author  paulson 
Tue, 16 Jul 2002 16:29:36 +0200  
changeset 13363  c26eeb000470 
parent 13352  3cd767f8d78b 
child 13385  31df66ca0780 
permissions  rwrr 
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header{*Early Instances of Separation and Strong Replacement*} 
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13324  3 
theory Separation = L_axioms + WF_absolute: 
13306  4 

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text{*This theory proves all instances needed for locale @{text "M_axioms"}*} 
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13306  7 
text{*Helps us solve for de Bruijn indices!*} 
8 
lemma nth_ConsI: "[nth(n,l) = x; n \<in> nat] ==> nth(succ(n), Cons(a,l)) = x" 

9 
by simp 

10 

13316  11 
lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI 
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lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats 
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fun_plus_iff_sats 
13306  14 

15 
lemma Collect_conj_in_DPow: 

16 
"[ {x\<in>A. P(x)} \<in> DPow(A); {x\<in>A. Q(x)} \<in> DPow(A) ] 

17 
==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)" 

18 
by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) 

19 

20 
lemma Collect_conj_in_DPow_Lset: 

21 
"[z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))] 

22 
==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))" 

23 
apply (frule mem_Lset_imp_subset_Lset) 

24 
apply (simp add: Collect_conj_in_DPow Collect_mem_eq 

25 
subset_Int_iff2 elem_subset_in_DPow) 

26 
done 

27 

28 
lemma separation_CollectI: 

29 
"(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))" 

30 
apply (unfold separation_def, clarify) 

31 
apply (rule_tac x="{x\<in>z. P(x)}" in rexI) 

32 
apply simp_all 

33 
done 

34 

35 
text{*Reduces the original comprehension to the reflected one*} 

36 
lemma reflection_imp_L_separation: 

37 
"[ \<forall>x\<in>Lset(j). P(x) <> Q(x); 

38 
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j)); 

39 
Ord(j); z \<in> Lset(j)] ==> L({x \<in> z . P(x)})" 

40 
apply (rule_tac i = "succ(j)" in L_I) 

41 
prefer 2 apply simp 

42 
apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}") 

43 
prefer 2 

44 
apply (blast dest: mem_Lset_imp_subset_Lset) 

45 
apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) 

46 
done 

47 

48 

13316  49 
subsection{*Separation for Intersection*} 
13306  50 

51 
lemma Inter_Reflects: 

13314  52 
"REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A > x \<in> y, 
53 
\<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A > x \<in> y]" 

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by (intro FOL_reflections) 
13306  55 

56 
lemma Inter_separation: 

57 
"L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A > x\<in>y)" 

58 
apply (rule separation_CollectI) 

59 
apply (rule_tac A="{A,z}" in subset_LsetE, blast ) 

60 
apply (rule ReflectsE [OF Inter_Reflects], assumption) 

61 
apply (drule subset_Lset_ltD, assumption) 

62 
apply (erule reflection_imp_L_separation) 

63 
apply (simp_all add: lt_Ord2, clarify) 

64 
apply (rule DPowI2) 

65 
apply (rule ball_iff_sats) 

66 
apply (rule imp_iff_sats) 

67 
apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) 

68 
apply (rule_tac i=0 and j=2 in mem_iff_sats) 

69 
apply (simp_all add: succ_Un_distrib [symmetric]) 

70 
done 

71 

13316  72 
subsection{*Separation for Cartesian Product*} 
13306  73 

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lemma cartprod_Reflects: 
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"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)), 
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\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B & 
13314  77 
pair(**Lset(i),x,y,z))]" 
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by (intro FOL_reflections function_reflections) 
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80 
lemma cartprod_separation: 

81 
"[ L(A); L(B) ] 

82 
==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))" 

83 
apply (rule separation_CollectI) 

84 
apply (rule_tac A="{A,B,z}" in subset_LsetE, blast ) 

85 
apply (rule ReflectsE [OF cartprod_Reflects], assumption) 

86 
apply (drule subset_Lset_ltD, assumption) 

87 
apply (erule reflection_imp_L_separation) 

88 
apply (simp_all add: lt_Ord2, clarify) 

89 
apply (rule DPowI2) 

90 
apply (rename_tac u) 

91 
apply (rule bex_iff_sats) 

92 
apply (rule conj_iff_sats) 

93 
apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all) 

13316  94 
apply (rule sep_rules  simp)+ 
13306  95 
apply (simp_all add: succ_Un_distrib [symmetric]) 
96 
done 

97 

13316  98 
subsection{*Separation for Image*} 
13306  99 

100 
lemma image_Reflects: 

13314  101 
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)), 
102 
\<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p))]" 

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by (intro FOL_reflections function_reflections) 
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105 
lemma image_separation: 

106 
"[ L(A); L(r) ] 

107 
==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))" 

108 
apply (rule separation_CollectI) 

109 
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 

110 
apply (rule ReflectsE [OF image_Reflects], assumption) 

111 
apply (drule subset_Lset_ltD, assumption) 

112 
apply (erule reflection_imp_L_separation) 

113 
apply (simp_all add: lt_Ord2, clarify) 

114 
apply (rule DPowI2) 

115 
apply (rule bex_iff_sats) 

116 
apply (rule conj_iff_sats) 

117 
apply (rule_tac env="[p,y,A,r]" in mem_iff_sats) 

13316  118 
apply (rule sep_rules  simp)+ 
13306  119 
apply (simp_all add: succ_Un_distrib [symmetric]) 
120 
done 

121 

122 

13316  123 
subsection{*Separation for Converse*} 
13306  124 

125 
lemma converse_Reflects: 

13314  126 
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)), 
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\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). 
13314  128 
pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]" 
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by (intro FOL_reflections function_reflections) 
13306  130 

131 
lemma converse_separation: 

132 
"L(r) ==> separation(L, 

133 
\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))" 

134 
apply (rule separation_CollectI) 

135 
apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 

136 
apply (rule ReflectsE [OF converse_Reflects], assumption) 

137 
apply (drule subset_Lset_ltD, assumption) 

138 
apply (erule reflection_imp_L_separation) 

139 
apply (simp_all add: lt_Ord2, clarify) 

140 
apply (rule DPowI2) 

141 
apply (rename_tac u) 

142 
apply (rule bex_iff_sats) 

143 
apply (rule conj_iff_sats) 

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apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all) 
13316  145 
apply (rule sep_rules  simp)+ 
13306  146 
apply (simp_all add: succ_Un_distrib [symmetric]) 
147 
done 

148 

149 

13316  150 
subsection{*Separation for Restriction*} 
13306  151 

152 
lemma restrict_Reflects: 

13314  153 
"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)), 
154 
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]" 

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by (intro FOL_reflections function_reflections) 
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157 
lemma restrict_separation: 

158 
"L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))" 

159 
apply (rule separation_CollectI) 

160 
apply (rule_tac A="{A,z}" in subset_LsetE, blast ) 

161 
apply (rule ReflectsE [OF restrict_Reflects], assumption) 

162 
apply (drule subset_Lset_ltD, assumption) 

163 
apply (erule reflection_imp_L_separation) 

164 
apply (simp_all add: lt_Ord2, clarify) 

165 
apply (rule DPowI2) 

166 
apply (rename_tac u) 

167 
apply (rule bex_iff_sats) 

168 
apply (rule conj_iff_sats) 

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apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all) 
13316  170 
apply (rule sep_rules  simp)+ 
13306  171 
apply (simp_all add: succ_Un_distrib [symmetric]) 
172 
done 

173 

174 

13316  175 
subsection{*Separation for Composition*} 
13306  176 

177 
lemma comp_Reflects: 

13314  178 
"REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. 
13306  179 
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & 
180 
xy\<in>s & yz\<in>r, 

181 
\<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i). 

182 
pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) & 

13314  183 
pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]" 
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by (intro FOL_reflections function_reflections) 
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186 
lemma comp_separation: 

187 
"[ L(r); L(s) ] 

188 
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. 

189 
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & 

190 
xy\<in>s & yz\<in>r)" 

191 
apply (rule separation_CollectI) 

192 
apply (rule_tac A="{r,s,z}" in subset_LsetE, blast ) 

193 
apply (rule ReflectsE [OF comp_Reflects], assumption) 

194 
apply (drule subset_Lset_ltD, assumption) 

195 
apply (erule reflection_imp_L_separation) 

196 
apply (simp_all add: lt_Ord2, clarify) 

197 
apply (rule DPowI2) 

198 
apply (rename_tac u) 

199 
apply (rule bex_iff_sats)+ 

200 
apply (rename_tac x y z) 

201 
apply (rule conj_iff_sats) 

202 
apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats) 

13316  203 
apply (rule sep_rules  simp)+ 
13306  204 
apply (simp_all add: succ_Un_distrib [symmetric]) 
205 
done 

206 

13316  207 
subsection{*Separation for Predecessors in an Order*} 
13306  208 

209 
lemma pred_Reflects: 

13314  210 
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p), 
211 
\<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]" 

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by (intro FOL_reflections function_reflections) 
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214 
lemma pred_separation: 

215 
"[ L(r); L(x) ] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))" 

216 
apply (rule separation_CollectI) 

217 
apply (rule_tac A="{r,x,z}" in subset_LsetE, blast ) 

218 
apply (rule ReflectsE [OF pred_Reflects], assumption) 

219 
apply (drule subset_Lset_ltD, assumption) 

220 
apply (erule reflection_imp_L_separation) 

221 
apply (simp_all add: lt_Ord2, clarify) 

222 
apply (rule DPowI2) 

223 
apply (rename_tac u) 

224 
apply (rule bex_iff_sats) 

225 
apply (rule conj_iff_sats) 

226 
apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats) 

13316  227 
apply (rule sep_rules  simp)+ 
13306  228 
apply (simp_all add: succ_Un_distrib [symmetric]) 
229 
done 

230 

231 

13316  232 
subsection{*Separation for the Membership Relation*} 
13306  233 

234 
lemma Memrel_Reflects: 

13314  235 
"REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y, 
236 
\<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]" 

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by (intro FOL_reflections function_reflections) 
13306  238 

239 
lemma Memrel_separation: 

240 
"separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)" 

241 
apply (rule separation_CollectI) 

242 
apply (rule_tac A="{z}" in subset_LsetE, blast ) 

243 
apply (rule ReflectsE [OF Memrel_Reflects], assumption) 

244 
apply (drule subset_Lset_ltD, assumption) 

245 
apply (erule reflection_imp_L_separation) 

246 
apply (simp_all add: lt_Ord2) 

247 
apply (rule DPowI2) 

248 
apply (rename_tac u) 

13316  249 
apply (rule bex_iff_sats conj_iff_sats)+ 
13306  250 
apply (rule_tac env = "[y,x,u]" in pair_iff_sats) 
13316  251 
apply (rule sep_rules  simp)+ 
13306  252 
apply (simp_all add: succ_Un_distrib [symmetric]) 
253 
done 

254 

255 

13316  256 
subsection{*Replacement for FunSpace*} 
13306  257 

258 
lemma funspace_succ_Reflects: 

13314  259 
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. 
13306  260 
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & 
261 
upair(L,cnbf,cnbf,z)), 

262 
\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i). 

263 
\<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i). 

264 
pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) & 

13314  265 
is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]" 
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by (intro FOL_reflections function_reflections) 
13306  267 

268 
lemma funspace_succ_replacement: 

269 
"L(n) ==> 

270 
strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. 

271 
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & 

272 
upair(L,cnbf,cnbf,z))" 

273 
apply (rule strong_replacementI) 

274 
apply (rule rallI) 

275 
apply (rule separation_CollectI) 

276 
apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) 

277 
apply (rule ReflectsE [OF funspace_succ_Reflects], assumption) 

278 
apply (drule subset_Lset_ltD, assumption) 

279 
apply (erule reflection_imp_L_separation) 

280 
apply (simp_all add: lt_Ord2) 

281 
apply (rule DPowI2) 

282 
apply (rename_tac u) 

283 
apply (rule bex_iff_sats) 

284 
apply (rule conj_iff_sats) 

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apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats) 
13316  286 
apply (rule sep_rules  simp)+ 
13306  287 
apply (simp_all add: succ_Un_distrib [symmetric]) 
288 
done 

289 

290 

13316  291 
subsection{*Separation for OrderIsomorphisms*} 
13306  292 

293 
lemma well_ord_iso_Reflects: 

13314  294 
"REFLECTS[\<lambda>x. x\<in>A > 
295 
(\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r), 

296 
\<lambda>i x. x\<in>A > (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). 

297 
fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]" 

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by (intro FOL_reflections function_reflections) 
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300 
lemma well_ord_iso_separation: 

301 
"[ L(A); L(f); L(r) ] 

302 
==> separation (L, \<lambda>x. x\<in>A > (\<exists>y[L]. (\<exists>p[L]. 

303 
fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))" 

304 
apply (rule separation_CollectI) 

305 
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) 

306 
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption) 

307 
apply (drule subset_Lset_ltD, assumption) 

308 
apply (erule reflection_imp_L_separation) 

309 
apply (simp_all add: lt_Ord2) 

310 
apply (rule DPowI2) 

311 
apply (rename_tac u) 

312 
apply (rule imp_iff_sats) 

313 
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats) 

13316  314 
apply (rule sep_rules  simp)+ 
315 
apply (simp_all add: succ_Un_distrib [symmetric]) 

316 
done 

317 

318 

319 
subsection{*Separation for @{term "obase"}*} 

320 

321 
lemma obase_reflects: 

322 
"REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 

323 
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & 

324 
order_isomorphism(L,par,r,x,mx,g), 

325 
\<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i). 

326 
ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & 

327 
order_isomorphism(**Lset(i),par,r,x,mx,g)]" 

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by (intro FOL_reflections function_reflections fun_plus_reflections) 
13316  329 

330 
lemma obase_separation: 

331 
{*part of the order type formalization*} 

332 
"[ L(A); L(r) ] 

333 
==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 

334 
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & 

335 
order_isomorphism(L,par,r,x,mx,g))" 

336 
apply (rule separation_CollectI) 

337 
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 

338 
apply (rule ReflectsE [OF obase_reflects], assumption) 

339 
apply (drule subset_Lset_ltD, assumption) 

340 
apply (erule reflection_imp_L_separation) 

341 
apply (simp_all add: lt_Ord2) 

342 
apply (rule DPowI2) 

343 
apply (rename_tac u) 

13306  344 
apply (rule bex_iff_sats) 
345 
apply (rule conj_iff_sats) 

13316  346 
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats) 
347 
apply (rule sep_rules  simp)+ 

348 
apply (simp_all add: succ_Un_distrib [symmetric]) 

349 
done 

350 

351 

13319  352 
subsection{*Separation for a Theorem about @{term "obase"}*} 
13316  353 

354 
lemma obase_equals_reflects: 

355 
"REFLECTS[\<lambda>x. x\<in>A > ~(\<exists>y[L]. \<exists>g[L]. 

356 
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 

357 
membership(L,y,my) & pred_set(L,A,x,r,pxr) & 

358 
order_isomorphism(L,pxr,r,y,my,g))), 

359 
\<lambda>i x. x\<in>A > ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). 

360 
ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). 

361 
membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) & 

362 
order_isomorphism(**Lset(i),pxr,r,y,my,g)))]" 

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363 
by (intro FOL_reflections function_reflections fun_plus_reflections) 
13316  364 

365 

366 
lemma obase_equals_separation: 

367 
"[ L(A); L(r) ] 

368 
==> separation (L, \<lambda>x. x\<in>A > ~(\<exists>y[L]. \<exists>g[L]. 

369 
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 

370 
membership(L,y,my) & pred_set(L,A,x,r,pxr) & 

371 
order_isomorphism(L,pxr,r,y,my,g))))" 

372 
apply (rule separation_CollectI) 

373 
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 

374 
apply (rule ReflectsE [OF obase_equals_reflects], assumption) 

375 
apply (drule subset_Lset_ltD, assumption) 

376 
apply (erule reflection_imp_L_separation) 

377 
apply (simp_all add: lt_Ord2) 

378 
apply (rule DPowI2) 

379 
apply (rename_tac u) 

380 
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+ 

381 
apply (rule_tac env = "[u,A,r]" in mem_iff_sats) 

382 
apply (rule sep_rules  simp)+ 

383 
apply (simp_all add: succ_Un_distrib [symmetric]) 

384 
done 

385 

386 

387 
subsection{*Replacement for @{term "omap"}*} 

388 

389 
lemma omap_reflects: 

390 
"REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 

391 
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 

392 
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)), 

393 
\<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). 

394 
\<exists>par \<in> Lset(i). 

395 
ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & 

396 
membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & 

397 
order_isomorphism(**Lset(i),par,r,x,mx,g))]" 

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398 
by (intro FOL_reflections function_reflections fun_plus_reflections) 
13316  399 

400 
lemma omap_replacement: 

401 
"[ L(A); L(r) ] 

402 
==> strong_replacement(L, 

403 
\<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 

404 
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 

405 
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))" 

406 
apply (rule strong_replacementI) 

407 
apply (rule rallI) 

408 
apply (rename_tac B) 

409 
apply (rule separation_CollectI) 

410 
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) 

411 
apply (rule ReflectsE [OF omap_reflects], assumption) 

412 
apply (drule subset_Lset_ltD, assumption) 

413 
apply (erule reflection_imp_L_separation) 

414 
apply (simp_all add: lt_Ord2) 

415 
apply (rule DPowI2) 

416 
apply (rename_tac u) 

417 
apply (rule bex_iff_sats conj_iff_sats)+ 

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418 
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats) 
13316  419 
apply (rule sep_rules  simp)+ 
13306  420 
apply (simp_all add: succ_Un_distrib [symmetric]) 
421 
done 

422 

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423 

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424 
subsection{*Separation for a Theorem about @{term "obase"}*} 
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425 

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426 
lemma is_recfun_reflects: 
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427 
"REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
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428 
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
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429 
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
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430 
fx \<noteq> gx), 
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431 
\<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i). 
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pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r & 
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433 
(\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) & 
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434 
fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]" 
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435 
by (intro FOL_reflections function_reflections fun_plus_reflections) 
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436 

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437 
lemma is_recfun_separation: 
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438 
{*for wellfounded recursion*} 
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439 
"[ L(r); L(f); L(g); L(a); L(b) ] 
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440 
==> separation(L, 
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441 
\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
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442 
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
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443 
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
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444 
fx \<noteq> gx))" 
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445 
apply (rule separation_CollectI) 
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446 
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast ) 
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447 
apply (rule ReflectsE [OF is_recfun_reflects], assumption) 
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448 
apply (drule subset_Lset_ltD, assumption) 
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449 
apply (erule reflection_imp_L_separation) 
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450 
apply (simp_all add: lt_Ord2) 
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451 
apply (rule DPowI2) 
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452 
apply (rename_tac u) 
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453 
apply (rule bex_iff_sats conj_iff_sats)+ 
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454 
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats) 
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455 
apply (rule sep_rules  simp)+ 
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456 
apply (simp_all add: succ_Un_distrib [symmetric]) 
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457 
done 
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458 

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459 

13363  460 
subsection{*Instantiating the locale @{text M_axioms}*} 
461 
text{*Separation (and Strong Replacement) for basic settheoretic constructions 

462 
such as intersection, Cartesian Product and image.*} 

463 

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464 
ML 
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465 
{* 
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466 
val Inter_separation = thm "Inter_separation"; 
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467 
val cartprod_separation = thm "cartprod_separation"; 
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468 
val image_separation = thm "image_separation"; 
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469 
val converse_separation = thm "converse_separation"; 
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470 
val restrict_separation = thm "restrict_separation"; 
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471 
val comp_separation = thm "comp_separation"; 
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472 
val pred_separation = thm "pred_separation"; 
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473 
val Memrel_separation = thm "Memrel_separation"; 
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474 
val funspace_succ_replacement = thm "funspace_succ_replacement"; 
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475 
val well_ord_iso_separation = thm "well_ord_iso_separation"; 
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476 
val obase_separation = thm "obase_separation"; 
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477 
val obase_equals_separation = thm "obase_equals_separation"; 
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478 
val omap_replacement = thm "omap_replacement"; 
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479 
val is_recfun_separation = thm "is_recfun_separation"; 
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480 

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481 
val m_axioms = 
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482 
[Inter_separation, cartprod_separation, image_separation, 
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483 
converse_separation, restrict_separation, comp_separation, 
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484 
pred_separation, Memrel_separation, funspace_succ_replacement, 
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485 
well_ord_iso_separation, obase_separation, 
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486 
obase_equals_separation, omap_replacement, is_recfun_separation] 
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487 

13363  488 
fun axioms_L th = 
489 
kill_flex_triv_prems (m_axioms MRS (triv_axioms_L th)); 

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490 

13363  491 
bind_thm ("cartprod_iff", axioms_L (thm "M_axioms.cartprod_iff")); 
492 
bind_thm ("cartprod_closed", axioms_L (thm "M_axioms.cartprod_closed")); 

493 
bind_thm ("sum_closed", axioms_L (thm "M_axioms.sum_closed")); 

494 
bind_thm ("M_converse_iff", axioms_L (thm "M_axioms.M_converse_iff")); 

495 
bind_thm ("converse_closed", axioms_L (thm "M_axioms.converse_closed")); 

496 
bind_thm ("converse_abs", axioms_L (thm "M_axioms.converse_abs")); 

497 
bind_thm ("image_closed", axioms_L (thm "M_axioms.image_closed")); 

498 
bind_thm ("vimage_abs", axioms_L (thm "M_axioms.vimage_abs")); 

499 
bind_thm ("vimage_closed", axioms_L (thm "M_axioms.vimage_closed")); 

500 
bind_thm ("domain_abs", axioms_L (thm "M_axioms.domain_abs")); 

501 
bind_thm ("domain_closed", axioms_L (thm "M_axioms.domain_closed")); 

502 
bind_thm ("range_abs", axioms_L (thm "M_axioms.range_abs")); 

503 
bind_thm ("range_closed", axioms_L (thm "M_axioms.range_closed")); 

504 
bind_thm ("field_abs", axioms_L (thm "M_axioms.field_abs")); 

505 
bind_thm ("field_closed", axioms_L (thm "M_axioms.field_closed")); 

506 
bind_thm ("relation_abs", axioms_L (thm "M_axioms.relation_abs")); 

507 
bind_thm ("function_abs", axioms_L (thm "M_axioms.function_abs")); 

508 
bind_thm ("apply_closed", axioms_L (thm "M_axioms.apply_closed")); 

509 
bind_thm ("apply_abs", axioms_L (thm "M_axioms.apply_abs")); 

510 
bind_thm ("typed_function_abs", axioms_L (thm "M_axioms.typed_function_abs")); 

511 
bind_thm ("injection_abs", axioms_L (thm "M_axioms.injection_abs")); 

512 
bind_thm ("surjection_abs", axioms_L (thm "M_axioms.surjection_abs")); 

513 
bind_thm ("bijection_abs", axioms_L (thm "M_axioms.bijection_abs")); 

514 
bind_thm ("M_comp_iff", axioms_L (thm "M_axioms.M_comp_iff")); 

515 
bind_thm ("comp_closed", axioms_L (thm "M_axioms.comp_closed")); 

516 
bind_thm ("composition_abs", axioms_L (thm "M_axioms.composition_abs")); 

517 
bind_thm ("restriction_is_function", axioms_L (thm "M_axioms.restriction_is_function")); 

518 
bind_thm ("restriction_abs", axioms_L (thm "M_axioms.restriction_abs")); 

519 
bind_thm ("M_restrict_iff", axioms_L (thm "M_axioms.M_restrict_iff")); 

520 
bind_thm ("restrict_closed", axioms_L (thm "M_axioms.restrict_closed")); 

521 
bind_thm ("Inter_abs", axioms_L (thm "M_axioms.Inter_abs")); 

522 
bind_thm ("Inter_closed", axioms_L (thm "M_axioms.Inter_closed")); 

523 
bind_thm ("Int_closed", axioms_L (thm "M_axioms.Int_closed")); 

524 
bind_thm ("finite_fun_closed", axioms_L (thm "M_axioms.finite_fun_closed")); 

525 
bind_thm ("is_funspace_abs", axioms_L (thm "M_axioms.is_funspace_abs")); 

526 
bind_thm ("succ_fun_eq2", axioms_L (thm "M_axioms.succ_fun_eq2")); 

527 
bind_thm ("funspace_succ", axioms_L (thm "M_axioms.funspace_succ")); 

528 
bind_thm ("finite_funspace_closed", axioms_L (thm "M_axioms.finite_funspace_closed")); 

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529 
*} 
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530 

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531 
ML 
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532 
{* 
13363  533 
bind_thm ("is_recfun_equal", axioms_L (thm "M_axioms.is_recfun_equal")); 
534 
bind_thm ("is_recfun_cut", axioms_L (thm "M_axioms.is_recfun_cut")); 

535 
bind_thm ("is_recfun_functional", axioms_L (thm "M_axioms.is_recfun_functional")); 

536 
bind_thm ("is_recfun_relativize", axioms_L (thm "M_axioms.is_recfun_relativize")); 

537 
bind_thm ("is_recfun_restrict", axioms_L (thm "M_axioms.is_recfun_restrict")); 

538 
bind_thm ("univalent_is_recfun", axioms_L (thm "M_axioms.univalent_is_recfun")); 

539 
bind_thm ("exists_is_recfun_indstep", axioms_L (thm "M_axioms.exists_is_recfun_indstep")); 

540 
bind_thm ("wellfounded_exists_is_recfun", axioms_L (thm "M_axioms.wellfounded_exists_is_recfun")); 

541 
bind_thm ("wf_exists_is_recfun", axioms_L (thm "M_axioms.wf_exists_is_recfun")); 

542 
bind_thm ("is_recfun_abs", axioms_L (thm "M_axioms.is_recfun_abs")); 

543 
bind_thm ("irreflexive_abs", axioms_L (thm "M_axioms.irreflexive_abs")); 

544 
bind_thm ("transitive_rel_abs", axioms_L (thm "M_axioms.transitive_rel_abs")); 

545 
bind_thm ("linear_rel_abs", axioms_L (thm "M_axioms.linear_rel_abs")); 

546 
bind_thm ("wellordered_is_trans_on", axioms_L (thm "M_axioms.wellordered_is_trans_on")); 

547 
bind_thm ("wellordered_is_linear", axioms_L (thm "M_axioms.wellordered_is_linear")); 

548 
bind_thm ("wellordered_is_wellfounded_on", axioms_L (thm "M_axioms.wellordered_is_wellfounded_on")); 

549 
bind_thm ("wellfounded_imp_wellfounded_on", axioms_L (thm "M_axioms.wellfounded_imp_wellfounded_on")); 

550 
bind_thm ("wellfounded_on_subset_A", axioms_L (thm "M_axioms.wellfounded_on_subset_A")); 

551 
bind_thm ("wellfounded_on_iff_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_iff_wellfounded")); 

552 
bind_thm ("wellfounded_on_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_imp_wellfounded")); 

553 
bind_thm ("wellfounded_on_field_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_field_imp_wellfounded")); 

554 
bind_thm ("wellfounded_iff_wellfounded_on_field", axioms_L (thm "M_axioms.wellfounded_iff_wellfounded_on_field")); 

555 
bind_thm ("wellfounded_induct", axioms_L (thm "M_axioms.wellfounded_induct")); 

556 
bind_thm ("wellfounded_on_induct", axioms_L (thm "M_axioms.wellfounded_on_induct")); 

557 
bind_thm ("wellfounded_on_induct2", axioms_L (thm "M_axioms.wellfounded_on_induct2")); 

558 
bind_thm ("linear_imp_relativized", axioms_L (thm "M_axioms.linear_imp_relativized")); 

559 
bind_thm ("trans_on_imp_relativized", axioms_L (thm "M_axioms.trans_on_imp_relativized")); 

560 
bind_thm ("wf_on_imp_relativized", axioms_L (thm "M_axioms.wf_on_imp_relativized")); 

561 
bind_thm ("wf_imp_relativized", axioms_L (thm "M_axioms.wf_imp_relativized")); 

562 
bind_thm ("well_ord_imp_relativized", axioms_L (thm "M_axioms.well_ord_imp_relativized")); 

563 
bind_thm ("order_isomorphism_abs", axioms_L (thm "M_axioms.order_isomorphism_abs")); 

564 
bind_thm ("pred_set_abs", axioms_L (thm "M_axioms.pred_set_abs")); 

13323
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565 
*} 
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566 

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567 
ML 
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568 
{* 
13363  569 
bind_thm ("pred_closed", axioms_L (thm "M_axioms.pred_closed")); 
570 
bind_thm ("membership_abs", axioms_L (thm "M_axioms.membership_abs")); 

571 
bind_thm ("M_Memrel_iff", axioms_L (thm "M_axioms.M_Memrel_iff")); 

572 
bind_thm ("Memrel_closed", axioms_L (thm "M_axioms.Memrel_closed")); 

573 
bind_thm ("wellordered_iso_predD", axioms_L (thm "M_axioms.wellordered_iso_predD")); 

574 
bind_thm ("wellordered_iso_pred_eq", axioms_L (thm "M_axioms.wellordered_iso_pred_eq")); 

575 
bind_thm ("wellfounded_on_asym", axioms_L (thm "M_axioms.wellfounded_on_asym")); 

576 
bind_thm ("wellordered_asym", axioms_L (thm "M_axioms.wellordered_asym")); 

577 
bind_thm ("ord_iso_pred_imp_lt", axioms_L (thm "M_axioms.ord_iso_pred_imp_lt")); 

578 
bind_thm ("obase_iff", axioms_L (thm "M_axioms.obase_iff")); 

579 
bind_thm ("omap_iff", axioms_L (thm "M_axioms.omap_iff")); 

580 
bind_thm ("omap_unique", axioms_L (thm "M_axioms.omap_unique")); 

581 
bind_thm ("omap_yields_Ord", axioms_L (thm "M_axioms.omap_yields_Ord")); 

582 
bind_thm ("otype_iff", axioms_L (thm "M_axioms.otype_iff")); 

583 
bind_thm ("otype_eq_range", axioms_L (thm "M_axioms.otype_eq_range")); 

584 
bind_thm ("Ord_otype", axioms_L (thm "M_axioms.Ord_otype")); 

585 
bind_thm ("domain_omap", axioms_L (thm "M_axioms.domain_omap")); 

586 
bind_thm ("omap_subset", axioms_L (thm "M_axioms.omap_subset")); 

587 
bind_thm ("omap_funtype", axioms_L (thm "M_axioms.omap_funtype")); 

588 
bind_thm ("wellordered_omap_bij", axioms_L (thm "M_axioms.wellordered_omap_bij")); 

589 
bind_thm ("omap_ord_iso", axioms_L (thm "M_axioms.omap_ord_iso")); 

590 
bind_thm ("Ord_omap_image_pred", axioms_L (thm "M_axioms.Ord_omap_image_pred")); 

591 
bind_thm ("restrict_omap_ord_iso", axioms_L (thm "M_axioms.restrict_omap_ord_iso")); 

592 
bind_thm ("obase_equals", axioms_L (thm "M_axioms.obase_equals")); 

593 
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype")); 

594 
bind_thm ("obase_exists", axioms_L (thm "M_axioms.obase_exists")); 

595 
bind_thm ("omap_exists", axioms_L (thm "M_axioms.omap_exists")); 

596 
bind_thm ("otype_exists", axioms_L (thm "M_axioms.otype_exists")); 

597 
bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype")); 

598 
bind_thm ("ordertype_exists", axioms_L (thm "M_axioms.ordertype_exists")); 

599 
bind_thm ("relativized_imp_well_ord", axioms_L (thm "M_axioms.relativized_imp_well_ord")); 

600 
bind_thm ("well_ord_abs", axioms_L (thm "M_axioms.well_ord_abs")); 

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601 
*} 
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602 

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603 
declare cartprod_closed [intro,simp] 
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604 
declare sum_closed [intro,simp] 
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605 
declare converse_closed [intro,simp] 
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606 
declare converse_abs [simp] 
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607 
declare image_closed [intro,simp] 
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608 
declare vimage_abs [simp] 
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609 
declare vimage_closed [intro,simp] 
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610 
declare domain_abs [simp] 
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611 
declare domain_closed [intro,simp] 
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612 
declare range_abs [simp] 
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613 
declare range_closed [intro,simp] 
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614 
declare field_abs [simp] 
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615 
declare field_closed [intro,simp] 
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616 
declare relation_abs [simp] 
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617 
declare function_abs [simp] 
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618 
declare apply_closed [intro,simp] 
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619 
declare typed_function_abs [simp] 
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620 
declare injection_abs [simp] 
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621 
declare surjection_abs [simp] 
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622 
declare bijection_abs [simp] 
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623 
declare comp_closed [intro,simp] 
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624 
declare composition_abs [simp] 
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625 
declare restriction_abs [simp] 
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626 
declare restrict_closed [intro,simp] 
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627 
declare Inter_abs [simp] 
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628 
declare Inter_closed [intro,simp] 
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629 
declare Int_closed [intro,simp] 
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630 
declare finite_fun_closed [rule_format] 
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631 
declare is_funspace_abs [simp] 
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632 
declare finite_funspace_closed [intro,simp] 
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633 

13306  634 
end 