src/HOL/BNF_Cardinal_Order_Relation.thy
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     1 (*  Title:      HOL/BNF_Cardinal_Order_Relation.thy

     2     Author:     Andrei Popescu, TU Muenchen

     3     Copyright   2012

     4

     5 Cardinal-order relations as needed by bounded natural functors.

     6 *)

     7

     8 section \<open>Cardinal-Order Relations as Needed by Bounded Natural Functors\<close>

     9

    10 theory BNF_Cardinal_Order_Relation

    11 imports Zorn BNF_Wellorder_Constructions

    12 begin

    13

    14 text\<open>In this section, we define cardinal-order relations to be minim well-orders

    15 on their field.  Then we define the cardinal of a set to be {\em some} cardinal-order

    16 relation on that set, which will be unique up to order isomorphism.  Then we study

    17 the connection between cardinals and:

    18 \begin{itemize}

    19 \item standard set-theoretic constructions: products,

    20 sums, unions, lists, powersets, set-of finite sets operator;

    21 \item finiteness and infiniteness (in particular, with the numeric cardinal operator

    22 for finite sets, \<open>card\<close>, from the theory \<open>Finite_Sets.thy\<close>).

    23 \end{itemize}

    24 %

    25 On the way, we define the canonical $\omega$ cardinal and finite cardinals.  We also

    26 define (again, up to order isomorphism) the successor of a cardinal, and show that

    27 any cardinal admits a successor.

    28

    29 Main results of this section are the existence of cardinal relations and the

    30 facts that, in the presence of infiniteness,

    31 most of the standard set-theoretic constructions (except for the powerset)

    32 {\em do not increase cardinality}.  In particular, e.g., the set of words/lists over

    33 any infinite set has the same cardinality (hence, is in bijection) with that set.

    34 \<close>

    35

    36

    37 subsection \<open>Cardinal orders\<close>

    38

    39 text\<open>A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the

    40 order-embedding relation, \<open>\<le>o\<close> (which is the same as being {\em minimal} w.r.t. the

    41 strict order-embedding relation, \<open><o\<close>), among all the well-orders on its field.\<close>

    42

    43 definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"

    44 where

    45 "card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"

    46

    47 abbreviation "Card_order r \<equiv> card_order_on (Field r) r"

    48 abbreviation "card_order r \<equiv> card_order_on UNIV r"

    49

    50 lemma card_order_on_well_order_on:

    51 assumes "card_order_on A r"

    52 shows "well_order_on A r"

    53 using assms unfolding card_order_on_def by simp

    54

    55 lemma card_order_on_Card_order:

    56 "card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"

    57 unfolding card_order_on_def using well_order_on_Field by blast

    58

    59 text\<open>The existence of a cardinal relation on any given set (which will mean

    60 that any set has a cardinal) follows from two facts:

    61 \begin{itemize}

    62 \item Zermelo's theorem (proved in \<open>Zorn.thy\<close> as theorem \<open>well_order_on\<close>),

    63 which states that on any given set there exists a well-order;

    64 \item The well-founded-ness of \<open><o\<close>, ensuring that then there exists a minimal

    65 such well-order, i.e., a cardinal order.

    66 \end{itemize}

    67 \<close>

    68

    69 theorem card_order_on: "\<exists>r. card_order_on A r"

    70 proof-

    71   obtain R where R_def: "R = {r. well_order_on A r}" by blast

    72   have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"

    73   using well_order_on[of A] R_def well_order_on_Well_order by blast

    74   hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"

    75   using  exists_minim_Well_order[of R] by auto

    76   thus ?thesis using R_def unfolding card_order_on_def by auto

    77 qed

    78

    79 lemma card_order_on_ordIso:

    80 assumes CO: "card_order_on A r" and CO': "card_order_on A r'"

    81 shows "r =o r'"

    82 using assms unfolding card_order_on_def

    83 using ordIso_iff_ordLeq by blast

    84

    85 lemma Card_order_ordIso:

    86 assumes CO: "Card_order r" and ISO: "r' =o r"

    87 shows "Card_order r'"

    88 using ISO unfolding ordIso_def

    89 proof(unfold card_order_on_def, auto)

    90   fix p' assume "well_order_on (Field r') p'"

    91   hence 0: "Well_order p' \<and> Field p' = Field r'"

    92   using well_order_on_Well_order by blast

    93   obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"

    94   using ISO unfolding ordIso_def by auto

    95   hence 3: "inj_on f (Field r') \<and> f  (Field r') = Field r"

    96   by (auto simp add: iso_iff embed_inj_on)

    97   let ?p = "dir_image p' f"

    98   have 4: "p' =o ?p \<and> Well_order ?p"

    99   using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)

   100   moreover have "Field ?p =  Field r"

   101   using 0 3 by (auto simp add: dir_image_Field)

   102   ultimately have "well_order_on (Field r) ?p" by auto

   103   hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto

   104   thus "r' \<le>o p'"

   105   using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast

   106 qed

   107

   108 lemma Card_order_ordIso2:

   109 assumes CO: "Card_order r" and ISO: "r =o r'"

   110 shows "Card_order r'"

   111 using assms Card_order_ordIso ordIso_symmetric by blast

   112

   113

   114 subsection \<open>Cardinal of a set\<close>

   115

   116 text\<open>We define the cardinal of set to be {\em some} cardinal order on that set.

   117 We shall prove that this notion is unique up to order isomorphism, meaning

   118 that order isomorphism shall be the true identity of cardinals.\<close>

   119

   120 definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )

   121 where "card_of A = (SOME r. card_order_on A r)"

   122

   123 lemma card_of_card_order_on: "card_order_on A |A|"

   124 unfolding card_of_def by (auto simp add: card_order_on someI_ex)

   125

   126 lemma card_of_well_order_on: "well_order_on A |A|"

   127 using card_of_card_order_on card_order_on_def by blast

   128

   129 lemma Field_card_of: "Field |A| = A"

   130 using card_of_card_order_on[of A] unfolding card_order_on_def

   131 using well_order_on_Field by blast

   132

   133 lemma card_of_Card_order: "Card_order |A|"

   134 by (simp only: card_of_card_order_on Field_card_of)

   135

   136 corollary ordIso_card_of_imp_Card_order:

   137 "r =o |A| \<Longrightarrow> Card_order r"

   138 using card_of_Card_order Card_order_ordIso by blast

   139

   140 lemma card_of_Well_order: "Well_order |A|"

   141 using card_of_Card_order unfolding card_order_on_def by auto

   142

   143 lemma card_of_refl: "|A| =o |A|"

   144 using card_of_Well_order ordIso_reflexive by blast

   145

   146 lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"

   147 using card_of_card_order_on unfolding card_order_on_def by blast

   148

   149 lemma card_of_ordIso:

   150 "(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"

   151 proof(auto)

   152   fix f assume *: "bij_betw f A B"

   153   then obtain r where "well_order_on B r \<and> |A| =o r"

   154   using Well_order_iso_copy card_of_well_order_on by blast

   155   hence "|B| \<le>o |A|" using card_of_least

   156   ordLeq_ordIso_trans ordIso_symmetric by blast

   157   moreover

   158   {let ?g = "inv_into A f"

   159    have "bij_betw ?g B A" using * bij_betw_inv_into by blast

   160    then obtain r where "well_order_on A r \<and> |B| =o r"

   161    using Well_order_iso_copy card_of_well_order_on by blast

   162    hence "|A| \<le>o |B|" using card_of_least

   163    ordLeq_ordIso_trans ordIso_symmetric by blast

   164   }

   165   ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast

   166 next

   167   assume "|A| =o |B|"

   168   then obtain f where "iso ( |A| ) ( |B| ) f"

   169   unfolding ordIso_def by auto

   170   hence "bij_betw f A B" unfolding iso_def Field_card_of by simp

   171   thus "\<exists>f. bij_betw f A B" by auto

   172 qed

   173

   174 lemma card_of_ordLeq:

   175 "(\<exists>f. inj_on f A \<and> f  A \<le> B) = ( |A| \<le>o |B| )"

   176 proof(auto)

   177   fix f assume *: "inj_on f A" and **: "f  A \<le> B"

   178   {assume "|B| <o |A|"

   179    hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast

   180    then obtain g where "embed ( |B| ) ( |A| ) g"

   181    unfolding ordLeq_def by auto

   182    hence 1: "inj_on g B \<and> g  B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]

   183    card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]

   184    embed_Field[of "|B|" "|A|" g] by auto

   185    obtain h where "bij_betw h A B"

   186    using * ** 1 Schroeder_Bernstein[of f] by fastforce

   187    hence "|A| =o |B|" using card_of_ordIso by blast

   188    hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto

   189   }

   190   thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]

   191   by (auto simp: card_of_Well_order)

   192 next

   193   assume *: "|A| \<le>o |B|"

   194   obtain f where "embed ( |A| ) ( |B| ) f"

   195   using * unfolding ordLeq_def by auto

   196   hence "inj_on f A \<and> f  A \<le> B" using embed_inj_on[of "|A|" "|B|" f]

   197   card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]

   198   embed_Field[of "|A|" "|B|" f] by auto

   199   thus "\<exists>f. inj_on f A \<and> f  A \<le> B" by auto

   200 qed

   201

   202 lemma card_of_ordLeq2:

   203 "A \<noteq> {} \<Longrightarrow> (\<exists>g. g  B = A) = ( |A| \<le>o |B| )"

   204 using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto

   205

   206 lemma card_of_ordLess:

   207 "(\<not>(\<exists>f. inj_on f A \<and> f  A \<le> B)) = ( |B| <o |A| )"

   208 proof-

   209   have "(\<not>(\<exists>f. inj_on f A \<and> f  A \<le> B)) = (\<not> |A| \<le>o |B| )"

   210   using card_of_ordLeq by blast

   211   also have "\<dots> = ( |B| <o |A| )"

   212   using card_of_Well_order[of A] card_of_Well_order[of B]

   213         not_ordLeq_iff_ordLess by blast

   214   finally show ?thesis .

   215 qed

   216

   217 lemma card_of_ordLess2:

   218 "B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f  A = B)) = ( |A| <o |B| )"

   219 using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto

   220

   221 lemma card_of_ordIsoI:

   222 assumes "bij_betw f A B"

   223 shows "|A| =o |B|"

   224 using assms unfolding card_of_ordIso[symmetric] by auto

   225

   226 lemma card_of_ordLeqI:

   227 assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"

   228 shows "|A| \<le>o |B|"

   229 using assms unfolding card_of_ordLeq[symmetric] by auto

   230

   231 lemma card_of_unique:

   232 "card_order_on A r \<Longrightarrow> r =o |A|"

   233 by (simp only: card_order_on_ordIso card_of_card_order_on)

   234

   235 lemma card_of_mono1:

   236 "A \<le> B \<Longrightarrow> |A| \<le>o |B|"

   237 using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce

   238

   239 lemma card_of_mono2:

   240 assumes "r \<le>o r'"

   241 shows "|Field r| \<le>o |Field r'|"

   242 proof-

   243   obtain f where

   244   1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"

   245   using assms unfolding ordLeq_def

   246   by (auto simp add: well_order_on_Well_order)

   247   hence "inj_on f (Field r) \<and> f  (Field r) \<le> Field r'"

   248   by (auto simp add: embed_inj_on embed_Field)

   249   thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast

   250 qed

   251

   252 lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"

   253 by (simp add: ordIso_iff_ordLeq card_of_mono2)

   254

   255 lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"

   256 using card_of_least card_of_well_order_on well_order_on_Well_order by blast

   257

   258 lemma card_of_Field_ordIso:

   259 assumes "Card_order r"

   260 shows "|Field r| =o r"

   261 proof-

   262   have "card_order_on (Field r) r"

   263   using assms card_order_on_Card_order by blast

   264   moreover have "card_order_on (Field r) |Field r|"

   265   using card_of_card_order_on by blast

   266   ultimately show ?thesis using card_order_on_ordIso by blast

   267 qed

   268

   269 lemma Card_order_iff_ordIso_card_of:

   270 "Card_order r = (r =o |Field r| )"

   271 using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast

   272

   273 lemma Card_order_iff_ordLeq_card_of:

   274 "Card_order r = (r \<le>o |Field r| )"

   275 proof-

   276   have "Card_order r = (r =o |Field r| )"

   277   unfolding Card_order_iff_ordIso_card_of by simp

   278   also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"

   279   unfolding ordIso_iff_ordLeq by simp

   280   also have "... = (r \<le>o |Field r| )"

   281   using card_of_Field_ordLess

   282   by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)

   283   finally show ?thesis .

   284 qed

   285

   286 lemma Card_order_iff_Restr_underS:

   287 assumes "Well_order r"

   288 shows "Card_order r = (\<forall>a \<in> Field r. Restr r (underS r a) <o |Field r| )"

   289 using assms unfolding Card_order_iff_ordLeq_card_of

   290 using ordLeq_iff_ordLess_Restr card_of_Well_order by blast

   291

   292 lemma card_of_underS:

   293 assumes r: "Card_order r" and a: "a : Field r"

   294 shows "|underS r a| <o r"

   295 proof-

   296   let ?A = "underS r a"  let ?r' = "Restr r ?A"

   297   have 1: "Well_order r"

   298   using r unfolding card_order_on_def by simp

   299   have "Well_order ?r'" using 1 Well_order_Restr by auto

   300   moreover have "card_order_on (Field ?r') |Field ?r'|"

   301   using card_of_card_order_on .

   302   ultimately have "|Field ?r'| \<le>o ?r'"

   303   unfolding card_order_on_def by simp

   304   moreover have "Field ?r' = ?A"

   305   using 1 wo_rel.underS_ofilter Field_Restr_ofilter

   306   unfolding wo_rel_def by fastforce

   307   ultimately have "|?A| \<le>o ?r'" by simp

   308   also have "?r' <o |Field r|"

   309   using 1 a r Card_order_iff_Restr_underS by blast

   310   also have "|Field r| =o r"

   311   using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto

   312   finally show ?thesis .

   313 qed

   314

   315 lemma ordLess_Field:

   316 assumes "r <o r'"

   317 shows "|Field r| <o r'"

   318 proof-

   319   have "well_order_on (Field r) r" using assms unfolding ordLess_def

   320   by (auto simp add: well_order_on_Well_order)

   321   hence "|Field r| \<le>o r" using card_of_least by blast

   322   thus ?thesis using assms ordLeq_ordLess_trans by blast

   323 qed

   324

   325 lemma internalize_card_of_ordLeq:

   326 "( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"

   327 proof

   328   assume "|A| \<le>o r"

   329   then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"

   330   using internalize_ordLeq[of "|A|" r] by blast

   331   hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast

   332   hence "|Field p| =o p" using card_of_Field_ordIso by blast

   333   hence "|A| =o |Field p| \<and> |Field p| \<le>o r"

   334   using 1 ordIso_equivalence ordIso_ordLeq_trans by blast

   335   thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast

   336 next

   337   assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"

   338   thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast

   339 qed

   340

   341 lemma internalize_card_of_ordLeq2:

   342 "( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"

   343 using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto

   344

   345

   346 subsection \<open>Cardinals versus set operations on arbitrary sets\<close>

   347

   348 text\<open>Here we embark in a long journey of simple results showing

   349 that the standard set-theoretic operations are well-behaved w.r.t. the notion of

   350 cardinal -- essentially, this means that they preserve the cardinal identity"

   351 \<open>=o\<close> and are monotonic w.r.t. \<open>\<le>o\<close>.

   352 \<close>

   353

   354 lemma card_of_empty: "|{}| \<le>o |A|"

   355 using card_of_ordLeq inj_on_id by blast

   356

   357 lemma card_of_empty1:

   358 assumes "Well_order r \<or> Card_order r"

   359 shows "|{}| \<le>o r"

   360 proof-

   361   have "Well_order r" using assms unfolding card_order_on_def by auto

   362   hence "|Field r| <=o r"

   363   using assms card_of_Field_ordLess by blast

   364   moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)

   365   ultimately show ?thesis using ordLeq_transitive by blast

   366 qed

   367

   368 corollary Card_order_empty:

   369 "Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)

   370

   371 lemma card_of_empty2:

   372 assumes LEQ: "|A| =o |{}|"

   373 shows "A = {}"

   374 using assms card_of_ordIso[of A] bij_betw_empty2 by blast

   375

   376 lemma card_of_empty3:

   377 assumes LEQ: "|A| \<le>o |{}|"

   378 shows "A = {}"

   379 using assms

   380 by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2

   381               ordLeq_Well_order_simp)

   382

   383 lemma card_of_empty_ordIso:

   384 "|{}::'a set| =o |{}::'b set|"

   385 using card_of_ordIso unfolding bij_betw_def inj_on_def by blast

   386

   387 lemma card_of_image:

   388 "|f  A| <=o |A|"

   389 proof(cases "A = {}", simp add: card_of_empty)

   390   assume "A \<noteq> {}"

   391   hence "f  A \<noteq> {}" by auto

   392   thus "|f  A| \<le>o |A|"

   393   using card_of_ordLeq2[of "f  A" A] by auto

   394 qed

   395

   396 lemma surj_imp_ordLeq:

   397 assumes "B \<subseteq> f  A"

   398 shows "|B| \<le>o |A|"

   399 proof-

   400   have "|B| <=o |f  A|" using assms card_of_mono1 by auto

   401   thus ?thesis using card_of_image ordLeq_transitive by blast

   402 qed

   403

   404 lemma card_of_singl_ordLeq:

   405 assumes "A \<noteq> {}"

   406 shows "|{b}| \<le>o |A|"

   407 proof-

   408   obtain a where *: "a \<in> A" using assms by auto

   409   let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"

   410   have "inj_on ?h {b} \<and> ?h  {b} \<le> A"

   411   using * unfolding inj_on_def by auto

   412   thus ?thesis unfolding card_of_ordLeq[symmetric] by (intro exI)

   413 qed

   414

   415 corollary Card_order_singl_ordLeq:

   416 "\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"

   417 using card_of_singl_ordLeq[of "Field r" b]

   418       card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast

   419

   420 lemma card_of_Pow: "|A| <o |Pow A|"

   421 using card_of_ordLess2[of "Pow A" A]  Cantors_paradox[of A]

   422       Pow_not_empty[of A] by auto

   423

   424 corollary Card_order_Pow:

   425 "Card_order r \<Longrightarrow> r <o |Pow(Field r)|"

   426 using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast

   427

   428 lemma card_of_Plus1: "|A| \<le>o |A <+> B|"

   429 proof-

   430   have "Inl  A \<le> A <+> B" by auto

   431   thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast

   432 qed

   433

   434 corollary Card_order_Plus1:

   435 "Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"

   436 using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast

   437

   438 lemma card_of_Plus2: "|B| \<le>o |A <+> B|"

   439 proof-

   440   have "Inr  B \<le> A <+> B" by auto

   441   thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast

   442 qed

   443

   444 corollary Card_order_Plus2:

   445 "Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"

   446 using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast

   447

   448 lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"

   449 proof-

   450   have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto

   451   thus ?thesis using card_of_ordIso by auto

   452 qed

   453

   454 lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"

   455 proof-

   456   have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto

   457   thus ?thesis using card_of_ordIso by auto

   458 qed

   459

   460 lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"

   461 proof-

   462   let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a

   463                                    | Inr b \<Rightarrow> Inl b"

   464   have "bij_betw ?f (A <+> B) (B <+> A)"

   465   unfolding bij_betw_def inj_on_def by force

   466   thus ?thesis using card_of_ordIso by blast

   467 qed

   468

   469 lemma card_of_Plus_assoc:

   470 fixes A :: "'a set" and B :: "'b set" and C :: "'c set"

   471 shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"

   472 proof -

   473   define f :: "('a + 'b) + 'c \<Rightarrow> 'a + 'b + 'c"

   474     where [abs_def]: "f k =

   475       (case k of

   476         Inl ab \<Rightarrow>

   477           (case ab of

   478             Inl a \<Rightarrow> Inl a

   479           | Inr b \<Rightarrow> Inr (Inl b))

   480       | Inr c \<Rightarrow> Inr (Inr c))"

   481     for k

   482   have "A <+> B <+> C \<subseteq> f  ((A <+> B) <+> C)"

   483   proof

   484     fix x assume x: "x \<in> A <+> B <+> C"

   485     show "x \<in> f  ((A <+> B) <+> C)"

   486     proof(cases x)

   487       case (Inl a)

   488       hence "a \<in> A" "x = f (Inl (Inl a))"

   489       using x unfolding f_def by auto

   490       thus ?thesis by auto

   491     next

   492       case (Inr bc) note 1 = Inr show ?thesis

   493       proof(cases bc)

   494         case (Inl b)

   495         hence "b \<in> B" "x = f (Inl (Inr b))"

   496         using x 1 unfolding f_def by auto

   497         thus ?thesis by auto

   498       next

   499         case (Inr c)

   500         hence "c \<in> C" "x = f (Inr c)"

   501         using x 1 unfolding f_def by auto

   502         thus ?thesis by auto

   503       qed

   504     qed

   505   qed

   506   hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"

   507     unfolding bij_betw_def inj_on_def f_def by fastforce

   508   thus ?thesis using card_of_ordIso by blast

   509 qed

   510

   511 lemma card_of_Plus_mono1:

   512 assumes "|A| \<le>o |B|"

   513 shows "|A <+> C| \<le>o |B <+> C|"

   514 proof-

   515   obtain f where 1: "inj_on f A \<and> f  A \<le> B"

   516   using assms card_of_ordLeq[of A] by fastforce

   517   obtain g where g_def:

   518   "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast

   519   have "inj_on g (A <+> C) \<and> g  (A <+> C) \<le> (B <+> C)"

   520   proof-

   521     {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and

   522                           "g d1 = g d2"

   523      hence "d1 = d2" using 1 unfolding inj_on_def g_def by force

   524     }

   525     moreover

   526     {fix d assume "d \<in> A <+> C"

   527      hence "g d \<in> B <+> C"  using 1

   528      by(case_tac d, auto simp add: g_def)

   529     }

   530     ultimately show ?thesis unfolding inj_on_def by auto

   531   qed

   532   thus ?thesis using card_of_ordLeq by blast

   533 qed

   534

   535 corollary ordLeq_Plus_mono1:

   536 assumes "r \<le>o r'"

   537 shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"

   538 using assms card_of_mono2 card_of_Plus_mono1 by blast

   539

   540 lemma card_of_Plus_mono2:

   541 assumes "|A| \<le>o |B|"

   542 shows "|C <+> A| \<le>o |C <+> B|"

   543 using assms card_of_Plus_mono1[of A B C]

   544       card_of_Plus_commute[of C A]  card_of_Plus_commute[of B C]

   545       ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]

   546 by blast

   547

   548 corollary ordLeq_Plus_mono2:

   549 assumes "r \<le>o r'"

   550 shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"

   551 using assms card_of_mono2 card_of_Plus_mono2 by blast

   552

   553 lemma card_of_Plus_mono:

   554 assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"

   555 shows "|A <+> C| \<le>o |B <+> D|"

   556 using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]

   557       ordLeq_transitive[of "|A <+> C|"] by blast

   558

   559 corollary ordLeq_Plus_mono:

   560 assumes "r \<le>o r'" and "p \<le>o p'"

   561 shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"

   562 using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast

   563

   564 lemma card_of_Plus_cong1:

   565 assumes "|A| =o |B|"

   566 shows "|A <+> C| =o |B <+> C|"

   567 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)

   568

   569 corollary ordIso_Plus_cong1:

   570 assumes "r =o r'"

   571 shows "|(Field r) <+> C| =o |(Field r') <+> C|"

   572 using assms card_of_cong card_of_Plus_cong1 by blast

   573

   574 lemma card_of_Plus_cong2:

   575 assumes "|A| =o |B|"

   576 shows "|C <+> A| =o |C <+> B|"

   577 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)

   578

   579 corollary ordIso_Plus_cong2:

   580 assumes "r =o r'"

   581 shows "|A <+> (Field r)| =o |A <+> (Field r')|"

   582 using assms card_of_cong card_of_Plus_cong2 by blast

   583

   584 lemma card_of_Plus_cong:

   585 assumes "|A| =o |B|" and "|C| =o |D|"

   586 shows "|A <+> C| =o |B <+> D|"

   587 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)

   588

   589 corollary ordIso_Plus_cong:

   590 assumes "r =o r'" and "p =o p'"

   591 shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"

   592 using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast

   593

   594 lemma card_of_Un_Plus_ordLeq:

   595 "|A \<union> B| \<le>o |A <+> B|"

   596 proof-

   597    let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"

   598    have "inj_on ?f (A \<union> B) \<and> ?f  (A \<union> B) \<le> A <+> B"

   599    unfolding inj_on_def by auto

   600    thus ?thesis using card_of_ordLeq by blast

   601 qed

   602

   603 lemma card_of_Times1:

   604 assumes "A \<noteq> {}"

   605 shows "|B| \<le>o |B \<times> A|"

   606 proof(cases "B = {}", simp add: card_of_empty)

   607   assume *: "B \<noteq> {}"

   608   have "fst (B \<times> A) = B" using assms by auto

   609   thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]

   610                      card_of_ordLeq[of B "B \<times> A"] * by blast

   611 qed

   612

   613 lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"

   614 proof-

   615   let ?f = "\<lambda>(a::'a,b::'b). (b,a)"

   616   have "bij_betw ?f (A \<times> B) (B \<times> A)"

   617   unfolding bij_betw_def inj_on_def by auto

   618   thus ?thesis using card_of_ordIso by blast

   619 qed

   620

   621 lemma card_of_Times2:

   622 assumes "A \<noteq> {}"   shows "|B| \<le>o |A \<times> B|"

   623 using assms card_of_Times1[of A B] card_of_Times_commute[of B A]

   624       ordLeq_ordIso_trans by blast

   625

   626 corollary Card_order_Times1:

   627 "\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"

   628 using card_of_Times1[of B] card_of_Field_ordIso

   629       ordIso_ordLeq_trans ordIso_symmetric by blast

   630

   631 corollary Card_order_Times2:

   632 "\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"

   633 using card_of_Times2[of A] card_of_Field_ordIso

   634       ordIso_ordLeq_trans ordIso_symmetric by blast

   635

   636 lemma card_of_Times3: "|A| \<le>o |A \<times> A|"

   637 using card_of_Times1[of A]

   638 by(cases "A = {}", simp add: card_of_empty, blast)

   639

   640 lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"

   641 proof-

   642   let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)

   643                                   |Inr a \<Rightarrow> (a,False)"

   644   have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"

   645   proof-

   646     {fix  c1 and c2 assume "?f c1 = ?f c2"

   647      hence "c1 = c2"

   648      by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)

   649     }

   650     moreover

   651     {fix c assume "c \<in> A <+> A"

   652      hence "?f c \<in> A \<times> (UNIV::bool set)"

   653      by(case_tac c, auto)

   654     }

   655     moreover

   656     {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"

   657      have "(a,bl) \<in> ?f  ( A <+> A)"

   658      proof(cases bl)

   659        assume bl hence "?f(Inl a) = (a,bl)" by auto

   660        thus ?thesis using * by force

   661      next

   662        assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto

   663        thus ?thesis using * by force

   664      qed

   665     }

   666     ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto

   667   qed

   668   thus ?thesis using card_of_ordIso by blast

   669 qed

   670

   671 lemma card_of_Times_mono1:

   672 assumes "|A| \<le>o |B|"

   673 shows "|A \<times> C| \<le>o |B \<times> C|"

   674 proof-

   675   obtain f where 1: "inj_on f A \<and> f  A \<le> B"

   676   using assms card_of_ordLeq[of A] by fastforce

   677   obtain g where g_def:

   678   "g = (\<lambda>(a,c::'c). (f a,c))" by blast

   679   have "inj_on g (A \<times> C) \<and> g  (A \<times> C) \<le> (B \<times> C)"

   680   using 1 unfolding inj_on_def using g_def by auto

   681   thus ?thesis using card_of_ordLeq by blast

   682 qed

   683

   684 corollary ordLeq_Times_mono1:

   685 assumes "r \<le>o r'"

   686 shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"

   687 using assms card_of_mono2 card_of_Times_mono1 by blast

   688

   689 lemma card_of_Times_mono2:

   690 assumes "|A| \<le>o |B|"

   691 shows "|C \<times> A| \<le>o |C \<times> B|"

   692 using assms card_of_Times_mono1[of A B C]

   693       card_of_Times_commute[of C A]  card_of_Times_commute[of B C]

   694       ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]

   695 by blast

   696

   697 corollary ordLeq_Times_mono2:

   698 assumes "r \<le>o r'"

   699 shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"

   700 using assms card_of_mono2 card_of_Times_mono2 by blast

   701

   702 lemma card_of_Sigma_mono1:

   703 assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"

   704 shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"

   705 proof-

   706   have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f  (A i) \<le> B i)"

   707   using assms by (auto simp add: card_of_ordLeq)

   708   with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f  (A i) \<le> B i"]

   709   obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i)  (A i) \<le> B i"

   710     by atomize_elim (auto intro: bchoice)

   711   obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast

   712   have "inj_on g (Sigma I A) \<and> g  (Sigma I A) \<le> (Sigma I B)"

   713   using 1 unfolding inj_on_def using g_def by force

   714   thus ?thesis using card_of_ordLeq by blast

   715 qed

   716

   717 lemma card_of_UNION_Sigma:

   718 "|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"

   719 using Ex_inj_on_UNION_Sigma [of A I] card_of_ordLeq by blast

   720

   721 lemma card_of_bool:

   722 assumes "a1 \<noteq> a2"

   723 shows "|UNIV::bool set| =o |{a1,a2}|"

   724 proof-

   725   let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"

   726   have "bij_betw ?f UNIV {a1,a2}"

   727   proof-

   728     {fix bl1 and bl2 assume "?f  bl1 = ?f bl2"

   729      hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)

   730     }

   731     moreover

   732     {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)

   733     }

   734     moreover

   735     {fix a assume *: "a \<in> {a1,a2}"

   736      have "a \<in> ?f  UNIV"

   737      proof(cases "a = a1")

   738        assume "a = a1"

   739        hence "?f True = a" by auto  thus ?thesis by blast

   740      next

   741        assume "a \<noteq> a1" hence "a = a2" using * by auto

   742        hence "?f False = a" by auto  thus ?thesis by blast

   743      qed

   744     }

   745     ultimately show ?thesis unfolding bij_betw_def inj_on_def by blast

   746   qed

   747   thus ?thesis using card_of_ordIso by blast

   748 qed

   749

   750 lemma card_of_Plus_Times_aux:

   751 assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and

   752         LEQ: "|A| \<le>o |B|"

   753 shows "|A <+> B| \<le>o |A \<times> B|"

   754 proof-

   755   have 1: "|UNIV::bool set| \<le>o |A|"

   756   using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]

   757         ordIso_ordLeq_trans[of "|UNIV::bool set|"] by blast

   758   (*  *)

   759   have "|A <+> B| \<le>o |B <+> B|"

   760   using LEQ card_of_Plus_mono1 by blast

   761   moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"

   762   using card_of_Plus_Times_bool by blast

   763   moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"

   764   using 1 by (simp add: card_of_Times_mono2)

   765   moreover have " |B \<times> A| =o |A \<times> B|"

   766   using card_of_Times_commute by blast

   767   ultimately show "|A <+> B| \<le>o |A \<times> B|"

   768   using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]

   769         ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]

   770         ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]

   771   by blast

   772 qed

   773

   774 lemma card_of_Plus_Times:

   775 assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and

   776         B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"

   777 shows "|A <+> B| \<le>o |A \<times> B|"

   778 proof-

   779   {assume "|A| \<le>o |B|"

   780    hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)

   781   }

   782   moreover

   783   {assume "|B| \<le>o |A|"

   784    hence "|B <+> A| \<le>o |B \<times> A|"

   785    using assms by (auto simp add: card_of_Plus_Times_aux)

   786    hence ?thesis

   787    using card_of_Plus_commute card_of_Times_commute

   788          ordIso_ordLeq_trans ordLeq_ordIso_trans by blast

   789   }

   790   ultimately show ?thesis

   791   using card_of_Well_order[of A] card_of_Well_order[of B]

   792         ordLeq_total[of "|A|"] by blast

   793 qed

   794

   795 lemma card_of_Times_Plus_distrib:

   796   "|A \<times> (B <+> C)| =o |A \<times> B <+> A \<times> C|" (is "|?RHS| =o |?LHS|")

   797 proof -

   798   let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"

   799   have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force

   800   thus ?thesis using card_of_ordIso by blast

   801 qed

   802

   803 lemma card_of_ordLeq_finite:

   804 assumes "|A| \<le>o |B|" and "finite B"

   805 shows "finite A"

   806 using assms unfolding ordLeq_def

   807 using embed_inj_on[of "|A|" "|B|"]  embed_Field[of "|A|" "|B|"]

   808       Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce

   809

   810 lemma card_of_ordLeq_infinite:

   811 assumes "|A| \<le>o |B|" and "\<not> finite A"

   812 shows "\<not> finite B"

   813 using assms card_of_ordLeq_finite by auto

   814

   815 lemma card_of_ordIso_finite:

   816 assumes "|A| =o |B|"

   817 shows "finite A = finite B"

   818 using assms unfolding ordIso_def iso_def[abs_def]

   819 by (auto simp: bij_betw_finite Field_card_of)

   820

   821 lemma card_of_ordIso_finite_Field:

   822 assumes "Card_order r" and "r =o |A|"

   823 shows "finite(Field r) = finite A"

   824 using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast

   825

   826

   827 subsection \<open>Cardinals versus set operations involving infinite sets\<close>

   828

   829 text\<open>Here we show that, for infinite sets, most set-theoretic constructions

   830 do not increase the cardinality.  The cornerstone for this is

   831 theorem \<open>Card_order_Times_same_infinite\<close>, which states that self-product

   832 does not increase cardinality -- the proof of this fact adapts a standard

   833 set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11

   834 at page 47 in @{cite "card-book"}. Then everything else follows fairly easily.\<close>

   835

   836 lemma infinite_iff_card_of_nat:

   837 "\<not> finite A \<longleftrightarrow> ( |UNIV::nat set| \<le>o |A| )"

   838 unfolding infinite_iff_countable_subset card_of_ordLeq ..

   839

   840 text\<open>The next two results correspond to the ZF fact that all infinite cardinals are

   841 limit ordinals:\<close>

   842

   843 lemma Card_order_infinite_not_under:

   844 assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"

   845 shows "\<not> (\<exists>a. Field r = under r a)"

   846 proof(auto)

   847   have 0: "Well_order r \<and> wo_rel r \<and> Refl r"

   848   using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto

   849   fix a assume *: "Field r = under r a"

   850   show False

   851   proof(cases "a \<in> Field r")

   852     assume Case1: "a \<notin> Field r"

   853     hence "under r a = {}" unfolding Field_def under_def by auto

   854     thus False using INF *  by auto

   855   next

   856     let ?r' = "Restr r (underS r a)"

   857     assume Case2: "a \<in> Field r"

   858     hence 1: "under r a = underS r a \<union> {a} \<and> a \<notin> underS r a"

   859     using 0 Refl_under_underS[of r a] underS_notIn[of a r] by blast

   860     have 2: "wo_rel.ofilter r (underS r a) \<and> underS r a < Field r"

   861     using 0 wo_rel.underS_ofilter * 1 Case2 by fast

   862     hence "?r' <o r" using 0 using ofilter_ordLess by blast

   863     moreover

   864     have "Field ?r' = underS r a \<and> Well_order ?r'"

   865     using  2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast

   866     ultimately have "|underS r a| <o r" using ordLess_Field[of ?r'] by auto

   867     moreover have "|under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto

   868     ultimately have "|underS r a| <o |under r a|"

   869     using ordIso_symmetric ordLess_ordIso_trans by blast

   870     moreover

   871     {have "\<exists>f. bij_betw f (under r a) (underS r a)"

   872      using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto

   873      hence "|under r a| =o |underS r a|" using card_of_ordIso by blast

   874     }

   875     ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast

   876   qed

   877 qed

   878

   879 lemma infinite_Card_order_limit:

   880 assumes r: "Card_order r" and "\<not>finite (Field r)"

   881 and a: "a : Field r"

   882 shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"

   883 proof-

   884   have "Field r \<noteq> under r a"

   885   using assms Card_order_infinite_not_under by blast

   886   moreover have "under r a \<le> Field r"

   887   using under_Field .

   888   ultimately have "under r a < Field r" by blast

   889   then obtain b where 1: "b \<in> Field r \<and> \<not> (b,a) \<in> r"

   890   unfolding under_def by blast

   891   moreover have ba: "b \<noteq> a"

   892   using 1 r unfolding card_order_on_def well_order_on_def

   893   linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto

   894   ultimately have "(a,b) : r"

   895   using a r unfolding card_order_on_def well_order_on_def linear_order_on_def

   896   total_on_def by blast

   897   thus ?thesis using 1 ba by auto

   898 qed

   899

   900 theorem Card_order_Times_same_infinite:

   901 assumes CO: "Card_order r" and INF: "\<not>finite(Field r)"

   902 shows "|Field r \<times> Field r| \<le>o r"

   903 proof-

   904   obtain phi where phi_def:

   905   "phi = (\<lambda>r::'a rel. Card_order r \<and> \<not>finite(Field r) \<and>

   906                       \<not> |Field r \<times> Field r| \<le>o r )" by blast

   907   have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"

   908   unfolding phi_def card_order_on_def by auto

   909   have Ft: "\<not>(\<exists>r. phi r)"

   910   proof

   911     assume "\<exists>r. phi r"

   912     hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"

   913     using temp1 by auto

   914     then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and

   915                    3: "Card_order r \<and> Well_order r"

   916     using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast

   917     let ?A = "Field r"  let ?r' = "bsqr r"

   918     have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"

   919     using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast

   920     have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"

   921     using card_of_Card_order card_of_Well_order by blast

   922     (*  *)

   923     have "r <o |?A \<times> ?A|"

   924     using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast

   925     moreover have "|?A \<times> ?A| \<le>o ?r'"

   926     using card_of_least[of "?A \<times> ?A"] 4 by auto

   927     ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto

   928     then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"

   929     unfolding ordLess_def embedS_def[abs_def]

   930     by (auto simp add: Field_bsqr)

   931     let ?B = "f  ?A"

   932     have "|?A| =o |?B|"

   933     using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast

   934     hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast

   935     (*  *)

   936     have "wo_rel.ofilter ?r' ?B"

   937     using 6 embed_Field_ofilter 3 4 by blast

   938     hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"

   939     using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto

   940     hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"

   941     using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast

   942     have "\<not> (\<exists>a. Field r = under r a)"

   943     using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto

   944     then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"

   945     using temp2 3 bsqr_ofilter[of r ?B] by blast

   946     hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast

   947     hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast

   948     let ?r1 = "Restr r A1"

   949     have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast

   950     moreover

   951     {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast

   952      hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast

   953     }

   954     ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast

   955     (*  *)

   956     have "\<not> finite (Field r)" using 1 unfolding phi_def by simp

   957     hence "\<not> finite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast

   958     hence "\<not> finite A1" using 9 finite_cartesian_product finite_subset by blast

   959     moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"

   960     using card_of_Card_order[of A1] card_of_Well_order[of A1]

   961     by (simp add: Field_card_of)

   962     moreover have "\<not> r \<le>o | A1 |"

   963     using temp4 11 3 using not_ordLeq_iff_ordLess by blast

   964     ultimately have "\<not> finite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"

   965     by (simp add: card_of_card_order_on)

   966     hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"

   967     using 2 unfolding phi_def by blast

   968     hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto

   969     hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast

   970     thus False using 11 not_ordLess_ordLeq by auto

   971   qed

   972   thus ?thesis using assms unfolding phi_def by blast

   973 qed

   974

   975 corollary card_of_Times_same_infinite:

   976 assumes "\<not>finite A"

   977 shows "|A \<times> A| =o |A|"

   978 proof-

   979   let ?r = "|A|"

   980   have "Field ?r = A \<and> Card_order ?r"

   981   using Field_card_of card_of_Card_order[of A] by fastforce

   982   hence "|A \<times> A| \<le>o |A|"

   983   using Card_order_Times_same_infinite[of ?r] assms by auto

   984   thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast

   985 qed

   986

   987 lemma card_of_Times_infinite:

   988 assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"

   989 shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"

   990 proof-

   991   have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"

   992   using assms by (simp add: card_of_Times1 card_of_Times2)

   993   moreover

   994   {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"

   995    using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast

   996    moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast

   997    ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"

   998    using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto

   999   }

  1000   ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)

  1001 qed

  1002

  1003 corollary Card_order_Times_infinite:

  1004 assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and

  1005         NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"

  1006 shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"

  1007 proof-

  1008   have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"

  1009   using assms by (simp add: card_of_Times_infinite card_of_mono2)

  1010   thus ?thesis

  1011   using assms card_of_Field_ordIso[of r]

  1012         ordIso_transitive[of "|Field r \<times> Field p|"]

  1013         ordIso_transitive[of _ "|Field r|"] by blast

  1014 qed

  1015

  1016 lemma card_of_Sigma_ordLeq_infinite:

  1017 assumes INF: "\<not>finite B" and

  1018         LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"

  1019 shows "|SIGMA i : I. A i| \<le>o |B|"

  1020 proof(cases "I = {}", simp add: card_of_empty)

  1021   assume *: "I \<noteq> {}"

  1022   have "|SIGMA i : I. A i| \<le>o |I \<times> B|"

  1023   using card_of_Sigma_mono1[OF LEQ] by blast

  1024   moreover have "|I \<times> B| =o |B|"

  1025   using INF * LEQ_I by (auto simp add: card_of_Times_infinite)

  1026   ultimately show ?thesis using ordLeq_ordIso_trans by blast

  1027 qed

  1028

  1029 lemma card_of_Sigma_ordLeq_infinite_Field:

  1030 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1031         LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"

  1032 shows "|SIGMA i : I. A i| \<le>o r"

  1033 proof-

  1034   let ?B  = "Field r"

  1035   have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso

  1036   ordIso_symmetric by blast

  1037   hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"

  1038   using LEQ_I LEQ ordLeq_ordIso_trans by blast+

  1039   hence  "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ

  1040   card_of_Sigma_ordLeq_infinite by blast

  1041   thus ?thesis using 1 ordLeq_ordIso_trans by blast

  1042 qed

  1043

  1044 lemma card_of_Times_ordLeq_infinite_Field:

  1045 "\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>

  1046  \<Longrightarrow> |A \<times> B| \<le>o r"

  1047 by(simp add: card_of_Sigma_ordLeq_infinite_Field)

  1048

  1049 lemma card_of_Times_infinite_simps:

  1050 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"

  1051 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"

  1052 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"

  1053 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"

  1054 by (auto simp add: card_of_Times_infinite ordIso_symmetric)

  1055

  1056 lemma card_of_UNION_ordLeq_infinite:

  1057 assumes INF: "\<not>finite B" and

  1058         LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"

  1059 shows "|\<Union>i \<in> I. A i| \<le>o |B|"

  1060 proof(cases "I = {}", simp add: card_of_empty)

  1061   assume *: "I \<noteq> {}"

  1062   have "|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"

  1063   using card_of_UNION_Sigma by blast

  1064   moreover have "|SIGMA i : I. A i| \<le>o |B|"

  1065   using assms card_of_Sigma_ordLeq_infinite by blast

  1066   ultimately show ?thesis using ordLeq_transitive by blast

  1067 qed

  1068

  1069 corollary card_of_UNION_ordLeq_infinite_Field:

  1070 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1071         LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"

  1072 shows "|\<Union>i \<in> I. A i| \<le>o r"

  1073 proof-

  1074   let ?B  = "Field r"

  1075   have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso

  1076   ordIso_symmetric by blast

  1077   hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"

  1078   using LEQ_I LEQ ordLeq_ordIso_trans by blast+

  1079   hence  "|\<Union>i \<in> I. A i| \<le>o |?B|" using INF LEQ

  1080   card_of_UNION_ordLeq_infinite by blast

  1081   thus ?thesis using 1 ordLeq_ordIso_trans by blast

  1082 qed

  1083

  1084 lemma card_of_Plus_infinite1:

  1085 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1086 shows "|A <+> B| =o |A|"

  1087 proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)

  1088   let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b"  let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"

  1089   assume *: "B \<noteq> {}"

  1090   then obtain b1 where 1: "b1 \<in> B" by blast

  1091   show ?thesis

  1092   proof(cases "B = {b1}")

  1093     assume Case1: "B = {b1}"

  1094     have 2: "bij_betw ?Inl A ((?Inl  A))"

  1095     unfolding bij_betw_def inj_on_def by auto

  1096     hence 3: "\<not>finite (?Inl  A)"

  1097     using INF bij_betw_finite[of ?Inl A] by blast

  1098     let ?A' = "?Inl  A \<union> {?Inr b1}"

  1099     obtain g where "bij_betw g (?Inl  A) ?A'"

  1100     using 3 infinite_imp_bij_betw2[of "?Inl  A"] by auto

  1101     moreover have "?A' = A <+> B" using Case1 by blast

  1102     ultimately have "bij_betw g (?Inl  A) (A <+> B)" by simp

  1103     hence "bij_betw (g \<circ> ?Inl) A (A <+> B)"

  1104     using 2 by (auto simp add: bij_betw_trans)

  1105     thus ?thesis using card_of_ordIso ordIso_symmetric by blast

  1106   next

  1107     assume Case2: "B \<noteq> {b1}"

  1108     with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce

  1109     obtain f where "inj_on f B \<and> f  B \<le> A"

  1110     using LEQ card_of_ordLeq[of B] by fastforce

  1111     with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"

  1112     unfolding inj_on_def by auto

  1113     with 3 have "|A <+> B| \<le>o |A \<times> B|"

  1114     by (auto simp add: card_of_Plus_Times)

  1115     moreover have "|A \<times> B| =o |A|"

  1116     using assms * by (simp add: card_of_Times_infinite_simps)

  1117     ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by blast

  1118     thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast

  1119   qed

  1120 qed

  1121

  1122 lemma card_of_Plus_infinite2:

  1123 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1124 shows "|B <+> A| =o |A|"

  1125 using assms card_of_Plus_commute card_of_Plus_infinite1

  1126 ordIso_equivalence by blast

  1127

  1128 lemma card_of_Plus_infinite:

  1129 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1130 shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"

  1131 using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)

  1132

  1133 corollary Card_order_Plus_infinite:

  1134 assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and

  1135         LEQ: "p \<le>o r"

  1136 shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"

  1137 proof-

  1138   have "| Field r <+> Field p | =o | Field r | \<and>

  1139         | Field p <+> Field r | =o | Field r |"

  1140   using assms by (simp add: card_of_Plus_infinite card_of_mono2)

  1141   thus ?thesis

  1142   using assms card_of_Field_ordIso[of r]

  1143         ordIso_transitive[of "|Field r <+> Field p|"]

  1144         ordIso_transitive[of _ "|Field r|"] by blast

  1145 qed

  1146

  1147

  1148 subsection \<open>The cardinal $\omega$ and the finite cardinals\<close>

  1149

  1150 text\<open>The cardinal $\omega$, of natural numbers, shall be the standard non-strict

  1151 order relation on

  1152 \<open>nat\<close>, that we abbreviate by \<open>natLeq\<close>.  The finite cardinals

  1153 shall be the restrictions of these relations to the numbers smaller than

  1154 fixed numbers \<open>n\<close>, that we abbreviate by \<open>natLeq_on n\<close>.\<close>

  1155

  1156 definition "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"

  1157 definition "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"

  1158

  1159 abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"

  1160 where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"

  1161

  1162 lemma infinite_cartesian_product:

  1163 assumes "\<not>finite A" "\<not>finite B"

  1164 shows "\<not>finite (A \<times> B)"

  1165 proof

  1166   assume "finite (A \<times> B)"

  1167   from assms(1) have "A \<noteq> {}" by auto

  1168   with \<open>finite (A \<times> B)\<close> have "finite B" using finite_cartesian_productD2 by auto

  1169   with assms(2) show False by simp

  1170 qed

  1171

  1172

  1173 subsubsection \<open>First as well-orders\<close>

  1174

  1175 lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"

  1176 by(unfold Field_def natLeq_def, auto)

  1177

  1178 lemma natLeq_Refl: "Refl natLeq"

  1179 unfolding refl_on_def Field_def natLeq_def by auto

  1180

  1181 lemma natLeq_trans: "trans natLeq"

  1182 unfolding trans_def natLeq_def by auto

  1183

  1184 lemma natLeq_Preorder: "Preorder natLeq"

  1185 unfolding preorder_on_def

  1186 by (auto simp add: natLeq_Refl natLeq_trans)

  1187

  1188 lemma natLeq_antisym: "antisym natLeq"

  1189 unfolding antisym_def natLeq_def by auto

  1190

  1191 lemma natLeq_Partial_order: "Partial_order natLeq"

  1192 unfolding partial_order_on_def

  1193 by (auto simp add: natLeq_Preorder natLeq_antisym)

  1194

  1195 lemma natLeq_Total: "Total natLeq"

  1196 unfolding total_on_def natLeq_def by auto

  1197

  1198 lemma natLeq_Linear_order: "Linear_order natLeq"

  1199 unfolding linear_order_on_def

  1200 by (auto simp add: natLeq_Partial_order natLeq_Total)

  1201

  1202 lemma natLeq_natLess_Id: "natLess = natLeq - Id"

  1203 unfolding natLeq_def natLess_def by auto

  1204

  1205 lemma natLeq_Well_order: "Well_order natLeq"

  1206 unfolding well_order_on_def

  1207 using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto

  1208

  1209 lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"

  1210 unfolding Field_def by auto

  1211

  1212 lemma natLeq_underS_less: "underS natLeq n = {x. x < n}"

  1213 unfolding underS_def natLeq_def by auto

  1214

  1215 lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"

  1216 unfolding natLeq_def by force

  1217

  1218 lemma Restr_natLeq2:

  1219 "Restr natLeq (underS natLeq n) = natLeq_on n"

  1220 by (auto simp add: Restr_natLeq natLeq_underS_less)

  1221

  1222 lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"

  1223 using Restr_natLeq[of n] natLeq_Well_order

  1224       Well_order_Restr[of natLeq "{x. x < n}"] by auto

  1225

  1226 corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"

  1227 using natLeq_on_Well_order Field_natLeq_on by auto

  1228

  1229 lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"

  1230 unfolding wo_rel_def using natLeq_on_Well_order .

  1231

  1232

  1233 subsubsection \<open>Then as cardinals\<close>

  1234

  1235 lemma natLeq_Card_order: "Card_order natLeq"

  1236 proof(auto simp add: natLeq_Well_order

  1237       Card_order_iff_Restr_underS Restr_natLeq2, simp add:  Field_natLeq)

  1238   fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)

  1239   moreover have "\<not>finite(UNIV::nat set)" by auto

  1240   ultimately show "natLeq_on n <o |UNIV::nat set|"

  1241   using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]

  1242         Field_card_of[of "UNIV::nat set"]

  1243         card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto

  1244 qed

  1245

  1246 corollary card_of_Field_natLeq:

  1247 "|Field natLeq| =o natLeq"

  1248 using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]

  1249       ordIso_symmetric[of natLeq] by blast

  1250

  1251 corollary card_of_nat:

  1252 "|UNIV::nat set| =o natLeq"

  1253 using Field_natLeq card_of_Field_natLeq by auto

  1254

  1255 corollary infinite_iff_natLeq_ordLeq:

  1256 "\<not>finite A = ( natLeq \<le>o |A| )"

  1257 using infinite_iff_card_of_nat[of A] card_of_nat

  1258       ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast

  1259

  1260 corollary finite_iff_ordLess_natLeq:

  1261 "finite A = ( |A| <o natLeq)"

  1262 using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess

  1263       card_of_Well_order natLeq_Well_order by blast

  1264

  1265

  1266 subsection \<open>The successor of a cardinal\<close>

  1267

  1268 text\<open>First we define \<open>isCardSuc r r'\<close>, the notion of \<open>r'\<close>

  1269 being a successor cardinal of \<open>r\<close>. Although the definition does

  1270 not require \<open>r\<close> to be a cardinal, only this case will be meaningful.\<close>

  1271

  1272 definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"

  1273 where

  1274 "isCardSuc r r' \<equiv>

  1275  Card_order r' \<and> r <o r' \<and>

  1276  (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"

  1277

  1278 text\<open>Now we introduce the cardinal-successor operator \<open>cardSuc\<close>,

  1279 by picking {\em some} cardinal-order relation fulfilling \<open>isCardSuc\<close>.

  1280 Again, the picked item shall be proved unique up to order-isomorphism.\<close>

  1281

  1282 definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"

  1283 where

  1284 "cardSuc r \<equiv> SOME r'. isCardSuc r r'"

  1285

  1286 lemma exists_minim_Card_order:

  1287 "\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"

  1288 unfolding card_order_on_def using exists_minim_Well_order by blast

  1289

  1290 lemma exists_isCardSuc:

  1291 assumes "Card_order r"

  1292 shows "\<exists>r'. isCardSuc r r'"

  1293 proof-

  1294   let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"

  1295   have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms

  1296   by (simp add: card_of_Card_order Card_order_Pow)

  1297   then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"

  1298   using exists_minim_Card_order[of ?R] by blast

  1299   thus ?thesis unfolding isCardSuc_def by auto

  1300 qed

  1301

  1302 lemma cardSuc_isCardSuc:

  1303 assumes "Card_order r"

  1304 shows "isCardSuc r (cardSuc r)"

  1305 unfolding cardSuc_def using assms

  1306 by (simp add: exists_isCardSuc someI_ex)

  1307

  1308 lemma cardSuc_Card_order:

  1309 "Card_order r \<Longrightarrow> Card_order(cardSuc r)"

  1310 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1311

  1312 lemma cardSuc_greater:

  1313 "Card_order r \<Longrightarrow> r <o cardSuc r"

  1314 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1315

  1316 lemma cardSuc_ordLeq:

  1317 "Card_order r \<Longrightarrow> r \<le>o cardSuc r"

  1318 using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast

  1319

  1320 text\<open>The minimality property of \<open>cardSuc\<close> originally present in its definition

  1321 is local to the type \<open>'a set rel\<close>, i.e., that of \<open>cardSuc r\<close>:\<close>

  1322

  1323 lemma cardSuc_least_aux:

  1324 "\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"

  1325 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1326

  1327 text\<open>But from this we can infer general minimality:\<close>

  1328

  1329 lemma cardSuc_least:

  1330 assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"

  1331 shows "cardSuc r \<le>o r'"

  1332 proof-

  1333   let ?p = "cardSuc r"

  1334   have 0: "Well_order ?p \<and> Well_order r'"

  1335   using assms cardSuc_Card_order unfolding card_order_on_def by blast

  1336   {assume "r' <o ?p"

  1337    then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"

  1338    using internalize_ordLess[of r' ?p] by blast

  1339    (*  *)

  1340    have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast

  1341    moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast

  1342    ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast

  1343    hence False using 2 not_ordLess_ordLeq by blast

  1344   }

  1345   thus ?thesis using 0 ordLess_or_ordLeq by blast

  1346 qed

  1347

  1348 lemma cardSuc_ordLess_ordLeq:

  1349 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1350 shows "(r <o r') = (cardSuc r \<le>o r')"

  1351 proof(auto simp add: assms cardSuc_least)

  1352   assume "cardSuc r \<le>o r'"

  1353   thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast

  1354 qed

  1355

  1356 lemma cardSuc_ordLeq_ordLess:

  1357 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1358 shows "(r' <o cardSuc r) = (r' \<le>o r)"

  1359 proof-

  1360   have "Well_order r \<and> Well_order r'"

  1361   using assms unfolding card_order_on_def by auto

  1362   moreover have "Well_order(cardSuc r)"

  1363   using assms cardSuc_Card_order card_order_on_def by blast

  1364   ultimately show ?thesis

  1365   using assms cardSuc_ordLess_ordLeq[of r r']

  1366   not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast

  1367 qed

  1368

  1369 lemma cardSuc_mono_ordLeq:

  1370 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1371 shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"

  1372 using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast

  1373

  1374 lemma cardSuc_invar_ordIso:

  1375 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1376 shows "(cardSuc r =o cardSuc r') = (r =o r')"

  1377 proof-

  1378   have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"

  1379   using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)

  1380   thus ?thesis

  1381   using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq

  1382   using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast

  1383 qed

  1384

  1385 lemma card_of_cardSuc_finite:

  1386 "finite(Field(cardSuc |A| )) = finite A"

  1387 proof

  1388   assume *: "finite (Field (cardSuc |A| ))"

  1389   have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"

  1390   using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast

  1391   hence "|A| \<le>o |Field(cardSuc |A| )|"

  1392   using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric

  1393   ordLeq_ordIso_trans by blast

  1394   thus "finite A" using * card_of_ordLeq_finite by blast

  1395 next

  1396   assume "finite A"

  1397   then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp

  1398   then show "finite (Field (cardSuc |A| ))"

  1399   proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])

  1400     show "cardSuc |A| \<le>o |Pow A|"

  1401       by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)

  1402   qed

  1403 qed

  1404

  1405 lemma cardSuc_finite:

  1406 assumes "Card_order r"

  1407 shows "finite (Field (cardSuc r)) = finite (Field r)"

  1408 proof-

  1409   let ?A = "Field r"

  1410   have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)

  1411   hence "cardSuc |?A| =o cardSuc r" using assms

  1412   by (simp add: card_of_Card_order cardSuc_invar_ordIso)

  1413   moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"

  1414   by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)

  1415   moreover

  1416   {have "|Field (cardSuc r) | =o cardSuc r"

  1417    using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)

  1418    hence "cardSuc r =o |Field (cardSuc r) |"

  1419    using ordIso_symmetric by blast

  1420   }

  1421   ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"

  1422   using ordIso_transitive by blast

  1423   hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"

  1424   using card_of_ordIso_finite by blast

  1425   thus ?thesis by (simp only: card_of_cardSuc_finite)

  1426 qed

  1427

  1428 lemma card_of_Plus_ordLess_infinite:

  1429 assumes INF: "\<not>finite C" and

  1430         LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"

  1431 shows "|A <+> B| <o |C|"

  1432 proof(cases "A = {} \<or> B = {}")

  1433   assume Case1: "A = {} \<or> B = {}"

  1434   hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"

  1435   using card_of_Plus_empty1 card_of_Plus_empty2 by blast

  1436   hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"

  1437   using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast

  1438   thus ?thesis using LESS1 LESS2

  1439        ordIso_ordLess_trans[of "|A <+> B|" "|A|"]

  1440        ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast

  1441 next

  1442   assume Case2: "\<not>(A = {} \<or> B = {})"

  1443   {assume *: "|C| \<le>o |A <+> B|"

  1444    hence "\<not>finite (A <+> B)" using INF card_of_ordLeq_finite by blast

  1445    hence 1: "\<not>finite A \<or> \<not>finite B" using finite_Plus by blast

  1446    {assume Case21: "|A| \<le>o |B|"

  1447     hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast

  1448     hence "|A <+> B| =o |B|" using Case2 Case21

  1449     by (auto simp add: card_of_Plus_infinite)

  1450     hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast

  1451    }

  1452    moreover

  1453    {assume Case22: "|B| \<le>o |A|"

  1454     hence "\<not>finite A" using 1 card_of_ordLeq_finite by blast

  1455     hence "|A <+> B| =o |A|" using Case2 Case22

  1456     by (auto simp add: card_of_Plus_infinite)

  1457     hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast

  1458    }

  1459    ultimately have False using ordLeq_total card_of_Well_order[of A]

  1460    card_of_Well_order[of B] by blast

  1461   }

  1462   thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]

  1463   card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto

  1464 qed

  1465

  1466 lemma card_of_Plus_ordLess_infinite_Field:

  1467 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1468         LESS1: "|A| <o r" and LESS2: "|B| <o r"

  1469 shows "|A <+> B| <o r"

  1470 proof-

  1471   let ?C  = "Field r"

  1472   have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso

  1473   ordIso_symmetric by blast

  1474   hence "|A| <o |?C|"  "|B| <o |?C|"

  1475   using LESS1 LESS2 ordLess_ordIso_trans by blast+

  1476   hence  "|A <+> B| <o |?C|" using INF

  1477   card_of_Plus_ordLess_infinite by blast

  1478   thus ?thesis using 1 ordLess_ordIso_trans by blast

  1479 qed

  1480

  1481 lemma card_of_Plus_ordLeq_infinite_Field:

  1482 assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"

  1483 and c: "Card_order r"

  1484 shows "|A <+> B| \<le>o r"

  1485 proof-

  1486   let ?r' = "cardSuc r"

  1487   have "Card_order ?r' \<and> \<not>finite (Field ?r')" using assms

  1488   by (simp add: cardSuc_Card_order cardSuc_finite)

  1489   moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c

  1490   by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)

  1491   ultimately have "|A <+> B| <o ?r'"

  1492   using card_of_Plus_ordLess_infinite_Field by blast

  1493   thus ?thesis using c r

  1494   by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)

  1495 qed

  1496

  1497 lemma card_of_Un_ordLeq_infinite_Field:

  1498 assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"

  1499 and "Card_order r"

  1500 shows "|A Un B| \<le>o r"

  1501 using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq

  1502 ordLeq_transitive by fast

  1503

  1504

  1505 subsection \<open>Regular cardinals\<close>

  1506

  1507 definition cofinal where

  1508 "cofinal A r \<equiv>

  1509  ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"

  1510

  1511 definition regularCard where

  1512 "regularCard r \<equiv>

  1513  \<forall>K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"

  1514

  1515 definition relChain where

  1516 "relChain r As \<equiv>

  1517  \<forall>i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"

  1518

  1519 lemma regularCard_UNION:

  1520 assumes r: "Card_order r"   "regularCard r"

  1521 and As: "relChain r As"

  1522 and Bsub: "B \<le> (UN i : Field r. As i)"

  1523 and cardB: "|B| <o r"

  1524 shows "EX i : Field r. B \<le> As i"

  1525 proof-

  1526   let ?phi = "\<lambda>b j. j \<in> Field r \<and> b \<in> As j"

  1527   have "\<forall>b\<in>B. \<exists>j. ?phi b j" using Bsub by blast

  1528   then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"

  1529   using bchoice[of B ?phi] by blast

  1530   let ?K = "f  B"

  1531   {assume 1: "\<And>i. i \<in> Field r \<Longrightarrow> \<not> B \<le> As i"

  1532    have 2: "cofinal ?K r"

  1533    unfolding cofinal_def proof auto

  1534      fix i assume i: "i : Field r"

  1535      with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast

  1536      hence "i \<noteq> f b \<and> \<not> (f b,i) \<in> r"

  1537      using As f unfolding relChain_def by auto

  1538      hence "i \<noteq> f b \<and> (i, f b) : r" using r

  1539      unfolding card_order_on_def well_order_on_def linear_order_on_def

  1540      total_on_def using i f b by auto

  1541      with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast

  1542    qed

  1543    moreover have "?K \<le> Field r" using f by blast

  1544    ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast

  1545    moreover

  1546    {

  1547     have "|?K| <=o |B|" using card_of_image .

  1548     hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast

  1549    }

  1550    ultimately have False using not_ordLess_ordIso by blast

  1551   }

  1552   thus ?thesis by blast

  1553 qed

  1554

  1555 lemma infinite_cardSuc_regularCard:

  1556 assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"

  1557 shows "regularCard (cardSuc r)"

  1558 proof-

  1559   let ?r' = "cardSuc r"

  1560   have r': "Card_order ?r'"

  1561   "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"

  1562   using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)

  1563   show ?thesis

  1564   unfolding regularCard_def proof auto

  1565     fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"

  1566     hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)

  1567     also have 22: "|Field ?r'| =o ?r'"

  1568     using r' by (simp add: card_of_Field_ordIso[of ?r'])

  1569     finally have "|K| \<le>o ?r'" .

  1570     moreover

  1571     {let ?L = "UN j : K. underS ?r' j"

  1572      let ?J = "Field r"

  1573      have rJ: "r =o |?J|"

  1574      using r_card card_of_Field_ordIso ordIso_symmetric by blast

  1575      assume "|K| <o ?r'"

  1576      hence "|K| <=o r" using r' card_of_Card_order[of K] by blast

  1577      hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast

  1578      moreover

  1579      {have "ALL j : K. |underS ?r' j| <o ?r'"

  1580       using r' 1 by (auto simp: card_of_underS)

  1581       hence "ALL j : K. |underS ?r' j| \<le>o r"

  1582       using r' card_of_Card_order by blast

  1583       hence "ALL j : K. |underS ?r' j| \<le>o |?J|"

  1584       using rJ ordLeq_ordIso_trans by blast

  1585      }

  1586      ultimately have "|?L| \<le>o |?J|"

  1587      using r_inf card_of_UNION_ordLeq_infinite by blast

  1588      hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast

  1589      hence "|?L| <o ?r'" using r' card_of_Card_order by blast

  1590      moreover

  1591      {

  1592       have "Field ?r' \<le> ?L"

  1593       using 2 unfolding underS_def cofinal_def by auto

  1594       hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)

  1595       hence "?r' \<le>o |?L|"

  1596       using 22 ordIso_ordLeq_trans ordIso_symmetric by blast

  1597      }

  1598      ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast

  1599      hence False using ordLess_irreflexive by blast

  1600     }

  1601     ultimately show "|K| =o ?r'"

  1602     unfolding ordLeq_iff_ordLess_or_ordIso by blast

  1603   qed

  1604 qed

  1605

  1606 lemma cardSuc_UNION:

  1607 assumes r: "Card_order r" and "\<not>finite (Field r)"

  1608 and As: "relChain (cardSuc r) As"

  1609 and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"

  1610 and cardB: "|B| <=o r"

  1611 shows "EX i : Field (cardSuc r). B \<le> As i"

  1612 proof-

  1613   let ?r' = "cardSuc r"

  1614   have "Card_order ?r' \<and> |B| <o ?r'"

  1615   using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order

  1616   card_of_Card_order by blast

  1617   moreover have "regularCard ?r'"

  1618   using assms by(simp add: infinite_cardSuc_regularCard)

  1619   ultimately show ?thesis

  1620   using As Bsub cardB regularCard_UNION by blast

  1621 qed

  1622

  1623

  1624 subsection \<open>Others\<close>

  1625

  1626 lemma card_of_Func_Times:

  1627 "|Func (A \<times> B) C| =o |Func A (Func B C)|"

  1628 unfolding card_of_ordIso[symmetric]

  1629 using bij_betw_curr by blast

  1630

  1631 lemma card_of_Pow_Func:

  1632 "|Pow A| =o |Func A (UNIV::bool set)|"

  1633 proof-

  1634   define F where [abs_def]: "F A' a =

  1635     (if a \<in> A then (if a \<in> A' then True else False) else undefined)" for A' a

  1636   have "bij_betw F (Pow A) (Func A (UNIV::bool set))"

  1637   unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)

  1638     fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"

  1639     thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: if_split_asm)

  1640   next

  1641     show "F  Pow A = Func A UNIV"

  1642     proof safe

  1643       fix f assume f: "f \<in> Func A (UNIV::bool set)"

  1644       show "f \<in> F  Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)

  1645         let ?A1 = "{a \<in> A. f a = True}"

  1646         show "f = F ?A1"

  1647           unfolding F_def apply(rule ext)

  1648           using f unfolding Func_def mem_Collect_eq by auto

  1649       qed auto

  1650     qed(unfold Func_def mem_Collect_eq F_def, auto)

  1651   qed

  1652   thus ?thesis unfolding card_of_ordIso[symmetric] by blast

  1653 qed

  1654

  1655 lemma card_of_Func_UNIV:

  1656 "|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"

  1657 apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)

  1658   let ?F = "\<lambda> f (a::'a). ((f a)::'b)"

  1659   show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"

  1660   unfolding bij_betw_def inj_on_def proof safe

  1661     fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"

  1662     hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto

  1663     then obtain f where f: "\<forall> a. h a = f a" by blast

  1664     hence "range f \<subseteq> B" using h unfolding Func_def by auto

  1665     thus "h \<in> (\<lambda>f a. f a)  {f. range f \<subseteq> B}" using f by auto

  1666   qed(unfold Func_def fun_eq_iff, auto)

  1667 qed

  1668

  1669 lemma Func_Times_Range:

  1670   "|Func A (B \<times> C)| =o |Func A B \<times> Func A C|" (is "|?LHS| =o |?RHS|")

  1671 proof -

  1672   let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,

  1673                   \<lambda>x. if x \<in> A then snd (fg x) else undefined)"

  1674   let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"

  1675   have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def

  1676   proof (intro conjI impI ballI equalityI subsetI)

  1677     fix f g assume *: "f \<in> Func A (B \<times> C)" "g \<in> Func A (B \<times> C)" "?F f = ?F g"

  1678     show "f = g"

  1679     proof

  1680       fix x from * have "fst (f x) = fst (g x) \<and> snd (f x) = snd (g x)"

  1681         by (case_tac "x \<in> A") (auto simp: Func_def fun_eq_iff split: if_splits)

  1682       then show "f x = g x" by (subst (1 2) surjective_pairing) simp

  1683     qed

  1684   next

  1685     fix fg assume "fg \<in> Func A B \<times> Func A C"

  1686     thus "fg \<in> ?F  Func A (B \<times> C)"

  1687       by (intro image_eqI[of _ _ "?G fg"]) (auto simp: Func_def)

  1688   qed (auto simp: Func_def fun_eq_iff)

  1689   thus ?thesis using card_of_ordIso by blast

  1690 qed

  1691

  1692 end
`