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 ~= {}"
   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   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 : Field r \<and> ~ (b,a) : 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 o ?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  ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
  1514 
  1515 definition relChain where
  1516 "relChain r As \<equiv>
  1517  ALL 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 = "%b j. j : Field r \<and> b : As j"
  1527   have "ALL b : B. EX 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: "!! i. i : Field r \<Longrightarrow> ~ 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> ~ (f b,i) : 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