author  paulson 
Tue, 08 Jun 2004 16:22:30 +0200  
changeset 14883  ca000a495448 
parent 14565  c6dc17aab88a 
child 16417  9bc16273c2d4 
permissions  rwrr 
1478  1 
(* Title: ZF/CardinalArith.thy 
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ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1994 University of Cambridge 
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*) 
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header{*Cardinal Arithmetic Without the Axiom of Choice*} 
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theory CardinalArith = Cardinal + OrderArith + ArithSimp + Finite: 
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constdefs 
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InfCard :: "i=>o" 
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"InfCard(i) == Card(i) & nat le i" 

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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" 

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

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rvimage(K*K, 

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

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

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12667  29 
jump_cardinal :: "i=>i" 
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{*This def is more complex than Kunen's but it more easily proved to 
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be a cardinal*} 

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"jump_cardinal(K) == 
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\<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}" 
12667  34 

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csucc :: "i=>i" 

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{*needed because @{term "jump_cardinal(K)"} might not be the successor 
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of @{term K}*} 

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"csucc(K) == LEAST L. Card(L) & K<L" 
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syntax (xsymbols) 
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"op +" :: "[i,i] => i" (infixl "\<oplus>" 65) 
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"op *" :: "[i,i] => i" (infixl "\<otimes>" 70) 

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syntax (HTML output) 
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"op +" :: "[i,i] => i" (infixl "\<oplus>" 65) 

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"op *" :: "[i,i] => i" (infixl "\<otimes>" 70) 

12667  46 

47 

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

49 
apply (rule CardI) 

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apply (simp add: Card_is_Ord) 

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apply (clarify dest!: ltD) 

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apply (drule bspec, assumption) 

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apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 

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apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll]) 

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apply (drule lesspoll_trans1, assumption) 

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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) 

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done 

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lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
12667  62 
by (blast intro: Card_Union) 
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lemma Card_OUN [simp,intro,TC]: 

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"(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))" 
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by (simp add: OUnion_def Card_0) 
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lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat" 
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apply (unfold lesspoll_def) 

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apply (rule conjI) 

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

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apply (rule notI) 

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apply (erule eqpollE) 

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apply (rule succ_lepoll_natE) 

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

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lepoll_trans, assumption) 
12776  77 
done 
78 

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

80 
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) 

83 
done 

84 

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lemma lesspoll_lemma: "[ ~ A \<prec> B; C \<prec> B ] ==> A  C \<noteq> 0" 
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apply (unfold lesspoll_def) 
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apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll] 

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intro!: eqpollI elim: notE 

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elim!: eqpollE lepoll_trans) 

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done 

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subsection{*Cardinal addition*} 
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text{*Note: Could omit proving the algebraic laws for cardinal addition and 
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multiplication. On finite cardinals these operations coincide with 

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addition and multiplication of natural numbers; on infinite cardinals they 

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

99 

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subsubsection{*Cardinal addition is commutative*} 
13216  101 

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

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apply (unfold eqpoll_def) 

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apply (rule exI) 

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

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apply auto 

107 
done 

108 

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

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apply (unfold cadd_def) 

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apply (rule sum_commute_eqpoll [THEN cardinal_cong]) 

112 
done 

113 

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subsubsection{*Cardinal addition is associative*} 
13216  115 

116 
lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)" 

117 
apply (unfold eqpoll_def) 

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apply (rule exI) 

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apply (rule sum_assoc_bij) 

120 
done 

121 

122 
(*Unconditional version requires AC*) 

123 
lemma well_ord_cadd_assoc: 

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

125 
==> (i + j) + k = i + (j + k)" 

126 
apply (unfold cadd_def) 

127 
apply (rule cardinal_cong) 

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apply (rule eqpoll_trans) 

129 
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 

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apply (blast intro: well_ord_radd ) 
13216  131 
apply (rule sum_assoc_eqpoll [THEN eqpoll_trans]) 
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apply (rule eqpoll_sym) 

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apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 

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apply (blast intro: well_ord_radd ) 
13216  135 
done 
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subsubsection{*0 is the identity for addition*} 
13216  138 

139 
lemma sum_0_eqpoll: "0+A \<approx> A" 

140 
apply (unfold eqpoll_def) 

141 
apply (rule exI) 

142 
apply (rule bij_0_sum) 

143 
done 

144 

145 
lemma cadd_0 [simp]: "Card(K) ==> 0 + K = K" 

146 
apply (unfold cadd_def) 

147 
apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq) 

148 
done 

149 

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subsubsection{*Addition by another cardinal*} 
13216  151 

152 
lemma sum_lepoll_self: "A \<lesssim> A+B" 

153 
apply (unfold lepoll_def inj_def) 

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

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apply simp 
13216  156 
done 
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158 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

159 

160 
lemma cadd_le_self: 

161 
"[ Card(K); Ord(L) ] ==> K le (K + L)" 

162 
apply (unfold cadd_def) 

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

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

167 
done 

168 

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subsubsection{*Monotonicity of addition*} 
13216  170 

171 
lemma sum_lepoll_mono: 

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"[ A \<lesssim> C; B \<lesssim> D ] ==> A + B \<lesssim> C + D" 
13216  173 
apply (unfold lepoll_def) 
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apply (elim exE) 
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apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI) 
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apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))" 
13216  177 
in lam_injective) 
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apply (typecheck add: inj_is_fun, auto) 
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done 
180 

181 
lemma cadd_le_mono: 

182 
"[ K' le K; L' le L ] ==> (K' + L') le (K + L)" 

183 
apply (unfold cadd_def) 

184 
apply (safe dest!: le_subset_iff [THEN iffD1]) 

185 
apply (rule well_ord_lepoll_imp_Card_le) 

186 
apply (blast intro: well_ord_radd well_ord_Memrel) 

187 
apply (blast intro: sum_lepoll_mono subset_imp_lepoll) 

188 
done 

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subsubsection{*Addition of finite cardinals is "ordinary" addition*} 
13216  191 

192 
lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)" 

193 
apply (unfold eqpoll_def) 

194 
apply (rule exI) 

195 
apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 

196 
and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective) 

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apply simp_all 
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apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+ 
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done 

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201 
(*Pulling the succ(...) outside the ... requires m, n: nat *) 

202 
(*Unconditional version requires AC*) 

203 
lemma cadd_succ_lemma: 

204 
"[ Ord(m); Ord(n) ] ==> succ(m) + n = succ(m + n)" 

205 
apply (unfold cadd_def) 

206 
apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans]) 

207 
apply (rule succ_eqpoll_cong [THEN cardinal_cong]) 

208 
apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym]) 

209 
apply (blast intro: well_ord_radd well_ord_Memrel) 

210 
done 

211 

212 
lemma nat_cadd_eq_add: "[ m: nat; n: nat ] ==> m + n = m#+n" 

13244  213 
apply (induct_tac m) 
13216  214 
apply (simp add: nat_into_Card [THEN cadd_0]) 
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apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq]) 

216 
done 

217 

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subsection{*Cardinal multiplication*} 
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subsubsection{*Cardinal multiplication is commutative*} 
13216  222 

223 
(*Easier to prove the two directions separately*) 

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

225 
apply (unfold eqpoll_def) 

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apply (rule exI) 

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

13216  229 
done 
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231 
lemma cmult_commute: "i * j = j * i" 

232 
apply (unfold cmult_def) 

233 
apply (rule prod_commute_eqpoll [THEN cardinal_cong]) 

234 
done 

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subsubsection{*Cardinal multiplication is associative*} 
13216  237 

238 
lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)" 

239 
apply (unfold eqpoll_def) 

240 
apply (rule exI) 

241 
apply (rule prod_assoc_bij) 

242 
done 

243 

244 
(*Unconditional version requires AC*) 

245 
lemma well_ord_cmult_assoc: 

246 
"[ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] 

247 
==> (i * j) * k = i * (j * k)" 

248 
apply (unfold cmult_def) 

249 
apply (rule cardinal_cong) 

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apply (rule eqpoll_trans) 
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apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 
252 
apply (blast intro: well_ord_rmult) 

253 
apply (rule prod_assoc_eqpoll [THEN eqpoll_trans]) 

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apply (rule eqpoll_sym) 
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apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 
256 
apply (blast intro: well_ord_rmult) 

257 
done 

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subsubsection{*Cardinal multiplication distributes over addition*} 
13216  260 

261 
lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)" 

262 
apply (unfold eqpoll_def) 

263 
apply (rule exI) 

264 
apply (rule sum_prod_distrib_bij) 

265 
done 

266 

267 
lemma well_ord_cadd_cmult_distrib: 

268 
"[ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] 

269 
==> (i + j) * k = (i * k) + (j * k)" 

270 
apply (unfold cadd_def cmult_def) 

271 
apply (rule cardinal_cong) 

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apply (rule eqpoll_trans) 
13216  273 
apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) 
274 
apply (blast intro: well_ord_radd) 

275 
apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans]) 

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apply (rule eqpoll_sym) 
13216  277 
apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
278 
well_ord_cardinal_eqpoll]) 

279 
apply (blast intro: well_ord_rmult)+ 

280 
done 

281 

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subsubsection{*Multiplication by 0 yields 0*} 
13216  283 

284 
lemma prod_0_eqpoll: "0*A \<approx> 0" 

285 
apply (unfold eqpoll_def) 

286 
apply (rule exI) 

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apply (rule lam_bijective, safe) 
13216  288 
done 
289 

290 
lemma cmult_0 [simp]: "0 * i = 0" 

13221  291 
by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong]) 
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subsubsection{*1 is the identity for multiplication*} 
13216  294 

295 
lemma prod_singleton_eqpoll: "{x}*A \<approx> A" 

296 
apply (unfold eqpoll_def) 

297 
apply (rule exI) 

298 
apply (rule singleton_prod_bij [THEN bij_converse_bij]) 

299 
done 

300 

301 
lemma cmult_1 [simp]: "Card(K) ==> 1 * K = K" 

302 
apply (unfold cmult_def succ_def) 

303 
apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq) 

304 
done 

305 

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subsection{*Some inequalities for multiplication*} 
13216  307 

308 
lemma prod_square_lepoll: "A \<lesssim> A*A" 

309 
apply (unfold lepoll_def inj_def) 

13221  310 
apply (rule_tac x = "lam x:A. <x,x>" in exI, simp) 
13216  311 
done 
312 

313 
(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) 

314 
lemma cmult_square_le: "Card(K) ==> K le K * K" 

315 
apply (unfold cmult_def) 

316 
apply (rule le_trans) 

317 
apply (rule_tac [2] well_ord_lepoll_imp_Card_le) 

318 
apply (rule_tac [3] prod_square_lepoll) 

13221  319 
apply (simp add: le_refl Card_is_Ord Card_cardinal_eq) 
320 
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) 

13216  321 
done 
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subsubsection{*Multiplication by a nonzero cardinal*} 
13216  324 

325 
lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B" 

326 
apply (unfold lepoll_def inj_def) 

13221  327 
apply (rule_tac x = "lam x:A. <x,b>" in exI, simp) 
13216  328 
done 
329 

330 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

331 
lemma cmult_le_self: 

332 
"[ Card(K); Ord(L); 0<L ] ==> K le (K * L)" 

333 
apply (unfold cmult_def) 

334 
apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le]) 

13221  335 
apply assumption 
13216  336 
apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord) 
337 
apply (blast intro: prod_lepoll_self ltD) 

338 
done 

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subsubsection{*Monotonicity of multiplication*} 
13216  341 

342 
lemma prod_lepoll_mono: 

343 
"[ A \<lesssim> C; B \<lesssim> D ] ==> A * B \<lesssim> C * D" 

344 
apply (unfold lepoll_def) 

13221  345 
apply (elim exE) 
13216  346 
apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI) 
347 
apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 

348 
in lam_injective) 

13221  349 
apply (typecheck add: inj_is_fun, auto) 
13216  350 
done 
351 

352 
lemma cmult_le_mono: 

353 
"[ K' le K; L' le L ] ==> (K' * L') le (K * L)" 

354 
apply (unfold cmult_def) 

355 
apply (safe dest!: le_subset_iff [THEN iffD1]) 

356 
apply (rule well_ord_lepoll_imp_Card_le) 

357 
apply (blast intro: well_ord_rmult well_ord_Memrel) 

358 
apply (blast intro: prod_lepoll_mono subset_imp_lepoll) 

359 
done 

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subsection{*Multiplication of finite cardinals is "ordinary" multiplication*} 
13216  362 

363 
lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B" 

364 
apply (unfold eqpoll_def) 

13221  365 
apply (rule exI) 
13216  366 
apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)" 
367 
and d = "case (%y. <A,y>, %z. z)" in lam_bijective) 

368 
apply safe 

369 
apply (simp_all add: succI2 if_type mem_imp_not_eq) 

370 
done 

371 

372 
(*Unconditional version requires AC*) 

373 
lemma cmult_succ_lemma: 

374 
"[ Ord(m); Ord(n) ] ==> succ(m) * n = n + (m * n)" 

375 
apply (unfold cmult_def cadd_def) 

376 
apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans]) 

377 
apply (rule cardinal_cong [symmetric]) 

378 
apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) 

379 
apply (blast intro: well_ord_rmult well_ord_Memrel) 

380 
done 

381 

382 
lemma nat_cmult_eq_mult: "[ m: nat; n: nat ] ==> m * n = m#*n" 

13244  383 
apply (induct_tac m) 
13221  384 
apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add) 
13216  385 
done 
386 

387 
lemma cmult_2: "Card(n) ==> 2 * n = n + n" 

13221  388 
by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0]) 
13216  389 

390 
lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B" 

13221  391 
apply (rule lepoll_trans) 
13216  392 
apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
393 
apply (erule prod_lepoll_mono) 

13221  394 
apply (rule lepoll_refl) 
13216  395 
done 
396 

397 
lemma lepoll_imp_sum_lepoll_prod: "[ A \<lesssim> B; 2 \<lesssim> A ] ==> A+B \<lesssim> A*B" 

13221  398 
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl) 
13216  399 

400 

13356  401 
subsection{*Infinite Cardinals are Limit Ordinals*} 
13216  402 

403 
(*This proof is modelled upon one assuming nat<=A, with injection 

404 
lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z 

405 
and inverse %y. if y:nat then nat_case(u, %z. z, y) else y. \ 

406 
If f: inj(nat,A) then range(f) behaves like the natural numbers.*) 

407 
lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A" 

408 
apply (unfold lepoll_def) 

409 
apply (erule exE) 

410 
apply (rule_tac x = 

411 
"lam z:cons (u,A). 

412 
if z=u then f`0 

413 
else if z: range (f) then f`succ (converse (f) `z) else z" 

414 
in exI) 

415 
apply (rule_tac d = 

416 
"%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 

417 
else y" 

418 
in lam_injective) 

419 
apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun) 

420 
apply (simp add: inj_is_fun [THEN apply_rangeI] 

421 
inj_converse_fun [THEN apply_rangeI] 

422 
inj_converse_fun [THEN apply_funtype]) 

423 
done 

424 

425 
lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A" 

426 
apply (erule nat_cons_lepoll [THEN eqpollI]) 

427 
apply (rule subset_consI [THEN subset_imp_lepoll]) 

428 
done 

429 

430 
(*Specialized version required below*) 

431 
lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A" 

432 
apply (unfold succ_def) 

433 
apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll]) 

434 
done 

435 

436 
lemma InfCard_nat: "InfCard(nat)" 

437 
apply (unfold InfCard_def) 

438 
apply (blast intro: Card_nat le_refl Card_is_Ord) 

439 
done 

440 

441 
lemma InfCard_is_Card: "InfCard(K) ==> Card(K)" 

442 
apply (unfold InfCard_def) 

443 
apply (erule conjunct1) 

444 
done 

445 

446 
lemma InfCard_Un: 

447 
"[ InfCard(K); Card(L) ] ==> InfCard(K Un L)" 

448 
apply (unfold InfCard_def) 

449 
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord) 

450 
done 

451 

452 
(*Kunen's Lemma 10.11*) 

453 
lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)" 

454 
apply (unfold InfCard_def) 

455 
apply (erule conjE) 

456 
apply (frule Card_is_Ord) 

457 
apply (rule ltI [THEN non_succ_LimitI]) 

458 
apply (erule le_imp_subset [THEN subsetD]) 

459 
apply (safe dest!: Limit_nat [THEN Limit_le_succD]) 

460 
apply (unfold Card_def) 

461 
apply (drule trans) 

462 
apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]) 

463 
apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl]) 

13221  464 
apply (rule le_eqI, assumption) 
13216  465 
apply (rule Ord_cardinal) 
466 
done 

467 

468 

469 
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) 

470 

471 
(*A general fact about ordermap*) 

472 
lemma ordermap_eqpoll_pred: 

13269  473 
"[ well_ord(A,r); x:A ] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)" 
13216  474 
apply (unfold eqpoll_def) 
475 
apply (rule exI) 

13221  476 
apply (simp add: ordermap_eq_image well_ord_is_wf) 
477 
apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 

478 
THEN bij_converse_bij]) 

13216  479 
apply (rule pred_subset) 
480 
done 

481 

14883  482 
subsubsection{*Establishing the wellordering*} 
13216  483 

484 
lemma csquare_lam_inj: 

485 
"Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)" 

486 
apply (unfold inj_def) 

487 
apply (force intro: lam_type Un_least_lt [THEN ltD] ltI) 

488 
done 

489 

490 
lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))" 

491 
apply (unfold csquare_rel_def) 

13221  492 
apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption) 
13216  493 
apply (blast intro: well_ord_rmult well_ord_Memrel) 
494 
done 

495 

14883  496 
subsubsection{*Characterising initial segments of the wellordering*} 
13216  497 

498 
lemma csquareD: 

499 
"[ <<x,y>, <z,z>> : csquare_rel(K); x<K; y<K; z<K ] ==> x le z & y le z" 

500 
apply (unfold csquare_rel_def) 

501 
apply (erule rev_mp) 

502 
apply (elim ltE) 

13221  503 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  504 
apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le) 
13221  505 
apply (simp_all add: lt_def succI2) 
13216  506 
done 
507 

508 
lemma pred_csquare_subset: 

13269  509 
"z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)" 
13216  510 
apply (unfold Order.pred_def) 
511 
apply (safe del: SigmaI succCI) 

512 
apply (erule csquareD [THEN conjE]) 

13221  513 
apply (unfold lt_def, auto) 
13216  514 
done 
515 

516 
lemma csquare_ltI: 

517 
"[ x<z; y<z; z<K ] ==> <<x,y>, <z,z>> : csquare_rel(K)" 

518 
apply (unfold csquare_rel_def) 

519 
apply (subgoal_tac "x<K & y<K") 

520 
prefer 2 apply (blast intro: lt_trans) 

521 
apply (elim ltE) 

13221  522 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  523 
done 
524 

525 
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) 

526 
lemma csquare_or_eqI: 

527 
"[ x le z; y le z; z<K ] ==> <<x,y>, <z,z>> : csquare_rel(K)  x=z & y=z" 

528 
apply (unfold csquare_rel_def) 

529 
apply (subgoal_tac "x<K & y<K") 

530 
prefer 2 apply (blast intro: lt_trans1) 

531 
apply (elim ltE) 

13221  532 
apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD) 
13216  533 
apply (elim succE) 
13221  534 
apply (simp_all add: subset_Un_iff [THEN iff_sym] 
535 
subset_Un_iff2 [THEN iff_sym] OrdmemD) 

13216  536 
done 
537 

14883  538 
subsubsection{*The cardinality of initial segments*} 
13216  539 

540 
lemma ordermap_z_lt: 

541 
"[ Limit(K); x<K; y<K; z=succ(x Un y) ] ==> 

542 
ordermap(K*K, csquare_rel(K)) ` <x,y> < 

543 
ordermap(K*K, csquare_rel(K)) ` <z,z>" 

544 
apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))") 

545 
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ 

13221  546 
Limit_is_Ord [THEN well_ord_csquare], clarify) 
13216  547 
apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI]) 
548 
apply (erule_tac [4] well_ord_is_wf) 

549 
apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+ 

550 
done 

551 

552 
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) 

553 
lemma ordermap_csquare_le: 

13221  554 
"[ Limit(K); x<K; y<K; z=succ(x Un y) ] 
555 
==>  ordermap(K*K, csquare_rel(K)) ` <x,y>  le succ(z) * succ(z)" 

13216  556 
apply (unfold cmult_def) 
557 
apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le]) 

558 
apply (rule Ord_cardinal [THEN well_ord_Memrel])+ 

559 
apply (subgoal_tac "z<K") 

560 
prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ) 

13221  561 
apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
562 
assumption+) 

13216  563 
apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) 
564 
apply (erule Limit_is_Ord [THEN well_ord_csquare]) 

565 
apply (blast intro: ltD) 

566 
apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans], 

567 
assumption) 

568 
apply (elim ltE) 

569 
apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll]) 

570 
apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+ 

571 
done 

572 

573 
(*Kunen: "... so the order type <= K" *) 

574 
lemma ordertype_csquare_le: 

575 
"[ InfCard(K); ALL y:K. InfCard(y) > y * y = y ] 

576 
==> ordertype(K*K, csquare_rel(K)) le K" 

577 
apply (frule InfCard_is_Card [THEN Card_is_Ord]) 

13221  578 
apply (rule all_lt_imp_le, assumption) 
13216  579 
apply (erule well_ord_csquare [THEN Ord_ordertype]) 
580 
apply (rule Card_lt_imp_lt) 

581 
apply (erule_tac [3] InfCard_is_Card) 

582 
apply (erule_tac [2] ltE) 

583 
apply (simp add: ordertype_unfold) 

584 
apply (safe elim!: ltE) 

585 
apply (subgoal_tac "Ord (xa) & Ord (ya)") 

13221  586 
prefer 2 apply (blast intro: Ord_in_Ord, clarify) 
13216  587 
(*??WHAT A MESS!*) 
588 
apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1], 

589 
(assumption  rule refl  erule ltI)+) 

13784  590 
apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2, 
13216  591 
simp_all add: Ord_Un Ord_nat) 
592 
prefer 2 (*case nat le (xa Un ya) *) 

593 
apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 

594 
le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un 

595 
ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD]) 

596 
(*the finite case: xa Un ya < nat *) 

13784  597 
apply (rule_tac j = nat in lt_trans2) 
13216  598 
apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type 
599 
nat_into_Card [THEN Card_cardinal_eq] Ord_nat) 

600 
apply (simp add: InfCard_def) 

601 
done 

602 

603 
(*Main result: Kunen's Theorem 10.12*) 

604 
lemma InfCard_csquare_eq: "InfCard(K) ==> K * K = K" 

605 
apply (frule InfCard_is_Card [THEN Card_is_Ord]) 

606 
apply (erule rev_mp) 

607 
apply (erule_tac i=K in trans_induct) 

608 
apply (rule impI) 

609 
apply (rule le_anti_sym) 

610 
apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le]) 

611 
apply (rule ordertype_csquare_le [THEN [2] le_trans]) 

13221  612 
apply (simp add: cmult_def Ord_cardinal_le 
613 
well_ord_csquare [THEN Ord_ordertype] 

614 
well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 

615 
THEN cardinal_cong], assumption+) 

13216  616 
done 
617 

618 
(*Corollary for arbitrary wellordered sets (all sets, assuming AC)*) 

619 
lemma well_ord_InfCard_square_eq: 

620 
"[ well_ord(A,r); InfCard(A) ] ==> A*A \<approx> A" 

621 
apply (rule prod_eqpoll_cong [THEN eqpoll_trans]) 

622 
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+ 

623 
apply (rule well_ord_cardinal_eqE) 

13221  624 
apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption) 
625 
apply (simp add: cmult_def [symmetric] InfCard_csquare_eq) 

13216  626 
done 
627 

13356  628 
lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K" 
629 
apply (rule well_ord_InfCard_square_eq) 

630 
apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 

631 
apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 

632 
done 

633 

634 
lemma Inf_Card_is_InfCard: "[ ~Finite(i); Card(i) ] ==> InfCard(i)" 

635 
by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord]) 

636 

14883  637 
subsubsection{*Toward's Kunen's Corollary 10.13 (1)*} 
13216  638 

639 
lemma InfCard_le_cmult_eq: "[ InfCard(K); L le K; 0<L ] ==> K * L = K" 

640 
apply (rule le_anti_sym) 

641 
prefer 2 

642 
apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card) 

643 
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) 

644 
apply (rule cmult_le_mono [THEN le_trans], assumption+) 

645 
apply (simp add: InfCard_csquare_eq) 

646 
done 

647 

648 
(*Corollary 10.13 (1), for cardinal multiplication*) 

649 
lemma InfCard_cmult_eq: "[ InfCard(K); InfCard(L) ] ==> K * L = K Un L" 

13784  650 
apply (rule_tac i = K and j = L in Ord_linear_le) 
13216  651 
apply (typecheck add: InfCard_is_Card Card_is_Ord) 
652 
apply (rule cmult_commute [THEN ssubst]) 

653 
apply (rule Un_commute [THEN ssubst]) 

13221  654 
apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
655 
subset_Un_iff2 [THEN iffD1] le_imp_subset) 

13216  656 
done 
657 

658 
lemma InfCard_cdouble_eq: "InfCard(K) ==> K + K = K" 

13221  659 
apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute) 
660 
apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ) 

13216  661 
done 
662 

663 
(*Corollary 10.13 (1), for cardinal addition*) 

664 
lemma InfCard_le_cadd_eq: "[ InfCard(K); L le K ] ==> K + L = K" 

665 
apply (rule le_anti_sym) 

666 
prefer 2 

667 
apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card) 

668 
apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl]) 

669 
apply (rule cadd_le_mono [THEN le_trans], assumption+) 

670 
apply (simp add: InfCard_cdouble_eq) 

671 
done 

672 

673 
lemma InfCard_cadd_eq: "[ InfCard(K); InfCard(L) ] ==> K + L = K Un L" 

13784  674 
apply (rule_tac i = K and j = L in Ord_linear_le) 
13216  675 
apply (typecheck add: InfCard_is_Card Card_is_Ord) 
676 
apply (rule cadd_commute [THEN ssubst]) 

677 
apply (rule Un_commute [THEN ssubst]) 

13221  678 
apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset) 
13216  679 
done 
680 

681 
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set 

682 
of all ntuples of elements of K. A better version for the Isabelle theory 

683 
might be InfCard(K) ==> list(K) = K. 

684 
*) 

685 

13356  686 
subsection{*For Every Cardinal Number There Exists A Greater One} 
687 

688 
text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*} 

13216  689 

690 
lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))" 

691 
apply (unfold jump_cardinal_def) 

692 
apply (rule Ord_is_Transset [THEN [2] OrdI]) 

693 
prefer 2 apply (blast intro!: Ord_ordertype) 

694 
apply (unfold Transset_def) 

695 
apply (safe del: subsetI) 

13221  696 
apply (simp add: ordertype_pred_unfold, safe) 
13216  697 
apply (rule UN_I) 
698 
apply (rule_tac [2] ReplaceI) 

699 
prefer 4 apply (blast intro: well_ord_subset elim!: predE)+ 

700 
done 

701 

702 
(*Allows selective unfolding. Less work than deriving intro/elim rules*) 

703 
lemma jump_cardinal_iff: 

704 
"i : jump_cardinal(K) <> 

705 
(EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))" 

706 
apply (unfold jump_cardinal_def) 

707 
apply (blast del: subsetI) 

708 
done 

709 

710 
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) 

711 
lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)" 

712 
apply (rule Ord_jump_cardinal [THEN [2] ltI]) 

713 
apply (rule jump_cardinal_iff [THEN iffD2]) 

714 
apply (rule_tac x="Memrel(K)" in exI) 

715 
apply (rule_tac x=K in exI) 

716 
apply (simp add: ordertype_Memrel well_ord_Memrel) 

717 
apply (simp add: Memrel_def subset_iff) 

718 
done 

719 

720 
(*The proof by contradiction: the bijection f yields a wellordering of X 

721 
whose ordertype is jump_cardinal(K). *) 

722 
lemma Card_jump_cardinal_lemma: 

723 
"[ well_ord(X,r); r <= K * K; X <= K; 

724 
f : bij(ordertype(X,r), jump_cardinal(K)) ] 

725 
==> jump_cardinal(K) : jump_cardinal(K)" 

726 
apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))") 

727 
prefer 2 apply (blast intro: comp_bij ordermap_bij) 

728 
apply (rule jump_cardinal_iff [THEN iffD2]) 

729 
apply (intro exI conjI) 

13221  730 
apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+) 
13216  731 
apply (erule bij_is_inj [THEN well_ord_rvimage]) 
732 
apply (rule Ord_jump_cardinal [THEN well_ord_Memrel]) 

733 
apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage] 

734 
ordertype_Memrel Ord_jump_cardinal) 

735 
done 

736 

737 
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) 

738 
lemma Card_jump_cardinal: "Card(jump_cardinal(K))" 

739 
apply (rule Ord_jump_cardinal [THEN CardI]) 

740 
apply (unfold eqpoll_def) 

741 
apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1]) 

742 
apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl]) 

743 
done 

744 

13356  745 
subsection{*Basic Properties of Successor Cardinals*} 
13216  746 

747 
lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)" 

748 
apply (unfold csucc_def) 

749 
apply (rule LeastI) 

750 
apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+ 

751 
done 

752 

753 
lemmas Card_csucc = csucc_basic [THEN conjunct1, standard] 

754 

755 
lemmas lt_csucc = csucc_basic [THEN conjunct2, standard] 

756 

757 
lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)" 

13221  758 
by (blast intro: Ord_0_le lt_csucc lt_trans1) 
13216  759 

760 
lemma csucc_le: "[ Card(L); K<L ] ==> csucc(K) le L" 

761 
apply (unfold csucc_def) 

762 
apply (rule Least_le) 

763 
apply (blast intro: Card_is_Ord)+ 

764 
done 

765 

766 
lemma lt_csucc_iff: "[ Ord(i); Card(K) ] ==> i < csucc(K) <> i le K" 

767 
apply (rule iffI) 

768 
apply (rule_tac [2] Card_lt_imp_lt) 

769 
apply (erule_tac [2] lt_trans1) 

770 
apply (simp_all add: lt_csucc Card_csucc Card_is_Ord) 

771 
apply (rule notI [THEN not_lt_imp_le]) 

13221  772 
apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption) 
13216  773 
apply (rule Ord_cardinal_le [THEN lt_trans1]) 
774 
apply (simp_all add: Ord_cardinal Card_is_Ord) 

775 
done 

776 

777 
lemma Card_lt_csucc_iff: 

778 
"[ Card(K'); Card(K) ] ==> K' < csucc(K) <> K' le K" 

13221  779 
by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord) 
13216  780 

781 
lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))" 

782 
by (simp add: InfCard_def Card_csucc Card_is_Ord 

783 
lt_csucc [THEN leI, THEN [2] le_trans]) 

784 

785 

14883  786 
subsubsection{*Removing elements from a finite set decreases its cardinality*} 
13216  787 

788 
lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A > ~ cons(x,A) \<lesssim> A" 

789 
apply (erule Fin_induct) 

13221  790 
apply (simp add: lepoll_0_iff) 
13216  791 
apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))") 
13221  792 
apply simp 
793 
apply (blast dest!: cons_lepoll_consD, blast) 

13216  794 
done 
795 

14883  796 
lemma Finite_imp_cardinal_cons [simp]: 
13221  797 
"[ Finite(A); a~:A ] ==> cons(a,A) = succ(A)" 
13216  798 
apply (unfold cardinal_def) 
799 
apply (rule Least_equality) 

800 
apply (fold cardinal_def) 

13221  801 
apply (simp add: succ_def) 
13216  802 
apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll 
803 
elim!: mem_irrefl dest!: Finite_imp_well_ord) 

804 
apply (blast intro: Card_cardinal Card_is_Ord) 

805 
apply (rule notI) 

13221  806 
apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE], 
807 
assumption, assumption) 

13216  808 
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) 
809 
apply (erule le_imp_lepoll [THEN lepoll_trans]) 

810 
apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll] 

811 
dest!: Finite_imp_well_ord) 

812 
done 

813 

814 

13221  815 
lemma Finite_imp_succ_cardinal_Diff: 
816 
"[ Finite(A); a:A ] ==> succ(A{a}) = A" 

13784  817 
apply (rule_tac b = A in cons_Diff [THEN subst], assumption) 
13221  818 
apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite]) 
819 
apply (simp add: cons_Diff) 

13216  820 
done 
821 

822 
lemma Finite_imp_cardinal_Diff: "[ Finite(A); a:A ] ==> A{a} < A" 

823 
apply (rule succ_leE) 

13221  824 
apply (simp add: Finite_imp_succ_cardinal_Diff) 
13216  825 
done 
826 

14883  827 
lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> A : nat" 
828 
apply (erule Finite_induct) 

829 
apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons) 

830 
done 

13216  831 

14883  832 
lemma card_Un_Int: 
833 
"[Finite(A); Finite(B)] ==> A #+ B = A Un B #+ A Int B" 

834 
apply (erule Finite_induct, simp) 

835 
apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left) 

836 
done 

837 

838 
lemma card_Un_disjoint: 

839 
"[Finite(A); Finite(B); A Int B = 0] ==> A Un B = A #+ B" 

840 
by (simp add: Finite_Un card_Un_Int) 

841 

842 
lemma card_partition [rule_format]: 

843 
"Finite(C) ==> 

844 
Finite (\<Union> C) > 

845 
(\<forall>c\<in>C. c = k) > 

846 
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 > c1 \<inter> c2 = 0) > 

847 
k #* C = \<Union> C" 

848 
apply (erule Finite_induct, auto) 

849 
apply (subgoal_tac " x \<inter> \<Union>B = 0") 

850 
apply (auto simp add: card_Un_disjoint Finite_Union 

851 
subset_Finite [of _ "\<Union> (cons(x,F))"]) 

852 
done 

853 

854 

855 
subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*} 

13216  856 

857 
lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard] 

858 

859 
lemma nat_sum_eqpoll_sum: "[ m:nat; n:nat ] ==> m + n \<approx> m #+ n" 

860 
apply (rule eqpoll_trans) 

861 
apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym]) 

862 
apply (erule nat_implies_well_ord)+ 

13221  863 
apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl) 
13216  864 
done 
865 

13221  866 
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat > i : nat  i=nat" 
867 
apply (erule trans_induct3, auto) 

13216  868 
apply (blast dest!: nat_le_Limit [THEN le_imp_subset]) 
869 
done 

870 

871 
lemma Ord_nat_subset_into_Card: "[ Ord(i); i <= nat ] ==> Card(i)" 

13221  872 
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card) 
13216  873 

874 
lemma Finite_Diff_sing_eq_diff_1: "[ Finite(A); x:A ] ==> A{x} = A # 1" 

875 
apply (rule succ_inject) 

876 
apply (rule_tac b = "A" in trans) 

13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset

877 
apply (simp add: Finite_imp_succ_cardinal_Diff) 
13216  878 
apply (subgoal_tac "1 \<lesssim> A") 
13221  879 
prefer 2 apply (blast intro: not_0_is_lepoll_1) 
880 
apply (frule Finite_imp_well_ord, clarify) 

13216  881 
apply (drule well_ord_lepoll_imp_Card_le) 
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset

882 
apply (auto simp add: cardinal_1) 
13216  883 
apply (rule trans) 
13615
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset

884 
apply (rule_tac [2] diff_succ) 
449a70d88b38
Numerous cosmetic changes, prompted by the new simplifier
paulson
parents:
13356
diff
changeset

885 
apply (auto simp add: Finite_cardinal_in_nat) 
13216  886 
done 
887 

13221  888 
lemma cardinal_lt_imp_Diff_not_0 [rule_format]: 
889 
"Finite(B) ==> ALL A. B<A > A  B ~= 0" 

890 
apply (erule Finite_induct, auto) 

891 
apply (case_tac "Finite (A)") 

892 
apply (subgoal_tac [2] "Finite (cons (x, B))") 

893 
apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite) 

894 
apply (auto simp add: Finite_0 Finite_cons) 

13216  895 
apply (subgoal_tac "B<A") 
13221  896 
prefer 2 apply (blast intro: lt_trans Ord_cardinal) 
13216  897 
apply (case_tac "x:A") 
13221  898 
apply (subgoal_tac [2] "A  cons (x, B) = A  B") 
899 
apply auto 

13216  900 
apply (subgoal_tac "A le cons (x, B) ") 
13221  901 
prefer 2 
13216  902 
apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
903 
intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll) 

904 
apply (auto simp add: Finite_imp_cardinal_cons) 

905 
apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff) 

906 
apply (blast intro: lt_trans) 

907 
done 

908 

909 

910 
ML{* 

911 
val InfCard_def = thm "InfCard_def" 

912 
val cmult_def = thm "cmult_def" 

913 
val cadd_def = thm "cadd_def" 

914 
val jump_cardinal_def = thm "jump_cardinal_def" 

915 
val csucc_def = thm "csucc_def" 

916 

917 
val sum_commute_eqpoll = thm "sum_commute_eqpoll"; 

918 
val cadd_commute = thm "cadd_commute"; 

919 
val sum_assoc_eqpoll = thm "sum_assoc_eqpoll"; 

920 
val well_ord_cadd_assoc = thm "well_ord_cadd_assoc"; 

921 
val sum_0_eqpoll = thm "sum_0_eqpoll"; 

922 
val cadd_0 = thm "cadd_0"; 

923 
val sum_lepoll_self = thm "sum_lepoll_self"; 

924 
val cadd_le_self = thm "cadd_le_self"; 

925 
val sum_lepoll_mono = thm "sum_lepoll_mono"; 

926 
val cadd_le_mono = thm "cadd_le_mono"; 

927 
val eq_imp_not_mem = thm "eq_imp_not_mem"; 

928 
val sum_succ_eqpoll = thm "sum_succ_eqpoll"; 

929 
val nat_cadd_eq_add = thm "nat_cadd_eq_add"; 

930 
val prod_commute_eqpoll = thm "prod_commute_eqpoll"; 

931 
val cmult_commute = thm "cmult_commute"; 

932 
val prod_assoc_eqpoll = thm "prod_assoc_eqpoll"; 

933 
val well_ord_cmult_assoc = thm "well_ord_cmult_assoc"; 

934 
val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll"; 

935 
val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib"; 

936 
val prod_0_eqpoll = thm "prod_0_eqpoll"; 

937 
val cmult_0 = thm "cmult_0"; 

938 
val prod_singleton_eqpoll = thm "prod_singleton_eqpoll"; 

939 
val cmult_1 = thm "cmult_1"; 

940 
val prod_lepoll_self = thm "prod_lepoll_self"; 

941 
val cmult_le_self = thm "cmult_le_self"; 

942 
val prod_lepoll_mono = thm "prod_lepoll_mono"; 

943 
val cmult_le_mono = thm "cmult_le_mono"; 

944 
val prod_succ_eqpoll = thm "prod_succ_eqpoll"; 

945 
val nat_cmult_eq_mult = thm "nat_cmult_eq_mult"; 

946 
val cmult_2 = thm "cmult_2"; 

947 
val sum_lepoll_prod = thm "sum_lepoll_prod"; 

948 
val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod"; 

949 
val nat_cons_lepoll = thm "nat_cons_lepoll"; 

950 
val nat_cons_eqpoll = thm "nat_cons_eqpoll"; 

951 
val nat_succ_eqpoll = thm "nat_succ_eqpoll"; 

952 
val InfCard_nat = thm "InfCard_nat"; 

953 
val InfCard_is_Card = thm "InfCard_is_Card"; 

954 
val InfCard_Un = thm "InfCard_Un"; 

955 
val InfCard_is_Limit = thm "InfCard_is_Limit"; 

956 
val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred"; 

957 
val ordermap_z_lt = thm "ordermap_z_lt"; 

958 
val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq"; 

959 
val InfCard_cmult_eq = thm "InfCard_cmult_eq"; 

960 
val InfCard_cdouble_eq = thm "InfCard_cdouble_eq"; 

961 
val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq"; 

962 
val InfCard_cadd_eq = thm "InfCard_cadd_eq"; 

963 
val Ord_jump_cardinal = thm "Ord_jump_cardinal"; 

964 
val jump_cardinal_iff = thm "jump_cardinal_iff"; 

965 
val K_lt_jump_cardinal = thm "K_lt_jump_cardinal"; 

966 
val Card_jump_cardinal = thm "Card_jump_cardinal"; 

967 
val csucc_basic = thm "csucc_basic"; 

968 
val Card_csucc = thm "Card_csucc"; 

969 
val lt_csucc = thm "lt_csucc"; 

970 
val Ord_0_lt_csucc = thm "Ord_0_lt_csucc"; 

971 
val csucc_le = thm "csucc_le"; 

972 
val lt_csucc_iff = thm "lt_csucc_iff"; 

973 
val Card_lt_csucc_iff = thm "Card_lt_csucc_iff"; 

974 
val InfCard_csucc = thm "InfCard_csucc"; 

975 
val Finite_into_Fin = thm "Finite_into_Fin"; 

976 
val Fin_into_Finite = thm "Fin_into_Finite"; 

977 
val Finite_Fin_iff = thm "Finite_Fin_iff"; 

978 
val Finite_Un = thm "Finite_Un"; 

979 
val Finite_Union = thm "Finite_Union"; 

980 
val Finite_induct = thm "Finite_induct"; 

981 
val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll"; 

982 
val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons"; 

983 
val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff"; 

984 
val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff"; 

985 
val nat_implies_well_ord = thm "nat_implies_well_ord"; 

986 
val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum"; 

987 
val Diff_sing_Finite = thm "Diff_sing_Finite"; 

988 
val Diff_Finite = thm "Diff_Finite"; 

989 
val Ord_subset_natD = thm "Ord_subset_natD"; 

990 
val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card"; 

991 
val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat"; 

992 
val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1"; 

993 
val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0"; 

994 
*} 

995 

437  996 
end 