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 -> 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
39 val make_clauses_tac : int -> tactic
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 ****)
70 (** First-order Resolution **)
72 fun typ_pair_of (ix, (sort,ty)) = (TVar (ix,sort), ty);
73 fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);
75 val Envir.Envir {asol = tenv0, iTs = tyenv0, ...} = Envir.empty 0
77 (*FIXME: currently does not "rename variables apart"*)
78 fun first_order_resolve thA thB =
79 let val thy = theory_of_thm thA
80 val tmA = concl_of thA
81 fun match pat = Pattern.first_order_match thy (pat,tmA) (tyenv0,tenv0)
82 val Const("==>",_) $ tmB $ _ = prop_of thB
83 val (tyenv,tenv) = match tmB
84 val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)
85 in thA RS (cterm_instantiate ct_pairs thB) end
86 handle _ => raise THM ("first_order_resolve", 0, [thA,thB]);
88 (*raises exception if no rules apply -- unlike RL*)
89 fun tryres (th, rls) =
90 let fun tryall [] = raise THM("tryres", 0, th::rls)
91 | tryall (rl::rls) = (th RS rl handle THM _ => tryall rls)
94 (*Permits forward proof from rules that discharge assumptions. The supplied proof state st,
95 e.g. from conj_forward, should have the form
96 "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"
97 and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)
98 fun forward_res nf st =
99 let fun forward_tacf [prem] = rtac (nf prem) 1
100 | forward_tacf prems =
101 error ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:\n" ^
104 cat_lines (map string_of_thm prems))
106 case Seq.pull (ALLGOALS (METAHYPS forward_tacf) st)
108 | NONE => raise THM("forward_res", 0, [st])
111 (*Are any of the logical connectives in "bs" present in the term?*)
113 let fun has (Const(a,_)) = false
114 | has (Const("Trueprop",_) $ p) = has p
115 | has (Const("Not",_) $ p) = has p
116 | has (Const("op |",_) $ p $ q) = member (op =) bs "op |" orelse has p orelse has q
117 | has (Const("op &",_) $ p $ q) = member (op =) bs "op &" orelse has p orelse has q
118 | has (Const("All",_) $ Abs(_,_,p)) = member (op =) bs "All" orelse has p
119 | has (Const("Ex",_) $ Abs(_,_,p)) = member (op =) bs "Ex" orelse has p
124 (**** Clause handling ****)
126 fun literals (Const("Trueprop",_) $ P) = literals P
127 | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
128 | literals (Const("Not",_) $ P) = [(false,P)]
129 | literals P = [(true,P)];
131 (*number of literals in a term*)
132 val nliterals = length o literals;
135 (*** Tautology Checking ***)
137 fun signed_lits_aux (Const ("op |", _) $ P $ Q) (poslits, neglits) =
138 signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
139 | signed_lits_aux (Const("Not",_) $ P) (poslits, neglits) = (poslits, P::neglits)
140 | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);
142 fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);
144 (*Literals like X=X are tautologous*)
145 fun taut_poslit (Const("op =",_) $ t $ u) = t aconv u
146 | taut_poslit (Const("True",_)) = true
147 | taut_poslit _ = false;
150 let val (poslits,neglits) = signed_lits th
151 in exists taut_poslit poslits
153 exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)
155 handle TERM _ => false; (*probably dest_Trueprop on a weird theorem*)
158 (*** To remove trivial negated equality literals from clauses ***)
160 (*They are typically functional reflexivity axioms and are the converses of
161 injectivity equivalences*)
163 val not_refl_disj_D = thm"meson_not_refl_disj_D";
165 (*Is either term a Var that does not properly occur in the other term?*)
166 fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))
167 | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))
168 | eliminable _ = false;
170 fun refl_clause_aux 0 th = th
171 | refl_clause_aux n th =
172 case HOLogic.dest_Trueprop (concl_of th) of
173 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _) =>
174 refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
175 | (Const ("op |", _) $ (Const("Not",_) $ (Const("op =",_) $ t $ u)) $ _) =>
177 then refl_clause_aux (n-1) (th RS not_refl_disj_D) (*Var inequation: delete*)
178 else refl_clause_aux (n-1) (th RS disj_comm) (*not between Vars: ignore*)
179 | (Const ("op |", _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
180 | _ => (*not a disjunction*) th;
182 fun notequal_lits_count (Const ("op |", _) $ P $ Q) =
183 notequal_lits_count P + notequal_lits_count Q
184 | notequal_lits_count (Const("Not",_) $ (Const("op =",_) $ _ $ _)) = 1
185 | notequal_lits_count _ = 0;
187 (*Simplify a clause by applying reflexivity to its negated equality literals*)
189 let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))
190 in zero_var_indexes (refl_clause_aux neqs th) end
191 handle TERM _ => th; (*probably dest_Trueprop on a weird theorem*)
194 (*** The basic CNF transformation ***)
196 val max_clauses = ref 40;
198 fun sum x y = if x < !max_clauses andalso y < !max_clauses then x+y else !max_clauses;
199 fun prod x y = if x < !max_clauses andalso y < !max_clauses then x*y else !max_clauses;
201 (*Estimate the number of clauses in order to detect infeasible theorems*)
202 fun signed_nclauses b (Const("Trueprop",_) $ t) = signed_nclauses b t
203 | signed_nclauses b (Const("Not",_) $ t) = signed_nclauses (not b) t
204 | signed_nclauses b (Const("op &",_) $ t $ u) =
205 if b then sum (signed_nclauses b t) (signed_nclauses b u)
206 else prod (signed_nclauses b t) (signed_nclauses b u)
207 | signed_nclauses b (Const("op |",_) $ t $ u) =
208 if b then prod (signed_nclauses b t) (signed_nclauses b u)
209 else sum (signed_nclauses b t) (signed_nclauses b u)
210 | signed_nclauses b (Const("op -->",_) $ t $ u) =
211 if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)
212 else sum (signed_nclauses (not b) t) (signed_nclauses b u)
213 | signed_nclauses b (Const("op =", Type ("fun", [T, _])) $ t $ u) =
214 if T = HOLogic.boolT then (*Boolean equality is if-and-only-if*)
215 if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))
216 (prod (signed_nclauses (not b) u) (signed_nclauses b t))
217 else sum (prod (signed_nclauses b t) (signed_nclauses b u))
218 (prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))
220 | signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
221 | signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
222 | signed_nclauses _ _ = 1; (* literal *)
224 val nclauses = signed_nclauses true;
226 fun too_many_clauses t = nclauses t >= !max_clauses;
228 (*Replaces universally quantified variables by FREE variables -- because
229 assumptions may not contain scheme variables. Later, call "generalize". *)
231 let val newname = gensym "mes_"
232 val spec' = read_instantiate [("x", newname)] spec
235 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
236 and then normalized via function nf. The normal form is given to resolve_tac,
237 presumably to instantiate a Boolean variable.*)
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 |", _) => (*disjunction*)
268 (METAHYPS (resop cnf_nil) 1) THEN
269 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
270 in Seq.list_of (tac (th RS disj_forward)) @ ths end
271 | _ => (*no work to do*) th::ths
272 and cnf_nil th = cnf_aux (th,[])
274 if too_many_clauses (concl_of th)
275 then (Output.debug ("cnf is ignoring: " ^ string_of_thm th); ths)
276 else cnf_aux (th,ths)
279 (*Convert all suitable free variables to schematic variables,
280 but don't discharge assumptions.*)
281 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
283 fun make_cnf skoths th = cnf skoths (th, []);
285 (*Generalization, removal of redundant equalities, removal of tautologies.*)
286 fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
289 (**** Removal of duplicate literals ****)
291 (*Forward proof, passing extra assumptions as theorems to the tactic*)
292 fun forward_res2 nf hyps st =
295 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
298 | NONE => raise THM("forward_res2", 0, [st]);
300 (*Remove duplicates in P|Q by assuming ~P in Q
301 rls (initially []) accumulates assumptions of the form P==>False*)
302 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
303 handle THM _ => tryres(th,rls)
304 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
305 [disj_FalseD1, disj_FalseD2, asm_rl])
308 (*Remove duplicate literals, if there are any*)
310 if has_duplicates (op =) (literals (prop_of th))
311 then nodups_aux [] th
315 (**** Generation of contrapositives ****)
317 fun is_left (Const ("Trueprop", _) $
318 (Const ("op |", _) $ (Const ("op |", _) $ _ $ _) $ _)) = true
321 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
323 if is_left (prop_of th) then assoc_right (th RS disj_assoc)
326 (*Must check for negative literal first!*)
327 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
329 (*For ordinary resolution. *)
330 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
332 (*Create a goal or support clause, conclusing False*)
333 fun make_goal th = (*Must check for negative literal first!*)
334 make_goal (tryres(th, clause_rules))
335 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
337 (*Sort clauses by number of literals*)
338 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
340 fun sort_clauses ths = sort (make_ord fewerlits) ths;
342 (*True if the given type contains bool anywhere*)
343 fun has_bool (Type("bool",_)) = true
344 | has_bool (Type(_, Ts)) = exists has_bool Ts
345 | has_bool _ = false;
347 (*Is the string the name of a connective? Really only | and Not can remain,
348 since this code expects to be called on a clause form.*)
349 val is_conn = member (op =)
350 ["Trueprop", "op &", "op |", "op -->", "Not",
351 "All", "Ex", "Ball", "Bex"];
353 (*True if the term contains a function--not a logical connective--where the type
354 of any argument contains bool.*)
355 val has_bool_arg_const =
357 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
359 (*Raises an exception if any Vars in the theorem mention type bool.
360 Also rejects functions whose arguments are Booleans or other functions.*)
362 not (exists (has_bool o fastype_of) (term_vars t) orelse
363 not (Term.is_first_order ["all","All","Ex"] t) orelse
364 has_bool_arg_const t orelse
367 fun rigid t = not (is_Var (head_of t));
369 fun ok4horn (Const ("Trueprop",_) $ (Const ("op |", _) $ t $ _)) = rigid t
370 | ok4horn (Const ("Trueprop",_) $ t) = rigid t
373 (*Create a meta-level Horn clause*)
374 fun make_horn crules th =
375 if ok4horn (concl_of th)
376 then make_horn crules (tryres(th,crules)) handle THM _ => th
379 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
380 is a HOL disjunction.*)
381 fun add_contras crules (th,hcs) =
382 let fun rots (0,th) = hcs
383 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
384 rots(k-1, assoc_right (th RS disj_comm))
385 in case nliterals(prop_of th) of
387 | n => rots(n, assoc_right th)
390 (*Use "theorem naming" to label the clauses*)
391 fun name_thms label =
392 let fun name1 (th, (k,ths)) =
393 (k-1, PureThy.put_name_hint (label ^ string_of_int k) th :: ths)
394 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
396 (*Is the given disjunction an all-negative support clause?*)
397 fun is_negative th = forall (not o #1) (literals (prop_of th));
399 val neg_clauses = List.filter is_negative;
402 (***** MESON PROOF PROCEDURE *****)
404 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
405 As) = rhyps(phi, A::As)
406 | rhyps (_, As) = As;
408 (** Detecting repeated assumptions in a subgoal **)
410 (*The stringtree detects repeated assumptions.*)
411 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
413 (*detects repetitions in a list of terms*)
414 fun has_reps [] = false
415 | has_reps [_] = false
416 | has_reps [t,u] = (t aconv u)
417 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
418 handle Net.INSERT => true;
420 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
421 fun TRYING_eq_assume_tac 0 st = Seq.single st
422 | TRYING_eq_assume_tac i st =
423 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
424 handle THM _ => TRYING_eq_assume_tac (i-1) st;
426 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
428 (*Loop checking: FAIL if trying to prove the same thing twice
429 -- if *ANY* subgoal has repeated literals*)
431 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
432 then Seq.empty else Seq.single st;
435 (* net_resolve_tac actually made it slower... *)
436 fun prolog_step_tac horns i =
437 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
438 TRYALL_eq_assume_tac;
440 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
441 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
443 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
446 (*Negation Normal Form*)
447 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
448 not_impD, not_iffD, not_allD, not_exD, not_notD];
450 fun ok4nnf (Const ("Trueprop",_) $ (Const ("Not", _) $ t)) = rigid t
451 | ok4nnf (Const ("Trueprop",_) $ t) = rigid t
455 if ok4nnf (concl_of th)
456 then make_nnf1 (tryres(th, nnf_rls))
458 forward_res make_nnf1
459 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
463 (*The simplification removes defined quantifiers and occurrences of True and False.
464 nnf_ss also includes the one-point simprocs,
465 which are needed to avoid the various one-point theorems from generating junk clauses.*)
467 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
468 if_False, if_cancel, if_eq_cancel, cases_simp];
469 val nnf_extra_simps =
470 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
473 HOL_basic_ss addsimps nnf_extra_simps
474 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
476 fun make_nnf th = case prems_of th of
477 [] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
480 | _ => raise THM ("make_nnf: premises in argument", 0, [th]);
482 (*Pull existential quantifiers to front. This accomplishes Skolemization for
483 clauses that arise from a subgoal.*)
485 if not (has_conns ["Ex"] (prop_of th)) then th
486 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
487 disj_exD, disj_exD1, disj_exD2])))
489 skolemize (forward_res skolemize
490 (tryres (th, [conj_forward, disj_forward, all_forward])))
491 handle THM _ => forward_res skolemize (th RS ex_forward);
494 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
495 The resulting clauses are HOL disjunctions.*)
496 fun make_clauses ths =
497 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
499 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
502 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
504 (*Could simply use nprems_of, which would count remaining subgoals -- no
505 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
507 fun best_prolog_tac sizef horns =
508 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
510 fun depth_prolog_tac horns =
511 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
513 (*Return all negative clauses, as possible goal clauses*)
514 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
516 fun skolemize_prems_tac prems =
517 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
520 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
521 fun expand_defs_tac st =
522 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
523 in PRIMITIVE (LocalDefs.expand defs) st end;
525 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
526 fun MESON cltac i st =
528 (EVERY [rtac ccontr 1,
530 EVERY1 [skolemize_prems_tac negs,
531 METAHYPS (cltac o make_clauses)]) 1,
532 expand_defs_tac]) i st
533 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
535 (** Best-first search versions **)
537 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
538 fun best_meson_tac sizef =
540 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
541 (has_fewer_prems 1, sizef)
542 (prolog_step_tac (make_horns cls) 1));
544 (*First, breaks the goal into independent units*)
545 val safe_best_meson_tac =
546 SELECT_GOAL (TRY Safe_tac THEN
547 TRYALL (best_meson_tac size_of_subgoals));
549 (** Depth-first search version **)
551 val depth_meson_tac =
552 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
553 depth_prolog_tac (make_horns cls)]);
556 (** Iterative deepening version **)
558 (*This version does only one inference per call;
559 having only one eq_assume_tac speeds it up!*)
560 fun prolog_step_tac' horns =
561 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
562 take_prefix Thm.no_prems horns
563 val nrtac = net_resolve_tac horns
564 in fn i => eq_assume_tac i ORELSE
565 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
566 ((assume_tac i APPEND nrtac i) THEN check_tac)
569 fun iter_deepen_prolog_tac horns =
570 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
572 fun iter_deepen_meson_tac ths = MESON
574 case (gocls (cls@ths)) of
575 [] => no_tac (*no goal clauses*)
577 let val horns = make_horns (cls@ths)
578 val _ = if !Output.show_debug_msgs
579 then Output.debug ("meson method called:\n" ^
580 space_implode "\n" (map string_of_thm (cls@ths)) ^
582 space_implode "\n" (map string_of_thm horns))
584 in THEN_ITER_DEEPEN (resolve_tac goes 1) (has_fewer_prems 1) (prolog_step_tac' horns)
588 fun meson_claset_tac ths cs =
589 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
591 val meson_tac = CLASET' (meson_claset_tac []);
594 (**** Code to support ordinary resolution, rather than Model Elimination ****)
596 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
597 with no contrapositives, for ordinary resolution.*)
599 (*Rules to convert the head literal into a negated assumption. If the head
600 literal is already negated, then using notEfalse instead of notEfalse'
601 prevents a double negation.*)
602 val notEfalse = read_instantiate [("R","False")] notE;
603 val notEfalse' = rotate_prems 1 notEfalse;
605 fun negated_asm_of_head th =
606 th RS notEfalse handle THM _ => th RS notEfalse';
608 (*Converting one clause*)
609 fun make_meta_clause th =
610 negated_asm_of_head (make_horn resolution_clause_rules th);
612 fun make_meta_clauses ths =
614 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
616 (*Permute a rule's premises to move the i-th premise to the last position.*)
618 let val n = nprems_of th
619 in if 1 <= i andalso i <= n
620 then Thm.permute_prems (i-1) 1 th
621 else raise THM("select_literal", i, [th])
624 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
626 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
628 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
629 fun select_literal i cl = negate_head (make_last i cl);
632 (*Top-level Skolemization. Allows part of the conversion to clauses to be
633 expressed as a tactic (or Isar method). Each assumption of the selected
634 goal is converted to NNF and then its existential quantifiers are pulled
635 to the front. Finally, all existential quantifiers are eliminated,
636 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
637 might generate many subgoals.*)
639 fun skolemize_tac i st =
640 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
643 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
644 THEN REPEAT (etac exE 1))),
645 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
647 handle Subscript => Seq.empty;
649 (*Top-level conversion to meta-level clauses. Each clause has
650 leading !!-bound universal variables, to express generality. To get
651 meta-clauses instead of disjunctions, uncomment "make_meta_clauses" below.*)
652 val make_clauses_tac =
655 let val ts = Logic.strip_assums_hyp prop
660 (map forall_intr_vars
661 ( (**make_meta_clauses**) (make_clauses hyps))) 1)),
662 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
666 (*** setup the special skoklemization methods ***)
668 (*No CHANGED_PROP here, since these always appear in the preamble*)
670 val skolemize_setup =
672 [("skolemize", Method.no_args (Method.SIMPLE_METHOD' skolemize_tac),
673 "Skolemization into existential quantifiers"),
674 ("make_clauses", Method.no_args (Method.SIMPLE_METHOD' make_clauses_tac),
675 "Conversion to !!-quantified meta-level clauses")];
679 structure BasicMeson: BASIC_MESON = Meson;