src/HOL/simpdata.ML
 author wenzelm Wed Sep 06 16:54:12 2000 +0200 (2000-09-06) changeset 9876 a069795f1060 parent 9875 c50349d252b7 child 9894 c8ff37b637a7 permissions -rw-r--r--
tuned;
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 val [prem] = goal (the_context ()) "x==y ==> x=y";
12 by (rewtac prem);
13 by (rtac refl 1);
14 qed "meta_eq_to_obj_eq";
16 Goal "(%s. f s) = f";
17 br refl 1;
18 qed "eta_contract_eq";
20 local
22   fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);
24 in
26 (*Make meta-equalities.  The operator below is Trueprop*)
28 fun mk_meta_eq r = r RS eq_reflection;
29 fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
31 val Eq_TrueI  = mk_meta_eq(prover  "P --> (P = True)"  RS mp);
32 val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
34 fun mk_eq th = case concl_of th of
35         Const("==",_)\$_\$_       => th
36     |   _\$(Const("op =",_)\$_\$_) => mk_meta_eq th
37     |   _\$(Const("Not",_)\$_)    => th RS Eq_FalseI
38     |   _                       => th RS Eq_TrueI;
39 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
41 fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);
43 (*Congruence rules for = (instead of ==)*)
44 fun mk_meta_cong rl =
45   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
46   handle THM _ =>
47   error("Premises and conclusion of congruence rules must be =-equalities");
49 val not_not = prover "(~ ~ P) = P";
51 val simp_thms = [not_not] @ map prover
52  [ "(x=x) = True",
53    "(~True) = False", "(~False) = True",
54    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
55    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",
56    "(True --> P) = P", "(False --> P) = True",
57    "(P --> True) = True", "(P --> P) = True",
58    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
59    "(P & True) = P", "(True & P) = P",
60    "(P & False) = False", "(False & P) = False",
61    "(P & P) = P", "(P & (P & Q)) = (P & Q)",
62    "(P & ~P) = False",    "(~P & P) = False",
63    "(P | True) = True", "(True | P) = True",
64    "(P | False) = P", "(False | P) = P",
65    "(P | P) = P", "(P | (P | Q)) = (P | Q)",
66    "(P | ~P) = True",    "(~P | P) = True",
67    "((~P) = (~Q)) = (P=Q)",
68    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",
69 (*two needed for the one-point-rule quantifier simplification procs*)
70    "(? x. x=t & P(x)) = P(t)",          (*essential for termination!!*)
71    "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)
73 val imp_cong = standard(impI RSN
74     (2, prove_goal (the_context ()) "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
75         (fn _=> [(Blast_tac 1)]) RS mp RS mp));
77 (*Miniscoping: pushing in existential quantifiers*)
78 val ex_simps = map prover
79                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",
80                  "(EX x. P & Q x)   = (P & (EX x. Q x))",
81                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",
82                  "(EX x. P | Q x)   = (P | (EX x. Q x))",
83                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
84                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];
86 (*Miniscoping: pushing in universal quantifiers*)
87 val all_simps = map prover
88                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",
89                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",
90                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",
91                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",
92                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
93                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
96 (* elimination of existential quantifiers in assumptions *)
98 val ex_all_equiv =
99   let val lemma1 = prove_goal (the_context ())
100         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
101         (fn prems => [resolve_tac prems 1, etac exI 1]);
102       val lemma2 = prove_goalw (the_context ()) [Ex_def]
103         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
104         (fn prems => [(REPEAT(resolve_tac prems 1))])
105   in equal_intr lemma1 lemma2 end;
107 end;
109 bind_thms ("ex_simps", ex_simps);
110 bind_thms ("all_simps", all_simps);
111 bind_thm ("not_not", not_not);
112 bind_thm ("imp_cong", imp_cong);
114 (* Elimination of True from asumptions: *)
116 val True_implies_equals = prove_goal (the_context ())
117  "(True ==> PROP P) == PROP P"
118 (fn _ => [rtac equal_intr_rule 1, atac 2,
119           METAHYPS (fn prems => resolve_tac prems 1) 1,
120           rtac TrueI 1]);
122 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [(Blast_tac 1)]);
124 prove "eq_commute" "(a=b) = (b=a)";
125 prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";
126 prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";
127 val eq_ac = [eq_commute, eq_left_commute, eq_assoc];
129 prove "neq_commute" "(a~=b) = (b~=a)";
131 prove "conj_commute" "(P&Q) = (Q&P)";
132 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
133 val conj_comms = [conj_commute, conj_left_commute];
134 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
136 prove "disj_commute" "(P|Q) = (Q|P)";
137 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
138 val disj_comms = [disj_commute, disj_left_commute];
139 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
141 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
142 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
144 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
145 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
147 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
148 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";
149 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
151 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
152 prove "imp_disj_not1" "(P --> Q | R) = (~Q --> P --> R)";
153 prove "imp_disj_not2" "(P --> Q | R) = (~R --> P --> Q)";
155 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";
156 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";
158 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
159 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
160 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
161 prove "not_iff" "(P~=Q) = (P = (~Q))";
162 prove "disj_not1" "(~P | Q) = (P --> Q)";
163 prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)
164 prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
166 prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
169 (*Avoids duplication of subgoals after split_if, when the true and false
170   cases boil down to the same thing.*)
171 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";
173 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";
174 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
175 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";
176 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
178 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
179 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
181 (* '&' congruence rule: not included by default!
182    May slow rewrite proofs down by as much as 50% *)
184 let val th = prove_goal (the_context ())
185                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
186                 (fn _=> [(Blast_tac 1)])
187 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
189 let val th = prove_goal (the_context ())
190                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
191                 (fn _=> [(Blast_tac 1)])
192 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
194 (* '|' congruence rule: not included by default! *)
196 let val th = prove_goal (the_context ())
197                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
198                 (fn _=> [(Blast_tac 1)])
199 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;
201 prove "eq_sym_conv" "(x=y) = (y=x)";
204 (** if-then-else rules **)
206 Goalw [if_def] "(if True then x else y) = x";
207 by (Blast_tac 1);
208 qed "if_True";
210 Goalw [if_def] "(if False then x else y) = y";
211 by (Blast_tac 1);
212 qed "if_False";
214 Goalw [if_def] "P ==> (if P then x else y) = x";
215 by (Blast_tac 1);
216 qed "if_P";
218 Goalw [if_def] "~P ==> (if P then x else y) = y";
219 by (Blast_tac 1);
220 qed "if_not_P";
222 Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";
223 by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);
224 by (stac if_P 2);
225 by (stac if_not_P 1);
226 by (ALLGOALS (Blast_tac));
227 qed "split_if";
229 Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";
230 by (stac split_if 1);
231 by (Blast_tac 1);
232 qed "split_if_asm";
234 bind_thms ("if_splits", [split_if, split_if_asm]);
236 Goal "(if c then x else x) = x";
237 by (stac split_if 1);
238 by (Blast_tac 1);
239 qed "if_cancel";
241 Goal "(if x = y then y else x) = x";
242 by (stac split_if 1);
243 by (Blast_tac 1);
244 qed "if_eq_cancel";
246 (*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
247 Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";
248 by (rtac split_if 1);
249 qed "if_bool_eq_conj";
251 (*And this form is useful for expanding IFs on the LEFT*)
252 Goal "(if P then Q else R) = ((P&Q) | (~P&R))";
253 by (stac split_if 1);
254 by (Blast_tac 1);
255 qed "if_bool_eq_disj";
258 (*** make simplification procedures for quantifier elimination ***)
260 structure Quantifier1 = Quantifier1Fun
261 (struct
262   (*abstract syntax*)
263   fun dest_eq((c as Const("op =",_)) \$ s \$ t) = Some(c,s,t)
264     | dest_eq _ = None;
265   fun dest_conj((c as Const("op &",_)) \$ s \$ t) = Some(c,s,t)
266     | dest_conj _ = None;
267   val conj = HOLogic.conj
268   val imp  = HOLogic.imp
269   (*rules*)
270   val iff_reflection = eq_reflection
271   val iffI = iffI
272   val sym  = sym
273   val conjI= conjI
274   val conjE= conjE
275   val impI = impI
276   val impE = impE
277   val mp   = mp
278   val exI  = exI
279   val exE  = exE
280   val allI = allI
281   val allE = allE
282 end);
284 local
285 val ex_pattern =
286   Thm.read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)",HOLogic.boolT)
288 val all_pattern =
289   Thm.read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)
291 in
292 val defEX_regroup =
293   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
294 val defALL_regroup =
295   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
296 end;
299 (*** Case splitting ***)
301 structure SplitterData =
302   struct
303   structure Simplifier = Simplifier
304   val mk_eq          = mk_eq
305   val meta_eq_to_iff = meta_eq_to_obj_eq
306   val iffD           = iffD2
307   val disjE          = disjE
308   val conjE          = conjE
309   val exE            = exE
310   val contrapos      = contrapos
311   val contrapos2     = contrapos2
312   val notnotD        = notnotD
313   end;
315 structure Splitter = SplitterFun(SplitterData);
317 val split_tac        = Splitter.split_tac;
318 val split_inside_tac = Splitter.split_inside_tac;
319 val split_asm_tac    = Splitter.split_asm_tac;
321 val op delsplits     = Splitter.delsplits;
323 val Delsplits        = Splitter.Delsplits;
325 (*In general it seems wrong to add distributive laws by default: they
326   might cause exponential blow-up.  But imp_disjL has been in for a while
327   and cannot be removed without affecting existing proofs.  Moreover,
328   rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
329   grounds that it allows simplification of R in the two cases.*)
331 fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
333 val mksimps_pairs =
334   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
335    ("All", [spec]), ("True", []), ("False", []),
336    ("If", [if_bool_eq_conj RS iffD1])];
338 (* ###FIXME: move to Provers/simplifier.ML
339 val mk_atomize:      (string * thm list) list -> thm -> thm list
340 *)
341 (* ###FIXME: move to Provers/simplifier.ML *)
342 fun mk_atomize pairs =
343   let fun atoms th =
344         (case concl_of th of
345            Const("Trueprop",_) \$ p =>
346              (case head_of p of
347                 Const(a,_) =>
348                   (case assoc(pairs,a) of
349                      Some(rls) => flat (map atoms ([th] RL rls))
350                    | None => [th])
351               | _ => [th])
352          | _ => [th])
353   in atoms end;
355 fun mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);
357 fun unsafe_solver_tac prems =
358   FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
359 val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
361 (*No premature instantiation of variables during simplification*)
362 fun safe_solver_tac prems =
363   FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
364          eq_assume_tac, ematch_tac [FalseE]];
365 val safe_solver = mk_solver "HOL safe" safe_solver_tac;
367 val HOL_basic_ss =
368   empty_ss setsubgoaler asm_simp_tac
369     setSSolver safe_solver
370     setSolver unsafe_solver
371     setmksimps (mksimps mksimps_pairs)
372     setmkeqTrue mk_eq_True
373     setmkcong mk_meta_cong;
375 val HOL_ss =
377      ([triv_forall_equality, (* prunes params *)
378        True_implies_equals, (* prune asms `True' *)
379        eta_contract_eq, (* prunes eta-expansions *)
380        if_True, if_False, if_cancel, if_eq_cancel,
381        imp_disjL, conj_assoc, disj_assoc,
382        de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
383        disj_not1, not_all, not_ex, cases_simp, Eps_eq, Eps_sym_eq,
384        thm"plus_ac0.zero", thm"plus_ac0_zero_right"]
385      @ ex_simps @ all_simps @ simp_thms)
390 (*Simplifies x assuming c and y assuming ~c*)
391 val prems = Goalw [if_def]
392   "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
393 \  (if b then x else y) = (if c then u else v)";
394 by (asm_simp_tac (HOL_ss addsimps prems) 1);
395 qed "if_cong";
397 (*Prevents simplification of x and y: faster and allows the execution
398   of functional programs. NOW THE DEFAULT.*)
399 Goal "b=c ==> (if b then x else y) = (if c then x else y)";
400 by (etac arg_cong 1);
401 qed "if_weak_cong";
403 (*Prevents simplification of t: much faster*)
404 Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
405 by (etac arg_cong 1);
406 qed "let_weak_cong";
408 Goal "f(if c then x else y) = (if c then f x else f y)";
409 by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
410 qed "if_distrib";
412 (*For expand_case_tac*)
413 val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
414 by (case_tac "P" 1);
415 by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
416 qed "expand_case";
418 (*Used in Auth proofs.  Typically P contains Vars that become instantiated
419   during unification.*)
420 fun expand_case_tac P i =
421     res_inst_tac [("P",P)] expand_case i THEN
422     Simp_tac (i+1) THEN
423     Simp_tac i;
425 (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
426   side of an equality.  Used in {Integ,Real}/simproc.ML*)
427 Goal "x=y ==> (x=z) = (y=z)";
428 by (asm_simp_tac HOL_ss 1);
429 qed "restrict_to_left";
431 (* default simpset *)
432 val simpsetup =
433   [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
435 (*** conversion of -->/! into ==>/!! ***)
437 local
438   val rules = [symmetric(thm"all_eq"),symmetric(thm"imp_eq"),Drule.norm_hhf_eq]
439   val ss = HOL_basic_ss addsimps rules
440 in
442 val rulify = zero_var_indexes o strip_shyps_warning o forall_elim_vars_safe o simplify ss;
444 fun qed_spec_mp name = ThmDatabase.ml_store_thm(name, rulify(result()));
446 fun qed_goal_spec_mp name thy s p =
447 	bind_thm (name, rulify (prove_goal thy s p));
449 fun qed_goalw_spec_mp name thy defs s p =
450 	bind_thm (name, rulify (prove_goalw thy defs s p));
452 end;
454 local
456 fun gen_rulify x =
457   Attrib.no_args (Drule.rule_attribute (fn _ => rulify)) x;
459 in
461 val rulify_attrib_setup =
463   [("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];
465 end;
467 (*** integration of simplifier with classical reasoner ***)
469 structure Clasimp = ClasimpFun
470  (structure Simplifier = Simplifier and Splitter = Splitter
471   and Classical  = Classical and Blast = Blast
472   val dest_Trueprop = HOLogic.dest_Trueprop
473   val iff_const = HOLogic.eq_const HOLogic.boolT
474   val not_const = HOLogic.not_const
475   val notE = notE val iffD1 = iffD1 val iffD2 = iffD2
476   val cla_make_elim = cla_make_elim);
477 open Clasimp;
479 val HOL_css = (HOL_cs, HOL_ss);
483 (*** A general refutation procedure ***)
485 (* Parameters:
487    test: term -> bool
488    tests if a term is at all relevant to the refutation proof;
489    if not, then it can be discarded. Can improve performance,
490    esp. if disjunctions can be discarded (no case distinction needed!).
492    prep_tac: int -> tactic
493    A preparation tactic to be applied to the goal once all relevant premises
494    have been moved to the conclusion.
496    ref_tac: int -> tactic
497    the actual refutation tactic. Should be able to deal with goals
498    [| A1; ...; An |] ==> False
499    where the Ai are atomic, i.e. no top-level &, | or EX
500 *)
502 fun refute_tac test prep_tac ref_tac =
503   let val nnf_simps =
504         [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
505          not_all,not_ex,not_not];
506       val nnf_simpset =
507         empty_ss setmkeqTrue mk_eq_True
508                  setmksimps (mksimps mksimps_pairs)