Theory BNF_Cardinal_Order_Relation

theory BNF_Cardinal_Order_Relation
imports Zorn BNF_Wellorder_Constructions
(*  Title:      HOL/BNF_Cardinal_Order_Relation.thy
    Author:     Andrei Popescu, TU Muenchen
    Copyright   2012

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

section ‹Cardinal-Order Relations as Needed by Bounded Natural Functors›

theory BNF_Cardinal_Order_Relation
imports Zorn BNF_Wellorder_Constructions
begin

text‹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, ‹card›, from the theory ‹Finite_Sets.thy›).
\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.
›


subsection ‹Cardinal orders›

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

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‹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 ‹Zorn.thy› as theorem ‹well_order_on›),
which states that on any given set there exists a well-order;
\item The well-founded-ness of ‹<o›, ensuring that then there exists a minimal
such well-order, i.e., a cardinal order.
\end{itemize}
›

theorem card_order_on: "∃r. card_order_on A r"
proof-
  obtain R where R_def: "R = {r. well_order_on A r}" by blast
  have 1: "R ≠ {} ∧ (∀r ∈ R. Well_order r)"
  using well_order_on[of A] R_def well_order_on_Well_order by blast
  hence "∃r ∈ R. ∀r' ∈ R. r ≤o r'"
  using  exists_minim_Well_order[of R] by auto
  thus ?thesis using R_def unfolding card_order_on_def by auto
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 ‹Cardinal of a set›

text‹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.›

definition card_of :: "'a set ⇒ 'a rel" ("|_|" )
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| <o |A|"
   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 by blast
   hence "|A| ≤o |B|" using 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|" f]
  card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
  embed_Field[of "|A|" "|B|" f] by auto
  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| <o |A| )"
proof-
  have "(¬(∃f. inj_on f A ∧ f ` A ≤ B)) = (¬ |A| ≤o |B| )"
  using card_of_ordLeq by blast
  also have "… = ( |B| <o |A| )"
  using card_of_Well_order[of A] card_of_Well_order[of B]
        not_ordLeq_iff_ordLess by blast
  finally show ?thesis .
qed

lemma card_of_ordLess2:
"B ≠ {} ⟹ (¬(∃f. f ` A = B)) = ( |A| <o |B| )"
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_ordLess: "Well_order r ⟹ |Field r| ≤o r"
using card_of_least card_of_well_order_on well_order_on_Well_order by blast

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_Field_ordLess
  by (auto simp: card_of_Field_ordLess 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) <o |Field r| )"
using assms unfolding Card_order_iff_ordLeq_card_of
using ordLeq_iff_ordLess_Restr card_of_Well_order by blast

lemma card_of_underS:
assumes r: "Card_order r" and a: "a : Field r"
shows "|underS r a| <o r"
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
  moreover have "card_order_on (Field ?r') |Field ?r'|"
  using card_of_card_order_on .
  ultimately have "|Field ?r'| ≤o ?r'"
  unfolding card_order_on_def by simp
  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' <o |Field 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 <o r'"
shows "|Field r| <o 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 ‹Cardinals versus set operations on arbitrary sets›

text‹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"
‹=o› and are monotonic w.r.t. ‹≤o›.
›

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_Field_ordLess 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 LEQ: "|A| =o |{}|"
shows "A = {}"
using assms card_of_ordIso[of A] bij_betw_empty2 by blast

lemma card_of_empty3:
assumes LEQ: "|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 = {}", simp add: card_of_empty)
  assume "A ~= {}"
  hence "f ` A ~= {}" by auto
  thus "|f ` A| ≤o |A|"
  using card_of_ordLeq2[of "f ` A" A] by auto
qed

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 = "λ 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| <o |Pow A|"
using card_of_ordLess2[of "Pow A" A]  Cantors_paradox[of A]
      Pow_not_empty[of A] by auto

corollary Card_order_Pow:
"Card_order r ⟹ r <o |Pow(Field r)|"
using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast

lemma card_of_Plus1: "|A| ≤o |A <+> B|"
proof-
  have "Inl ` A ≤ A <+> B" by auto
  thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
qed

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

lemma card_of_Plus2: "|B| ≤o |A <+> B|"
proof-
  have "Inr ` B ≤ A <+> B" by auto
  thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
qed

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 = "λ(c::'a + 'b). 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 ⇒
          (case ab of
            Inl a ⇒ Inl a
          | Inr b ⇒ Inr (Inl b))
      | Inr c ⇒ 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) note 1 = Inr show ?thesis
      proof(cases bc)
        case (Inl b)
        hence "b ∈ B" "x = f (Inl (Inr b))"
        using x 1 unfolding f_def by auto
        thus ?thesis by auto
      next
        case (Inr c)
        hence "c ∈ C" "x = f (Inr c)"
        using x 1 unfolding f_def by auto
        thus ?thesis by auto
      qed
    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 1: "inj_on f A ∧ f ` A ≤ B"
  using assms card_of_ordLeq[of A] by fastforce
  obtain g where g_def:
  "g = (λd. case d of Inl a ⇒ Inl(f a) | Inr (c::'c) ⇒ Inr c)" by blast
  have "inj_on g (A <+> C) ∧ g ` (A <+> C) ≤ (B <+> C)"
  proof-
    {fix d1 and d2 assume "d1 ∈ A <+> C ∧ d2 ∈ A <+> C" and
                          "g d1 = g d2"
     hence "d1 = d2" using 1 unfolding inj_on_def g_def by force
    }
    moreover
    {fix d assume "d ∈ A <+> C"
     hence "g d ∈ B <+> C"  using 1
     by(case_tac d, auto simp add: g_def)
    }
    ultimately show ?thesis unfolding inj_on_def by auto
  qed
  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 assms card_of_Plus_mono1[of A B C]
      card_of_Plus_commute[of C A]  card_of_Plus_commute[of B C]
      ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
by blast

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[of "|A <+> C|"] 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 = "λ 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 = {}", simp add: card_of_empty)
  assume *: "B ≠ {}"
  have "fst `(B × A) = B" using assms by auto
  thus ?thesis using inj_on_iff_surj[of B "B × A"]
                     card_of_ordLeq[of B "B × A"] * by blast
qed

lemma card_of_Times_commute: "|A × B| =o |B × A|"
proof-
  let ?f = "λ(a::'a,b::'b). (b,a)"
  have "bij_betw ?f (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, blast)

lemma card_of_Plus_Times_bool: "|A <+> A| =o |A × (UNIV::bool set)|"
proof-
  let ?f = "λc::'a + 'a. case c of Inl a ⇒ (a,True)
                                  |Inr a ⇒ (a,False)"
  have "bij_betw ?f (A <+> A) (A × (UNIV::bool set))"
  proof-
    {fix  c1 and c2 assume "?f c1 = ?f c2"
     hence "c1 = c2"
     by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
    }
    moreover
    {fix c assume "c ∈ A <+> A"
     hence "?f c ∈ A × (UNIV::bool set)"
     by(case_tac c, auto)
    }
    moreover
    {fix a bl assume *: "(a,bl) ∈ A × (UNIV::bool set)"
     have "(a,bl) ∈ ?f ` ( A <+> A)"
     proof(cases bl)
       assume bl hence "?f(Inl a) = (a,bl)" by auto
       thus ?thesis using * by force
     next
       assume "¬ bl" hence "?f(Inr a) = (a,bl)" by auto
       thus ?thesis using * by force
     qed
    }
    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 1: "inj_on f A ∧ f ` A ≤ B"
  using assms card_of_ordLeq[of A] by fastforce
  obtain g where g_def:
  "g = (λ(a,c::'c). (f a,c))" by blast
  have "inj_on g (A × C) ∧ g ` (A × C) ≤ (B × C)"
  using 1 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]
      card_of_Times_commute[of C A]  card_of_Times_commute[of B C]
      ordIso_ordLeq_trans[of "|C × A|"] ordLeq_ordIso_trans[of "|C × A|"]
by blast

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 "λ i f. i ∈ I ⟶ inj_on f (A i) ∧ f ` (A i) ≤ B i"]
  obtain F where 1: "∀i ∈ I. inj_on (F i) (A i) ∧ (F i) ` (A i) ≤ B i"
    by atomize_elim (auto intro: bchoice)
  obtain g where g_def: "g = (λ(i,a::'b). (i,F i a))" by blast
  have "inj_on g (Sigma I A) ∧ g ` (Sigma I A) ≤ (Sigma I B)"
  using 1 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 = "λ bl. case bl of True ⇒ a1 | False ⇒ a2"
  have "bij_betw ?f UNIV {a1,a2}"
  proof-
    {fix bl1 and bl2 assume "?f  bl1 = ?f bl2"
     hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
    }
    moreover
    {fix bl have "?f bl ∈ {a1,a2}" by (case_tac bl, auto)
    }
    moreover
    {fix a assume *: "a ∈ {a1,a2}"
     have "a ∈ ?f ` UNIV"
     proof(cases "a = a1")
       assume "a = a1"
       hence "?f True = a" by auto  thus ?thesis by blast
     next
       assume "a ≠ a1" hence "a = a2" using * by auto
       hence "?f False = a" by auto  thus ?thesis by blast
     qed
    }
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by blast
  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]
        ordIso_ordLeq_trans[of "|UNIV::bool set|"] by blast
  (*  *)
  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|"
  using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B × (UNIV::bool set)|"]
        ordLeq_transitive[of "|A <+> B|" "|B × (UNIV::bool set)|" "|B × A|"]
        ordLeq_ordIso_trans[of "|A <+> B|" "|B × A|" "|A × B|"]
  by blast
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 = "λ(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 ‹Cardinals versus set operations involving infinite sets›

text‹Here we show that, for infinite sets, most set-theoretic constructions
do not increase the cardinality.  The cornerstone for this is
theorem ‹Card_order_Times_same_infinite›, 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 "card-book"}. Then everything else follows fairly easily.›

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

text‹The next two results correspond to the ZF fact that all infinite cardinals are
limit ordinals:›

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' <o 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| <o r" 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| <o |under 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 "EX 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 have "under r a < Field r" by blast
  then obtain b where 1: "b : Field r ∧ ~ (b,a) : r"
  unfolding under_def by blast
  moreover have ba: "b ≠ a"
  using 1 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 1 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-
  obtain phi where phi_def:
  "phi = (λr::'a rel. Card_order r ∧ ¬finite(Field r) ∧
                      ¬ |Field r × Field r| ≤o r )" by blast
  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 <o |?A × ?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 <o ?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 <o r" 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| <o r" 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[of r]
        ordIso_transitive[of "|Field r × Field p|"]
        ordIso_transitive[of _ "|Field r|"] by blast
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 = {}", simp add: card_of_empty)
  assume *: "I ≠ {}"
  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 * LEQ_I by (auto simp add: card_of_Times_infinite)
  ultimately show ?thesis using ordLeq_ordIso_trans by blast
qed

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 = {}", simp add: card_of_empty)
  assume *: "I ≠ {}"
  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

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 = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
  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}")
    assume Case1: "B = {b1}"
    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 Case1 by blast
    ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
    hence "bij_betw (g o ?Inl) A (A <+> B)"
    using 2 by (auto simp add: bij_betw_trans)
    thus ?thesis using card_of_ordIso ordIso_symmetric by blast
  next
    assume Case2: "B ≠ {b1}"
    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[of r]
        ordIso_transitive[of "|Field r <+> Field p|"]
        ordIso_transitive[of _ "|Field r|"] by blast
qed


subsection ‹The cardinal $\omega$ and the finite cardinals›

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

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)"
proof
  assume "finite (A × B)"
  from assms(1) have "A ≠ {}" by auto
  with ‹finite (A × B)› have "finite B" using finite_cartesian_productD2 by auto
  with assms(2) show False by simp
qed


subsubsection ‹First as well-orders›

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
by (auto simp add: natLeq_Refl natLeq_trans)

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 ‹Then as cardinals›

lemma natLeq_Card_order: "Card_order natLeq"
proof(auto simp add: natLeq_Well_order
      Card_order_iff_Restr_underS Restr_natLeq2, simp add:  Field_natLeq)
  fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
  moreover have "¬finite(UNIV::nat set)" by auto
  ultimately show "natLeq_on n <o |UNIV::nat set|"
  using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
        Field_card_of[of "UNIV::nat set"]
        card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
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| <o natLeq)"
using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
      card_of_Well_order natLeq_Well_order by blast


subsection ‹The successor of a cardinal›

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

definition isCardSuc :: "'a rel ⇒ 'a set rel ⇒ bool"
where
"isCardSuc r r' ≡
 Card_order r' ∧ r <o r' ∧
 (∀(r''::'a set rel). Card_order r'' ∧ r <o r'' ⟶ r' ≤o r'')"

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

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 <o 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 <o cardSuc 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‹The minimality property of ‹cardSuc› originally present in its definition
is local to the type ‹'a set rel›, i.e., that of ‹cardSuc r›:›

lemma cardSuc_least_aux:
"⟦Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'⟧ ⟹ cardSuc r ≤o r'"
using cardSuc_isCardSuc unfolding isCardSuc_def by blast

text‹But from this we can infer general minimality:›

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' <o ?p"
   then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' ∧ r'' <o ?p"
   using internalize_ordLess[of r' ?p] by blast
   (*  *)
   have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
   moreover have "r <o 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') = (cardSuc r ≤o r')"
proof(auto simp add: assms cardSuc_least)
  assume "cardSuc r ≤o r'"
  thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
qed

lemma cardSuc_ordLeq_ordLess:
assumes CARD: "Card_order r" and CARD': "Card_order r'"
shows "(r' <o cardSuc r) = (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[of r r']
  not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
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
  then show "finite (Field (cardSuc |A| ))"
  proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])
    show "cardSuc |A| ≤o |Pow A|"
      by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)
  qed
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 card_of_Plus_ordLess_infinite:
assumes INF: "¬finite C" and
        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
shows "|A <+> B| <o |C|"
proof(cases "A = {} ∨ B = {}")
  assume Case1: "A = {} ∨ B = {}"
  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
  assume Case2: "¬(A = {} ∨ B = {})"
  {assume *: "|C| ≤o |A <+> B|"
   hence "¬finite (A <+> B)" using INF card_of_ordLeq_finite by blast
   hence 1: "¬finite A ∨ ¬finite B" using finite_Plus by blast
   {assume Case21: "|A| ≤o |B|"
    hence "¬finite B" using 1 card_of_ordLeq_finite by blast
    hence "|A <+> B| =o |B|" using Case2 Case21
    by (auto simp add: card_of_Plus_infinite)
    hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
   }
   moreover
   {assume Case22: "|B| ≤o |A|"
    hence "¬finite A" using 1 card_of_ordLeq_finite by blast
    hence "|A <+> B| =o |A|" using Case2 Case22
    by (auto simp add: card_of_Plus_infinite)
    hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
   }
   ultimately have False using ordLeq_total card_of_Well_order[of A]
   card_of_Well_order[of B] by blast
  }
  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| <o r" and LESS2: "|B| <o r"
shows "|A <+> B| <o r"
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| <o |?C|"  "|B| <o |?C|"
  using LESS1 LESS2 ordLess_ordIso_trans by blast+
  hence  "|A <+> B| <o |?C|" 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| <o ?r'" and "|B| <o ?r'" using A B c
  by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
  ultimately have "|A <+> B| <o ?r'"
  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


subsection ‹Regular cardinals›

definition cofinal where
"cofinal A r ≡
 ALL a : Field r. EX b : A. a ≠ b ∧ (a,b) : r"

definition regularCard where
"regularCard r ≡
 ALL K. K ≤ Field r ∧ cofinal K r ⟶ |K| =o r"

definition relChain where
"relChain r As ≡
 ALL 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 ≤ (UN i : Field r. As i)"
and cardB: "|B| <o r"
shows "EX i : Field r. B ≤ As i"
proof-
  let ?phi = "%b j. j : Field r ∧ b : As j"
  have "ALL b : B. EX 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 auto
     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∈B. i ≠ f b ∧ (i, f 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| <=o |B|" using card_of_image .
    hence "|?K| <o r" using cardB ordLeq_ordLess_trans 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'"
  "!! p. Card_order p ⟶ (p ≤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 = "UN 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| <o ?r'"
     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 "ALL j : K. |underS ?r' j| <o ?r'"
      using r' 1 by (auto simp: card_of_underS)
      hence "ALL j : K. |underS ?r' j| ≤o r"
      using r' card_of_Card_order by blast
      hence "ALL 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| <o ?r'" 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| <o |?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 ≤ (UN i : Field (cardSuc r). As i)"
and cardB: "|B| <=o r"
shows "EX i : Field (cardSuc r). B ≤ As i"
proof-
  let ?r' = "cardSuc r"
  have "Card_order ?r' ∧ |B| <o ?r'"
  using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
  card_of_Card_order by blast
  moreover have "regularCard ?r'"
  using assms by(simp add: infinite_cardSuc_regularCard)
  ultimately show ?thesis
  using As Bsub cardB regularCard_UNION by blast
qed


subsection ‹Others›

lemma card_of_Func_Times:
"|Func (A × B) C| =o |Func A (Func B C)|"
unfolding card_of_ordIso[symmetric]
using bij_betw_curr by blast

lemma card_of_Pow_Func:
"|Pow A| =o |Func A (UNIV::bool set)|"
proof-
  define F where [abs_def]: "F A' a =
    (if a ∈ A then (if a ∈ A' then True else False) else undefined)" for A' a
  have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
  unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
    fix A1 A2 assume "A1 ∈ Pow A" "A2 ∈ Pow A" "F A1 = F A2"
    thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: if_split_asm)
  next
    show "F ` Pow A = Func A UNIV"
    proof safe
      fix f assume f: "f ∈ Func A (UNIV::bool set)"
      show "f ∈ F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
        let ?A1 = "{a ∈ A. f a = True}"
        show "f = F ?A1"
          unfolding F_def apply(rule ext)
          using f unfolding Func_def mem_Collect_eq by auto
      qed auto
    qed(unfold Func_def mem_Collect_eq F_def, auto)
  qed
  thus ?thesis unfolding card_of_ordIso[symmetric] by blast
qed

lemma card_of_Func_UNIV:
"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a ⇒ 'b. range f ⊆ B}|"
apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
  let ?F = "λ f (a::'a). ((f a)::'b)"
  show "bij_betw ?F {f. range f ⊆ B} (Func UNIV B)"
  unfolding bij_betw_def inj_on_def proof safe
    fix h :: "'a ⇒ 'b" assume h: "h ∈ Func UNIV B"
    hence "∀ a. ∃ b. h a = b" unfolding Func_def by auto
    then obtain f where f: "∀ a. h a = f a" by blast
    hence "range f ⊆ B" using h unfolding Func_def by auto
    thus "h ∈ (λf a. f a) ` {f. range f ⊆ B}" using f by auto
  qed(unfold Func_def fun_eq_iff, auto)
qed

lemma Func_Times_Range:
  "|Func A (B × C)| =o |Func A B × Func A C|" (is "|?LHS| =o |?RHS|")
proof -
  let ?F = "λfg. (λx. if x ∈ A then fst (fg x) else undefined,
                  λx. if x ∈ A then snd (fg x) else undefined)"
  let ?G = "λ(f, g) x. if x ∈ A then (f x, g x) else undefined"
  have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def
  proof (intro conjI impI ballI equalityI subsetI)
    fix f g assume *: "f ∈ Func A (B × C)" "g ∈ Func A (B × C)" "?F f = ?F g"
    show "f = g"
    proof
      fix x from * have "fst (f x) = fst (g x) ∧ snd (f x) = snd (g x)"
        by (case_tac "x ∈ A") (auto simp: Func_def fun_eq_iff split: if_splits)
      then show "f x = g x" by (subst (1 2) surjective_pairing) simp
    qed
  next
    fix fg assume "fg ∈ Func A B × Func A C"
    thus "fg ∈ ?F ` Func A (B × C)"
      by (intro image_eqI[of _ _ "?G fg"]) (auto simp: Func_def)
  qed (auto simp: Func_def fun_eq_iff)
  thus ?thesis using card_of_ordIso by blast
qed

end