author  paulson 
Tue, 02 Jul 2002 13:28:08 +0200  
changeset 13269  3ba9be497c33 
parent 13244  7b37e218f298 
child 13328  703de709a64b 
permissions  rwrr 
1478  1 
(* Title: ZF/CardinalArith.thy 
437  2 
ID: $Id$ 
1478  3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
437  4 
Copyright 1994 University of Cambridge 
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13216  6 
Cardinal arithmetic  WITHOUT the Axiom of Choice 
7 

8 
Note: Could omit proving the algebraic laws for cardinal addition and 

9 
multiplication. On finite cardinals these operations coincide with 

10 
addition and multiplication of natural numbers; on infinite cardinals they 

11 
coincide with union (maximum). Either way we get most laws for free. 

437  12 
*) 
13 

12667  14 
theory CardinalArith = Cardinal + OrderArith + ArithSimp + Finite: 
467  15 

12667  16 
constdefs 
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12667  18 
InfCard :: "i=>o" 
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"InfCard(i) == Card(i) & nat le i" 

437  20 

12667  21 
cmult :: "[i,i]=>i" (infixl "*" 70) 
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"i * j == i*j" 

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cadd :: "[i,i]=>i" (infixl "+" 65) 

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"i + j == i+j" 

437  26 

12667  27 
csquare_rel :: "i=>i" 
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"csquare_rel(K) == 

29 
rvimage(K*K, 

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lam <x,y>:K*K. <x Un y, x, y>, 

31 
rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))" 

437  32 

484  33 
(*This def is more complex than Kunen's but it more easily proved to 
34 
be a cardinal*) 

12667  35 
jump_cardinal :: "i=>i" 
36 
"jump_cardinal(K) == 

1155  37 
UN X:Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" 
12667  38 

484  39 
(*needed because jump_cardinal(K) might not be the successor of K*) 
12667  40 
csucc :: "i=>i" 
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"csucc(K) == LEAST L. Card(L) & K<L" 

484  42 

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a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
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syntax (xsymbols) 
12667  44 
"op +" :: "[i,i] => i" (infixl "\<oplus>" 65) 
45 
"op *" :: "[i,i] => i" (infixl "\<otimes>" 70) 

46 

47 

48 
lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))" 

49 
apply (rule CardI) 

50 
apply (simp add: Card_is_Ord) 

51 
apply (clarify dest!: ltD) 

52 
apply (drule bspec, assumption) 

53 
apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 

54 
apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll]) 

55 
apply (drule lesspoll_trans1, assumption) 

13216  56 
apply (subgoal_tac "B \<lesssim> \<Union>A") 
12667  57 
apply (drule lesspoll_trans1, assumption, blast) 
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apply (blast intro: subset_imp_lepoll) 

59 
done 

60 

61 
lemma Card_UN: 

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"(!!x. x:A ==> Card(K(x))) ==> Card(UN x:A. K(x))" 

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by (blast intro: Card_Union) 

64 

65 
lemma Card_OUN [simp,intro,TC]: 

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"(!!x. x:A ==> Card(K(x))) ==> Card(UN x<A. K(x))" 

67 
by (simp add: OUnion_def Card_0) 

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9754ba005b64
Xsymbols for ordinal, cardinal, integer arithmetic
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12776  69 
lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat" 
70 
apply (unfold lesspoll_def) 

71 
apply (rule conjI) 

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apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat) 

73 
apply (rule notI) 

74 
apply (erule eqpollE) 

75 
apply (rule succ_lepoll_natE) 

76 
apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 

12820  77 
lepoll_trans, assumption) 
12776  78 
done 
79 

80 
lemma in_Card_imp_lesspoll: "[ Card(K); b \<in> K ] ==> b \<prec> K" 

81 
apply (unfold lesspoll_def) 

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apply (simp add: Card_iff_initial) 

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apply (fast intro!: le_imp_lepoll ltI leI) 

84 
done 

85 

86 
lemma lesspoll_lemma: 

87 
"[ ~ A \<prec> B; C \<prec> B ] ==> A  C \<noteq> 0" 

88 
apply (unfold lesspoll_def) 

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apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll] 

90 
intro!: eqpollI elim: notE 

91 
elim!: eqpollE lepoll_trans) 

92 
done 

93 

13216  94 

95 
(*** Cardinal addition ***) 

96 

97 
(** Cardinal addition is commutative **) 

98 

99 
lemma sum_commute_eqpoll: "A+B \<approx> B+A" 

100 
apply (unfold eqpoll_def) 

101 
apply (rule exI) 

102 
apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective) 

103 
apply auto 

104 
done 

105 

106 
lemma cadd_commute: "i + j = j + i" 

107 
apply (unfold cadd_def) 

108 
apply (rule sum_commute_eqpoll [THEN cardinal_cong]) 

109 
done 

110 

111 
(** Cardinal addition is associative **) 

112 

113 
lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)" 

114 
apply (unfold eqpoll_def) 

115 
apply (rule exI) 

116 
apply (rule sum_assoc_bij) 

117 
done 

118 

119 
(*Unconditional version requires AC*) 

120 
lemma well_ord_cadd_assoc: 

121 
"[ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] 

122 
==> (i + j) + k = i + (j + k)" 

123 
apply (unfold cadd_def) 

124 
apply (rule cardinal_cong) 

125 
apply (rule eqpoll_trans) 

126 
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 

13221  127 
apply (blast intro: well_ord_radd ) 
13216  128 
apply (rule sum_assoc_eqpoll [THEN eqpoll_trans]) 
129 
apply (rule eqpoll_sym) 

130 
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 

13221  131 
apply (blast intro: well_ord_radd ) 
13216  132 
done 
133 

134 
(** 0 is the identity for addition **) 

135 

136 
lemma sum_0_eqpoll: "0+A \<approx> A" 

137 
apply (unfold eqpoll_def) 

138 
apply (rule exI) 

139 
apply (rule bij_0_sum) 

140 
done 

141 

142 
lemma cadd_0 [simp]: "Card(K) ==> 0 + K = K" 

143 
apply (unfold cadd_def) 

144 
apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq) 

145 
done 

146 

147 
(** Addition by another cardinal **) 

148 

149 
lemma sum_lepoll_self: "A \<lesssim> A+B" 

150 
apply (unfold lepoll_def inj_def) 

151 
apply (rule_tac x = "lam x:A. Inl (x) " in exI) 

13221  152 
apply simp 
13216  153 
done 
154 

155 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

156 

157 
lemma cadd_le_self: 

158 
"[ Card(K); Ord(L) ] ==> K le (K + L)" 

159 
apply (unfold cadd_def) 

13221  160 
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le], 
161 
assumption) 

13216  162 
apply (rule_tac [2] sum_lepoll_self) 
163 
apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord) 

164 
done 

165 

166 
(** Monotonicity of addition **) 

167 

168 
lemma sum_lepoll_mono: 

13221  169 
"[ A \<lesssim> C; B \<lesssim> D ] ==> A + B \<lesssim> C + D" 
13216  170 
apply (unfold lepoll_def) 
13221  171 
apply (elim exE) 
13216  172 
apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI) 
13221  173 
apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))" 
13216  174 
in lam_injective) 
13221  175 
apply (typecheck add: inj_is_fun, auto) 
13216  176 
done 
177 

178 
lemma cadd_le_mono: 

179 
"[ K' le K; L' le L ] ==> (K' + L') le (K + L)" 

180 
apply (unfold cadd_def) 

181 
apply (safe dest!: le_subset_iff [THEN iffD1]) 

182 
apply (rule well_ord_lepoll_imp_Card_le) 

183 
apply (blast intro: well_ord_radd well_ord_Memrel) 

184 
apply (blast intro: sum_lepoll_mono subset_imp_lepoll) 

185 
done 

186 

187 
(** Addition of finite cardinals is "ordinary" addition **) 

188 

189 
lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)" 

190 
apply (unfold eqpoll_def) 

191 
apply (rule exI) 

192 
apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 

193 
and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective) 

13221  194 
apply simp_all 
13216  195 
apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+ 
196 
done 

197 

198 
(*Pulling the succ(...) outside the ... requires m, n: nat *) 

199 
(*Unconditional version requires AC*) 

200 
lemma cadd_succ_lemma: 

201 
"[ Ord(m); Ord(n) ] ==> succ(m) + n = succ(m + n)" 

202 
apply (unfold cadd_def) 

203 
apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans]) 

204 
apply (rule succ_eqpoll_cong [THEN cardinal_cong]) 

205 
apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym]) 

206 
apply (blast intro: well_ord_radd well_ord_Memrel) 

207 
done 

208 

209 
lemma nat_cadd_eq_add: "[ m: nat; n: nat ] ==> m + n = m#+n" 

13244  210 
apply (induct_tac m) 
13216  211 
apply (simp add: nat_into_Card [THEN cadd_0]) 
212 
apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq]) 

213 
done 

214 

215 

216 
(*** Cardinal multiplication ***) 

217 

218 
(** Cardinal multiplication is commutative **) 

219 

220 
(*Easier to prove the two directions separately*) 

221 
lemma prod_commute_eqpoll: "A*B \<approx> B*A" 

222 
apply (unfold eqpoll_def) 

223 
apply (rule exI) 

13221  224 
apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
225 
auto) 

13216  226 
done 
227 

228 
lemma cmult_commute: "i * j = j * i" 

229 
apply (unfold cmult_def) 

230 
apply (rule prod_commute_eqpoll [THEN cardinal_cong]) 

231 
done 

232 

233 
(** Cardinal multiplication is associative **) 

234 

235 
lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)" 

236 
apply (unfold eqpoll_def) 

237 
apply (rule exI) 

238 
apply (rule prod_assoc_bij) 

239 
done 

240 

241 
(*Unconditional version requires AC*) 

242 
lemma well_ord_cmult_assoc: 

243 
"[ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] 

244 
==> (i * j) * k = i * (j * k)" 

245 
apply (unfold cmult_def) 

246 
apply (rule cardinal_cong) 

13221  247 
apply (rule eqpoll_trans) 
13216  248 
apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 
249 
apply (blast intro: well_ord_rmult) 

250 
apply (rule prod_assoc_eqpoll [THEN eqpoll_trans]) 

13221  251 
apply (rule eqpoll_sym) 
13216  252 
apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 
253 
apply (blast intro: well_ord_rmult) 

254 
done 

255 

256 
(** Cardinal multiplication distributes over addition **) 

257 

258 
lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)" 

259 
apply (unfold eqpoll_def) 

260 
apply (rule exI) 

261 
apply (rule sum_prod_distrib_bij) 

262 
done 

263 

264 
lemma well_ord_cadd_cmult_distrib: 

265 
"[ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] 

266 
==> (i + j) * k = (i * k) + (j * k)" 

267 
apply (unfold cadd_def cmult_def) 

268 
apply (rule cardinal_cong) 

13221  269 
apply (rule eqpoll_trans) 
13216  270 
apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 
271 
apply (blast intro: well_ord_radd) 

272 
apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans]) 

13221  273 
apply (rule eqpoll_sym) 
13216  274 
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
275 
well_ord_cardinal_eqpoll]) 

276 
apply (blast intro: well_ord_rmult)+ 

277 
done 

278 

279 
(** Multiplication by 0 yields 0 **) 

280 

281 
lemma prod_0_eqpoll: "0*A \<approx> 0" 

282 
apply (unfold eqpoll_def) 

283 
apply (rule exI) 

13221  284 
apply (rule lam_bijective, safe) 
13216  285 
done 
286 

287 
lemma cmult_0 [simp]: "0 * i = 0" 

13221  288 
by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong]) 
13216  289 

290 
(** 1 is the identity for multiplication **) 

291 

292 
lemma prod_singleton_eqpoll: "{x}*A \<approx> A" 

293 
apply (unfold eqpoll_def) 

294 
apply (rule exI) 

295 
apply (rule singleton_prod_bij [THEN bij_converse_bij]) 

296 
done 

297 

298 
lemma cmult_1 [simp]: "Card(K) ==> 1 * K = K" 

299 
apply (unfold cmult_def succ_def) 

300 
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq) 

301 
done 

302 

303 
(*** Some inequalities for multiplication ***) 

304 

305 
lemma prod_square_lepoll: "A \<lesssim> A*A" 

306 
apply (unfold lepoll_def inj_def) 

13221  307 
apply (rule_tac x = "lam x:A. <x,x>" in exI, simp) 
13216  308 
done 
309 

310 
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) 

311 
lemma cmult_square_le: "Card(K) ==> K le K * K" 

312 
apply (unfold cmult_def) 

313 
apply (rule le_trans) 

314 
apply (rule_tac [2] well_ord_lepoll_imp_Card_le) 

315 
apply (rule_tac [3] prod_square_lepoll) 

13221  316 
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq) 
317 
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) 

13216  318 
done 
319 

320 
(** Multiplication by a nonzero cardinal **) 

321 

322 
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B" 

323 
apply (unfold lepoll_def inj_def) 

13221  324 
apply (rule_tac x = "lam x:A. <x,b>" in exI, simp) 
13216  325 
done 
326 

327 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

328 
lemma cmult_le_self: 

329 
"[ Card(K); Ord(L); 0<L ] ==> K le (K * L)" 

330 
apply (unfold cmult_def) 

331 
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le]) 

13221  332 
apply assumption 
13216  333 
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) 
334 
apply (blast intro: prod_lepoll_self ltD) 

335 
done 

336 

337 
(** Monotonicity of multiplication **) 

338 

339 
lemma prod_lepoll_mono: 

340 
"[ A \<lesssim> C; B \<lesssim> D ] ==> A * B \<lesssim> C * D" 

341 
apply (unfold lepoll_def) 

13221  342 
apply (elim exE) 
13216  343 
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI) 
344 
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 

345 
in lam_injective) 

13221  346 
apply (typecheck add: inj_is_fun, auto) 
13216  347 
done 
348 

349 
lemma cmult_le_mono: 

350 
"[ K' le K; L' le L ] ==> (K' * L') le (K * L)" 

351 
apply (unfold cmult_def) 

352 
apply (safe dest!: le_subset_iff [THEN iffD1]) 

353 
apply (rule well_ord_lepoll_imp_Card_le) 

354 
apply (blast intro: well_ord_rmult well_ord_Memrel) 

355 
apply (blast intro: prod_lepoll_mono subset_imp_lepoll) 

356 
done 

357 

358 
(*** Multiplication of finite cardinals is "ordinary" multiplication ***) 

359 

360 
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B" 

361 
apply (unfold eqpoll_def) 

13221  362 
apply (rule exI) 
13216  363 
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)" 
364 
and d = "case (%y. <A,y>, %z. z)" in lam_bijective) 

365 
apply safe 

366 
apply (simp_all add: succI2 if_type mem_imp_not_eq) 

367 
done 

368 

369 
(*Unconditional version requires AC*) 

370 
lemma cmult_succ_lemma: 

371 
"[ Ord(m); Ord(n) ] ==> succ(m) * n = n + (m * n)" 

372 
apply (unfold cmult_def cadd_def) 

373 
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans]) 

374 
apply (rule cardinal_cong [symmetric]) 

375 
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 

376 
apply (blast intro: well_ord_rmult well_ord_Memrel) 

377 
done 

378 

379 
lemma nat_cmult_eq_mult: "[ m: nat; n: nat ] ==> m * n = m#*n" 

13244  380 
apply (induct_tac m) 
13221  381 
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add) 
13216  382 
done 
383 

384 
lemma cmult_2: "Card(n) ==> 2 * n = n + n" 

13221  385 
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0]) 
13216  386 

387 
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B" 

13221  388 
apply (rule lepoll_trans) 
13216  389 
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
390 
apply (erule prod_lepoll_mono) 

13221  391 
apply (rule lepoll_refl) 
13216  392 
done 
393 

394 
lemma lepoll_imp_sum_lepoll_prod: "[ A \<lesssim> B; 2 \<lesssim> A ] ==> A+B \<lesssim> A*B" 

13221  395 
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl) 
13216  396 

397 

398 
(*** Infinite Cardinals are Limit Ordinals ***) 

399 

400 
(*This proof is modelled upon one assuming nat<=A, with injection 

401 
lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z 

402 
and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \ 

403 
If f: inj(nat,A) then range(f) behaves like the natural numbers.*) 

404 
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A" 

405 
apply (unfold lepoll_def) 

406 
apply (erule exE) 

407 
apply (rule_tac x = 

408 
"lam z:cons (u,A). 

409 
if z=u then f`0 

410 
else if z: range (f) then f`succ (converse (f) `z) else z" 

411 
in exI) 

412 
apply (rule_tac d = 

413 
"%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 

414 
else y" 

415 
in lam_injective) 

416 
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun) 

417 
apply (simp add: inj_is_fun [THEN apply_rangeI] 

418 
inj_converse_fun [THEN apply_rangeI] 

419 
inj_converse_fun [THEN apply_funtype]) 

420 
done 

421 

422 
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A" 

423 
apply (erule nat_cons_lepoll [THEN eqpollI]) 

424 
apply (rule subset_consI [THEN subset_imp_lepoll]) 

425 
done 

426 

427 
(*Specialized version required below*) 

428 
lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A" 

429 
apply (unfold succ_def) 

430 
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll]) 

431 
done 

432 

433 
lemma InfCard_nat: "InfCard(nat)" 

434 
apply (unfold InfCard_def) 

435 
apply (blast intro: Card_nat le_refl Card_is_Ord) 

436 
done 

437 

438 
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)" 

439 
apply (unfold InfCard_def) 

440 
apply (erule conjunct1) 

441 
done 

442 

443 
lemma InfCard_Un: 

444 
"[ InfCard(K); Card(L) ] ==> InfCard(K Un L)" 

445 
apply (unfold InfCard_def) 

446 
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord) 

447 
done 

448 

449 
(*Kunen's Lemma 10.11*) 

450 
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)" 

451 
apply (unfold InfCard_def) 

452 
apply (erule conjE) 

453 
apply (frule Card_is_Ord) 

454 
apply (rule ltI [THEN non_succ_LimitI]) 

455 
apply (erule le_imp_subset [THEN subsetD]) 

456 
apply (safe dest!: Limit_nat [THEN Limit_le_succD]) 

457 
apply (unfold Card_def) 

458 
apply (drule trans) 

459 
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]) 

460 
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl]) 

13221  461 
apply (rule le_eqI, assumption) 
13216  462 
apply (rule Ord_cardinal) 
463 
done 

464 

465 

466 
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) 

467 

468 
(*A general fact about ordermap*) 

469 
lemma ordermap_eqpoll_pred: 

13269  470 
"[ well_ord(A,r); x:A ] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)" 
13216  471 
apply (unfold eqpoll_def) 
472 
apply (rule exI) 

13221  473 
apply (simp add: ordermap_eq_image well_ord_is_wf) 
474 
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 

475 
THEN bij_converse_bij]) 

13216  476 
apply (rule pred_subset) 
477 
done 

478 

479 
(** Establishing the wellordering **) 

480 

481 
lemma csquare_lam_inj: 

482 
"Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)" 

483 
apply (unfold inj_def) 

484 
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI) 

485 
done 

486 

487 
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))" 

488 
apply (unfold csquare_rel_def) 

13221  489 
apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption) 
13216  490 
apply (blast intro: well_ord_rmult well_ord_Memrel) 
491 
done 

492 

493 
(** Characterising initial segments of the wellordering **) 

494 

495 
lemma csquareD: 

496 
"[ <<x,y>, <z,z>> : csquare_rel(K); x<K; y<K; z<K ] ==> x le z & y le z" 

497 
apply (unfold csquare_rel_def) 

498 
apply (erule rev_mp) 

499 
apply (elim ltE) 

13221  500 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  501 
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le) 
13221  502 
apply (simp_all add: lt_def succI2) 
13216  503 
done 
504 

505 
lemma pred_csquare_subset: 

13269  506 
"z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)" 
13216  507 
apply (unfold Order.pred_def) 
508 
apply (safe del: SigmaI succCI) 

509 
apply (erule csquareD [THEN conjE]) 

13221  510 
apply (unfold lt_def, auto) 
13216  511 
done 
512 

513 
lemma csquare_ltI: 

514 
"[ x<z; y<z; z<K ] ==> <<x,y>, <z,z>> : csquare_rel(K)" 

515 
apply (unfold csquare_rel_def) 

516 
apply (subgoal_tac "x<K & y<K") 

517 
prefer 2 apply (blast intro: lt_trans) 

518 
apply (elim ltE) 

13221  519 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  520 
done 
521 

522 
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) 

523 
lemma csquare_or_eqI: 

524 
"[ x le z; y le z; z<K ] ==> <<x,y>, <z,z>> : csquare_rel(K)  x=z & y=z" 

525 
apply (unfold csquare_rel_def) 

526 
apply (subgoal_tac "x<K & y<K") 

527 
prefer 2 apply (blast intro: lt_trans1) 

528 
apply (elim ltE) 

13221  529 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  530 
apply (elim succE) 
13221  531 
apply (simp_all add: subset_Un_iff [THEN iff_sym] 
532 
subset_Un_iff2 [THEN iff_sym] OrdmemD) 

13216  533 
done 
534 

535 
(** The cardinality of initial segments **) 

536 

537 
lemma ordermap_z_lt: 

538 
"[ Limit(K); x<K; y<K; z=succ(x Un y) ] ==> 

539 
ordermap(K*K, csquare_rel(K)) ` <x,y> < 

540 
ordermap(K*K, csquare_rel(K)) ` <z,z>" 

541 
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))") 

542 
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ 

13221  543 
Limit_is_Ord [THEN well_ord_csquare], clarify) 
13216  544 
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI]) 
545 
apply (erule_tac [4] well_ord_is_wf) 

546 
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+ 

547 
done 

548 

549 
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) 

550 
lemma ordermap_csquare_le: 

13221  551 
"[ Limit(K); x<K; y<K; z=succ(x Un y) ] 
552 
==>  ordermap(K*K, csquare_rel(K)) ` <x,y>  le succ(z) * succ(z)" 

13216  553 
apply (unfold cmult_def) 
554 
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le]) 

555 
apply (rule Ord_cardinal [THEN well_ord_Memrel])+ 

556 
apply (subgoal_tac "z<K") 

557 
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ) 

13221  558 
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
559 
assumption+) 

13216  560 
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) 
561 
apply (erule Limit_is_Ord [THEN well_ord_csquare]) 

562 
apply (blast intro: ltD) 

563 
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans], 

564 
assumption) 

565 
apply (elim ltE) 

566 
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll]) 

567 
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+ 

568 
done 

569 

570 
(*Kunen: "... so the order type <= K" *) 

571 
lemma ordertype_csquare_le: 

572 
"[ InfCard(K); ALL y:K. InfCard(y) > y * y = y ] 

573 
==> ordertype(K*K, csquare_rel(K)) le K" 

574 
apply (frule InfCard_is_Card [THEN Card_is_Ord]) 

13221  575 
apply (rule all_lt_imp_le, assumption) 
13216  576 
apply (erule well_ord_csquare [THEN Ord_ordertype]) 
577 
apply (rule Card_lt_imp_lt) 

578 
apply (erule_tac [3] InfCard_is_Card) 

579 
apply (erule_tac [2] ltE) 

580 
apply (simp add: ordertype_unfold) 

581 
apply (safe elim!: ltE) 

582 
apply (subgoal_tac "Ord (xa) & Ord (ya)") 

13221  583 
prefer 2 apply (blast intro: Ord_in_Ord, clarify) 
13216  584 
(*??WHAT A MESS!*) 
585 
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1], 

586 
(assumption  rule refl  erule ltI)+) 

587 
apply (rule_tac i = "xa Un ya" and j = "nat" in Ord_linear2, 

588 
simp_all add: Ord_Un Ord_nat) 

589 
prefer 2 (*case nat le (xa Un ya) *) 

590 
apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 

591 
le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un 

592 
ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD]) 

593 
(*the finite case: xa Un ya < nat *) 

594 
apply (rule_tac j = "nat" in lt_trans2) 

595 
apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type 

596 
nat_into_Card [THEN Card_cardinal_eq] Ord_nat) 

597 
apply (simp add: InfCard_def) 

598 
done 

599 

600 
(*Main result: Kunen's Theorem 10.12*) 

601 
lemma InfCard_csquare_eq: "InfCard(K) ==> K * K = K" 

602 
apply (frule InfCard_is_Card [THEN Card_is_Ord]) 

603 
apply (erule rev_mp) 

604 
apply (erule_tac i=K in trans_induct) 

605 
apply (rule impI) 

606 
apply (rule le_anti_sym) 

607 
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le]) 

608 
apply (rule ordertype_csquare_le [THEN [2] le_trans]) 

13221  609 
apply (simp add: cmult_def Ord_cardinal_le 
610 
well_ord_csquare [THEN Ord_ordertype] 

611 
well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 

612 
THEN cardinal_cong], assumption+) 

13216  613 
done 
614 

615 
(*Corollary for arbitrary wellordered sets (all sets, assuming AC)*) 

616 
lemma well_ord_InfCard_square_eq: 

617 
"[ well_ord(A,r); InfCard(A) ] ==> A*A \<approx> A" 

618 
apply (rule prod_eqpoll_cong [THEN eqpoll_trans]) 

619 
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+ 

620 
apply (rule well_ord_cardinal_eqE) 

13221  621 
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption) 
622 
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq) 

13216  623 
done 
624 

625 
(** Toward's Kunen's Corollary 10.13 (1) **) 

626 

627 
lemma InfCard_le_cmult_eq: "[ InfCard(K); L le K; 0<L ] ==> K * L = K" 

628 
apply (rule le_anti_sym) 

629 
prefer 2 

630 
apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card) 

631 
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) 

632 
apply (rule cmult_le_mono [THEN le_trans], assumption+) 

633 
apply (simp add: InfCard_csquare_eq) 

634 
done 

635 

636 
(*Corollary 10.13 (1), for cardinal multiplication*) 

637 
lemma InfCard_cmult_eq: "[ InfCard(K); InfCard(L) ] ==> K * L = K Un L" 

638 
apply (rule_tac i = "K" and j = "L" in Ord_linear_le) 

639 
apply (typecheck add: InfCard_is_Card Card_is_Ord) 

640 
apply (rule cmult_commute [THEN ssubst]) 

641 
apply (rule Un_commute [THEN ssubst]) 

13221  642 
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
643 
subset_Un_iff2 [THEN iffD1] le_imp_subset) 

13216  644 
done 
645 

646 
lemma InfCard_cdouble_eq: "InfCard(K) ==> K + K = K" 

13221  647 
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute) 
648 
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ) 

13216  649 
done 
650 

651 
(*Corollary 10.13 (1), for cardinal addition*) 

652 
lemma InfCard_le_cadd_eq: "[ InfCard(K); L le K ] ==> K + L = K" 

653 
apply (rule le_anti_sym) 

654 
prefer 2 

655 
apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card) 

656 
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) 

657 
apply (rule cadd_le_mono [THEN le_trans], assumption+) 

658 
apply (simp add: InfCard_cdouble_eq) 

659 
done 

660 

661 
lemma InfCard_cadd_eq: "[ InfCard(K); InfCard(L) ] ==> K + L = K Un L" 

662 
apply (rule_tac i = "K" and j = "L" in Ord_linear_le) 

663 
apply (typecheck add: InfCard_is_Card Card_is_Ord) 

664 
apply (rule cadd_commute [THEN ssubst]) 

665 
apply (rule Un_commute [THEN ssubst]) 

13221  666 
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset) 
13216  667 
done 
668 

669 
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set 

670 
of all ntuples of elements of K. A better version for the Isabelle theory 

671 
might be InfCard(K) ==> list(K) = K. 

672 
*) 

673 

674 
(*** For every cardinal number there exists a greater one 

675 
[Kunen's Theorem 10.16, which would be trivial using AC] ***) 

676 

677 
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))" 

678 
apply (unfold jump_cardinal_def) 

679 
apply (rule Ord_is_Transset [THEN [2] OrdI]) 

680 
prefer 2 apply (blast intro!: Ord_ordertype) 

681 
apply (unfold Transset_def) 

682 
apply (safe del: subsetI) 

13221  683 
apply (simp add: ordertype_pred_unfold, safe) 
13216  684 
apply (rule UN_I) 
685 
apply (rule_tac [2] ReplaceI) 

686 
prefer 4 apply (blast intro: well_ord_subset elim!: predE)+ 

687 
done 

688 

689 
(*Allows selective unfolding. Less work than deriving intro/elim rules*) 

690 
lemma jump_cardinal_iff: 

691 
"i : jump_cardinal(K) <> 

692 
(EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))" 

693 
apply (unfold jump_cardinal_def) 

694 
apply (blast del: subsetI) 

695 
done 

696 

697 
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) 

698 
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)" 

699 
apply (rule Ord_jump_cardinal [THEN [2] ltI]) 

700 
apply (rule jump_cardinal_iff [THEN iffD2]) 

701 
apply (rule_tac x="Memrel(K)" in exI) 

702 
apply (rule_tac x=K in exI) 

703 
apply (simp add: ordertype_Memrel well_ord_Memrel) 

704 
apply (simp add: Memrel_def subset_iff) 

705 
done 

706 

707 
(*The proof by contradiction: the bijection f yields a wellordering of X 

708 
whose ordertype is jump_cardinal(K). *) 

709 
lemma Card_jump_cardinal_lemma: 

710 
"[ well_ord(X,r); r <= K * K; X <= K; 

711 
f : bij(ordertype(X,r), jump_cardinal(K)) ] 

712 
==> jump_cardinal(K) : jump_cardinal(K)" 

713 
apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))") 

714 
prefer 2 apply (blast intro: comp_bij ordermap_bij) 

715 
apply (rule jump_cardinal_iff [THEN iffD2]) 

716 
apply (intro exI conjI) 

13221  717 
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+) 
13216  718 
apply (erule bij_is_inj [THEN well_ord_rvimage]) 
719 
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel]) 

720 
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage] 

721 
ordertype_Memrel Ord_jump_cardinal) 

722 
done 

723 

724 
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) 

725 
lemma Card_jump_cardinal: "Card(jump_cardinal(K))" 

726 
apply (rule Ord_jump_cardinal [THEN CardI]) 

727 
apply (unfold eqpoll_def) 

728 
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1]) 

729 
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl]) 

730 
done 

731 

732 
(*** Basic properties of successor cardinals ***) 

733 

734 
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)" 

735 
apply (unfold csucc_def) 

736 
apply (rule LeastI) 

737 
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+ 

738 
done 

739 

740 
lemmas Card_csucc = csucc_basic [THEN conjunct1, standard] 

741 

742 
lemmas lt_csucc = csucc_basic [THEN conjunct2, standard] 

743 

744 
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)" 

13221  745 
by (blast intro: Ord_0_le lt_csucc lt_trans1) 
13216  746 

747 
lemma csucc_le: "[ Card(L); K<L ] ==> csucc(K) le L" 

748 
apply (unfold csucc_def) 

749 
apply (rule Least_le) 

750 
apply (blast intro: Card_is_Ord)+ 

751 
done 

752 

753 
lemma lt_csucc_iff: "[ Ord(i); Card(K) ] ==> i < csucc(K) <> i le K" 

754 
apply (rule iffI) 

755 
apply (rule_tac [2] Card_lt_imp_lt) 

756 
apply (erule_tac [2] lt_trans1) 

757 
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord) 

758 
apply (rule notI [THEN not_lt_imp_le]) 

13221  759 
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption) 
13216  760 
apply (rule Ord_cardinal_le [THEN lt_trans1]) 
761 
apply (simp_all add: Ord_cardinal Card_is_Ord) 

762 
done 

763 

764 
lemma Card_lt_csucc_iff: 

765 
"[ Card(K'); Card(K) ] ==> K' < csucc(K) <> K' le K" 

13221  766 
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord) 
13216  767 

768 
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))" 

769 
by (simp add: InfCard_def Card_csucc Card_is_Ord 

770 
lt_csucc [THEN leI, THEN [2] le_trans]) 

771 

772 

773 
(** Removing elements from a finite set decreases its cardinality **) 

774 

775 
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A > ~ cons(x,A) \<lesssim> A" 

776 
apply (erule Fin_induct) 

13221  777 
apply (simp add: lepoll_0_iff) 
13216  778 
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))") 
13221  779 
apply simp 
780 
apply (blast dest!: cons_lepoll_consD, blast) 

13216  781 
done 
782 

13221  783 
lemma Finite_imp_cardinal_cons: 
784 
"[ Finite(A); a~:A ] ==> cons(a,A) = succ(A)" 

13216  785 
apply (unfold cardinal_def) 
786 
apply (rule Least_equality) 

787 
apply (fold cardinal_def) 

13221  788 
apply (simp add: succ_def) 
13216  789 
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll 
790 
elim!: mem_irrefl dest!: Finite_imp_well_ord) 

791 
apply (blast intro: Card_cardinal Card_is_Ord) 

792 
apply (rule notI) 

13221  793 
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE], 
794 
assumption, assumption) 

13216  795 
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) 
796 
apply (erule le_imp_lepoll [THEN lepoll_trans]) 

797 
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll] 

798 
dest!: Finite_imp_well_ord) 

799 
done 

800 

801 

13221  802 
lemma Finite_imp_succ_cardinal_Diff: 
803 
"[ Finite(A); a:A ] ==> succ(A{a}) = A" 

804 
apply (rule_tac b = "A" in cons_Diff [THEN subst], assumption) 

805 
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite]) 

806 
apply (simp add: cons_Diff) 

13216  807 
done 
808 

809 
lemma Finite_imp_cardinal_Diff: "[ Finite(A); a:A ] ==> A{a} < A" 

810 
apply (rule succ_leE) 

13221  811 
apply (simp add: Finite_imp_succ_cardinal_Diff) 
13216  812 
done 
813 

814 

815 
(** Theorems by Krzysztof Grabczewski, proofs by lcp **) 

816 

817 
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard] 

818 

819 
lemma nat_sum_eqpoll_sum: "[ m:nat; n:nat ] ==> m + n \<approx> m #+ n" 

820 
apply (rule eqpoll_trans) 

821 
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym]) 

822 
apply (erule nat_implies_well_ord)+ 

13221  823 
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl) 
13216  824 
done 
825 

13221  826 
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat > i : nat  i=nat" 
827 
apply (erule trans_induct3, auto) 

13216  828 
apply (blast dest!: nat_le_Limit [THEN le_imp_subset]) 
829 
done 

830 

831 
lemma Ord_nat_subset_into_Card: "[ Ord(i); i <= nat ] ==> Card(i)" 

13221  832 
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card) 
13216  833 

834 
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> A : nat" 

835 
apply (erule Finite_induct) 

836 
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons) 

837 
done 

838 

839 
lemma Finite_Diff_sing_eq_diff_1: "[ Finite(A); x:A ] ==> A{x} = A # 1" 

840 
apply (rule succ_inject) 

841 
apply (rule_tac b = "A" in trans) 

13221  842 
apply (simp add: Finite_imp_succ_cardinal_Diff) 
13216  843 
apply (subgoal_tac "1 \<lesssim> A") 
13221  844 
prefer 2 apply (blast intro: not_0_is_lepoll_1) 
845 
apply (frule Finite_imp_well_ord, clarify) 

13216  846 
apply (rotate_tac 1) 
847 
apply (drule well_ord_lepoll_imp_Card_le) 

848 
apply (auto simp add: cardinal_1) 

849 
apply (rule trans) 

850 
apply (rule_tac [2] diff_succ) 

851 
apply (auto simp add: Finite_cardinal_in_nat) 

852 
done 

853 

13221  854 
lemma cardinal_lt_imp_Diff_not_0 [rule_format]: 
855 
"Finite(B) ==> ALL A. B<A > A  B ~= 0" 

856 
apply (erule Finite_induct, auto) 

13216  857 
apply (simp_all add: Finite_imp_cardinal_cons) 
13221  858 
apply (case_tac "Finite (A)") 
859 
apply (subgoal_tac [2] "Finite (cons (x, B))") 

860 
apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite) 

861 
apply (auto simp add: Finite_0 Finite_cons) 

13216  862 
apply (subgoal_tac "B<A") 
13221  863 
prefer 2 apply (blast intro: lt_trans Ord_cardinal) 
13216  864 
apply (case_tac "x:A") 
13221  865 
apply (subgoal_tac [2] "A  cons (x, B) = A  B") 
866 
apply auto 

13216  867 
apply (subgoal_tac "A le cons (x, B) ") 
13221  868 
prefer 2 
13216  869 
apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
870 
intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll) 

871 
apply (auto simp add: Finite_imp_cardinal_cons) 

872 
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff) 

873 
apply (blast intro: lt_trans) 

874 
done 

875 

876 

877 
ML{* 

878 
val InfCard_def = thm "InfCard_def" 

879 
val cmult_def = thm "cmult_def" 

880 
val cadd_def = thm "cadd_def" 

881 
val jump_cardinal_def = thm "jump_cardinal_def" 

882 
val csucc_def = thm "csucc_def" 

883 

884 
val sum_commute_eqpoll = thm "sum_commute_eqpoll"; 

885 
val cadd_commute = thm "cadd_commute"; 

886 
val sum_assoc_eqpoll = thm "sum_assoc_eqpoll"; 

887 
val well_ord_cadd_assoc = thm "well_ord_cadd_assoc"; 

888 
val sum_0_eqpoll = thm "sum_0_eqpoll"; 

889 
val cadd_0 = thm "cadd_0"; 

890 
val sum_lepoll_self = thm "sum_lepoll_self"; 

891 
val cadd_le_self = thm "cadd_le_self"; 

892 
val sum_lepoll_mono = thm "sum_lepoll_mono"; 

893 
val cadd_le_mono = thm "cadd_le_mono"; 

894 
val eq_imp_not_mem = thm "eq_imp_not_mem"; 

895 
val sum_succ_eqpoll = thm "sum_succ_eqpoll"; 

896 
val nat_cadd_eq_add = thm "nat_cadd_eq_add"; 

897 
val prod_commute_eqpoll = thm "prod_commute_eqpoll"; 

898 
val cmult_commute = thm "cmult_commute"; 

899 
val prod_assoc_eqpoll = thm "prod_assoc_eqpoll"; 

900 
val well_ord_cmult_assoc = thm "well_ord_cmult_assoc"; 

901 
val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll"; 

902 
val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib"; 

903 
val prod_0_eqpoll = thm "prod_0_eqpoll"; 

904 
val cmult_0 = thm "cmult_0"; 

905 
val prod_singleton_eqpoll = thm "prod_singleton_eqpoll"; 

906 
val cmult_1 = thm "cmult_1"; 

907 
val prod_lepoll_self = thm "prod_lepoll_self"; 

908 
val cmult_le_self = thm "cmult_le_self"; 

909 
val prod_lepoll_mono = thm "prod_lepoll_mono"; 

910 
val cmult_le_mono = thm "cmult_le_mono"; 

911 
val prod_succ_eqpoll = thm "prod_succ_eqpoll"; 

912 
val nat_cmult_eq_mult = thm "nat_cmult_eq_mult"; 

913 
val cmult_2 = thm "cmult_2"; 

914 
val sum_lepoll_prod = thm "sum_lepoll_prod"; 

915 
val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod"; 

916 
val nat_cons_lepoll = thm "nat_cons_lepoll"; 

917 
val nat_cons_eqpoll = thm "nat_cons_eqpoll"; 

918 
val nat_succ_eqpoll = thm "nat_succ_eqpoll"; 

919 
val InfCard_nat = thm "InfCard_nat"; 

920 
val InfCard_is_Card = thm "InfCard_is_Card"; 

921 
val InfCard_Un = thm "InfCard_Un"; 

922 
val InfCard_is_Limit = thm "InfCard_is_Limit"; 

923 
val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred"; 

924 
val ordermap_z_lt = thm "ordermap_z_lt"; 

925 
val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq"; 

926 
val InfCard_cmult_eq = thm "InfCard_cmult_eq"; 

927 
val InfCard_cdouble_eq = thm "InfCard_cdouble_eq"; 

928 
val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq"; 

929 
val InfCard_cadd_eq = thm "InfCard_cadd_eq"; 

930 
val Ord_jump_cardinal = thm "Ord_jump_cardinal"; 

931 
val jump_cardinal_iff = thm "jump_cardinal_iff"; 

932 
val K_lt_jump_cardinal = thm "K_lt_jump_cardinal"; 

933 
val Card_jump_cardinal = thm "Card_jump_cardinal"; 

934 
val csucc_basic = thm "csucc_basic"; 

935 
val Card_csucc = thm "Card_csucc"; 

936 
val lt_csucc = thm "lt_csucc"; 

937 
val Ord_0_lt_csucc = thm "Ord_0_lt_csucc"; 

938 
val csucc_le = thm "csucc_le"; 

939 
val lt_csucc_iff = thm "lt_csucc_iff"; 

940 
val Card_lt_csucc_iff = thm "Card_lt_csucc_iff"; 

941 
val InfCard_csucc = thm "InfCard_csucc"; 

942 
val Finite_into_Fin = thm "Finite_into_Fin"; 

943 
val Fin_into_Finite = thm "Fin_into_Finite"; 

944 
val Finite_Fin_iff = thm "Finite_Fin_iff"; 

945 
val Finite_Un = thm "Finite_Un"; 

946 
val Finite_Union = thm "Finite_Union"; 

947 
val Finite_induct = thm "Finite_induct"; 

948 
val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll"; 

949 
val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons"; 

950 
val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff"; 

951 
val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff"; 

952 
val nat_implies_well_ord = thm "nat_implies_well_ord"; 

953 
val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum"; 

954 
val Diff_sing_Finite = thm "Diff_sing_Finite"; 

955 
val Diff_Finite = thm "Diff_Finite"; 

956 
val Ord_subset_natD = thm "Ord_subset_natD"; 

957 
val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card"; 

958 
val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat"; 

959 
val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1"; 

960 
val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0"; 

961 
*} 

962 

437  963 
end 