src/ZF/CardinalArith.thy
author wenzelm
Sun Oct 07 21:19:31 2007 +0200 (2007-10-07)
changeset 24893 b8ef7afe3a6b
parent 16417 9bc16273c2d4
child 27517 c055e1d49285
permissions -rw-r--r--
modernized specifications;
removed legacy ML bindings;
     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 definition
    13   InfCard       :: "i=>o"  where
    14     "InfCard(i) == Card(i) & nat le i"
    15 
    16 definition
    17   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where
    18     "i |*| j == |i*j|"
    19   
    20 definition
    21   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where
    22     "i |+| j == |i+j|"
    23 
    24 definition
    25   csquare_rel   :: "i=>i"  where
    26     "csquare_rel(K) ==   
    27 	  rvimage(K*K,   
    28 		  lam <x,y>:K*K. <x Un y, x, y>, 
    29 		  rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
    30 
    31 definition
    32   jump_cardinal :: "i=>i"  where
    33     --{*This def is more complex than Kunen's but it more easily proved to
    34         be a cardinal*}
    35     "jump_cardinal(K) ==   
    36          \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
    37   
    38 definition
    39   csucc         :: "i=>i"  where
    40     --{*needed because @{term "jump_cardinal(K)"} might not be the successor
    41         of @{term K}*}
    42     "csucc(K) == LEAST L. Card(L) & K<L"
    43 
    44 notation (xsymbols output)
    45   cadd  (infixl "\<oplus>" 65) and
    46   cmult  (infixl "\<otimes>" 70)
    47 
    48 notation (HTML output)
    49   cadd  (infixl "\<oplus>" 65) and
    50   cmult  (infixl "\<otimes>" 70)
    51 
    52 
    53 lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
    54 apply (rule CardI) 
    55  apply (simp add: Card_is_Ord) 
    56 apply (clarify dest!: ltD)
    57 apply (drule bspec, assumption) 
    58 apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
    59 apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
    60 apply (drule lesspoll_trans1, assumption) 
    61 apply (subgoal_tac "B \<lesssim> \<Union>A")
    62  apply (drule lesspoll_trans1, assumption, blast) 
    63 apply (blast intro: subset_imp_lepoll) 
    64 done
    65 
    66 lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
    67 by (blast intro: Card_Union) 
    68 
    69 lemma Card_OUN [simp,intro,TC]:
    70      "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
    71 by (simp add: OUnion_def Card_0) 
    72 
    73 lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
    74 apply (unfold lesspoll_def)
    75 apply (rule conjI)
    76 apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
    77 apply (rule notI)
    78 apply (erule eqpollE)
    79 apply (rule succ_lepoll_natE)
    80 apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
    81                     lepoll_trans, assumption) 
    82 done
    83 
    84 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
    85 apply (unfold lesspoll_def)
    86 apply (simp add: Card_iff_initial)
    87 apply (fast intro!: le_imp_lepoll ltI leI)
    88 done
    89 
    90 lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
    91 apply (unfold lesspoll_def)
    92 apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
    93             intro!: eqpollI elim: notE 
    94             elim!: eqpollE lepoll_trans)
    95 done
    96 
    97 
    98 subsection{*Cardinal addition*}
    99 
   100 text{*Note: Could omit proving the algebraic laws for cardinal addition and
   101 multiplication.  On finite cardinals these operations coincide with
   102 addition and multiplication of natural numbers; on infinite cardinals they
   103 coincide with union (maximum).  Either way we get most laws for free.*}
   104 
   105 subsubsection{*Cardinal addition is commutative*}
   106 
   107 lemma sum_commute_eqpoll: "A+B \<approx> B+A"
   108 apply (unfold eqpoll_def)
   109 apply (rule exI)
   110 apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
   111 apply auto
   112 done
   113 
   114 lemma cadd_commute: "i |+| j = j |+| i"
   115 apply (unfold cadd_def)
   116 apply (rule sum_commute_eqpoll [THEN cardinal_cong])
   117 done
   118 
   119 subsubsection{*Cardinal addition is associative*}
   120 
   121 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
   122 apply (unfold eqpoll_def)
   123 apply (rule exI)
   124 apply (rule sum_assoc_bij)
   125 done
   126 
   127 (*Unconditional version requires AC*)
   128 lemma well_ord_cadd_assoc: 
   129     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   130      ==> (i |+| j) |+| k = i |+| (j |+| k)"
   131 apply (unfold cadd_def)
   132 apply (rule cardinal_cong)
   133 apply (rule eqpoll_trans)
   134  apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   135  apply (blast intro: well_ord_radd ) 
   136 apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
   137 apply (rule eqpoll_sym)
   138 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   139 apply (blast intro: well_ord_radd ) 
   140 done
   141 
   142 subsubsection{*0 is the identity for addition*}
   143 
   144 lemma sum_0_eqpoll: "0+A \<approx> A"
   145 apply (unfold eqpoll_def)
   146 apply (rule exI)
   147 apply (rule bij_0_sum)
   148 done
   149 
   150 lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
   151 apply (unfold cadd_def)
   152 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   153 done
   154 
   155 subsubsection{*Addition by another cardinal*}
   156 
   157 lemma sum_lepoll_self: "A \<lesssim> A+B"
   158 apply (unfold lepoll_def inj_def)
   159 apply (rule_tac x = "lam x:A. Inl (x) " in exI)
   160 apply simp
   161 done
   162 
   163 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   164 
   165 lemma cadd_le_self: 
   166     "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
   167 apply (unfold cadd_def)
   168 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
   169        assumption)
   170 apply (rule_tac [2] sum_lepoll_self)
   171 apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
   172 done
   173 
   174 subsubsection{*Monotonicity of addition*}
   175 
   176 lemma sum_lepoll_mono: 
   177      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
   178 apply (unfold lepoll_def)
   179 apply (elim exE)
   180 apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   181 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
   182        in lam_injective)
   183 apply (typecheck add: inj_is_fun, auto)
   184 done
   185 
   186 lemma cadd_le_mono:
   187     "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
   188 apply (unfold cadd_def)
   189 apply (safe dest!: le_subset_iff [THEN iffD1])
   190 apply (rule well_ord_lepoll_imp_Card_le)
   191 apply (blast intro: well_ord_radd well_ord_Memrel)
   192 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
   193 done
   194 
   195 subsubsection{*Addition of finite cardinals is "ordinary" addition*}
   196 
   197 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
   198 apply (unfold eqpoll_def)
   199 apply (rule exI)
   200 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
   201             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
   202    apply simp_all
   203 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
   204 done
   205 
   206 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
   207 (*Unconditional version requires AC*)
   208 lemma cadd_succ_lemma:
   209     "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
   210 apply (unfold cadd_def)
   211 apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
   212 apply (rule succ_eqpoll_cong [THEN cardinal_cong])
   213 apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
   214 apply (blast intro: well_ord_radd well_ord_Memrel)
   215 done
   216 
   217 lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
   218 apply (induct_tac m)
   219 apply (simp add: nat_into_Card [THEN cadd_0])
   220 apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
   221 done
   222 
   223 
   224 subsection{*Cardinal multiplication*}
   225 
   226 subsubsection{*Cardinal multiplication is commutative*}
   227 
   228 (*Easier to prove the two directions separately*)
   229 lemma prod_commute_eqpoll: "A*B \<approx> B*A"
   230 apply (unfold eqpoll_def)
   231 apply (rule exI)
   232 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
   233        auto) 
   234 done
   235 
   236 lemma cmult_commute: "i |*| j = j |*| i"
   237 apply (unfold cmult_def)
   238 apply (rule prod_commute_eqpoll [THEN cardinal_cong])
   239 done
   240 
   241 subsubsection{*Cardinal multiplication is associative*}
   242 
   243 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
   244 apply (unfold eqpoll_def)
   245 apply (rule exI)
   246 apply (rule prod_assoc_bij)
   247 done
   248 
   249 (*Unconditional version requires AC*)
   250 lemma well_ord_cmult_assoc:
   251     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   252      ==> (i |*| j) |*| k = i |*| (j |*| k)"
   253 apply (unfold cmult_def)
   254 apply (rule cardinal_cong)
   255 apply (rule eqpoll_trans) 
   256  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   257  apply (blast intro: well_ord_rmult)
   258 apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
   259 apply (rule eqpoll_sym) 
   260 apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   261 apply (blast intro: well_ord_rmult)
   262 done
   263 
   264 subsubsection{*Cardinal multiplication distributes over addition*}
   265 
   266 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
   267 apply (unfold eqpoll_def)
   268 apply (rule exI)
   269 apply (rule sum_prod_distrib_bij)
   270 done
   271 
   272 lemma well_ord_cadd_cmult_distrib:
   273     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   274      ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
   275 apply (unfold cadd_def cmult_def)
   276 apply (rule cardinal_cong)
   277 apply (rule eqpoll_trans) 
   278  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   279 apply (blast intro: well_ord_radd)
   280 apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
   281 apply (rule eqpoll_sym) 
   282 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
   283                                 well_ord_cardinal_eqpoll])
   284 apply (blast intro: well_ord_rmult)+
   285 done
   286 
   287 subsubsection{*Multiplication by 0 yields 0*}
   288 
   289 lemma prod_0_eqpoll: "0*A \<approx> 0"
   290 apply (unfold eqpoll_def)
   291 apply (rule exI)
   292 apply (rule lam_bijective, safe)
   293 done
   294 
   295 lemma cmult_0 [simp]: "0 |*| i = 0"
   296 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
   297 
   298 subsubsection{*1 is the identity for multiplication*}
   299 
   300 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
   301 apply (unfold eqpoll_def)
   302 apply (rule exI)
   303 apply (rule singleton_prod_bij [THEN bij_converse_bij])
   304 done
   305 
   306 lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
   307 apply (unfold cmult_def succ_def)
   308 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   309 done
   310 
   311 subsection{*Some inequalities for multiplication*}
   312 
   313 lemma prod_square_lepoll: "A \<lesssim> A*A"
   314 apply (unfold lepoll_def inj_def)
   315 apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
   316 done
   317 
   318 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
   319 lemma cmult_square_le: "Card(K) ==> K le K |*| K"
   320 apply (unfold cmult_def)
   321 apply (rule le_trans)
   322 apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
   323 apply (rule_tac [3] prod_square_lepoll)
   324 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
   325 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   326 done
   327 
   328 subsubsection{*Multiplication by a non-zero cardinal*}
   329 
   330 lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
   331 apply (unfold lepoll_def inj_def)
   332 apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
   333 done
   334 
   335 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   336 lemma cmult_le_self:
   337     "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
   338 apply (unfold cmult_def)
   339 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
   340   apply assumption
   341  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   342 apply (blast intro: prod_lepoll_self ltD)
   343 done
   344 
   345 subsubsection{*Monotonicity of multiplication*}
   346 
   347 lemma prod_lepoll_mono:
   348      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
   349 apply (unfold lepoll_def)
   350 apply (elim exE)
   351 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
   352 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
   353        in lam_injective)
   354 apply (typecheck add: inj_is_fun, auto)
   355 done
   356 
   357 lemma cmult_le_mono:
   358     "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
   359 apply (unfold cmult_def)
   360 apply (safe dest!: le_subset_iff [THEN iffD1])
   361 apply (rule well_ord_lepoll_imp_Card_le)
   362  apply (blast intro: well_ord_rmult well_ord_Memrel)
   363 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
   364 done
   365 
   366 subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
   367 
   368 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
   369 apply (unfold eqpoll_def)
   370 apply (rule exI)
   371 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
   372             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
   373 apply safe
   374 apply (simp_all add: succI2 if_type mem_imp_not_eq)
   375 done
   376 
   377 (*Unconditional version requires AC*)
   378 lemma cmult_succ_lemma:
   379     "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
   380 apply (unfold cmult_def cadd_def)
   381 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
   382 apply (rule cardinal_cong [symmetric])
   383 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   384 apply (blast intro: well_ord_rmult well_ord_Memrel)
   385 done
   386 
   387 lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
   388 apply (induct_tac m)
   389 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
   390 done
   391 
   392 lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
   393 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
   394 
   395 lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
   396 apply (rule lepoll_trans) 
   397 apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
   398 apply (erule prod_lepoll_mono) 
   399 apply (rule lepoll_refl) 
   400 done
   401 
   402 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
   403 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
   404 
   405 
   406 subsection{*Infinite Cardinals are Limit Ordinals*}
   407 
   408 (*This proof is modelled upon one assuming nat<=A, with injection
   409   lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
   410   and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
   411   If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
   412 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
   413 apply (unfold lepoll_def)
   414 apply (erule exE)
   415 apply (rule_tac x = 
   416           "lam z:cons (u,A).
   417              if z=u then f`0 
   418              else if z: range (f) then f`succ (converse (f) `z) else z" 
   419        in exI)
   420 apply (rule_tac d =
   421           "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
   422                               else y" 
   423        in lam_injective)
   424 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
   425 apply (simp add: inj_is_fun [THEN apply_rangeI]
   426                  inj_converse_fun [THEN apply_rangeI]
   427                  inj_converse_fun [THEN apply_funtype])
   428 done
   429 
   430 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
   431 apply (erule nat_cons_lepoll [THEN eqpollI])
   432 apply (rule subset_consI [THEN subset_imp_lepoll])
   433 done
   434 
   435 (*Specialized version required below*)
   436 lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
   437 apply (unfold succ_def)
   438 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
   439 done
   440 
   441 lemma InfCard_nat: "InfCard(nat)"
   442 apply (unfold InfCard_def)
   443 apply (blast intro: Card_nat le_refl Card_is_Ord)
   444 done
   445 
   446 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
   447 apply (unfold InfCard_def)
   448 apply (erule conjunct1)
   449 done
   450 
   451 lemma InfCard_Un:
   452     "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
   453 apply (unfold InfCard_def)
   454 apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
   455 done
   456 
   457 (*Kunen's Lemma 10.11*)
   458 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
   459 apply (unfold InfCard_def)
   460 apply (erule conjE)
   461 apply (frule Card_is_Ord)
   462 apply (rule ltI [THEN non_succ_LimitI])
   463 apply (erule le_imp_subset [THEN subsetD])
   464 apply (safe dest!: Limit_nat [THEN Limit_le_succD])
   465 apply (unfold Card_def)
   466 apply (drule trans)
   467 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
   468 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
   469 apply (rule le_eqI, assumption)
   470 apply (rule Ord_cardinal)
   471 done
   472 
   473 
   474 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
   475 
   476 (*A general fact about ordermap*)
   477 lemma ordermap_eqpoll_pred:
   478     "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
   479 apply (unfold eqpoll_def)
   480 apply (rule exI)
   481 apply (simp add: ordermap_eq_image well_ord_is_wf)
   482 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
   483                            THEN bij_converse_bij])
   484 apply (rule pred_subset)
   485 done
   486 
   487 subsubsection{*Establishing the well-ordering*}
   488 
   489 lemma csquare_lam_inj:
   490      "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
   491 apply (unfold inj_def)
   492 apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
   493 done
   494 
   495 lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
   496 apply (unfold csquare_rel_def)
   497 apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
   498 apply (blast intro: well_ord_rmult well_ord_Memrel)
   499 done
   500 
   501 subsubsection{*Characterising initial segments of the well-ordering*}
   502 
   503 lemma csquareD:
   504  "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
   505 apply (unfold csquare_rel_def)
   506 apply (erule rev_mp)
   507 apply (elim ltE)
   508 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   509 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
   510 apply (simp_all add: lt_def succI2)
   511 done
   512 
   513 lemma pred_csquare_subset: 
   514     "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
   515 apply (unfold Order.pred_def)
   516 apply (safe del: SigmaI succCI)
   517 apply (erule csquareD [THEN conjE])
   518 apply (unfold lt_def, auto) 
   519 done
   520 
   521 lemma csquare_ltI:
   522  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
   523 apply (unfold csquare_rel_def)
   524 apply (subgoal_tac "x<K & y<K")
   525  prefer 2 apply (blast intro: lt_trans) 
   526 apply (elim ltE)
   527 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   528 done
   529 
   530 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
   531 lemma csquare_or_eqI:
   532  "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
   533 apply (unfold csquare_rel_def)
   534 apply (subgoal_tac "x<K & y<K")
   535  prefer 2 apply (blast intro: lt_trans1) 
   536 apply (elim ltE)
   537 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   538 apply (elim succE)
   539 apply (simp_all add: subset_Un_iff [THEN iff_sym] 
   540                      subset_Un_iff2 [THEN iff_sym] OrdmemD)
   541 done
   542 
   543 subsubsection{*The cardinality of initial segments*}
   544 
   545 lemma ordermap_z_lt:
   546       "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
   547           ordermap(K*K, csquare_rel(K)) ` <x,y> <
   548           ordermap(K*K, csquare_rel(K)) ` <z,z>"
   549 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
   550 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
   551                               Limit_is_Ord [THEN well_ord_csquare], clarify) 
   552 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
   553 apply (erule_tac [4] well_ord_is_wf)
   554 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
   555 done
   556 
   557 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
   558 lemma ordermap_csquare_le:
   559   "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
   560    ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
   561 apply (unfold cmult_def)
   562 apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
   563 apply (rule Ord_cardinal [THEN well_ord_Memrel])+
   564 apply (subgoal_tac "z<K")
   565  prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
   566 apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
   567        assumption+)
   568 apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   569 apply (erule Limit_is_Ord [THEN well_ord_csquare])
   570 apply (blast intro: ltD)
   571 apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
   572             assumption)
   573 apply (elim ltE)
   574 apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
   575 apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
   576 done
   577 
   578 (*Kunen: "... so the order type <= K" *)
   579 lemma ordertype_csquare_le:
   580      "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
   581       ==> ordertype(K*K, csquare_rel(K)) le K"
   582 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   583 apply (rule all_lt_imp_le, assumption)
   584 apply (erule well_ord_csquare [THEN Ord_ordertype])
   585 apply (rule Card_lt_imp_lt)
   586 apply (erule_tac [3] InfCard_is_Card)
   587 apply (erule_tac [2] ltE)
   588 apply (simp add: ordertype_unfold)
   589 apply (safe elim!: ltE)
   590 apply (subgoal_tac "Ord (xa) & Ord (ya)")
   591  prefer 2 apply (blast intro: Ord_in_Ord, clarify)
   592 (*??WHAT A MESS!*)  
   593 apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
   594        (assumption | rule refl | erule ltI)+) 
   595 apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
   596        simp_all add: Ord_Un Ord_nat)
   597 prefer 2 (*case nat le (xa Un ya) *)
   598  apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
   599                   le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
   600                 ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
   601 (*the finite case: xa Un ya < nat *)
   602 apply (rule_tac j = nat in lt_trans2)
   603  apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
   604                   nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
   605 apply (simp add: InfCard_def)
   606 done
   607 
   608 (*Main result: Kunen's Theorem 10.12*)
   609 lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
   610 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   611 apply (erule rev_mp)
   612 apply (erule_tac i=K in trans_induct) 
   613 apply (rule impI)
   614 apply (rule le_anti_sym)
   615 apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
   616 apply (rule ordertype_csquare_le [THEN [2] le_trans])
   617 apply (simp add: cmult_def Ord_cardinal_le   
   618                  well_ord_csquare [THEN Ord_ordertype]
   619                  well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
   620                                    THEN cardinal_cong], assumption+)
   621 done
   622 
   623 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
   624 lemma well_ord_InfCard_square_eq:
   625      "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
   626 apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
   627 apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
   628 apply (rule well_ord_cardinal_eqE)
   629 apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
   630 apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
   631 done
   632 
   633 lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
   634 apply (rule well_ord_InfCard_square_eq)  
   635  apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
   636 apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
   637 done
   638 
   639 lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
   640 by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
   641 
   642 subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
   643 
   644 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
   645 apply (rule le_anti_sym)
   646  prefer 2
   647  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
   648 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   649 apply (rule cmult_le_mono [THEN le_trans], assumption+)
   650 apply (simp add: InfCard_csquare_eq)
   651 done
   652 
   653 (*Corollary 10.13 (1), for cardinal multiplication*)
   654 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
   655 apply (rule_tac i = K and j = L in Ord_linear_le)
   656 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   657 apply (rule cmult_commute [THEN ssubst])
   658 apply (rule Un_commute [THEN ssubst])
   659 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
   660                      subset_Un_iff2 [THEN iffD1] le_imp_subset)
   661 done
   662 
   663 lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
   664 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
   665 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
   666 done
   667 
   668 (*Corollary 10.13 (1), for cardinal addition*)
   669 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
   670 apply (rule le_anti_sym)
   671  prefer 2
   672  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
   673 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   674 apply (rule cadd_le_mono [THEN le_trans], assumption+)
   675 apply (simp add: InfCard_cdouble_eq)
   676 done
   677 
   678 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
   679 apply (rule_tac i = K and j = L in Ord_linear_le)
   680 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   681 apply (rule cadd_commute [THEN ssubst])
   682 apply (rule Un_commute [THEN ssubst])
   683 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
   684 done
   685 
   686 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
   687   of all n-tuples of elements of K.  A better version for the Isabelle theory
   688   might be  InfCard(K) ==> |list(K)| = K.
   689 *)
   690 
   691 subsection{*For Every Cardinal Number There Exists A Greater One}
   692 
   693 text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
   694 
   695 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
   696 apply (unfold jump_cardinal_def)
   697 apply (rule Ord_is_Transset [THEN [2] OrdI])
   698  prefer 2 apply (blast intro!: Ord_ordertype)
   699 apply (unfold Transset_def)
   700 apply (safe del: subsetI)
   701 apply (simp add: ordertype_pred_unfold, safe)
   702 apply (rule UN_I)
   703 apply (rule_tac [2] ReplaceI)
   704    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
   705 done
   706 
   707 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
   708 lemma jump_cardinal_iff:
   709      "i : jump_cardinal(K) <->
   710       (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
   711 apply (unfold jump_cardinal_def)
   712 apply (blast del: subsetI) 
   713 done
   714 
   715 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
   716 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
   717 apply (rule Ord_jump_cardinal [THEN [2] ltI])
   718 apply (rule jump_cardinal_iff [THEN iffD2])
   719 apply (rule_tac x="Memrel(K)" in exI)
   720 apply (rule_tac x=K in exI)  
   721 apply (simp add: ordertype_Memrel well_ord_Memrel)
   722 apply (simp add: Memrel_def subset_iff)
   723 done
   724 
   725 (*The proof by contradiction: the bijection f yields a wellordering of X
   726   whose ordertype is jump_cardinal(K).  *)
   727 lemma Card_jump_cardinal_lemma:
   728      "[| well_ord(X,r);  r <= K * K;  X <= K;
   729          f : bij(ordertype(X,r), jump_cardinal(K)) |]
   730       ==> jump_cardinal(K) : jump_cardinal(K)"
   731 apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
   732  prefer 2 apply (blast intro: comp_bij ordermap_bij)
   733 apply (rule jump_cardinal_iff [THEN iffD2])
   734 apply (intro exI conjI)
   735 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
   736 apply (erule bij_is_inj [THEN well_ord_rvimage])
   737 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
   738 apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
   739                  ordertype_Memrel Ord_jump_cardinal)
   740 done
   741 
   742 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
   743 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
   744 apply (rule Ord_jump_cardinal [THEN CardI])
   745 apply (unfold eqpoll_def)
   746 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
   747 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
   748 done
   749 
   750 subsection{*Basic Properties of Successor Cardinals*}
   751 
   752 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
   753 apply (unfold csucc_def)
   754 apply (rule LeastI)
   755 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
   756 done
   757 
   758 lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
   759 
   760 lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
   761 
   762 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
   763 by (blast intro: Ord_0_le lt_csucc lt_trans1)
   764 
   765 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
   766 apply (unfold csucc_def)
   767 apply (rule Least_le)
   768 apply (blast intro: Card_is_Ord)+
   769 done
   770 
   771 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
   772 apply (rule iffI)
   773 apply (rule_tac [2] Card_lt_imp_lt)
   774 apply (erule_tac [2] lt_trans1)
   775 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
   776 apply (rule notI [THEN not_lt_imp_le])
   777 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
   778 apply (rule Ord_cardinal_le [THEN lt_trans1])
   779 apply (simp_all add: Ord_cardinal Card_is_Ord) 
   780 done
   781 
   782 lemma Card_lt_csucc_iff:
   783      "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
   784 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
   785 
   786 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
   787 by (simp add: InfCard_def Card_csucc Card_is_Ord 
   788               lt_csucc [THEN leI, THEN [2] le_trans])
   789 
   790 
   791 subsubsection{*Removing elements from a finite set decreases its cardinality*}
   792 
   793 lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
   794 apply (erule Fin_induct)
   795 apply (simp add: lepoll_0_iff)
   796 apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
   797 apply simp
   798 apply (blast dest!: cons_lepoll_consD, blast)
   799 done
   800 
   801 lemma Finite_imp_cardinal_cons [simp]:
   802      "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
   803 apply (unfold cardinal_def)
   804 apply (rule Least_equality)
   805 apply (fold cardinal_def)
   806 apply (simp add: succ_def)
   807 apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
   808              elim!: mem_irrefl  dest!: Finite_imp_well_ord)
   809 apply (blast intro: Card_cardinal Card_is_Ord)
   810 apply (rule notI)
   811 apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
   812        assumption, assumption)
   813 apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   814 apply (erule le_imp_lepoll [THEN lepoll_trans])
   815 apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
   816              dest!: Finite_imp_well_ord)
   817 done
   818 
   819 
   820 lemma Finite_imp_succ_cardinal_Diff:
   821      "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
   822 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
   823 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
   824 apply (simp add: cons_Diff)
   825 done
   826 
   827 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
   828 apply (rule succ_leE)
   829 apply (simp add: Finite_imp_succ_cardinal_Diff)
   830 done
   831 
   832 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
   833 apply (erule Finite_induct)
   834 apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
   835 done
   836 
   837 lemma card_Un_Int:
   838      "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
   839 apply (erule Finite_induct, simp) 
   840 apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
   841 done
   842 
   843 lemma card_Un_disjoint: 
   844      "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
   845 by (simp add: Finite_Un card_Un_Int)
   846 
   847 lemma card_partition [rule_format]:
   848      "Finite(C) ==>  
   849         Finite (\<Union> C) -->  
   850         (\<forall>c\<in>C. |c| = k) -->   
   851         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
   852         k #* |C| = |\<Union> C|"
   853 apply (erule Finite_induct, auto)
   854 apply (subgoal_tac " x \<inter> \<Union>B = 0")  
   855 apply (auto simp add: card_Un_disjoint Finite_Union
   856        subset_Finite [of _ "\<Union> (cons(x,F))"])
   857 done
   858 
   859 
   860 subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
   861 
   862 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
   863 
   864 lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
   865 apply (rule eqpoll_trans)
   866 apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
   867 apply (erule nat_implies_well_ord)+
   868 apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
   869 done
   870 
   871 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
   872 apply (erule trans_induct3, auto)
   873 apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
   874 done
   875 
   876 lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
   877 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
   878 
   879 lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
   880 apply (rule succ_inject)
   881 apply (rule_tac b = "|A|" in trans)
   882  apply (simp add: Finite_imp_succ_cardinal_Diff)
   883 apply (subgoal_tac "1 \<lesssim> A")
   884  prefer 2 apply (blast intro: not_0_is_lepoll_1)
   885 apply (frule Finite_imp_well_ord, clarify)
   886 apply (drule well_ord_lepoll_imp_Card_le)
   887  apply (auto simp add: cardinal_1)
   888 apply (rule trans)
   889  apply (rule_tac [2] diff_succ)
   890   apply (auto simp add: Finite_cardinal_in_nat)
   891 done
   892 
   893 lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
   894      "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
   895 apply (erule Finite_induct, auto)
   896 apply (case_tac "Finite (A)")
   897  apply (subgoal_tac [2] "Finite (cons (x, B))")
   898   apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
   899    apply (auto simp add: Finite_0 Finite_cons)
   900 apply (subgoal_tac "|B|<|A|")
   901  prefer 2 apply (blast intro: lt_trans Ord_cardinal)
   902 apply (case_tac "x:A")
   903  apply (subgoal_tac [2] "A - cons (x, B) = A - B")
   904   apply auto
   905 apply (subgoal_tac "|A| le |cons (x, B) |")
   906  prefer 2
   907  apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
   908               intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
   909 apply (auto simp add: Finite_imp_cardinal_cons)
   910 apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
   911 apply (blast intro: lt_trans)
   912 done
   913 
   914 
   915 ML{*
   916 val InfCard_def = thm "InfCard_def"
   917 val cmult_def = thm "cmult_def"
   918 val cadd_def = thm "cadd_def"
   919 val jump_cardinal_def = thm "jump_cardinal_def"
   920 val csucc_def = thm "csucc_def"
   921 
   922 val sum_commute_eqpoll = thm "sum_commute_eqpoll";
   923 val cadd_commute = thm "cadd_commute";
   924 val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
   925 val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
   926 val sum_0_eqpoll = thm "sum_0_eqpoll";
   927 val cadd_0 = thm "cadd_0";
   928 val sum_lepoll_self = thm "sum_lepoll_self";
   929 val cadd_le_self = thm "cadd_le_self";
   930 val sum_lepoll_mono = thm "sum_lepoll_mono";
   931 val cadd_le_mono = thm "cadd_le_mono";
   932 val eq_imp_not_mem = thm "eq_imp_not_mem";
   933 val sum_succ_eqpoll = thm "sum_succ_eqpoll";
   934 val nat_cadd_eq_add = thm "nat_cadd_eq_add";
   935 val prod_commute_eqpoll = thm "prod_commute_eqpoll";
   936 val cmult_commute = thm "cmult_commute";
   937 val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
   938 val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
   939 val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
   940 val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
   941 val prod_0_eqpoll = thm "prod_0_eqpoll";
   942 val cmult_0 = thm "cmult_0";
   943 val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
   944 val cmult_1 = thm "cmult_1";
   945 val prod_lepoll_self = thm "prod_lepoll_self";
   946 val cmult_le_self = thm "cmult_le_self";
   947 val prod_lepoll_mono = thm "prod_lepoll_mono";
   948 val cmult_le_mono = thm "cmult_le_mono";
   949 val prod_succ_eqpoll = thm "prod_succ_eqpoll";
   950 val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
   951 val cmult_2 = thm "cmult_2";
   952 val sum_lepoll_prod = thm "sum_lepoll_prod";
   953 val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
   954 val nat_cons_lepoll = thm "nat_cons_lepoll";
   955 val nat_cons_eqpoll = thm "nat_cons_eqpoll";
   956 val nat_succ_eqpoll = thm "nat_succ_eqpoll";
   957 val InfCard_nat = thm "InfCard_nat";
   958 val InfCard_is_Card = thm "InfCard_is_Card";
   959 val InfCard_Un = thm "InfCard_Un";
   960 val InfCard_is_Limit = thm "InfCard_is_Limit";
   961 val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
   962 val ordermap_z_lt = thm "ordermap_z_lt";
   963 val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
   964 val InfCard_cmult_eq = thm "InfCard_cmult_eq";
   965 val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
   966 val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
   967 val InfCard_cadd_eq = thm "InfCard_cadd_eq";
   968 val Ord_jump_cardinal = thm "Ord_jump_cardinal";
   969 val jump_cardinal_iff = thm "jump_cardinal_iff";
   970 val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
   971 val Card_jump_cardinal = thm "Card_jump_cardinal";
   972 val csucc_basic = thm "csucc_basic";
   973 val Card_csucc = thm "Card_csucc";
   974 val lt_csucc = thm "lt_csucc";
   975 val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
   976 val csucc_le = thm "csucc_le";
   977 val lt_csucc_iff = thm "lt_csucc_iff";
   978 val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
   979 val InfCard_csucc = thm "InfCard_csucc";
   980 val Finite_into_Fin = thm "Finite_into_Fin";
   981 val Fin_into_Finite = thm "Fin_into_Finite";
   982 val Finite_Fin_iff = thm "Finite_Fin_iff";
   983 val Finite_Un = thm "Finite_Un";
   984 val Finite_Union = thm "Finite_Union";
   985 val Finite_induct = thm "Finite_induct";
   986 val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
   987 val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
   988 val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
   989 val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
   990 val nat_implies_well_ord = thm "nat_implies_well_ord";
   991 val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
   992 val Diff_sing_Finite = thm "Diff_sing_Finite";
   993 val Diff_Finite = thm "Diff_Finite";
   994 val Ord_subset_natD = thm "Ord_subset_natD";
   995 val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
   996 val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
   997 val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
   998 val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
   999 *}
  1000 
  1001 end