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 20;
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 =",_) $ t $ u) =
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))
218 | signed_nclauses b (Const("Ex", _) $ Abs (_,_,t)) = signed_nclauses b t
219 | signed_nclauses b (Const("All",_) $ Abs (_,_,t)) = signed_nclauses b t
220 | signed_nclauses _ _ = 1; (* literal *)
222 val nclauses = signed_nclauses true;
224 fun too_many_clauses t = nclauses t >= !max_clauses;
226 (*Replaces universally quantified variables by FREE variables -- because
227 assumptions may not contain scheme variables. Later, call "generalize". *)
229 let val newname = gensym "mes_"
230 val spec' = read_instantiate [("x", newname)] spec
233 (*Used with METAHYPS below. There is one assumption, which gets bound to prem
234 and then normalized via function nf. The normal form is given to resolve_tac,
235 presumably to instantiate a Boolean variable.*)
236 fun resop nf [prem] = resolve_tac (nf prem) 1;
238 (*Any need to extend this list with
239 "HOL.type_class","Code_Generator.eq_class","ProtoPure.term"?*)
241 exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "prop"]);
243 fun apply_skolem_ths (th, rls) =
244 let fun tryall [] = raise THM("apply_skolem_ths", 0, th::rls)
245 | tryall (rl::rls) = (first_order_resolve th rl handle THM _ => tryall rls)
248 (*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
249 Strips universal quantifiers and breaks up conjunctions.
250 Eliminates existential quantifiers using skoths: Skolemization theorems.*)
251 fun cnf skoths (th,ths) =
252 let fun cnf_aux (th,ths) =
253 if not (HOLogic.is_Trueprop (prop_of th)) then ths (*meta-level: ignore*)
254 else if not (has_conns ["All","Ex","op &"] (prop_of th))
255 then th::ths (*no work to do, terminate*)
256 else case head_of (HOLogic.dest_Trueprop (concl_of th)) of
257 Const ("op &", _) => (*conjunction*)
258 cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
259 | Const ("All", _) => (*universal quantifier*)
260 cnf_aux (freeze_spec th, ths)
262 (*existential quantifier: Insert Skolem functions*)
263 cnf_aux (apply_skolem_ths (th,skoths), ths)
264 | Const ("op |", _) => (*disjunction*)
266 (METAHYPS (resop cnf_nil) 1) THEN
267 (fn st' => st' |> METAHYPS (resop cnf_nil) 1)
268 in Seq.list_of (tac (th RS disj_forward)) @ ths end
269 | _ => (*no work to do*) th::ths
270 and cnf_nil th = cnf_aux (th,[])
272 if too_many_clauses (concl_of th)
273 then (Output.debug ("cnf is ignoring: " ^ string_of_thm th); ths)
274 else cnf_aux (th,ths)
277 (*Convert all suitable free variables to schematic variables,
278 but don't discharge assumptions.*)
279 fun generalize th = Thm.varifyT (forall_elim_vars 0 (forall_intr_frees th));
281 fun make_cnf skoths th = cnf skoths (th, []);
283 (*Generalization, removal of redundant equalities, removal of tautologies.*)
284 fun finish_cnf ths = filter (not o is_taut) (map (refl_clause o generalize) ths);
287 (**** Removal of duplicate literals ****)
289 (*Forward proof, passing extra assumptions as theorems to the tactic*)
290 fun forward_res2 nf hyps st =
293 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
296 | NONE => raise THM("forward_res2", 0, [st]);
298 (*Remove duplicates in P|Q by assuming ~P in Q
299 rls (initially []) accumulates assumptions of the form P==>False*)
300 fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
301 handle THM _ => tryres(th,rls)
302 handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
303 [disj_FalseD1, disj_FalseD2, asm_rl])
306 (*Remove duplicate literals, if there are any*)
308 if has_duplicates (op =) (literals (prop_of th))
309 then nodups_aux [] th
313 (**** Generation of contrapositives ****)
315 (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
316 fun assoc_right th = assoc_right (th RS disj_assoc)
319 (*Must check for negative literal first!*)
320 val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
322 (*For ordinary resolution. *)
323 val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];
325 (*Create a goal or support clause, conclusing False*)
326 fun make_goal th = (*Must check for negative literal first!*)
327 make_goal (tryres(th, clause_rules))
328 handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
330 (*Sort clauses by number of literals*)
331 fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
333 fun sort_clauses ths = sort (make_ord fewerlits) ths;
335 (*True if the given type contains bool anywhere*)
336 fun has_bool (Type("bool",_)) = true
337 | has_bool (Type(_, Ts)) = exists has_bool Ts
338 | has_bool _ = false;
340 (*Is the string the name of a connective? Really only | and Not can remain,
341 since this code expects to be called on a clause form.*)
342 val is_conn = member (op =)
343 ["Trueprop", "op &", "op |", "op -->", "Not",
344 "All", "Ex", "Ball", "Bex"];
346 (*True if the term contains a function--not a logical connective--where the type
347 of any argument contains bool.*)
348 val has_bool_arg_const =
350 (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
352 (*Raises an exception if any Vars in the theorem mention type bool; they
353 could cause make_horn to loop! Also rejects functions whose arguments are
354 Booleans or other functions.*)
356 not (exists (has_bool o fastype_of) (term_vars t) orelse
357 not (Term.is_first_order ["all","All","Ex"] t) orelse
358 has_bool_arg_const t orelse
361 (*Create a meta-level Horn clause*)
362 fun make_horn crules th = make_horn crules (tryres(th,crules))
365 (*Generate Horn clauses for all contrapositives of a clause. The input, th,
366 is a HOL disjunction.*)
367 fun add_contras crules (th,hcs) =
368 let fun rots (0,th) = hcs
369 | rots (k,th) = zero_var_indexes (make_horn crules th) ::
370 rots(k-1, assoc_right (th RS disj_comm))
371 in case nliterals(prop_of th) of
373 | n => rots(n, assoc_right th)
376 (*Use "theorem naming" to label the clauses*)
377 fun name_thms label =
378 let fun name1 (th, (k,ths)) =
379 (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
381 in fn ths => #2 (foldr name1 (length ths, []) ths) end;
383 (*Is the given disjunction an all-negative support clause?*)
384 fun is_negative th = forall (not o #1) (literals (prop_of th));
386 val neg_clauses = List.filter is_negative;
389 (***** MESON PROOF PROCEDURE *****)
391 fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
392 As) = rhyps(phi, A::As)
393 | rhyps (_, As) = As;
395 (** Detecting repeated assumptions in a subgoal **)
397 (*The stringtree detects repeated assumptions.*)
398 fun ins_term (net,t) = Net.insert_term (op aconv) (t,t) net;
400 (*detects repetitions in a list of terms*)
401 fun has_reps [] = false
402 | has_reps [_] = false
403 | has_reps [t,u] = (t aconv u)
404 | has_reps ts = (Library.foldl ins_term (Net.empty, ts); false)
405 handle Net.INSERT => true;
407 (*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
408 fun TRYING_eq_assume_tac 0 st = Seq.single st
409 | TRYING_eq_assume_tac i st =
410 TRYING_eq_assume_tac (i-1) (eq_assumption i st)
411 handle THM _ => TRYING_eq_assume_tac (i-1) st;
413 fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;
415 (*Loop checking: FAIL if trying to prove the same thing twice
416 -- if *ANY* subgoal has repeated literals*)
418 if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
419 then Seq.empty else Seq.single st;
422 (* net_resolve_tac actually made it slower... *)
423 fun prolog_step_tac horns i =
424 (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
425 TRYALL_eq_assume_tac;
427 (*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
428 fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
430 fun size_of_subgoals st = foldr addconcl 0 (prems_of st);
433 (*Negation Normal Form*)
434 val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
435 not_impD, not_iffD, not_allD, not_exD, not_notD];
437 fun make_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
439 forward_res make_nnf1
440 (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
443 (*The simplification removes defined quantifiers and occurrences of True and False.
444 nnf_ss also includes the one-point simprocs,
445 which are needed to avoid the various one-point theorems from generating junk clauses.*)
447 [simp_implies_def, Ex1_def, Ball_def, Bex_def, if_True,
448 if_False, if_cancel, if_eq_cancel, cases_simp];
449 val nnf_extra_simps =
450 thms"split_ifs" @ ex_simps @ all_simps @ simp_thms;
453 HOL_basic_ss addsimps nnf_extra_simps
454 addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc];
456 fun make_nnf th = case prems_of th of
457 [] => th |> rewrite_rule (map safe_mk_meta_eq nnf_simps)
458 |> simplify nnf_ss (*But this doesn't simplify premises...*)
460 | _ => raise THM ("make_nnf: premises in argument", 0, [th]);
462 (*Pull existential quantifiers to front. This accomplishes Skolemization for
463 clauses that arise from a subgoal.*)
465 if not (has_conns ["Ex"] (prop_of th)) then th
466 else (skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
467 disj_exD, disj_exD1, disj_exD2])))
469 skolemize (forward_res skolemize
470 (tryres (th, [conj_forward, disj_forward, all_forward])))
471 handle THM _ => forward_res skolemize (th RS ex_forward);
474 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.
475 The resulting clauses are HOL disjunctions.*)
476 fun make_clauses ths =
477 (sort_clauses (map (generalize o nodups) (foldr (cnf[]) [] ths)));
479 (*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)
482 (distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) [] ths));
484 (*Could simply use nprems_of, which would count remaining subgoals -- no
485 discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
487 fun best_prolog_tac sizef horns =
488 BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
490 fun depth_prolog_tac horns =
491 DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
493 (*Return all negative clauses, as possible goal clauses*)
494 fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
496 fun skolemize_prems_tac prems =
497 cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
500 (*Expand all definitions (presumably of Skolem functions) in a proof state.*)
501 fun expand_defs_tac st =
502 let val defs = filter (can dest_equals) (#hyps (crep_thm st))
503 in PRIMITIVE (LocalDefs.def_export false defs) st end;
505 (*Basis of all meson-tactics. Supplies cltac with clauses: HOL disjunctions*)
506 fun MESON cltac i st =
508 (EVERY [rtac ccontr 1,
510 EVERY1 [skolemize_prems_tac negs,
511 METAHYPS (cltac o make_clauses)]) 1,
512 expand_defs_tac]) i st
513 handle THM _ => no_tac st; (*probably from make_meta_clause, not first-order*)
515 (** Best-first search versions **)
517 (*ths is a list of additional clauses (HOL disjunctions) to use.*)
518 fun best_meson_tac sizef =
520 THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
521 (has_fewer_prems 1, sizef)
522 (prolog_step_tac (make_horns cls) 1));
524 (*First, breaks the goal into independent units*)
525 val safe_best_meson_tac =
526 SELECT_GOAL (TRY Safe_tac THEN
527 TRYALL (best_meson_tac size_of_subgoals));
529 (** Depth-first search version **)
531 val depth_meson_tac =
532 MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
533 depth_prolog_tac (make_horns cls)]);
536 (** Iterative deepening version **)
538 (*This version does only one inference per call;
539 having only one eq_assume_tac speeds it up!*)
540 fun prolog_step_tac' horns =
541 let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
542 take_prefix Thm.no_prems horns
543 val nrtac = net_resolve_tac horns
544 in fn i => eq_assume_tac i ORELSE
545 match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
546 ((assume_tac i APPEND nrtac i) THEN check_tac)
549 fun iter_deepen_prolog_tac horns =
550 ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
552 fun iter_deepen_meson_tac ths =
554 case (gocls (cls@ths)) of
555 [] => no_tac (*no goal clauses*)
557 (THEN_ITER_DEEPEN (resolve_tac goes 1)
559 (prolog_step_tac' (make_horns (cls@ths)))));
561 fun meson_claset_tac ths cs =
562 SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL (iter_deepen_meson_tac ths));
564 val meson_tac = CLASET' (meson_claset_tac []);
567 (**** Code to support ordinary resolution, rather than Model Elimination ****)
569 (*Convert a list of clauses (disjunctions) to meta-level clauses (==>),
570 with no contrapositives, for ordinary resolution.*)
572 (*Rules to convert the head literal into a negated assumption. If the head
573 literal is already negated, then using notEfalse instead of notEfalse'
574 prevents a double negation.*)
575 val notEfalse = read_instantiate [("R","False")] notE;
576 val notEfalse' = rotate_prems 1 notEfalse;
578 fun negated_asm_of_head th =
579 th RS notEfalse handle THM _ => th RS notEfalse';
581 (*Converting one clause*)
582 fun make_meta_clause th =
583 if is_fol_term (prop_of th)
584 then negated_asm_of_head (make_horn resolution_clause_rules th)
585 else raise THM("make_meta_clause", 0, [th]);
587 fun make_meta_clauses ths =
589 (distinct Drule.eq_thm_prop (map make_meta_clause ths));
591 (*Permute a rule's premises to move the i-th premise to the last position.*)
593 let val n = nprems_of th
594 in if 1 <= i andalso i <= n
595 then Thm.permute_prems (i-1) 1 th
596 else raise THM("select_literal", i, [th])
599 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
601 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
603 (*Maps the clause [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
604 fun select_literal i cl = negate_head (make_last i cl);
607 (*Top-level Skolemization. Allows part of the conversion to clauses to be
608 expressed as a tactic (or Isar method). Each assumption of the selected
609 goal is converted to NNF and then its existential quantifiers are pulled
610 to the front. Finally, all existential quantifiers are eliminated,
611 leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
612 might generate many subgoals.*)
614 fun skolemize_tac i st =
615 let val ts = Logic.strip_assums_hyp (List.nth (prems_of st, i-1))
618 (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
619 THEN REPEAT (etac exE 1))),
620 REPEAT_DETERM_N (length ts) o (etac thin_rl)] i st
622 handle Subscript => Seq.empty;
624 (*Top-level conversion to meta-level clauses. Each clause has
625 leading !!-bound universal variables, to express generality. To get
626 disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
627 val make_clauses_tac =
630 let val ts = Logic.strip_assums_hyp prop
635 (map forall_intr_vars
636 (make_meta_clauses (make_clauses hyps))) 1)),
637 REPEAT_DETERM_N (length ts) o (etac thin_rl)]
641 (*** setup the special skoklemization methods ***)
643 (*No CHANGED_PROP here, since these always appear in the preamble*)
645 val skolemize_meth = Method.SIMPLE_METHOD' HEADGOAL skolemize_tac;
647 val make_clauses_meth = Method.SIMPLE_METHOD' HEADGOAL make_clauses_tac;
649 val skolemize_setup =
651 [("skolemize", Method.no_args skolemize_meth,
652 "Skolemization into existential quantifiers"),
653 ("make_clauses", Method.no_args make_clauses_meth,
654 "Conversion to !!-quantified meta-level clauses")];
658 structure BasicMeson: BASIC_MESON = Meson;