src/HOL/simpdata.ML
 author wenzelm Wed Mar 15 18:47:28 2000 +0100 (2000-03-15) changeset 8473 2798d2f71ec2 parent 8114 09a7a180cc99 child 8641 978db2870862 permissions -rw-r--r--
splitter setup;
1 (*  Title:      HOL/simpdata.ML
2     ID:         \$Id\$
3     Author:     Tobias Nipkow
4     Copyright   1991  University of Cambridge
6 Instantiation of the generic simplifier for HOL.
7 *)
9 section "Simplifier";
11 (*** Addition of rules to simpsets and clasets simultaneously ***)	(* FIXME move to Provers/clasimp.ML? *)
13 infix 4 addIffs delIffs;
15 (*Takes UNCONDITIONAL theorems of the form A<->B to
16         the Safe Intr     rule B==>A and
17         the Safe Destruct rule A==>B.
18   Also ~A goes to the Safe Elim rule A ==> ?R
19   Failing other cases, A is added as a Safe Intr rule*)
20 local
21   val iff_const = HOLogic.eq_const HOLogic.boolT;
23   fun addIff ((cla, simp), th) =
24       (case HOLogic.dest_Trueprop (#prop (rep_thm th)) of
25                 (Const("Not", _) \$ A) =>
26                     cla addSEs [zero_var_indexes (th RS notE)]
27               | (con \$ _ \$ _) =>
28                     if con = iff_const
29                     then cla addSIs [zero_var_indexes (th RS iffD2)]
30                               addSDs [zero_var_indexes (th RS iffD1)]
31                     else  cla addSIs [th]
32               | _ => cla addSIs [th],
33        simp addsimps [th])
34       handle TERM _ => error ("AddIffs: theorem must be unconditional\n" ^
35                          string_of_thm th);
37   fun delIff ((cla, simp), th) =
38       (case HOLogic.dest_Trueprop (#prop (rep_thm th)) of
39                 (Const ("Not", _) \$ A) =>
40                     cla delrules [zero_var_indexes (th RS notE)]
41               | (con \$ _ \$ _) =>
42                     if con = iff_const
43                     then cla delrules [zero_var_indexes (th RS iffD2),
44                                        make_elim (zero_var_indexes (th RS iffD1))]
45                     else cla delrules [th]
46               | _ => cla delrules [th],
47        simp delsimps [th])
48       handle TERM _ => (warning("DelIffs: ignoring conditional theorem\n" ^
49                           string_of_thm th); (cla, simp));
51   fun store_clasimp (cla, simp) = (claset_ref () := cla; simpset_ref () := simp)
52 in
53 val op addIffs = foldl addIff;
54 val op delIffs = foldl delIff;
55 fun AddIffs thms = store_clasimp ((claset (), simpset ()) addIffs thms);
56 fun DelIffs thms = store_clasimp ((claset (), simpset ()) delIffs thms);
57 end;
60 (* "iff" attribute *)
62 local
63   fun change_global_css f (thy, th) =
64     let
65       val cs_ref = Classical.claset_ref_of thy;
66       val ss_ref = Simplifier.simpset_ref_of thy;
67       val (cs', ss') = f ((! cs_ref, ! ss_ref), [th]);
68     in cs_ref := cs'; ss_ref := ss'; (thy, th) end;
70   fun change_local_css f (ctxt, th) =
71     let
72       val cs = Classical.get_local_claset ctxt;
73       val ss = Simplifier.get_local_simpset ctxt;
74       val (cs', ss') = f ((cs, ss), [th]);
75       val ctxt' =
76         ctxt
77         |> Classical.put_local_claset cs'
78         |> Simplifier.put_local_simpset ss';
79     in (ctxt', th) end;
80 in
82 val iff_add_global = change_global_css (op addIffs);
83 val iff_add_local = change_local_css (op addIffs);
85 val iff_attrib_setup =
87     "add rules to simpset and claset simultaneously")]];
89 end;
92 val [prem] = goal (the_context ()) "x==y ==> x=y";
93 by (rewtac prem);
94 by (rtac refl 1);
95 qed "meta_eq_to_obj_eq";
97 local
99   fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);
101 in
103 (*Make meta-equalities.  The operator below is Trueprop*)
105 fun mk_meta_eq r = r RS eq_reflection;
107 val Eq_TrueI  = mk_meta_eq(prover  "P --> (P = True)"  RS mp);
108 val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
110 fun mk_eq th = case concl_of th of
111         Const("==",_)\$_\$_       => th
112     |   _\$(Const("op =",_)\$_\$_) => mk_meta_eq th
113     |   _\$(Const("Not",_)\$_)    => th RS Eq_FalseI
114     |   _                       => th RS Eq_TrueI;
115 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
117 fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);
119 fun mk_meta_cong rl =
120   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
121   handle THM _ =>
122   error("Premises and conclusion of congruence rules must be =-equalities");
124 val not_not = prover "(~ ~ P) = P";
126 val simp_thms = [not_not] @ map prover
127  [ "(x=x) = True",
128    "(~True) = False", "(~False) = True",
129    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
130    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
131    "(True --> P) = P", "(False --> P) = True",
132    "(P --> True) = True", "(P --> P) = True",
133    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
134    "(P & True) = P", "(True & P) = P",
135    "(P & False) = False", "(False & P) = False",
136    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
137    "(P & ~P) = False",    "(~P & P) = False",
138    "(P | True) = True", "(True | P) = True",
139    "(P | False) = P", "(False | P) = P",
140    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
141    "(P | ~P) = True",    "(~P | P) = True",
142    "((~P) = (~Q)) = (P=Q)",
143    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",
144 (*two needed for the one-point-rule quantifier simplification procs*)
145    "(? x. x=t & P(x)) = P(t)",		(*essential for termination!!*)
146    "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)
148 (* Add congruence rules for = (instead of ==) *)
150 (* ###FIXME: Move to simplifier,
151    taking mk_meta_cong as input, eliminating addeqcongs and deleqcongs *)
152 infix 4 addcongs delcongs;
153 fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs);
154 fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs);
155 fun Addcongs congs = (simpset_ref() := simpset() addcongs congs);
156 fun Delcongs congs = (simpset_ref() := simpset() delcongs congs);
159 val imp_cong = impI RSN
160     (2, prove_goal (the_context ()) "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
161         (fn _=> [(Blast_tac 1)]) RS mp RS mp);
163 (*Miniscoping: pushing in existential quantifiers*)
164 val ex_simps = map prover
165                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",
166                  "(EX x. P & Q x)   = (P & (EX x. Q x))",
167                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",
168                  "(EX x. P | Q x)   = (P | (EX x. Q x))",
169                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
170                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];
172 (*Miniscoping: pushing in universal quantifiers*)
173 val all_simps = map prover
174                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",
175                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",
176                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",
177                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",
178                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
179                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
182 (* elimination of existential quantifiers in assumptions *)
184 val ex_all_equiv =
185   let val lemma1 = prove_goal (the_context ())
186         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
187         (fn prems => [resolve_tac prems 1, etac exI 1]);
188       val lemma2 = prove_goalw (the_context ()) [Ex_def]
189         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
190         (fn prems => [(REPEAT(resolve_tac prems 1))])
191   in equal_intr lemma1 lemma2 end;
193 end;
195 bind_thms ("ex_simps", ex_simps);
196 bind_thms ("all_simps", all_simps);
197 bind_thm ("not_not", not_not);
199 (* Elimination of True from asumptions: *)
201 val True_implies_equals = prove_goal (the_context ())
202  "(True ==> PROP P) == PROP P"
203 (fn _ => [rtac equal_intr_rule 1, atac 2,
204           METAHYPS (fn prems => resolve_tac prems 1) 1,
205           rtac TrueI 1]);
207 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [(Blast_tac 1)]);
209 prove "eq_commute" "(a=b)=(b=a)";
210 prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";
211 prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";
212 val eq_ac = [eq_commute, eq_left_commute, eq_assoc];
214 prove "conj_commute" "(P&Q) = (Q&P)";
215 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
216 val conj_comms = [conj_commute, conj_left_commute];
217 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
219 prove "disj_commute" "(P|Q) = (Q|P)";
220 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
221 val disj_comms = [disj_commute, disj_left_commute];
222 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
224 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
225 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
227 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
228 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
230 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
231 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
232 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
234 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
235 prove "imp_disj_not1" "(P --> Q | R) = (~Q --> P --> R)";
236 prove "imp_disj_not2" "(P --> Q | R) = (~R --> P --> Q)";
238 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
239 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
241 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
242 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
243 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
244 prove "not_iff" "(P~=Q) = (P = (~Q))";
245 prove "disj_not1" "(~P | Q) = (P --> Q)";
246 prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
247 prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
249 prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
252 (*Avoids duplication of subgoals after split_if, when the true and false
253   cases boil down to the same thing.*)
254 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
256 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
257 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
258 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";
259 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
261 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
262 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
264 (* '&' congruence rule: not included by default!
265    May slow rewrite proofs down by as much as 50% *)
267 let val th = prove_goal (the_context ())
268                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
269                 (fn _=> [(Blast_tac 1)])
270 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
272 let val th = prove_goal (the_context ())
273                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
274                 (fn _=> [(Blast_tac 1)])
275 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
277 (* '|' congruence rule: not included by default! *)
279 let val th = prove_goal (the_context ())
280                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
281                 (fn _=> [(Blast_tac 1)])
282 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
284 prove "eq_sym_conv" "(x=y) = (y=x)";
287 (** if-then-else rules **)
289 Goalw [if_def] "(if True then x else y) = x";
290 by (Blast_tac 1);
291 qed "if_True";
293 Goalw [if_def] "(if False then x else y) = y";
294 by (Blast_tac 1);
295 qed "if_False";
297 Goalw [if_def] "P ==> (if P then x else y) = x";
298 by (Blast_tac 1);
299 qed "if_P";
301 Goalw [if_def] "~P ==> (if P then x else y) = y";
302 by (Blast_tac 1);
303 qed "if_not_P";
305 Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";
306 by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);
307 by (stac if_P 2);
308 by (stac if_not_P 1);
309 by (ALLGOALS (Blast_tac));
310 qed "split_if";
312 (* for backwards compatibility: *)
313 val expand_if = split_if;
315 Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";
316 by (stac split_if 1);
317 by (Blast_tac 1);
318 qed "split_if_asm";
320 Goal "(if c then x else x) = x";
321 by (stac split_if 1);
322 by (Blast_tac 1);
323 qed "if_cancel";
325 Goal "(if x = y then y else x) = x";
326 by (stac split_if 1);
327 by (Blast_tac 1);
328 qed "if_eq_cancel";
330 (*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
331 Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";
332 by (rtac split_if 1);
333 qed "if_bool_eq_conj";
335 (*And this form is useful for expanding IFs on the LEFT*)
336 Goal "(if P then Q else R) = ((P&Q) | (~P&R))";
337 by (stac split_if 1);
338 by (Blast_tac 1);
339 qed "if_bool_eq_disj";
342 (*** make simplification procedures for quantifier elimination ***)
344 structure Quantifier1 = Quantifier1Fun(
345 struct
346   (*abstract syntax*)
347   fun dest_eq((c as Const("op =",_)) \$ s \$ t) = Some(c,s,t)
348     | dest_eq _ = None;
349   fun dest_conj((c as Const("op &",_)) \$ s \$ t) = Some(c,s,t)
350     | dest_conj _ = None;
351   val conj = HOLogic.conj
352   val imp  = HOLogic.imp
353   (*rules*)
354   val iff_reflection = eq_reflection
355   val iffI = iffI
356   val sym  = sym
357   val conjI= conjI
358   val conjE= conjE
359   val impI = impI
360   val impE = impE
361   val mp   = mp
362   val exI  = exI
363   val exE  = exE
364   val allI = allI
365   val allE = allE
366 end);
368 local
369 val ex_pattern =
370   Thm.read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)",HOLogic.boolT)
372 val all_pattern =
373   Thm.read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
375 in
376 val defEX_regroup =
377   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
378 val defALL_regroup =
379   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
380 end;
383 (*** Case splitting ***)
385 structure SplitterData =
386   struct
387   structure Simplifier = Simplifier
388   val mk_eq          = mk_eq
389   val meta_eq_to_iff = meta_eq_to_obj_eq
390   val iffD           = iffD2
391   val disjE          = disjE
392   val conjE          = conjE
393   val exE            = exE
394   val contrapos      = contrapos
395   val contrapos2     = contrapos2
396   val notnotD        = notnotD
397   end;
399 structure Splitter = SplitterFun(SplitterData);
401 val split_tac        = Splitter.split_tac;
402 val split_inside_tac = Splitter.split_inside_tac;
403 val split_asm_tac    = Splitter.split_asm_tac;
405 val op delsplits     = Splitter.delsplits;
407 val Delsplits        = Splitter.Delsplits;
409 (*In general it seems wrong to add distributive laws by default: they
410   might cause exponential blow-up.  But imp_disjL has been in for a while
411   and cannot be removed without affecting existing proofs.  Moreover,
412   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
413   grounds that it allows simplification of R in the two cases.*)
415 fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
417 val mksimps_pairs =
418   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
419    ("All", [spec]), ("True", []), ("False", []),
420    ("If", [if_bool_eq_conj RS iffD1])];
422 (* ###FIXME: move to Provers/simplifier.ML
423 val mk_atomize:      (string * thm list) list -> thm -> thm list
424 *)
425 (* ###FIXME: move to Provers/simplifier.ML *)
426 fun mk_atomize pairs =
427   let fun atoms th =
428         (case concl_of th of
429            Const("Trueprop",_) \$ p =>
430              (case head_of p of
431                 Const(a,_) =>
432                   (case assoc(pairs,a) of
433                      Some(rls) => flat (map atoms ([th] RL rls))
434                    | None => [th])
435               | _ => [th])
436          | _ => [th])
437   in atoms end;
439 fun mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);
441 fun unsafe_solver_tac prems =
442   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
443 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
445 (*No premature instantiation of variables during simplification*)
446 fun safe_solver_tac prems =
447   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
448          eq_assume_tac, ematch_tac [FalseE]];
449 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
451 val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
452 			    setSSolver safe_solver
453 			    setSolver  unsafe_solver
454 			    setmksimps (mksimps mksimps_pairs)
455 			    setmkeqTrue mk_eq_True;
457 val HOL_ss =
459      ([triv_forall_equality, (* prunes params *)
460        True_implies_equals, (* prune asms `True' *)
461        if_True, if_False, if_cancel, if_eq_cancel,
462        imp_disjL, conj_assoc, disj_assoc,
463        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
464        disj_not1, not_all, not_ex, cases_simp, Eps_eq, Eps_sym_eq]
465      @ ex_simps @ all_simps @ simp_thms)
470 (*Simplifies x assuming c and y assuming ~c*)
471 val prems = Goalw [if_def]
472   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
473 \  (if b then x else y) = (if c then u else v)";
474 by (asm_simp_tac (HOL_ss addsimps prems) 1);
475 qed "if_cong";
477 (*Prevents simplification of x and y: faster and allows the execution
478   of functional programs. NOW THE DEFAULT.*)
479 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
480 by (etac arg_cong 1);
481 qed "if_weak_cong";
483 (*Prevents simplification of t: much faster*)
484 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
485 by (etac arg_cong 1);
486 qed "let_weak_cong";
488 Goal "f(if c then x else y) = (if c then f x else f y)";
489 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
490 qed "if_distrib";
492 (*For expand_case_tac*)
493 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
494 by (case_tac "P" 1);
495 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
496 qed "expand_case";
498 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
499   during unification.*)
500 fun expand_case_tac P i =
501     res_inst_tac [("P",P)] expand_case i THEN
502     Simp_tac (i+1) THEN
503     Simp_tac i;
505 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
506   side of an equality.  Used in {Integ,Real}/simproc.ML*)
507 Goal "x=y ==> (x=z) = (y=z)";
508 by (asm_simp_tac HOL_ss 1);
509 qed "restrict_to_left";
511 (* default simpset *)
512 val simpsetup =
513     [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong];
514 		thy)];
517 (*** integration of simplifier with classical reasoner ***)
519 structure Clasimp = ClasimpFun
520  (structure Simplifier = Simplifier and Splitter = Splitter
521    and Classical  = Classical and Blast = Blast);
522 open Clasimp;
524 val HOL_css = (HOL_cs, HOL_ss);
527 (*** A general refutation procedure ***)
529 (* Parameters:
531    test: term -> bool
532    tests if a term is at all relevant to the refutation proof;
533    if not, then it can be discarded. Can improve performance,
534    esp. if disjunctions can be discarded (no case distinction needed!).
536    prep_tac: int -> tactic
537    A preparation tactic to be applied to the goal once all relevant premises
538    have been moved to the conclusion.
540    ref_tac: int -> tactic
541    the actual refutation tactic. Should be able to deal with goals
542    [| A1; ...; An |] ==> False
543    where the Ai are atomic, i.e. no top-level &, | or ?
544 *)
546 fun refute_tac test prep_tac ref_tac =
547   let val nnf_simps =
548         [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
549          not_all,not_ex,not_not];
550       val nnf_simpset =
551         empty_ss setmkeqTrue mk_eq_True
552                  setmksimps (mksimps mksimps_pairs)