author  wenzelm 
Wed, 27 Mar 2013 16:38:25 +0100  
changeset 51553  63327f679cff 
parent 46953  2b6e55924af3 
child 51717  9e7d1c139569 
permissions  rwrr 
41777  1 
(* Title: ZF/OrdQuant.thy 
2469  2 
Authors: Krzysztof Grabczewski and L C Paulson 
3 
*) 

4 

13253  5 
header {*Special quantifiers*} 
6 

16417  7 
theory OrdQuant imports Ordinal begin 
2469  8 

13253  9 
subsection {*Quantifiers and union operator for ordinals*} 
10 

24893  11 
definition 
2469  12 
(* Ordinal Quantifiers *) 
24893  13 
oall :: "[i, i => o] => o" where 
46820  14 
"oall(A, P) == \<forall>x. x<A \<longrightarrow> P(x)" 
13298  15 

24893  16 
definition 
17 
oex :: "[i, i => o] => o" where 

46820  18 
"oex(A, P) == \<exists>x. x<A & P(x)" 
2469  19 

24893  20 
definition 
2469  21 
(* Ordinal Union *) 
24893  22 
OUnion :: "[i, i => i] => i" where 
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

23 
"OUnion(i,B) == {z: \<Union>x\<in>i. B(x). Ord(i)}" 
13298  24 

2469  25 
syntax 
35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

26 
"_oall" :: "[idt, i, o] => o" ("(3ALL _<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

27 
"_oex" :: "[idt, i, o] => o" ("(3EX _<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

28 
"_OUNION" :: "[idt, i, i] => i" ("(3UN _<_./ _)" 10) 
2469  29 

30 
translations 

24893  31 
"ALL x<a. P" == "CONST oall(a, %x. P)" 
32 
"EX x<a. P" == "CONST oex(a, %x. P)" 

33 
"UN x<a. B" == "CONST OUnion(a, %x. B)" 

2469  34 

12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
6093
diff
changeset

35 
syntax (xsymbols) 
35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

36 
"_oall" :: "[idt, i, o] => o" ("(3\<forall>_<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

37 
"_oex" :: "[idt, i, o] => o" ("(3\<exists>_<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

38 
"_OUNION" :: "[idt, i, i] => i" ("(3\<Union>_<_./ _)" 10) 
14565  39 
syntax (HTML output) 
35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

40 
"_oall" :: "[idt, i, o] => o" ("(3\<forall>_<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

41 
"_oex" :: "[idt, i, o] => o" ("(3\<exists>_<_./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

42 
"_OUNION" :: "[idt, i, i] => i" ("(3\<Union>_<_./ _)" 10) 
12620  43 

44 

13302  45 
subsubsection {*simplification of the new quantifiers*} 
12825  46 

47 

13169  48 
(*MOST IMPORTANT that this is added to the simpset BEFORE Ord_atomize 
13298  49 
is proved. Ord_atomize would convert this rule to 
12825  50 
x < 0 ==> P(x) == True, which causes dire effects!*) 
46820  51 
lemma [simp]: "(\<forall>x<0. P(x))" 
13298  52 
by (simp add: oall_def) 
12825  53 

46820  54 
lemma [simp]: "~(\<exists>x<0. P(x))" 
13298  55 
by (simp add: oex_def) 
12825  56 

46820  57 
lemma [simp]: "(\<forall>x<succ(i). P(x)) <> (Ord(i) \<longrightarrow> P(i) & (\<forall>x<i. P(x)))" 
13298  58 
apply (simp add: oall_def le_iff) 
59 
apply (blast intro: lt_Ord2) 

12825  60 
done 
61 

46820  62 
lemma [simp]: "(\<exists>x<succ(i). P(x)) <> (Ord(i) & (P(i)  (\<exists>x<i. P(x))))" 
13298  63 
apply (simp add: oex_def le_iff) 
64 
apply (blast intro: lt_Ord2) 

12825  65 
done 
66 

13302  67 
subsubsection {*Union over ordinals*} 
13118  68 

12620  69 
lemma Ord_OUN [intro,simp]: 
13162
660a71e712af
New theorems from Constructible, and moving some Isar material from Main
paulson
parents:
13149
diff
changeset

70 
"[ !!x. x<A ==> Ord(B(x)) ] ==> Ord(\<Union>x<A. B(x))" 
13298  71 
by (simp add: OUnion_def ltI Ord_UN) 
12620  72 

73 
lemma OUN_upper_lt: 

13162
660a71e712af
New theorems from Constructible, and moving some Isar material from Main
paulson
parents:
13149
diff
changeset

74 
"[ a<A; i < b(a); Ord(\<Union>x<A. b(x)) ] ==> i < (\<Union>x<A. b(x))" 
12620  75 
by (unfold OUnion_def lt_def, blast ) 
76 

77 
lemma OUN_upper_le: 

13162
660a71e712af
New theorems from Constructible, and moving some Isar material from Main
paulson
parents:
13149
diff
changeset

78 
"[ a<A; i\<le>b(a); Ord(\<Union>x<A. b(x)) ] ==> i \<le> (\<Union>x<A. b(x))" 
12820  79 
apply (unfold OUnion_def, auto) 
12620  80 
apply (rule UN_upper_le ) 
13298  81 
apply (auto simp add: lt_def) 
12620  82 
done 
2469  83 

13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

84 
lemma Limit_OUN_eq: "Limit(i) ==> (\<Union>x<i. x) = i" 
12620  85 
by (simp add: OUnion_def Limit_Union_eq Limit_is_Ord) 
86 

46820  87 
(* No < version of this theorem: consider that @{term"(\<Union>i\<in>nat.i)=nat"}! *) 
12620  88 
lemma OUN_least: 
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

89 
"(!!x. x<A ==> B(x) \<subseteq> C) ==> (\<Union>x<A. B(x)) \<subseteq> C" 
12620  90 
by (simp add: OUnion_def UN_least ltI) 
91 

92 
lemma OUN_least_le: 

13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

93 
"[ Ord(i); !!x. x<A ==> b(x) \<le> i ] ==> (\<Union>x<A. b(x)) \<le> i" 
12620  94 
by (simp add: OUnion_def UN_least_le ltI Ord_0_le) 
95 

96 
lemma le_implies_OUN_le_OUN: 

13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

97 
"[ !!x. x<A ==> c(x) \<le> d(x) ] ==> (\<Union>x<A. c(x)) \<le> (\<Union>x<A. d(x))" 
12620  98 
by (blast intro: OUN_least_le OUN_upper_le le_Ord2 Ord_OUN) 
99 

100 
lemma OUN_UN_eq: 

46953  101 
"(!!x. x \<in> A ==> Ord(B(x))) 
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

102 
==> (\<Union>z < (\<Union>x\<in>A. B(x)). C(z)) = (\<Union>x\<in>A. \<Union>z < B(x). C(z))" 
13298  103 
by (simp add: OUnion_def) 
12620  104 

105 
lemma OUN_Union_eq: 

46953  106 
"(!!x. x \<in> X ==> Ord(x)) 
46820  107 
==> (\<Union>z < \<Union>(X). C(z)) = (\<Union>x\<in>X. \<Union>z < x. C(z))" 
13298  108 
by (simp add: OUnion_def) 
12620  109 

46820  110 
(*So that rule_format will get rid of this quantifier...*) 
12763  111 
lemma atomize_oall [symmetric, rulify]: 
46820  112 
"(!!x. x<A ==> P(x)) == Trueprop (\<forall>x<A. P(x))" 
12763  113 
by (simp add: oall_def atomize_all atomize_imp) 
114 

13302  115 
subsubsection {*universal quantifier for ordinals*} 
13169  116 

117 
lemma oallI [intro!]: 

46820  118 
"[ !!x. x<A ==> P(x) ] ==> \<forall>x<A. P(x)" 
13298  119 
by (simp add: oall_def) 
13169  120 

46820  121 
lemma ospec: "[ \<forall>x<A. P(x); x<A ] ==> P(x)" 
13298  122 
by (simp add: oall_def) 
13169  123 

124 
lemma oallE: 

46820  125 
"[ \<forall>x<A. P(x); P(x) ==> Q; ~x<A ==> Q ] ==> Q" 
13298  126 
by (simp add: oall_def, blast) 
13169  127 

128 
lemma rev_oallE [elim]: 

46820  129 
"[ \<forall>x<A. P(x); ~x<A ==> Q; P(x) ==> Q ] ==> Q" 
13298  130 
by (simp add: oall_def, blast) 
13169  131 

132 

46820  133 
(*Trival rewrite rule. @{term"(\<forall>x<a.P)<>P"} holds only if a is not 0!*) 
134 
lemma oall_simp [simp]: "(\<forall>x<a. True) <> True" 

13170  135 
by blast 
13169  136 

137 
(*Congruence rule for rewriting*) 

138 
lemma oall_cong [cong]: 

13298  139 
"[ a=a'; !!x. x<a' ==> P(x) <> P'(x) ] 
13289
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

140 
==> oall(a, %x. P(x)) <> oall(a', %x. P'(x))" 
13169  141 
by (simp add: oall_def) 
142 

143 

13302  144 
subsubsection {*existential quantifier for ordinals*} 
13169  145 

146 
lemma oexI [intro]: 

46820  147 
"[ P(x); x<A ] ==> \<exists>x<A. P(x)" 
13298  148 
apply (simp add: oex_def, blast) 
13169  149 
done 
150 

46820  151 
(*Not of the general form for such rules... *) 
13169  152 
lemma oexCI: 
46820  153 
"[ \<forall>x<A. ~P(x) ==> P(a); a<A ] ==> \<exists>x<A. P(x)" 
13298  154 
apply (simp add: oex_def, blast) 
13169  155 
done 
156 

157 
lemma oexE [elim!]: 

46820  158 
"[ \<exists>x<A. P(x); !!x. [ x<A; P(x) ] ==> Q ] ==> Q" 
13298  159 
apply (simp add: oex_def, blast) 
13169  160 
done 
161 

162 
lemma oex_cong [cong]: 

13298  163 
"[ a=a'; !!x. x<a' ==> P(x) <> P'(x) ] 
13289
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

164 
==> oex(a, %x. P(x)) <> oex(a', %x. P'(x))" 
13169  165 
apply (simp add: oex_def cong add: conj_cong) 
166 
done 

167 

168 

13302  169 
subsubsection {*Rules for OrdinalIndexed Unions*} 
13169  170 

46953  171 
lemma OUN_I [intro]: "[ a<i; b \<in> B(a) ] ==> b: (\<Union>z<i. B(z))" 
13170  172 
by (unfold OUnion_def lt_def, blast) 
13169  173 

174 
lemma OUN_E [elim!]: 

46953  175 
"[ b \<in> (\<Union>z<i. B(z)); !!a.[ b \<in> B(a); a<i ] ==> R ] ==> R" 
13170  176 
apply (unfold OUnion_def lt_def, blast) 
13169  177 
done 
178 

46820  179 
lemma OUN_iff: "b \<in> (\<Union>x<i. B(x)) <> (\<exists>x<i. b \<in> B(x))" 
13170  180 
by (unfold OUnion_def oex_def lt_def, blast) 
13169  181 

182 
lemma OUN_cong [cong]: 

13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13462
diff
changeset

183 
"[ i=j; !!x. x<j ==> C(x)=D(x) ] ==> (\<Union>x<i. C(x)) = (\<Union>x<j. D(x))" 
13169  184 
by (simp add: OUnion_def lt_def OUN_iff) 
185 

13298  186 
lemma lt_induct: 
46820  187 
"[ i<k; !!x.[ x<k; \<forall>y<x. P(y) ] ==> P(x) ] ==> P(i)" 
13169  188 
apply (simp add: lt_def oall_def) 
13298  189 
apply (erule conjE) 
190 
apply (erule Ord_induct, assumption, blast) 

13169  191 
done 
192 

13253  193 

194 
subsection {*Quantification over a class*} 

195 

24893  196 
definition 
197 
"rall" :: "[i=>o, i=>o] => o" where 

46820  198 
"rall(M, P) == \<forall>x. M(x) \<longrightarrow> P(x)" 
13253  199 

24893  200 
definition 
201 
"rex" :: "[i=>o, i=>o] => o" where 

46820  202 
"rex(M, P) == \<exists>x. M(x) & P(x)" 
13253  203 

204 
syntax 

35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

205 
"_rall" :: "[pttrn, i=>o, o] => o" ("(3ALL _[_]./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

206 
"_rex" :: "[pttrn, i=>o, o] => o" ("(3EX _[_]./ _)" 10) 
13253  207 

208 
syntax (xsymbols) 

35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

209 
"_rall" :: "[pttrn, i=>o, o] => o" ("(3\<forall>_[_]./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

210 
"_rex" :: "[pttrn, i=>o, o] => o" ("(3\<exists>_[_]./ _)" 10) 
14565  211 
syntax (HTML output) 
35112
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

212 
"_rall" :: "[pttrn, i=>o, o] => o" ("(3\<forall>_[_]./ _)" 10) 
ff6f60e6ab85
numeral syntax: clarify parse trees vs. actual terms;
wenzelm
parents:
32010
diff
changeset

213 
"_rex" :: "[pttrn, i=>o, o] => o" ("(3\<exists>_[_]./ _)" 10) 
13253  214 

215 
translations 

24893  216 
"ALL x[M]. P" == "CONST rall(M, %x. P)" 
217 
"EX x[M]. P" == "CONST rex(M, %x. P)" 

13253  218 

13298  219 

220 
subsubsection{*Relativized universal quantifier*} 

13253  221 

46820  222 
lemma rallI [intro!]: "[ !!x. M(x) ==> P(x) ] ==> \<forall>x[M]. P(x)" 
13253  223 
by (simp add: rall_def) 
224 

46820  225 
lemma rspec: "[ \<forall>x[M]. P(x); M(x) ] ==> P(x)" 
13253  226 
by (simp add: rall_def) 
227 

228 
(*Instantiates x first: better for automatic theorem proving?*) 

13298  229 
lemma rev_rallE [elim]: 
46820  230 
"[ \<forall>x[M]. P(x); ~ M(x) ==> Q; P(x) ==> Q ] ==> Q" 
13298  231 
by (simp add: rall_def, blast) 
13253  232 

46820  233 
lemma rallE: "[ \<forall>x[M]. P(x); P(x) ==> Q; ~ M(x) ==> Q ] ==> Q" 
13253  234 
by blast 
235 

236 
(*Trival rewrite rule; (ALL x[M].P)<>P holds only if A is nonempty!*) 

237 
lemma rall_triv [simp]: "(ALL x[M]. P) <> ((EX x. M(x)) > P)" 

238 
by (simp add: rall_def) 

239 

240 
(*Congruence rule for rewriting*) 

241 
lemma rall_cong [cong]: 

46820  242 
"(!!x. M(x) ==> P(x) <> P'(x)) ==> (\<forall>x[M]. P(x)) <> (\<forall>x[M]. P'(x))" 
13253  243 
by (simp add: rall_def) 
244 

13298  245 

246 
subsubsection{*Relativized existential quantifier*} 

13253  247 

46820  248 
lemma rexI [intro]: "[ P(x); M(x) ] ==> \<exists>x[M]. P(x)" 
13253  249 
by (simp add: rex_def, blast) 
250 

251 
(*The best argument order when there is only one M(x)*) 

46820  252 
lemma rev_rexI: "[ M(x); P(x) ] ==> \<exists>x[M]. P(x)" 
13253  253 
by blast 
254 

46820  255 
(*Not of the general form for such rules... *) 
256 
lemma rexCI: "[ \<forall>x[M]. ~P(x) ==> P(a); M(a) ] ==> \<exists>x[M]. P(x)" 

13253  257 
by blast 
258 

46820  259 
lemma rexE [elim!]: "[ \<exists>x[M]. P(x); !!x. [ M(x); P(x) ] ==> Q ] ==> Q" 
13253  260 
by (simp add: rex_def, blast) 
261 

262 
(*We do not even have (EX x[M]. True) <> True unless A is nonempty!!*) 

263 
lemma rex_triv [simp]: "(EX x[M]. P) <> ((EX x. M(x)) & P)" 

264 
by (simp add: rex_def) 

265 

266 
lemma rex_cong [cong]: 

46820  267 
"(!!x. M(x) ==> P(x) <> P'(x)) ==> (\<exists>x[M]. P(x)) <> (\<exists>x[M]. P'(x))" 
13253  268 
by (simp add: rex_def cong: conj_cong) 
269 

13289
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

270 
lemma rall_is_ball [simp]: "(\<forall>x[%z. z\<in>A]. P(x)) <> (\<forall>x\<in>A. P(x))" 
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

271 
by blast 
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

272 

53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

273 
lemma rex_is_bex [simp]: "(\<exists>x[%z. z\<in>A]. P(x)) <> (\<exists>x\<in>A. P(x))" 
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

274 
by blast 
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

275 

46820  276 
lemma atomize_rall: "(!!x. M(x) ==> P(x)) == Trueprop (\<forall>x[M]. P(x))"; 
13253  277 
by (simp add: rall_def atomize_all atomize_imp) 
278 

279 
declare atomize_rall [symmetric, rulify] 

280 

281 
lemma rall_simps1: 

46820  282 
"(\<forall>x[M]. P(x) & Q) <> (\<forall>x[M]. P(x)) & ((\<forall>x[M]. False)  Q)" 
283 
"(\<forall>x[M]. P(x)  Q) <> ((\<forall>x[M]. P(x))  Q)" 

284 
"(\<forall>x[M]. P(x) \<longrightarrow> Q) <> ((\<exists>x[M]. P(x)) \<longrightarrow> Q)" 

285 
"(~(\<forall>x[M]. P(x))) <> (\<exists>x[M]. ~P(x))" 

13253  286 
by blast+ 
287 

288 
lemma rall_simps2: 

46820  289 
"(\<forall>x[M]. P & Q(x)) <> ((\<forall>x[M]. False)  P) & (\<forall>x[M]. Q(x))" 
290 
"(\<forall>x[M]. P  Q(x)) <> (P  (\<forall>x[M]. Q(x)))" 

291 
"(\<forall>x[M]. P \<longrightarrow> Q(x)) <> (P \<longrightarrow> (\<forall>x[M]. Q(x)))" 

13253  292 
by blast+ 
293 

13289
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

294 
lemmas rall_simps [simp] = rall_simps1 rall_simps2 
13253  295 

296 
lemma rall_conj_distrib: 

46820  297 
"(\<forall>x[M]. P(x) & Q(x)) <> ((\<forall>x[M]. P(x)) & (\<forall>x[M]. Q(x)))" 
13253  298 
by blast 
299 

300 
lemma rex_simps1: 

46820  301 
"(\<exists>x[M]. P(x) & Q) <> ((\<exists>x[M]. P(x)) & Q)" 
302 
"(\<exists>x[M]. P(x)  Q) <> (\<exists>x[M]. P(x))  ((\<exists>x[M]. True) & Q)" 

303 
"(\<exists>x[M]. P(x) \<longrightarrow> Q) <> ((\<forall>x[M]. P(x)) \<longrightarrow> ((\<exists>x[M]. True) & Q))" 

304 
"(~(\<exists>x[M]. P(x))) <> (\<forall>x[M]. ~P(x))" 

13253  305 
by blast+ 
306 

307 
lemma rex_simps2: 

46820  308 
"(\<exists>x[M]. P & Q(x)) <> (P & (\<exists>x[M]. Q(x)))" 
309 
"(\<exists>x[M]. P  Q(x)) <> ((\<exists>x[M]. True) & P)  (\<exists>x[M]. Q(x))" 

310 
"(\<exists>x[M]. P \<longrightarrow> Q(x)) <> (((\<forall>x[M]. False)  P) \<longrightarrow> (\<exists>x[M]. Q(x)))" 

13253  311 
by blast+ 
312 

13289
53e201efdaa2
miniscoping for classbounded quantifiers (rall and rex)
paulson
parents:
13253
diff
changeset

313 
lemmas rex_simps [simp] = rex_simps1 rex_simps2 
13253  314 

315 
lemma rex_disj_distrib: 

46820  316 
"(\<exists>x[M]. P(x)  Q(x)) <> ((\<exists>x[M]. P(x))  (\<exists>x[M]. Q(x)))" 
13253  317 
by blast 
318 

319 

13298  320 
subsubsection{*Onepoint rule for bounded quantifiers*} 
13253  321 

46820  322 
lemma rex_triv_one_point1 [simp]: "(\<exists>x[M]. x=a) <> ( M(a))" 
13253  323 
by blast 
324 

46820  325 
lemma rex_triv_one_point2 [simp]: "(\<exists>x[M]. a=x) <> ( M(a))" 
13253  326 
by blast 
327 

46820  328 
lemma rex_one_point1 [simp]: "(\<exists>x[M]. x=a & P(x)) <> ( M(a) & P(a))" 
13253  329 
by blast 
330 

46820  331 
lemma rex_one_point2 [simp]: "(\<exists>x[M]. a=x & P(x)) <> ( M(a) & P(a))" 
13253  332 
by blast 
333 

46820  334 
lemma rall_one_point1 [simp]: "(\<forall>x[M]. x=a \<longrightarrow> P(x)) <> ( M(a) \<longrightarrow> P(a))" 
13253  335 
by blast 
336 

46820  337 
lemma rall_one_point2 [simp]: "(\<forall>x[M]. a=x \<longrightarrow> P(x)) <> ( M(a) \<longrightarrow> P(a))" 
13253  338 
by blast 
339 

340 

13298  341 
subsubsection{*Sets as Classes*} 
342 

24893  343 
definition 
344 
setclass :: "[i,i] => o" ("##_" [40] 40) where 

46820  345 
"setclass(A) == %x. x \<in> A" 
13298  346 

46820  347 
lemma setclass_iff [simp]: "setclass(A,x) <> x \<in> A" 
13362  348 
by (simp add: setclass_def) 
13298  349 

13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13615
diff
changeset

350 
lemma rall_setclass_is_ball [simp]: "(\<forall>x[##A]. P(x)) <> (\<forall>x\<in>A. P(x))" 
13298  351 
by auto 
352 

13807
a28a8fbc76d4
changed ** to ## to avoid conflict with new comment syntax
paulson
parents:
13615
diff
changeset

353 
lemma rex_setclass_is_bex [simp]: "(\<exists>x[##A]. P(x)) <> (\<exists>x\<in>A. P(x))" 
13298  354 
by auto 
355 

356 

13169  357 
ML 
358 
{* 

359 
val Ord_atomize = 

24893  360 
atomize ([("OrdQuant.oall", [@{thm ospec}]),("OrdQuant.rall", [@{thm rspec}])]@ 
13298  361 
ZF_conn_pairs, 
13253  362 
ZF_mem_pairs); 
26339  363 
*} 
364 
declaration {* fn _ => 

45625
750c5a47400b
modernized some oldstyle infix operations, which were left over from the time of ML proof scripts;
wenzelm
parents:
42459
diff
changeset

365 
Simplifier.map_ss (Simplifier.set_mksimps (K (map mk_eq o Ord_atomize o gen_all))) 
13169  366 
*} 
367 

13462  368 
text {* Setting up the onepointrule simproc *} 
13253  369 

46820  370 
simproc_setup defined_rex ("\<exists>x[M]. P(x) & Q(x)") = {* 
42455  371 
let 
372 
val unfold_rex_tac = unfold_tac @{thms rex_def}; 

373 
fun prove_rex_tac ss = unfold_rex_tac ss THEN Quantifier1.prove_one_point_ex_tac; 

42459  374 
in fn _ => fn ss => Quantifier1.rearrange_bex (prove_rex_tac ss) ss end 
42455  375 
*} 
13253  376 

46820  377 
simproc_setup defined_rall ("\<forall>x[M]. P(x) \<longrightarrow> Q(x)") = {* 
42455  378 
let 
379 
val unfold_rall_tac = unfold_tac @{thms rall_def}; 

380 
fun prove_rall_tac ss = unfold_rall_tac ss THEN Quantifier1.prove_one_point_all_tac; 

42459  381 
in fn _ => fn ss => Quantifier1.rearrange_ball (prove_rall_tac ss) ss end 
13253  382 
*} 
383 

2469  384 
end 