author  paulson 
Wed, 26 Jun 2002 18:31:20 +0200  
changeset 13251  74cb2af8811e 
parent 13247  e3c289f0724b 
child 13269  3ba9be497c33 
permissions  rwrr 
13223  1 
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" 

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"irreflexive(M,A,r) == \<forall>x\<in>A. M(x) > <x,x> \<notin> r" 

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

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"transitive_rel(M,A,r) == 

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\<forall>x\<in>A. M(x) > (\<forall>y\<in>A. M(y) > (\<forall>z\<in>A. M(z) > 

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<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) == 

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\<forall>x\<in>A. M(x) > (\<forall>y\<in>A. M(y) > <x,y>\<in>r  x=y  <y,x>\<in>r)" 

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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(x) > ~ empty(M,x) 

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

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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) == 

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\<forall>x. M(x) > ~ empty(M,x) > subset(M,x,A) 

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

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wellordered :: "[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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"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 ) 

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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 ) 

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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)" 

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by (auto simp add: wellordered_def ) 

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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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subsubsection {*Wellfounded relations*} 

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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 spec, blast) 

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done 

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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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(*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 spec) 
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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)); 

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\<forall>x\<in>A. M(x) & (\<forall>y\<in>A. <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_on_def) 

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apply (drule_tac x="{z\<in>A. z\<in>A > ~P(z)}" in spec) 

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apply (blast intro: transM) 

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done 

107 

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

109 
hypothesis by removing the restriction to @{term A}.*} 

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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; 

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

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

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==> P(a)"; 

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by (rule wellfounded_on_induct, assumption+, blast) 

116 

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

119 

120 
lemma (in M_axioms) linear_imp_relativized: 

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

122 
by (simp add: linear_def linear_rel_def) 

123 

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lemma (in M_axioms) trans_on_imp_relativized: 

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

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by (unfold transitive_rel_def trans_on_def, blast) 

127 

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lemma (in M_axioms) wf_on_imp_relativized: 

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

130 
apply (simp add: wellfounded_on_def wf_def wf_on_def, clarify) 

131 
apply (drule_tac x="x" in spec, blast) 

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done 

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134 
lemma (in M_axioms) wf_imp_relativized: 

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"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) 

138 
done 

139 

140 
lemma (in M_axioms) well_ord_imp_relativized: 

141 
"well_ord(A,r) ==> wellordered(M,A,r)" 

142 
by (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def 

143 
linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized) 

144 

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

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148 
lemma (in M_axioms) order_isomorphism_abs [simp]: 

149 
"[ M(A); M(B); M(f) ] 

150 
==> order_isomorphism(M,A,r,B,s,f) <> f \<in> ord_iso(A,r,B,s)" 

151 
by (simp add: typed_apply_abs [OF bij_is_fun] apply_closed 

152 
order_isomorphism_def ord_iso_def) 

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lemma (in M_axioms) pred_set_abs [simp]: 

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

157 
apply (simp add: pred_set_def Order.pred_def) 

158 
apply (blast dest: transM) 

159 
done 

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lemma (in M_axioms) pred_closed [intro,simp]: 
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"[ M(A); M(r); M(x) ] ==> M(Order.pred(A,x,r))" 
163 
apply (simp add: Order.pred_def) 

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apply (insert pred_separation [of r x], simp) 
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done 
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lemma (in M_axioms) membership_abs [simp]: 

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

169 
apply (simp add: membership_def Memrel_def, safe) 

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apply (rule equalityI) 

171 
apply clarify 

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apply (frule transM, assumption) 

173 
apply blast 

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apply clarify 

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

176 
apply (blast dest: transM) 

177 
apply auto 

178 
done 

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lemma (in M_axioms) M_Memrel_iff: 

181 
"M(A) ==> 

182 
Memrel(A) = {z \<in> A*A. \<exists>x. M(x) \<and> (\<exists>y. M(y) \<and> z = \<langle>x,y\<rangle> \<and> x \<in> y)}" 

183 
apply (simp add: Memrel_def) 

184 
apply (blast dest: transM) 

185 
done 

186 

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lemma (in M_axioms) Memrel_closed [intro,simp]: 
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"M(A) ==> M(Memrel(A))" 
189 
apply (simp add: M_Memrel_iff) 

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apply (insert Memrel_separation, simp) 
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done 
192 

193 

194 
subsection {* Main results of Kunen, Chapter 1 section 6 *} 

195 

196 
text{*Subset properties proved outside the locale*} 

197 

198 
lemma linear_rel_subset: 

199 
"[ linear_rel(M,A,r); B<=A ] ==> linear_rel(M,B,r)" 

200 
by (unfold linear_rel_def, blast) 

201 

202 
lemma transitive_rel_subset: 

203 
"[ transitive_rel(M,A,r); B<=A ] ==> transitive_rel(M,B,r)" 

204 
by (unfold transitive_rel_def, blast) 

205 

206 
lemma wellfounded_on_subset: 

207 
"[ wellfounded_on(M,A,r); B<=A ] ==> wellfounded_on(M,B,r)" 

208 
by (unfold wellfounded_on_def subset_def, blast) 

209 

210 
lemma wellordered_subset: 

211 
"[ wellordered(M,A,r); B<=A ] ==> wellordered(M,B,r)" 

212 
apply (unfold wellordered_def) 

213 
apply (blast intro: linear_rel_subset transitive_rel_subset 

214 
wellfounded_on_subset) 

215 
done 

216 

217 
text{*Inductive argument for Kunen's Lemma 6.1, etc. 

218 
Simple proof from Halmos, page 72*} 

219 
lemma (in M_axioms) wellordered_iso_subset_lemma: 

220 
"[ wellordered(M,A,r); f \<in> ord_iso(A,r, A',r); A'<= A; y \<in> A; 

221 
M(A); M(f); M(r) ] ==> ~ <f`y, y> \<in> r" 

222 
apply (unfold wellordered_def ord_iso_def) 

223 
apply (elim conjE CollectE) 

224 
apply (erule wellfounded_on_induct, assumption+) 

225 
apply (insert well_ord_iso_separation [of A f r]) 

226 
apply (simp add: typed_apply_abs [OF bij_is_fun] apply_closed, clarify) 

227 
apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast) 

228 
done 

229 

230 

231 
text{*Kunen's Lemma 6.1: there's no orderisomorphism to an initial segment 

232 
of a wellordering*} 

233 
lemma (in M_axioms) wellordered_iso_predD: 

234 
"[ wellordered(M,A,r); f \<in> ord_iso(A, r, Order.pred(A,x,r), r); 

235 
M(A); M(f); M(r) ] ==> x \<notin> A" 

236 
apply (rule notI) 

237 
apply (frule wellordered_iso_subset_lemma, assumption) 

238 
apply (auto elim: predE) 

239 
(*Now we know ~ (f`x < x) *) 

240 
apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) 

241 
(*Now we also know f`x \<in> pred(A,x,r); contradiction! *) 

242 
apply (simp add: Order.pred_def) 

243 
done 

244 

245 

246 
lemma (in M_axioms) wellordered_iso_pred_eq_lemma: 

247 
"[ f \<in> \<langle>Order.pred(A,y,r), r\<rangle> \<cong> \<langle>Order.pred(A,x,r), r\<rangle>; 

248 
wellordered(M,A,r); x\<in>A; y\<in>A; M(A); M(f); M(r) ] ==> <x,y> \<notin> r" 

249 
apply (frule wellordered_is_trans_on, assumption) 

250 
apply (rule notI) 

251 
apply (drule_tac x2=y and x=x and r2=r in 

252 
wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD]) 

253 
apply (simp add: trans_pred_pred_eq) 

254 
apply (blast intro: predI dest: transM)+ 

255 
done 

256 

257 

258 
text{*Simple consequence of Lemma 6.1*} 

259 
lemma (in M_axioms) wellordered_iso_pred_eq: 

260 
"[ wellordered(M,A,r); 

261 
f \<in> ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r); 

262 
M(A); M(f); M(r); a\<in>A; c\<in>A ] ==> a=c" 

263 
apply (frule wellordered_is_trans_on, assumption) 

264 
apply (frule wellordered_is_linear, assumption) 

265 
apply (erule_tac x=a and y=c in linearE, auto) 

266 
apply (drule ord_iso_sym) 

267 
(*two symmetric cases*) 

268 
apply (blast dest: wellordered_iso_pred_eq_lemma)+ 

269 
done 

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271 
lemma (in M_axioms) wellfounded_on_asym: 

272 
"[ wellfounded_on(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 

273 
apply (simp add: wellfounded_on_def) 

274 
apply (drule_tac x="{x,a}" in spec) 

275 
apply (simp add: cons_closed) 

276 
apply (blast dest: transM) 

277 
done 

278 

279 
lemma (in M_axioms) wellordered_asym: 

280 
"[ wellordered(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 

281 
by (simp add: wellordered_def, blast dest: wellfounded_on_asym) 

282 

283 

284 
text{*Surely a shorter proof using lemmas in @{text Order}? 

285 
Like well_ord_iso_preserving?*} 

286 
lemma (in M_axioms) ord_iso_pred_imp_lt: 

287 
"[ f \<in> ord_iso(Order.pred(A,x,r), r, i, Memrel(i)); 

288 
g \<in> ord_iso(Order.pred(A,y,r), r, j, Memrel(j)); 

289 
wellordered(M,A,r); x \<in> A; y \<in> A; M(A); M(r); M(f); M(g); M(j); 

290 
Ord(i); Ord(j); \<langle>x,y\<rangle> \<in> r ] 

291 
==> i < j" 

292 
apply (frule wellordered_is_trans_on, assumption) 

293 
apply (frule_tac y=y in transM, assumption) 

294 
apply (rule_tac i=i and j=j in Ord_linear_lt, auto) 

295 
txt{*case @{term "i=j"} yields a contradiction*} 

296 
apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in 

297 
wellordered_iso_predD [THEN notE]) 

298 
apply (blast intro: wellordered_subset [OF _ pred_subset]) 

299 
apply (simp add: trans_pred_pred_eq) 

300 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

301 
apply (simp_all add: pred_iff pred_closed converse_closed comp_closed) 

302 
txt{*case @{term "j<i"} also yields a contradiction*} 

303 
apply (frule restrict_ord_iso2, assumption+) 

304 
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun]) 

305 
apply (frule apply_type, blast intro: ltD) 

306 
{*thus @{term "converse(f)`j \<in> Order.pred(A,x,r)"}*} 

307 
apply (simp add: pred_iff) 

308 
apply (subgoal_tac 

309 
"\<exists>h. M(h) & h \<in> ord_iso(Order.pred(A,y,r), r, 

310 
Order.pred(A, converse(f)`j, r), r)") 

311 
apply (clarify, frule wellordered_iso_pred_eq, assumption+) 

312 
apply (blast dest: wellordered_asym) 

313 
apply (intro exI conjI) 

314 
prefer 2 apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+ 

315 
done 

316 

317 

318 
lemma ord_iso_converse1: 

319 
"[ f: ord_iso(A,r,B,s); <b, f`a>: s; a:A; b:B ] 

320 
==> <converse(f) ` b, a> : r" 

321 
apply (frule ord_iso_converse, assumption+) 

322 
apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype]) 

323 
apply (simp add: left_inverse_bij [OF ord_iso_is_bij]) 

324 
done 

325 

326 

327 
subsection {* Order Types: A Direct Construction by Replacement*} 

328 

329 
text{*This follows Kunen's Theorem I 7.6, page 17.*} 

330 

331 
constdefs 

332 

333 
obase :: "[i=>o,i,i,i] => o" 

334 
{*the domain of @{text om}, eventually shown to equal @{text A}*} 

335 
"obase(M,A,r,z) == 

336 
\<forall>a. M(a) > 

337 
(a \<in> z <> 

338 
(a\<in>A & (\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) & 

339 
membership(M,x,mx) & pred_set(M,A,a,r,par) & 

340 
order_isomorphism(M,par,r,x,mx,g))))" 

341 

342 

343 
omap :: "[i=>o,i,i,i] => o" 

344 
{*the function that maps wosets to order types*} 

345 
"omap(M,A,r,f) == 

346 
\<forall>z. M(z) > 

347 
(z \<in> f <> 

348 
(\<exists>a\<in>A. M(a) & 

349 
(\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) & 

350 
pair(M,a,x,z) & membership(M,x,mx) & 

351 
pred_set(M,A,a,r,par) & 

352 
order_isomorphism(M,par,r,x,mx,g))))" 

353 

354 

355 
otype :: "[i=>o,i,i,i] => o" {*the order types themselves*} 

356 
"otype(M,A,r,i) == \<exists>f. M(f) & omap(M,A,r,f) & is_range(M,f,i)" 

357 

358 

359 

360 
lemma (in M_axioms) obase_iff: 

361 
"[ M(A); M(r); M(z) ] 

362 
==> obase(M,A,r,z) <> 

363 
z = {a\<in>A. \<exists>x g. M(x) & M(g) & Ord(x) & 

364 
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}" 

365 
apply (simp add: obase_def Memrel_closed pred_closed) 

366 
apply (rule iffI) 

367 
prefer 2 apply blast 

368 
apply (rule equalityI) 

369 
apply (clarify, frule transM, assumption, rotate_tac 1, simp) 

370 
apply (clarify, frule transM, assumption, force) 

371 
done 

372 

373 
text{*Can also be proved with the premise @{term "M(z)"} instead of 

374 
@{term "M(f)"}, but that version is less useful.*} 

375 
lemma (in M_axioms) omap_iff: 

376 
"[ omap(M,A,r,f); M(A); M(r); M(f) ] 

377 
==> z \<in> f <> 

378 
(\<exists>a\<in>A. \<exists>x g. M(x) & M(g) & z = <a,x> & Ord(x) & 

379 
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" 

380 
apply (rotate_tac 1) 

381 
apply (simp add: omap_def Memrel_closed pred_closed) 

382 
apply (rule iffI) 

383 
apply (drule_tac x=z in spec, blast dest: transM)+ 

384 
done 

385 

386 
lemma (in M_axioms) omap_unique: 

387 
"[ omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') ] ==> f' = f" 

388 
apply (rule equality_iffI) 

389 
apply (simp add: omap_iff) 

390 
done 

391 

392 
lemma (in M_axioms) omap_yields_Ord: 

393 
"[ omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) ] ==> Ord(x)" 

394 
apply (simp add: omap_def, blast) 

395 
done 

396 

397 
lemma (in M_axioms) otype_iff: 

398 
"[ otype(M,A,r,i); M(A); M(r); M(i) ] 

399 
==> x \<in> i <> 

400 
(\<exists>a\<in>A. \<exists>g. M(x) & M(g) & Ord(x) & 

401 
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" 

402 
apply (simp add: otype_def, auto) 

403 
apply (blast dest: transM) 

404 
apply (blast dest!: omap_iff intro: transM) 

405 
apply (rename_tac a g) 

406 
apply (rule_tac a=a in rangeI) 

407 
apply (frule transM, assumption) 

408 
apply (simp add: omap_iff, blast) 

409 
done 

410 

411 
lemma (in M_axioms) otype_eq_range: 

412 
"[ omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) ] ==> i = range(f)" 

413 
apply (auto simp add: otype_def omap_iff) 

414 
apply (blast dest: omap_unique) 

415 
done 

416 

417 

418 
lemma (in M_axioms) Ord_otype: 

419 
"[ otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) ] ==> Ord(i)" 

420 
apply (rotate_tac 1) 

421 
apply (rule OrdI) 

422 
prefer 2 

423 
apply (simp add: Ord_def otype_def omap_def) 

424 
apply clarify 

425 
apply (frule pair_components_in_M, assumption) 

426 
apply blast 

427 
apply (auto simp add: Transset_def otype_iff) 

428 
apply (blast intro: transM) 

429 
apply (rename_tac y a g) 

430 
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun, 

431 
THEN apply_funtype], assumption) 

432 
apply (rule_tac x="converse(g)`y" in bexI) 

433 
apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption) 

434 
apply (safe elim!: predE) 

435 
apply (intro conjI exI) 

436 
prefer 3 

437 
apply (blast intro: restrict_ord_iso ord_iso_sym ltI) 

438 
apply (blast intro: transM) 

439 
apply (blast intro: Ord_in_Ord) 

440 
done 

441 

442 
lemma (in M_axioms) domain_omap: 

443 
"[ omap(M,A,r,f); obase(M,A,r,B); M(A); M(r); M(B); M(f) ] 

444 
==> domain(f) = B" 

445 
apply (rotate_tac 2) 

446 
apply (simp add: domain_closed obase_iff) 

447 
apply (rule equality_iffI) 

448 
apply (simp add: domain_iff omap_iff, blast) 

449 
done 

450 

451 
lemma (in M_axioms) omap_subset: 

452 
"[ omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

453 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<subseteq> B * i" 

454 
apply (rotate_tac 3, clarify) 

455 
apply (simp add: omap_iff obase_iff) 

456 
apply (force simp add: otype_iff) 

457 
done 

458 

459 
lemma (in M_axioms) omap_funtype: 

460 
"[ omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

461 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> B > i" 

462 
apply (rotate_tac 3) 

463 
apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff) 

464 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

465 
done 

466 

467 

468 
lemma (in M_axioms) wellordered_omap_bij: 

469 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

470 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> bij(B,i)" 

471 
apply (insert omap_funtype [of A r f B i]) 

472 
apply (auto simp add: bij_def inj_def) 

473 
prefer 2 apply (blast intro: fun_is_surj dest: otype_eq_range) 

474 
apply (frule_tac a="w" in apply_Pair, assumption) 

475 
apply (frule_tac a="x" in apply_Pair, assumption) 

476 
apply (simp add: omap_iff) 

477 
apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans) 

478 
done 

479 

480 

481 
text{*This is not the final result: we must show @{term "oB(A,r) = A"}*} 

482 
lemma (in M_axioms) omap_ord_iso: 

483 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

484 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> ord_iso(B,r,i,Memrel(i))" 

485 
apply (rule ord_isoI) 

486 
apply (erule wellordered_omap_bij, assumption+) 

487 
apply (insert omap_funtype [of A r f B i], simp) 

488 
apply (frule_tac a="x" in apply_Pair, assumption) 

489 
apply (frule_tac a="y" in apply_Pair, assumption) 

490 
apply (auto simp add: omap_iff) 

491 
txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*} 

492 
apply (blast intro: ltD ord_iso_pred_imp_lt) 

493 
txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*} 

494 
apply (rename_tac x y g ga) 

495 
apply (frule wellordered_is_linear, assumption, 

496 
erule_tac x=x and y=y in linearE, assumption+) 

497 
txt{*the case @{term "x=y"} leads to immediate contradiction*} 

498 
apply (blast elim: mem_irrefl) 

499 
txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*} 

500 
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym) 

501 
done 

502 

503 
lemma (in M_axioms) Ord_omap_image_pred: 

504 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

505 
M(A); M(r); M(f); M(B); M(i); b \<in> A ] ==> Ord(f `` Order.pred(A,b,r))" 

506 
apply (frule wellordered_is_trans_on, assumption) 

507 
apply (rule OrdI) 

508 
prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast) 

509 
txt{*Hard part is to show that the image is a transitive set.*} 

510 
apply (rotate_tac 3) 

511 
apply (simp add: Transset_def, clarify) 

512 
apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f B i]]) 

513 
apply (rename_tac c j, clarify) 

514 
apply (frule omap_funtype [of A r f B, THEN apply_funtype], assumption+) 

515 
apply (subgoal_tac "j : i") 

516 
prefer 2 apply (blast intro: Ord_trans Ord_otype) 

517 
apply (subgoal_tac "converse(f) ` j : B") 

518 
prefer 2 

519 
apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij, 

520 
THEN bij_is_fun, THEN apply_funtype]) 

521 
apply (rule_tac x="converse(f) ` j" in bexI) 

522 
apply (simp add: right_inverse_bij [OF wellordered_omap_bij]) 

523 
apply (intro predI conjI) 

524 
apply (erule_tac b=c in trans_onD) 

525 
apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f B i]]) 

526 
apply (auto simp add: obase_iff) 

527 
done 

528 

529 
lemma (in M_axioms) restrict_omap_ord_iso: 

530 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

531 
D \<subseteq> B; M(A); M(r); M(f); M(B); M(i) ] 

532 
==> restrict(f,D) \<in> (\<langle>D,r\<rangle> \<cong> \<langle>f``D, Memrel(f``D)\<rangle>)" 

533 
apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f B i]], 

534 
assumption+) 

535 
apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel]) 

536 
apply (blast dest: subsetD [OF omap_subset]) 

537 
apply (drule ord_iso_sym, simp) 

538 
done 

539 

540 
lemma (in M_axioms) obase_equals: 

541 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

542 
M(A); M(r); M(f); M(B); M(i) ] ==> B = A" 

543 
apply (rotate_tac 4) 

544 
apply (rule equalityI, force simp add: obase_iff, clarify) 

545 
apply (subst obase_iff [of A r B, THEN iffD1], assumption+, simp) 

546 
apply (frule wellordered_is_wellfounded_on, assumption) 

547 
apply (erule wellfounded_on_induct, assumption+) 

548 
apply (insert obase_equals_separation, simp add: Memrel_closed pred_closed, clarify) 

549 
apply (rename_tac b) 

550 
apply (subgoal_tac "Order.pred(A,b,r) <= B") 

551 
prefer 2 apply (force simp add: pred_iff obase_iff) 

552 
apply (intro conjI exI) 

553 
prefer 4 apply (blast intro: restrict_omap_ord_iso) 

554 
apply (blast intro: Ord_omap_image_pred)+ 

555 
done 

556 

557 

558 

559 
text{*Main result: @{term om} gives the orderisomorphism 

560 
@{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *} 

561 
theorem (in M_axioms) omap_ord_iso_otype: 

562 
"[ wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); 

563 
M(A); M(r); M(f); M(B); M(i) ] ==> f \<in> ord_iso(A, r, i, Memrel(i))" 

564 
apply (frule omap_ord_iso, assumption+) 

565 
apply (frule obase_equals, assumption+, blast) 

566 
done 

567 

568 
lemma (in M_axioms) obase_exists: 

569 
"[ M(A); M(r) ] ==> \<exists>z. M(z) & obase(M,A,r,z)" 

570 
apply (simp add: obase_def) 

571 
apply (insert obase_separation [of A r]) 

572 
apply (simp add: separation_def) 

573 
done 

574 

575 
lemma (in M_axioms) omap_exists: 

576 
"[ M(A); M(r) ] ==> \<exists>z. M(z) & omap(M,A,r,z)" 

577 
apply (insert obase_exists [of A r]) 

578 
apply (simp add: omap_def) 

579 
apply (insert omap_replacement [of A r]) 

580 
apply (simp add: strong_replacement_def, clarify) 

581 
apply (drule_tac x=z in spec, clarify) 

582 
apply (simp add: Memrel_closed pred_closed obase_iff) 

583 
apply (erule impE) 

584 
apply (clarsimp simp add: univalent_def) 

585 
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify) 

586 
apply (rule_tac x=Y in exI) 

587 
apply (simp add: Memrel_closed pred_closed obase_iff, blast) 

588 
done 

589 

590 
lemma (in M_axioms) otype_exists: 

591 
"[ wellordered(M,A,r); M(A); M(r) ] ==> \<exists>i. M(i) & otype(M,A,r,i)" 

592 
apply (insert omap_exists [of A r]) 

593 
apply (simp add: otype_def, clarify) 

594 
apply (rule_tac x="range(z)" in exI) 

595 
apply blast 

596 
done 

597 

598 
theorem (in M_axioms) omap_ord_iso_otype: 

599 
"[ wellordered(M,A,r); M(A); M(r) ] 

600 
==> \<exists>f. M(f) & (\<exists>i. M(i) & Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" 

601 
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) 

602 
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) 

603 
apply (rule Ord_otype) 

604 
apply (force simp add: otype_def range_closed) 

605 
apply (simp_all add: wellordered_is_trans_on) 

606 
done 

607 

608 
lemma (in M_axioms) ordertype_exists: 

609 
"[ wellordered(M,A,r); M(A); M(r) ] 

610 
==> \<exists>f. M(f) & (\<exists>i. M(i) & Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" 

611 
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) 

612 
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) 

613 
apply (rule Ord_otype) 

614 
apply (force simp add: otype_def range_closed) 

615 
apply (simp_all add: wellordered_is_trans_on) 

616 
done 

617 

618 

619 
lemma (in M_axioms) relativized_imp_well_ord: 

620 
"[ wellordered(M,A,r); M(A); M(r) ] ==> well_ord(A,r)" 

621 
apply (insert ordertype_exists [of A r], simp) 

622 
apply (blast intro: well_ord_ord_iso well_ord_Memrel ) 

623 
done 

624 

625 
subsection {*Kunen's theorem 5.4, poage 127*} 

626 

627 
text{*(a) The notion of Wellordering is absolute*} 

628 
theorem (in M_axioms) well_ord_abs [simp]: 

629 
"[ M(A); M(r) ] ==> wellordered(M,A,r) <> well_ord(A,r)" 

630 
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord) 

631 

632 

633 
text{*(b) Order types are absolute*} 

634 
lemma (in M_axioms) 

635 
"[ wellordered(M,A,r); f \<in> ord_iso(A, r, i, Memrel(i)); 

636 
M(A); M(r); M(f); M(i); Ord(i) ] ==> i = ordertype(A,r)" 

637 
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso 

638 
Ord_iso_implies_eq ord_iso_sym ord_iso_trans) 

639 

640 
end 