author  lcp 
Thu, 08 Dec 1994 14:06:16 +0100  
changeset 767  a4fce3b94065 
parent 760  f0200e91b272 
child 782  200a16083201 
permissions  rwrr 
437  1 
(* Title: ZF/CardinalArith.ML 
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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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Cardinal arithmetic  WITHOUT the Axiom of Choice 

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*) 

8 

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open CardinalArith; 

10 

484  11 
(*** Elementary properties ***) 
467  12 

437  13 
(*Use AC to discharge first premise*) 
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goal CardinalArith.thy 

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"!!A B. [ well_ord(B,r); A lepoll B ] ==> A le B"; 

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by (res_inst_tac [("i","A"),("j","B")] Ord_linear_le 1); 

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by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI])); 

18 
by (rtac (eqpollI RS cardinal_cong) 1 THEN assume_tac 1); 

19 
by (rtac lepoll_trans 1); 

20 
by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym RS eqpoll_imp_lepoll) 1); 

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by (assume_tac 1); 

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by (etac (le_imp_subset RS subset_imp_lepoll RS lepoll_trans) 1); 

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by (rtac eqpoll_imp_lepoll 1); 

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by (rewtac lepoll_def); 

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by (etac exE 1); 

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by (rtac well_ord_cardinal_eqpoll 1); 

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by (etac well_ord_rvimage 1); 

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by (assume_tac 1); 

767  29 
qed "well_ord_lepoll_imp_Card_le"; 
437  30 

484  31 
(*Each element of Fin(A) is equivalent to a natural number*) 
32 
goal CardinalArith.thy 

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"!!X A. X: Fin(A) ==> EX n:nat. X eqpoll n"; 

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by (eresolve_tac [Fin_induct] 1); 

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by (fast_tac (ZF_cs addIs [eqpoll_refl, nat_0I]) 1); 

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by (fast_tac (ZF_cs addSIs [cons_eqpoll_cong, 

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rewrite_rule [succ_def] nat_succI] 

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addSEs [mem_irrefl]) 1); 

760  39 
qed "Fin_eqpoll"; 
484  40 

437  41 
(*** Cardinal addition ***) 
42 

43 
(** Cardinal addition is commutative **) 

44 

45 
goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A"; 

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by (rtac exI 1); 

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by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] 

48 
lam_bijective 1); 

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by (safe_tac (ZF_cs addSEs [sumE])); 

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by (ALLGOALS (asm_simp_tac case_ss)); 

760  51 
qed "sum_commute_eqpoll"; 
437  52 

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goalw CardinalArith.thy [cadd_def] "i + j = j + i"; 

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by (rtac (sum_commute_eqpoll RS cardinal_cong) 1); 

760  55 
qed "cadd_commute"; 
437  56 

57 
(** Cardinal addition is associative **) 

58 

59 
goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)"; 

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by (rtac exI 1); 

61 
by (res_inst_tac [("c", "case(case(Inl, %y.Inr(Inl(y))), %y. Inr(Inr(y)))"), 

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("d", "case(%x.Inl(Inl(x)), case(%x.Inl(Inr(x)), Inr))")] 

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lam_bijective 1); 

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by (ALLGOALS (asm_simp_tac (case_ss setloop etac sumE))); 

760  65 
qed "sum_assoc_eqpoll"; 
437  66 

67 
(*Unconditional version requires AC*) 

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goalw CardinalArith.thy [cadd_def] 

484  69 
"!!i j k. [ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] ==> \ 
437  70 
\ (i + j) + k = i + (j + k)"; 
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by (rtac cardinal_cong 1); 

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br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS 

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eqpoll_trans) 1; 

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by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2); 

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br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS 

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eqpoll_sym) 2; 

484  77 
by (REPEAT (ares_tac [well_ord_radd] 1)); 
760  78 
qed "well_ord_cadd_assoc"; 
437  79 

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

81 

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goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A"; 

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by (rtac exI 1); 

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by (res_inst_tac [("c", "case(%x.x, %y.y)"), ("d", "Inr")] 

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lam_bijective 1); 

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by (ALLGOALS (asm_simp_tac (case_ss setloop eresolve_tac [sumE,emptyE]))); 

760  87 
qed "sum_0_eqpoll"; 
437  88 

484  89 
goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 + K = K"; 
437  90 
by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong, 
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Card_cardinal_eq]) 1); 

760  92 
qed "cadd_0"; 
437  93 

767  94 
(** Addition by another cardinal **) 
95 

96 
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A+B"; 

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by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1); 

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by (asm_simp_tac (sum_ss addsimps [lam_type]) 1); 

99 
val sum_lepoll_self = result(); 

100 

101 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

102 
goalw CardinalArith.thy [cadd_def] 

103 
"!!K. [ Card(K); Ord(L) ] ==> K le (K + L)"; 

104 
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); 

105 
by (rtac sum_lepoll_self 3); 

106 
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1)); 

107 
val cadd_le_self = result(); 

108 

109 
(** Monotonicity of addition **) 

110 

111 
goalw CardinalArith.thy [lepoll_def] 

112 
"!!A B C D. [ A lepoll C; B lepoll D ] ==> A + B lepoll C + D"; 

113 
by (REPEAT (etac exE 1)); 

114 
by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")] 

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exI 1); 

116 
by (res_inst_tac 

117 
[("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")] 

118 
lam_injective 1); 

119 
by (typechk_tac ([inj_is_fun,case_type, InlI, InrI] @ ZF_typechecks)); 

120 
by (eresolve_tac [sumE] 1); 

121 
by (ALLGOALS (asm_simp_tac (sum_ss addsimps [left_inverse]))); 

122 
val sum_lepoll_mono = result(); 

123 

124 
goalw CardinalArith.thy [cadd_def] 

125 
"!!K. [ K' le K; L' le L ] ==> (K' + L') le (K + L)"; 

126 
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); 

127 
by (resolve_tac [well_ord_lepoll_imp_Card_le] 1); 

128 
by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2)); 

129 
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); 

130 
val cadd_le_mono = result(); 

131 

437  132 
(** Addition of finite cardinals is "ordinary" addition **) 
133 

134 
goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)"; 

135 
by (rtac exI 1); 

136 
by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"), 

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("d", "%z.if(z=A+B,Inl(A),z)")] 

138 
lam_bijective 1); 

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by (ALLGOALS 

140 
(asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq] 

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setloop eresolve_tac [sumE,succE]))); 

760  142 
qed "sum_succ_eqpoll"; 
437  143 

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

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(*Unconditional version requires AC*) 

146 
goalw CardinalArith.thy [cadd_def] 

147 
"!!m n. [ Ord(m); Ord(n) ] ==> succ(m) + n = succ(m + n)"; 

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by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1); 

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by (rtac (succ_eqpoll_cong RS cardinal_cong) 1); 

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by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1); 

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by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); 

760  152 
qed "cadd_succ_lemma"; 
437  153 

154 
val [mnat,nnat] = goal CardinalArith.thy 

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

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by (cut_facts_tac [nnat] 1); 

157 
by (nat_ind_tac "m" [mnat] 1); 

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by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1); 

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by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma, 

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nat_into_Card RS Card_cardinal_eq]) 1); 

760  161 
qed "nat_cadd_eq_add"; 
437  162 

163 

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(*** Cardinal multiplication ***) 

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(** Cardinal multiplication is commutative **) 

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(*Easier to prove the two directions separately*) 

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goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A"; 

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by (rtac exI 1); 

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by (res_inst_tac [("c", "split(%x y.<y,x>)"), ("d", "split(%x y.<y,x>)")] 

172 
lam_bijective 1); 

173 
by (safe_tac ZF_cs); 

174 
by (ALLGOALS (asm_simp_tac ZF_ss)); 

760  175 
qed "prod_commute_eqpoll"; 
437  176 

177 
goalw CardinalArith.thy [cmult_def] "i * j = j * i"; 

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by (rtac (prod_commute_eqpoll RS cardinal_cong) 1); 

760  179 
qed "cmult_commute"; 
437  180 

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(** Cardinal multiplication is associative **) 

182 

183 
goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)"; 

184 
by (rtac exI 1); 

185 
by (res_inst_tac [("c", "split(%w z. split(%x y. <x,<y,z>>, w))"), 

186 
("d", "split(%x. split(%y z. <<x,y>, z>))")] 

187 
lam_bijective 1); 

188 
by (safe_tac ZF_cs); 

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by (ALLGOALS (asm_simp_tac ZF_ss)); 

760  190 
qed "prod_assoc_eqpoll"; 
437  191 

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(*Unconditional version requires AC*) 

193 
goalw CardinalArith.thy [cmult_def] 

484  194 
"!!i j k. [ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] ==> \ 
437  195 
\ (i * j) * k = i * (j * k)"; 
196 
by (rtac cardinal_cong 1); 

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br ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS 

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eqpoll_trans) 1; 

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by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2); 

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br ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS 

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eqpoll_sym) 2; 

484  202 
by (REPEAT (ares_tac [well_ord_rmult] 1)); 
760  203 
qed "well_ord_cmult_assoc"; 
437  204 

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(** Cardinal multiplication distributes over addition **) 

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goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)"; 

208 
by (rtac exI 1); 

209 
by (res_inst_tac 

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[("c", "split(%x z. case(%y.Inl(<y,z>), %y.Inr(<y,z>), x))"), 

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("d", "case(split(%x y.<Inl(x),y>), split(%x y.<Inr(x),y>))")] 

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lam_bijective 1); 

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by (safe_tac (ZF_cs addSEs [sumE])); 

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by (ALLGOALS (asm_simp_tac case_ss)); 

760  215 
qed "sum_prod_distrib_eqpoll"; 
437  216 

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(** Multiplication by 0 yields 0 **) 

218 

219 
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0"; 

220 
by (rtac exI 1); 

221 
by (rtac lam_bijective 1); 

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by (safe_tac ZF_cs); 

760  223 
qed "prod_0_eqpoll"; 
437  224 

225 
goalw CardinalArith.thy [cmult_def] "0 * i = 0"; 

226 
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, 

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Card_0 RS Card_cardinal_eq]) 1); 

760  228 
qed "cmult_0"; 
437  229 

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

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232 
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A"; 

233 
by (rtac exI 1); 

234 
by (res_inst_tac [("c", "snd"), ("d", "%z.<x,z>")] lam_bijective 1); 

235 
by (safe_tac ZF_cs); 

236 
by (ALLGOALS (asm_simp_tac ZF_ss)); 

760  237 
qed "prod_singleton_eqpoll"; 
437  238 

484  239 
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 * K = K"; 
437  240 
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, 
241 
Card_cardinal_eq]) 1); 

760  242 
qed "cmult_1"; 
437  243 

767  244 
(*** Some inequalities for multiplication ***) 
245 

246 
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A"; 

247 
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1); 

248 
by (simp_tac (ZF_ss addsimps [lam_type]) 1); 

249 
qed "prod_square_lepoll"; 

250 

251 
(*Could probably weaken the premise to well_ord(K,r), or removing using AC*) 

252 
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K * K"; 

253 
by (rtac le_trans 1); 

254 
by (rtac well_ord_lepoll_imp_Card_le 2); 

255 
by (rtac prod_square_lepoll 3); 

256 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2)); 

257 
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1); 

258 
qed "cmult_square_le"; 

259 

260 
(** Multiplication by a nonzero cardinal **) 

261 

262 
goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B"; 

263 
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1); 

264 
by (asm_simp_tac (ZF_ss addsimps [lam_type]) 1); 

265 
val prod_lepoll_self = result(); 

266 

267 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

268 
goalw CardinalArith.thy [cmult_def] 

269 
"!!K. [ Card(K); Ord(L); 0<L ] ==> K le (K * L)"; 

270 
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); 

271 
by (rtac prod_lepoll_self 3); 

272 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1)); 

273 
val cmult_le_self = result(); 

274 

275 
(** Monotonicity of multiplication **) 

276 

277 
(*Equivalent to KG's lepoll_SigmaI*) 

278 
goalw CardinalArith.thy [lepoll_def] 

279 
"!!A B C D. [ A lepoll C; B lepoll D ] ==> A * B lepoll C * D"; 

280 
by (REPEAT (etac exE 1)); 

281 
by (res_inst_tac [("x", "lam z:A*B. split(%w y.<f`w, fa`y>, z)")] exI 1); 

282 
by (res_inst_tac [("d", "split(%w y.<converse(f)`w, converse(fa)`y>)")] 

283 
lam_injective 1); 

284 
by (typechk_tac (inj_is_fun::ZF_typechecks)); 

285 
by (eresolve_tac [SigmaE] 1); 

286 
by (asm_simp_tac (ZF_ss addsimps [left_inverse]) 1); 

287 
val prod_lepoll_mono = result(); 

288 

289 
goalw CardinalArith.thy [cmult_def] 

290 
"!!K. [ K' le K; L' le L ] ==> (K' * L') le (K * L)"; 

291 
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); 

292 
by (resolve_tac [well_ord_lepoll_imp_Card_le] 1); 

293 
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2)); 

294 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

295 
val cmult_le_mono = result(); 

296 

297 
(*** Multiplication of finite cardinals is "ordinary" multiplication ***) 

437  298 

299 
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B"; 

300 
by (rtac exI 1); 

301 
by (res_inst_tac [("c", "split(%x y. if(x=A, Inl(y), Inr(<x,y>)))"), 

302 
("d", "case(%y. <A,y>, %z.z)")] 

303 
lam_bijective 1); 

304 
by (safe_tac (ZF_cs addSEs [sumE])); 

305 
by (ALLGOALS 

306 
(asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq]))); 

760  307 
qed "prod_succ_eqpoll"; 
437  308 

309 
(*Unconditional version requires AC*) 

310 
goalw CardinalArith.thy [cmult_def, cadd_def] 

311 
"!!m n. [ Ord(m); Ord(n) ] ==> succ(m) * n = n + (m * n)"; 

312 
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1); 

313 
by (rtac (cardinal_cong RS sym) 1); 

314 
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1); 

315 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

760  316 
qed "cmult_succ_lemma"; 
437  317 

318 
val [mnat,nnat] = goal CardinalArith.thy 

319 
"[ m: nat; n: nat ] ==> m * n = m#*n"; 

320 
by (cut_facts_tac [nnat] 1); 

321 
by (nat_ind_tac "m" [mnat] 1); 

322 
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1); 

323 
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma, 

324 
nat_cadd_eq_add]) 1); 

760  325 
qed "nat_cmult_eq_mult"; 
437  326 

767  327 
(*Needs Krzysztof Grabczewski's macro "2" == "succ(1)"*) 
328 
goal CardinalArith.thy "!!m n. Card(n) ==> succ(1) * n = n + n"; 

329 
by (asm_simp_tac 

330 
(ZF_ss addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord, 

331 
read_instantiate [("j","0")] cadd_commute, cadd_0]) 1); 

332 
val cmult_2 = result(); 

333 

437  334 

335 
(*** Infinite Cardinals are Limit Ordinals ***) 

336 

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(*This proof is modelled upon one assuming nat<=A, with injection 
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lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z)) and inverse 
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%y. if(y:nat, nat_case(u,%z.z,y), y). If f: inj(nat,A) then 
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range(f) behaves like the natural numbers.*) 
516  341 
goalw CardinalArith.thy [lepoll_def] 
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"!!i. nat lepoll A ==> cons(u,A) lepoll A"; 
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by (etac exE 1); 
516  344 
by (res_inst_tac [("x", 
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"lam z:cons(u,A). if(z=u, f`0, \ 
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\ if(z: range(f), f`succ(converse(f)`z), z))")] exI 1); 
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by (res_inst_tac [("d", "%y. if(y: range(f), \ 
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\ nat_case(u, %z.f`z, converse(f)`y), y)")] 
516  349 
lam_injective 1); 
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by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type] 
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addIs [inj_is_fun, inj_converse_fun]) 1); 
516  352 
by (asm_simp_tac 
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(ZF_ss addsimps [inj_is_fun RS apply_rangeI, 
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inj_converse_fun RS apply_rangeI, 
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inj_converse_fun RS apply_funtype, 
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left_inverse, right_inverse, nat_0I, nat_succI, 
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nat_case_0, nat_case_succ] 
516  358 
setloop split_tac [expand_if]) 1); 
760  359 
qed "nat_cons_lepoll"; 
516  360 

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goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A"; 
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by (etac (nat_cons_lepoll RS eqpollI) 1); 
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changeset

363 
by (rtac (subset_consI RS subset_imp_lepoll) 1); 
760  364 
qed "nat_cons_eqpoll"; 
437  365 

571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

366 
(*Specialized version required below*) 
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

367 
goalw CardinalArith.thy [succ_def] "!!i. nat <= A ==> succ(A) eqpoll A"; 
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

368 
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1); 
760  369 
qed "nat_succ_eqpoll"; 
437  370 

488  371 
goalw CardinalArith.thy [InfCard_def] "InfCard(nat)"; 
372 
by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1); 

760  373 
qed "InfCard_nat"; 
488  374 

484  375 
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)"; 
437  376 
by (etac conjunct1 1); 
760  377 
qed "InfCard_is_Card"; 
437  378 

523  379 
goalw CardinalArith.thy [InfCard_def] 
380 
"!!K L. [ InfCard(K); Card(L) ] ==> InfCard(K Un L)"; 

381 
by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 

382 
Card_is_Ord]) 1); 

760  383 
qed "InfCard_Un"; 
523  384 

437  385 
(*Kunen's Lemma 10.11*) 
484  386 
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)"; 
437  387 
by (etac conjE 1); 
388 
by (rtac (ltI RS non_succ_LimitI) 1); 

389 
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1); 

390 
by (etac Card_is_Ord 1); 

391 
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD])); 

392 
by (forward_tac [Card_is_Ord RS Ord_succD] 1); 

393 
by (rewtac Card_def); 

394 
by (res_inst_tac [("i", "succ(y)")] lt_irrefl 1); 

395 
by (dtac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1); 

396 
(*Tricky combination of substitutions; backtracking needed*) 

397 
by (etac ssubst 1 THEN etac ssubst 1 THEN rtac Ord_cardinal_le 1); 

398 
by (assume_tac 1); 

760  399 
qed "InfCard_is_Limit"; 
437  400 

401 

402 

403 
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) 

404 

405 
(*A general fact about ordermap*) 

406 
goalw Cardinal.thy [eqpoll_def] 

407 
"!!A. [ well_ord(A,r); x:A ] ==> ordermap(A,r)`x eqpoll pred(A,x,r)"; 

408 
by (rtac exI 1); 

409 
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1); 

467  410 
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1); 
437  411 
by (rtac pred_subset 1); 
760  412 
qed "ordermap_eqpoll_pred"; 
437  413 

414 
(** Establishing the wellordering **) 

415 

416 
goalw CardinalArith.thy [inj_def] 

484  417 
"!!K. Ord(K) ==> \ 
418 
\ (lam z:K*K. split(%x y. <x Un y, <x, y>>, z)) : inj(K*K, K*K*K)"; 

437  419 
by (safe_tac ZF_cs); 
420 
by (fast_tac (ZF_cs addIs [lam_type, Un_least_lt RS ltD, ltI] 

421 
addSEs [split_type]) 1); 

422 
by (asm_full_simp_tac ZF_ss 1); 

760  423 
qed "csquare_lam_inj"; 
437  424 

425 
goalw CardinalArith.thy [csquare_rel_def] 

484  426 
"!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))"; 
437  427 
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1); 
428 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

760  429 
qed "well_ord_csquare"; 
437  430 

431 
(** Characterising initial segments of the wellordering **) 

432 

433 
goalw CardinalArith.thy [csquare_rel_def] 

484  434 
"!!K. [ x<K; y<K; z<K ] ==> \ 
435 
\ <<x,y>, <z,z>> : csquare_rel(K) > x le z & y le z"; 

437  436 
by (REPEAT (etac ltE 1)); 
437 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

438 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

439 
by (safe_tac (ZF_cs addSEs [mem_irrefl] 

440 
addSIs [Un_upper1_le, Un_upper2_le])); 

441 
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ]))); 

760  442 
qed "csquareD_lemma"; 
437  443 
val csquareD = csquareD_lemma RS mp > standard; 
444 

445 
goalw CardinalArith.thy [pred_def] 

484  446 
"!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"; 
437  447 
by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*) 
448 
by (rtac (csquareD RS conjE) 1); 

449 
by (rewtac lt_def); 

450 
by (assume_tac 4); 

451 
by (ALLGOALS (fast_tac ZF_cs)); 

760  452 
qed "pred_csquare_subset"; 
437  453 

454 
goalw CardinalArith.thy [csquare_rel_def] 

484  455 
"!!K. [ x<z; y<z; z<K ] ==> \ 
456 
\ <<x,y>, <z,z>> : csquare_rel(K)"; 

457 
by (subgoals_tac ["x<K", "y<K"] 1); 

437  458 
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2)); 
459 
by (REPEAT (etac ltE 1)); 

460 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

461 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

760  462 
qed "csquare_ltI"; 
437  463 

464 
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) 

465 
goalw CardinalArith.thy [csquare_rel_def] 

484  466 
"!!K. [ x le z; y le z; z<K ] ==> \ 
467 
\ <<x,y>, <z,z>> : csquare_rel(K)  x=z & y=z"; 

468 
by (subgoals_tac ["x<K", "y<K"] 1); 

437  469 
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2)); 
470 
by (REPEAT (etac ltE 1)); 

471 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

472 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

473 
by (REPEAT_FIRST (etac succE)); 

474 
by (ALLGOALS 

475 
(asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, 

476 
subset_Un_iff2 RS iff_sym, OrdmemD]))); 

760  477 
qed "csquare_or_eqI"; 
437  478 

479 
(** The cardinality of initial segments **) 

480 

481 
goal CardinalArith.thy 

484  482 
"!!K. [ InfCard(K); x<K; y<K; z=succ(x Un y) ] ==> \ 
483 
\ ordermap(K*K, csquare_rel(K)) ` <x,y> lepoll \ 

484 
\ ordermap(K*K, csquare_rel(K)) ` <z,z>"; 

485 
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1); 

437  486 
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 2); 
487 
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, Limit_has_succ]) 2); 

488 
by (rtac (OrdmemD RS subset_imp_lepoll) 1); 

467  489 
by (res_inst_tac [("z1","z")] (csquare_ltI RS ordermap_mono) 1); 
437  490 
by (etac well_ord_is_wf 4); 
491 
by (ALLGOALS 

492 
(fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 

493 
addSEs [ltE]))); 

760  494 
qed "ordermap_z_lepoll"; 
437  495 

484  496 
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) 
437  497 
goalw CardinalArith.thy [cmult_def] 
484  498 
"!!K. [ InfCard(K); x<K; y<K; z=succ(x Un y) ] ==> \ 
499 
\  ordermap(K*K, csquare_rel(K)) ` <x,y>  le succ(z) * succ(z)"; 

767  500 
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1); 
437  501 
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1)); 
484  502 
by (subgoals_tac ["z<K"] 1); 
437  503 
by (fast_tac (ZF_cs addSIs [Un_least_lt, InfCard_is_Limit, 
504 
Limit_has_succ]) 2); 

505 
by (rtac (ordermap_z_lepoll RS lepoll_trans) 1); 

506 
by (REPEAT_SOME assume_tac); 

507 
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1); 

508 
by (etac (InfCard_is_Card RS Card_is_Ord RS well_ord_csquare) 1); 

509 
by (fast_tac (ZF_cs addIs [ltD]) 1); 

510 
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN 

511 
assume_tac 1); 

512 
by (REPEAT_FIRST (etac ltE)); 

513 
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1); 

514 
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll))); 

760  515 
qed "ordermap_csquare_le"; 
437  516 

484  517 
(*Kunen: "... so the order type <= K" *) 
437  518 
goal CardinalArith.thy 
484  519 
"!!K. [ InfCard(K); ALL y:K. InfCard(y) > y * y = y ] ==> \ 
520 
\ ordertype(K*K, csquare_rel(K)) le K"; 

437  521 
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); 
522 
by (rtac all_lt_imp_le 1); 

523 
by (assume_tac 1); 

524 
by (etac (well_ord_csquare RS Ord_ordertype) 1); 

525 
by (rtac Card_lt_imp_lt 1); 

526 
by (etac InfCard_is_Card 3); 

527 
by (etac ltE 2 THEN assume_tac 2); 

528 
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1); 

529 
by (safe_tac (ZF_cs addSEs [ltE])); 

530 
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1); 

531 
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2)); 

532 
by (rtac (ordermap_csquare_le RS lt_trans1) 1 THEN 

533 
REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1)); 

534 
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN 

535 
REPEAT (ares_tac [Ord_Un, Ord_nat] 1)); 

536 
(*the finite case: xb Un y < nat *) 

537 
by (res_inst_tac [("j", "nat")] lt_trans2 1); 

538 
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2); 

539 
by (asm_full_simp_tac 

540 
(ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type, 

541 
nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1); 

542 
(*case nat le (xb Un y), equivalently InfCard(xb Un y) *) 

543 
by (asm_full_simp_tac 

544 
(ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong, 

545 
le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 

546 
Ord_Un, ltI, nat_le_cardinal, 

547 
Ord_cardinal_le RS lt_trans1 RS ltD]) 1); 

760  548 
qed "ordertype_csquare_le"; 
437  549 

550 
(*This lemma can easily be generalized to premise well_ord(A*A,r) *) 

551 
goalw CardinalArith.thy [cmult_def] 

484  552 
"!!K. Ord(K) ==> K * K = ordertype(K*K, csquare_rel(K))"; 
437  553 
by (rtac cardinal_cong 1); 
554 
by (rewtac eqpoll_def); 

555 
by (rtac exI 1); 

467  556 
by (etac (well_ord_csquare RS ordermap_bij) 1); 
760  557 
qed "csquare_eq_ordertype"; 
437  558 

559 
(*Main result: Kunen's Theorem 10.12*) 

484  560 
goal CardinalArith.thy "!!K. InfCard(K) ==> K * K = K"; 
437  561 
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); 
562 
by (etac rev_mp 1); 

484  563 
by (trans_ind_tac "K" [] 1); 
437  564 
by (rtac impI 1); 
565 
by (rtac le_anti_sym 1); 

566 
by (etac (InfCard_is_Card RS cmult_square_le) 2); 

567 
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1); 

568 
by (assume_tac 2); 

569 
by (assume_tac 2); 

570 
by (asm_simp_tac 

571 
(ZF_ss addsimps [csquare_eq_ordertype, Ord_cardinal_le, 

572 
well_ord_csquare RS Ord_ordertype]) 1); 

760  573 
qed "InfCard_csquare_eq"; 
484  574 

767  575 
(*Corollary for arbitrary wellordered sets (all sets, assuming AC)*) 
484  576 
goal CardinalArith.thy 
577 
"!!A. [ well_ord(A,r); InfCard(A) ] ==> A*A eqpoll A"; 

578 
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1); 

579 
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1)); 

580 
by (resolve_tac [well_ord_cardinal_eqE] 1); 

581 
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1)); 

582 
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1); 

760  583 
qed "well_ord_InfCard_square_eq"; 
484  584 

767  585 
(** Toward's Kunen's Corollary 10.13 (1) **) 
586 

587 
goal CardinalArith.thy "!!K. [ InfCard(K); L le K; 0<L ] ==> K * L = K"; 

588 
by (resolve_tac [le_anti_sym] 1); 

589 
by (eresolve_tac [ltE] 2 THEN 

590 
REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2)); 

591 
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); 

592 
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); 

593 
by (asm_simp_tac (ZF_ss addsimps [InfCard_csquare_eq]) 1); 

594 
val InfCard_le_cmult_eq = result(); 

595 

596 
(*Corollary 10.13 (1), for cardinal multiplication*) 

597 
goal CardinalArith.thy 

598 
"!!K. [ InfCard(K); InfCard(L) ] ==> K * L = K Un L"; 

599 
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); 

600 
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); 

601 
by (resolve_tac [cmult_commute RS ssubst] 1); 

602 
by (resolve_tac [Un_commute RS ssubst] 1); 

603 
by (ALLGOALS 

604 
(asm_simp_tac 

605 
(ZF_ss addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq, 

606 
subset_Un_iff2 RS iffD1, le_imp_subset]))); 

607 
val InfCard_cmult_eq = result(); 

608 

609 
(*This proof appear to be the simplest!*) 

610 
goal CardinalArith.thy "!!K. InfCard(K) ==> K + K = K"; 

611 
by (asm_simp_tac 

612 
(ZF_ss addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1); 

613 
by (resolve_tac [InfCard_le_cmult_eq] 1); 

614 
by (typechk_tac [Ord_0, le_refl, leI]); 

615 
by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]); 

616 
val InfCard_cdouble_eq = result(); 

617 

618 
(*Corollary 10.13 (1), for cardinal addition*) 

619 
goal CardinalArith.thy "!!K. [ InfCard(K); L le K ] ==> K + L = K"; 

620 
by (resolve_tac [le_anti_sym] 1); 

621 
by (eresolve_tac [ltE] 2 THEN 

622 
REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2)); 

623 
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); 

624 
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); 

625 
by (asm_simp_tac (ZF_ss addsimps [InfCard_cdouble_eq]) 1); 

626 
val InfCard_le_cadd_eq = result(); 

627 

628 
goal CardinalArith.thy 

629 
"!!K. [ InfCard(K); InfCard(L) ] ==> K + L = K Un L"; 

630 
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); 

631 
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); 

632 
by (resolve_tac [cadd_commute RS ssubst] 1); 

633 
by (resolve_tac [Un_commute RS ssubst] 1); 

634 
by (ALLGOALS 

635 
(asm_simp_tac 

636 
(ZF_ss addsimps [InfCard_le_cadd_eq, 

637 
subset_Un_iff2 RS iffD1, le_imp_subset]))); 

638 
val InfCard_cadd_eq = result(); 

639 

640 
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set 

641 
of all ntuples of elements of K. A better version for the Isabelle theory 

642 
might be InfCard(K) ==> list(K) = K. 

643 
*) 

484  644 

645 
(*** For every cardinal number there exists a greater one 

646 
[Kunen's Theorem 10.16, which would be trivial using AC] ***) 

647 

648 
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))"; 

649 
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1); 

650 
by (safe_tac (ZF_cs addSIs [Ord_ordertype])); 

651 
bw Transset_def; 

652 
by (safe_tac ZF_cs); 

653 
by (rtac (ordertype_subset RS exE) 1 THEN REPEAT (assume_tac 1)); 

654 
by (resolve_tac [UN_I] 1); 

655 
by (resolve_tac [ReplaceI] 2); 

656 
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset]))); 

760  657 
qed "Ord_jump_cardinal"; 
484  658 

659 
(*Allows selective unfolding. Less work than deriving intro/elim rules*) 

660 
goalw CardinalArith.thy [jump_cardinal_def] 

661 
"i : jump_cardinal(K) <> \ 

662 
\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"; 

663 
by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*) 

760  664 
qed "jump_cardinal_iff"; 
484  665 

666 
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) 

667 
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)"; 

668 
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1); 

669 
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); 

670 
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel])); 

671 
by (resolve_tac [subset_refl] 2); 

672 
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1); 

673 
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1); 

760  674 
qed "K_lt_jump_cardinal"; 
484  675 

676 
(*The proof by contradiction: the bijection f yields a wellordering of X 

677 
whose ordertype is jump_cardinal(K). *) 

678 
goal CardinalArith.thy 

679 
"!!K. [ well_ord(X,r); r <= K * K; X <= K; \ 

680 
\ f : bij(ordertype(X,r), jump_cardinal(K)) \ 

681 
\ ] ==> jump_cardinal(K) : jump_cardinal(K)"; 

682 
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1); 

683 
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2)); 

684 
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); 

685 
by (REPEAT_FIRST (resolve_tac [exI, conjI])); 

686 
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1); 

687 
by (REPEAT (assume_tac 1)); 

688 
by (etac (bij_is_inj RS well_ord_rvimage) 1); 

689 
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1); 

690 
by (asm_simp_tac 

691 
(ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 

692 
ordertype_Memrel, Ord_jump_cardinal]) 1); 

760  693 
qed "Card_jump_cardinal_lemma"; 
484  694 

695 
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) 

696 
goal CardinalArith.thy "Card(jump_cardinal(K))"; 

697 
by (rtac (Ord_jump_cardinal RS CardI) 1); 

698 
by (rewrite_goals_tac [eqpoll_def]); 

699 
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1])); 

700 
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1)); 

760  701 
qed "Card_jump_cardinal"; 
484  702 

703 
(*** Basic properties of successor cardinals ***) 

704 

705 
goalw CardinalArith.thy [csucc_def] 

706 
"!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)"; 

707 
by (rtac LeastI 1); 

708 
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal, 

709 
Ord_jump_cardinal] 1)); 

760  710 
qed "csucc_basic"; 
484  711 

712 
val Card_csucc = csucc_basic RS conjunct1 > standard; 

713 

714 
val lt_csucc = csucc_basic RS conjunct2 > standard; 

715 

517  716 
goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)"; 
717 
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1); 

718 
by (REPEAT (assume_tac 1)); 

760  719 
qed "Ord_0_lt_csucc"; 
517  720 

484  721 
goalw CardinalArith.thy [csucc_def] 
722 
"!!K L. [ Card(L); K<L ] ==> csucc(K) le L"; 

723 
by (rtac Least_le 1); 

724 
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1)); 

760  725 
qed "csucc_le"; 
484  726 

727 
goal CardinalArith.thy 

728 
"!!K. [ Ord(i); Card(K) ] ==> i < csucc(K) <> i le K"; 

729 
by (resolve_tac [iffI] 1); 

730 
by (resolve_tac [Card_lt_imp_lt] 2); 

731 
by (eresolve_tac [lt_trans1] 2); 

732 
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2)); 

733 
by (resolve_tac [notI RS not_lt_imp_le] 1); 

734 
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1); 

735 
by (assume_tac 1); 

736 
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1); 

737 
by (REPEAT (ares_tac [Ord_cardinal] 1 

738 
ORELSE eresolve_tac [ltE, Card_is_Ord] 1)); 

760  739 
qed "lt_csucc_iff"; 
484  740 

741 
goal CardinalArith.thy 

742 
"!!K' K. [ Card(K'); Card(K) ] ==> K' < csucc(K) <> K' le K"; 

743 
by (asm_simp_tac 

744 
(ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1); 

760  745 
qed "Card_lt_csucc_iff"; 
488  746 

747 
goalw CardinalArith.thy [InfCard_def] 

748 
"!!K. InfCard(K) ==> InfCard(csucc(K))"; 

749 
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, 

750 
lt_csucc RS leI RSN (2,le_trans)]) 1); 

760  751 
qed "InfCard_csucc"; 
517  752 

753 
val Limit_csucc = InfCard_csucc RS InfCard_is_Limit > standard; 