src/HOL/BNF_Cardinal_Order_Relation.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 58127 b7cab82f488e child 58623 2db1df2c8467 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
     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 header {* Cardinal-Order Relations as Needed by Bounded Natural Functors *}

     9

    10 theory BNF_Cardinal_Order_Relation

    11 imports Zorn BNF_Wellorder_Constructions

    12 begin

    13

    14 text{* 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, @{text "card"}, from the theory @{text "Finite_Sets.thy"}).

    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 *}

    35

    36

    37 subsection {* Cardinal orders *}

    38

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

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

    41 strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}

    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{* 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 @{text "Zorn.thy"} as theorem @{text "well_order_on"}),

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

    64 \item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal

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

    66 \end{itemize}

    67 *}

    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 {* Cardinal of a set *}

   115

   116 text{* 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. *}

   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 Cantor_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 {* Cardinals versus set operations on arbitrary sets *}

   347

   348 text{* 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 @{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.

   352 *}

   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 ~= {}"

   391   hence "f  A ~= {}" 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   def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).

   474   case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a

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

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

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

   478   proof

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

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

   481     proof(cases x)

   482       case (Inl a)

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

   484       using x unfolding f_def by auto

   485       thus ?thesis by auto

   486     next

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

   488       proof(cases bc)

   489         case (Inl b)

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

   491         using x 1 unfolding f_def by auto

   492         thus ?thesis by auto

   493       next

   494         case (Inr c)

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

   496         using x 1 unfolding f_def by auto

   497         thus ?thesis by auto

   498       qed

   499     qed

   500   qed

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

   502   unfolding bij_betw_def inj_on_def f_def by fastforce

   503   thus ?thesis using card_of_ordIso by blast

   504 qed

   505

   506 lemma card_of_Plus_mono1:

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

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

   509 proof-

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

   511   using assms card_of_ordLeq[of A] by fastforce

   512   obtain g where g_def:

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

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

   515   proof-

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

   517                           "g d1 = g d2"

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

   519     }

   520     moreover

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

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

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

   524     }

   525     ultimately show ?thesis unfolding inj_on_def by auto

   526   qed

   527   thus ?thesis using card_of_ordLeq by blast

   528 qed

   529

   530 corollary ordLeq_Plus_mono1:

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

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

   533 using assms card_of_mono2 card_of_Plus_mono1 by blast

   534

   535 lemma card_of_Plus_mono2:

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

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

   538 using assms card_of_Plus_mono1[of A B C]

   539       card_of_Plus_commute[of C A]  card_of_Plus_commute[of B C]

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

   541 by blast

   542

   543 corollary ordLeq_Plus_mono2:

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

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

   546 using assms card_of_mono2 card_of_Plus_mono2 by blast

   547

   548 lemma card_of_Plus_mono:

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

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

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

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

   553

   554 corollary ordLeq_Plus_mono:

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

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

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

   558

   559 lemma card_of_Plus_cong1:

   560 assumes "|A| =o |B|"

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

   562 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)

   563

   564 corollary ordIso_Plus_cong1:

   565 assumes "r =o r'"

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

   567 using assms card_of_cong card_of_Plus_cong1 by blast

   568

   569 lemma card_of_Plus_cong2:

   570 assumes "|A| =o |B|"

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

   572 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)

   573

   574 corollary ordIso_Plus_cong2:

   575 assumes "r =o r'"

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

   577 using assms card_of_cong card_of_Plus_cong2 by blast

   578

   579 lemma card_of_Plus_cong:

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

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

   582 using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)

   583

   584 corollary ordIso_Plus_cong:

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

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

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

   588

   589 lemma card_of_Un_Plus_ordLeq:

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

   591 proof-

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

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

   594    unfolding inj_on_def by auto

   595    thus ?thesis using card_of_ordLeq by blast

   596 qed

   597

   598 lemma card_of_Times1:

   599 assumes "A \<noteq> {}"

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

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

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

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

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

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

   606 qed

   607

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

   609 proof-

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

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

   612   unfolding bij_betw_def inj_on_def by auto

   613   thus ?thesis using card_of_ordIso by blast

   614 qed

   615

   616 lemma card_of_Times2:

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

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

   619       ordLeq_ordIso_trans by blast

   620

   621 corollary Card_order_Times1:

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

   623 using card_of_Times1[of B] card_of_Field_ordIso

   624       ordIso_ordLeq_trans ordIso_symmetric by blast

   625

   626 corollary Card_order_Times2:

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

   628 using card_of_Times2[of A] card_of_Field_ordIso

   629       ordIso_ordLeq_trans ordIso_symmetric by blast

   630

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

   632 using card_of_Times1[of A]

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

   634

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

   636 proof-

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

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

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

   640   proof-

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

   642      hence "c1 = c2"

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

   644     }

   645     moreover

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

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

   648      by(case_tac c, auto)

   649     }

   650     moreover

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

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

   653      proof(cases bl)

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

   655        thus ?thesis using * by force

   656      next

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

   658        thus ?thesis using * by force

   659      qed

   660     }

   661     ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto

   662   qed

   663   thus ?thesis using card_of_ordIso by blast

   664 qed

   665

   666 lemma card_of_Times_mono1:

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

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

   669 proof-

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

   671   using assms card_of_ordLeq[of A] by fastforce

   672   obtain g where g_def:

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

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

   675   using 1 unfolding inj_on_def using g_def by auto

   676   thus ?thesis using card_of_ordLeq by blast

   677 qed

   678

   679 corollary ordLeq_Times_mono1:

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

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

   682 using assms card_of_mono2 card_of_Times_mono1 by blast

   683

   684 lemma card_of_Times_mono2:

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

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

   687 using assms card_of_Times_mono1[of A B C]

   688       card_of_Times_commute[of C A]  card_of_Times_commute[of B C]

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

   690 by blast

   691

   692 corollary ordLeq_Times_mono2:

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

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

   695 using assms card_of_mono2 card_of_Times_mono2 by blast

   696

   697 lemma card_of_Sigma_mono1:

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

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

   700 proof-

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

   702   using assms by (auto simp add: card_of_ordLeq)

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

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

   705     by atomize_elim (auto intro: bchoice)

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

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

   708   using 1 unfolding inj_on_def using g_def by force

   709   thus ?thesis using card_of_ordLeq by blast

   710 qed

   711

   712 lemma card_of_UNION_Sigma:

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

   714 using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by blast

   715

   716 lemma card_of_bool:

   717 assumes "a1 \<noteq> a2"

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

   719 proof-

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

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

   722   proof-

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

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

   725     }

   726     moreover

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

   728     }

   729     moreover

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

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

   732      proof(cases "a = a1")

   733        assume "a = a1"

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

   735      next

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

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

   738      qed

   739     }

   740     ultimately show ?thesis unfolding bij_betw_def inj_on_def by blast

   741   qed

   742   thus ?thesis using card_of_ordIso by blast

   743 qed

   744

   745 lemma card_of_Plus_Times_aux:

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

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

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

   749 proof-

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

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

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

   753   (*  *)

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

   755   using LEQ card_of_Plus_mono1 by blast

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

   757   using card_of_Plus_Times_bool by blast

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

   759   using 1 by (simp add: card_of_Times_mono2)

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

   761   using card_of_Times_commute by blast

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

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

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

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

   766   by blast

   767 qed

   768

   769 lemma card_of_Plus_Times:

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

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

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

   773 proof-

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

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

   776   }

   777   moreover

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

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

   780    using assms by (auto simp add: card_of_Plus_Times_aux)

   781    hence ?thesis

   782    using card_of_Plus_commute card_of_Times_commute

   783          ordIso_ordLeq_trans ordLeq_ordIso_trans by blast

   784   }

   785   ultimately show ?thesis

   786   using card_of_Well_order[of A] card_of_Well_order[of B]

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

   788 qed

   789

   790 lemma card_of_Times_Plus_distrib:

   791   "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")

   792 proof -

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

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

   795   thus ?thesis using card_of_ordIso by blast

   796 qed

   797

   798 lemma card_of_ordLeq_finite:

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

   800 shows "finite A"

   801 using assms unfolding ordLeq_def

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

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

   804

   805 lemma card_of_ordLeq_infinite:

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

   807 shows "\<not> finite B"

   808 using assms card_of_ordLeq_finite by auto

   809

   810 lemma card_of_ordIso_finite:

   811 assumes "|A| =o |B|"

   812 shows "finite A = finite B"

   813 using assms unfolding ordIso_def iso_def[abs_def]

   814 by (auto simp: bij_betw_finite Field_card_of)

   815

   816 lemma card_of_ordIso_finite_Field:

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

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

   819 using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast

   820

   821

   822 subsection {* Cardinals versus set operations involving infinite sets *}

   823

   824 text{* Here we show that, for infinite sets, most set-theoretic constructions

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

   826 theorem @{text "Card_order_Times_same_infinite"}, which states that self-product

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

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

   829 at page 47 in \cite{card-book}. Then everything else follows fairly easily. *}

   830

   831 lemma infinite_iff_card_of_nat:

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

   833 unfolding infinite_iff_countable_subset card_of_ordLeq ..

   834

   835 text{* The next two results correspond to the ZF fact that all infinite cardinals are

   836 limit ordinals: *}

   837

   838 lemma Card_order_infinite_not_under:

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

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

   841 proof(auto)

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

   843   using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto

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

   845   show False

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

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

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

   849     thus False using INF *  by auto

   850   next

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

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

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

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

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

   856     using 0 wo_rel.underS_ofilter * 1 Case2 by fast

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

   858     moreover

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

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

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

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

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

   864     using ordIso_symmetric ordLess_ordIso_trans by blast

   865     moreover

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

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

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

   869     }

   870     ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast

   871   qed

   872 qed

   873

   874 lemma infinite_Card_order_limit:

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

   876 and a: "a : Field r"

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

   878 proof-

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

   880   using assms Card_order_infinite_not_under by blast

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

   882   using under_Field .

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

   884   then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"

   885   unfolding under_def by blast

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

   887   using 1 r unfolding card_order_on_def well_order_on_def

   888   linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto

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

   890   using a r unfolding card_order_on_def well_order_on_def linear_order_on_def

   891   total_on_def by blast

   892   thus ?thesis using 1 ba by auto

   893 qed

   894

   895 theorem Card_order_Times_same_infinite:

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

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

   898 proof-

   899   obtain phi where phi_def:

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

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

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

   903   unfolding phi_def card_order_on_def by auto

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

   905   proof

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

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

   908     using temp1 by auto

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

   910                    3: "Card_order r \<and> Well_order r"

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

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

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

   914     using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast

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

   916     using card_of_Card_order card_of_Well_order by blast

   917     (*  *)

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

   919     using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast

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

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

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

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

   924     unfolding ordLess_def embedS_def[abs_def]

   925     by (auto simp add: Field_bsqr)

   926     let ?B = "f  ?A"

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

   928     using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast

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

   930     (*  *)

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

   932     using 6 embed_Field_ofilter 3 4 by blast

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

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

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

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

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

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

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

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

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

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

   943     let ?r1 = "Restr r A1"

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

   945     moreover

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

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

   948     }

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

   950     (*  *)

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

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

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

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

   955     using card_of_Card_order[of A1] card_of_Well_order[of A1]

   956     by (simp add: Field_card_of)

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

   958     using temp4 11 3 using not_ordLeq_iff_ordLess by blast

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

   960     by (simp add: card_of_card_order_on)

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

   962     using 2 unfolding phi_def by blast

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

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

   965     thus False using 11 not_ordLess_ordLeq by auto

   966   qed

   967   thus ?thesis using assms unfolding phi_def by blast

   968 qed

   969

   970 corollary card_of_Times_same_infinite:

   971 assumes "\<not>finite A"

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

   973 proof-

   974   let ?r = "|A|"

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

   976   using Field_card_of card_of_Card_order[of A] by fastforce

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

   978   using Card_order_Times_same_infinite[of ?r] assms by auto

   979   thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast

   980 qed

   981

   982 lemma card_of_Times_infinite:

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

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

   985 proof-

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

   987   using assms by (simp add: card_of_Times1 card_of_Times2)

   988   moreover

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

   990    using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast

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

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

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

   994   }

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

   996 qed

   997

   998 corollary Card_order_Times_infinite:

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

  1000         NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"

  1001 shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"

  1002 proof-

  1003   have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"

  1004   using assms by (simp add: card_of_Times_infinite card_of_mono2)

  1005   thus ?thesis

  1006   using assms card_of_Field_ordIso[of r]

  1007         ordIso_transitive[of "|Field r \<times> Field p|"]

  1008         ordIso_transitive[of _ "|Field r|"] by blast

  1009 qed

  1010

  1011 lemma card_of_Sigma_ordLeq_infinite:

  1012 assumes INF: "\<not>finite B" and

  1013         LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"

  1014 shows "|SIGMA i : I. A i| \<le>o |B|"

  1015 proof(cases "I = {}", simp add: card_of_empty)

  1016   assume *: "I \<noteq> {}"

  1017   have "|SIGMA i : I. A i| \<le>o |I \<times> B|"

  1018   using card_of_Sigma_mono1[OF LEQ] by blast

  1019   moreover have "|I \<times> B| =o |B|"

  1020   using INF * LEQ_I by (auto simp add: card_of_Times_infinite)

  1021   ultimately show ?thesis using ordLeq_ordIso_trans by blast

  1022 qed

  1023

  1024 lemma card_of_Sigma_ordLeq_infinite_Field:

  1025 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1026         LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"

  1027 shows "|SIGMA i : I. A i| \<le>o r"

  1028 proof-

  1029   let ?B  = "Field r"

  1030   have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso

  1031   ordIso_symmetric by blast

  1032   hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"

  1033   using LEQ_I LEQ ordLeq_ordIso_trans by blast+

  1034   hence  "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ

  1035   card_of_Sigma_ordLeq_infinite by blast

  1036   thus ?thesis using 1 ordLeq_ordIso_trans by blast

  1037 qed

  1038

  1039 lemma card_of_Times_ordLeq_infinite_Field:

  1040 "\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>

  1041  \<Longrightarrow> |A <*> B| \<le>o r"

  1042 by(simp add: card_of_Sigma_ordLeq_infinite_Field)

  1043

  1044 lemma card_of_Times_infinite_simps:

  1045 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"

  1046 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"

  1047 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"

  1048 "\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"

  1049 by (auto simp add: card_of_Times_infinite ordIso_symmetric)

  1050

  1051 lemma card_of_UNION_ordLeq_infinite:

  1052 assumes INF: "\<not>finite B" and

  1053         LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"

  1054 shows "|\<Union> i \<in> I. A i| \<le>o |B|"

  1055 proof(cases "I = {}", simp add: card_of_empty)

  1056   assume *: "I \<noteq> {}"

  1057   have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"

  1058   using card_of_UNION_Sigma by blast

  1059   moreover have "|SIGMA i : I. A i| \<le>o |B|"

  1060   using assms card_of_Sigma_ordLeq_infinite by blast

  1061   ultimately show ?thesis using ordLeq_transitive by blast

  1062 qed

  1063

  1064 corollary card_of_UNION_ordLeq_infinite_Field:

  1065 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1066         LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"

  1067 shows "|\<Union> i \<in> I. A i| \<le>o r"

  1068 proof-

  1069   let ?B  = "Field r"

  1070   have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso

  1071   ordIso_symmetric by blast

  1072   hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"

  1073   using LEQ_I LEQ ordLeq_ordIso_trans by blast+

  1074   hence  "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ

  1075   card_of_UNION_ordLeq_infinite by blast

  1076   thus ?thesis using 1 ordLeq_ordIso_trans by blast

  1077 qed

  1078

  1079 lemma card_of_Plus_infinite1:

  1080 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1081 shows "|A <+> B| =o |A|"

  1082 proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)

  1083   let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b"  let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"

  1084   assume *: "B \<noteq> {}"

  1085   then obtain b1 where 1: "b1 \<in> B" by blast

  1086   show ?thesis

  1087   proof(cases "B = {b1}")

  1088     assume Case1: "B = {b1}"

  1089     have 2: "bij_betw ?Inl A ((?Inl  A))"

  1090     unfolding bij_betw_def inj_on_def by auto

  1091     hence 3: "\<not>finite (?Inl  A)"

  1092     using INF bij_betw_finite[of ?Inl A] by blast

  1093     let ?A' = "?Inl  A \<union> {?Inr b1}"

  1094     obtain g where "bij_betw g (?Inl  A) ?A'"

  1095     using 3 infinite_imp_bij_betw2[of "?Inl  A"] by auto

  1096     moreover have "?A' = A <+> B" using Case1 by blast

  1097     ultimately have "bij_betw g (?Inl  A) (A <+> B)" by simp

  1098     hence "bij_betw (g o ?Inl) A (A <+> B)"

  1099     using 2 by (auto simp add: bij_betw_trans)

  1100     thus ?thesis using card_of_ordIso ordIso_symmetric by blast

  1101   next

  1102     assume Case2: "B \<noteq> {b1}"

  1103     with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce

  1104     obtain f where "inj_on f B \<and> f  B \<le> A"

  1105     using LEQ card_of_ordLeq[of B] by fastforce

  1106     with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"

  1107     unfolding inj_on_def by auto

  1108     with 3 have "|A <+> B| \<le>o |A \<times> B|"

  1109     by (auto simp add: card_of_Plus_Times)

  1110     moreover have "|A \<times> B| =o |A|"

  1111     using assms * by (simp add: card_of_Times_infinite_simps)

  1112     ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by blast

  1113     thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast

  1114   qed

  1115 qed

  1116

  1117 lemma card_of_Plus_infinite2:

  1118 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1119 shows "|B <+> A| =o |A|"

  1120 using assms card_of_Plus_commute card_of_Plus_infinite1

  1121 ordIso_equivalence by blast

  1122

  1123 lemma card_of_Plus_infinite:

  1124 assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"

  1125 shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"

  1126 using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)

  1127

  1128 corollary Card_order_Plus_infinite:

  1129 assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and

  1130         LEQ: "p \<le>o r"

  1131 shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"

  1132 proof-

  1133   have "| Field r <+> Field p | =o | Field r | \<and>

  1134         | Field p <+> Field r | =o | Field r |"

  1135   using assms by (simp add: card_of_Plus_infinite card_of_mono2)

  1136   thus ?thesis

  1137   using assms card_of_Field_ordIso[of r]

  1138         ordIso_transitive[of "|Field r <+> Field p|"]

  1139         ordIso_transitive[of _ "|Field r|"] by blast

  1140 qed

  1141

  1142

  1143 subsection {* The cardinal $\omega$ and the finite cardinals *}

  1144

  1145 text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict

  1146 order relation on

  1147 @{text "nat"}, that we abbreviate by @{text "natLeq"}.  The finite cardinals

  1148 shall be the restrictions of these relations to the numbers smaller than

  1149 fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}

  1150

  1151 definition "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"

  1152 definition "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"

  1153

  1154 abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"

  1155 where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"

  1156

  1157 lemma infinite_cartesian_product:

  1158 assumes "\<not>finite A" "\<not>finite B"

  1159 shows "\<not>finite (A \<times> B)"

  1160 proof

  1161   assume "finite (A \<times> B)"

  1162   from assms(1) have "A \<noteq> {}" by auto

  1163   with finite (A \<times> B) have "finite B" using finite_cartesian_productD2 by auto

  1164   with assms(2) show False by simp

  1165 qed

  1166

  1167

  1168 subsubsection {* First as well-orders *}

  1169

  1170 lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"

  1171 by(unfold Field_def natLeq_def, auto)

  1172

  1173 lemma natLeq_Refl: "Refl natLeq"

  1174 unfolding refl_on_def Field_def natLeq_def by auto

  1175

  1176 lemma natLeq_trans: "trans natLeq"

  1177 unfolding trans_def natLeq_def by auto

  1178

  1179 lemma natLeq_Preorder: "Preorder natLeq"

  1180 unfolding preorder_on_def

  1181 by (auto simp add: natLeq_Refl natLeq_trans)

  1182

  1183 lemma natLeq_antisym: "antisym natLeq"

  1184 unfolding antisym_def natLeq_def by auto

  1185

  1186 lemma natLeq_Partial_order: "Partial_order natLeq"

  1187 unfolding partial_order_on_def

  1188 by (auto simp add: natLeq_Preorder natLeq_antisym)

  1189

  1190 lemma natLeq_Total: "Total natLeq"

  1191 unfolding total_on_def natLeq_def by auto

  1192

  1193 lemma natLeq_Linear_order: "Linear_order natLeq"

  1194 unfolding linear_order_on_def

  1195 by (auto simp add: natLeq_Partial_order natLeq_Total)

  1196

  1197 lemma natLeq_natLess_Id: "natLess = natLeq - Id"

  1198 unfolding natLeq_def natLess_def by auto

  1199

  1200 lemma natLeq_Well_order: "Well_order natLeq"

  1201 unfolding well_order_on_def

  1202 using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto

  1203

  1204 lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"

  1205 unfolding Field_def by auto

  1206

  1207 lemma natLeq_underS_less: "underS natLeq n = {x. x < n}"

  1208 unfolding underS_def natLeq_def by auto

  1209

  1210 lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"

  1211 unfolding natLeq_def by force

  1212

  1213 lemma Restr_natLeq2:

  1214 "Restr natLeq (underS natLeq n) = natLeq_on n"

  1215 by (auto simp add: Restr_natLeq natLeq_underS_less)

  1216

  1217 lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"

  1218 using Restr_natLeq[of n] natLeq_Well_order

  1219       Well_order_Restr[of natLeq "{x. x < n}"] by auto

  1220

  1221 corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"

  1222 using natLeq_on_Well_order Field_natLeq_on by auto

  1223

  1224 lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"

  1225 unfolding wo_rel_def using natLeq_on_Well_order .

  1226

  1227

  1228 subsubsection {* Then as cardinals *}

  1229

  1230 lemma natLeq_Card_order: "Card_order natLeq"

  1231 proof(auto simp add: natLeq_Well_order

  1232       Card_order_iff_Restr_underS Restr_natLeq2, simp add:  Field_natLeq)

  1233   fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)

  1234   moreover have "\<not>finite(UNIV::nat set)" by auto

  1235   ultimately show "natLeq_on n <o |UNIV::nat set|"

  1236   using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]

  1237         Field_card_of[of "UNIV::nat set"]

  1238         card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto

  1239 qed

  1240

  1241 corollary card_of_Field_natLeq:

  1242 "|Field natLeq| =o natLeq"

  1243 using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]

  1244       ordIso_symmetric[of natLeq] by blast

  1245

  1246 corollary card_of_nat:

  1247 "|UNIV::nat set| =o natLeq"

  1248 using Field_natLeq card_of_Field_natLeq by auto

  1249

  1250 corollary infinite_iff_natLeq_ordLeq:

  1251 "\<not>finite A = ( natLeq \<le>o |A| )"

  1252 using infinite_iff_card_of_nat[of A] card_of_nat

  1253       ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast

  1254

  1255 corollary finite_iff_ordLess_natLeq:

  1256 "finite A = ( |A| <o natLeq)"

  1257 using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess

  1258       card_of_Well_order natLeq_Well_order by blast

  1259

  1260

  1261 subsection {* The successor of a cardinal *}

  1262

  1263 text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}

  1264 being a successor cardinal of @{text "r"}. Although the definition does

  1265 not require @{text "r"} to be a cardinal, only this case will be meaningful. *}

  1266

  1267 definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"

  1268 where

  1269 "isCardSuc r r' \<equiv>

  1270  Card_order r' \<and> r <o r' \<and>

  1271  (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"

  1272

  1273 text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},

  1274 by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.

  1275 Again, the picked item shall be proved unique up to order-isomorphism. *}

  1276

  1277 definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"

  1278 where

  1279 "cardSuc r \<equiv> SOME r'. isCardSuc r r'"

  1280

  1281 lemma exists_minim_Card_order:

  1282 "\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"

  1283 unfolding card_order_on_def using exists_minim_Well_order by blast

  1284

  1285 lemma exists_isCardSuc:

  1286 assumes "Card_order r"

  1287 shows "\<exists>r'. isCardSuc r r'"

  1288 proof-

  1289   let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"

  1290   have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms

  1291   by (simp add: card_of_Card_order Card_order_Pow)

  1292   then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"

  1293   using exists_minim_Card_order[of ?R] by blast

  1294   thus ?thesis unfolding isCardSuc_def by auto

  1295 qed

  1296

  1297 lemma cardSuc_isCardSuc:

  1298 assumes "Card_order r"

  1299 shows "isCardSuc r (cardSuc r)"

  1300 unfolding cardSuc_def using assms

  1301 by (simp add: exists_isCardSuc someI_ex)

  1302

  1303 lemma cardSuc_Card_order:

  1304 "Card_order r \<Longrightarrow> Card_order(cardSuc r)"

  1305 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1306

  1307 lemma cardSuc_greater:

  1308 "Card_order r \<Longrightarrow> r <o cardSuc r"

  1309 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1310

  1311 lemma cardSuc_ordLeq:

  1312 "Card_order r \<Longrightarrow> r \<le>o cardSuc r"

  1313 using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast

  1314

  1315 text{* The minimality property of @{text "cardSuc"} originally present in its definition

  1316 is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}

  1317

  1318 lemma cardSuc_least_aux:

  1319 "\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"

  1320 using cardSuc_isCardSuc unfolding isCardSuc_def by blast

  1321

  1322 text{* But from this we can infer general minimality: *}

  1323

  1324 lemma cardSuc_least:

  1325 assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"

  1326 shows "cardSuc r \<le>o r'"

  1327 proof-

  1328   let ?p = "cardSuc r"

  1329   have 0: "Well_order ?p \<and> Well_order r'"

  1330   using assms cardSuc_Card_order unfolding card_order_on_def by blast

  1331   {assume "r' <o ?p"

  1332    then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"

  1333    using internalize_ordLess[of r' ?p] by blast

  1334    (*  *)

  1335    have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast

  1336    moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast

  1337    ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast

  1338    hence False using 2 not_ordLess_ordLeq by blast

  1339   }

  1340   thus ?thesis using 0 ordLess_or_ordLeq by blast

  1341 qed

  1342

  1343 lemma cardSuc_ordLess_ordLeq:

  1344 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1345 shows "(r <o r') = (cardSuc r \<le>o r')"

  1346 proof(auto simp add: assms cardSuc_least)

  1347   assume "cardSuc r \<le>o r'"

  1348   thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast

  1349 qed

  1350

  1351 lemma cardSuc_ordLeq_ordLess:

  1352 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1353 shows "(r' <o cardSuc r) = (r' \<le>o r)"

  1354 proof-

  1355   have "Well_order r \<and> Well_order r'"

  1356   using assms unfolding card_order_on_def by auto

  1357   moreover have "Well_order(cardSuc r)"

  1358   using assms cardSuc_Card_order card_order_on_def by blast

  1359   ultimately show ?thesis

  1360   using assms cardSuc_ordLess_ordLeq[of r r']

  1361   not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast

  1362 qed

  1363

  1364 lemma cardSuc_mono_ordLeq:

  1365 assumes CARD: "Card_order r" and CARD': "Card_order r'"

  1366 shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"

  1367 using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast

  1368

  1369 lemma cardSuc_invar_ordIso:

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

  1371 shows "(cardSuc r =o cardSuc r') = (r =o r')"

  1372 proof-

  1373   have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"

  1374   using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)

  1375   thus ?thesis

  1376   using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq

  1377   using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast

  1378 qed

  1379

  1380 lemma card_of_cardSuc_finite:

  1381 "finite(Field(cardSuc |A| )) = finite A"

  1382 proof

  1383   assume *: "finite (Field (cardSuc |A| ))"

  1384   have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"

  1385   using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast

  1386   hence "|A| \<le>o |Field(cardSuc |A| )|"

  1387   using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric

  1388   ordLeq_ordIso_trans by blast

  1389   thus "finite A" using * card_of_ordLeq_finite by blast

  1390 next

  1391   assume "finite A"

  1392   then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp

  1393   then show "finite (Field (cardSuc |A| ))"

  1394   proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])

  1395     show "cardSuc |A| \<le>o |Pow A|"

  1396       by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)

  1397   qed

  1398 qed

  1399

  1400 lemma cardSuc_finite:

  1401 assumes "Card_order r"

  1402 shows "finite (Field (cardSuc r)) = finite (Field r)"

  1403 proof-

  1404   let ?A = "Field r"

  1405   have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)

  1406   hence "cardSuc |?A| =o cardSuc r" using assms

  1407   by (simp add: card_of_Card_order cardSuc_invar_ordIso)

  1408   moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"

  1409   by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)

  1410   moreover

  1411   {have "|Field (cardSuc r) | =o cardSuc r"

  1412    using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)

  1413    hence "cardSuc r =o |Field (cardSuc r) |"

  1414    using ordIso_symmetric by blast

  1415   }

  1416   ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"

  1417   using ordIso_transitive by blast

  1418   hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"

  1419   using card_of_ordIso_finite by blast

  1420   thus ?thesis by (simp only: card_of_cardSuc_finite)

  1421 qed

  1422

  1423 lemma card_of_Plus_ordLess_infinite:

  1424 assumes INF: "\<not>finite C" and

  1425         LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"

  1426 shows "|A <+> B| <o |C|"

  1427 proof(cases "A = {} \<or> B = {}")

  1428   assume Case1: "A = {} \<or> B = {}"

  1429   hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"

  1430   using card_of_Plus_empty1 card_of_Plus_empty2 by blast

  1431   hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"

  1432   using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast

  1433   thus ?thesis using LESS1 LESS2

  1434        ordIso_ordLess_trans[of "|A <+> B|" "|A|"]

  1435        ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast

  1436 next

  1437   assume Case2: "\<not>(A = {} \<or> B = {})"

  1438   {assume *: "|C| \<le>o |A <+> B|"

  1439    hence "\<not>finite (A <+> B)" using INF card_of_ordLeq_finite by blast

  1440    hence 1: "\<not>finite A \<or> \<not>finite B" using finite_Plus by blast

  1441    {assume Case21: "|A| \<le>o |B|"

  1442     hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast

  1443     hence "|A <+> B| =o |B|" using Case2 Case21

  1444     by (auto simp add: card_of_Plus_infinite)

  1445     hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast

  1446    }

  1447    moreover

  1448    {assume Case22: "|B| \<le>o |A|"

  1449     hence "\<not>finite A" using 1 card_of_ordLeq_finite by blast

  1450     hence "|A <+> B| =o |A|" using Case2 Case22

  1451     by (auto simp add: card_of_Plus_infinite)

  1452     hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast

  1453    }

  1454    ultimately have False using ordLeq_total card_of_Well_order[of A]

  1455    card_of_Well_order[of B] by blast

  1456   }

  1457   thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]

  1458   card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto

  1459 qed

  1460

  1461 lemma card_of_Plus_ordLess_infinite_Field:

  1462 assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and

  1463         LESS1: "|A| <o r" and LESS2: "|B| <o r"

  1464 shows "|A <+> B| <o r"

  1465 proof-

  1466   let ?C  = "Field r"

  1467   have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso

  1468   ordIso_symmetric by blast

  1469   hence "|A| <o |?C|"  "|B| <o |?C|"

  1470   using LESS1 LESS2 ordLess_ordIso_trans by blast+

  1471   hence  "|A <+> B| <o |?C|" using INF

  1472   card_of_Plus_ordLess_infinite by blast

  1473   thus ?thesis using 1 ordLess_ordIso_trans by blast

  1474 qed

  1475

  1476 lemma card_of_Plus_ordLeq_infinite_Field:

  1477 assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"

  1478 and c: "Card_order r"

  1479 shows "|A <+> B| \<le>o r"

  1480 proof-

  1481   let ?r' = "cardSuc r"

  1482   have "Card_order ?r' \<and> \<not>finite (Field ?r')" using assms

  1483   by (simp add: cardSuc_Card_order cardSuc_finite)

  1484   moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c

  1485   by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)

  1486   ultimately have "|A <+> B| <o ?r'"

  1487   using card_of_Plus_ordLess_infinite_Field by blast

  1488   thus ?thesis using c r

  1489   by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)

  1490 qed

  1491

  1492 lemma card_of_Un_ordLeq_infinite_Field:

  1493 assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"

  1494 and "Card_order r"

  1495 shows "|A Un B| \<le>o r"

  1496 using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq

  1497 ordLeq_transitive by fast

  1498

  1499

  1500 subsection {* Regular cardinals *}

  1501

  1502 definition cofinal where

  1503 "cofinal A r \<equiv>

  1504  ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"

  1505

  1506 definition regularCard where

  1507 "regularCard r \<equiv>

  1508  ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"

  1509

  1510 definition relChain where

  1511 "relChain r As \<equiv>

  1512  ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"

  1513

  1514 lemma regularCard_UNION:

  1515 assumes r: "Card_order r"   "regularCard r"

  1516 and As: "relChain r As"

  1517 and Bsub: "B \<le> (UN i : Field r. As i)"

  1518 and cardB: "|B| <o r"

  1519 shows "EX i : Field r. B \<le> As i"

  1520 proof-

  1521   let ?phi = "%b j. j : Field r \<and> b : As j"

  1522   have "ALL b : B. EX j. ?phi b j" using Bsub by blast

  1523   then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"

  1524   using bchoice[of B ?phi] by blast

  1525   let ?K = "f  B"

  1526   {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"

  1527    have 2: "cofinal ?K r"

  1528    unfolding cofinal_def proof auto

  1529      fix i assume i: "i : Field r"

  1530      with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast

  1531      hence "i \<noteq> f b \<and> ~ (f b,i) : r"

  1532      using As f unfolding relChain_def by auto

  1533      hence "i \<noteq> f b \<and> (i, f b) : r" using r

  1534      unfolding card_order_on_def well_order_on_def linear_order_on_def

  1535      total_on_def using i f b by auto

  1536      with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast

  1537    qed

  1538    moreover have "?K \<le> Field r" using f by blast

  1539    ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast

  1540    moreover

  1541    {

  1542     have "|?K| <=o |B|" using card_of_image .

  1543     hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast

  1544    }

  1545    ultimately have False using not_ordLess_ordIso by blast

  1546   }

  1547   thus ?thesis by blast

  1548 qed

  1549

  1550 lemma infinite_cardSuc_regularCard:

  1551 assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"

  1552 shows "regularCard (cardSuc r)"

  1553 proof-

  1554   let ?r' = "cardSuc r"

  1555   have r': "Card_order ?r'"

  1556   "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"

  1557   using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)

  1558   show ?thesis

  1559   unfolding regularCard_def proof auto

  1560     fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"

  1561     hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)

  1562     also have 22: "|Field ?r'| =o ?r'"

  1563     using r' by (simp add: card_of_Field_ordIso[of ?r'])

  1564     finally have "|K| \<le>o ?r'" .

  1565     moreover

  1566     {let ?L = "UN j : K. underS ?r' j"

  1567      let ?J = "Field r"

  1568      have rJ: "r =o |?J|"

  1569      using r_card card_of_Field_ordIso ordIso_symmetric by blast

  1570      assume "|K| <o ?r'"

  1571      hence "|K| <=o r" using r' card_of_Card_order[of K] by blast

  1572      hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast

  1573      moreover

  1574      {have "ALL j : K. |underS ?r' j| <o ?r'"

  1575       using r' 1 by (auto simp: card_of_underS)

  1576       hence "ALL j : K. |underS ?r' j| \<le>o r"

  1577       using r' card_of_Card_order by blast

  1578       hence "ALL j : K. |underS ?r' j| \<le>o |?J|"

  1579       using rJ ordLeq_ordIso_trans by blast

  1580      }

  1581      ultimately have "|?L| \<le>o |?J|"

  1582      using r_inf card_of_UNION_ordLeq_infinite by blast

  1583      hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast

  1584      hence "|?L| <o ?r'" using r' card_of_Card_order by blast

  1585      moreover

  1586      {

  1587       have "Field ?r' \<le> ?L"

  1588       using 2 unfolding underS_def cofinal_def by auto

  1589       hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)

  1590       hence "?r' \<le>o |?L|"

  1591       using 22 ordIso_ordLeq_trans ordIso_symmetric by blast

  1592      }

  1593      ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast

  1594      hence False using ordLess_irreflexive by blast

  1595     }

  1596     ultimately show "|K| =o ?r'"

  1597     unfolding ordLeq_iff_ordLess_or_ordIso by blast

  1598   qed

  1599 qed

  1600

  1601 lemma cardSuc_UNION:

  1602 assumes r: "Card_order r" and "\<not>finite (Field r)"

  1603 and As: "relChain (cardSuc r) As"

  1604 and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"

  1605 and cardB: "|B| <=o r"

  1606 shows "EX i : Field (cardSuc r). B \<le> As i"

  1607 proof-

  1608   let ?r' = "cardSuc r"

  1609   have "Card_order ?r' \<and> |B| <o ?r'"

  1610   using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order

  1611   card_of_Card_order by blast

  1612   moreover have "regularCard ?r'"

  1613   using assms by(simp add: infinite_cardSuc_regularCard)

  1614   ultimately show ?thesis

  1615   using As Bsub cardB regularCard_UNION by blast

  1616 qed

  1617

  1618

  1619 subsection {* Others *}

  1620

  1621 lemma card_of_Func_Times:

  1622 "|Func (A <*> B) C| =o |Func A (Func B C)|"

  1623 unfolding card_of_ordIso[symmetric]

  1624 using bij_betw_curr by blast

  1625

  1626 lemma card_of_Pow_Func:

  1627 "|Pow A| =o |Func A (UNIV::bool set)|"

  1628 proof-

  1629   def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)

  1630                             else undefined"

  1631   have "bij_betw F (Pow A) (Func A (UNIV::bool set))"

  1632   unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)

  1633     fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"

  1634     thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)

  1635   next

  1636     show "F  Pow A = Func A UNIV"

  1637     proof safe

  1638       fix f assume f: "f \<in> Func A (UNIV::bool set)"

  1639       show "f \<in> F  Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)

  1640         let ?A1 = "{a \<in> A. f a = True}"

  1641         show "f = F ?A1" unfolding F_def apply(rule ext)

  1642         using f unfolding Func_def mem_Collect_eq by auto

  1643       qed auto

  1644     qed(unfold Func_def mem_Collect_eq F_def, auto)

  1645   qed

  1646   thus ?thesis unfolding card_of_ordIso[symmetric] by blast

  1647 qed

  1648

  1649 lemma card_of_Func_UNIV:

  1650 "|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"

  1651 apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)

  1652   let ?F = "\<lambda> f (a::'a). ((f a)::'b)"

  1653   show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"

  1654   unfolding bij_betw_def inj_on_def proof safe

  1655     fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"

  1656     hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto

  1657     then obtain f where f: "\<forall> a. h a = f a" by blast

  1658     hence "range f \<subseteq> B" using h unfolding Func_def by auto

  1659     thus "h \<in> (\<lambda>f a. f a)  {f. range f \<subseteq> B}" using f by auto

  1660   qed(unfold Func_def fun_eq_iff, auto)

  1661 qed

  1662

  1663 lemma Func_Times_Range:

  1664   "|Func A (B <*> C)| =o |Func A B <*> Func A C|" (is "|?LHS| =o |?RHS|")

  1665 proof -

  1666   let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,

  1667                   \<lambda>x. if x \<in> A then snd (fg x) else undefined)"

  1668   let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"

  1669   have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def

  1670   proof (intro conjI impI ballI equalityI subsetI)

  1671     fix f g assume *: "f \<in> Func A (B \<times> C)" "g \<in> Func A (B \<times> C)" "?F f = ?F g"

  1672     show "f = g"

  1673     proof

  1674       fix x from * have "fst (f x) = fst (g x) \<and> snd (f x) = snd (g x)"

  1675         by (case_tac "x \<in> A") (auto simp: Func_def fun_eq_iff split: if_splits)

  1676       then show "f x = g x" by (subst (1 2) surjective_pairing) simp

  1677     qed

  1678   next

  1679     fix fg assume "fg \<in> Func A B \<times> Func A C"

  1680     thus "fg \<in> ?F  Func A (B \<times> C)"

  1681       by (intro image_eqI[of _ _ "?G fg"]) (auto simp: Func_def)

  1682   qed (auto simp: Func_def fun_eq_iff)

  1683   thus ?thesis using card_of_ordIso by blast

  1684 qed

  1685

  1686 end
`