src/ZF/CardinalArith.thy
author paulson
Mon Jun 24 11:59:14 2002 +0200 (2002-06-24)
changeset 13244 7b37e218f298
parent 13221 e29378f347e4
child 13269 3ba9be497c33
permissions -rw-r--r--
moving some results around
     1 (*  Title:      ZF/CardinalArith.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Cardinal arithmetic -- WITHOUT the Axiom of Choice
     7 
     8 Note: Could omit proving the algebraic laws for cardinal addition and
     9 multiplication.  On finite cardinals these operations coincide with
    10 addition and multiplication of natural numbers; on infinite cardinals they
    11 coincide with union (maximum).  Either way we get most laws for free.
    12 *)
    13 
    14 theory CardinalArith = Cardinal + OrderArith + ArithSimp + Finite:
    15 
    16 constdefs
    17 
    18   InfCard       :: "i=>o"
    19     "InfCard(i) == Card(i) & nat le i"
    20 
    21   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)
    22     "i |*| j == |i*j|"
    23   
    24   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)
    25     "i |+| j == |i+j|"
    26 
    27   csquare_rel   :: "i=>i"
    28     "csquare_rel(K) ==   
    29 	  rvimage(K*K,   
    30 		  lam <x,y>:K*K. <x Un y, x, y>, 
    31 		  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
    32 
    33   (*This def is more complex than Kunen's but it more easily proved to
    34     be a cardinal*)
    35   jump_cardinal :: "i=>i"
    36     "jump_cardinal(K) ==   
    37          UN X:Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
    38   
    39   (*needed because jump_cardinal(K) might not be the successor of K*)
    40   csucc         :: "i=>i"
    41     "csucc(K) == LEAST L. Card(L) & K<L"
    42 
    43 syntax (xsymbols)
    44   "op |+|"     :: "[i,i] => i"          (infixl "\<oplus>" 65)
    45   "op |*|"     :: "[i,i] => i"          (infixl "\<otimes>" 70)
    46 
    47 
    48 lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
    49 apply (rule CardI) 
    50  apply (simp add: Card_is_Ord) 
    51 apply (clarify dest!: ltD)
    52 apply (drule bspec, assumption) 
    53 apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
    54 apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
    55 apply (drule lesspoll_trans1, assumption) 
    56 apply (subgoal_tac "B \<lesssim> \<Union>A")
    57  apply (drule lesspoll_trans1, assumption, blast) 
    58 apply (blast intro: subset_imp_lepoll) 
    59 done
    60 
    61 lemma Card_UN:
    62      "(!!x. x:A ==> Card(K(x))) ==> Card(UN x:A. K(x))" 
    63 by (blast intro: Card_Union) 
    64 
    65 lemma Card_OUN [simp,intro,TC]:
    66      "(!!x. x:A ==> Card(K(x))) ==> Card(UN x<A. K(x))"
    67 by (simp add: OUnion_def Card_0) 
    68 
    69 lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
    70 apply (unfold lesspoll_def)
    71 apply (rule conjI)
    72 apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
    73 apply (rule notI)
    74 apply (erule eqpollE)
    75 apply (rule succ_lepoll_natE)
    76 apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
    77                     lepoll_trans, assumption) 
    78 done
    79 
    80 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
    81 apply (unfold lesspoll_def)
    82 apply (simp add: Card_iff_initial)
    83 apply (fast intro!: le_imp_lepoll ltI leI)
    84 done
    85 
    86 lemma lesspoll_lemma: 
    87         "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
    88 apply (unfold lesspoll_def)
    89 apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
    90             intro!: eqpollI elim: notE 
    91             elim!: eqpollE lepoll_trans)
    92 done
    93 
    94 
    95 (*** Cardinal addition ***)
    96 
    97 (** Cardinal addition is commutative **)
    98 
    99 lemma sum_commute_eqpoll: "A+B \<approx> B+A"
   100 apply (unfold eqpoll_def)
   101 apply (rule exI)
   102 apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
   103 apply auto
   104 done
   105 
   106 lemma cadd_commute: "i |+| j = j |+| i"
   107 apply (unfold cadd_def)
   108 apply (rule sum_commute_eqpoll [THEN cardinal_cong])
   109 done
   110 
   111 (** Cardinal addition is associative **)
   112 
   113 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
   114 apply (unfold eqpoll_def)
   115 apply (rule exI)
   116 apply (rule sum_assoc_bij)
   117 done
   118 
   119 (*Unconditional version requires AC*)
   120 lemma well_ord_cadd_assoc: 
   121     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   122      ==> (i |+| j) |+| k = i |+| (j |+| k)"
   123 apply (unfold cadd_def)
   124 apply (rule cardinal_cong)
   125 apply (rule eqpoll_trans)
   126  apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   127  apply (blast intro: well_ord_radd ) 
   128 apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
   129 apply (rule eqpoll_sym)
   130 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   131 apply (blast intro: well_ord_radd ) 
   132 done
   133 
   134 (** 0 is the identity for addition **)
   135 
   136 lemma sum_0_eqpoll: "0+A \<approx> A"
   137 apply (unfold eqpoll_def)
   138 apply (rule exI)
   139 apply (rule bij_0_sum)
   140 done
   141 
   142 lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
   143 apply (unfold cadd_def)
   144 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   145 done
   146 
   147 (** Addition by another cardinal **)
   148 
   149 lemma sum_lepoll_self: "A \<lesssim> A+B"
   150 apply (unfold lepoll_def inj_def)
   151 apply (rule_tac x = "lam x:A. Inl (x) " in exI)
   152 apply simp
   153 done
   154 
   155 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   156 
   157 lemma cadd_le_self: 
   158     "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
   159 apply (unfold cadd_def)
   160 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
   161        assumption)
   162 apply (rule_tac [2] sum_lepoll_self)
   163 apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
   164 done
   165 
   166 (** Monotonicity of addition **)
   167 
   168 lemma sum_lepoll_mono: 
   169      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
   170 apply (unfold lepoll_def)
   171 apply (elim exE)
   172 apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   173 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
   174        in lam_injective)
   175 apply (typecheck add: inj_is_fun, auto)
   176 done
   177 
   178 lemma cadd_le_mono:
   179     "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
   180 apply (unfold cadd_def)
   181 apply (safe dest!: le_subset_iff [THEN iffD1])
   182 apply (rule well_ord_lepoll_imp_Card_le)
   183 apply (blast intro: well_ord_radd well_ord_Memrel)
   184 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
   185 done
   186 
   187 (** Addition of finite cardinals is "ordinary" addition **)
   188 
   189 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
   190 apply (unfold eqpoll_def)
   191 apply (rule exI)
   192 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
   193             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
   194    apply simp_all
   195 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
   196 done
   197 
   198 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
   199 (*Unconditional version requires AC*)
   200 lemma cadd_succ_lemma:
   201     "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
   202 apply (unfold cadd_def)
   203 apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
   204 apply (rule succ_eqpoll_cong [THEN cardinal_cong])
   205 apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
   206 apply (blast intro: well_ord_radd well_ord_Memrel)
   207 done
   208 
   209 lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
   210 apply (induct_tac m)
   211 apply (simp add: nat_into_Card [THEN cadd_0])
   212 apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
   213 done
   214 
   215 
   216 (*** Cardinal multiplication ***)
   217 
   218 (** Cardinal multiplication is commutative **)
   219 
   220 (*Easier to prove the two directions separately*)
   221 lemma prod_commute_eqpoll: "A*B \<approx> B*A"
   222 apply (unfold eqpoll_def)
   223 apply (rule exI)
   224 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
   225        auto) 
   226 done
   227 
   228 lemma cmult_commute: "i |*| j = j |*| i"
   229 apply (unfold cmult_def)
   230 apply (rule prod_commute_eqpoll [THEN cardinal_cong])
   231 done
   232 
   233 (** Cardinal multiplication is associative **)
   234 
   235 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
   236 apply (unfold eqpoll_def)
   237 apply (rule exI)
   238 apply (rule prod_assoc_bij)
   239 done
   240 
   241 (*Unconditional version requires AC*)
   242 lemma well_ord_cmult_assoc:
   243     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   244      ==> (i |*| j) |*| k = i |*| (j |*| k)"
   245 apply (unfold cmult_def)
   246 apply (rule cardinal_cong)
   247 apply (rule eqpoll_trans) 
   248  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   249  apply (blast intro: well_ord_rmult)
   250 apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
   251 apply (rule eqpoll_sym) 
   252 apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   253 apply (blast intro: well_ord_rmult)
   254 done
   255 
   256 (** Cardinal multiplication distributes over addition **)
   257 
   258 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
   259 apply (unfold eqpoll_def)
   260 apply (rule exI)
   261 apply (rule sum_prod_distrib_bij)
   262 done
   263 
   264 lemma well_ord_cadd_cmult_distrib:
   265     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   266      ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
   267 apply (unfold cadd_def cmult_def)
   268 apply (rule cardinal_cong)
   269 apply (rule eqpoll_trans) 
   270  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   271 apply (blast intro: well_ord_radd)
   272 apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
   273 apply (rule eqpoll_sym) 
   274 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
   275                                 well_ord_cardinal_eqpoll])
   276 apply (blast intro: well_ord_rmult)+
   277 done
   278 
   279 (** Multiplication by 0 yields 0 **)
   280 
   281 lemma prod_0_eqpoll: "0*A \<approx> 0"
   282 apply (unfold eqpoll_def)
   283 apply (rule exI)
   284 apply (rule lam_bijective, safe)
   285 done
   286 
   287 lemma cmult_0 [simp]: "0 |*| i = 0"
   288 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
   289 
   290 (** 1 is the identity for multiplication **)
   291 
   292 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
   293 apply (unfold eqpoll_def)
   294 apply (rule exI)
   295 apply (rule singleton_prod_bij [THEN bij_converse_bij])
   296 done
   297 
   298 lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
   299 apply (unfold cmult_def succ_def)
   300 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   301 done
   302 
   303 (*** Some inequalities for multiplication ***)
   304 
   305 lemma prod_square_lepoll: "A \<lesssim> A*A"
   306 apply (unfold lepoll_def inj_def)
   307 apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
   308 done
   309 
   310 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
   311 lemma cmult_square_le: "Card(K) ==> K le K |*| K"
   312 apply (unfold cmult_def)
   313 apply (rule le_trans)
   314 apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
   315 apply (rule_tac [3] prod_square_lepoll)
   316 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
   317 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   318 done
   319 
   320 (** Multiplication by a non-zero cardinal **)
   321 
   322 lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
   323 apply (unfold lepoll_def inj_def)
   324 apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
   325 done
   326 
   327 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   328 lemma cmult_le_self:
   329     "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
   330 apply (unfold cmult_def)
   331 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
   332   apply assumption
   333  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   334 apply (blast intro: prod_lepoll_self ltD)
   335 done
   336 
   337 (** Monotonicity of multiplication **)
   338 
   339 lemma prod_lepoll_mono:
   340      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
   341 apply (unfold lepoll_def)
   342 apply (elim exE)
   343 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
   344 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
   345        in lam_injective)
   346 apply (typecheck add: inj_is_fun, auto)
   347 done
   348 
   349 lemma cmult_le_mono:
   350     "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
   351 apply (unfold cmult_def)
   352 apply (safe dest!: le_subset_iff [THEN iffD1])
   353 apply (rule well_ord_lepoll_imp_Card_le)
   354  apply (blast intro: well_ord_rmult well_ord_Memrel)
   355 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
   356 done
   357 
   358 (*** Multiplication of finite cardinals is "ordinary" multiplication ***)
   359 
   360 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
   361 apply (unfold eqpoll_def)
   362 apply (rule exI)
   363 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
   364             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
   365 apply safe
   366 apply (simp_all add: succI2 if_type mem_imp_not_eq)
   367 done
   368 
   369 (*Unconditional version requires AC*)
   370 lemma cmult_succ_lemma:
   371     "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
   372 apply (unfold cmult_def cadd_def)
   373 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
   374 apply (rule cardinal_cong [symmetric])
   375 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   376 apply (blast intro: well_ord_rmult well_ord_Memrel)
   377 done
   378 
   379 lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
   380 apply (induct_tac m)
   381 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
   382 done
   383 
   384 lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
   385 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
   386 
   387 lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
   388 apply (rule lepoll_trans) 
   389 apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
   390 apply (erule prod_lepoll_mono) 
   391 apply (rule lepoll_refl) 
   392 done
   393 
   394 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
   395 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
   396 
   397 
   398 (*** Infinite Cardinals are Limit Ordinals ***)
   399 
   400 (*This proof is modelled upon one assuming nat<=A, with injection
   401   lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
   402   and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
   403   If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
   404 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
   405 apply (unfold lepoll_def)
   406 apply (erule exE)
   407 apply (rule_tac x = 
   408           "lam z:cons (u,A).
   409              if z=u then f`0 
   410              else if z: range (f) then f`succ (converse (f) `z) else z" 
   411        in exI)
   412 apply (rule_tac d =
   413           "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
   414                               else y" 
   415        in lam_injective)
   416 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
   417 apply (simp add: inj_is_fun [THEN apply_rangeI]
   418                  inj_converse_fun [THEN apply_rangeI]
   419                  inj_converse_fun [THEN apply_funtype])
   420 done
   421 
   422 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
   423 apply (erule nat_cons_lepoll [THEN eqpollI])
   424 apply (rule subset_consI [THEN subset_imp_lepoll])
   425 done
   426 
   427 (*Specialized version required below*)
   428 lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
   429 apply (unfold succ_def)
   430 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
   431 done
   432 
   433 lemma InfCard_nat: "InfCard(nat)"
   434 apply (unfold InfCard_def)
   435 apply (blast intro: Card_nat le_refl Card_is_Ord)
   436 done
   437 
   438 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
   439 apply (unfold InfCard_def)
   440 apply (erule conjunct1)
   441 done
   442 
   443 lemma InfCard_Un:
   444     "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
   445 apply (unfold InfCard_def)
   446 apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
   447 done
   448 
   449 (*Kunen's Lemma 10.11*)
   450 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
   451 apply (unfold InfCard_def)
   452 apply (erule conjE)
   453 apply (frule Card_is_Ord)
   454 apply (rule ltI [THEN non_succ_LimitI])
   455 apply (erule le_imp_subset [THEN subsetD])
   456 apply (safe dest!: Limit_nat [THEN Limit_le_succD])
   457 apply (unfold Card_def)
   458 apply (drule trans)
   459 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
   460 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
   461 apply (rule le_eqI, assumption)
   462 apply (rule Ord_cardinal)
   463 done
   464 
   465 
   466 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
   467 
   468 (*A general fact about ordermap*)
   469 lemma ordermap_eqpoll_pred:
   470     "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> pred(A,x,r)"
   471 apply (unfold eqpoll_def)
   472 apply (rule exI)
   473 apply (simp add: ordermap_eq_image well_ord_is_wf)
   474 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
   475                            THEN bij_converse_bij])
   476 apply (rule pred_subset)
   477 done
   478 
   479 (** Establishing the well-ordering **)
   480 
   481 lemma csquare_lam_inj:
   482      "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
   483 apply (unfold inj_def)
   484 apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
   485 done
   486 
   487 lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
   488 apply (unfold csquare_rel_def)
   489 apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
   490 apply (blast intro: well_ord_rmult well_ord_Memrel)
   491 done
   492 
   493 (** Characterising initial segments of the well-ordering **)
   494 
   495 lemma csquareD:
   496  "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
   497 apply (unfold csquare_rel_def)
   498 apply (erule rev_mp)
   499 apply (elim ltE)
   500 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   501 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
   502 apply (simp_all add: lt_def succI2)
   503 done
   504 
   505 lemma pred_csquare_subset: 
   506     "z<K ==> pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
   507 apply (unfold Order.pred_def)
   508 apply (safe del: SigmaI succCI)
   509 apply (erule csquareD [THEN conjE])
   510 apply (unfold lt_def, auto) 
   511 done
   512 
   513 lemma csquare_ltI:
   514  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
   515 apply (unfold csquare_rel_def)
   516 apply (subgoal_tac "x<K & y<K")
   517  prefer 2 apply (blast intro: lt_trans) 
   518 apply (elim ltE)
   519 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   520 done
   521 
   522 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
   523 lemma csquare_or_eqI:
   524  "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
   525 apply (unfold csquare_rel_def)
   526 apply (subgoal_tac "x<K & y<K")
   527  prefer 2 apply (blast intro: lt_trans1) 
   528 apply (elim ltE)
   529 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   530 apply (elim succE)
   531 apply (simp_all add: subset_Un_iff [THEN iff_sym] 
   532                      subset_Un_iff2 [THEN iff_sym] OrdmemD)
   533 done
   534 
   535 (** The cardinality of initial segments **)
   536 
   537 lemma ordermap_z_lt:
   538       "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
   539           ordermap(K*K, csquare_rel(K)) ` <x,y> <
   540           ordermap(K*K, csquare_rel(K)) ` <z,z>"
   541 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
   542 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
   543                               Limit_is_Ord [THEN well_ord_csquare], clarify) 
   544 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
   545 apply (erule_tac [4] well_ord_is_wf)
   546 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
   547 done
   548 
   549 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
   550 lemma ordermap_csquare_le:
   551   "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
   552    ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
   553 apply (unfold cmult_def)
   554 apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
   555 apply (rule Ord_cardinal [THEN well_ord_Memrel])+
   556 apply (subgoal_tac "z<K")
   557  prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
   558 apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
   559        assumption+)
   560 apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   561 apply (erule Limit_is_Ord [THEN well_ord_csquare])
   562 apply (blast intro: ltD)
   563 apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
   564             assumption)
   565 apply (elim ltE)
   566 apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
   567 apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
   568 done
   569 
   570 (*Kunen: "... so the order type <= K" *)
   571 lemma ordertype_csquare_le:
   572      "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
   573       ==> ordertype(K*K, csquare_rel(K)) le K"
   574 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   575 apply (rule all_lt_imp_le, assumption)
   576 apply (erule well_ord_csquare [THEN Ord_ordertype])
   577 apply (rule Card_lt_imp_lt)
   578 apply (erule_tac [3] InfCard_is_Card)
   579 apply (erule_tac [2] ltE)
   580 apply (simp add: ordertype_unfold)
   581 apply (safe elim!: ltE)
   582 apply (subgoal_tac "Ord (xa) & Ord (ya)")
   583  prefer 2 apply (blast intro: Ord_in_Ord, clarify)
   584 (*??WHAT A MESS!*)  
   585 apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
   586        (assumption | rule refl | erule ltI)+) 
   587 apply (rule_tac i = "xa Un ya" and j = "nat" in Ord_linear2,
   588        simp_all add: Ord_Un Ord_nat)
   589 prefer 2 (*case nat le (xa Un ya) *)
   590  apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
   591                   le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
   592                 ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
   593 (*the finite case: xa Un ya < nat *)
   594 apply (rule_tac j = "nat" in lt_trans2)
   595  apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
   596                   nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
   597 apply (simp add: InfCard_def)
   598 done
   599 
   600 (*Main result: Kunen's Theorem 10.12*)
   601 lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
   602 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   603 apply (erule rev_mp)
   604 apply (erule_tac i=K in trans_induct) 
   605 apply (rule impI)
   606 apply (rule le_anti_sym)
   607 apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
   608 apply (rule ordertype_csquare_le [THEN [2] le_trans])
   609 apply (simp add: cmult_def Ord_cardinal_le   
   610                  well_ord_csquare [THEN Ord_ordertype]
   611                  well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
   612                                    THEN cardinal_cong], assumption+)
   613 done
   614 
   615 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
   616 lemma well_ord_InfCard_square_eq:
   617      "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
   618 apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
   619 apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
   620 apply (rule well_ord_cardinal_eqE)
   621 apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
   622 apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
   623 done
   624 
   625 (** Toward's Kunen's Corollary 10.13 (1) **)
   626 
   627 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
   628 apply (rule le_anti_sym)
   629  prefer 2
   630  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
   631 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   632 apply (rule cmult_le_mono [THEN le_trans], assumption+)
   633 apply (simp add: InfCard_csquare_eq)
   634 done
   635 
   636 (*Corollary 10.13 (1), for cardinal multiplication*)
   637 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
   638 apply (rule_tac i = "K" and j = "L" in Ord_linear_le)
   639 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   640 apply (rule cmult_commute [THEN ssubst])
   641 apply (rule Un_commute [THEN ssubst])
   642 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
   643                      subset_Un_iff2 [THEN iffD1] le_imp_subset)
   644 done
   645 
   646 lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
   647 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
   648 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
   649 done
   650 
   651 (*Corollary 10.13 (1), for cardinal addition*)
   652 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
   653 apply (rule le_anti_sym)
   654  prefer 2
   655  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
   656 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   657 apply (rule cadd_le_mono [THEN le_trans], assumption+)
   658 apply (simp add: InfCard_cdouble_eq)
   659 done
   660 
   661 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
   662 apply (rule_tac i = "K" and j = "L" in Ord_linear_le)
   663 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   664 apply (rule cadd_commute [THEN ssubst])
   665 apply (rule Un_commute [THEN ssubst])
   666 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
   667 done
   668 
   669 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
   670   of all n-tuples of elements of K.  A better version for the Isabelle theory
   671   might be  InfCard(K) ==> |list(K)| = K.
   672 *)
   673 
   674 (*** For every cardinal number there exists a greater one
   675      [Kunen's Theorem 10.16, which would be trivial using AC] ***)
   676 
   677 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
   678 apply (unfold jump_cardinal_def)
   679 apply (rule Ord_is_Transset [THEN [2] OrdI])
   680  prefer 2 apply (blast intro!: Ord_ordertype)
   681 apply (unfold Transset_def)
   682 apply (safe del: subsetI)
   683 apply (simp add: ordertype_pred_unfold, safe)
   684 apply (rule UN_I)
   685 apply (rule_tac [2] ReplaceI)
   686    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
   687 done
   688 
   689 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
   690 lemma jump_cardinal_iff:
   691      "i : jump_cardinal(K) <->
   692       (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
   693 apply (unfold jump_cardinal_def)
   694 apply (blast del: subsetI) 
   695 done
   696 
   697 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
   698 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
   699 apply (rule Ord_jump_cardinal [THEN [2] ltI])
   700 apply (rule jump_cardinal_iff [THEN iffD2])
   701 apply (rule_tac x="Memrel(K)" in exI)
   702 apply (rule_tac x=K in exI)  
   703 apply (simp add: ordertype_Memrel well_ord_Memrel)
   704 apply (simp add: Memrel_def subset_iff)
   705 done
   706 
   707 (*The proof by contradiction: the bijection f yields a wellordering of X
   708   whose ordertype is jump_cardinal(K).  *)
   709 lemma Card_jump_cardinal_lemma:
   710      "[| well_ord(X,r);  r <= K * K;  X <= K;
   711          f : bij(ordertype(X,r), jump_cardinal(K)) |]
   712       ==> jump_cardinal(K) : jump_cardinal(K)"
   713 apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
   714  prefer 2 apply (blast intro: comp_bij ordermap_bij)
   715 apply (rule jump_cardinal_iff [THEN iffD2])
   716 apply (intro exI conjI)
   717 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
   718 apply (erule bij_is_inj [THEN well_ord_rvimage])
   719 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
   720 apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
   721                  ordertype_Memrel Ord_jump_cardinal)
   722 done
   723 
   724 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
   725 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
   726 apply (rule Ord_jump_cardinal [THEN CardI])
   727 apply (unfold eqpoll_def)
   728 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
   729 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
   730 done
   731 
   732 (*** Basic properties of successor cardinals ***)
   733 
   734 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
   735 apply (unfold csucc_def)
   736 apply (rule LeastI)
   737 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
   738 done
   739 
   740 lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
   741 
   742 lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
   743 
   744 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
   745 by (blast intro: Ord_0_le lt_csucc lt_trans1)
   746 
   747 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
   748 apply (unfold csucc_def)
   749 apply (rule Least_le)
   750 apply (blast intro: Card_is_Ord)+
   751 done
   752 
   753 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
   754 apply (rule iffI)
   755 apply (rule_tac [2] Card_lt_imp_lt)
   756 apply (erule_tac [2] lt_trans1)
   757 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
   758 apply (rule notI [THEN not_lt_imp_le])
   759 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
   760 apply (rule Ord_cardinal_le [THEN lt_trans1])
   761 apply (simp_all add: Ord_cardinal Card_is_Ord) 
   762 done
   763 
   764 lemma Card_lt_csucc_iff:
   765      "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
   766 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
   767 
   768 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
   769 by (simp add: InfCard_def Card_csucc Card_is_Ord 
   770               lt_csucc [THEN leI, THEN [2] le_trans])
   771 
   772 
   773 (** Removing elements from a finite set decreases its cardinality **)
   774 
   775 lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
   776 apply (erule Fin_induct)
   777 apply (simp add: lepoll_0_iff)
   778 apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
   779 apply simp
   780 apply (blast dest!: cons_lepoll_consD, blast)
   781 done
   782 
   783 lemma Finite_imp_cardinal_cons:
   784      "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
   785 apply (unfold cardinal_def)
   786 apply (rule Least_equality)
   787 apply (fold cardinal_def)
   788 apply (simp add: succ_def)
   789 apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
   790              elim!: mem_irrefl  dest!: Finite_imp_well_ord)
   791 apply (blast intro: Card_cardinal Card_is_Ord)
   792 apply (rule notI)
   793 apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
   794        assumption, assumption)
   795 apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   796 apply (erule le_imp_lepoll [THEN lepoll_trans])
   797 apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
   798              dest!: Finite_imp_well_ord)
   799 done
   800 
   801 
   802 lemma Finite_imp_succ_cardinal_Diff:
   803      "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
   804 apply (rule_tac b = "A" in cons_Diff [THEN subst], assumption)
   805 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
   806 apply (simp add: cons_Diff)
   807 done
   808 
   809 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
   810 apply (rule succ_leE)
   811 apply (simp add: Finite_imp_succ_cardinal_Diff)
   812 done
   813 
   814 
   815 (** Theorems by Krzysztof Grabczewski, proofs by lcp **)
   816 
   817 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
   818 
   819 lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
   820 apply (rule eqpoll_trans)
   821 apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
   822 apply (erule nat_implies_well_ord)+
   823 apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
   824 done
   825 
   826 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
   827 apply (erule trans_induct3, auto)
   828 apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
   829 done
   830 
   831 lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
   832 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
   833 
   834 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
   835 apply (erule Finite_induct)
   836 apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
   837 done
   838 
   839 lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
   840 apply (rule succ_inject)
   841 apply (rule_tac b = "|A|" in trans)
   842 apply (simp add: Finite_imp_succ_cardinal_Diff)
   843 apply (subgoal_tac "1 \<lesssim> A")
   844  prefer 2 apply (blast intro: not_0_is_lepoll_1)
   845 apply (frule Finite_imp_well_ord, clarify)
   846 apply (rotate_tac -1)
   847 apply (drule well_ord_lepoll_imp_Card_le)
   848 apply (auto simp add: cardinal_1)
   849 apply (rule trans)
   850 apply (rule_tac [2] diff_succ)
   851 apply (auto simp add: Finite_cardinal_in_nat)
   852 done
   853 
   854 lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
   855      "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
   856 apply (erule Finite_induct, auto)
   857 apply (simp_all add: Finite_imp_cardinal_cons)
   858 apply (case_tac "Finite (A)")
   859  apply (subgoal_tac [2] "Finite (cons (x, B))")
   860   apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
   861    apply (auto simp add: Finite_0 Finite_cons)
   862 apply (subgoal_tac "|B|<|A|")
   863  prefer 2 apply (blast intro: lt_trans Ord_cardinal)
   864 apply (case_tac "x:A")
   865  apply (subgoal_tac [2] "A - cons (x, B) = A - B")
   866   apply auto
   867 apply (subgoal_tac "|A| le |cons (x, B) |")
   868  prefer 2
   869  apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
   870               intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
   871 apply (auto simp add: Finite_imp_cardinal_cons)
   872 apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
   873 apply (blast intro: lt_trans)
   874 done
   875 
   876 
   877 ML{*
   878 val InfCard_def = thm "InfCard_def"
   879 val cmult_def = thm "cmult_def"
   880 val cadd_def = thm "cadd_def"
   881 val jump_cardinal_def = thm "jump_cardinal_def"
   882 val csucc_def = thm "csucc_def"
   883 
   884 val sum_commute_eqpoll = thm "sum_commute_eqpoll";
   885 val cadd_commute = thm "cadd_commute";
   886 val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
   887 val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
   888 val sum_0_eqpoll = thm "sum_0_eqpoll";
   889 val cadd_0 = thm "cadd_0";
   890 val sum_lepoll_self = thm "sum_lepoll_self";
   891 val cadd_le_self = thm "cadd_le_self";
   892 val sum_lepoll_mono = thm "sum_lepoll_mono";
   893 val cadd_le_mono = thm "cadd_le_mono";
   894 val eq_imp_not_mem = thm "eq_imp_not_mem";
   895 val sum_succ_eqpoll = thm "sum_succ_eqpoll";
   896 val nat_cadd_eq_add = thm "nat_cadd_eq_add";
   897 val prod_commute_eqpoll = thm "prod_commute_eqpoll";
   898 val cmult_commute = thm "cmult_commute";
   899 val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
   900 val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
   901 val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
   902 val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
   903 val prod_0_eqpoll = thm "prod_0_eqpoll";
   904 val cmult_0 = thm "cmult_0";
   905 val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
   906 val cmult_1 = thm "cmult_1";
   907 val prod_lepoll_self = thm "prod_lepoll_self";
   908 val cmult_le_self = thm "cmult_le_self";
   909 val prod_lepoll_mono = thm "prod_lepoll_mono";
   910 val cmult_le_mono = thm "cmult_le_mono";
   911 val prod_succ_eqpoll = thm "prod_succ_eqpoll";
   912 val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
   913 val cmult_2 = thm "cmult_2";
   914 val sum_lepoll_prod = thm "sum_lepoll_prod";
   915 val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
   916 val nat_cons_lepoll = thm "nat_cons_lepoll";
   917 val nat_cons_eqpoll = thm "nat_cons_eqpoll";
   918 val nat_succ_eqpoll = thm "nat_succ_eqpoll";
   919 val InfCard_nat = thm "InfCard_nat";
   920 val InfCard_is_Card = thm "InfCard_is_Card";
   921 val InfCard_Un = thm "InfCard_Un";
   922 val InfCard_is_Limit = thm "InfCard_is_Limit";
   923 val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
   924 val ordermap_z_lt = thm "ordermap_z_lt";
   925 val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
   926 val InfCard_cmult_eq = thm "InfCard_cmult_eq";
   927 val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
   928 val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
   929 val InfCard_cadd_eq = thm "InfCard_cadd_eq";
   930 val Ord_jump_cardinal = thm "Ord_jump_cardinal";
   931 val jump_cardinal_iff = thm "jump_cardinal_iff";
   932 val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
   933 val Card_jump_cardinal = thm "Card_jump_cardinal";
   934 val csucc_basic = thm "csucc_basic";
   935 val Card_csucc = thm "Card_csucc";
   936 val lt_csucc = thm "lt_csucc";
   937 val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
   938 val csucc_le = thm "csucc_le";
   939 val lt_csucc_iff = thm "lt_csucc_iff";
   940 val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
   941 val InfCard_csucc = thm "InfCard_csucc";
   942 val Finite_into_Fin = thm "Finite_into_Fin";
   943 val Fin_into_Finite = thm "Fin_into_Finite";
   944 val Finite_Fin_iff = thm "Finite_Fin_iff";
   945 val Finite_Un = thm "Finite_Un";
   946 val Finite_Union = thm "Finite_Union";
   947 val Finite_induct = thm "Finite_induct";
   948 val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
   949 val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
   950 val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
   951 val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
   952 val nat_implies_well_ord = thm "nat_implies_well_ord";
   953 val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
   954 val Diff_sing_Finite = thm "Diff_sing_Finite";
   955 val Diff_Finite = thm "Diff_Finite";
   956 val Ord_subset_natD = thm "Ord_subset_natD";
   957 val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
   958 val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
   959 val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
   960 val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
   961 *}
   962 
   963 end