src/ZF/OrdQuant.thy
 author paulson Tue Jul 16 16:28:49 2002 +0200 (2002-07-16) changeset 13362 cd7f9ea58338 parent 13339 0f89104dd377 child 13462 56610e2ba220 permissions -rw-r--r--
tweaked definition of setclass
1 (*  Title:      ZF/AC/OrdQuant.thy
2     ID:         \$Id\$
3     Authors:    Krzysztof Grabczewski and L C Paulson
4 *)
8 theory OrdQuant = Ordinal:
10 subsection {*Quantifiers and union operator for ordinals*}
12 constdefs
14   (* Ordinal Quantifiers *)
15   oall :: "[i, i => o] => o"
16     "oall(A, P) == ALL x. x<A --> P(x)"
18   oex :: "[i, i => o] => o"
19     "oex(A, P)  == EX x. x<A & P(x)"
21   (* Ordinal Union *)
22   OUnion :: "[i, i => i] => i"
23     "OUnion(i,B) == {z: UN x:i. B(x). Ord(i)}"
25 syntax
26   "@oall"     :: "[idt, i, o] => o"        ("(3ALL _<_./ _)" 10)
27   "@oex"      :: "[idt, i, o] => o"        ("(3EX _<_./ _)" 10)
28   "@OUNION"   :: "[idt, i, i] => i"        ("(3UN _<_./ _)" 10)
30 translations
31   "ALL x<a. P"  == "oall(a, %x. P)"
32   "EX x<a. P"   == "oex(a, %x. P)"
33   "UN x<a. B"   == "OUnion(a, %x. B)"
35 syntax (xsymbols)
36   "@oall"     :: "[idt, i, o] => o"        ("(3\<forall>_<_./ _)" 10)
37   "@oex"      :: "[idt, i, o] => o"        ("(3\<exists>_<_./ _)" 10)
38   "@OUNION"   :: "[idt, i, i] => i"        ("(3\<Union>_<_./ _)" 10)
41 subsubsection {*simplification of the new quantifiers*}
44 (*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize
45   is proved.  Ord_atomize would convert this rule to
46     x < 0 ==> P(x) == True, which causes dire effects!*)
47 lemma [simp]: "(ALL x<0. P(x))"
50 lemma [simp]: "~(EX x<0. P(x))"
53 lemma [simp]: "(ALL x<succ(i). P(x)) <-> (Ord(i) --> P(i) & (ALL x<i. P(x)))"
54 apply (simp add: oall_def le_iff)
55 apply (blast intro: lt_Ord2)
56 done
58 lemma [simp]: "(EX x<succ(i). P(x)) <-> (Ord(i) & (P(i) | (EX x<i. P(x))))"
59 apply (simp add: oex_def le_iff)
60 apply (blast intro: lt_Ord2)
61 done
63 subsubsection {*Union over ordinals*}
65 lemma Ord_OUN [intro,simp]:
66      "[| !!x. x<A ==> Ord(B(x)) |] ==> Ord(\<Union>x<A. B(x))"
67 by (simp add: OUnion_def ltI Ord_UN)
69 lemma OUN_upper_lt:
70      "[| a<A;  i < b(a);  Ord(\<Union>x<A. b(x)) |] ==> i < (\<Union>x<A. b(x))"
71 by (unfold OUnion_def lt_def, blast )
73 lemma OUN_upper_le:
74      "[| a<A;  i\<le>b(a);  Ord(\<Union>x<A. b(x)) |] ==> i \<le> (\<Union>x<A. b(x))"
75 apply (unfold OUnion_def, auto)
76 apply (rule UN_upper_le )
77 apply (auto simp add: lt_def)
78 done
80 lemma Limit_OUN_eq: "Limit(i) ==> (UN x<i. x) = i"
81 by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord)
83 (* No < version; consider (UN i:nat.i)=nat *)
84 lemma OUN_least:
85      "(!!x. x<A ==> B(x) \<subseteq> C) ==> (UN x<A. B(x)) \<subseteq> C"
86 by (simp add: OUnion_def UN_least ltI)
88 (* No < version; consider (UN i:nat.i)=nat *)
89 lemma OUN_least_le:
90      "[| Ord(i);  !!x. x<A ==> b(x) \<le> i |] ==> (UN x<A. b(x)) \<le> i"
91 by (simp add: OUnion_def UN_least_le ltI Ord_0_le)
93 lemma le_implies_OUN_le_OUN:
94      "[| !!x. x<A ==> c(x) \<le> d(x) |] ==> (UN x<A. c(x)) \<le> (UN x<A. d(x))"
95 by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN)
97 lemma OUN_UN_eq:
98      "(!!x. x:A ==> Ord(B(x)))
99       ==> (UN z < (UN x:A. B(x)). C(z)) = (UN  x:A. UN z < B(x). C(z))"
102 lemma OUN_Union_eq:
103      "(!!x. x:X ==> Ord(x))
104       ==> (UN z < Union(X). C(z)) = (UN x:X. UN z < x. C(z))"
107 (*So that rule_format will get rid of ALL x<A...*)
108 lemma atomize_oall [symmetric, rulify]:
109      "(!!x. x<A ==> P(x)) == Trueprop (ALL x<A. P(x))"
110 by (simp add: oall_def atomize_all atomize_imp)
112 subsubsection {*universal quantifier for ordinals*}
114 lemma oallI [intro!]:
115     "[| !!x. x<A ==> P(x) |] ==> ALL x<A. P(x)"
118 lemma ospec: "[| ALL x<A. P(x);  x<A |] ==> P(x)"
121 lemma oallE:
122     "[| ALL x<A. P(x);  P(x) ==> Q;  ~x<A ==> Q |] ==> Q"
123 by (simp add: oall_def, blast)
125 lemma rev_oallE [elim]:
126     "[| ALL x<A. P(x);  ~x<A ==> Q;  P(x) ==> Q |] ==> Q"
127 by (simp add: oall_def, blast)
130 (*Trival rewrite rule;   (ALL x<a.P)<->P holds only if a is not 0!*)
131 lemma oall_simp [simp]: "(ALL x<a. True) <-> True"
132 by blast
134 (*Congruence rule for rewriting*)
135 lemma oall_cong [cong]:
136     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
137      ==> oall(a, %x. P(x)) <-> oall(a', %x. P'(x))"
141 subsubsection {*existential quantifier for ordinals*}
143 lemma oexI [intro]:
144     "[| P(x);  x<A |] ==> EX x<A. P(x)"
145 apply (simp add: oex_def, blast)
146 done
148 (*Not of the general form for such rules; ~EX has become ALL~ *)
149 lemma oexCI:
150    "[| ALL x<A. ~P(x) ==> P(a);  a<A |] ==> EX x<A. P(x)"
151 apply (simp add: oex_def, blast)
152 done
154 lemma oexE [elim!]:
155     "[| EX x<A. P(x);  !!x. [| x<A; P(x) |] ==> Q |] ==> Q"
156 apply (simp add: oex_def, blast)
157 done
159 lemma oex_cong [cong]:
160     "[| a=a';  !!x. x<a' ==> P(x) <-> P'(x) |]
161      ==> oex(a, %x. P(x)) <-> oex(a', %x. P'(x))"
163 done
166 subsubsection {*Rules for Ordinal-Indexed Unions*}
168 lemma OUN_I [intro]: "[| a<i;  b: B(a) |] ==> b: (UN z<i. B(z))"
169 by (unfold OUnion_def lt_def, blast)
171 lemma OUN_E [elim!]:
172     "[| b : (UN z<i. B(z));  !!a.[| b: B(a);  a<i |] ==> R |] ==> R"
173 apply (unfold OUnion_def lt_def, blast)
174 done
176 lemma OUN_iff: "b : (UN x<i. B(x)) <-> (EX x<i. b : B(x))"
177 by (unfold OUnion_def oex_def lt_def, blast)
179 lemma OUN_cong [cong]:
180     "[| i=j;  !!x. x<j ==> C(x)=D(x) |] ==> (UN x<i. C(x)) = (UN x<j. D(x))"
181 by (simp add: OUnion_def lt_def OUN_iff)
183 lemma lt_induct:
184     "[| i<k;  !!x.[| x<k;  ALL y<x. P(y) |] ==> P(x) |]  ==>  P(i)"
185 apply (simp add: lt_def oall_def)
186 apply (erule conjE)
187 apply (erule Ord_induct, assumption, blast)
188 done
191 subsection {*Quantification over a class*}
193 constdefs
194   "rall"     :: "[i=>o, i=>o] => o"
195     "rall(M, P) == ALL x. M(x) --> P(x)"
197   "rex"      :: "[i=>o, i=>o] => o"
198     "rex(M, P) == EX x. M(x) & P(x)"
200 syntax
201   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3ALL _[_]./ _)" 10)
202   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3EX _[_]./ _)" 10)
204 syntax (xsymbols)
205   "@rall"     :: "[pttrn, i=>o, o] => o"        ("(3\<forall>_[_]./ _)" 10)
206   "@rex"      :: "[pttrn, i=>o, o] => o"        ("(3\<exists>_[_]./ _)" 10)
208 translations
209   "ALL x[M]. P"  == "rall(M, %x. P)"
210   "EX x[M]. P"   == "rex(M, %x. P)"
213 subsubsection{*Relativized universal quantifier*}
215 lemma rallI [intro!]: "[| !!x. M(x) ==> P(x) |] ==> ALL x[M]. P(x)"
218 lemma rspec: "[| ALL x[M]. P(x); M(x) |] ==> P(x)"
221 (*Instantiates x first: better for automatic theorem proving?*)
222 lemma rev_rallE [elim]:
223     "[| ALL x[M]. P(x);  ~ M(x) ==> Q;  P(x) ==> Q |] ==> Q"
224 by (simp add: rall_def, blast)
226 lemma rallE: "[| ALL x[M]. P(x);  P(x) ==> Q;  ~ M(x) ==> Q |] ==> Q"
227 by blast
229 (*Trival rewrite rule;   (ALL x[M].P)<->P holds only if A is nonempty!*)
230 lemma rall_triv [simp]: "(ALL x[M]. P) <-> ((EX x. M(x)) --> P)"
233 (*Congruence rule for rewriting*)
234 lemma rall_cong [cong]:
235     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (ALL x[M]. P(x)) <-> (ALL x[M]. P'(x))"
239 subsubsection{*Relativized existential quantifier*}
241 lemma rexI [intro]: "[| P(x); M(x) |] ==> EX x[M]. P(x)"
242 by (simp add: rex_def, blast)
244 (*The best argument order when there is only one M(x)*)
245 lemma rev_rexI: "[| M(x);  P(x) |] ==> EX x[M]. P(x)"
246 by blast
248 (*Not of the general form for such rules; ~EX has become ALL~ *)
249 lemma rexCI: "[| ALL x[M]. ~P(x) ==> P(a); M(a) |] ==> EX x[M]. P(x)"
250 by blast
252 lemma rexE [elim!]: "[| EX x[M]. P(x);  !!x. [| M(x); P(x) |] ==> Q |] ==> Q"
253 by (simp add: rex_def, blast)
255 (*We do not even have (EX x[M]. True) <-> True unless A is nonempty!!*)
256 lemma rex_triv [simp]: "(EX x[M]. P) <-> ((EX x. M(x)) & P)"
259 lemma rex_cong [cong]:
260     "(!!x. M(x) ==> P(x) <-> P'(x)) ==> (EX x[M]. P(x)) <-> (EX x[M]. P'(x))"
261 by (simp add: rex_def cong: conj_cong)
263 lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
264 by blast
266 lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
267 by blast
269 lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (ALL x[M]. P(x))";
270 by (simp add: rall_def atomize_all atomize_imp)
272 declare atomize_rall [symmetric, rulify]
274 lemma rall_simps1:
275      "(ALL x[M]. P(x) & Q)   <-> (ALL x[M]. P(x)) & ((ALL x[M]. False) | Q)"
276      "(ALL x[M]. P(x) | Q)   <-> ((ALL x[M]. P(x)) | Q)"
277      "(ALL x[M]. P(x) --> Q) <-> ((EX x[M]. P(x)) --> Q)"
278      "(~(ALL x[M]. P(x))) <-> (EX x[M]. ~P(x))"
279 by blast+
281 lemma rall_simps2:
282      "(ALL x[M]. P & Q(x))   <-> ((ALL x[M]. False) | P) & (ALL x[M]. Q(x))"
283      "(ALL x[M]. P | Q(x))   <-> (P | (ALL x[M]. Q(x)))"
284      "(ALL x[M]. P --> Q(x)) <-> (P --> (ALL x[M]. Q(x)))"
285 by blast+
287 lemmas rall_simps [simp] = rall_simps1 rall_simps2
289 lemma rall_conj_distrib:
290     "(ALL x[M]. P(x) & Q(x)) <-> ((ALL x[M]. P(x)) & (ALL x[M]. Q(x)))"
291 by blast
293 lemma rex_simps1:
294      "(EX x[M]. P(x) & Q) <-> ((EX x[M]. P(x)) & Q)"
295      "(EX x[M]. P(x) | Q) <-> (EX x[M]. P(x)) | ((EX x[M]. True) & Q)"
296      "(EX x[M]. P(x) --> Q) <-> ((ALL x[M]. P(x)) --> ((EX x[M]. True) & Q))"
297      "(~(EX x[M]. P(x))) <-> (ALL x[M]. ~P(x))"
298 by blast+
300 lemma rex_simps2:
301      "(EX x[M]. P & Q(x)) <-> (P & (EX x[M]. Q(x)))"
302      "(EX x[M]. P | Q(x)) <-> ((EX x[M]. True) & P) | (EX x[M]. Q(x))"
303      "(EX x[M]. P --> Q(x)) <-> (((ALL x[M]. False) | P) --> (EX x[M]. Q(x)))"
304 by blast+
306 lemmas rex_simps [simp] = rex_simps1 rex_simps2
308 lemma rex_disj_distrib:
309     "(EX x[M]. P(x) | Q(x)) <-> ((EX x[M]. P(x)) | (EX x[M]. Q(x)))"
310 by blast
313 subsubsection{*One-point rule for bounded quantifiers*}
315 lemma rex_triv_one_point1 [simp]: "(EX x[M]. x=a) <-> ( M(a))"
316 by blast
318 lemma rex_triv_one_point2 [simp]: "(EX x[M]. a=x) <-> ( M(a))"
319 by blast
321 lemma rex_one_point1 [simp]: "(EX x[M]. x=a & P(x)) <-> ( M(a) & P(a))"
322 by blast
324 lemma rex_one_point2 [simp]: "(EX x[M]. a=x & P(x)) <-> ( M(a) & P(a))"
325 by blast
327 lemma rall_one_point1 [simp]: "(ALL x[M]. x=a --> P(x)) <-> ( M(a) --> P(a))"
328 by blast
330 lemma rall_one_point2 [simp]: "(ALL x[M]. a=x --> P(x)) <-> ( M(a) --> P(a))"
331 by blast
334 subsubsection{*Sets as Classes*}
336 constdefs setclass :: "[i,i] => o"       ("**_" [40] 40)
337    "setclass(A) == %x. x : A"
339 lemma setclass_iff [simp]: "setclass(A,x) <-> x : A"
342 lemma rall_setclass_is_ball [simp]: "(\<forall>x[**A]. P(x)) <-> (\<forall>x\<in>A. P(x))"
343 by auto
345 lemma rex_setclass_is_bex [simp]: "(\<exists>x[**A]. P(x)) <-> (\<exists>x\<in>A. P(x))"
346 by auto
349 ML
350 {*
351 val oall_def = thm "oall_def"
352 val oex_def = thm "oex_def"
353 val OUnion_def = thm "OUnion_def"
355 val oallI = thm "oallI";
356 val ospec = thm "ospec";
357 val oallE = thm "oallE";
358 val rev_oallE = thm "rev_oallE";
359 val oall_simp = thm "oall_simp";
360 val oall_cong = thm "oall_cong";
361 val oexI = thm "oexI";
362 val oexCI = thm "oexCI";
363 val oexE = thm "oexE";
364 val oex_cong = thm "oex_cong";
365 val OUN_I = thm "OUN_I";
366 val OUN_E = thm "OUN_E";
367 val OUN_iff = thm "OUN_iff";
368 val OUN_cong = thm "OUN_cong";
369 val lt_induct = thm "lt_induct";
371 val rall_def = thm "rall_def"
372 val rex_def = thm "rex_def"
374 val rallI = thm "rallI";
375 val rspec = thm "rspec";
376 val rallE = thm "rallE";
377 val rev_oallE = thm "rev_oallE";
378 val rall_cong = thm "rall_cong";
379 val rexI = thm "rexI";
380 val rexCI = thm "rexCI";
381 val rexE = thm "rexE";
382 val rex_cong = thm "rex_cong";
384 val Ord_atomize =
385     atomize ([("OrdQuant.oall", [ospec]),("OrdQuant.rall", [rspec])]@
386                  ZF_conn_pairs,
387              ZF_mem_pairs);
388 simpset_ref() := simpset() setmksimps (map mk_eq o Ord_atomize o gen_all);
389 *}
391 text{*Setting up the one-point-rule simproc*}
392 ML
393 {*
395 let
396 val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
397                                 ("EX x[M]. P(x) & Q(x)", FOLogic.oT)
399 val prove_rex_tac = rewtac rex_def THEN
400                     Quantifier1.prove_one_point_ex_tac;
402 val rearrange_bex = Quantifier1.rearrange_bex prove_rex_tac;
404 val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
405                                  ("ALL x[M]. P(x) --> Q(x)", FOLogic.oT)
407 val prove_rall_tac = rewtac rall_def THEN
408                      Quantifier1.prove_one_point_all_tac;
410 val rearrange_ball = Quantifier1.rearrange_ball prove_rall_tac;
412 val defREX_regroup = mk_simproc "defined REX" [ex_pattern] rearrange_bex;
413 val defRALL_regroup = mk_simproc "defined RALL" [all_pattern] rearrange_ball;
414 in