# 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

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

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
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'|"

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
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]
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]
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 |"
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"

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|"
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"
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
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

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')"
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
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
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
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
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'"
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
`