src/ZF/CardinalArith.thy
 author wenzelm Tue Jul 31 19:40:22 2007 +0200 (2007-07-31) changeset 24091 109f19a13872 parent 16417 9bc16273c2d4 child 24893 b8ef7afe3a6b permissions -rw-r--r--
```     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
```