src/ZF/CardinalArith.thy
author haftmann
Fri Jun 17 16:12:49 2005 +0200 (2005-06-17)
changeset 16417 9bc16273c2d4
parent 14883 ca000a495448
child 24893 b8ef7afe3a6b
permissions -rw-r--r--
migrated theory headers to new format
     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 *)
     7 
     8 header{*Cardinal Arithmetic Without the Axiom of Choice*}
     9 
    10 theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin
    11 
    12 constdefs
    13 
    14   InfCard       :: "i=>o"
    15     "InfCard(i) == Card(i) & nat le i"
    16 
    17   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)
    18     "i |*| j == |i*j|"
    19   
    20   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)
    21     "i |+| j == |i+j|"
    22 
    23   csquare_rel   :: "i=>i"
    24     "csquare_rel(K) ==   
    25 	  rvimage(K*K,   
    26 		  lam <x,y>:K*K. <x Un y, x, y>, 
    27 		  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
    28 
    29   jump_cardinal :: "i=>i"
    30     --{*This def is more complex than Kunen's but it more easily proved to
    31         be a cardinal*}
    32     "jump_cardinal(K) ==   
    33          \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
    34   
    35   csucc         :: "i=>i"
    36     --{*needed because @{term "jump_cardinal(K)"} might not be the successor
    37         of @{term K}*}
    38     "csucc(K) == LEAST L. Card(L) & K<L"
    39 
    40 syntax (xsymbols)
    41   "op |+|"     :: "[i,i] => i"          (infixl "\<oplus>" 65)
    42   "op |*|"     :: "[i,i] => i"          (infixl "\<otimes>" 70)
    43 syntax (HTML output)
    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: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
    62 by (blast intro: Card_Union) 
    63 
    64 lemma Card_OUN [simp,intro,TC]:
    65      "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
    66 by (simp add: OUnion_def Card_0) 
    67 
    68 lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
    69 apply (unfold lesspoll_def)
    70 apply (rule conjI)
    71 apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
    72 apply (rule notI)
    73 apply (erule eqpollE)
    74 apply (rule succ_lepoll_natE)
    75 apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
    76                     lepoll_trans, assumption) 
    77 done
    78 
    79 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
    80 apply (unfold lesspoll_def)
    81 apply (simp add: Card_iff_initial)
    82 apply (fast intro!: le_imp_lepoll ltI leI)
    83 done
    84 
    85 lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
    86 apply (unfold lesspoll_def)
    87 apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
    88             intro!: eqpollI elim: notE 
    89             elim!: eqpollE lepoll_trans)
    90 done
    91 
    92 
    93 subsection{*Cardinal addition*}
    94 
    95 text{*Note: Could omit proving the algebraic laws for cardinal addition and
    96 multiplication.  On finite cardinals these operations coincide with
    97 addition and multiplication of natural numbers; on infinite cardinals they
    98 coincide with union (maximum).  Either way we get most laws for free.*}
    99 
   100 subsubsection{*Cardinal addition is commutative*}
   101 
   102 lemma sum_commute_eqpoll: "A+B \<approx> B+A"
   103 apply (unfold eqpoll_def)
   104 apply (rule exI)
   105 apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
   106 apply auto
   107 done
   108 
   109 lemma cadd_commute: "i |+| j = j |+| i"
   110 apply (unfold cadd_def)
   111 apply (rule sum_commute_eqpoll [THEN cardinal_cong])
   112 done
   113 
   114 subsubsection{*Cardinal addition is associative*}
   115 
   116 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
   117 apply (unfold eqpoll_def)
   118 apply (rule exI)
   119 apply (rule sum_assoc_bij)
   120 done
   121 
   122 (*Unconditional version requires AC*)
   123 lemma well_ord_cadd_assoc: 
   124     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   125      ==> (i |+| j) |+| k = i |+| (j |+| k)"
   126 apply (unfold cadd_def)
   127 apply (rule cardinal_cong)
   128 apply (rule eqpoll_trans)
   129  apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   130  apply (blast intro: well_ord_radd ) 
   131 apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
   132 apply (rule eqpoll_sym)
   133 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   134 apply (blast intro: well_ord_radd ) 
   135 done
   136 
   137 subsubsection{*0 is the identity for addition*}
   138 
   139 lemma sum_0_eqpoll: "0+A \<approx> A"
   140 apply (unfold eqpoll_def)
   141 apply (rule exI)
   142 apply (rule bij_0_sum)
   143 done
   144 
   145 lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
   146 apply (unfold cadd_def)
   147 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   148 done
   149 
   150 subsubsection{*Addition by another cardinal*}
   151 
   152 lemma sum_lepoll_self: "A \<lesssim> A+B"
   153 apply (unfold lepoll_def inj_def)
   154 apply (rule_tac x = "lam x:A. Inl (x) " in exI)
   155 apply simp
   156 done
   157 
   158 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   159 
   160 lemma cadd_le_self: 
   161     "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
   162 apply (unfold cadd_def)
   163 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
   164        assumption)
   165 apply (rule_tac [2] sum_lepoll_self)
   166 apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
   167 done
   168 
   169 subsubsection{*Monotonicity of addition*}
   170 
   171 lemma sum_lepoll_mono: 
   172      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
   173 apply (unfold lepoll_def)
   174 apply (elim exE)
   175 apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   176 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
   177        in lam_injective)
   178 apply (typecheck add: inj_is_fun, auto)
   179 done
   180 
   181 lemma cadd_le_mono:
   182     "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
   183 apply (unfold cadd_def)
   184 apply (safe dest!: le_subset_iff [THEN iffD1])
   185 apply (rule well_ord_lepoll_imp_Card_le)
   186 apply (blast intro: well_ord_radd well_ord_Memrel)
   187 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
   188 done
   189 
   190 subsubsection{*Addition of finite cardinals is "ordinary" addition*}
   191 
   192 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
   193 apply (unfold eqpoll_def)
   194 apply (rule exI)
   195 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
   196             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
   197    apply simp_all
   198 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
   199 done
   200 
   201 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
   202 (*Unconditional version requires AC*)
   203 lemma cadd_succ_lemma:
   204     "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
   205 apply (unfold cadd_def)
   206 apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
   207 apply (rule succ_eqpoll_cong [THEN cardinal_cong])
   208 apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
   209 apply (blast intro: well_ord_radd well_ord_Memrel)
   210 done
   211 
   212 lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
   213 apply (induct_tac m)
   214 apply (simp add: nat_into_Card [THEN cadd_0])
   215 apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
   216 done
   217 
   218 
   219 subsection{*Cardinal multiplication*}
   220 
   221 subsubsection{*Cardinal multiplication is commutative*}
   222 
   223 (*Easier to prove the two directions separately*)
   224 lemma prod_commute_eqpoll: "A*B \<approx> B*A"
   225 apply (unfold eqpoll_def)
   226 apply (rule exI)
   227 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
   228        auto) 
   229 done
   230 
   231 lemma cmult_commute: "i |*| j = j |*| i"
   232 apply (unfold cmult_def)
   233 apply (rule prod_commute_eqpoll [THEN cardinal_cong])
   234 done
   235 
   236 subsubsection{*Cardinal multiplication is associative*}
   237 
   238 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
   239 apply (unfold eqpoll_def)
   240 apply (rule exI)
   241 apply (rule prod_assoc_bij)
   242 done
   243 
   244 (*Unconditional version requires AC*)
   245 lemma well_ord_cmult_assoc:
   246     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   247      ==> (i |*| j) |*| k = i |*| (j |*| k)"
   248 apply (unfold cmult_def)
   249 apply (rule cardinal_cong)
   250 apply (rule eqpoll_trans) 
   251  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   252  apply (blast intro: well_ord_rmult)
   253 apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
   254 apply (rule eqpoll_sym) 
   255 apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   256 apply (blast intro: well_ord_rmult)
   257 done
   258 
   259 subsubsection{*Cardinal multiplication distributes over addition*}
   260 
   261 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
   262 apply (unfold eqpoll_def)
   263 apply (rule exI)
   264 apply (rule sum_prod_distrib_bij)
   265 done
   266 
   267 lemma well_ord_cadd_cmult_distrib:
   268     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   269      ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
   270 apply (unfold cadd_def cmult_def)
   271 apply (rule cardinal_cong)
   272 apply (rule eqpoll_trans) 
   273  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   274 apply (blast intro: well_ord_radd)
   275 apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
   276 apply (rule eqpoll_sym) 
   277 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
   278                                 well_ord_cardinal_eqpoll])
   279 apply (blast intro: well_ord_rmult)+
   280 done
   281 
   282 subsubsection{*Multiplication by 0 yields 0*}
   283 
   284 lemma prod_0_eqpoll: "0*A \<approx> 0"
   285 apply (unfold eqpoll_def)
   286 apply (rule exI)
   287 apply (rule lam_bijective, safe)
   288 done
   289 
   290 lemma cmult_0 [simp]: "0 |*| i = 0"
   291 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
   292 
   293 subsubsection{*1 is the identity for multiplication*}
   294 
   295 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
   296 apply (unfold eqpoll_def)
   297 apply (rule exI)
   298 apply (rule singleton_prod_bij [THEN bij_converse_bij])
   299 done
   300 
   301 lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
   302 apply (unfold cmult_def succ_def)
   303 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   304 done
   305 
   306 subsection{*Some inequalities for multiplication*}
   307 
   308 lemma prod_square_lepoll: "A \<lesssim> A*A"
   309 apply (unfold lepoll_def inj_def)
   310 apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
   311 done
   312 
   313 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
   314 lemma cmult_square_le: "Card(K) ==> K le K |*| K"
   315 apply (unfold cmult_def)
   316 apply (rule le_trans)
   317 apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
   318 apply (rule_tac [3] prod_square_lepoll)
   319 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
   320 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   321 done
   322 
   323 subsubsection{*Multiplication by a non-zero cardinal*}
   324 
   325 lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
   326 apply (unfold lepoll_def inj_def)
   327 apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
   328 done
   329 
   330 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   331 lemma cmult_le_self:
   332     "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
   333 apply (unfold cmult_def)
   334 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
   335   apply assumption
   336  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   337 apply (blast intro: prod_lepoll_self ltD)
   338 done
   339 
   340 subsubsection{*Monotonicity of multiplication*}
   341 
   342 lemma prod_lepoll_mono:
   343      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
   344 apply (unfold lepoll_def)
   345 apply (elim exE)
   346 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
   347 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
   348        in lam_injective)
   349 apply (typecheck add: inj_is_fun, auto)
   350 done
   351 
   352 lemma cmult_le_mono:
   353     "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
   354 apply (unfold cmult_def)
   355 apply (safe dest!: le_subset_iff [THEN iffD1])
   356 apply (rule well_ord_lepoll_imp_Card_le)
   357  apply (blast intro: well_ord_rmult well_ord_Memrel)
   358 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
   359 done
   360 
   361 subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
   362 
   363 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
   364 apply (unfold eqpoll_def)
   365 apply (rule exI)
   366 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
   367             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
   368 apply safe
   369 apply (simp_all add: succI2 if_type mem_imp_not_eq)
   370 done
   371 
   372 (*Unconditional version requires AC*)
   373 lemma cmult_succ_lemma:
   374     "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
   375 apply (unfold cmult_def cadd_def)
   376 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
   377 apply (rule cardinal_cong [symmetric])
   378 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   379 apply (blast intro: well_ord_rmult well_ord_Memrel)
   380 done
   381 
   382 lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
   383 apply (induct_tac m)
   384 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
   385 done
   386 
   387 lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
   388 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
   389 
   390 lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
   391 apply (rule lepoll_trans) 
   392 apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
   393 apply (erule prod_lepoll_mono) 
   394 apply (rule lepoll_refl) 
   395 done
   396 
   397 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
   398 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
   399 
   400 
   401 subsection{*Infinite Cardinals are Limit Ordinals*}
   402 
   403 (*This proof is modelled upon one assuming nat<=A, with injection
   404   lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
   405   and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
   406   If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
   407 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
   408 apply (unfold lepoll_def)
   409 apply (erule exE)
   410 apply (rule_tac x = 
   411           "lam z:cons (u,A).
   412              if z=u then f`0 
   413              else if z: range (f) then f`succ (converse (f) `z) else z" 
   414        in exI)
   415 apply (rule_tac d =
   416           "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
   417                               else y" 
   418        in lam_injective)
   419 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
   420 apply (simp add: inj_is_fun [THEN apply_rangeI]
   421                  inj_converse_fun [THEN apply_rangeI]
   422                  inj_converse_fun [THEN apply_funtype])
   423 done
   424 
   425 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
   426 apply (erule nat_cons_lepoll [THEN eqpollI])
   427 apply (rule subset_consI [THEN subset_imp_lepoll])
   428 done
   429 
   430 (*Specialized version required below*)
   431 lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
   432 apply (unfold succ_def)
   433 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
   434 done
   435 
   436 lemma InfCard_nat: "InfCard(nat)"
   437 apply (unfold InfCard_def)
   438 apply (blast intro: Card_nat le_refl Card_is_Ord)
   439 done
   440 
   441 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
   442 apply (unfold InfCard_def)
   443 apply (erule conjunct1)
   444 done
   445 
   446 lemma InfCard_Un:
   447     "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
   448 apply (unfold InfCard_def)
   449 apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
   450 done
   451 
   452 (*Kunen's Lemma 10.11*)
   453 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
   454 apply (unfold InfCard_def)
   455 apply (erule conjE)
   456 apply (frule Card_is_Ord)
   457 apply (rule ltI [THEN non_succ_LimitI])
   458 apply (erule le_imp_subset [THEN subsetD])
   459 apply (safe dest!: Limit_nat [THEN Limit_le_succD])
   460 apply (unfold Card_def)
   461 apply (drule trans)
   462 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
   463 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
   464 apply (rule le_eqI, assumption)
   465 apply (rule Ord_cardinal)
   466 done
   467 
   468 
   469 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
   470 
   471 (*A general fact about ordermap*)
   472 lemma ordermap_eqpoll_pred:
   473     "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
   474 apply (unfold eqpoll_def)
   475 apply (rule exI)
   476 apply (simp add: ordermap_eq_image well_ord_is_wf)
   477 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
   478                            THEN bij_converse_bij])
   479 apply (rule pred_subset)
   480 done
   481 
   482 subsubsection{*Establishing the well-ordering*}
   483 
   484 lemma csquare_lam_inj:
   485      "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
   486 apply (unfold inj_def)
   487 apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
   488 done
   489 
   490 lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
   491 apply (unfold csquare_rel_def)
   492 apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
   493 apply (blast intro: well_ord_rmult well_ord_Memrel)
   494 done
   495 
   496 subsubsection{*Characterising initial segments of the well-ordering*}
   497 
   498 lemma csquareD:
   499  "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
   500 apply (unfold csquare_rel_def)
   501 apply (erule rev_mp)
   502 apply (elim ltE)
   503 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   504 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
   505 apply (simp_all add: lt_def succI2)
   506 done
   507 
   508 lemma pred_csquare_subset: 
   509     "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
   510 apply (unfold Order.pred_def)
   511 apply (safe del: SigmaI succCI)
   512 apply (erule csquareD [THEN conjE])
   513 apply (unfold lt_def, auto) 
   514 done
   515 
   516 lemma csquare_ltI:
   517  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
   518 apply (unfold csquare_rel_def)
   519 apply (subgoal_tac "x<K & y<K")
   520  prefer 2 apply (blast intro: lt_trans) 
   521 apply (elim ltE)
   522 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   523 done
   524 
   525 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
   526 lemma csquare_or_eqI:
   527  "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
   528 apply (unfold csquare_rel_def)
   529 apply (subgoal_tac "x<K & y<K")
   530  prefer 2 apply (blast intro: lt_trans1) 
   531 apply (elim ltE)
   532 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   533 apply (elim succE)
   534 apply (simp_all add: subset_Un_iff [THEN iff_sym] 
   535                      subset_Un_iff2 [THEN iff_sym] OrdmemD)
   536 done
   537 
   538 subsubsection{*The cardinality of initial segments*}
   539 
   540 lemma ordermap_z_lt:
   541       "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
   542           ordermap(K*K, csquare_rel(K)) ` <x,y> <
   543           ordermap(K*K, csquare_rel(K)) ` <z,z>"
   544 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
   545 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
   546                               Limit_is_Ord [THEN well_ord_csquare], clarify) 
   547 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
   548 apply (erule_tac [4] well_ord_is_wf)
   549 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
   550 done
   551 
   552 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
   553 lemma ordermap_csquare_le:
   554   "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
   555    ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
   556 apply (unfold cmult_def)
   557 apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
   558 apply (rule Ord_cardinal [THEN well_ord_Memrel])+
   559 apply (subgoal_tac "z<K")
   560  prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
   561 apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
   562        assumption+)
   563 apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   564 apply (erule Limit_is_Ord [THEN well_ord_csquare])
   565 apply (blast intro: ltD)
   566 apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
   567             assumption)
   568 apply (elim ltE)
   569 apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
   570 apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
   571 done
   572 
   573 (*Kunen: "... so the order type <= K" *)
   574 lemma ordertype_csquare_le:
   575      "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
   576       ==> ordertype(K*K, csquare_rel(K)) le K"
   577 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   578 apply (rule all_lt_imp_le, assumption)
   579 apply (erule well_ord_csquare [THEN Ord_ordertype])
   580 apply (rule Card_lt_imp_lt)
   581 apply (erule_tac [3] InfCard_is_Card)
   582 apply (erule_tac [2] ltE)
   583 apply (simp add: ordertype_unfold)
   584 apply (safe elim!: ltE)
   585 apply (subgoal_tac "Ord (xa) & Ord (ya)")
   586  prefer 2 apply (blast intro: Ord_in_Ord, clarify)
   587 (*??WHAT A MESS!*)  
   588 apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
   589        (assumption | rule refl | erule ltI)+) 
   590 apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
   591        simp_all add: Ord_Un Ord_nat)
   592 prefer 2 (*case nat le (xa Un ya) *)
   593  apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
   594                   le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
   595                 ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
   596 (*the finite case: xa Un ya < nat *)
   597 apply (rule_tac j = nat in lt_trans2)
   598  apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
   599                   nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
   600 apply (simp add: InfCard_def)
   601 done
   602 
   603 (*Main result: Kunen's Theorem 10.12*)
   604 lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
   605 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   606 apply (erule rev_mp)
   607 apply (erule_tac i=K in trans_induct) 
   608 apply (rule impI)
   609 apply (rule le_anti_sym)
   610 apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
   611 apply (rule ordertype_csquare_le [THEN [2] le_trans])
   612 apply (simp add: cmult_def Ord_cardinal_le   
   613                  well_ord_csquare [THEN Ord_ordertype]
   614                  well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
   615                                    THEN cardinal_cong], assumption+)
   616 done
   617 
   618 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
   619 lemma well_ord_InfCard_square_eq:
   620      "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
   621 apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
   622 apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
   623 apply (rule well_ord_cardinal_eqE)
   624 apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
   625 apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
   626 done
   627 
   628 lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
   629 apply (rule well_ord_InfCard_square_eq)  
   630  apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
   631 apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
   632 done
   633 
   634 lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
   635 by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
   636 
   637 subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
   638 
   639 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
   640 apply (rule le_anti_sym)
   641  prefer 2
   642  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
   643 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   644 apply (rule cmult_le_mono [THEN le_trans], assumption+)
   645 apply (simp add: InfCard_csquare_eq)
   646 done
   647 
   648 (*Corollary 10.13 (1), for cardinal multiplication*)
   649 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
   650 apply (rule_tac i = K and j = L in Ord_linear_le)
   651 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   652 apply (rule cmult_commute [THEN ssubst])
   653 apply (rule Un_commute [THEN ssubst])
   654 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
   655                      subset_Un_iff2 [THEN iffD1] le_imp_subset)
   656 done
   657 
   658 lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
   659 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
   660 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
   661 done
   662 
   663 (*Corollary 10.13 (1), for cardinal addition*)
   664 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
   665 apply (rule le_anti_sym)
   666  prefer 2
   667  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
   668 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   669 apply (rule cadd_le_mono [THEN le_trans], assumption+)
   670 apply (simp add: InfCard_cdouble_eq)
   671 done
   672 
   673 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
   674 apply (rule_tac i = K and j = L in Ord_linear_le)
   675 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   676 apply (rule cadd_commute [THEN ssubst])
   677 apply (rule Un_commute [THEN ssubst])
   678 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
   679 done
   680 
   681 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
   682   of all n-tuples of elements of K.  A better version for the Isabelle theory
   683   might be  InfCard(K) ==> |list(K)| = K.
   684 *)
   685 
   686 subsection{*For Every Cardinal Number There Exists A Greater One}
   687 
   688 text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
   689 
   690 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
   691 apply (unfold jump_cardinal_def)
   692 apply (rule Ord_is_Transset [THEN [2] OrdI])
   693  prefer 2 apply (blast intro!: Ord_ordertype)
   694 apply (unfold Transset_def)
   695 apply (safe del: subsetI)
   696 apply (simp add: ordertype_pred_unfold, safe)
   697 apply (rule UN_I)
   698 apply (rule_tac [2] ReplaceI)
   699    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
   700 done
   701 
   702 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
   703 lemma jump_cardinal_iff:
   704      "i : jump_cardinal(K) <->
   705       (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
   706 apply (unfold jump_cardinal_def)
   707 apply (blast del: subsetI) 
   708 done
   709 
   710 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
   711 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
   712 apply (rule Ord_jump_cardinal [THEN [2] ltI])
   713 apply (rule jump_cardinal_iff [THEN iffD2])
   714 apply (rule_tac x="Memrel(K)" in exI)
   715 apply (rule_tac x=K in exI)  
   716 apply (simp add: ordertype_Memrel well_ord_Memrel)
   717 apply (simp add: Memrel_def subset_iff)
   718 done
   719 
   720 (*The proof by contradiction: the bijection f yields a wellordering of X
   721   whose ordertype is jump_cardinal(K).  *)
   722 lemma Card_jump_cardinal_lemma:
   723      "[| well_ord(X,r);  r <= K * K;  X <= K;
   724          f : bij(ordertype(X,r), jump_cardinal(K)) |]
   725       ==> jump_cardinal(K) : jump_cardinal(K)"
   726 apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
   727  prefer 2 apply (blast intro: comp_bij ordermap_bij)
   728 apply (rule jump_cardinal_iff [THEN iffD2])
   729 apply (intro exI conjI)
   730 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
   731 apply (erule bij_is_inj [THEN well_ord_rvimage])
   732 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
   733 apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
   734                  ordertype_Memrel Ord_jump_cardinal)
   735 done
   736 
   737 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
   738 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
   739 apply (rule Ord_jump_cardinal [THEN CardI])
   740 apply (unfold eqpoll_def)
   741 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
   742 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
   743 done
   744 
   745 subsection{*Basic Properties of Successor Cardinals*}
   746 
   747 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
   748 apply (unfold csucc_def)
   749 apply (rule LeastI)
   750 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
   751 done
   752 
   753 lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
   754 
   755 lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
   756 
   757 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
   758 by (blast intro: Ord_0_le lt_csucc lt_trans1)
   759 
   760 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
   761 apply (unfold csucc_def)
   762 apply (rule Least_le)
   763 apply (blast intro: Card_is_Ord)+
   764 done
   765 
   766 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
   767 apply (rule iffI)
   768 apply (rule_tac [2] Card_lt_imp_lt)
   769 apply (erule_tac [2] lt_trans1)
   770 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
   771 apply (rule notI [THEN not_lt_imp_le])
   772 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
   773 apply (rule Ord_cardinal_le [THEN lt_trans1])
   774 apply (simp_all add: Ord_cardinal Card_is_Ord) 
   775 done
   776 
   777 lemma Card_lt_csucc_iff:
   778      "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
   779 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
   780 
   781 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
   782 by (simp add: InfCard_def Card_csucc Card_is_Ord 
   783               lt_csucc [THEN leI, THEN [2] le_trans])
   784 
   785 
   786 subsubsection{*Removing elements from a finite set decreases its cardinality*}
   787 
   788 lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
   789 apply (erule Fin_induct)
   790 apply (simp add: lepoll_0_iff)
   791 apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
   792 apply simp
   793 apply (blast dest!: cons_lepoll_consD, blast)
   794 done
   795 
   796 lemma Finite_imp_cardinal_cons [simp]:
   797      "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
   798 apply (unfold cardinal_def)
   799 apply (rule Least_equality)
   800 apply (fold cardinal_def)
   801 apply (simp add: succ_def)
   802 apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
   803              elim!: mem_irrefl  dest!: Finite_imp_well_ord)
   804 apply (blast intro: Card_cardinal Card_is_Ord)
   805 apply (rule notI)
   806 apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
   807        assumption, assumption)
   808 apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   809 apply (erule le_imp_lepoll [THEN lepoll_trans])
   810 apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
   811              dest!: Finite_imp_well_ord)
   812 done
   813 
   814 
   815 lemma Finite_imp_succ_cardinal_Diff:
   816      "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
   817 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
   818 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
   819 apply (simp add: cons_Diff)
   820 done
   821 
   822 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
   823 apply (rule succ_leE)
   824 apply (simp add: Finite_imp_succ_cardinal_Diff)
   825 done
   826 
   827 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
   828 apply (erule Finite_induct)
   829 apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
   830 done
   831 
   832 lemma card_Un_Int:
   833      "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
   834 apply (erule Finite_induct, simp) 
   835 apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
   836 done
   837 
   838 lemma card_Un_disjoint: 
   839      "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
   840 by (simp add: Finite_Un card_Un_Int)
   841 
   842 lemma card_partition [rule_format]:
   843      "Finite(C) ==>  
   844         Finite (\<Union> C) -->  
   845         (\<forall>c\<in>C. |c| = k) -->   
   846         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
   847         k #* |C| = |\<Union> C|"
   848 apply (erule Finite_induct, auto)
   849 apply (subgoal_tac " x \<inter> \<Union>B = 0")  
   850 apply (auto simp add: card_Un_disjoint Finite_Union
   851        subset_Finite [of _ "\<Union> (cons(x,F))"])
   852 done
   853 
   854 
   855 subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
   856 
   857 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
   858 
   859 lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
   860 apply (rule eqpoll_trans)
   861 apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
   862 apply (erule nat_implies_well_ord)+
   863 apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
   864 done
   865 
   866 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
   867 apply (erule trans_induct3, auto)
   868 apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
   869 done
   870 
   871 lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
   872 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
   873 
   874 lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
   875 apply (rule succ_inject)
   876 apply (rule_tac b = "|A|" in trans)
   877  apply (simp add: Finite_imp_succ_cardinal_Diff)
   878 apply (subgoal_tac "1 \<lesssim> A")
   879  prefer 2 apply (blast intro: not_0_is_lepoll_1)
   880 apply (frule Finite_imp_well_ord, clarify)
   881 apply (drule well_ord_lepoll_imp_Card_le)
   882  apply (auto simp add: cardinal_1)
   883 apply (rule trans)
   884  apply (rule_tac [2] diff_succ)
   885   apply (auto simp add: Finite_cardinal_in_nat)
   886 done
   887 
   888 lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
   889      "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
   890 apply (erule Finite_induct, auto)
   891 apply (case_tac "Finite (A)")
   892  apply (subgoal_tac [2] "Finite (cons (x, B))")
   893   apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
   894    apply (auto simp add: Finite_0 Finite_cons)
   895 apply (subgoal_tac "|B|<|A|")
   896  prefer 2 apply (blast intro: lt_trans Ord_cardinal)
   897 apply (case_tac "x:A")
   898  apply (subgoal_tac [2] "A - cons (x, B) = A - B")
   899   apply auto
   900 apply (subgoal_tac "|A| le |cons (x, B) |")
   901  prefer 2
   902  apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
   903               intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
   904 apply (auto simp add: Finite_imp_cardinal_cons)
   905 apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
   906 apply (blast intro: lt_trans)
   907 done
   908 
   909 
   910 ML{*
   911 val InfCard_def = thm "InfCard_def"
   912 val cmult_def = thm "cmult_def"
   913 val cadd_def = thm "cadd_def"
   914 val jump_cardinal_def = thm "jump_cardinal_def"
   915 val csucc_def = thm "csucc_def"
   916 
   917 val sum_commute_eqpoll = thm "sum_commute_eqpoll";
   918 val cadd_commute = thm "cadd_commute";
   919 val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
   920 val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
   921 val sum_0_eqpoll = thm "sum_0_eqpoll";
   922 val cadd_0 = thm "cadd_0";
   923 val sum_lepoll_self = thm "sum_lepoll_self";
   924 val cadd_le_self = thm "cadd_le_self";
   925 val sum_lepoll_mono = thm "sum_lepoll_mono";
   926 val cadd_le_mono = thm "cadd_le_mono";
   927 val eq_imp_not_mem = thm "eq_imp_not_mem";
   928 val sum_succ_eqpoll = thm "sum_succ_eqpoll";
   929 val nat_cadd_eq_add = thm "nat_cadd_eq_add";
   930 val prod_commute_eqpoll = thm "prod_commute_eqpoll";
   931 val cmult_commute = thm "cmult_commute";
   932 val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
   933 val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
   934 val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
   935 val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
   936 val prod_0_eqpoll = thm "prod_0_eqpoll";
   937 val cmult_0 = thm "cmult_0";
   938 val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
   939 val cmult_1 = thm "cmult_1";
   940 val prod_lepoll_self = thm "prod_lepoll_self";
   941 val cmult_le_self = thm "cmult_le_self";
   942 val prod_lepoll_mono = thm "prod_lepoll_mono";
   943 val cmult_le_mono = thm "cmult_le_mono";
   944 val prod_succ_eqpoll = thm "prod_succ_eqpoll";
   945 val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
   946 val cmult_2 = thm "cmult_2";
   947 val sum_lepoll_prod = thm "sum_lepoll_prod";
   948 val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
   949 val nat_cons_lepoll = thm "nat_cons_lepoll";
   950 val nat_cons_eqpoll = thm "nat_cons_eqpoll";
   951 val nat_succ_eqpoll = thm "nat_succ_eqpoll";
   952 val InfCard_nat = thm "InfCard_nat";
   953 val InfCard_is_Card = thm "InfCard_is_Card";
   954 val InfCard_Un = thm "InfCard_Un";
   955 val InfCard_is_Limit = thm "InfCard_is_Limit";
   956 val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
   957 val ordermap_z_lt = thm "ordermap_z_lt";
   958 val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
   959 val InfCard_cmult_eq = thm "InfCard_cmult_eq";
   960 val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
   961 val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
   962 val InfCard_cadd_eq = thm "InfCard_cadd_eq";
   963 val Ord_jump_cardinal = thm "Ord_jump_cardinal";
   964 val jump_cardinal_iff = thm "jump_cardinal_iff";
   965 val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
   966 val Card_jump_cardinal = thm "Card_jump_cardinal";
   967 val csucc_basic = thm "csucc_basic";
   968 val Card_csucc = thm "Card_csucc";
   969 val lt_csucc = thm "lt_csucc";
   970 val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
   971 val csucc_le = thm "csucc_le";
   972 val lt_csucc_iff = thm "lt_csucc_iff";
   973 val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
   974 val InfCard_csucc = thm "InfCard_csucc";
   975 val Finite_into_Fin = thm "Finite_into_Fin";
   976 val Fin_into_Finite = thm "Fin_into_Finite";
   977 val Finite_Fin_iff = thm "Finite_Fin_iff";
   978 val Finite_Un = thm "Finite_Un";
   979 val Finite_Union = thm "Finite_Union";
   980 val Finite_induct = thm "Finite_induct";
   981 val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
   982 val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
   983 val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
   984 val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
   985 val nat_implies_well_ord = thm "nat_implies_well_ord";
   986 val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
   987 val Diff_sing_Finite = thm "Diff_sing_Finite";
   988 val Diff_Finite = thm "Diff_Finite";
   989 val Ord_subset_natD = thm "Ord_subset_natD";
   990 val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
   991 val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
   992 val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
   993 val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
   994 *}
   995 
   996 end