) $ t) = signed_nclauses b t
247 | signed_nclauses b (Const("Not",_) $ t) = signed_nclauses (not b) t
248 | signed_nclauses b (Const("op &",_) $ t $ u) =
249 if b then sum (signed_nclauses b t) (signed_nclauses b u)
250 else prod (signed_nclauses b t) (signed_nclauses b u)
251 | signed_nclauses b (Const("op |",_) $ t $ u) =
252 if b then prod (signed_nclauses b t) (signed_nclauses b u)
253 else sum (signed_nclauses b t) (signed_nclauses b u)
254 | signed_nclauses b (Const("op -->",_) $ t $ u) =
255 if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)
256 else sum (signed_nclauses (not b) t) (signed_nclauses b u)
257 | signed_nclauses b (Const("op =", Type ("fun", [T, _])) $ t $ u) =
258 if T = HOLogic.boolT then (*Boolean equality is if-and-only-if*)
259 if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))
260 (prod (signed_nclauses (not b) u) (signed_nclauses b t))
261 else sum (prod (signed_nclauses b t) (signed_nclauses b u))
262 (prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))
264 | signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
265 | signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
266 | signed_nclauses _ _ = 1; (* literal *)
268 val nclauses = signed_nclauses true;
270 fun too_many_clauses t = nclauses t >= max_clauses;
272 (*Replaces universally quantified variables by FREE variables -- because
273 assumptions may not contain scheme variables. Later, generalize using Variable.export. *)
275 val spec_var = Thm.dest_arg (Thm.dest_arg (#2 (Thm.dest_implies (Thm.cprop_of spec))));
276 val spec_varT = #T (Thm.rep_cterm spec_var);
277 fun name_of (Const ("All", _) $ Abs(x,_,_)) = x | name_of _ = Name.uu;
279 fun freeze_spec th ctxt =
281 val cert = Thm.cterm_of (ProofContext.theory_of ctxt);
282 val ([x], ctxt') = Variable.variant_fixes [name_of (HOLogic.dest_Trueprop (concl_of th))] ctxt;
283 val spec' = Thm.instantiate ([], [(spec_var, cert (Free (x, spec_varT)))]) spec;
284 in (th RS spec', ctxt') end
287 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
288 and then normalized via function nf. The normal form is given to resolve_tac,
289 instantiate a Boolean variable created by resolution with disj_forward. Since
290 (nf prem) returns a LIST of theorems, we can backtrack to get all combinations.*)
291 fun resop nf [prem] = resolve_tac (nf prem) 1;
293 (*Any need to extend this list with
294 "HOL.type_class","HOL.eq_class","ProtoPure.term"?*)
296 exists_Const (member (op =) ["==", "==>", "all", "prop"] o #1);
298 fun apply_skolem_ths (th, rls) =
299 let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
300 | tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
303 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
304 Strips universal quantifiers and breaks up conjunctions.
305 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
306 fun cnf skoths ctxt (th,ths) =
307 let val ctxtr = ref ctxt
308 fun cnf_aux (th,ths) =
309 if not (can HOLogic.dest_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
310 else if not (has_conns ["All","Ex","op &"] (prop_of th))
311 then nodups th :: ths (*no work to do, terminate*)
312 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
313 Const ("op &", _) => (*conjunction*)
314 cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
315 | Const ("All", _) => (*universal quantifier*)
316 let val (th',ctxt') = freeze_spec th (!ctxtr)
317 in ctxtr := ctxt'; cnf_aux (th', ths) end
319 (*existential quantifier: Insert Skolem functions*)
320 cnf_aux (apply_skolem_ths (th,skoths), ths)
321 | Const ("op |", _) =>
322 (*Disjunction of P, Q: Create new goal of proving ?P | ?Q and solve it using
323 all combinations of converting P, Q to CNF.*)
325 (METAHYPS (resop cnf_nil) 1) THEN
326 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
327 in Seq.list_of (tac (th RS disj_forward)) @ ths end
328 | _ => nodups th :: ths (*no work to do*)
329 and cnf_nil th = cnf_aux (th,[])
331 if too_many_clauses (concl_of th)
332 then (Output.debug (fn () => ("cnf is ignoring: " ^ string_of_thm th)); ths)
333 else cnf_aux (th,ths)
334 in (cls, !ctxtr) end;
336 fun make_cnf skoths th ctxt = cnf skoths ctxt (th, []);
338 (*Generalization, removal of redundant equalities, removal of tautologies.*)
339 fun finish_cnf ths = filter (not o is_taut) (map refl_clause ths);
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 (*Returns false 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, @{simproc neq}, @{simproc let_simp}];
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 (rename_bvs_RS th ex_forward);
528 fun skolemize_nnf_list [] = []
529 | skolemize_nnf_list (th::ths) =
530 skolemize (make_nnf th) :: skolemize_nnf_list ths
532 (warning ("Failed to Skolemize " ^ string_of_thm th);
533 skolemize_nnf_list ths);
535 fun add_clauses (th,cls) =
536 let val ctxt0 = Variable.thm_context th
537 val (cnfs,ctxt) = make_cnf [] th ctxt0
538 in Variable.export ctxt ctxt0 cnfs @ cls end;
540 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
541 The resulting clauses are HOL disjunctions.*)
542 fun make_clauses ths = sort_clauses (foldr add_clauses [] ths);
544 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
547 (distinct Thm.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
549 (*Could simply use nprems_of, which would count remaining subgoals -- no
550 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
552 fun best_prolog_tac sizef horns =
553 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
555 fun depth_prolog_tac horns =
556 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
558 (*Return all negative clauses, as possible goal clauses*)
559 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
561 fun skolemize_prems_tac prems =
562 cut_facts_tac (skolemize_nnf_list prems) THEN'
565 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions.
566 Function mkcl converts theorems to clauses.*)
567 fun MESON mkcl cltac i st =
569 (EVERY [ObjectLogic.atomize_prems_tac 1,
572 EVERY1 [skolemize_prems_tac negs,
573 METAHYPS (cltac o mkcl)]) 1]) i st
574 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
576 (** Best-first search versions **)
578 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
579 fun best_meson_tac sizef =
582 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
583 (has_fewer_prems 1, sizef)
584 (prolog_step_tac (make_horns cls) 1));
586 (*First, breaks the goal into independent units*)
587 val safe_best_meson_tac =
588 SELECT_GOAL (TRY (CLASET safe_tac) THEN
589 TRYALL (best_meson_tac size_of_subgoals));
591 (** Depth-first search version **)
593 val depth_meson_tac =
595 (fn cls => EVERY [resolve_tac (gocls cls) 1, depth_prolog_tac (make_horns cls)]);
598 (** Iterative deepening version **)
600 (*This version does only one inference per call;
601 having only one eq_assume_tac speeds it up!*)
602 fun prolog_step_tac' horns =
603 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
604 take_prefix Thm.no_prems horns
605 val nrtac = net_resolve_tac horns
606 in fn i => eq_assume_tac i ORELSE
607 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
608 ((assume_tac i APPEND nrtac i) THEN check_tac)
611 fun iter_deepen_prolog_tac horns =
612 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
614 fun iter_deepen_meson_tac ths = MESON make_clauses
616 case (gocls (cls@ths)) of
617 [] => no_tac (*no goal clauses*)
619 let val horns = make_horns (cls@ths)
621 Output.debug (fn () => ("meson method called:\n" ^
622 space_implode "\n" (map string_of_thm (cls@ths)) ^
624 space_implode "\n" (map string_of_thm horns)))
625 in THEN_ITER_DEEPEN (resolve_tac goes 1) (has_fewer_prems 1) (prolog_step_tac' horns)
629 fun meson_claset_tac ths cs =
630 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
632 val meson_tac = CLASET' (meson_claset_tac []);
635 (**** Code to support ordinary resolution, rather than Model Elimination ****)
637 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
638 with no contrapositives, for ordinary resolution.*)
640 (*Rules to convert the head literal into a negated assumption. If the head
641 literal is already negated, then using notEfalse instead of notEfalse'
642 prevents a double negation.*)
643 val notEfalse = read_instantiate [("R","False")] notE;
644 val notEfalse' = rotate_prems 1 notEfalse;
646 fun negated_asm_of_head th =
647 th RS notEfalse handle THM _ => th RS notEfalse';
649 (*Converting one clause*)
650 val make_meta_clause =
651 zero_var_indexes o negated_asm_of_head o make_horn resolution_clause_rules;
653 fun make_meta_clauses ths =
655 (distinct Thm.eq_thm_prop (map make_meta_clause ths));
657 (*Permute a rule's premises to move the i-th premise to the last position.*)
659 let val n = nprems_of th
660 in if 1 <= i andalso i <= n
661 then Thm.permute_prems (i-1) 1 th
662 else raise THM("select_literal", i, [th])
665 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
667 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
669 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
670 fun select_literal i cl = negate_head (make_last i cl);
673 (*Top-level Skolemization. Allows part of the conversion to clauses to be
674 expressed as a tactic (or Isar method). Each assumption of the selected
675 goal is converted to NNF and then its existential quantifiers are pulled
676 to the front. Finally, all existential quantifiers are eliminated,
677 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
678 might generate many subgoals.*)
680 fun skolemize_tac i st =
681 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
684 (fn hyps => (cut_facts_tac (skolemize_nnf_list hyps) 1
685 THEN REPEAT (etac exE 1))),
686 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
688 handle Subscript => Seq.empty;