author  paulson 
Tue, 26 Mar 1996 11:42:36 +0100  
changeset 1609  5324067d993f 
parent 1461  6bcb44e4d6e5 
child 1622  4b0608ce6150 
permissions  rwrr 
1461  1 
(* Title: ZF/CardinalArith.ML 
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ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
437  4 
Copyright 1994 University of Cambridge 
5 

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Cardinal arithmetic  WITHOUT the Axiom of Choice 

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846  8 
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 
846  11 
coincide with union (maximum). Either way we get most laws for free. 
437  12 
*) 
13 

14 
open CardinalArith; 

15 

16 
(*** Cardinal addition ***) 

17 

18 
(** Cardinal addition is commutative **) 

19 

20 
goalw CardinalArith.thy [eqpoll_def] "A+B eqpoll B+A"; 

21 
by (rtac exI 1); 

22 
by (res_inst_tac [("c", "case(Inr, Inl)"), ("d", "case(Inr, Inl)")] 

23 
lam_bijective 1); 

24 
by (safe_tac (ZF_cs addSEs [sumE])); 

25 
by (ALLGOALS (asm_simp_tac case_ss)); 

760  26 
qed "sum_commute_eqpoll"; 
437  27 

28 
goalw CardinalArith.thy [cadd_def] "i + j = j + i"; 

29 
by (rtac (sum_commute_eqpoll RS cardinal_cong) 1); 

760  30 
qed "cadd_commute"; 
437  31 

32 
(** Cardinal addition is associative **) 

33 

34 
goalw CardinalArith.thy [eqpoll_def] "(A+B)+C eqpoll A+(B+C)"; 

35 
by (rtac exI 1); 

1461  36 
by (rtac sum_assoc_bij 1); 
760  37 
qed "sum_assoc_eqpoll"; 
437  38 

39 
(*Unconditional version requires AC*) 

40 
goalw CardinalArith.thy [cadd_def] 

1461  41 
"!!i j k. [ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] ==> \ 
437  42 
\ (i + j) + k = i + (j + k)"; 
43 
by (rtac cardinal_cong 1); 

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by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS sum_eqpoll_cong RS 
1461  45 
eqpoll_trans) 1); 
437  46 
by (rtac (sum_assoc_eqpoll RS eqpoll_trans) 2); 
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by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS 
1461  48 
eqpoll_sym) 2); 
484  49 
by (REPEAT (ares_tac [well_ord_radd] 1)); 
760  50 
qed "well_ord_cadd_assoc"; 
437  51 

52 
(** 0 is the identity for addition **) 

53 

54 
goalw CardinalArith.thy [eqpoll_def] "0+A eqpoll A"; 

55 
by (rtac exI 1); 

846  56 
by (rtac bij_0_sum 1); 
760  57 
qed "sum_0_eqpoll"; 
437  58 

484  59 
goalw CardinalArith.thy [cadd_def] "!!K. Card(K) ==> 0 + K = K"; 
437  60 
by (asm_simp_tac (ZF_ss addsimps [sum_0_eqpoll RS cardinal_cong, 
1461  61 
Card_cardinal_eq]) 1); 
760  62 
qed "cadd_0"; 
437  63 

767  64 
(** Addition by another cardinal **) 
65 

66 
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A+B"; 

67 
by (res_inst_tac [("x", "lam x:A. Inl(x)")] exI 1); 

68 
by (asm_simp_tac (sum_ss addsimps [lam_type]) 1); 

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qed "sum_lepoll_self"; 
767  70 

71 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

72 
goalw CardinalArith.thy [cadd_def] 

73 
"!!K. [ Card(K); Ord(L) ] ==> K le (K + L)"; 

74 
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); 

75 
by (rtac sum_lepoll_self 3); 

76 
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel, Card_is_Ord] 1)); 

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qed "cadd_le_self"; 
767  78 

79 
(** Monotonicity of addition **) 

80 

81 
goalw CardinalArith.thy [lepoll_def] 

82 
"!!A B C D. [ A lepoll C; B lepoll D ] ==> A + B lepoll C + D"; 

83 
by (REPEAT (etac exE 1)); 

84 
by (res_inst_tac [("x", "lam z:A+B. case(%w. Inl(f`w), %y. Inr(fa`y), z)")] 

85 
exI 1); 

86 
by (res_inst_tac 

87 
[("d", "case(%w. Inl(converse(f)`w), %y. Inr(converse(fa)`y))")] 

88 
lam_injective 1); 

846  89 
by (typechk_tac ([inj_is_fun, case_type, InlI, InrI] @ ZF_typechecks)); 
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by (etac sumE 1); 
767  91 
by (ALLGOALS (asm_simp_tac (sum_ss addsimps [left_inverse]))); 
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qed "sum_lepoll_mono"; 
767  93 

94 
goalw CardinalArith.thy [cadd_def] 

95 
"!!K. [ K' le K; L' le L ] ==> (K' + L') le (K + L)"; 

96 
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); 

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by (rtac well_ord_lepoll_imp_Card_le 1); 
767  98 
by (REPEAT (ares_tac [sum_lepoll_mono, subset_imp_lepoll] 2)); 
99 
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); 

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qed "cadd_le_mono"; 
767  101 

437  102 
(** Addition of finite cardinals is "ordinary" addition **) 
103 

104 
goalw CardinalArith.thy [eqpoll_def] "succ(A)+B eqpoll succ(A+B)"; 

105 
by (rtac exI 1); 

106 
by (res_inst_tac [("c", "%z.if(z=Inl(A),A+B,z)"), 

1461  107 
("d", "%z.if(z=A+B,Inl(A),z)")] 
437  108 
lam_bijective 1); 
109 
by (ALLGOALS 

110 
(asm_simp_tac (case_ss addsimps [succI2, mem_imp_not_eq] 

1461  111 
setloop eresolve_tac [sumE,succE]))); 
760  112 
qed "sum_succ_eqpoll"; 
437  113 

114 
(*Pulling the succ(...) outside the ... requires m, n: nat *) 

115 
(*Unconditional version requires AC*) 

116 
goalw CardinalArith.thy [cadd_def] 

117 
"!!m n. [ Ord(m); Ord(n) ] ==> succ(m) + n = succ(m + n)"; 

118 
by (rtac (sum_succ_eqpoll RS cardinal_cong RS trans) 1); 

119 
by (rtac (succ_eqpoll_cong RS cardinal_cong) 1); 

120 
by (rtac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1); 

121 
by (REPEAT (ares_tac [well_ord_radd, well_ord_Memrel] 1)); 

760  122 
qed "cadd_succ_lemma"; 
437  123 

124 
val [mnat,nnat] = goal CardinalArith.thy 

125 
"[ m: nat; n: nat ] ==> m + n = m#+n"; 

126 
by (cut_facts_tac [nnat] 1); 

127 
by (nat_ind_tac "m" [mnat] 1); 

128 
by (asm_simp_tac (arith_ss addsimps [nat_into_Card RS cadd_0]) 1); 

129 
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cadd_succ_lemma, 

1461  130 
nat_into_Card RS Card_cardinal_eq]) 1); 
760  131 
qed "nat_cadd_eq_add"; 
437  132 

133 

134 
(*** Cardinal multiplication ***) 

135 

136 
(** Cardinal multiplication is commutative **) 

137 

138 
(*Easier to prove the two directions separately*) 

139 
goalw CardinalArith.thy [eqpoll_def] "A*B eqpoll B*A"; 

140 
by (rtac exI 1); 

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by (res_inst_tac [("c", "%<x,y>.<y,x>"), ("d", "%<x,y>.<y,x>")] 
437  142 
lam_bijective 1); 
143 
by (safe_tac ZF_cs); 

144 
by (ALLGOALS (asm_simp_tac ZF_ss)); 

760  145 
qed "prod_commute_eqpoll"; 
437  146 

147 
goalw CardinalArith.thy [cmult_def] "i * j = j * i"; 

148 
by (rtac (prod_commute_eqpoll RS cardinal_cong) 1); 

760  149 
qed "cmult_commute"; 
437  150 

151 
(** Cardinal multiplication is associative **) 

152 

153 
goalw CardinalArith.thy [eqpoll_def] "(A*B)*C eqpoll A*(B*C)"; 

154 
by (rtac exI 1); 

1461  155 
by (rtac prod_assoc_bij 1); 
760  156 
qed "prod_assoc_eqpoll"; 
437  157 

158 
(*Unconditional version requires AC*) 

159 
goalw CardinalArith.thy [cmult_def] 

1461  160 
"!!i j k. [ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] ==> \ 
437  161 
\ (i * j) * k = i * (j * k)"; 
162 
by (rtac cardinal_cong 1); 

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by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS 
1461  164 
eqpoll_trans) 1); 
437  165 
by (rtac (prod_assoc_eqpoll RS eqpoll_trans) 2); 
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by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS prod_eqpoll_cong RS 
1461  167 
eqpoll_sym) 2); 
484  168 
by (REPEAT (ares_tac [well_ord_rmult] 1)); 
760  169 
qed "well_ord_cmult_assoc"; 
437  170 

171 
(** Cardinal multiplication distributes over addition **) 

172 

173 
goalw CardinalArith.thy [eqpoll_def] "(A+B)*C eqpoll (A*C)+(B*C)"; 

174 
by (rtac exI 1); 

1461  175 
by (rtac sum_prod_distrib_bij 1); 
760  176 
qed "sum_prod_distrib_eqpoll"; 
437  177 

846  178 
goalw CardinalArith.thy [cadd_def, cmult_def] 
1461  179 
"!!i j k. [ well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) ] ==> \ 
846  180 
\ (i + j) * k = (i * k) + (j * k)"; 
181 
by (rtac cardinal_cong 1); 

182 
by (rtac ([well_ord_cardinal_eqpoll, eqpoll_refl] MRS prod_eqpoll_cong RS 

1461  183 
eqpoll_trans) 1); 
846  184 
by (rtac (sum_prod_distrib_eqpoll RS eqpoll_trans) 2); 
185 
by (rtac ([well_ord_cardinal_eqpoll, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong RS 

1461  186 
eqpoll_sym) 2); 
846  187 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_radd] 1)); 
188 
qed "well_ord_cadd_cmult_distrib"; 

189 

437  190 
(** Multiplication by 0 yields 0 **) 
191 

192 
goalw CardinalArith.thy [eqpoll_def] "0*A eqpoll 0"; 

193 
by (rtac exI 1); 

194 
by (rtac lam_bijective 1); 

195 
by (safe_tac ZF_cs); 

760  196 
qed "prod_0_eqpoll"; 
437  197 

198 
goalw CardinalArith.thy [cmult_def] "0 * i = 0"; 

199 
by (asm_simp_tac (ZF_ss addsimps [prod_0_eqpoll RS cardinal_cong, 

1461  200 
Card_0 RS Card_cardinal_eq]) 1); 
760  201 
qed "cmult_0"; 
437  202 

203 
(** 1 is the identity for multiplication **) 

204 

205 
goalw CardinalArith.thy [eqpoll_def] "{x}*A eqpoll A"; 

206 
by (rtac exI 1); 

846  207 
by (resolve_tac [singleton_prod_bij RS bij_converse_bij] 1); 
760  208 
qed "prod_singleton_eqpoll"; 
437  209 

484  210 
goalw CardinalArith.thy [cmult_def, succ_def] "!!K. Card(K) ==> 1 * K = K"; 
437  211 
by (asm_simp_tac (ZF_ss addsimps [prod_singleton_eqpoll RS cardinal_cong, 
1461  212 
Card_cardinal_eq]) 1); 
760  213 
qed "cmult_1"; 
437  214 

767  215 
(*** Some inequalities for multiplication ***) 
216 

217 
goalw CardinalArith.thy [lepoll_def, inj_def] "A lepoll A*A"; 

218 
by (res_inst_tac [("x", "lam x:A. <x,x>")] exI 1); 

219 
by (simp_tac (ZF_ss addsimps [lam_type]) 1); 

220 
qed "prod_square_lepoll"; 

221 

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(*Could probably weaken the premise to well_ord(K,r), or remove using AC*) 
767  223 
goalw CardinalArith.thy [cmult_def] "!!K. Card(K) ==> K le K * K"; 
224 
by (rtac le_trans 1); 

225 
by (rtac well_ord_lepoll_imp_Card_le 2); 

226 
by (rtac prod_square_lepoll 3); 

227 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord] 2)); 

228 
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1); 

229 
qed "cmult_square_le"; 

230 

231 
(** Multiplication by a nonzero cardinal **) 

232 

233 
goalw CardinalArith.thy [lepoll_def, inj_def] "!!b. b: B ==> A lepoll A*B"; 

234 
by (res_inst_tac [("x", "lam x:A. <x,b>")] exI 1); 

235 
by (asm_simp_tac (ZF_ss addsimps [lam_type]) 1); 

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236 
qed "prod_lepoll_self"; 
767  237 

238 
(*Could probably weaken the premises to well_ord(K,r), or removing using AC*) 

239 
goalw CardinalArith.thy [cmult_def] 

240 
"!!K. [ Card(K); Ord(L); 0<L ] ==> K le (K * L)"; 

241 
by (rtac ([Card_cardinal_le, well_ord_lepoll_imp_Card_le] MRS le_trans) 1); 

242 
by (rtac prod_lepoll_self 3); 

243 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel, Card_is_Ord, ltD] 1)); 

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244 
qed "cmult_le_self"; 
767  245 

246 
(** Monotonicity of multiplication **) 

247 

248 
goalw CardinalArith.thy [lepoll_def] 

249 
"!!A B C D. [ A lepoll C; B lepoll D ] ==> A * B lepoll C * D"; 

250 
by (REPEAT (etac exE 1)); 

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251 
by (res_inst_tac [("x", "lam <w,y>:A*B. <f`w, fa`y>")] exI 1); 
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252 
by (res_inst_tac [("d", "%<w,y>.<converse(f)`w, converse(fa)`y>")] 
1461  253 
lam_injective 1); 
767  254 
by (typechk_tac (inj_is_fun::ZF_typechecks)); 
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by (etac SigmaE 1); 
767  256 
by (asm_simp_tac (ZF_ss addsimps [left_inverse]) 1); 
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257 
qed "prod_lepoll_mono"; 
767  258 

259 
goalw CardinalArith.thy [cmult_def] 

260 
"!!K. [ K' le K; L' le L ] ==> (K' * L') le (K * L)"; 

261 
by (safe_tac (ZF_cs addSDs [le_subset_iff RS iffD1])); 

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262 
by (rtac well_ord_lepoll_imp_Card_le 1); 
767  263 
by (REPEAT (ares_tac [prod_lepoll_mono, subset_imp_lepoll] 2)); 
264 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

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265 
qed "cmult_le_mono"; 
767  266 

267 
(*** Multiplication of finite cardinals is "ordinary" multiplication ***) 

437  268 

269 
goalw CardinalArith.thy [eqpoll_def] "succ(A)*B eqpoll B + A*B"; 

270 
by (rtac exI 1); 

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271 
by (res_inst_tac [("c", "%<x,y>. if(x=A, Inl(y), Inr(<x,y>))"), 
1461  272 
("d", "case(%y. <A,y>, %z.z)")] 
437  273 
lam_bijective 1); 
274 
by (safe_tac (ZF_cs addSEs [sumE])); 

275 
by (ALLGOALS 

276 
(asm_simp_tac (case_ss addsimps [succI2, if_type, mem_imp_not_eq]))); 

760  277 
qed "prod_succ_eqpoll"; 
437  278 

279 
(*Unconditional version requires AC*) 

280 
goalw CardinalArith.thy [cmult_def, cadd_def] 

281 
"!!m n. [ Ord(m); Ord(n) ] ==> succ(m) * n = n + (m * n)"; 

282 
by (rtac (prod_succ_eqpoll RS cardinal_cong RS trans) 1); 

283 
by (rtac (cardinal_cong RS sym) 1); 

284 
by (rtac ([eqpoll_refl, well_ord_cardinal_eqpoll] MRS sum_eqpoll_cong) 1); 

285 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

760  286 
qed "cmult_succ_lemma"; 
437  287 

288 
val [mnat,nnat] = goal CardinalArith.thy 

289 
"[ m: nat; n: nat ] ==> m * n = m#*n"; 

290 
by (cut_facts_tac [nnat] 1); 

291 
by (nat_ind_tac "m" [mnat] 1); 

292 
by (asm_simp_tac (arith_ss addsimps [cmult_0]) 1); 

293 
by (asm_simp_tac (arith_ss addsimps [nat_into_Ord, cmult_succ_lemma, 

1461  294 
nat_cadd_eq_add]) 1); 
760  295 
qed "nat_cmult_eq_mult"; 
437  296 

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297 
goal CardinalArith.thy "!!m n. Card(n) ==> 2 * n = n + n"; 
767  298 
by (asm_simp_tac 
299 
(ZF_ss addsimps [Ord_0, Ord_succ, cmult_0, cmult_succ_lemma, Card_is_Ord, 

1461  300 
read_instantiate [("j","0")] cadd_commute, cadd_0]) 1); 
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301 
qed "cmult_2"; 
767  302 

437  303 

304 
(*** Infinite Cardinals are Limit Ordinals ***) 

305 

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306 
(*This proof is modelled upon one assuming nat<=A, with injection 
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307 
lam z:cons(u,A). if(z=u, 0, if(z : nat, succ(z), z)) and inverse 
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308 
%y. if(y:nat, nat_case(u,%z.z,y), y). If f: inj(nat,A) then 
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309 
range(f) behaves like the natural numbers.*) 
516  310 
goalw CardinalArith.thy [lepoll_def] 
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311 
"!!i. nat lepoll A ==> cons(u,A) lepoll A"; 
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312 
by (etac exE 1); 
516  313 
by (res_inst_tac [("x", 
1461  314 
"lam z:cons(u,A). if(z=u, f`0, \ 
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315 
\ if(z: range(f), f`succ(converse(f)`z), z))")] exI 1); 
1461  316 
by (res_inst_tac [("d", "%y. if(y: range(f), \ 
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317 
\ nat_case(u, %z.f`z, converse(f)`y), y)")] 
516  318 
lam_injective 1); 
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319 
by (fast_tac (ZF_cs addSIs [if_type, nat_0I, nat_succI, apply_type] 
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523
diff
changeset

320 
addIs [inj_is_fun, inj_converse_fun]) 1); 
516  321 
by (asm_simp_tac 
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

322 
(ZF_ss addsimps [inj_is_fun RS apply_rangeI, 
1461  323 
inj_converse_fun RS apply_rangeI, 
324 
inj_converse_fun RS apply_funtype, 

325 
left_inverse, right_inverse, nat_0I, nat_succI, 

326 
nat_case_0, nat_case_succ] 

516  327 
setloop split_tac [expand_if]) 1); 
760  328 
qed "nat_cons_lepoll"; 
516  329 

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

330 
goal CardinalArith.thy "!!i. nat lepoll A ==> cons(u,A) eqpoll A"; 
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

331 
by (etac (nat_cons_lepoll RS eqpollI) 1); 
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

332 
by (rtac (subset_consI RS subset_imp_lepoll) 1); 
760  333 
qed "nat_cons_eqpoll"; 
437  334 

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

335 
(*Specialized version required below*) 
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
523
diff
changeset

336 
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

337 
by (eresolve_tac [subset_imp_lepoll RS nat_cons_eqpoll] 1); 
760  338 
qed "nat_succ_eqpoll"; 
437  339 

488  340 
goalw CardinalArith.thy [InfCard_def] "InfCard(nat)"; 
341 
by (fast_tac (ZF_cs addIs [Card_nat, le_refl, Card_is_Ord]) 1); 

760  342 
qed "InfCard_nat"; 
488  343 

484  344 
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Card(K)"; 
437  345 
by (etac conjunct1 1); 
760  346 
qed "InfCard_is_Card"; 
437  347 

523  348 
goalw CardinalArith.thy [InfCard_def] 
349 
"!!K L. [ InfCard(K); Card(L) ] ==> InfCard(K Un L)"; 

350 
by (asm_simp_tac (ZF_ss addsimps [Card_Un, Un_upper1_le RSN (2,le_trans), 

1461  351 
Card_is_Ord]) 1); 
760  352 
qed "InfCard_Un"; 
523  353 

437  354 
(*Kunen's Lemma 10.11*) 
484  355 
goalw CardinalArith.thy [InfCard_def] "!!K. InfCard(K) ==> Limit(K)"; 
437  356 
by (etac conjE 1); 
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

357 
by (forward_tac [Card_is_Ord] 1); 
437  358 
by (rtac (ltI RS non_succ_LimitI) 1); 
359 
by (etac ([asm_rl, nat_0I] MRS (le_imp_subset RS subsetD)) 1); 

360 
by (safe_tac (ZF_cs addSDs [Limit_nat RS Limit_le_succD])); 

361 
by (rewtac Card_def); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

362 
by (dtac trans 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

363 
by (etac (le_imp_subset RS nat_succ_eqpoll RS cardinal_cong) 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

364 
by (etac (Ord_succD RS Ord_cardinal_le RS lt_trans2 RS lt_irrefl) 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

365 
by (REPEAT (ares_tac [le_eqI, Ord_cardinal] 1)); 
760  366 
qed "InfCard_is_Limit"; 
437  367 

368 

369 
(*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***) 

370 

371 
(*A general fact about ordermap*) 

372 
goalw Cardinal.thy [eqpoll_def] 

373 
"!!A. [ well_ord(A,r); x:A ] ==> ordermap(A,r)`x eqpoll pred(A,x,r)"; 

374 
by (rtac exI 1); 

375 
by (asm_simp_tac (ZF_ss addsimps [ordermap_eq_image, well_ord_is_wf]) 1); 

467  376 
by (etac (ordermap_bij RS bij_is_inj RS restrict_bij RS bij_converse_bij) 1); 
437  377 
by (rtac pred_subset 1); 
760  378 
qed "ordermap_eqpoll_pred"; 
437  379 

380 
(** Establishing the wellordering **) 

381 

382 
goalw CardinalArith.thy [inj_def] 

1090
8ab69b3e396b
Changed some definitions and proofs to use patternmatching.
lcp
parents:
1075
diff
changeset

383 
"!!K. Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"; 
989  384 
by (fast_tac (ZF_cs addss ZF_ss 
1461  385 
addIs [lam_type, Un_least_lt RS ltD, ltI]) 1); 
760  386 
qed "csquare_lam_inj"; 
437  387 

388 
goalw CardinalArith.thy [csquare_rel_def] 

484  389 
"!!K. Ord(K) ==> well_ord(K*K, csquare_rel(K))"; 
437  390 
by (rtac (csquare_lam_inj RS well_ord_rvimage) 1); 
391 
by (REPEAT (ares_tac [well_ord_rmult, well_ord_Memrel] 1)); 

760  392 
qed "well_ord_csquare"; 
437  393 

394 
(** Characterising initial segments of the wellordering **) 

395 

396 
goalw CardinalArith.thy [csquare_rel_def] 

484  397 
"!!K. [ x<K; y<K; z<K ] ==> \ 
398 
\ <<x,y>, <z,z>> : csquare_rel(K) > x le z & y le z"; 

437  399 
by (REPEAT (etac ltE 1)); 
400 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

401 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

402 
by (safe_tac (ZF_cs addSEs [mem_irrefl] 

403 
addSIs [Un_upper1_le, Un_upper2_le])); 

404 
by (ALLGOALS (asm_simp_tac (ZF_ss addsimps [lt_def, succI2, Ord_succ]))); 

800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset

405 
val csquareD_lemma = result(); 
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset

406 

23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset

407 
bind_thm ("csquareD", csquareD_lemma RS mp); 
437  408 

409 
goalw CardinalArith.thy [pred_def] 

484  410 
"!!K. z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"; 
1461  411 
by (safe_tac (lemmas_cs addSEs [SigmaE])); (*avoids using succCI*) 
437  412 
by (rtac (csquareD RS conjE) 1); 
413 
by (rewtac lt_def); 

414 
by (assume_tac 4); 

415 
by (ALLGOALS (fast_tac ZF_cs)); 

760  416 
qed "pred_csquare_subset"; 
437  417 

418 
goalw CardinalArith.thy [csquare_rel_def] 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

419 
"!!K. [ x<z; y<z; z<K ] ==> <<x,y>, <z,z>> : csquare_rel(K)"; 
484  420 
by (subgoals_tac ["x<K", "y<K"] 1); 
437  421 
by (REPEAT (eresolve_tac [asm_rl, lt_trans] 2)); 
422 
by (REPEAT (etac ltE 1)); 

423 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

424 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

760  425 
qed "csquare_ltI"; 
437  426 

427 
(*Part of the traditional proof. UNUSED since it's harder to prove & apply *) 

428 
goalw CardinalArith.thy [csquare_rel_def] 

484  429 
"!!K. [ x le z; y le z; z<K ] ==> \ 
430 
\ <<x,y>, <z,z>> : csquare_rel(K)  x=z & y=z"; 

431 
by (subgoals_tac ["x<K", "y<K"] 1); 

437  432 
by (REPEAT (eresolve_tac [asm_rl, lt_trans1] 2)); 
433 
by (REPEAT (etac ltE 1)); 

434 
by (asm_simp_tac (ZF_ss addsimps [rvimage_iff, rmult_iff, Memrel_iff, 

435 
Un_absorb, Un_least_mem_iff, ltD]) 1); 

436 
by (REPEAT_FIRST (etac succE)); 

437 
by (ALLGOALS 

438 
(asm_simp_tac (ZF_ss addsimps [subset_Un_iff RS iff_sym, 

1461  439 
subset_Un_iff2 RS iff_sym, OrdmemD]))); 
760  440 
qed "csquare_or_eqI"; 
437  441 

442 
(** The cardinality of initial segments **) 

443 

444 
goal CardinalArith.thy 

846  445 
"!!K. [ Limit(K); x<K; y<K; z=succ(x Un y) ] ==> \ 
1461  446 
\ ordermap(K*K, csquare_rel(K)) ` <x,y> < \ 
484  447 
\ ordermap(K*K, csquare_rel(K)) ` <z,z>"; 
448 
by (subgoals_tac ["z<K", "well_ord(K*K, csquare_rel(K))"] 1); 

846  449 
by (etac (Limit_is_Ord RS well_ord_csquare) 2); 
450 
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2); 

870  451 
by (rtac (csquare_ltI RS ordermap_mono RS ltI) 1); 
437  452 
by (etac well_ord_is_wf 4); 
453 
by (ALLGOALS 

454 
(fast_tac (ZF_cs addSIs [Un_upper1_le, Un_upper2_le, Ord_ordermap] 

455 
addSEs [ltE]))); 

870  456 
qed "ordermap_z_lt"; 
437  457 

484  458 
(*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) 
437  459 
goalw CardinalArith.thy [cmult_def] 
846  460 
"!!K. [ Limit(K); x<K; y<K; z=succ(x Un y) ] ==> \ 
484  461 
\  ordermap(K*K, csquare_rel(K)) ` <x,y>  le succ(z) * succ(z)"; 
767  462 
by (rtac (well_ord_rmult RS well_ord_lepoll_imp_Card_le) 1); 
437  463 
by (REPEAT (ares_tac [Ord_cardinal, well_ord_Memrel] 1)); 
484  464 
by (subgoals_tac ["z<K"] 1); 
846  465 
by (fast_tac (ZF_cs addSIs [Un_least_lt, Limit_has_succ]) 2); 
1609  466 
by (rtac (ordermap_z_lt RS leI RS le_imp_lepoll RS lepoll_trans) 1); 
437  467 
by (REPEAT_SOME assume_tac); 
468 
by (rtac (ordermap_eqpoll_pred RS eqpoll_imp_lepoll RS lepoll_trans) 1); 

846  469 
by (etac (Limit_is_Ord RS well_ord_csquare) 1); 
437  470 
by (fast_tac (ZF_cs addIs [ltD]) 1); 
471 
by (rtac (pred_csquare_subset RS subset_imp_lepoll RS lepoll_trans) 1 THEN 

472 
assume_tac 1); 

473 
by (REPEAT_FIRST (etac ltE)); 

474 
by (rtac (prod_eqpoll_cong RS eqpoll_sym RS eqpoll_imp_lepoll) 1); 

475 
by (REPEAT_FIRST (etac (Ord_succ RS Ord_cardinal_eqpoll))); 

760  476 
qed "ordermap_csquare_le"; 
437  477 

484  478 
(*Kunen: "... so the order type <= K" *) 
437  479 
goal CardinalArith.thy 
484  480 
"!!K. [ InfCard(K); ALL y:K. InfCard(y) > y * y = y ] ==> \ 
481 
\ ordertype(K*K, csquare_rel(K)) le K"; 

437  482 
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); 
483 
by (rtac all_lt_imp_le 1); 

484 
by (assume_tac 1); 

485 
by (etac (well_ord_csquare RS Ord_ordertype) 1); 

486 
by (rtac Card_lt_imp_lt 1); 

487 
by (etac InfCard_is_Card 3); 

488 
by (etac ltE 2 THEN assume_tac 2); 

489 
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_unfold]) 1); 

490 
by (safe_tac (ZF_cs addSEs [ltE])); 

491 
by (subgoals_tac ["Ord(xb)", "Ord(y)"] 1); 

492 
by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 2)); 

846  493 
by (rtac (InfCard_is_Limit RS ordermap_csquare_le RS lt_trans1) 1 THEN 
437  494 
REPEAT (ares_tac [refl] 1 ORELSE etac ltI 1)); 
495 
by (res_inst_tac [("i","xb Un y"), ("j","nat")] Ord_linear2 1 THEN 

496 
REPEAT (ares_tac [Ord_Un, Ord_nat] 1)); 

497 
(*the finite case: xb Un y < nat *) 

498 
by (res_inst_tac [("j", "nat")] lt_trans2 1); 

499 
by (asm_full_simp_tac (FOL_ss addsimps [InfCard_def]) 2); 

500 
by (asm_full_simp_tac 

501 
(ZF_ss addsimps [lt_def, nat_cmult_eq_mult, nat_succI, mult_type, 

1461  502 
nat_into_Card RS Card_cardinal_eq, Ord_nat]) 1); 
846  503 
(*case nat le (xb Un y) *) 
437  504 
by (asm_full_simp_tac 
505 
(ZF_ss addsimps [le_imp_subset RS nat_succ_eqpoll RS cardinal_cong, 

1461  506 
le_succ_iff, InfCard_def, Card_cardinal, Un_least_lt, 
507 
Ord_Un, ltI, nat_le_cardinal, 

508 
Ord_cardinal_le RS lt_trans1 RS ltD]) 1); 

760  509 
qed "ordertype_csquare_le"; 
437  510 

511 
(*Main result: Kunen's Theorem 10.12*) 

484  512 
goal CardinalArith.thy "!!K. InfCard(K) ==> K * K = K"; 
437  513 
by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1); 
514 
by (etac rev_mp 1); 

484  515 
by (trans_ind_tac "K" [] 1); 
437  516 
by (rtac impI 1); 
517 
by (rtac le_anti_sym 1); 

518 
by (etac (InfCard_is_Card RS cmult_square_le) 2); 

519 
by (rtac (ordertype_csquare_le RSN (2, le_trans)) 1); 

520 
by (assume_tac 2); 

521 
by (assume_tac 2); 

522 
by (asm_simp_tac 

846  523 
(ZF_ss addsimps [cmult_def, Ord_cardinal_le, 
1461  524 
well_ord_csquare RS ordermap_bij RS 
525 
bij_imp_eqpoll RS cardinal_cong, 

437  526 
well_ord_csquare RS Ord_ordertype]) 1); 
760  527 
qed "InfCard_csquare_eq"; 
484  528 

767  529 
(*Corollary for arbitrary wellordered sets (all sets, assuming AC)*) 
484  530 
goal CardinalArith.thy 
531 
"!!A. [ well_ord(A,r); InfCard(A) ] ==> A*A eqpoll A"; 

532 
by (resolve_tac [prod_eqpoll_cong RS eqpoll_trans] 1); 

533 
by (REPEAT (etac (well_ord_cardinal_eqpoll RS eqpoll_sym) 1)); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

534 
by (rtac well_ord_cardinal_eqE 1); 
484  535 
by (REPEAT (ares_tac [Ord_cardinal, well_ord_rmult, well_ord_Memrel] 1)); 
536 
by (asm_simp_tac (ZF_ss addsimps [symmetric cmult_def, InfCard_csquare_eq]) 1); 

760  537 
qed "well_ord_InfCard_square_eq"; 
484  538 

767  539 
(** Toward's Kunen's Corollary 10.13 (1) **) 
540 

541 
goal CardinalArith.thy "!!K. [ InfCard(K); L le K; 0<L ] ==> K * L = K"; 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

542 
by (rtac le_anti_sym 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

543 
by (etac ltE 2 THEN 
767  544 
REPEAT (ares_tac [cmult_le_self, InfCard_is_Card] 2)); 
545 
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); 

546 
by (resolve_tac [cmult_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); 

547 
by (asm_simp_tac (ZF_ss addsimps [InfCard_csquare_eq]) 1); 

782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset

548 
qed "InfCard_le_cmult_eq"; 
767  549 

550 
(*Corollary 10.13 (1), for cardinal multiplication*) 

551 
goal CardinalArith.thy 

552 
"!!K. [ InfCard(K); InfCard(L) ] ==> K * L = K Un L"; 

553 
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); 

554 
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); 

555 
by (resolve_tac [cmult_commute RS ssubst] 1); 

556 
by (resolve_tac [Un_commute RS ssubst] 1); 

557 
by (ALLGOALS 

558 
(asm_simp_tac 

559 
(ZF_ss addsimps [InfCard_is_Limit RS Limit_has_0, InfCard_le_cmult_eq, 

1461  560 
subset_Un_iff2 RS iffD1, le_imp_subset]))); 
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset

561 
qed "InfCard_cmult_eq"; 
767  562 

563 
(*This proof appear to be the simplest!*) 

564 
goal CardinalArith.thy "!!K. InfCard(K) ==> K + K = K"; 

565 
by (asm_simp_tac 

566 
(ZF_ss addsimps [cmult_2 RS sym, InfCard_is_Card, cmult_commute]) 1); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

567 
by (rtac InfCard_le_cmult_eq 1); 
767  568 
by (typechk_tac [Ord_0, le_refl, leI]); 
569 
by (typechk_tac [InfCard_is_Limit, Limit_has_0, Limit_has_succ]); 

782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset

570 
qed "InfCard_cdouble_eq"; 
767  571 

572 
(*Corollary 10.13 (1), for cardinal addition*) 

573 
goal CardinalArith.thy "!!K. [ InfCard(K); L le K ] ==> K + L = K"; 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

574 
by (rtac le_anti_sym 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

575 
by (etac ltE 2 THEN 
767  576 
REPEAT (ares_tac [cadd_le_self, InfCard_is_Card] 2)); 
577 
by (forward_tac [InfCard_is_Card RS Card_is_Ord RS le_refl] 1); 

578 
by (resolve_tac [cadd_le_mono RS le_trans] 1 THEN REPEAT (assume_tac 1)); 

579 
by (asm_simp_tac (ZF_ss addsimps [InfCard_cdouble_eq]) 1); 

782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset

580 
qed "InfCard_le_cadd_eq"; 
767  581 

582 
goal CardinalArith.thy 

583 
"!!K. [ InfCard(K); InfCard(L) ] ==> K + L = K Un L"; 

584 
by (res_inst_tac [("i","K"),("j","L")] Ord_linear_le 1); 

585 
by (typechk_tac [InfCard_is_Card, Card_is_Ord]); 

586 
by (resolve_tac [cadd_commute RS ssubst] 1); 

587 
by (resolve_tac [Un_commute RS ssubst] 1); 

588 
by (ALLGOALS 

589 
(asm_simp_tac 

590 
(ZF_ss addsimps [InfCard_le_cadd_eq, 

1461  591 
subset_Un_iff2 RS iffD1, le_imp_subset]))); 
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
767
diff
changeset

592 
qed "InfCard_cadd_eq"; 
767  593 

594 
(*The other part, Corollary 10.13 (2), refers to the cardinality of the set 

595 
of all ntuples of elements of K. A better version for the Isabelle theory 

596 
might be InfCard(K) ==> list(K) = K. 

597 
*) 

484  598 

599 
(*** For every cardinal number there exists a greater one 

600 
[Kunen's Theorem 10.16, which would be trivial using AC] ***) 

601 

602 
goalw CardinalArith.thy [jump_cardinal_def] "Ord(jump_cardinal(K))"; 

603 
by (rtac (Ord_is_Transset RSN (2,OrdI)) 1); 

1075
848bf2e18dff
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
989
diff
changeset

604 
by (fast_tac (ZF_cs addSIs [Ord_ordertype]) 2); 
823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

605 
by (rewtac Transset_def); 
1075
848bf2e18dff
Modified proofs for new claset primitives. The problem is that they enforce
lcp
parents:
989
diff
changeset

606 
by (safe_tac subset_cs); 
846  607 
by (asm_full_simp_tac (ZF_ss addsimps [ordertype_pred_unfold]) 1); 
608 
by (safe_tac ZF_cs); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

609 
by (rtac UN_I 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

610 
by (rtac ReplaceI 2); 
846  611 
by (ALLGOALS (fast_tac (ZF_cs addSEs [well_ord_subset, predE]))); 
760  612 
qed "Ord_jump_cardinal"; 
484  613 

614 
(*Allows selective unfolding. Less work than deriving intro/elim rules*) 

615 
goalw CardinalArith.thy [jump_cardinal_def] 

616 
"i : jump_cardinal(K) <> \ 

617 
\ (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"; 

1461  618 
by (fast_tac subset_cs 1); (*It's vital to avoid reasoning about <=*) 
760  619 
qed "jump_cardinal_iff"; 
484  620 

621 
(*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*) 

622 
goal CardinalArith.thy "!!K. Ord(K) ==> K < jump_cardinal(K)"; 

623 
by (resolve_tac [Ord_jump_cardinal RSN (2,ltI)] 1); 

624 
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); 

625 
by (REPEAT_FIRST (ares_tac [exI, conjI, well_ord_Memrel])); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

626 
by (rtac subset_refl 2); 
484  627 
by (asm_simp_tac (ZF_ss addsimps [Memrel_def, subset_iff]) 1); 
628 
by (asm_simp_tac (ZF_ss addsimps [ordertype_Memrel]) 1); 

760  629 
qed "K_lt_jump_cardinal"; 
484  630 

631 
(*The proof by contradiction: the bijection f yields a wellordering of X 

632 
whose ordertype is jump_cardinal(K). *) 

633 
goal CardinalArith.thy 

1461  634 
"!!K. [ well_ord(X,r); r <= K * K; X <= K; \ 
635 
\ f : bij(ordertype(X,r), jump_cardinal(K)) \ 

636 
\ ] ==> jump_cardinal(K) : jump_cardinal(K)"; 

484  637 
by (subgoal_tac "f O ordermap(X,r): bij(X, jump_cardinal(K))" 1); 
638 
by (REPEAT (ares_tac [comp_bij, ordermap_bij] 2)); 

639 
by (resolve_tac [jump_cardinal_iff RS iffD2] 1); 

640 
by (REPEAT_FIRST (resolve_tac [exI, conjI])); 

641 
by (rtac ([rvimage_type, Sigma_mono] MRS subset_trans) 1); 

642 
by (REPEAT (assume_tac 1)); 

643 
by (etac (bij_is_inj RS well_ord_rvimage) 1); 

644 
by (rtac (Ord_jump_cardinal RS well_ord_Memrel) 1); 

645 
by (asm_simp_tac 

646 
(ZF_ss addsimps [well_ord_Memrel RSN (2, bij_ordertype_vimage), 

1461  647 
ordertype_Memrel, Ord_jump_cardinal]) 1); 
760  648 
qed "Card_jump_cardinal_lemma"; 
484  649 

650 
(*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*) 

651 
goal CardinalArith.thy "Card(jump_cardinal(K))"; 

652 
by (rtac (Ord_jump_cardinal RS CardI) 1); 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

653 
by (rewtac eqpoll_def); 
484  654 
by (safe_tac (ZF_cs addSDs [ltD, jump_cardinal_iff RS iffD1])); 
655 
by (REPEAT (ares_tac [Card_jump_cardinal_lemma RS mem_irrefl] 1)); 

760  656 
qed "Card_jump_cardinal"; 
484  657 

658 
(*** Basic properties of successor cardinals ***) 

659 

660 
goalw CardinalArith.thy [csucc_def] 

661 
"!!K. Ord(K) ==> Card(csucc(K)) & K < csucc(K)"; 

662 
by (rtac LeastI 1); 

663 
by (REPEAT (ares_tac [conjI, Card_jump_cardinal, K_lt_jump_cardinal, 

1461  664 
Ord_jump_cardinal] 1)); 
760  665 
qed "csucc_basic"; 
484  666 

800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset

667 
bind_thm ("Card_csucc", csucc_basic RS conjunct1); 
484  668 

800
23f55b829ccb
Limit_csucc: moved to InfDatatype and proved explicitly in
lcp
parents:
782
diff
changeset

669 
bind_thm ("lt_csucc", csucc_basic RS conjunct2); 
484  670 

517  671 
goal CardinalArith.thy "!!K. Ord(K) ==> 0 < csucc(K)"; 
672 
by (resolve_tac [[Ord_0_le, lt_csucc] MRS lt_trans1] 1); 

673 
by (REPEAT (assume_tac 1)); 

760  674 
qed "Ord_0_lt_csucc"; 
517  675 

484  676 
goalw CardinalArith.thy [csucc_def] 
677 
"!!K L. [ Card(L); K<L ] ==> csucc(K) le L"; 

678 
by (rtac Least_le 1); 

679 
by (REPEAT (ares_tac [conjI, Card_is_Ord] 1)); 

760  680 
qed "csucc_le"; 
484  681 

682 
goal CardinalArith.thy 

683 
"!!K. [ Ord(i); Card(K) ] ==> i < csucc(K) <> i le K"; 

823
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

684 
by (rtac iffI 1); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

685 
by (rtac Card_lt_imp_lt 2); 
33dc37d46296
Changed succ(1) to 2 in cmult_2; Simplified proof of InfCard_is_Limit
lcp
parents:
800
diff
changeset

686 
by (etac lt_trans1 2); 
484  687 
by (REPEAT (ares_tac [lt_csucc, Card_csucc, Card_is_Ord] 2)); 
688 
by (resolve_tac [notI RS not_lt_imp_le] 1); 

689 
by (resolve_tac [Card_cardinal RS csucc_le RS lt_trans1 RS lt_irrefl] 1); 

690 
by (assume_tac 1); 

691 
by (resolve_tac [Ord_cardinal_le RS lt_trans1] 1); 

692 
by (REPEAT (ares_tac [Ord_cardinal] 1 

693 
ORELSE eresolve_tac [ltE, Card_is_Ord] 1)); 

760  694 
qed "lt_csucc_iff"; 
484  695 

696 
goal CardinalArith.thy 

697 
"!!K' K. [ Card(K'); Card(K) ] ==> K' < csucc(K) <> K' le K"; 

698 
by (asm_simp_tac 

699 
(ZF_ss addsimps [lt_csucc_iff, Card_cardinal_eq, Card_is_Ord]) 1); 

760  700 
qed "Card_lt_csucc_iff"; 
488  701 

702 
goalw CardinalArith.thy [InfCard_def] 

703 
"!!K. InfCard(K) ==> InfCard(csucc(K))"; 

704 
by (asm_simp_tac (ZF_ss addsimps [Card_csucc, Card_is_Ord, 

1461  705 
lt_csucc RS leI RSN (2,le_trans)]) 1); 
760  706 
qed "InfCard_csucc"; 
517  707 

1609  708 

709 
(*** Finite sets ***) 

710 

711 
goal CardinalArith.thy 

712 
"!!n. n: nat ==> ALL A. A eqpoll n > A : Fin(A)"; 

713 
by (eresolve_tac [nat_induct] 1); 

714 
by (simp_tac (ZF_ss addsimps (eqpoll_0_iff::Fin.intrs)) 1); 

715 
by (step_tac ZF_cs 1); 

716 
by (subgoal_tac "EX u. u:A" 1); 

717 
by (eresolve_tac [exE] 1); 

718 
by (resolve_tac [Diff_sing_eqpoll RS revcut_rl] 1); 

719 
by (assume_tac 2); 

720 
by (assume_tac 1); 

721 
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1); 

722 
by (assume_tac 1); 

723 
by (resolve_tac [Fin.consI] 1); 

724 
by (fast_tac ZF_cs 1); 

725 
by (deepen_tac (ZF_cs addIs [Fin_mono RS subsetD]) 0 1); (*SLOW*) 

726 
(*Now for the lemma assumed above*) 

727 
by (rewrite_goals_tac [eqpoll_def]); 

728 
by (fast_tac (ZF_cs addSEs [bij_converse_bij RS bij_is_fun RS apply_type]) 1); 

729 
val lemma = result(); 

730 

731 
goalw CardinalArith.thy [Finite_def] "!!A. Finite(A) ==> A : Fin(A)"; 

732 
by (fast_tac (ZF_cs addIs [lemma RS spec RS mp]) 1); 

733 
qed "Finite_into_Fin"; 

734 

735 
goal CardinalArith.thy "!!A. A : Fin(U) ==> Finite(A)"; 

736 
by (fast_tac (ZF_cs addSIs [Finite_0, Finite_cons] addEs [Fin.induct]) 1); 

737 
qed "Fin_into_Finite"; 

738 

739 
goal CardinalArith.thy "Finite(A) <> A : Fin(A)"; 

740 
by (fast_tac (ZF_cs addIs [Finite_into_Fin] addEs [Fin_into_Finite]) 1); 

741 
qed "Finite_Fin_iff"; 

742 

743 
goal CardinalArith.thy 

744 
"!!A. [ Finite(A); Finite(B) ] ==> Finite(A Un B)"; 

745 
by (fast_tac (ZF_cs addSIs [Fin_into_Finite, Fin_UnI] 

746 
addSDs [Finite_into_Fin] 

747 
addSEs [Un_upper1 RS Fin_mono RS subsetD, 

748 
Un_upper2 RS Fin_mono RS subsetD]) 1); 

749 
qed "Finite_Un"; 

750 

751 

752 
(** Removing elements from a finite set decreases its cardinality **) 

753 

754 
goal CardinalArith.thy 

755 
"!!A. A: Fin(U) ==> x~:A > ~ cons(x,A) lepoll A"; 

756 
by (eresolve_tac [Fin_induct] 1); 

757 
by (simp_tac (ZF_ss addsimps [lepoll_0_iff]) 1); 

758 
by (subgoal_tac "cons(x,cons(xa,y)) = cons(xa,cons(x,y))" 1); 

759 
by (asm_simp_tac ZF_ss 1); 

760 
by (fast_tac (ZF_cs addSDs [cons_lepoll_consD]) 1); 

761 
by (fast_tac eq_cs 1); 

762 
qed "Fin_imp_not_cons_lepoll"; 

763 

764 
goal CardinalArith.thy 

765 
"!!a A. [ Finite(A); a~:A ] ==> cons(a,A) = succ(A)"; 

766 
by (rewrite_goals_tac [cardinal_def]); 

767 
by (resolve_tac [Least_equality] 1); 

768 
by (fold_tac [cardinal_def]); 

769 
by (simp_tac (ZF_ss addsimps [succ_def]) 1); 

770 
by (fast_tac (ZF_cs addIs [cons_eqpoll_cong, well_ord_cardinal_eqpoll] 

771 
addSEs [mem_irrefl] 

772 
addSDs [Finite_imp_well_ord]) 1); 

773 
by (fast_tac (ZF_cs addIs [Ord_succ, Card_cardinal, Card_is_Ord]) 1); 

774 
by (resolve_tac [notI] 1); 

775 
by (resolve_tac [Finite_into_Fin RS Fin_imp_not_cons_lepoll RS mp RS notE] 1); 

776 
by (assume_tac 1); 

777 
by (assume_tac 1); 

778 
by (eresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_trans] 1); 

779 
by (eresolve_tac [le_imp_lepoll RS lepoll_trans] 1); 

780 
by (fast_tac (ZF_cs addIs [well_ord_cardinal_eqpoll RS eqpoll_imp_lepoll] 

781 
addSDs [Finite_imp_well_ord]) 1); 

782 
qed "Finite_imp_cardinal_cons"; 

783 

784 

785 
goal CardinalArith.thy "!!a A. [ Finite(A); a:A ] ==> A{a} < A"; 

786 
by (res_inst_tac [("b", "A")] (cons_Diff RS subst) 1); 

787 
by (assume_tac 1); 

788 
by (asm_simp_tac (ZF_ss addsimps [Finite_imp_cardinal_cons, 

789 
Diff_subset RS subset_imp_lepoll RS 

790 
lepoll_Finite]) 1); 

791 
by (asm_simp_tac (ZF_ss addsimps [cons_Diff, Ord_cardinal RS le_refl]) 1); 

792 
qed "Finite_imp_cardinal_Diff"; 

793 

794 

795 
(** Thanks to Krzysztof Grabczewski **) 

796 

797 
val nat_implies_well_ord = nat_into_Ord RS well_ord_Memrel; 

798 

799 
goal CardinalArith.thy "!!m n. [ m:nat; n:nat ] ==> m + n eqpoll m #+ n"; 

800 
by (rtac eqpoll_trans 1); 

801 
by (eresolve_tac [nat_implies_well_ord RS ( 

802 
nat_implies_well_ord RSN (2, 

803 
well_ord_radd RS well_ord_cardinal_eqpoll)) RS eqpoll_sym] 1 

804 
THEN (assume_tac 1)); 

805 
by (eresolve_tac [nat_cadd_eq_add RS subst] 1 THEN (assume_tac 1)); 

806 
by (asm_full_simp_tac (ZF_ss addsimps [cadd_def, eqpoll_refl]) 1); 

807 
qed "nat_sum_eqpoll_sum"; 

808 

809 
goal Nat.thy "!!m. [ m le n; n:nat ] ==> m:nat"; 

810 
by (fast_tac (ZF_cs addSDs [nat_succI RS (Ord_nat RSN (2, OrdmemD))] 

811 
addSEs [ltE]) 1); 

812 
qed "le_in_nat"; 

813 