author  paulson 
Fri, 16 Aug 2002 16:41:48 +0200  
changeset 13505  52a16cb7fefb 
parent 13428  99e52e78eb65 
child 13513  b9e14471629c 
permissions  rwrr 
13505  1 
(* Title: ZF/Constructible/Wellorderings.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 2002 University of Cambridge 

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*) 

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header {*Relativized Wellorderings*} 
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theory Wellorderings = Relative: 

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text{*We define functions analogous to @{term ordermap} @{term ordertype} 

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but without using recursion. Instead, there is a direct appeal 

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to Replacement. This will be the basis for a version relativized 

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to some class @{text M}. The main result is Theorem I 7.6 in Kunen, 

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page 17.*} 

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subsection{*Wellorderings*} 

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constdefs 

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irreflexive :: "[i=>o,i,i]=>o" 

13299  22 
"irreflexive(M,A,r) == \<forall>x[M]. x\<in>A > <x,x> \<notin> r" 
13223  23 

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transitive_rel :: "[i=>o,i,i]=>o" 

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"transitive_rel(M,A,r) == 

13299  26 
\<forall>x[M]. x\<in>A > (\<forall>y[M]. y\<in>A > (\<forall>z[M]. z\<in>A > 
13223  27 
<x,y>\<in>r > <y,z>\<in>r > <x,z>\<in>r))" 
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linear_rel :: "[i=>o,i,i]=>o" 

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"linear_rel(M,A,r) == 

13299  31 
\<forall>x[M]. x\<in>A > (\<forall>y[M]. y\<in>A > <x,y>\<in>r  x=y  <y,x>\<in>r)" 
13223  32 

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wellfounded :: "[i=>o,i]=>o" 

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{*EVERY nonempty set has an @{text r}minimal element*} 

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"wellfounded(M,r) == 

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\<forall>x[M]. ~ empty(M,x) 
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> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & <z,y> \<in> r))" 

13223  38 
wellfounded_on :: "[i=>o,i,i]=>o" 
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{*every nonempty SUBSET OF @{text A} has an @{text r}minimal element*} 

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"wellfounded_on(M,A,r) == 

13299  41 
\<forall>x[M]. ~ empty(M,x) > subset(M,x,A) 
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> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & <z,y> \<in> r))" 

13223  43 

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wellordered :: "[i=>o,i,i]=>o" 

45 
{*every nonempty subset of @{text A} has an @{text r}minimal element*} 

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"wellordered(M,A,r) == 

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transitive_rel(M,A,r) & linear_rel(M,A,r) & wellfounded_on(M,A,r)" 

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subsubsection {*Trivial absoluteness proofs*} 

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lemma (in M_axioms) irreflexive_abs [simp]: 

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"M(A) ==> irreflexive(M,A,r) <> irrefl(A,r)" 

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by (simp add: irreflexive_def irrefl_def) 

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lemma (in M_axioms) transitive_rel_abs [simp]: 

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"M(A) ==> transitive_rel(M,A,r) <> trans[A](r)" 

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by (simp add: transitive_rel_def trans_on_def) 

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lemma (in M_axioms) linear_rel_abs [simp]: 

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"M(A) ==> linear_rel(M,A,r) <> linear(A,r)" 

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by (simp add: linear_rel_def linear_def) 

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lemma (in M_axioms) wellordered_is_trans_on: 

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"[ wellordered(M,A,r); M(A) ] ==> trans[A](r)" 

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by (auto simp add: wellordered_def) 
13223  67 

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lemma (in M_axioms) wellordered_is_linear: 

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"[ wellordered(M,A,r); M(A) ] ==> linear(A,r)" 

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by (auto simp add: wellordered_def) 
13223  71 

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lemma (in M_axioms) wellordered_is_wellfounded_on: 

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"[ wellordered(M,A,r); M(A) ] ==> wellfounded_on(M,A,r)" 

13505  74 
by (auto simp add: wellordered_def) 
13223  75 

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lemma (in M_axioms) wellfounded_imp_wellfounded_on: 

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"[ wellfounded(M,r); M(A) ] ==> wellfounded_on(M,A,r)" 

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by (auto simp add: wellfounded_def wellfounded_on_def) 

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lemma (in M_axioms) wellfounded_on_subset_A: 
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"[ wellfounded_on(M,A,r); B<=A ] ==> wellfounded_on(M,B,r)" 

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by (simp add: wellfounded_on_def, blast) 

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13223  84 

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subsubsection {*Wellfounded relations*} 

86 

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lemma (in M_axioms) wellfounded_on_iff_wellfounded: 

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"wellfounded_on(M,A,r) <> wellfounded(M, r \<inter> A*A)" 

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apply (simp add: wellfounded_on_def wellfounded_def, safe) 

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apply blast 

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apply (drule_tac x=x in rspec, assumption, blast) 
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done 
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13247  94 
lemma (in M_axioms) wellfounded_on_imp_wellfounded: 
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"[wellfounded_on(M,A,r); r \<subseteq> A*A] ==> wellfounded(M,r)" 

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by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff) 

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13269  98 
lemma (in M_axioms) wellfounded_on_field_imp_wellfounded: 
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"wellfounded_on(M, field(r), r) ==> wellfounded(M,r)" 

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by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast) 

101 

102 
lemma (in M_axioms) wellfounded_iff_wellfounded_on_field: 

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"M(r) ==> wellfounded(M,r) <> wellfounded_on(M, field(r), r)" 

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by (blast intro: wellfounded_imp_wellfounded_on 

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wellfounded_on_field_imp_wellfounded) 

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(*Consider the least z in domain(r) such that P(z) does not hold...*) 
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lemma (in M_axioms) wellfounded_induct: 
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"[ wellfounded(M,r); M(a); M(r); separation(M, \<lambda>x. ~P(x)); 
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\<forall>x. M(x) & (\<forall>y. <y,x> \<in> r > P(y)) > P(x) ] 
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==> P(a)"; 
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apply (simp (no_asm_use) add: wellfounded_def) 
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apply (drule_tac x="{z \<in> domain(r). ~P(z)}" in rspec) 
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apply (blast dest: transM)+ 

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done 
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lemma (in M_axioms) wellfounded_on_induct: 
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"[ a\<in>A; wellfounded_on(M,A,r); M(A); 

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separation(M, \<lambda>x. x\<in>A > ~P(x)); 

120 
\<forall>x\<in>A. M(x) & (\<forall>y\<in>A. <y,x> \<in> r > P(y)) > P(x) ] 

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==> P(a)"; 

122 
apply (simp (no_asm_use) add: wellfounded_on_def) 

13299  123 
apply (drule_tac x="{z\<in>A. z\<in>A > ~P(z)}" in rspec) 
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apply (blast intro: transM)+ 

13223  125 
done 
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127 
text{*The assumption @{term "r \<subseteq> A*A"} justifies strengthening the induction 

128 
hypothesis by removing the restriction to @{term A}.*} 

129 
lemma (in M_axioms) wellfounded_on_induct2: 

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"[ a\<in>A; wellfounded_on(M,A,r); M(A); r \<subseteq> A*A; 

131 
separation(M, \<lambda>x. x\<in>A > ~P(x)); 

132 
\<forall>x\<in>A. M(x) & (\<forall>y. <y,x> \<in> r > P(y)) > P(x) ] 

133 
==> P(a)"; 

134 
by (rule wellfounded_on_induct, assumption+, blast) 

135 

136 

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subsubsection {*Kunen's lemma IV 3.14, page 123*} 

138 

139 
lemma (in M_axioms) linear_imp_relativized: 

140 
"linear(A,r) ==> linear_rel(M,A,r)" 

141 
by (simp add: linear_def linear_rel_def) 

142 

143 
lemma (in M_axioms) trans_on_imp_relativized: 

144 
"trans[A](r) ==> transitive_rel(M,A,r)" 

145 
by (unfold transitive_rel_def trans_on_def, blast) 

146 

147 
lemma (in M_axioms) wf_on_imp_relativized: 

148 
"wf[A](r) ==> wellfounded_on(M,A,r)" 

149 
apply (simp add: wellfounded_on_def wf_def wf_on_def, clarify) 

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apply (drule_tac x=x in spec, blast) 
13223  151 
done 
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lemma (in M_axioms) wf_imp_relativized: 

154 
"wf(r) ==> wellfounded(M,r)" 

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apply (simp add: wellfounded_def wf_def, clarify) 

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apply (drule_tac x=x in spec, blast) 
13223  157 
done 
158 

159 
lemma (in M_axioms) well_ord_imp_relativized: 

160 
"well_ord(A,r) ==> wellordered(M,A,r)" 

161 
by (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def 

162 
linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized) 

163 

164 

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subsection{* Relativized versions of orderisomorphisms and order types *} 

166 

167 
lemma (in M_axioms) order_isomorphism_abs [simp]: 

168 
"[ M(A); M(B); M(f) ] 

169 
==> order_isomorphism(M,A,r,B,s,f) <> f \<in> ord_iso(A,r,B,s)" 

13352  170 
by (simp add: apply_closed order_isomorphism_def ord_iso_def) 
13223  171 

172 
lemma (in M_axioms) pred_set_abs [simp]: 

173 
"[ M(r); M(B) ] ==> pred_set(M,A,x,r,B) <> B = Order.pred(A,x,r)" 

174 
apply (simp add: pred_set_def Order.pred_def) 

175 
apply (blast dest: transM) 

176 
done 

177 

13245  178 
lemma (in M_axioms) pred_closed [intro,simp]: 
13223  179 
"[ M(A); M(r); M(x) ] ==> M(Order.pred(A,x,r))" 
180 
apply (simp add: Order.pred_def) 

13245  181 
apply (insert pred_separation [of r x], simp) 
13223  182 
done 
183 

184 
lemma (in M_axioms) membership_abs [simp]: 

185 
"[ M(r); M(A) ] ==> membership(M,A,r) <> r = Memrel(A)" 

186 
apply (simp add: membership_def Memrel_def, safe) 

187 
apply (rule equalityI) 

188 
apply clarify 

189 
apply (frule transM, assumption) 

190 
apply blast 

191 
apply clarify 

192 
apply (subgoal_tac "M(<xb,ya>)", blast) 

193 
apply (blast dest: transM) 

194 
apply auto 

195 
done 

196 

197 
lemma (in M_axioms) M_Memrel_iff: 

198 
"M(A) ==> 

13298  199 
Memrel(A) = {z \<in> A*A. \<exists>x[M]. \<exists>y[M]. z = \<langle>x,y\<rangle> & x \<in> y}" 
13223  200 
apply (simp add: Memrel_def) 
201 
apply (blast dest: transM) 

202 
done 

203 

13245  204 
lemma (in M_axioms) Memrel_closed [intro,simp]: 
13223  205 
"M(A) ==> M(Memrel(A))" 
206 
apply (simp add: M_Memrel_iff) 

13245  207 
apply (insert Memrel_separation, simp) 
13223  208 
done 
209 

210 

211 
subsection {* Main results of Kunen, Chapter 1 section 6 *} 

212 

213 
text{*Subset properties proved outside the locale*} 

214 

215 
lemma linear_rel_subset: 

216 
"[ linear_rel(M,A,r); B<=A ] ==> linear_rel(M,B,r)" 

217 
by (unfold linear_rel_def, blast) 

218 

219 
lemma transitive_rel_subset: 

220 
"[ transitive_rel(M,A,r); B<=A ] ==> transitive_rel(M,B,r)" 

221 
by (unfold transitive_rel_def, blast) 

222 

223 
lemma wellfounded_on_subset: 

224 
"[ wellfounded_on(M,A,r); B<=A ] ==> wellfounded_on(M,B,r)" 

225 
by (unfold wellfounded_on_def subset_def, blast) 

226 

227 
lemma wellordered_subset: 

228 
"[ wellordered(M,A,r); B<=A ] ==> wellordered(M,B,r)" 

229 
apply (unfold wellordered_def) 

230 
apply (blast intro: linear_rel_subset transitive_rel_subset 

231 
wellfounded_on_subset) 

232 
done 

233 

234 
text{*Inductive argument for Kunen's Lemma 6.1, etc. 

235 
Simple proof from Halmos, page 72*} 

236 
lemma (in M_axioms) wellordered_iso_subset_lemma: 

237 
"[ wellordered(M,A,r); f \<in> ord_iso(A,r, A',r); A'<= A; y \<in> A; 

238 
M(A); M(f); M(r) ] ==> ~ <f`y, y> \<in> r" 

239 
apply (unfold wellordered_def ord_iso_def) 

240 
apply (elim conjE CollectE) 

241 
apply (erule wellfounded_on_induct, assumption+) 

242 
apply (insert well_ord_iso_separation [of A f r]) 

13352  243 
apply (simp, clarify) 
13223  244 
apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast) 
245 
done 

246 

247 

248 
text{*Kunen's Lemma 6.1: there's no orderisomorphism to an initial segment 

249 
of a wellordering*} 

250 
lemma (in M_axioms) wellordered_iso_predD: 

251 
"[ wellordered(M,A,r); f \<in> ord_iso(A, r, Order.pred(A,x,r), r); 

252 
M(A); M(f); M(r) ] ==> x \<notin> A" 

253 
apply (rule notI) 

254 
apply (frule wellordered_iso_subset_lemma, assumption) 

255 
apply (auto elim: predE) 

256 
(*Now we know ~ (f`x < x) *) 

257 
apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) 

258 
(*Now we also know f`x \<in> pred(A,x,r); contradiction! *) 

259 
apply (simp add: Order.pred_def) 

260 
done 

261 

262 

263 
lemma (in M_axioms) wellordered_iso_pred_eq_lemma: 

264 
"[ f \<in> \<langle>Order.pred(A,y,r), r\<rangle> \<cong> \<langle>Order.pred(A,x,r), r\<rangle>; 

265 
wellordered(M,A,r); x\<in>A; y\<in>A; M(A); M(f); M(r) ] ==> <x,y> \<notin> r" 

266 
apply (frule wellordered_is_trans_on, assumption) 

267 
apply (rule notI) 

268 
apply (drule_tac x2=y and x=x and r2=r in 

269 
wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD]) 

270 
apply (simp add: trans_pred_pred_eq) 

271 
apply (blast intro: predI dest: transM)+ 

272 
done 

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274 

275 
text{*Simple consequence of Lemma 6.1*} 

276 
lemma (in M_axioms) wellordered_iso_pred_eq: 

277 
"[ wellordered(M,A,r); 

278 
f \<in> ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r); 

279 
M(A); M(f); M(r); a\<in>A; c\<in>A ] ==> a=c" 

280 
apply (frule wellordered_is_trans_on, assumption) 

281 
apply (frule wellordered_is_linear, assumption) 

282 
apply (erule_tac x=a and y=c in linearE, auto) 

283 
apply (drule ord_iso_sym) 

284 
(*two symmetric cases*) 

285 
apply (blast dest: wellordered_iso_pred_eq_lemma)+ 

286 
done 

287 

288 
lemma (in M_axioms) wellfounded_on_asym: 

289 
"[ wellfounded_on(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 

290 
apply (simp add: wellfounded_on_def) 

13299  291 
apply (drule_tac x="{x,a}" in rspec) 
292 
apply (blast dest: transM)+ 

13223  293 
done 
294 

295 
lemma (in M_axioms) wellordered_asym: 

296 
"[ wellordered(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 

297 
by (simp add: wellordered_def, blast dest: wellfounded_on_asym) 

298 

299 

300 
text{*Surely a shorter proof using lemmas in @{text Order}? 

13295  301 
Like @{text well_ord_iso_preserving}?*} 
13223  302 
lemma (in M_axioms) ord_iso_pred_imp_lt: 
303 
"[ f \<in> ord_iso(Order.pred(A,x,r), r, i, Memrel(i)); 

304 
g \<in> ord_iso(Order.pred(A,y,r), r, j, Memrel(j)); 

305 
wellordered(M,A,r); x \<in> A; y \<in> A; M(A); M(r); M(f); M(g); M(j); 

306 
Ord(i); Ord(j); \<langle>x,y\<rangle> \<in> r ] 

307 
==> i < j" 

308 
apply (frule wellordered_is_trans_on, assumption) 

309 
apply (frule_tac y=y in transM, assumption) 

310 
apply (rule_tac i=i and j=j in Ord_linear_lt, auto) 

311 
txt{*case @{term "i=j"} yields a contradiction*} 

312 
apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in 

313 
wellordered_iso_predD [THEN notE]) 

314 
apply (blast intro: wellordered_subset [OF _ pred_subset]) 

315 
apply (simp add: trans_pred_pred_eq) 

316 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

317 
apply (simp_all add: pred_iff pred_closed converse_closed comp_closed) 

318 
txt{*case @{term "j<i"} also yields a contradiction*} 

319 
apply (frule restrict_ord_iso2, assumption+) 

320 
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun]) 

321 
apply (frule apply_type, blast intro: ltD) 

322 
{*thus @{term "converse(f)`j \<in> Order.pred(A,x,r)"}*} 

323 
apply (simp add: pred_iff) 

324 
apply (subgoal_tac 

13299  325 
"\<exists>h[M]. h \<in> ord_iso(Order.pred(A,y,r), r, 
13223  326 
Order.pred(A, converse(f)`j, r), r)") 
327 
apply (clarify, frule wellordered_iso_pred_eq, assumption+) 

328 
apply (blast dest: wellordered_asym) 

13299  329 
apply (intro rexI) 
330 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+ 

13223  331 
done 
332 

333 

334 
lemma ord_iso_converse1: 

335 
"[ f: ord_iso(A,r,B,s); <b, f`a>: s; a:A; b:B ] 

336 
==> <converse(f) ` b, a> : r" 

337 
apply (frule ord_iso_converse, assumption+) 

338 
apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype]) 

339 
apply (simp add: left_inverse_bij [OF ord_iso_is_bij]) 

340 
done 

341 

342 

343 
subsection {* Order Types: A Direct Construction by Replacement*} 

344 

345 
text{*This follows Kunen's Theorem I 7.6, page 17.*} 

346 

347 
constdefs 

348 

349 
obase :: "[i=>o,i,i,i] => o" 

350 
{*the domain of @{text om}, eventually shown to equal @{text A}*} 

351 
"obase(M,A,r,z) == 

13293  352 
\<forall>a[M]. 
353 
a \<in> z <> 

13306  354 
(a\<in>A & (\<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M]. 
355 
ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) & 

356 
order_isomorphism(M,par,r,x,mx,g)))" 

13223  357 

358 

359 
omap :: "[i=>o,i,i,i] => o" 

360 
{*the function that maps wosets to order types*} 

361 
"omap(M,A,r,f) == 

13293  362 
\<forall>z[M]. 
363 
z \<in> f <> 

13299  364 
(\<exists>a[M]. a\<in>A & 
13306  365 
(\<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M]. 
366 
ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) & 

367 
pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g)))" 

13223  368 

369 

370 
otype :: "[i=>o,i,i,i] => o" {*the order types themselves*} 

13299  371 
"otype(M,A,r,i) == \<exists>f[M]. omap(M,A,r,f) & is_range(M,f,i)" 
13223  372 

373 

374 

375 
lemma (in M_axioms) obase_iff: 

376 
"[ M(A); M(r); M(z) ] 

377 
==> obase(M,A,r,z) <> 

13306  378 
z = {a\<in>A. \<exists>x[M]. \<exists>g[M]. Ord(x) & 
13223  379 
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}" 
380 
apply (simp add: obase_def Memrel_closed pred_closed) 

381 
apply (rule iffI) 

382 
prefer 2 apply blast 

383 
apply (rule equalityI) 

384 
apply (clarify, frule transM, assumption, rotate_tac 1, simp) 

385 
apply (clarify, frule transM, assumption, force) 

386 
done 

387 

388 
text{*Can also be proved with the premise @{term "M(z)"} instead of 

389 
@{term "M(f)"}, but that version is less useful.*} 

390 
lemma (in M_axioms) omap_iff: 

391 
"[ omap(M,A,r,f); M(A); M(r); M(f) ] 

392 
==> z \<in> f <> 

13306  393 
(\<exists>a\<in>A. \<exists>x[M]. \<exists>g[M]. z = <a,x> & Ord(x) & 
394 
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" 

13223  395 
apply (rotate_tac 1) 
396 
apply (simp add: omap_def Memrel_closed pred_closed) 

13293  397 
apply (rule iffI) 
398 
apply (drule_tac [2] x=z in rspec) 

399 
apply (drule_tac x=z in rspec) 

400 
apply (blast dest: transM)+ 

13223  401 
done 
402 

403 
lemma (in M_axioms) omap_unique: 

404 
"[ omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') ] ==> f' = f" 

405 
apply (rule equality_iffI) 

406 
apply (simp add: omap_iff) 

407 
done 

408 

409 
lemma (in M_axioms) omap_yields_Ord: 

410 
"[ omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) ] ==> Ord(x)" 

411 
apply (simp add: omap_def, blast) 

412 
done 

413 

414 
lemma (in M_axioms) otype_iff: 

415 
"[ otype(M,A,r,i); M(A); M(r); M(i) ] 

416 
==> x \<in> i <> 

13306  417 
(M(x) & Ord(x) & 
418 
(\<exists>a\<in>A. \<exists>g[M]. g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))))" 

419 
apply (auto simp add: omap_iff otype_def) 

420 
apply (blast intro: transM) 

421 
apply (rule rangeI) 

13223  422 
apply (frule transM, assumption) 
423 
apply (simp add: omap_iff, blast) 

424 
done 

425 

426 
lemma (in M_axioms) otype_eq_range: 

13306  427 
"[ omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) ] 
428 
==> i = range(f)" 

13223  429 
apply (auto simp add: otype_def omap_iff) 
430 
apply (blast dest: omap_unique) 

431 
done 

432 

433 

434 
lemma (in M_axioms) Ord_otype: 

435 
"[ otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) ] ==> Ord(i)" 

436 
apply (rotate_tac 1) 

437 
apply (rule OrdI) 

438 
prefer 2 

439 
apply (simp add: Ord_def otype_def omap_def) 

440 
apply clarify 

441 
apply (frule pair_components_in_M, assumption) 

442 
apply blast 

443 
apply (auto simp add: Transset_def otype_iff) 

13306  444 
apply (blast intro: transM) 
445 
apply (blast intro: Ord_in_Ord) 

13223  446 
apply (rename_tac y a g) 
447 
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun, 

448 
THEN apply_funtype], assumption) 

449 
apply (rule_tac x="converse(g)`y" in bexI) 

450 
apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption) 

451 
apply (safe elim!: predE) 

13306  452 
apply (blast intro: restrict_ord_iso ord_iso_sym ltI dest: transM) 
13223  453 
done 
454 

455 
lemma (in M_axioms) domain_omap: 

456 
"[ omap(M,A,r,f); obase(M,A,r,B); M(A); M(r); M(B); M(f) ] 

457 
==> domain(f) = B" 

458 
apply (rotate_tac 2) 

459 
apply (simp add: domain_closed obase_iff) 

460 
apply (rule equality_iffI) 

461 
apply (simp add: domain_iff omap_iff, blast) 

462 
done 

463 

464 
lemma (in M_axioms) omap_subset: 

465 
"[ omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

466 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<subseteq> B * i" 

467 
apply (rotate_tac 3, clarify) 

468 
apply (simp add: omap_iff obase_iff) 

469 
apply (force simp add: otype_iff) 

470 
done 

471 

472 
lemma (in M_axioms) omap_funtype: 

473 
"[ omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

474 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> B > i" 

475 
apply (rotate_tac 3) 

476 
apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff) 

477 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

478 
done 

479 

480 

481 
lemma (in M_axioms) wellordered_omap_bij: 

482 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

483 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> bij(B,i)" 

484 
apply (insert omap_funtype [of A r f B i]) 

485 
apply (auto simp add: bij_def inj_def) 

486 
prefer 2 apply (blast intro: fun_is_surj dest: otype_eq_range) 

13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13306
diff
changeset

487 
apply (frule_tac a=w in apply_Pair, assumption) 
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13306
diff
changeset

488 
apply (frule_tac a=x in apply_Pair, assumption) 
13223  489 
apply (simp add: omap_iff) 
490 
apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans) 

491 
done 

492 

493 

494 
text{*This is not the final result: we must show @{term "oB(A,r) = A"}*} 

495 
lemma (in M_axioms) omap_ord_iso: 

496 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

497 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> ord_iso(B,r,i,Memrel(i))" 

498 
apply (rule ord_isoI) 

499 
apply (erule wellordered_omap_bij, assumption+) 

500 
apply (insert omap_funtype [of A r f B i], simp) 

13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13306
diff
changeset

501 
apply (frule_tac a=x in apply_Pair, assumption) 
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13306
diff
changeset

502 
apply (frule_tac a=y in apply_Pair, assumption) 
13223  503 
apply (auto simp add: omap_iff) 
504 
txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*} 

505 
apply (blast intro: ltD ord_iso_pred_imp_lt) 

506 
txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*} 

507 
apply (rename_tac x y g ga) 

508 
apply (frule wellordered_is_linear, assumption, 

509 
erule_tac x=x and y=y in linearE, assumption+) 

510 
txt{*the case @{term "x=y"} leads to immediate contradiction*} 

511 
apply (blast elim: mem_irrefl) 

512 
txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*} 

513 
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym) 

514 
done 

515 

516 
lemma (in M_axioms) Ord_omap_image_pred: 

517 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

518 
M(A); M(r); M(f); M(B); M(i); b \<in> A ] ==> Ord(f `` Order.pred(A,b,r))" 

519 
apply (frule wellordered_is_trans_on, assumption) 

520 
apply (rule OrdI) 

521 
prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast) 

522 
txt{*Hard part is to show that the image is a transitive set.*} 

523 
apply (rotate_tac 3) 

524 
apply (simp add: Transset_def, clarify) 

525 
apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f B i]]) 

526 
apply (rename_tac c j, clarify) 

527 
apply (frule omap_funtype [of A r f B, THEN apply_funtype], assumption+) 

528 
apply (subgoal_tac "j : i") 

529 
prefer 2 apply (blast intro: Ord_trans Ord_otype) 

530 
apply (subgoal_tac "converse(f) ` j : B") 

531 
prefer 2 

532 
apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij, 

533 
THEN bij_is_fun, THEN apply_funtype]) 

534 
apply (rule_tac x="converse(f) ` j" in bexI) 

535 
apply (simp add: right_inverse_bij [OF wellordered_omap_bij]) 

536 
apply (intro predI conjI) 

537 
apply (erule_tac b=c in trans_onD) 

538 
apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f B i]]) 

539 
apply (auto simp add: obase_iff) 

540 
done 

541 

542 
lemma (in M_axioms) restrict_omap_ord_iso: 

543 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

544 
D \<subseteq> B; M(A); M(r); M(f); M(B); M(i) ] 

545 
==> restrict(f,D) \<in> (\<langle>D,r\<rangle> \<cong> \<langle>f``D, Memrel(f``D)\<rangle>)" 

546 
apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f B i]], 

547 
assumption+) 

548 
apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel]) 

549 
apply (blast dest: subsetD [OF omap_subset]) 

550 
apply (drule ord_iso_sym, simp) 

551 
done 

552 

553 
lemma (in M_axioms) obase_equals: 

554 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

555 
M(A); M(r); M(f); M(B); M(i) ] ==> B = A" 

556 
apply (rotate_tac 4) 

557 
apply (rule equalityI, force simp add: obase_iff, clarify) 

558 
apply (subst obase_iff [of A r B, THEN iffD1], assumption+, simp) 

559 
apply (frule wellordered_is_wellfounded_on, assumption) 

560 
apply (erule wellfounded_on_induct, assumption+) 

13306  561 
apply (frule obase_equals_separation [of A r], assumption) 
562 
apply (simp, clarify) 

13223  563 
apply (rename_tac b) 
564 
apply (subgoal_tac "Order.pred(A,b,r) <= B") 

13306  565 
apply (blast intro!: restrict_omap_ord_iso Ord_omap_image_pred) 
566 
apply (force simp add: pred_iff obase_iff) 

13223  567 
done 
568 

569 

570 

571 
text{*Main result: @{term om} gives the orderisomorphism 

572 
@{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *} 

573 
theorem (in M_axioms) omap_ord_iso_otype: 

574 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

575 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> ord_iso(A, r, i, Memrel(i))" 

576 
apply (frule omap_ord_iso, assumption+) 

577 
apply (frule obase_equals, assumption+, blast) 

13293  578 
done 
13223  579 

580 
lemma (in M_axioms) obase_exists: 

13293  581 
"[ M(A); M(r) ] ==> \<exists>z[M]. obase(M,A,r,z)" 
13223  582 
apply (simp add: obase_def) 
583 
apply (insert obase_separation [of A r]) 

584 
apply (simp add: separation_def) 

585 
done 

586 

587 
lemma (in M_axioms) omap_exists: 

13293  588 
"[ M(A); M(r) ] ==> \<exists>z[M]. omap(M,A,r,z)" 
13223  589 
apply (insert obase_exists [of A r]) 
590 
apply (simp add: omap_def) 

591 
apply (insert omap_replacement [of A r]) 

592 
apply (simp add: strong_replacement_def, clarify) 

13299  593 
apply (drule_tac x=x in rspec, clarify) 
13223  594 
apply (simp add: Memrel_closed pred_closed obase_iff) 
595 
apply (erule impE) 

596 
apply (clarsimp simp add: univalent_def) 

597 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify) 

13293  598 
apply (rule_tac x=Y in rexI) 
599 
apply (simp add: Memrel_closed pred_closed obase_iff, blast, assumption) 

13223  600 
done 
601 

13293  602 
declare rall_simps [simp] rex_simps [simp] 
603 

13223  604 
lemma (in M_axioms) otype_exists: 
13299  605 
"[ wellordered(M,A,r); M(A); M(r) ] ==> \<exists>i[M]. otype(M,A,r,i)" 
13293  606 
apply (insert omap_exists [of A r]) 
607 
apply (simp add: otype_def, safe) 

13299  608 
apply (rule_tac x="range(x)" in rexI) 
609 
apply blast+ 

13223  610 
done 
611 

13428  612 
theorem (in M_axioms) omap_ord_iso_otype': 
13223  613 
"[ wellordered(M,A,r); M(A); M(r) ] 
13299  614 
==> \<exists>f[M]. (\<exists>i[M]. Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" 
13223  615 
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) 
13299  616 
apply (rename_tac i) 
13223  617 
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) 
618 
apply (rule Ord_otype) 

619 
apply (force simp add: otype_def range_closed) 

620 
apply (simp_all add: wellordered_is_trans_on) 

621 
done 

622 

623 
lemma (in M_axioms) ordertype_exists: 

624 
"[ wellordered(M,A,r); M(A); M(r) ] 

13299  625 
==> \<exists>f[M]. (\<exists>i[M]. Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" 
13223  626 
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) 
13299  627 
apply (rename_tac i) 
13428  628 
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype') 
13223  629 
apply (rule Ord_otype) 
630 
apply (force simp add: otype_def range_closed) 

631 
apply (simp_all add: wellordered_is_trans_on) 

632 
done 

633 

634 

635 
lemma (in M_axioms) relativized_imp_well_ord: 

636 
"[ wellordered(M,A,r); M(A); M(r) ] ==> well_ord(A,r)" 

637 
apply (insert ordertype_exists [of A r], simp) 

13505  638 
apply (blast intro: well_ord_ord_iso well_ord_Memrel) 
13223  639 
done 
640 

641 
subsection {*Kunen's theorem 5.4, poage 127*} 

642 

643 
text{*(a) The notion of Wellordering is absolute*} 

644 
theorem (in M_axioms) well_ord_abs [simp]: 

645 
"[ M(A); M(r) ] ==> wellordered(M,A,r) <> well_ord(A,r)" 

646 
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord) 

647 

648 

649 
text{*(b) Order types are absolute*} 

650 
lemma (in M_axioms) 

651 
"[ wellordered(M,A,r); f \<in> ord_iso(A, r, i, Memrel(i)); 

652 
M(A); M(r); M(f); M(i); Ord(i) ] ==> i = ordertype(A,r)" 

653 
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso 

654 
Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

655 

656 
end 