src/ZF/CardinalArith.thy
author wenzelm
Sat Oct 17 14:43:18 2009 +0200 (2009-10-17)
changeset 32960 69916a850301
parent 27517 c055e1d49285
child 39159 0dec18004e75
permissions -rw-r--r--
eliminated hard tabulators, guessing at each author's individual tab-width;
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     1 (*  Title:      ZF/CardinalArith.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1994  University of Cambridge
     4 *)
     5 
     6 header{*Cardinal Arithmetic Without the Axiom of Choice*}
     7 
     8 theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin
     9 
    10 definition
    11   InfCard       :: "i=>o"  where
    12     "InfCard(i) == Card(i) & nat le i"
    13 
    14 definition
    15   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where
    16     "i |*| j == |i*j|"
    17   
    18 definition
    19   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where
    20     "i |+| j == |i+j|"
    21 
    22 definition
    23   csquare_rel   :: "i=>i"  where
    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 definition
    30   jump_cardinal :: "i=>i"  where
    31     --{*This def is more complex than Kunen's but it more easily proved to
    32         be a cardinal*}
    33     "jump_cardinal(K) ==   
    34          \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
    35   
    36 definition
    37   csucc         :: "i=>i"  where
    38     --{*needed because @{term "jump_cardinal(K)"} might not be the successor
    39         of @{term K}*}
    40     "csucc(K) == LEAST L. Card(L) & K<L"
    41 
    42 notation (xsymbols output)
    43   cadd  (infixl "\<oplus>" 65) and
    44   cmult  (infixl "\<otimes>" 70)
    45 
    46 notation (HTML output)
    47   cadd  (infixl "\<oplus>" 65) and
    48   cmult  (infixl "\<otimes>" 70)
    49 
    50 
    51 lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))"
    52 apply (rule CardI) 
    53  apply (simp add: Card_is_Ord) 
    54 apply (clarify dest!: ltD)
    55 apply (drule bspec, assumption) 
    56 apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) 
    57 apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
    58 apply (drule lesspoll_trans1, assumption) 
    59 apply (subgoal_tac "B \<lesssim> \<Union>A")
    60  apply (drule lesspoll_trans1, assumption, blast) 
    61 apply (blast intro: subset_imp_lepoll) 
    62 done
    63 
    64 lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))" 
    65 by (blast intro: Card_Union) 
    66 
    67 lemma Card_OUN [simp,intro,TC]:
    68      "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"
    69 by (simp add: OUnion_def Card_0) 
    70 
    71 lemma n_lesspoll_nat: "n \<in> nat ==> n \<prec> nat"
    72 apply (unfold lesspoll_def)
    73 apply (rule conjI)
    74 apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat)
    75 apply (rule notI)
    76 apply (erule eqpollE)
    77 apply (rule succ_lepoll_natE)
    78 apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] 
    79                     lepoll_trans, assumption) 
    80 done
    81 
    82 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"
    83 apply (unfold lesspoll_def)
    84 apply (simp add: Card_iff_initial)
    85 apply (fast intro!: le_imp_lepoll ltI leI)
    86 done
    87 
    88 lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C \<noteq> 0"
    89 apply (unfold lesspoll_def)
    90 apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll]
    91             intro!: eqpollI elim: notE 
    92             elim!: eqpollE lepoll_trans)
    93 done
    94 
    95 
    96 subsection{*Cardinal addition*}
    97 
    98 text{*Note: Could omit proving the algebraic laws for cardinal addition and
    99 multiplication.  On finite cardinals these operations coincide with
   100 addition and multiplication of natural numbers; on infinite cardinals they
   101 coincide with union (maximum).  Either way we get most laws for free.*}
   102 
   103 subsubsection{*Cardinal addition is commutative*}
   104 
   105 lemma sum_commute_eqpoll: "A+B \<approx> B+A"
   106 apply (unfold eqpoll_def)
   107 apply (rule exI)
   108 apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective)
   109 apply auto
   110 done
   111 
   112 lemma cadd_commute: "i |+| j = j |+| i"
   113 apply (unfold cadd_def)
   114 apply (rule sum_commute_eqpoll [THEN cardinal_cong])
   115 done
   116 
   117 subsubsection{*Cardinal addition is associative*}
   118 
   119 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"
   120 apply (unfold eqpoll_def)
   121 apply (rule exI)
   122 apply (rule sum_assoc_bij)
   123 done
   124 
   125 (*Unconditional version requires AC*)
   126 lemma well_ord_cadd_assoc: 
   127     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   128      ==> (i |+| j) |+| k = i |+| (j |+| k)"
   129 apply (unfold cadd_def)
   130 apply (rule cardinal_cong)
   131 apply (rule eqpoll_trans)
   132  apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   133  apply (blast intro: well_ord_radd ) 
   134 apply (rule sum_assoc_eqpoll [THEN eqpoll_trans])
   135 apply (rule eqpoll_sym)
   136 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   137 apply (blast intro: well_ord_radd ) 
   138 done
   139 
   140 subsubsection{*0 is the identity for addition*}
   141 
   142 lemma sum_0_eqpoll: "0+A \<approx> A"
   143 apply (unfold eqpoll_def)
   144 apply (rule exI)
   145 apply (rule bij_0_sum)
   146 done
   147 
   148 lemma cadd_0 [simp]: "Card(K) ==> 0 |+| K = K"
   149 apply (unfold cadd_def)
   150 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   151 done
   152 
   153 subsubsection{*Addition by another cardinal*}
   154 
   155 lemma sum_lepoll_self: "A \<lesssim> A+B"
   156 apply (unfold lepoll_def inj_def)
   157 apply (rule_tac x = "lam x:A. Inl (x) " in exI)
   158 apply simp
   159 done
   160 
   161 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   162 
   163 lemma cadd_le_self: 
   164     "[| Card(K);  Ord(L) |] ==> K le (K |+| L)"
   165 apply (unfold cadd_def)
   166 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],
   167        assumption)
   168 apply (rule_tac [2] sum_lepoll_self)
   169 apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)
   170 done
   171 
   172 subsubsection{*Monotonicity of addition*}
   173 
   174 lemma sum_lepoll_mono: 
   175      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"
   176 apply (unfold lepoll_def)
   177 apply (elim exE)
   178 apply (rule_tac x = "lam z:A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)
   179 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"
   180        in lam_injective)
   181 apply (typecheck add: inj_is_fun, auto)
   182 done
   183 
   184 lemma cadd_le_mono:
   185     "[| K' le K;  L' le L |] ==> (K' |+| L') le (K |+| L)"
   186 apply (unfold cadd_def)
   187 apply (safe dest!: le_subset_iff [THEN iffD1])
   188 apply (rule well_ord_lepoll_imp_Card_le)
   189 apply (blast intro: well_ord_radd well_ord_Memrel)
   190 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)
   191 done
   192 
   193 subsubsection{*Addition of finite cardinals is "ordinary" addition*}
   194 
   195 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"
   196 apply (unfold eqpoll_def)
   197 apply (rule exI)
   198 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z" 
   199             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)
   200    apply simp_all
   201 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+
   202 done
   203 
   204 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)
   205 (*Unconditional version requires AC*)
   206 lemma cadd_succ_lemma:
   207     "[| Ord(m);  Ord(n) |] ==> succ(m) |+| n = |succ(m |+| n)|"
   208 apply (unfold cadd_def)
   209 apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])
   210 apply (rule succ_eqpoll_cong [THEN cardinal_cong])
   211 apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
   212 apply (blast intro: well_ord_radd well_ord_Memrel)
   213 done
   214 
   215 lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m |+| n = m#+n"
   216 apply (induct_tac m)
   217 apply (simp add: nat_into_Card [THEN cadd_0])
   218 apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])
   219 done
   220 
   221 
   222 subsection{*Cardinal multiplication*}
   223 
   224 subsubsection{*Cardinal multiplication is commutative*}
   225 
   226 (*Easier to prove the two directions separately*)
   227 lemma prod_commute_eqpoll: "A*B \<approx> B*A"
   228 apply (unfold eqpoll_def)
   229 apply (rule exI)
   230 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective, 
   231        auto) 
   232 done
   233 
   234 lemma cmult_commute: "i |*| j = j |*| i"
   235 apply (unfold cmult_def)
   236 apply (rule prod_commute_eqpoll [THEN cardinal_cong])
   237 done
   238 
   239 subsubsection{*Cardinal multiplication is associative*}
   240 
   241 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"
   242 apply (unfold eqpoll_def)
   243 apply (rule exI)
   244 apply (rule prod_assoc_bij)
   245 done
   246 
   247 (*Unconditional version requires AC*)
   248 lemma well_ord_cmult_assoc:
   249     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   250      ==> (i |*| j) |*| k = i |*| (j |*| k)"
   251 apply (unfold cmult_def)
   252 apply (rule cardinal_cong)
   253 apply (rule eqpoll_trans) 
   254  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   255  apply (blast intro: well_ord_rmult)
   256 apply (rule prod_assoc_eqpoll [THEN eqpoll_trans])
   257 apply (rule eqpoll_sym) 
   258 apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   259 apply (blast intro: well_ord_rmult)
   260 done
   261 
   262 subsubsection{*Cardinal multiplication distributes over addition*}
   263 
   264 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"
   265 apply (unfold eqpoll_def)
   266 apply (rule exI)
   267 apply (rule sum_prod_distrib_bij)
   268 done
   269 
   270 lemma well_ord_cadd_cmult_distrib:
   271     "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |]
   272      ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
   273 apply (unfold cadd_def cmult_def)
   274 apply (rule cardinal_cong)
   275 apply (rule eqpoll_trans) 
   276  apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl])
   277 apply (blast intro: well_ord_radd)
   278 apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans])
   279 apply (rule eqpoll_sym) 
   280 apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll 
   281                                 well_ord_cardinal_eqpoll])
   282 apply (blast intro: well_ord_rmult)+
   283 done
   284 
   285 subsubsection{*Multiplication by 0 yields 0*}
   286 
   287 lemma prod_0_eqpoll: "0*A \<approx> 0"
   288 apply (unfold eqpoll_def)
   289 apply (rule exI)
   290 apply (rule lam_bijective, safe)
   291 done
   292 
   293 lemma cmult_0 [simp]: "0 |*| i = 0"
   294 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])
   295 
   296 subsubsection{*1 is the identity for multiplication*}
   297 
   298 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"
   299 apply (unfold eqpoll_def)
   300 apply (rule exI)
   301 apply (rule singleton_prod_bij [THEN bij_converse_bij])
   302 done
   303 
   304 lemma cmult_1 [simp]: "Card(K) ==> 1 |*| K = K"
   305 apply (unfold cmult_def succ_def)
   306 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)
   307 done
   308 
   309 subsection{*Some inequalities for multiplication*}
   310 
   311 lemma prod_square_lepoll: "A \<lesssim> A*A"
   312 apply (unfold lepoll_def inj_def)
   313 apply (rule_tac x = "lam x:A. <x,x>" in exI, simp)
   314 done
   315 
   316 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)
   317 lemma cmult_square_le: "Card(K) ==> K le K |*| K"
   318 apply (unfold cmult_def)
   319 apply (rule le_trans)
   320 apply (rule_tac [2] well_ord_lepoll_imp_Card_le)
   321 apply (rule_tac [3] prod_square_lepoll)
   322 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)
   323 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   324 done
   325 
   326 subsubsection{*Multiplication by a non-zero cardinal*}
   327 
   328 lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"
   329 apply (unfold lepoll_def inj_def)
   330 apply (rule_tac x = "lam x:A. <x,b>" in exI, simp)
   331 done
   332 
   333 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)
   334 lemma cmult_le_self:
   335     "[| Card(K);  Ord(L);  0<L |] ==> K le (K |*| L)"
   336 apply (unfold cmult_def)
   337 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])
   338   apply assumption
   339  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)
   340 apply (blast intro: prod_lepoll_self ltD)
   341 done
   342 
   343 subsubsection{*Monotonicity of multiplication*}
   344 
   345 lemma prod_lepoll_mono:
   346      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"
   347 apply (unfold lepoll_def)
   348 apply (elim exE)
   349 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)
   350 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>" 
   351        in lam_injective)
   352 apply (typecheck add: inj_is_fun, auto)
   353 done
   354 
   355 lemma cmult_le_mono:
   356     "[| K' le K;  L' le L |] ==> (K' |*| L') le (K |*| L)"
   357 apply (unfold cmult_def)
   358 apply (safe dest!: le_subset_iff [THEN iffD1])
   359 apply (rule well_ord_lepoll_imp_Card_le)
   360  apply (blast intro: well_ord_rmult well_ord_Memrel)
   361 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)
   362 done
   363 
   364 subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}
   365 
   366 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"
   367 apply (unfold eqpoll_def)
   368 apply (rule exI)
   369 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"
   370             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)
   371 apply safe
   372 apply (simp_all add: succI2 if_type mem_imp_not_eq)
   373 done
   374 
   375 (*Unconditional version requires AC*)
   376 lemma cmult_succ_lemma:
   377     "[| Ord(m);  Ord(n) |] ==> succ(m) |*| n = n |+| (m |*| n)"
   378 apply (unfold cmult_def cadd_def)
   379 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])
   380 apply (rule cardinal_cong [symmetric])
   381 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])
   382 apply (blast intro: well_ord_rmult well_ord_Memrel)
   383 done
   384 
   385 lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m |*| n = m#*n"
   386 apply (induct_tac m)
   387 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)
   388 done
   389 
   390 lemma cmult_2: "Card(n) ==> 2 |*| n = n |+| n"
   391 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])
   392 
   393 lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B"
   394 apply (rule lepoll_trans) 
   395 apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) 
   396 apply (erule prod_lepoll_mono) 
   397 apply (rule lepoll_refl) 
   398 done
   399 
   400 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"
   401 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
   402 
   403 
   404 subsection{*Infinite Cardinals are Limit Ordinals*}
   405 
   406 (*This proof is modelled upon one assuming nat<=A, with injection
   407   lam z:cons(u,A). if z=u then 0 else if z : nat then succ(z) else z
   408   and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \
   409   If f: inj(nat,A) then range(f) behaves like the natural numbers.*)
   410 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"
   411 apply (unfold lepoll_def)
   412 apply (erule exE)
   413 apply (rule_tac x = 
   414           "lam z:cons (u,A).
   415              if z=u then f`0 
   416              else if z: range (f) then f`succ (converse (f) `z) else z" 
   417        in exI)
   418 apply (rule_tac d =
   419           "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y) 
   420                               else y" 
   421        in lam_injective)
   422 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)
   423 apply (simp add: inj_is_fun [THEN apply_rangeI]
   424                  inj_converse_fun [THEN apply_rangeI]
   425                  inj_converse_fun [THEN apply_funtype])
   426 done
   427 
   428 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"
   429 apply (erule nat_cons_lepoll [THEN eqpollI])
   430 apply (rule subset_consI [THEN subset_imp_lepoll])
   431 done
   432 
   433 (*Specialized version required below*)
   434 lemma nat_succ_eqpoll: "nat <= A ==> succ(A) \<approx> A"
   435 apply (unfold succ_def)
   436 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])
   437 done
   438 
   439 lemma InfCard_nat: "InfCard(nat)"
   440 apply (unfold InfCard_def)
   441 apply (blast intro: Card_nat le_refl Card_is_Ord)
   442 done
   443 
   444 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"
   445 apply (unfold InfCard_def)
   446 apply (erule conjunct1)
   447 done
   448 
   449 lemma InfCard_Un:
   450     "[| InfCard(K);  Card(L) |] ==> InfCard(K Un L)"
   451 apply (unfold InfCard_def)
   452 apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)
   453 done
   454 
   455 (*Kunen's Lemma 10.11*)
   456 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"
   457 apply (unfold InfCard_def)
   458 apply (erule conjE)
   459 apply (frule Card_is_Ord)
   460 apply (rule ltI [THEN non_succ_LimitI])
   461 apply (erule le_imp_subset [THEN subsetD])
   462 apply (safe dest!: Limit_nat [THEN Limit_le_succD])
   463 apply (unfold Card_def)
   464 apply (drule trans)
   465 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])
   466 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])
   467 apply (rule le_eqI, assumption)
   468 apply (rule Ord_cardinal)
   469 done
   470 
   471 
   472 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)
   473 
   474 (*A general fact about ordermap*)
   475 lemma ordermap_eqpoll_pred:
   476     "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"
   477 apply (unfold eqpoll_def)
   478 apply (rule exI)
   479 apply (simp add: ordermap_eq_image well_ord_is_wf)
   480 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij, 
   481                            THEN bij_converse_bij])
   482 apply (rule pred_subset)
   483 done
   484 
   485 subsubsection{*Establishing the well-ordering*}
   486 
   487 lemma csquare_lam_inj:
   488      "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)"
   489 apply (unfold inj_def)
   490 apply (force intro: lam_type Un_least_lt [THEN ltD] ltI)
   491 done
   492 
   493 lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))"
   494 apply (unfold csquare_rel_def)
   495 apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption)
   496 apply (blast intro: well_ord_rmult well_ord_Memrel)
   497 done
   498 
   499 subsubsection{*Characterising initial segments of the well-ordering*}
   500 
   501 lemma csquareD:
   502  "[| <<x,y>, <z,z>> : csquare_rel(K);  x<K;  y<K;  z<K |] ==> x le z & y le z"
   503 apply (unfold csquare_rel_def)
   504 apply (erule rev_mp)
   505 apply (elim ltE)
   506 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   507 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)
   508 apply (simp_all add: lt_def succI2)
   509 done
   510 
   511 lemma pred_csquare_subset: 
   512     "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) <= succ(z)*succ(z)"
   513 apply (unfold Order.pred_def)
   514 apply (safe del: SigmaI succCI)
   515 apply (erule csquareD [THEN conjE])
   516 apply (unfold lt_def, auto) 
   517 done
   518 
   519 lemma csquare_ltI:
   520  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> : csquare_rel(K)"
   521 apply (unfold csquare_rel_def)
   522 apply (subgoal_tac "x<K & y<K")
   523  prefer 2 apply (blast intro: lt_trans) 
   524 apply (elim ltE)
   525 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   526 done
   527 
   528 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)
   529 lemma csquare_or_eqI:
   530  "[| x le z;  y le z;  z<K |] ==> <<x,y>, <z,z>> : csquare_rel(K) | x=z & y=z"
   531 apply (unfold csquare_rel_def)
   532 apply (subgoal_tac "x<K & y<K")
   533  prefer 2 apply (blast intro: lt_trans1) 
   534 apply (elim ltE)
   535 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)
   536 apply (elim succE)
   537 apply (simp_all add: subset_Un_iff [THEN iff_sym] 
   538                      subset_Un_iff2 [THEN iff_sym] OrdmemD)
   539 done
   540 
   541 subsubsection{*The cardinality of initial segments*}
   542 
   543 lemma ordermap_z_lt:
   544       "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |] ==>
   545           ordermap(K*K, csquare_rel(K)) ` <x,y> <
   546           ordermap(K*K, csquare_rel(K)) ` <z,z>"
   547 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")
   548 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ
   549                               Limit_is_Ord [THEN well_ord_csquare], clarify) 
   550 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])
   551 apply (erule_tac [4] well_ord_is_wf)
   552 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+
   553 done
   554 
   555 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *)
   556 lemma ordermap_csquare_le:
   557   "[| Limit(K);  x<K;  y<K;  z=succ(x Un y) |]
   558    ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le  |succ(z)| |*| |succ(z)|"
   559 apply (unfold cmult_def)
   560 apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le])
   561 apply (rule Ord_cardinal [THEN well_ord_Memrel])+
   562 apply (subgoal_tac "z<K")
   563  prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ)
   564 apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], 
   565        assumption+)
   566 apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   567 apply (erule Limit_is_Ord [THEN well_ord_csquare])
   568 apply (blast intro: ltD)
   569 apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans],
   570             assumption)
   571 apply (elim ltE)
   572 apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
   573 apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+
   574 done
   575 
   576 (*Kunen: "... so the order type <= K" *)
   577 lemma ordertype_csquare_le:
   578      "[| InfCard(K);  ALL y:K. InfCard(y) --> y |*| y = y |] 
   579       ==> ordertype(K*K, csquare_rel(K)) le K"
   580 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   581 apply (rule all_lt_imp_le, assumption)
   582 apply (erule well_ord_csquare [THEN Ord_ordertype])
   583 apply (rule Card_lt_imp_lt)
   584 apply (erule_tac [3] InfCard_is_Card)
   585 apply (erule_tac [2] ltE)
   586 apply (simp add: ordertype_unfold)
   587 apply (safe elim!: ltE)
   588 apply (subgoal_tac "Ord (xa) & Ord (ya)")
   589  prefer 2 apply (blast intro: Ord_in_Ord, clarify)
   590 (*??WHAT A MESS!*)  
   591 apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],
   592        (assumption | rule refl | erule ltI)+) 
   593 apply (rule_tac i = "xa Un ya" and j = nat in Ord_linear2,
   594        simp_all add: Ord_Un Ord_nat)
   595 prefer 2 (*case nat le (xa Un ya) *)
   596  apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong] 
   597                   le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un
   598                 ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])
   599 (*the finite case: xa Un ya < nat *)
   600 apply (rule_tac j = nat in lt_trans2)
   601  apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
   602                   nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)
   603 apply (simp add: InfCard_def)
   604 done
   605 
   606 (*Main result: Kunen's Theorem 10.12*)
   607 lemma InfCard_csquare_eq: "InfCard(K) ==> K |*| K = K"
   608 apply (frule InfCard_is_Card [THEN Card_is_Ord])
   609 apply (erule rev_mp)
   610 apply (erule_tac i=K in trans_induct) 
   611 apply (rule impI)
   612 apply (rule le_anti_sym)
   613 apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])
   614 apply (rule ordertype_csquare_le [THEN [2] le_trans])
   615 apply (simp add: cmult_def Ord_cardinal_le   
   616                  well_ord_csquare [THEN Ord_ordertype]
   617                  well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll, 
   618                                    THEN cardinal_cong], assumption+)
   619 done
   620 
   621 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)
   622 lemma well_ord_InfCard_square_eq:
   623      "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"
   624 apply (rule prod_eqpoll_cong [THEN eqpoll_trans])
   625 apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+
   626 apply (rule well_ord_cardinal_eqE)
   627 apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)
   628 apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)
   629 done
   630 
   631 lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"
   632 apply (rule well_ord_InfCard_square_eq)  
   633  apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel]) 
   634 apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq]) 
   635 done
   636 
   637 lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"
   638 by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])
   639 
   640 subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}
   641 
   642 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L le K;  0<L |] ==> K |*| L = K"
   643 apply (rule le_anti_sym)
   644  prefer 2
   645  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)
   646 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   647 apply (rule cmult_le_mono [THEN le_trans], assumption+)
   648 apply (simp add: InfCard_csquare_eq)
   649 done
   650 
   651 (*Corollary 10.13 (1), for cardinal multiplication*)
   652 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K |*| L = K Un L"
   653 apply (rule_tac i = K and j = L in Ord_linear_le)
   654 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   655 apply (rule cmult_commute [THEN ssubst])
   656 apply (rule Un_commute [THEN ssubst])
   657 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq 
   658                      subset_Un_iff2 [THEN iffD1] le_imp_subset)
   659 done
   660 
   661 lemma InfCard_cdouble_eq: "InfCard(K) ==> K |+| K = K"
   662 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)
   663 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)
   664 done
   665 
   666 (*Corollary 10.13 (1), for cardinal addition*)
   667 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L le K |] ==> K |+| L = K"
   668 apply (rule le_anti_sym)
   669  prefer 2
   670  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)
   671 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])
   672 apply (rule cadd_le_mono [THEN le_trans], assumption+)
   673 apply (simp add: InfCard_cdouble_eq)
   674 done
   675 
   676 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K |+| L = K Un L"
   677 apply (rule_tac i = K and j = L in Ord_linear_le)
   678 apply (typecheck add: InfCard_is_Card Card_is_Ord)
   679 apply (rule cadd_commute [THEN ssubst])
   680 apply (rule Un_commute [THEN ssubst])
   681 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)
   682 done
   683 
   684 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set
   685   of all n-tuples of elements of K.  A better version for the Isabelle theory
   686   might be  InfCard(K) ==> |list(K)| = K.
   687 *)
   688 
   689 subsection{*For Every Cardinal Number There Exists A Greater One*}
   690 
   691 text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}
   692 
   693 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"
   694 apply (unfold jump_cardinal_def)
   695 apply (rule Ord_is_Transset [THEN [2] OrdI])
   696  prefer 2 apply (blast intro!: Ord_ordertype)
   697 apply (unfold Transset_def)
   698 apply (safe del: subsetI)
   699 apply (simp add: ordertype_pred_unfold, safe)
   700 apply (rule UN_I)
   701 apply (rule_tac [2] ReplaceI)
   702    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+
   703 done
   704 
   705 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)
   706 lemma jump_cardinal_iff:
   707      "i : jump_cardinal(K) <->
   708       (EX r X. r <= K*K & X <= K & well_ord(X,r) & i = ordertype(X,r))"
   709 apply (unfold jump_cardinal_def)
   710 apply (blast del: subsetI) 
   711 done
   712 
   713 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)
   714 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"
   715 apply (rule Ord_jump_cardinal [THEN [2] ltI])
   716 apply (rule jump_cardinal_iff [THEN iffD2])
   717 apply (rule_tac x="Memrel(K)" in exI)
   718 apply (rule_tac x=K in exI)  
   719 apply (simp add: ordertype_Memrel well_ord_Memrel)
   720 apply (simp add: Memrel_def subset_iff)
   721 done
   722 
   723 (*The proof by contradiction: the bijection f yields a wellordering of X
   724   whose ordertype is jump_cardinal(K).  *)
   725 lemma Card_jump_cardinal_lemma:
   726      "[| well_ord(X,r);  r <= K * K;  X <= K;
   727          f : bij(ordertype(X,r), jump_cardinal(K)) |]
   728       ==> jump_cardinal(K) : jump_cardinal(K)"
   729 apply (subgoal_tac "f O ordermap (X,r) : bij (X, jump_cardinal (K))")
   730  prefer 2 apply (blast intro: comp_bij ordermap_bij)
   731 apply (rule jump_cardinal_iff [THEN iffD2])
   732 apply (intro exI conjI)
   733 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)
   734 apply (erule bij_is_inj [THEN well_ord_rvimage])
   735 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])
   736 apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]
   737                  ordertype_Memrel Ord_jump_cardinal)
   738 done
   739 
   740 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)
   741 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"
   742 apply (rule Ord_jump_cardinal [THEN CardI])
   743 apply (unfold eqpoll_def)
   744 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])
   745 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])
   746 done
   747 
   748 subsection{*Basic Properties of Successor Cardinals*}
   749 
   750 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"
   751 apply (unfold csucc_def)
   752 apply (rule LeastI)
   753 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+
   754 done
   755 
   756 lemmas Card_csucc = csucc_basic [THEN conjunct1, standard]
   757 
   758 lemmas lt_csucc = csucc_basic [THEN conjunct2, standard]
   759 
   760 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"
   761 by (blast intro: Ord_0_le lt_csucc lt_trans1)
   762 
   763 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) le L"
   764 apply (unfold csucc_def)
   765 apply (rule Least_le)
   766 apply (blast intro: Card_is_Ord)+
   767 done
   768 
   769 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| le K"
   770 apply (rule iffI)
   771 apply (rule_tac [2] Card_lt_imp_lt)
   772 apply (erule_tac [2] lt_trans1)
   773 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)
   774 apply (rule notI [THEN not_lt_imp_le])
   775 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)
   776 apply (rule Ord_cardinal_le [THEN lt_trans1])
   777 apply (simp_all add: Ord_cardinal Card_is_Ord) 
   778 done
   779 
   780 lemma Card_lt_csucc_iff:
   781      "[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' le K"
   782 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)
   783 
   784 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"
   785 by (simp add: InfCard_def Card_csucc Card_is_Ord 
   786               lt_csucc [THEN leI, THEN [2] le_trans])
   787 
   788 
   789 subsubsection{*Removing elements from a finite set decreases its cardinality*}
   790 
   791 lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x~:A --> ~ cons(x,A) \<lesssim> A"
   792 apply (erule Fin_induct)
   793 apply (simp add: lepoll_0_iff)
   794 apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")
   795 apply simp
   796 apply (blast dest!: cons_lepoll_consD, blast)
   797 done
   798 
   799 lemma Finite_imp_cardinal_cons [simp]:
   800      "[| Finite(A);  a~:A |] ==> |cons(a,A)| = succ(|A|)"
   801 apply (unfold cardinal_def)
   802 apply (rule Least_equality)
   803 apply (fold cardinal_def)
   804 apply (simp add: succ_def)
   805 apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll
   806              elim!: mem_irrefl  dest!: Finite_imp_well_ord)
   807 apply (blast intro: Card_cardinal Card_is_Ord)
   808 apply (rule notI)
   809 apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],
   810        assumption, assumption)
   811 apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])
   812 apply (erule le_imp_lepoll [THEN lepoll_trans])
   813 apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]
   814              dest!: Finite_imp_well_ord)
   815 done
   816 
   817 
   818 lemma Finite_imp_succ_cardinal_Diff:
   819      "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"
   820 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)
   821 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])
   822 apply (simp add: cons_Diff)
   823 done
   824 
   825 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"
   826 apply (rule succ_leE)
   827 apply (simp add: Finite_imp_succ_cardinal_Diff)
   828 done
   829 
   830 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| : nat"
   831 apply (erule Finite_induct)
   832 apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)
   833 done
   834 
   835 lemma card_Un_Int:
   836      "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A Un B| #+ |A Int B|"
   837 apply (erule Finite_induct, simp) 
   838 apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)
   839 done
   840 
   841 lemma card_Un_disjoint: 
   842      "[|Finite(A); Finite(B); A Int B = 0|] ==> |A Un B| = |A| #+ |B|" 
   843 by (simp add: Finite_Un card_Un_Int)
   844 
   845 lemma card_partition [rule_format]:
   846      "Finite(C) ==>  
   847         Finite (\<Union> C) -->  
   848         (\<forall>c\<in>C. |c| = k) -->   
   849         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 --> c1 \<inter> c2 = 0) -->  
   850         k #* |C| = |\<Union> C|"
   851 apply (erule Finite_induct, auto)
   852 apply (subgoal_tac " x \<inter> \<Union>B = 0")  
   853 apply (auto simp add: card_Un_disjoint Finite_Union
   854        subset_Finite [of _ "\<Union> (cons(x,F))"])
   855 done
   856 
   857 
   858 subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}
   859 
   860 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel, standard]
   861 
   862 lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"
   863 apply (rule eqpoll_trans)
   864 apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
   865 apply (erule nat_implies_well_ord)+
   866 apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)
   867 done
   868 
   869 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i <= nat --> i : nat | i=nat"
   870 apply (erule trans_induct3, auto)
   871 apply (blast dest!: nat_le_Limit [THEN le_imp_subset])
   872 done
   873 
   874 lemma Ord_nat_subset_into_Card: "[| Ord(i); i <= nat |] ==> Card(i)"
   875 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
   876 
   877 lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"
   878 apply (rule succ_inject)
   879 apply (rule_tac b = "|A|" in trans)
   880  apply (simp add: Finite_imp_succ_cardinal_Diff)
   881 apply (subgoal_tac "1 \<lesssim> A")
   882  prefer 2 apply (blast intro: not_0_is_lepoll_1)
   883 apply (frule Finite_imp_well_ord, clarify)
   884 apply (drule well_ord_lepoll_imp_Card_le)
   885  apply (auto simp add: cardinal_1)
   886 apply (rule trans)
   887  apply (rule_tac [2] diff_succ)
   888   apply (auto simp add: Finite_cardinal_in_nat)
   889 done
   890 
   891 lemma cardinal_lt_imp_Diff_not_0 [rule_format]:
   892      "Finite(B) ==> ALL A. |B|<|A| --> A - B ~= 0"
   893 apply (erule Finite_induct, auto)
   894 apply (case_tac "Finite (A)")
   895  apply (subgoal_tac [2] "Finite (cons (x, B))")
   896   apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)
   897    apply (auto simp add: Finite_0 Finite_cons)
   898 apply (subgoal_tac "|B|<|A|")
   899  prefer 2 apply (blast intro: lt_trans Ord_cardinal)
   900 apply (case_tac "x:A")
   901  apply (subgoal_tac [2] "A - cons (x, B) = A - B")
   902   apply auto
   903 apply (subgoal_tac "|A| le |cons (x, B) |")
   904  prefer 2
   905  apply (blast dest: Finite_cons [THEN Finite_imp_well_ord] 
   906               intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)
   907 apply (auto simp add: Finite_imp_cardinal_cons)
   908 apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)
   909 apply (blast intro: lt_trans)
   910 done
   911 
   912 
   913 ML{*
   914 val InfCard_def = thm "InfCard_def"
   915 val cmult_def = thm "cmult_def"
   916 val cadd_def = thm "cadd_def"
   917 val jump_cardinal_def = thm "jump_cardinal_def"
   918 val csucc_def = thm "csucc_def"
   919 
   920 val sum_commute_eqpoll = thm "sum_commute_eqpoll";
   921 val cadd_commute = thm "cadd_commute";
   922 val sum_assoc_eqpoll = thm "sum_assoc_eqpoll";
   923 val well_ord_cadd_assoc = thm "well_ord_cadd_assoc";
   924 val sum_0_eqpoll = thm "sum_0_eqpoll";
   925 val cadd_0 = thm "cadd_0";
   926 val sum_lepoll_self = thm "sum_lepoll_self";
   927 val cadd_le_self = thm "cadd_le_self";
   928 val sum_lepoll_mono = thm "sum_lepoll_mono";
   929 val cadd_le_mono = thm "cadd_le_mono";
   930 val eq_imp_not_mem = thm "eq_imp_not_mem";
   931 val sum_succ_eqpoll = thm "sum_succ_eqpoll";
   932 val nat_cadd_eq_add = thm "nat_cadd_eq_add";
   933 val prod_commute_eqpoll = thm "prod_commute_eqpoll";
   934 val cmult_commute = thm "cmult_commute";
   935 val prod_assoc_eqpoll = thm "prod_assoc_eqpoll";
   936 val well_ord_cmult_assoc = thm "well_ord_cmult_assoc";
   937 val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll";
   938 val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib";
   939 val prod_0_eqpoll = thm "prod_0_eqpoll";
   940 val cmult_0 = thm "cmult_0";
   941 val prod_singleton_eqpoll = thm "prod_singleton_eqpoll";
   942 val cmult_1 = thm "cmult_1";
   943 val prod_lepoll_self = thm "prod_lepoll_self";
   944 val cmult_le_self = thm "cmult_le_self";
   945 val prod_lepoll_mono = thm "prod_lepoll_mono";
   946 val cmult_le_mono = thm "cmult_le_mono";
   947 val prod_succ_eqpoll = thm "prod_succ_eqpoll";
   948 val nat_cmult_eq_mult = thm "nat_cmult_eq_mult";
   949 val cmult_2 = thm "cmult_2";
   950 val sum_lepoll_prod = thm "sum_lepoll_prod";
   951 val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod";
   952 val nat_cons_lepoll = thm "nat_cons_lepoll";
   953 val nat_cons_eqpoll = thm "nat_cons_eqpoll";
   954 val nat_succ_eqpoll = thm "nat_succ_eqpoll";
   955 val InfCard_nat = thm "InfCard_nat";
   956 val InfCard_is_Card = thm "InfCard_is_Card";
   957 val InfCard_Un = thm "InfCard_Un";
   958 val InfCard_is_Limit = thm "InfCard_is_Limit";
   959 val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred";
   960 val ordermap_z_lt = thm "ordermap_z_lt";
   961 val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq";
   962 val InfCard_cmult_eq = thm "InfCard_cmult_eq";
   963 val InfCard_cdouble_eq = thm "InfCard_cdouble_eq";
   964 val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq";
   965 val InfCard_cadd_eq = thm "InfCard_cadd_eq";
   966 val Ord_jump_cardinal = thm "Ord_jump_cardinal";
   967 val jump_cardinal_iff = thm "jump_cardinal_iff";
   968 val K_lt_jump_cardinal = thm "K_lt_jump_cardinal";
   969 val Card_jump_cardinal = thm "Card_jump_cardinal";
   970 val csucc_basic = thm "csucc_basic";
   971 val Card_csucc = thm "Card_csucc";
   972 val lt_csucc = thm "lt_csucc";
   973 val Ord_0_lt_csucc = thm "Ord_0_lt_csucc";
   974 val csucc_le = thm "csucc_le";
   975 val lt_csucc_iff = thm "lt_csucc_iff";
   976 val Card_lt_csucc_iff = thm "Card_lt_csucc_iff";
   977 val InfCard_csucc = thm "InfCard_csucc";
   978 val Finite_into_Fin = thm "Finite_into_Fin";
   979 val Fin_into_Finite = thm "Fin_into_Finite";
   980 val Finite_Fin_iff = thm "Finite_Fin_iff";
   981 val Finite_Un = thm "Finite_Un";
   982 val Finite_Union = thm "Finite_Union";
   983 val Finite_induct = thm "Finite_induct";
   984 val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll";
   985 val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons";
   986 val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff";
   987 val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff";
   988 val nat_implies_well_ord = thm "nat_implies_well_ord";
   989 val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum";
   990 val Diff_sing_Finite = thm "Diff_sing_Finite";
   991 val Diff_Finite = thm "Diff_Finite";
   992 val Ord_subset_natD = thm "Ord_subset_natD";
   993 val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card";
   994 val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat";
   995 val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1";
   996 val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0";
   997 *}
   998 
   999 end