Quelle BNF_Cardinal_Order_Relation.thy Sprache: Isabelle

(*  Title:      HOL/BNF_Cardinal_Order_Relation.thy
    Author:     Andrei Popescu, TU Muenchen
    Author:     Jan van Brügge, TU Muenchen
    Copyright   2012, 2022

Cardinal-order relations as needed by bounded natural functors.
*)

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

theory BNF_Cardinal_Order_Relation
  imports Zorn BNF_Wellorder_Constructions
begin

text\<open>In this section, we define cardinal-order relations to be minim well-orders
on their field.  Then we define the cardinal of a set to be {\em some} cardinal-order
relation on that set, which will be unique up to order isomorphism.  Then we study
the connection between cardinals and:
\begin{itemize}
\item standard set-theoretic constructions: products,
sums, unions, lists, powersets, set-of finite sets operator;
\item finiteness and infiniteness (in particular, with the numeric cardinal operator
for finite sets, \<open>card\<close>, from the theory \<open>Finite_Sets.thy\<close>).
\end{itemize}
%
On the way, we define the canonical $\omega$ cardinal and finite cardinals.  We also
define (again, up to order isomorphism) the successor of a cardinal, and show that
any cardinal admits a successor.

Main results of this section are the existence of cardinal relations and the
facts that, in the presence of infiniteness,
most of the standard set-theoretic constructions (except for the powerset)
{\em do not increase cardinality}.  In particular, e.g., the set of words/lists over
any infinite set has the same cardinality (hence, is in bijection) with that set.
\<close>

subsection \<open>Cardinal orders\<close>

text\<open>A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the
order-embedding relation, \<open>\<le>o\<close> (which is the same as being {\em minimal} w.r.t. the
strict order-embedding relation, \<open><o\<close>), among all the well-orders on its field.\<close>

definition card_order_on :: "'a set \ 'a rel \ bool"
  where
    "card_order_on A r \ well_order_on A r \ (\r'. well_order_on A r' \ r \o r')"

abbreviation "Card_order r \ card_order_on (Field r) r"
abbreviation "card_order r \ card_order_on UNIV r"

lemma card_order_on_well_order_on:
  assumes "card_order_on A r"
  shows "well_order_on A r"
  using assms unfolding card_order_on_def by simp

lemma card_order_on_Card_order:
  "card_order_on A r \ A = Field r \ Card_order r"
  unfolding card_order_on_def using well_order_on_Field by blast

text\<open>The existence of a cardinal relation on any given set (which will mean
that any set has a cardinal) follows from two facts:
\begin{itemize}
\item Zermelo's theorem (proved in \<open>Zorn.thy\<close> as theorem \<open>well_order_on\<close>),
which states that on any given set there exists a well-order;
\item The well-founded-ness of \<open><o\<close>, ensuring that then there exists a minimal
such well-order, i.e., a cardinal order.
\end{itemize}
\<close>

theorem card_order_on: "\r. card_order_on A r"
proof -
  define R where "R \ {r. well_order_on A r}"
  have "R \ {} \ (\r \ R. Well_order r)"
    using well_order_on[of A] R_def well_order_on_Well_order by blast
  with exists_minim_Well_order[of R] show ?thesis
    by (auto simp: R_def card_order_on_def)
qed

lemma card_order_on_ordIso:
  assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
  shows "r =o r'"
  using assms unfolding card_order_on_def
  using ordIso_iff_ordLeq by blast

lemma Card_order_ordIso:
  assumes CO: "Card_order r" and ISO: "r' =o r"
  shows "Card_order r'"
  using ISO unfolding ordIso_def
proof(unfold card_order_on_def, auto)
  fix p' assume "well_order_on (Field r') p'"
  hence 0: "Well_order p' \ Field p' = Field r'"
    using well_order_on_Well_order by blast
  obtain f where 1: "iso r' r f" and 2: "Well_order r \ Well_order r'"
    using ISO unfolding ordIso_def by auto
  hence 3: "inj_on f (Field r') \ f ` (Field r') = Field r"
    by (auto simp add: iso_iff embed_inj_on)
  let ?p = "dir_image p' f"
  have 4: "p' =o ?p \ Well_order ?p"
    using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
  moreover have "Field ?p = Field r"
    using 0 3 by (auto simp add: dir_image_Field)
  ultimately have "well_order_on (Field r) ?p" by auto
  hence "r \o ?p" using CO unfolding card_order_on_def by auto
  thus "r' \o p'"
    using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
qed

lemma Card_order_ordIso2:
  assumes CO: "Card_order r" and ISO: "r =o r'"
  shows "Card_order r'"
  using assms Card_order_ordIso ordIso_symmetric by blast

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

text\<open>We define the cardinal of set to be {\em some} cardinal order on that set.
We shall prove that this notion is unique up to order isomorphism, meaning
that order isomorphism shall be the true identity of cardinals.\<close>

definition card_of :: "'a set \ 'a rel" ($\open_block notation=\mixfix card_of\\|_|)$
  where "card_of A = (SOME r. card_order_on A r)"

lemma card_of_card_order_on: "card_order_on A |A|"
  unfolding card_of_def by (auto simp add: card_order_on someI_ex)

lemma card_of_well_order_on: "well_order_on A |A|"
  using card_of_card_order_on card_order_on_def by blast

lemma Field_card_of: "Field |A| = A"
  using card_of_card_order_on[of A] unfolding card_order_on_def
  using well_order_on_Field by blast

lemma card_of_Card_order: "Card_order |A|"
  by (simp only: card_of_card_order_on Field_card_of)

corollary ordIso_card_of_imp_Card_order:
  "r =o |A| \ Card_order r"
  using card_of_Card_order Card_order_ordIso by blast

lemma card_of_Well_order: "Well_order |A|"
  using card_of_Card_order unfolding card_order_on_def by auto

lemma card_of_refl: "|A| =o |A|"
  using card_of_Well_order ordIso_reflexive by blast

lemma card_of_least: "well_order_on A r \ |A| \o r"
  using card_of_card_order_on unfolding card_order_on_def by blast

lemma card_of_ordIso:
  "(\f. bij_betw f A B) = ( |A| =o |B| )"
proof(auto)
  fix f assume *: "bij_betw f A B"
  then obtain r where "well_order_on B r \ |A| =o r"
    using Well_order_iso_copy card_of_well_order_on by blast
  hence "|B| \o |A|" using card_of_least
      ordLeq_ordIso_trans ordIso_symmetric by blast
  moreover
  {let ?g = "inv_into A f"
    have "bij_betw ?g B A" using * bij_betw_inv_into by blast
    then obtain r where "well_order_on A r \ |B| =o r"
      using Well_order_iso_copy card_of_well_order_on by blast
    hence "|A| \o |B|"
      using card_of_least ordLeq_ordIso_trans ordIso_symmetric by blast
  }
  ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
next
  assume "|A| =o |B|"
  then obtain f where "iso ( |A| ) ( |B| ) f"
    unfolding ordIso_def by auto
  hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
  thus "\f. bij_betw f A B" by auto
qed

lemma card_of_ordLeq:
  "(\f. inj_on f A \ f ` A \ B) = ( |A| \o |B| )"
proof(auto)
  fix f assume *: "inj_on f A" and **: "f ` A \ B"
  {assume "|B|
    hence "|B| \o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
    then obtain g where "embed ( |B| ) ( |A| ) g"
      unfolding ordLeq_def by auto
    hence 1: "inj_on g B \ g ` B \ A" using embed_inj_on[of "|B|" "|A|" "g"]
        card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
        embed_Field[of "|B|" "|A|" g] by auto
    obtain h where "bij_betw h A B"
      using * ** 1 Schroeder_Bernstein[of f] by fastforce
    hence "|A| \o |B|" using card_of_ordIso ordIso_iff_ordLeq by auto
  }
  thus "|A| \o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
    by (auto simp: card_of_Well_order)
next
  assume *: "|A| \o |B|"
  obtain f where "embed |A| |B| f"
    using * unfolding ordLeq_def by auto
  hence "inj_on f A \ f ` A \ B"
    using embed_inj_on[of "|A|" "|B|"] card_of_Well_order embed_Field[of "|A|" "|B|"]
    by (auto simp: Field_card_of)
  thus "\f. inj_on f A \ f ` A \ B" by auto
qed

lemma card_of_ordLeq2:
  "A \ {} \ (\g. g ` B = A) = ( |A| \o |B| )"
  using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto

lemma card_of_ordLess:
  "(\(\f. inj_on f A \ f ` A \ B)) = ( |B|
proof -
  have "(\(\f. inj_on f A \ f ` A \ B)) = (\ |A| \o |B| )"
    using card_of_ordLeq by blast
  also have "\ = ( |B|
    using  not_ordLeq_iff_ordLess by (auto intro: card_of_Well_order)
  finally show ?thesis .
qed

lemma card_of_ordLess2:
  "B \ {} \ (\(\f. f ` A = B)) = ( |A|
  using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto

lemma card_of_ordIsoI:
  assumes "bij_betw f A B"
  shows "|A| =o |B|"
  using assms unfolding card_of_ordIso[symmetric] by auto

lemma card_of_ordLeqI:
  assumes "inj_on f A" and "\ a. a \ A \ f a \ B"
  shows "|A| \o |B|"
  using assms unfolding card_of_ordLeq[symmetric] by auto

lemma card_of_unique:
  "card_order_on A r \ r =o |A|"
  by (simp only: card_order_on_ordIso card_of_card_order_on)

lemma card_of_mono1:
  "A \ B \ |A| \o |B|"
  using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce

lemma card_of_mono2:
  assumes "r \o r'"
  shows "|Field r| \o |Field r'|"
proof -
  obtain f where
    1: "well_order_on (Field r) r \ well_order_on (Field r) r \ embed r r' f"
    using assms unfolding ordLeq_def
    by (auto simp add: well_order_on_Well_order)
  hence "inj_on f (Field r) \ f ` (Field r) \ Field r'"
    by (auto simp add: embed_inj_on embed_Field)
  thus "|Field r| \o |Field r'|" using card_of_ordLeq by blast
qed

lemma card_of_cong: "r =o r' \ |Field r| =o |Field r'|"
  by (simp add: ordIso_iff_ordLeq card_of_mono2)

lemma card_of_Field_ordIso:
  assumes "Card_order r"
  shows "|Field r| =o r"
proof -
  have "card_order_on (Field r) r"
    using assms card_order_on_Card_order by blast
  moreover have "card_order_on (Field r) |Field r|"
    using card_of_card_order_on by blast
  ultimately show ?thesis using card_order_on_ordIso by blast
qed

lemma Card_order_iff_ordIso_card_of:
  "Card_order r = (r =o |Field r| )"
  using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast

lemma Card_order_iff_ordLeq_card_of:
  "Card_order r = (r \o |Field r| )"
proof -
  have "Card_order r = (r =o |Field r| )"
    unfolding Card_order_iff_ordIso_card_of by simp
  also have "\ = (r \o |Field r| \ |Field r| \o r)"
    unfolding ordIso_iff_ordLeq by simp
  also have "\ = (r \o |Field r| )"
    using card_of_least
    by (auto simp: card_of_least ordLeq_Well_order_simp)
  finally show ?thesis .
qed

lemma Card_order_iff_Restr_underS:
  assumes "Well_order r"
  shows "Card_order r = (\a \ Field r. Restr r (underS r a)
  using assms ordLeq_iff_ordLess_Restr card_of_Well_order
  unfolding Card_order_iff_ordLeq_card_of  by blast

lemma card_of_underS:
  assumes r: "Card_order r" and a: "a \ Field r"
  shows "|underS r a|
proof -
  let ?A = "underS r a"  let ?r' = "Restr r ?A"
  have 1: "Well_order r"
    using r unfolding card_order_on_def by simp
  have "Well_order ?r'" using 1 Well_order_Restr by auto
  with card_of_card_order_on have "|Field ?r'| \o ?r'"
    unfolding card_order_on_def by auto
  moreover have "Field ?r' = ?A"
    using 1 wo_rel.underS_ofilter Field_Restr_ofilter
    unfolding wo_rel_def by fastforce
  ultimately have "|?A| \o ?r'" by simp
  also have "?r'
    using 1 a r Card_order_iff_Restr_underS by blast
  also have "|Field r| =o r"
    using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
  finally show ?thesis .
qed

lemma ordLess_Field:
  assumes "r
  shows "|Field r|
proof -
  have "well_order_on (Field r) r" using assms unfolding ordLess_def
    by (auto simp add: well_order_on_Well_order)
  hence "|Field r| \o r" using card_of_least by blast
  thus ?thesis using assms ordLeq_ordLess_trans by blast
qed

lemma internalize_card_of_ordLeq:
  "( |A| \o r) = (\B \ Field r. |A| =o |B| \ |B| \o r)"
proof
  assume "|A| \o r"
  then obtain p where 1: "Field p \ Field r \ |A| =o p \ p \o r"
    using internalize_ordLeq[of "|A|" r] by blast
  hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
  hence "|Field p| =o p" using card_of_Field_ordIso by blast
  hence "|A| =o |Field p| \ |Field p| \o r"
    using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
  thus "\B \ Field r. |A| =o |B| \ |B| \o r" using 1 by blast
next
  assume "\B \ Field r. |A| =o |B| \ |B| \o r"
  thus "|A| \o r" using ordIso_ordLeq_trans by blast
qed

lemma internalize_card_of_ordLeq2:
  "( |A| \o |C| ) = (\B \ C. |A| =o |B| \ |B| \o |C| )"
  using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto

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

text\<open>Here we embark in a long journey of simple results showing
that the standard set-theoretic operations are well-behaved w.r.t. the notion of
cardinal -- essentially, this means that they preserve the ``cardinal identity"
\<open>=o\<close> and are monotonic w.r.t. \<open>\<le>o\<close>.
\<close>

lemma card_of_empty: "|{}| \o |A|"
  using card_of_ordLeq inj_on_id by blast

lemma card_of_empty1:
  assumes "Well_order r \ Card_order r"
  shows "|{}| \o r"
proof -
  have "Well_order r" using assms unfolding card_order_on_def by auto
  hence "|Field r| \o r"
    using assms card_of_least by blast
  moreover have "|{}| \o |Field r|" by (simp add: card_of_empty)
  ultimately show ?thesis using ordLeq_transitive by blast
qed

corollary Card_order_empty:
  "Card_order r \ |{}| \o r" by (simp add: card_of_empty1)

lemma card_of_empty2:
  assumes "|A| =o |{}|"
  shows "A = {}"
  using assms card_of_ordIso[of A] bij_betw_empty2 by blast

lemma card_of_empty3:
  assumes "|A| \o |{}|"
  shows "A = {}"
  using assms
  by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
      ordLeq_Well_order_simp)

lemma card_of_empty_ordIso:
  "|{}::'a set| =o |{}::'b set|"
  using card_of_ordIso unfolding bij_betw_def inj_on_def by blast

lemma card_of_image:
  "|f ` A| \o |A|"
proof(cases "A = {}")
  case False
  hence "f ` A \ {}" by auto
  thus ?thesis
    using card_of_ordLeq2[of "f ` A" A] by auto
qed (simp add: card_of_empty)

lemma surj_imp_ordLeq:
  assumes "B \ f ` A"
  shows "|B| \o |A|"
proof -
  have "|B| \o |f ` A|" using assms card_of_mono1 by auto
  thus ?thesis using card_of_image ordLeq_transitive by blast
qed

lemma card_of_singl_ordLeq:
  assumes "A \ {}"
  shows "|{b}| \o |A|"
proof -
  obtain a where *: "a \ A" using assms by auto
  let ?h = "\ b'::'b. if b' = b then a else undefined"
  have "inj_on ?h {b} \ ?h ` {b} \ A"
    using * unfolding inj_on_def by auto
  thus ?thesis unfolding card_of_ordLeq[symmetric] by (intro exI)
qed

corollary Card_order_singl_ordLeq:
  "\Card_order r; Field r \ {}\ \ |{b}| \o r"
  using card_of_singl_ordLeq[of "Field r" b]
    card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast

lemma card_of_Pow: "|A|
  using card_of_ordLess2[of "Pow A" A]  Cantors_theorem[of A]
    Pow_not_empty[of A] by auto

corollary Card_order_Pow:
  "Card_order r \ r
  using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast

lemma card_of_Plus1: "|A| \o |A <+> B|" and card_of_Plus2: "|B| \o |A <+> B|"
  using card_of_ordLeq by force+

corollary Card_order_Plus1:
  "Card_order r \ r \o |(Field r) <+> B|"
  using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast

corollary Card_order_Plus2:
  "Card_order r \ r \o |A <+> (Field r)|"
  using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast

lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
proof -
  have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
  thus ?thesis using card_of_ordIso by auto
qed

lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
proof -
  have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
  thus ?thesis using card_of_ordIso by auto
qed

lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
proof -
  let ?f = "\c. case c of Inl a \ Inr a | Inr b \ Inl b"
  have "bij_betw ?f (A <+> B) (B <+> A)"
    unfolding bij_betw_def inj_on_def by force
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_Plus_assoc:
  fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
  shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
proof -
  define f :: "('a + 'b) + 'c \ 'a + 'b + 'c"
    where [abs_def]: "f k =
      (case k of
        Inl ab \<Rightarrow>
          (case ab of
            Inl a \<Rightarrow> Inl a
          | Inr b \<Rightarrow> Inr (Inl b))
      | Inr c \<Rightarrow> Inr (Inr c))"
    for k
  have "A <+> B <+> C \ f ` ((A <+> B) <+> C)"
  proof
    fix x assume x: "x \ A <+> B <+> C"
    show "x \ f ` ((A <+> B) <+> C)"
    proof(cases x)
      case (Inl a)
      hence "a \ A" "x = f (Inl (Inl a))"
        using x unfolding f_def by auto
      thus ?thesis by auto
    next
      case (Inr bc) with x show ?thesis
        by (cases bc) (force simp: f_def)+
    qed
  qed
  hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
    unfolding bij_betw_def inj_on_def f_def by fastforce
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_Plus_mono1:
  assumes "|A| \o |B|"
  shows "|A <+> C| \o |B <+> C|"
proof -
  obtain f where f: "inj_on f A \ f ` A \ B"
    using assms card_of_ordLeq[of A] by fastforce
  define g where "g \ \d. case d of Inl a \ Inl(f a) | Inr (c::'c) \ Inr c"
  have "inj_on g (A <+> C) \ g ` (A <+> C) \ (B <+> C)"
    using f unfolding inj_on_def g_def by force
  thus ?thesis using card_of_ordLeq by blast
qed

corollary ordLeq_Plus_mono1:
  assumes "r \o r'"
  shows "|(Field r) <+> C| \o |(Field r') <+> C|"
  using assms card_of_mono2 card_of_Plus_mono1 by blast

lemma card_of_Plus_mono2:
  assumes "|A| \o |B|"
  shows "|C <+> A| \o |C <+> B|"
  using card_of_Plus_mono1[OF assms]
  by (blast intro: card_of_Plus_commute ordIso_ordLeq_trans ordLeq_ordIso_trans)

corollary ordLeq_Plus_mono2:
  assumes "r \o r'"
  shows "|A <+> (Field r)| \o |A <+> (Field r')|"
  using assms card_of_mono2 card_of_Plus_mono2 by blast

lemma card_of_Plus_mono:
  assumes "|A| \o |B|" and "|C| \o |D|"
  shows "|A <+> C| \o |B <+> D|"
  using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
    ordLeq_transitive by blast

corollary ordLeq_Plus_mono:
  assumes "r \o r'" and "p \o p'"
  shows "|(Field r) <+> (Field p)| \o |(Field r') <+> (Field p')|"
  using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast

lemma card_of_Plus_cong1:
  assumes "|A| =o |B|"
  shows "|A <+> C| =o |B <+> C|"
  using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)

corollary ordIso_Plus_cong1:
  assumes "r =o r'"
  shows "|(Field r) <+> C| =o |(Field r') <+> C|"
  using assms card_of_cong card_of_Plus_cong1 by blast

lemma card_of_Plus_cong2:
  assumes "|A| =o |B|"
  shows "|C <+> A| =o |C <+> B|"
  using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)

corollary ordIso_Plus_cong2:
  assumes "r =o r'"
  shows "|A <+> (Field r)| =o |A <+> (Field r')|"
  using assms card_of_cong card_of_Plus_cong2 by blast

lemma card_of_Plus_cong:
  assumes "|A| =o |B|" and "|C| =o |D|"
  shows "|A <+> C| =o |B <+> D|"
  using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)

corollary ordIso_Plus_cong:
  assumes "r =o r'" and "p =o p'"
  shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
  using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast

lemma card_of_Un_Plus_ordLeq:
  "|A \ B| \o |A <+> B|"
proof -
  let ?f = "\ c. if c \ A then Inl c else Inr c"
  have "inj_on ?f (A \ B) \ ?f ` (A \ B) \ A <+> B"
    unfolding inj_on_def by auto
  thus ?thesis using card_of_ordLeq by blast
qed

lemma card_of_Times1:
  assumes "A \ {}"
  shows "|B| \o |B \ A|"
proof(cases "B = {}")
  case False
  have "fst `(B \ A) = B" using assms by auto
  thus ?thesis using inj_on_iff_surj[of B "B \ A"]
      card_of_ordLeq False by blast
qed (simp add: card_of_empty)

lemma card_of_Times_commute: "|A \ B| =o |B \ A|"
proof -
  have "bij_betw (\(a,b). (b,a)) (A \ B) (B \ A)"
    unfolding bij_betw_def inj_on_def by auto
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_Times2:
  assumes "A \ {}" shows "|B| \o |A \ B|"
  using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
    ordLeq_ordIso_trans by blast

corollary Card_order_Times1:
  "\Card_order r; B \ {}\ \ r \o |(Field r) \ B|"
  using card_of_Times1[of B] card_of_Field_ordIso
    ordIso_ordLeq_trans ordIso_symmetric by blast

corollary Card_order_Times2:
  "\Card_order r; A \ {}\ \ r \o |A \ (Field r)|"
  using card_of_Times2[of A] card_of_Field_ordIso
    ordIso_ordLeq_trans ordIso_symmetric by blast

lemma card_of_Times3: "|A| \o |A \ A|"
  using card_of_Times1[of A]
  by(cases "A = {}", simp add: card_of_empty)

lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \ (UNIV::bool set)|"
proof -
  let ?f = "\c::'a + 'a. case c of Inl a \ (a,True)
                                  |Inr a \<Rightarrow> (a,False)"
  have "bij_betw ?f (A <+> A) (A \ (UNIV::bool set))"
  proof -
    have "\c1 c2. ?f c1 = ?f c2 \ c1 = c2"
      by (force split: sum.split_asm)
    moreover
    have "\c. c \ A <+> A \ ?f c \ A \ (UNIV::bool set)"
      by (force split: sum.split_asm)
    moreover
    {fix a bl assume "(a,bl) \ A \ (UNIV::bool set)"
      hence "(a,bl) \ ?f ` ( A <+> A)"
      by (cases bl) (force split: sum.split_asm)+
    }
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
  qed
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_Times_mono1:
  assumes "|A| \o |B|"
  shows "|A \ C| \o |B \ C|"
proof -
  obtain f where f: "inj_on f A \ f ` A \ B"
    using assms card_of_ordLeq[of A] by fastforce
  define g where "g \ (\(a,c::'c). (f a,c))"
  have "inj_on g (A \ C) \ g ` (A \ C) \ (B \ C)"
    using f unfolding inj_on_def using g_def by auto
  thus ?thesis using card_of_ordLeq by blast
qed

corollary ordLeq_Times_mono1:
  assumes "r \o r'"
  shows "|(Field r) \ C| \o |(Field r') \ C|"
  using assms card_of_mono2 card_of_Times_mono1 by blast

lemma card_of_Times_mono2:
  assumes "|A| \o |B|"
  shows "|C \ A| \o |C \ B|"
  using assms card_of_Times_mono1[of A B C]
  by (blast intro: card_of_Times_commute ordIso_ordLeq_trans ordLeq_ordIso_trans)

corollary ordLeq_Times_mono2:
  assumes "r \o r'"
  shows "|A \ (Field r)| \o |A \ (Field r')|"
  using assms card_of_mono2 card_of_Times_mono2 by blast

lemma card_of_Sigma_mono1:
  assumes "\i \ I. |A i| \o |B i|"
  shows "|SIGMA i : I. A i| \o |SIGMA i : I. B i|"
proof -
  have "\i. i \ I \ (\f. inj_on f (A i) \ f ` (A i) \ B i)"
    using assms by (auto simp add: card_of_ordLeq)
  with choice[of "\ i f. i \ I \ inj_on f (A i) \ f ` (A i) \ B i"]
  obtain F where F: "\i \ I. inj_on (F i) (A i) \ (F i) ` (A i) \ B i"
    by atomize_elim (auto intro: bchoice)
  define g where "g \ (\(i,a::'b). (i,F i a))"
  have "inj_on g (Sigma I A) \ g ` (Sigma I A) \ (Sigma I B)"
    using F unfolding inj_on_def using g_def by force
  thus ?thesis using card_of_ordLeq by blast
qed

lemma card_of_UNION_Sigma:
  "|\i \ I. A i| \o |SIGMA i : I. A i|"
  using Ex_inj_on_UNION_Sigma [of A I] card_of_ordLeq by blast

lemma card_of_bool:
  assumes "a1 \ a2"
  shows "|UNIV::bool set| =o |{a1,a2}|"
proof -
  let ?f = "\ bl. if bl then a1 else a2"
  have "bij_betw ?f UNIV {a1,a2}"
  proof -
    have "\bl1 bl2. ?f bl1 = ?f bl2 \ bl1 = bl2"
      using assms by (force split: if_split_asm)
    moreover
    have "\bl. ?f bl \ {a1,a2}"
      using assms by (force split: if_split_asm)
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by force
  qed
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_Plus_Times_aux:
  assumes A2: "a1 \ a2 \ {a1,a2} \ A" and
    LEQ: "|A| \o |B|"
  shows "|A <+> B| \o |A \ B|"
proof -
  have 1: "|UNIV::bool set| \o |A|"
    using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
    by (blast intro: ordIso_ordLeq_trans)
  have "|A <+> B| \o |B <+> B|"
    using LEQ card_of_Plus_mono1 by blast
  moreover have "|B <+> B| =o |B \ (UNIV::bool set)|"
    using card_of_Plus_Times_bool by blast
  moreover have "|B \ (UNIV::bool set)| \o |B \ A|"
    using 1 by (simp add: card_of_Times_mono2)
  moreover have " |B \ A| =o |A \ B|"
    using card_of_Times_commute by blast
  ultimately show "|A <+> B| \o |A \ B|"
    by (blast intro: ordLeq_transitive dest: ordLeq_ordIso_trans)
qed

lemma card_of_Plus_Times:
  assumes A2: "a1 \ a2 \ {a1,a2} \ A" and B2: "b1 \ b2 \ {b1,b2} \ B"
  shows "|A <+> B| \o |A \ B|"
proof -
  {assume "|A| \o |B|"
    hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
  }
  moreover
  {assume "|B| \o |A|"
    hence "|B <+> A| \o |B \ A|"
      using assms by (auto simp add: card_of_Plus_Times_aux)
    hence ?thesis
      using card_of_Plus_commute card_of_Times_commute
        ordIso_ordLeq_trans ordLeq_ordIso_trans by blast
  }
  ultimately show ?thesis
    using card_of_Well_order[of A] card_of_Well_order[of B]
      ordLeq_total[of "|A|"] by blast
qed

lemma card_of_Times_Plus_distrib:
  "|A \ (B <+> C)| =o |A \ B <+> A \ C|" (is "|?RHS| =o |?LHS|")
proof -
  let ?f = "\(a, bc). case bc of Inl b \ Inl (a, b) | Inr c \ Inr (a, c)"
  have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
  thus ?thesis using card_of_ordIso by blast
qed

lemma card_of_ordLeq_finite:
  assumes "|A| \o |B|" and "finite B"
  shows "finite A"
  using assms unfolding ordLeq_def
  using embed_inj_on[of "|A|" "|B|"]  embed_Field[of "|A|" "|B|"]
    Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce

lemma card_of_ordLeq_infinite:
  assumes "|A| \o |B|" and "\ finite A"
  shows "\ finite B"
  using assms card_of_ordLeq_finite by auto

lemma card_of_ordIso_finite:
  assumes "|A| =o |B|"
  shows "finite A = finite B"
  using assms unfolding ordIso_def iso_def[abs_def]
  by (auto simp: bij_betw_finite Field_card_of)

lemma card_of_ordIso_finite_Field:
  assumes "Card_order r" and "r =o |A|"
  shows "finite(Field r) = finite A"
  using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast

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

text\<open>Here we show that, for infinite sets, most set-theoretic constructions
do not increase the cardinality.  The cornerstone for this is
theorem \<open>Card_order_Times_same_infinite\<close>, which states that self-product
does not increase cardinality -- the proof of this fact adapts a standard
set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
at page 47 in \<^cite>\<open>"card-book"\<close>. Then everything else follows fairly easily.\<close>

lemma infinite_iff_card_of_nat:
  "\ finite A \ ( |UNIV::nat set| \o |A| )"
  unfolding infinite_iff_countable_subset card_of_ordLeq ..

text\<open>The next two results correspond to the ZF fact that all infinite cardinals are
limit ordinals:\<close>

lemma Card_order_infinite_not_under:
  assumes CARD: "Card_order r" and INF: "\finite (Field r)"
  shows "\ (\a. Field r = under r a)"
proof(auto)
  have 0: "Well_order r \ wo_rel r \ Refl r"
    using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
  fix a assume *: "Field r = under r a"
  show False
  proof(cases "a \ Field r")
    assume Case1: "a \ Field r"
    hence "under r a = {}" unfolding Field_def under_def by auto
    thus False using INF *  by auto
  next
    let ?r' = "Restr r (underS r a)"
    assume Case2: "a \ Field r"
    hence 1: "under r a = underS r a \ {a} \ a \ underS r a"
      using 0 Refl_under_underS[of r a] underS_notIn[of a r] by blast
    have 2: "wo_rel.ofilter r (underS r a) \ underS r a < Field r"
      using 0 wo_rel.underS_ofilter * 1 Case2 by fast
    hence "?r' using 0 using ofilter_ordLess by blast
    moreover
    have "Field ?r' = underS r a \ Well_order ?r'"
      using  2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
    ultimately have "|underS r a| using ordLess_Field[of ?r'] by auto
    moreover have "|under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
    ultimately have "|underS r a|
      using ordIso_symmetric ordLess_ordIso_trans by blast
    moreover
    {have "\f. bij_betw f (under r a) (underS r a)"
        using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
      hence "|under r a| =o |underS r a|" using card_of_ordIso by blast
    }
    ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
  qed
qed

lemma infinite_Card_order_limit:
  assumes r: "Card_order r" and "\finite (Field r)"
    and a: "a \ Field r"
  shows "\b \ Field r. a \ b \ (a,b) \ r"
proof -
  have "Field r \ under r a"
    using assms Card_order_infinite_not_under by blast
  moreover have "under r a \ Field r"
    using under_Field .
  ultimately obtain b where b: "b \ Field r \ \ (b,a) \ r"
    unfolding under_def by blast
  moreover have ba: "b \ a"
    using b r unfolding card_order_on_def well_order_on_def
      linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
  ultimately have "(a,b) \ r"
    using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
      total_on_def by blast
  thus ?thesis using b ba by auto
qed

theorem Card_order_Times_same_infinite:
  assumes CO: "Card_order r" and INF: "\finite(Field r)"
  shows "|Field r \ Field r| \o r"
proof -
  define phi where
    "phi \ \r::'a rel. Card_order r \ \finite(Field r) \ \ |Field r \ Field r| \o r"
  have temp1: "\r. phi r \ Well_order r"
    unfolding phi_def card_order_on_def by auto
  have Ft: "\(\r. phi r)"
  proof
    assume "\r. phi r"
    hence "{r. phi r} \ {} \ {r. phi r} \ {r. Well_order r}"
      using temp1 by auto
    then obtain r where 1: "phi r" and 2: "\r'. phi r' \ r \o r'" and
      3: "Card_order r \ Well_order r"
      using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
    let ?A = "Field r"  let ?r' = "bsqr r"
    have 4: "Well_order ?r' \ Field ?r' = ?A \ ?A \ |?A| =o r"
      using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
    have 5: "Card_order |?A \ ?A| \ Well_order |?A \ ?A|"
      using card_of_Card_order card_of_Well_order by blast
        (*  *)
    have "r ?A|"
      using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
    moreover have "|?A \ ?A| \o ?r'"
      using card_of_least[of "?A \ ?A"] 4 by auto
    ultimately have "r using ordLess_ordLeq_trans by auto
    then obtain f where 6: "embed r ?r' f" and 7: "\ bij_betw f ?A (?A \ ?A)"
      unfolding ordLess_def embedS_def[abs_def]
      by (auto simp add: Field_bsqr)
    let ?B = "f ` ?A"
    have "|?A| =o |?B|"
      using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
    hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
        (*  *)
    have "wo_rel.ofilter ?r' ?B"
      using 6 embed_Field_ofilter 3 4 by blast
    hence "wo_rel.ofilter ?r' ?B \ ?B \ ?A \ ?A \ ?B \ Field ?r'"
      using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
    hence temp2: "wo_rel.ofilter ?r' ?B \ ?B < ?A \ ?A"
      using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
    have "\ (\a. Field r = under r a)"
      using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
    then obtain A1 where temp3: "wo_rel.ofilter r A1 \ A1 < ?A" and 9: "?B \ A1 \ A1"
      using temp2 3 bsqr_ofilter[of r ?B] by blast
    hence "|?B| \o |A1 \ A1|" using card_of_mono1 by blast
    hence 10: "r \o |A1 \ A1|" using 8 ordIso_ordLeq_trans by blast
    let ?r1 = "Restr r A1"
    have "?r1 using temp3 ofilter_ordLess 3 by blast
    moreover
    {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
      hence "|A1| \o ?r1" using 3 Well_order_Restr card_of_least by blast
    }
    ultimately have 11: "|A1| using ordLeq_ordLess_trans by blast
        (*  *)
    have "\ finite (Field r)" using 1 unfolding phi_def by simp
    hence "\ finite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
    hence "\ finite A1" using 9 finite_cartesian_product finite_subset by blast
    moreover have temp4: "Field |A1| = A1 \ Well_order |A1| \ Card_order |A1|"
      using card_of_Card_order[of A1] card_of_Well_order[of A1]
      by (simp add: Field_card_of)
    moreover have "\ r \o | A1 |"
      using temp4 11 3 using not_ordLeq_iff_ordLess by blast
    ultimately have "\ finite(Field |A1| ) \ Card_order |A1| \ \ r \o | A1 |"
      by (simp add: card_of_card_order_on)
    hence "|Field |A1| \ Field |A1| | \o |A1|"
      using 2 unfolding phi_def by blast
    hence "|A1 \ A1 | \o |A1|" using temp4 by auto
    hence "r \o |A1|" using 10 ordLeq_transitive by blast
    thus False using 11 not_ordLess_ordLeq by auto
  qed
  thus ?thesis using assms unfolding phi_def by blast
qed

corollary card_of_Times_same_infinite:
  assumes "\finite A"
  shows "|A \ A| =o |A|"
proof -
  let ?r = "|A|"
  have "Field ?r = A \ Card_order ?r"
    using Field_card_of card_of_Card_order[of A] by fastforce
  hence "|A \ A| \o |A|"
    using Card_order_Times_same_infinite[of ?r] assms by auto
  thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
qed

lemma card_of_Times_infinite:
  assumes INF: "\finite A" and NE: "B \ {}" and LEQ: "|B| \o |A|"
  shows "|A \ B| =o |A| \ |B \ A| =o |A|"
proof -
  have "|A| \o |A \ B| \ |A| \o |B \ A|"
    using assms by (simp add: card_of_Times1 card_of_Times2)
  moreover
  {have "|A \ B| \o |A \ A| \ |B \ A| \o |A \ A|"
      using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
    moreover have "|A \ A| =o |A|" using INF card_of_Times_same_infinite by blast
    ultimately have "|A \ B| \o |A| \ |B \ A| \o |A|"
      using ordLeq_ordIso_trans[of "|A \ B|"] ordLeq_ordIso_trans[of "|B \ A|"] by auto
  }
  ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
qed

corollary Card_order_Times_infinite:
  assumes INF: "\finite(Field r)" and CARD: "Card_order r" and
    NE: "Field p \ {}" and LEQ: "p \o r"
  shows "| (Field r) \ (Field p) | =o r \ | (Field p) \ (Field r) | =o r"
proof -
  have "|Field r \ Field p| =o |Field r| \ |Field p \ Field r| =o |Field r|"
    using assms by (simp add: card_of_Times_infinite card_of_mono2)
  thus ?thesis
    using assms card_of_Field_ordIso by (blast intro: ordIso_transitive)
qed

lemma card_of_Sigma_ordLeq_infinite:
  assumes INF: "\finite B" and
    LEQ_I: "|I| \o |B|" and LEQ: "\i \ I. |A i| \o |B|"
  shows "|SIGMA i : I. A i| \o |B|"
proof(cases "I = {}")
  case False
  have "|SIGMA i : I. A i| \o |I \ B|"
    using card_of_Sigma_mono1[OF LEQ] by blast
  moreover have "|I \ B| =o |B|"
    using INF False LEQ_I by (auto simp add: card_of_Times_infinite)
  ultimately show ?thesis using ordLeq_ordIso_trans by blast
qed (simp add: card_of_empty)

lemma card_of_Sigma_ordLeq_infinite_Field:
  assumes INF: "\finite (Field r)" and r: "Card_order r" and
    LEQ_I: "|I| \o r" and LEQ: "\i \ I. |A i| \o r"
  shows "|SIGMA i : I. A i| \o r"
proof -
  let ?B  = "Field r"
  have 1: "r =o |?B| \ |?B| =o r"
    using r card_of_Field_ordIso ordIso_symmetric by blast
  hence "|I| \o |?B|" "\i \ I. |A i| \o |?B|"
    using LEQ_I LEQ ordLeq_ordIso_trans by blast+
  hence  "|SIGMA i : I. A i| \o |?B|" using INF LEQ
      card_of_Sigma_ordLeq_infinite by blast
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
qed

lemma card_of_Times_ordLeq_infinite_Field:
  "\\finite (Field r); |A| \o r; |B| \o r; Card_order r\ \ |A \ B| \o r"
  by(simp add: card_of_Sigma_ordLeq_infinite_Field)

lemma card_of_Times_infinite_simps:
  "\\finite A; B \ {}; |B| \o |A|\ \ |A \ B| =o |A|"
  "\\finite A; B \ {}; |B| \o |A|\ \ |A| =o |A \ B|"
  "\\finite A; B \ {}; |B| \o |A|\ \ |B \ A| =o |A|"
  "\\finite A; B \ {}; |B| \o |A|\ \ |A| =o |B \ A|"
  by (auto simp add: card_of_Times_infinite ordIso_symmetric)

lemma card_of_UNION_ordLeq_infinite:
  assumes INF: "\finite B" and LEQ_I: "|I| \o |B|" and LEQ: "\i \ I. |A i| \o |B|"
  shows "|\i \ I. A i| \o |B|"
proof(cases "I = {}")
  case False
  have "|\i \ I. A i| \o |SIGMA i : I. A i|"
    using card_of_UNION_Sigma by blast
  moreover have "|SIGMA i : I. A i| \o |B|"
    using assms card_of_Sigma_ordLeq_infinite by blast
  ultimately show ?thesis using ordLeq_transitive by blast
qed (simp add: card_of_empty)

corollary card_of_UNION_ordLeq_infinite_Field:
  assumes INF: "\finite (Field r)" and r: "Card_order r" and
    LEQ_I: "|I| \o r" and LEQ: "\i \ I. |A i| \o r"
  shows "|\i \ I. A i| \o r"
proof -
  let ?B  = "Field r"
  have 1: "r =o |?B| \ |?B| =o r"
    using r card_of_Field_ordIso ordIso_symmetric by blast
  hence "|I| \o |?B|" "\i \ I. |A i| \o |?B|"
    using LEQ_I LEQ ordLeq_ordIso_trans by blast+
  hence  "|\i \ I. A i| \o |?B|" using INF LEQ
      card_of_UNION_ordLeq_infinite by blast
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
qed

lemma card_of_Plus_infinite1:
  assumes INF: "\finite A" and LEQ: "|B| \o |A|"
  shows "|A <+> B| =o |A|"
proof(cases "B = {}")
  case True
  then show ?thesis
    by (simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
next
  case False
  let ?Inl = "Inl::'a \ 'a + 'b" let ?Inr = "Inr::'b \ 'a + 'b"
  assume *: "B \ {}"
  then obtain b1 where 1: "b1 \ B" by blast
  show ?thesis
  proof(cases "B = {b1}")
    case True
    have 2: "bij_betw ?Inl A ((?Inl ` A))"
      unfolding bij_betw_def inj_on_def by auto
    hence 3: "\finite (?Inl ` A)"
      using INF bij_betw_finite[of ?Inl A] by blast
    let ?A' = "?Inl ` A \ {?Inr b1}"
    obtain g where "bij_betw g (?Inl ` A) ?A'"
      using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
    moreover have "?A' = A <+> B" using True by blast
    ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
    hence "bij_betw (g \ ?Inl) A (A <+> B)"
      using 2 by (auto simp add: bij_betw_trans)
    thus ?thesis using card_of_ordIso ordIso_symmetric by blast
  next
    case False
    with * 1 obtain b2 where 3: "b1 \ b2 \ {b1,b2} \ B" by fastforce
    obtain f where "inj_on f B \ f ` B \ A"
      using LEQ card_of_ordLeq[of B] by fastforce
    with 3 have "f b1 \ f b2 \ {f b1, f b2} \ A"
      unfolding inj_on_def by auto
    with 3 have "|A <+> B| \o |A \ B|"
      by (auto simp add: card_of_Plus_Times)
    moreover have "|A \ B| =o |A|"
      using assms * by (simp add: card_of_Times_infinite_simps)
    ultimately have "|A <+> B| \o |A|" using ordLeq_ordIso_trans by blast
    thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
  qed
qed

lemma card_of_Plus_infinite2:
  assumes INF: "\finite A" and LEQ: "|B| \o |A|"
  shows "|B <+> A| =o |A|"
  using assms card_of_Plus_commute card_of_Plus_infinite1
    ordIso_equivalence by blast

lemma card_of_Plus_infinite:
  assumes INF: "\finite A" and LEQ: "|B| \o |A|"
  shows "|A <+> B| =o |A| \ |B <+> A| =o |A|"
  using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)

corollary Card_order_Plus_infinite:
  assumes INF: "\finite(Field r)" and CARD: "Card_order r" and
    LEQ: "p \o r"
  shows "| (Field r) <+> (Field p) | =o r \ | (Field p) <+> (Field r) | =o r"
proof -
  have "| Field r <+> Field p | =o | Field r | \
        | Field p <+> Field r | =o | Field r |"
    using assms by (simp add: card_of_Plus_infinite card_of_mono2)
  thus ?thesis
    using assms card_of_Field_ordIso by (blast intro: ordIso_transitive)

qed

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

text\<open>The cardinal $\omega$, of natural numbers, shall be the standard non-strict
order relation on
\<open>nat\<close>, that we abbreviate by \<open>natLeq\<close>.  The finite cardinals
shall be the restrictions of these relations to the numbers smaller than
fixed numbers \<open>n\<close>, that we abbreviate by \<open>natLeq_on n\<close>.\<close>

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

abbreviation natLeq_on :: "nat \ (nat * nat) set"
  where "natLeq_on n \ {(x,y). x < n \ y < n \ x \ y}"

lemma infinite_cartesian_product:
  assumes "\finite A" "\finite B"
  shows "\finite (A \ B)"
using assms finite_cartesian_productD2 by auto

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

lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
  by(unfold Field_def natLeq_def, auto)

lemma natLeq_Refl: "Refl natLeq"
  unfolding refl_on_def Field_def natLeq_def by auto

lemma natLeq_trans: "trans natLeq"
  unfolding trans_def natLeq_def by auto

lemma natLeq_Preorder: "Preorder natLeq"
  unfolding preorder_on_def
proof (intro conjI)
  show "natLeq \ Field natLeq \ Field natLeq"
    unfolding natLeq_def Field_def by blast
next
  show "Refl natLeq"
    using natLeq_Refl .
next
  show "trans natLeq"
    using natLeq_trans .
qed

lemma natLeq_antisym: "antisym natLeq"
  unfolding antisym_def natLeq_def by auto

lemma natLeq_Partial_order: "Partial_order natLeq"
  unfolding partial_order_on_def
  by (auto simp add: natLeq_Preorder natLeq_antisym)

lemma natLeq_Total: "Total natLeq"
  unfolding total_on_def natLeq_def by auto

lemma natLeq_Linear_order: "Linear_order natLeq"
  unfolding linear_order_on_def
  by (auto simp add: natLeq_Partial_order natLeq_Total)

lemma natLeq_natLess_Id: "natLess = natLeq - Id"
  unfolding natLeq_def natLess_def by auto

lemma natLeq_Well_order: "Well_order natLeq"
  unfolding well_order_on_def
  using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto

lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"
  unfolding Field_def by auto

lemma natLeq_underS_less: "underS natLeq n = {x. x < n}"
  unfolding underS_def natLeq_def by auto

lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"
  unfolding natLeq_def by force

lemma Restr_natLeq2:
  "Restr natLeq (underS natLeq n) = natLeq_on n"
  by (auto simp add: Restr_natLeq natLeq_underS_less)

lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
  using Restr_natLeq[of n] natLeq_Well_order
    Well_order_Restr[of natLeq "{x. x < n}"] by auto

corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"
  using natLeq_on_Well_order Field_natLeq_on by auto

lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
  unfolding wo_rel_def using natLeq_on_Well_order .

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

lemma natLeq_Card_order: "Card_order natLeq"
proof -
  have "natLeq_on n for n
  proof -
    have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
    moreover have "\finite(UNIV::nat set)" by auto
    ultimately show ?thesis
      using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
        card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order
      by (force simp add: Field_card_of)
  qed
  then show ?thesis
    apply (simp add: natLeq_Well_order Card_order_iff_Restr_underS Restr_natLeq2)
    apply (force simp add: Field_natLeq)
    done
qed

corollary card_of_Field_natLeq:
  "|Field natLeq| =o natLeq"
  using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
    ordIso_symmetric[of natLeq] by blast

corollary card_of_nat:
  "|UNIV::nat set| =o natLeq"
  using Field_natLeq card_of_Field_natLeq by auto

corollary infinite_iff_natLeq_ordLeq:
  "\finite A = ( natLeq \o |A| )"
  using infinite_iff_card_of_nat[of A] card_of_nat
    ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast

corollary finite_iff_ordLess_natLeq:
  "finite A = ( |A|
  using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
    card_of_Well_order natLeq_Well_order by blast

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

text\<open>First we define \<open>isCardSuc r r'\<close>, the notion of \<open>r'\<close>
being a successor cardinal of \<open>r\<close>. Although the definition does
not require \<open>r\<close> to be a cardinal, only this case will be meaningful.\<close>

definition isCardSuc :: "'a rel \ 'a set rel \ bool"
  where
    "isCardSuc r r' \
         Card_order r' \ r
         (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"

text\<open>Now we introduce the cardinal-successor operator \<open>cardSuc\<close>,
by picking {\em some} cardinal-order relation fulfilling \<open>isCardSuc\<close>.
Again, the picked item shall be proved unique up to order-isomorphism.\<close>

definition cardSuc :: "'a rel \ 'a set rel"
  where "cardSuc r \ SOME r'. isCardSuc r r'"

lemma exists_minim_Card_order:
  "\R \ {}; \r \ R. Card_order r\ \ \r \ R. \r' \ R. r \o r'"
  unfolding card_order_on_def using exists_minim_Well_order by blast

lemma exists_isCardSuc:
  assumes "Card_order r"
  shows "\r'. isCardSuc r r'"
proof -
  let ?R = "{(r'::'a set rel). Card_order r' \ r
  have "|Pow(Field r)| \ ?R \ (\r \ ?R. Card_order r)" using assms
    by (simp add: card_of_Card_order Card_order_Pow)
  then obtain r where "r \ ?R \ (\r' \ ?R. r \o r')"
    using exists_minim_Card_order[of ?R] by blast
  thus ?thesis unfolding isCardSuc_def by auto
qed

lemma cardSuc_isCardSuc:
  assumes "Card_order r"
  shows "isCardSuc r (cardSuc r)"
  unfolding cardSuc_def using assms
  by (simp add: exists_isCardSuc someI_ex)

lemma cardSuc_Card_order:
  "Card_order r \ Card_order(cardSuc r)"
  using cardSuc_isCardSuc unfolding isCardSuc_def by blast

lemma cardSuc_greater:
  "Card_order r \ r
  using cardSuc_isCardSuc unfolding isCardSuc_def by blast

lemma cardSuc_ordLeq:
  "Card_order r \ r \o cardSuc r"
  using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast

text\<open>The minimality property of \<open>cardSuc\<close> originally present in its definition
is local to the type \<open>'a set rel\<close>, i.e., that of \<open>cardSuc r\<close>:\<close>

lemma cardSuc_least_aux:
  "\Card_order (r::'a rel); Card_order (r'::'a set rel); r \ cardSuc r \o r'"
  using cardSuc_isCardSuc unfolding isCardSuc_def by blast

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

lemma cardSuc_least:
  assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
  shows "cardSuc r \o r'"
proof -
  let ?p = "cardSuc r"
  have 0: "Well_order ?p \ Well_order r'"
    using assms cardSuc_Card_order unfolding card_order_on_def by blast
  { assume "r'
    then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \ r''
      using internalize_ordLess[of r' ?p] by blast
        (*  *)
    have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
    moreover have "r using LESS 2 ordLess_ordIso_trans by blast
    ultimately have "?p \o r''" using cardSuc_least_aux CARD by blast
    hence False using 2 not_ordLess_ordLeq by blast
  }
  thus ?thesis using 0 ordLess_or_ordLeq by blast
qed

lemma cardSuc_ordLess_ordLeq:
  assumes CARD: "Card_order r" and CARD': "Card_order r'"
  shows "(r o r')"
proof
  show "cardSuc r \o r' \ r
    using assms cardSuc_greater ordLess_ordLeq_trans by blast
qed (auto simp add: assms cardSuc_least)

lemma cardSuc_ordLeq_ordLess:
  assumes CARD: "Card_order r" and CARD': "Card_order r'"
  shows "(r' o r)"
proof -
  have "Well_order r \ Well_order r'"
    using assms unfolding card_order_on_def by auto
  moreover have "Well_order(cardSuc r)"
    using assms cardSuc_Card_order card_order_on_def by blast
  ultimately show ?thesis
    using assms cardSuc_ordLess_ordLeq by (blast dest: not_ordLeq_iff_ordLess)
qed

lemma cardSuc_mono_ordLeq:
  assumes CARD: "Card_order r" and CARD': "Card_order r'"
  shows "(cardSuc r \o cardSuc r') = (r \o r')"
  using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast

lemma cardSuc_invar_ordIso:
  assumes CARD: "Card_order r" and CARD': "Card_order r'"
  shows "(cardSuc r =o cardSuc r') = (r =o r')"
proof -
  have 0: "Well_order r \ Well_order r' \ Well_order(cardSuc r) \ Well_order(cardSuc r')"
    using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
  thus ?thesis
    using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
    using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
qed

lemma card_of_cardSuc_finite:
  "finite(Field(cardSuc |A| )) = finite A"
proof
  assume *: "finite (Field (cardSuc |A| ))"
  have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
    using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
  hence "|A| \o |Field(cardSuc |A| )|"
    using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
      ordLeq_ordIso_trans by blast
  thus "finite A" using * card_of_ordLeq_finite by blast
next
  assume "finite A"
  then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp
  moreover
  have "cardSuc |A| \o |Pow A|"
    by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)
  ultimately show "finite (Field (cardSuc |A| ))"
    by (blast intro: card_of_ordLeq_finite card_of_mono2)
qed

lemma cardSuc_finite:
  assumes "Card_order r"
  shows "finite (Field (cardSuc r)) = finite (Field r)"
proof -
  let ?A = "Field r"
  have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
  hence "cardSuc |?A| =o cardSuc r" using assms
    by (simp add: card_of_Card_order cardSuc_invar_ordIso)
  moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
    by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
  moreover
  { have "|Field (cardSuc r) | =o cardSuc r"
      using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
    hence "cardSuc r =o |Field (cardSuc r) |"
      using ordIso_symmetric by blast
  }
  ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
    using ordIso_transitive by blast
  hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
    using card_of_ordIso_finite by blast
  thus ?thesis by (simp only: card_of_cardSuc_finite)
qed

lemma Field_cardSuc_not_empty:
  assumes "Card_order r"
  shows "Field (cardSuc r) \ {}"
proof
  assume "Field(cardSuc r) = {}"
  then have "|Field(cardSuc r)| \o r" using assms Card_order_empty[of r] by auto
  then have "cardSuc r \o r" using assms card_of_Field_ordIso
      cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast
  then show False using cardSuc_greater not_ordLess_ordLeq assms by blast
qed

typedef 'a suc = "Field (cardSuc |UNIV :: 'a set| )"
  using Field_cardSuc_not_empty card_of_Card_order by blast

definition card_suc :: "'a rel \ 'a suc rel" where
  "card_suc \ \_. map_prod Abs_suc Abs_suc ` cardSuc |UNIV :: 'a set|"

lemma Field_card_suc: "Field (card_suc r) = UNIV"
proof -
  let ?r = "cardSuc |UNIV|"
  let ?ar = "\x. Abs_suc (Rep_suc x)"
  have 1: "\P. (\x\Field ?r. P x) = (\x. P (Rep_suc x))" using Rep_suc_induct Rep_suc by blast
  have 2: "\P. (\x\Field ?r. P x) = (\x. P (Rep_suc x))" using Rep_suc_cases Rep_suc by blast
  have 3: "\A a b. (a, b) \ A \ (Abs_suc a, Abs_suc b) \ map_prod Abs_suc Abs_suc ` A" unfolding map_prod_def by auto
  have "\x\Field ?r. (\b\Field ?r. (x, b) \ ?r) \ (\a\Field ?r. (a, x) \ ?r)"
    unfolding Field_def Range.simps Domain.simps Un_iff by blast
  then have "\x. (\b. (Rep_suc x, Rep_suc b) \ ?r) \ (\a. (Rep_suc a, Rep_suc x) \ ?r)" unfolding 1 2 .
  then have "\x. (\b. (?ar x, ?ar b) \ map_prod Abs_suc Abs_suc ` ?r) \ (\a. (?ar a, ?ar x) \ map_prod Abs_suc Abs_suc ` ?r)" using 3 by fast
  then have "\x. (\b. (x, b) \ card_suc r) \ (\a. (a, x) \ card_suc r)" unfolding card_suc_def Rep_suc_inverse .
  then show ?thesis unfolding Field_def Domain.simps Range.simps set_eq_iff Un_iff eqTrueI[OF UNIV_I] ex_simps simp_thms .
qed

lemma card_suc_alt: "card_suc r = dir_image (cardSuc |UNIV :: 'a set| ) Abs_suc"
  unfolding card_suc_def dir_image_def by auto

lemma cardSuc_Well_order: "Card_order r \ Well_order(cardSuc r)"
  using cardSuc_Card_order unfolding card_order_on_def by blast

lemma cardSuc_ordIso_card_suc:
  assumes "card_order r"
  shows "cardSuc r =o card_suc (r :: 'a rel)"
proof -
  have "cardSuc (r :: 'a rel) =o cardSuc |UNIV :: 'a set|"
    using cardSuc_invar_ordIso[THEN iffD2, OF _ card_of_Card_order card_of_unique[OF assms]] assms
    by (simp add: card_order_on_Card_order)
  also have "cardSuc |UNIV :: 'a set| =o card_suc (r :: 'a rel)"
    unfolding card_suc_alt
    by (rule dir_image_ordIso) (simp_all add: inj_on_def Abs_suc_inject cardSuc_Well_order card_of_Card_order)
  finally show ?thesis .
qed

lemma Card_order_card_suc: "card_order r \ Card_order (card_suc r)"
  using cardSuc_Card_order[THEN Card_order_ordIso2[OF _ cardSuc_ordIso_card_suc]] card_order_on_Card_order by blast

lemma card_order_card_suc: "card_order r \ card_order (card_suc r)"
  using Card_order_card_suc arg_cong2[OF Field_card_suc refl, of "card_order_on"] by blast

lemma card_suc_greater: "card_order r \ r
  using ordLess_ordIso_trans[OF cardSuc_greater cardSuc_ordIso_card_suc] card_order_on_Card_order by blast

lemma card_of_Plus_ordLess_infinite:
  assumes INF: "\finite C" and LESS1: "|A|
  shows "|A <+> B|
proof(cases "A = {} \ B = {}")
  case True
  hence "|A| =o |A <+> B| \ |B| =o |A <+> B|"
    using card_of_Plus_empty1 card_of_Plus_empty2 by blast
  hence "|A <+> B| =o |A| \ |A <+> B| =o |B|"
    using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
  thus ?thesis using LESS1 LESS2
      ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
      ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
next
  case False
  have False if "|C| \o |A <+> B|"
  proof -
    have \<section>: "\<not>finite A \<or> \<not>finite B"
      using that INF card_of_ordLeq_finite finite_Plus by blast
    consider  "|A| \o |B|" | "|B| \o |A|"
      using ordLeq_total [OF card_of_Well_order card_of_Well_order] by blast
    then show False
    proof cases
      case 1
      hence "\finite B" using \

card_of_ordLeq_finite by blast
      hence "|A <+> B| =o |B|" using False 1
        by (auto simp add: card_of_Plus_infinite)
      thus False using LESS2 not_ordLess_ordLeq that ordLeq_ordIso_trans by blast
    next
      case 2
      hence "\finite A" using \

card_of_ordLeq_finite by blast
      hence "|A <+> B| =o |A|" using False 2
        by (auto simp add: card_of_Plus_infinite)
      thus False using LESS1 not_ordLess_ordLeq that ordLeq_ordIso_trans by blast
    qed
  qed
  thus ?thesis
    using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
          card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
qed

lemma card_of_Plus_ordLess_infinite_Field:
  assumes INF: "\finite (Field r)" and r: "Card_order r" and
    LESS1: "|A| and LESS2: "|B|
  shows "|A <+> B|
proof -
  let ?C  = "Field r"
  have 1: "r =o |?C| \ |?C| =o r"
    using r card_of_Field_ordIso ordIso_symmetric by blast
  hence "|A|   "|B|
    using LESS1 LESS2 ordLess_ordIso_trans by blast+
  hence  "|A <+> B| using INF
      card_of_Plus_ordLess_infinite by blast
  thus ?thesis using 1 ordLess_ordIso_trans by blast
qed

lemma card_of_Plus_ordLeq_infinite_Field:
  assumes r: "\finite (Field r)" and A: "|A| \o r" and B: "|B| \o r"
    and c: "Card_order r"
  shows "|A <+> B| \o r"
proof -
  let ?r' = "cardSuc r"
  have "Card_order ?r' \ \finite (Field ?r')" using assms
    by (simp add: cardSuc_Card_order cardSuc_finite)
  moreover have "|A| and "|B| using A B c
    by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
  ultimately have "|A <+> B|
    using card_of_Plus_ordLess_infinite_Field by blast
  thus ?thesis using c r
    by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
qed

lemma card_of_Un_ordLeq_infinite_Field:
  assumes C: "\finite (Field r)" and A: "|A| \o r" and B: "|B| \o r"
    and "Card_order r"
  shows "|A Un B| \o r"
  using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
    ordLeq_transitive by fast

lemma card_of_Un_ordLess_infinite:
  assumes INF: "\finite C" and
    LESS1: "|A| and LESS2: "|B|
  shows "|A \ B|
  using assms card_of_Plus_ordLess_infinite card_of_Un_Plus_ordLeq
    ordLeq_ordLess_trans by blast

lemma card_of_Un_ordLess_infinite_Field:
  assumes INF: "\finite (Field r)" and r: "Card_order r" and
    LESS1: "|A| and LESS2: "|B|
  shows "|A Un B|
proof -
  let ?C  = "Field r"
  have 1: "r =o |?C| \ |?C| =o r" using r card_of_Field_ordIso
      ordIso_symmetric by blast
  hence "|A|   "|B|
    using LESS1 LESS2 ordLess_ordIso_trans by blast+
  hence  "|A Un B| using INF
      card_of_Un_ordLess_infinite by blast
  thus ?thesis using 1 ordLess_ordIso_trans by blast
qed

subsection \<open>Regular cardinals\<close>

definition cofinal where
  "cofinal A r \ \a \ Field r. \b \ A. a \ b \ (a,b) \ r"

definition regularCard where
  "regularCard r \ \K. K \ Field r \ cofinal K r \ |K| =o r"

definition relChain where
  "relChain r As \ \i j. (i,j) \ r \ As i \ As j"

lemma regularCard_UNION:
  assumes r: "Card_order r"   "regularCard r"
    and As: "relChain r As"
    and Bsub: "B \ (\i \ Field r. As i)"
    and cardB: "|B|
  shows "\i \ Field r. B \ As i"
proof -
  let ?phi = "\b j. j \ Field r \ b \ As j"
  have "\b\B. \j. ?phi b j" using Bsub by blast
  then obtain f where f: "\b. b \ B \ ?phi b (f b)"
    using bchoice[of B ?phi] by blast
  let ?K = "f ` B"
  {assume 1: "\i. i \ Field r \ \ B \ As i"
    have 2: "cofinal ?K r"
      unfolding cofinal_def
    proof (intro strip)
      fix i assume i: "i \ Field r"
      with 1 obtain b where b: "b \ B \ b \ As i" by blast
      hence "i \ f b \ \ (f b,i) \ r"
        using As f unfolding relChain_def by auto
      hence "i \ f b \ (i, f b) \ r" using r
        unfolding card_order_on_def well_order_on_def linear_order_on_def
          total_on_def using i f b by auto
      with b show "\b \ f`B. i \ b \ (i,b) \ r" by blast
    qed
    moreover have "?K \ Field r" using f by blast
    ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast
    moreover
    have "|?K| using cardB ordLeq_ordLess_trans card_of_image by blast
    ultimately have False using not_ordLess_ordIso by blast
  }
  thus ?thesis by blast
qed

lemma infinite_cardSuc_regularCard:
  assumes r_inf: "\finite (Field r)" and r_card: "Card_order r"
  shows "regularCard (cardSuc r)"
proof -
  let ?r' = "cardSuc r"
  have r': "Card_order ?r'" "\<And>p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
    using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
  show ?thesis
    unfolding regularCard_def proof auto
    fix K assume 1: "K \ Field ?r'" and 2: "cofinal K ?r'"
    hence "|K| \o |Field ?r'|" by (simp only: card_of_mono1)
    also have 22: "|Field ?r'| =o ?r'"
      using r' by (simp add: card_of_Field_ordIso[of ?r'])
    finally have "|K| \o ?r'" .
    moreover
    { let ?L = "\ j \ K. underS ?r' j"
      let ?J = "Field r"
      have rJ: "r =o |?J|"
        using r_card card_of_Field_ordIso ordIso_symmetric by blast
      assume "|K|
      hence "|K| \o r" using r' card_of_Card_order[of K] by blast
      hence "|K| \o |?J|" using rJ ordLeq_ordIso_trans by blast
      moreover
      {have "\j\K. |underS ?r' j|
          using r' 1 by (auto simp: card_of_underS)
        hence "\j\K. |underS ?r' j| \o r"
          using r' card_of_Card_order by blast
        hence "\j\K. |underS ?r' j| \o |?J|"
          using rJ ordLeq_ordIso_trans by blast
      }
      ultimately have "|?L| \o |?J|"
        using r_inf card_of_UNION_ordLeq_infinite by blast
      hence "|?L| \o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
      hence "|?L| using r' card_of_Card_order by blast
      moreover
      {
        have "Field ?r' \ ?L"
          using 2 unfolding underS_def cofinal_def by auto
        hence "|Field ?r'| \o |?L|" by (simp add: card_of_mono1)
        hence "?r' \o |?L|"
          using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
      }
      ultimately have "|?L| using ordLess_ordLeq_trans by blast
      hence False using ordLess_irreflexive by blast
    }
    ultimately show "|K| =o ?r'"
      unfolding ordLeq_iff_ordLess_or_ordIso by blast
  qed
qed

lemma cardSuc_UNION:
  assumes r: "Card_order r" and "\finite (Field r)"
    and As: "relChain (cardSuc r) As"
    and Bsub: "B \ (\ i \ Field (cardSuc r). As i)"
    and cardB: "|B| \o r"
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