--- a/src/HOL/Cardinals/Cardinal_Arithmetic.thy Mon Nov 18 18:04:44 2013 +0100
+++ b/src/HOL/Cardinals/Cardinal_Arithmetic.thy Mon Nov 18 18:04:44 2013 +0100
@@ -8,7 +8,7 @@
header {* Cardinal Arithmetic *}
theory Cardinal_Arithmetic
-imports Cardinal_Order_Relation_Base
+imports Cardinal_Order_Relation_LFP
begin
text {*
--- a/src/HOL/Cardinals/Cardinal_Order_Relation.thy Mon Nov 18 18:04:44 2013 +0100
+++ b/src/HOL/Cardinals/Cardinal_Order_Relation.thy Mon Nov 18 18:04:44 2013 +0100
@@ -8,7 +8,7 @@
header {* Cardinal-Order Relations *}
theory Cardinal_Order_Relation
-imports Cardinal_Order_Relation_Base Constructions_on_Wellorders
+imports Cardinal_Order_Relation_LFP Constructions_on_Wellorders
begin
declare
--- a/src/HOL/Cardinals/Cardinal_Order_Relation_Base.thy Mon Nov 18 18:04:44 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2438 +0,0 @@
-(* Title: HOL/Cardinals/Cardinal_Order_Relation_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Cardinal-order relations (base).
-*)
-
-header {* Cardinal-Order Relations (Base) *}
-
-theory Cardinal_Order_Relation_Base
-imports Constructions_on_Wellorders_Base
-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, @{text "card"}, from the theory @{text "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, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
-strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
-
-definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
-where
-"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
-
-
-abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
-abbreviation "card_order r \<equiv> 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 \<Longrightarrow> A = Field r \<and> Card_order r"
-unfolding card_order_on_def using rel.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 @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
-which states that on any given set there exists a well-order;
-\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
-such well-order, i.e., a cardinal order.
-\end{itemize}
-*}
-
-
-theorem card_order_on: "\<exists>r. card_order_on A r"
-proof-
- obtain R where R_def: "R = {r. well_order_on A r}" by blast
- have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
- using well_order_on[of A] R_def rel.well_order_on_Well_order by blast
- hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>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' \<and> Field p' = Field r'"
- using rel.well_order_on_Well_order by blast
- obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
- using ISO unfolding ordIso_def by auto
- hence 3: "inj_on f (Field r') \<and> 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 \<and> 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_Field2 order_on_defs)
- ultimately have "well_order_on (Field r) ?p" by auto
- hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
- thus "r' \<le>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 \<Rightarrow> '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 rel.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| \<Longrightarrow> 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 \<Longrightarrow> |A| \<le>o r"
-using card_of_card_order_on unfolding card_order_on_def by blast
-
-
-lemma card_of_ordIso:
-"(\<exists>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 \<and> |A| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
- hence "|B| \<le>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 \<and> |B| =o r"
- using Well_order_iso_copy card_of_well_order_on by blast
- hence "|A| \<le>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 "\<exists>f. bij_betw f A B" by auto
-qed
-
-
-lemma card_of_ordLeq:
-"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
-proof(auto)
- fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
- {assume "|B| <o |A|"
- hence "|B| \<le>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 \<and> g ` B \<le> 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 Cantor_Bernstein[of f] by fastforce
- hence "|A| =o |B|" using card_of_ordIso by blast
- hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
- }
- thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
- by (auto simp: card_of_Well_order)
-next
- assume *: "|A| \<le>o |B|"
- obtain f where "embed ( |A| ) ( |B| ) f"
- using * unfolding ordLeq_def by auto
- hence "inj_on f A \<and> f ` A \<le> 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 "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
-qed
-
-
-lemma card_of_ordLeq2:
-"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
-using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
-
-
-lemma card_of_ordLess:
-"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
-proof-
- have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
- using card_of_ordLeq by blast
- also have "\<dots> = ( |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 \<noteq> {} \<Longrightarrow> (\<not>(\<exists>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 "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
-shows "|A| \<le>o |B|"
-using assms unfolding card_of_ordLeq[symmetric] by auto
-
-
-lemma card_of_unique:
-"card_order_on A r \<Longrightarrow> r =o |A|"
-by (simp only: card_order_on_ordIso card_of_card_order_on)
-
-
-lemma card_of_mono1:
-"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
-using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
-
-
-lemma card_of_mono2:
-assumes "r \<le>o r'"
-shows "|Field r| \<le>o |Field r'|"
-proof-
- obtain f where
- 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
- using assms unfolding ordLeq_def
- by (auto simp add: rel.well_order_on_Well_order)
- hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
- by (auto simp add: embed_inj_on embed_Field)
- thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
-qed
-
-
-lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
-by (simp add: ordIso_iff_ordLeq card_of_mono2)
-
-
-lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
-using card_of_least card_of_well_order_on rel.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 \<le>o |Field r| )"
-proof-
- have "Card_order r = (r =o |Field r| )"
- unfolding Card_order_iff_ordIso_card_of by simp
- also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
- unfolding ordIso_iff_ordLeq by simp
- also have "... = (r \<le>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 = (\<forall>a \<in> Field r. Restr r (rel.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 "|rel.underS r a| <o r"
-proof-
- let ?A = "rel.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'| \<le>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| \<le>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: rel.well_order_on_Well_order)
- hence "|Field r| \<le>o r" using card_of_least by blast
- thus ?thesis using assms ordLeq_ordLess_trans by blast
-qed
-
-
-lemma internalize_card_of_ordLeq:
-"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
-proof
- assume "|A| \<le>o r"
- then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>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| \<and> |Field p| \<le>o r"
- using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
- thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
-next
- assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
- thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
-qed
-
-
-lemma internalize_card_of_ordLeq2:
-"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>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"
-@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
-*}
-
-
-lemma card_of_empty: "|{}| \<le>o |A|"
-using card_of_ordLeq inj_on_id by blast
-
-
-lemma card_of_empty1:
-assumes "Well_order r \<or> Card_order r"
-shows "|{}| \<le>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 "|{}| \<le>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 \<Longrightarrow> |{}| \<le>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| \<le>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| \<le>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_ordLeqI2:
-assumes "B \<subseteq> f ` A"
-shows "|B| \<le>o |A|"
-using assms by (metis surj_imp_ordLeq)
-
-
-lemma card_of_singl_ordLeq:
-assumes "A \<noteq> {}"
-shows "|{b}| \<le>o |A|"
-proof-
- obtain a where *: "a \<in> A" using assms by auto
- let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
- have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
- using * unfolding inj_on_def by auto
- thus ?thesis using card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_singl_ordLeq:
-"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>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
-
-
-lemma infinite_Pow:
-assumes "infinite A"
-shows "infinite (Pow A)"
-proof-
- have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
- thus ?thesis by (metis assms finite_Pow_iff)
-qed
-
-
-corollary Card_order_Pow:
-"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
-using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
-
-
-corollary card_of_set_type: "|UNIV::'a set| <o |UNIV::'a set set|"
-using card_of_Pow[of "UNIV::'a set"] by simp
-
-
-lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
-proof-
- have "Inl ` A \<le> A <+> B" by auto
- thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_Plus1:
-"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
-using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
-proof-
- have "Inr ` B \<le> A <+> B" by auto
- thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
-qed
-
-
-corollary Card_order_Plus2:
-"Card_order r \<Longrightarrow> r \<le>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 = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
- | Inr b \<Rightarrow> 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 -
- def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
- case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
- |Inr b \<Rightarrow> Inr (Inl b))
- |Inr c \<Rightarrow> Inr (Inr c)"
- have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
- proof
- fix x assume x: "x \<in> A <+> B <+> C"
- show "x \<in> f ` ((A <+> B) <+> C)"
- proof(cases x)
- case (Inl a)
- hence "a \<in> 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 \<in> B" "x = f (Inl (Inr b))"
- using x 1 unfolding f_def by auto
- thus ?thesis by auto
- next
- case (Inr c)
- hence "c \<in> 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 force
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Plus_mono1:
-assumes "|A| \<le>o |B|"
-shows "|A <+> C| \<le>o |B <+> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
- have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
- proof-
- {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
- "g d1 = g d2"
- hence "d1 = d2" using 1 unfolding inj_on_def
- by(case_tac d1, case_tac d2, auto simp add: g_def)
- }
- moreover
- {fix d assume "d \<in> A <+> C"
- hence "g d \<in> 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 metis
-qed
-
-
-corollary ordLeq_Plus_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
-using assms card_of_mono2 card_of_Plus_mono1 by blast
-
-
-lemma card_of_Plus_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C <+> A| \<le>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 \<le>o r'"
-shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
-using assms card_of_mono2 card_of_Plus_mono2 by blast
-
-
-lemma card_of_Plus_mono:
-assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
-shows "|A <+> C| \<le>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 \<le>o r'" and "p \<le>o p'"
-shows "|(Field r) <+> (Field p)| \<le>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_Un1:
-shows "|A| \<le>o |A \<union> B| "
-using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
-
-
-lemma card_of_diff:
-shows "|A - B| \<le>o |A|"
-using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
-
-
-lemma card_of_Un_Plus_ordLeq:
-"|A \<union> B| \<le>o |A <+> B|"
-proof-
- let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
- have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
- unfolding inj_on_def by auto
- thus ?thesis using card_of_ordLeq by blast
-qed
-
-
-lemma card_of_Times1:
-assumes "A \<noteq> {}"
-shows "|B| \<le>o |B \<times> A|"
-proof(cases "B = {}", simp add: card_of_empty)
- assume *: "B \<noteq> {}"
- have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
- thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
- card_of_ordLeq[of B "B \<times> A"] * by blast
-qed
-
-
-corollary Card_order_Times1:
-"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
-using card_of_Times1[of B] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
-proof-
- let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
- have "bij_betw ?f (A \<times> B) (B \<times> 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 \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
-using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
- ordLeq_ordIso_trans by blast
-
-
-corollary Card_order_Times2:
-"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
-using card_of_Times2[of A] card_of_Field_ordIso
- ordIso_ordLeq_trans ordIso_symmetric by blast
-
-
-lemma card_of_Times3: "|A| \<le>o |A \<times> 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 \<times> (UNIV::bool set)|"
-proof-
- let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
- |Inr a \<Rightarrow> (a,False)"
- have "bij_betw ?f (A <+> A) (A \<times> (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 \<in> A <+> A"
- hence "?f c \<in> A \<times> (UNIV::bool set)"
- by(case_tac c, auto)
- }
- moreover
- {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
- have "(a,bl) \<in> ?f ` ( A <+> A)"
- proof(cases bl)
- assume bl hence "?f(Inl a) = (a,bl)" by auto
- thus ?thesis using * by force
- next
- assume "\<not> 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| \<le>o |B|"
-shows "|A \<times> C| \<le>o |B \<times> C|"
-proof-
- obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
- using assms card_of_ordLeq[of A] by fastforce
- obtain g where g_def:
- "g = (\<lambda>(a,c::'c). (f a,c))" by blast
- have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
- using 1 unfolding inj_on_def using g_def by auto
- thus ?thesis using card_of_ordLeq by metis
-qed
-
-
-corollary ordLeq_Times_mono1:
-assumes "r \<le>o r'"
-shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
-using assms card_of_mono2 card_of_Times_mono1 by blast
-
-
-lemma card_of_Times_mono2:
-assumes "|A| \<le>o |B|"
-shows "|C \<times> A| \<le>o |C \<times> 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 \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
-by blast
-
-
-corollary ordLeq_Times_mono2:
-assumes "r \<le>o r'"
-shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
-using assms card_of_mono2 card_of_Times_mono2 by blast
-
-
-lemma card_of_Times_cong1:
-assumes "|A| =o |B|"
-shows "|A \<times> C| =o |B \<times> C|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
-
-
-lemma card_of_Times_cong2:
-assumes "|A| =o |B|"
-shows "|C \<times> A| =o |C \<times> B|"
-using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
-
-
-corollary ordIso_Times_cong2:
-assumes "r =o r'"
-shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
-using assms card_of_cong card_of_Times_cong2 by blast
-
-
-lemma card_of_Sigma_mono1:
-assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
-shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
-proof-
- have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
- using assms by (auto simp add: card_of_ordLeq)
- with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
- obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by metis
- obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
- have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
- using 1 unfolding inj_on_def using g_def by force
- thus ?thesis using card_of_ordLeq by metis
-qed
-
-
-corollary card_of_Sigma_Times:
-"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
-using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
-
-
-lemma card_of_UNION_Sigma:
-"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
-using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
-
-
-lemma card_of_bool:
-assumes "a1 \<noteq> a2"
-shows "|UNIV::bool set| =o |{a1,a2}|"
-proof-
- let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> 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 \<in> {a1,a2}" by (case_tac bl, auto)
- }
- moreover
- {fix a assume *: "a \<in> {a1,a2}"
- have "a \<in> ?f ` UNIV"
- proof(cases "a = a1")
- assume "a = a1"
- hence "?f True = a" by auto thus ?thesis by blast
- next
- assume "a \<noteq> 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 (metis image_subsetI order_eq_iff subsetI)
- qed
- thus ?thesis using card_of_ordIso by blast
-qed
-
-
-lemma card_of_Plus_Times_aux:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- LEQ: "|A| \<le>o |B|"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
- have 1: "|UNIV::bool set| \<le>o |A|"
- using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
- ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
- (* *)
- have "|A <+> B| \<le>o |B <+> B|"
- using LEQ card_of_Plus_mono1 by blast
- moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
- using card_of_Plus_Times_bool by blast
- moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
- using 1 by (simp add: card_of_Times_mono2)
- moreover have " |B \<times> A| =o |A \<times> B|"
- using card_of_Times_commute by blast
- ultimately show "|A <+> B| \<le>o |A \<times> B|"
- using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
- ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
- ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
- by blast
-qed
-
-
-lemma card_of_Plus_Times:
-assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
- B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
-shows "|A <+> B| \<le>o |A \<times> B|"
-proof-
- {assume "|A| \<le>o |B|"
- hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
- }
- moreover
- {assume "|B| \<le>o |A|"
- hence "|B <+> A| \<le>o |B \<times> 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 metis
- }
- ultimately show ?thesis
- using card_of_Well_order[of A] card_of_Well_order[of B]
- ordLeq_total[of "|A|"] by metis
-qed
-
-
-lemma card_of_ordLeq_finite:
-assumes "|A| \<le>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| \<le>o |B|" and "infinite A"
-shows "infinite 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 @{text "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:
-"infinite A = ( |UNIV::nat set| \<le>o |A| )"
-by (auto simp add: infinite_iff_countable_subset card_of_ordLeq)
-
-
-lemma finite_iff_cardOf_nat:
-"finite A = ( |A| <o |UNIV :: nat set| )"
-using infinite_iff_card_of_nat[of A]
-not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
-by (fastforce simp: card_of_Well_order)
-
-lemma finite_ordLess_infinite2:
-assumes "finite A" and "infinite B"
-shows "|A| <o |B|"
-using assms
-finite_ordLess_infinite[of "|A|" "|B|"]
-card_of_Well_order[of A] card_of_Well_order[of B]
-Field_card_of[of A] Field_card_of[of B] by auto
-
-
-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: "infinite (Field r)"
-shows "\<not> (\<exists>a. Field r = rel.under r a)"
-proof(auto)
- have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
- using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
- fix a assume *: "Field r = rel.under r a"
- show False
- proof(cases "a \<in> Field r")
- assume Case1: "a \<notin> Field r"
- hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
- thus False using INF * by auto
- next
- let ?r' = "Restr r (rel.underS r a)"
- assume Case2: "a \<in> Field r"
- hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
- using 0 rel.Refl_under_underS rel.underS_notIn by fastforce
- have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
- using 0 wo_rel.underS_ofilter * 1 Case2 by auto
- hence "?r' <o r" using 0 using ofilter_ordLess by blast
- moreover
- have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
- using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
- ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
- moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
- ultimately have "|rel.underS r a| <o |rel.under r a|"
- using ordIso_symmetric ordLess_ordIso_trans by blast
- moreover
- {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
- using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
- hence "|rel.under r a| =o |rel.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 "infinite (Field r)"
-and a: "a : Field r"
-shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
-proof-
- have "Field r \<noteq> rel.under r a"
- using assms Card_order_infinite_not_under by blast
- moreover have "rel.under r a \<le> Field r"
- using rel.under_Field .
- ultimately have "rel.under r a < Field r" by blast
- then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
- unfolding rel.under_def by blast
- moreover have ba: "b \<noteq> 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: "infinite(Field r)"
-shows "|Field r \<times> Field r| \<le>o r"
-proof-
- obtain phi where phi_def:
- "phi = (\<lambda>r::'a rel. Card_order r \<and> infinite(Field r) \<and>
- \<not> |Field r \<times> Field r| \<le>o r )" by blast
- have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
- unfolding phi_def card_order_on_def by auto
- have Ft: "\<not>(\<exists>r. phi r)"
- proof
- assume "\<exists>r. phi r"
- hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
- using temp1 by auto
- then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
- 3: "Card_order r \<and> 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' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
- using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
- have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
- using card_of_Card_order card_of_Well_order by blast
- (* *)
- have "r <o |?A \<times> ?A|"
- using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
- moreover have "|?A \<times> ?A| \<le>o ?r'"
- using card_of_least[of "?A \<times> ?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: "\<not> bij_betw f ?A (?A \<times> ?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 \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
- using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
- hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
- using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
- have "\<not> (\<exists>a. Field r = rel.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 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
- using temp2 3 bsqr_ofilter[of r ?B] by blast
- hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
- hence 10: "r \<le>o |A1 \<times> 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| \<le>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 "infinite (Field r)" using 1 unfolding phi_def by simp
- hence "infinite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
- hence "infinite A1" using 9 infinite_super finite_cartesian_product by blast
- moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
- using card_of_Card_order[of A1] card_of_Well_order[of A1]
- by (simp add: Field_card_of)
- moreover have "\<not> r \<le>o | A1 |"
- using temp4 11 3 using not_ordLeq_iff_ordLess by blast
- ultimately have "infinite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
- by (simp add: card_of_card_order_on)
- hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
- using 2 unfolding phi_def by blast
- hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
- hence "r \<le>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 "infinite A"
-shows "|A \<times> A| =o |A|"
-proof-
- let ?r = "|A|"
- have "Field ?r = A \<and> Card_order ?r"
- using Field_card_of card_of_Card_order[of A] by fastforce
- hence "|A \<times> A| \<le>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: "infinite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
-shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
-proof-
- have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
- using assms by (simp add: card_of_Times1 card_of_Times2)
- moreover
- {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
- using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
- moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
- ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
- using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
- }
- ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
-qed
-
-
-corollary card_of_Times_infinite_simps:
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
-"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
-by (auto simp add: card_of_Times_infinite ordIso_symmetric)
-
-
-corollary Card_order_Times_infinite:
-assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
- NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
-shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
-proof-
- have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> 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 \<times> Field p|"]
- ordIso_transitive[of _ "|Field r|"] by blast
-qed
-
-
-lemma card_of_Sigma_ordLeq_infinite:
-assumes INF: "infinite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|SIGMA i : I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
- have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
- using LEQ card_of_Sigma_Times by blast
- moreover have "|I \<times> 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: "infinite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|SIGMA i : I. A i| \<le>o r"
-proof-
- let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
- hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
- hence "|SIGMA i : I. A i| \<le>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:
-"\<lbrakk>infinite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
- \<Longrightarrow> |A <*> B| \<le>o r"
-by(simp add: card_of_Sigma_ordLeq_infinite_Field)
-
-
-lemma card_of_UNION_ordLeq_infinite:
-assumes INF: "infinite B" and
- LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
-shows "|\<Union> i \<in> I. A i| \<le>o |B|"
-proof(cases "I = {}", simp add: card_of_empty)
- assume *: "I \<noteq> {}"
- have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
- using card_of_UNION_Sigma by blast
- moreover have "|SIGMA i : I. A i| \<le>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: "infinite (Field r)" and r: "Card_order r" and
- LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
-shows "|\<Union> i \<in> I. A i| \<le>o r"
-proof-
- let ?B = "Field r"
- have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
- ordIso_symmetric by blast
- hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
- using LEQ_I LEQ ordLeq_ordIso_trans by blast+
- hence "|\<Union> i \<in> I. A i| \<le>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: "infinite A" and LEQ: "|B| \<le>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 \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
- assume *: "B \<noteq> {}"
- then obtain b1 where 1: "b1 \<in> 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: "infinite (?Inl ` A)"
- using INF bij_betw_finite[of ?Inl A] by blast
- let ?A' = "?Inl ` A \<union> {?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 \<noteq> {b1}"
- with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
- obtain f where "inj_on f B \<and> f ` B \<le> A"
- using LEQ card_of_ordLeq[of B] by fastforce
- with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
- unfolding inj_on_def by auto
- with 3 have "|A <+> B| \<le>o |A \<times> B|"
- by (auto simp add: card_of_Plus_Times)
- moreover have "|A \<times> B| =o |A|"
- using assms * by (simp add: card_of_Times_infinite_simps)
- ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
- thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
- qed
-qed
-
-
-lemma card_of_Plus_infinite2:
-assumes INF: "infinite A" and LEQ: "|B| \<le>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: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
-using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
-
-
-corollary Card_order_Plus_infinite:
-assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
- LEQ: "p \<le>o r"
-shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
-proof-
- have "| Field r <+> Field p | =o | Field r | \<and>
- | 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
-
-
-lemma card_of_Un_infinite:
-assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
-shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
-proof-
- have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
- moreover have "|A <+> B| =o |A|"
- using assms by (metis card_of_Plus_infinite)
- ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
- hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
- thus ?thesis using Un_commute[of B A] by auto
-qed
-
-
-lemma card_of_Un_diff_infinite:
-assumes INF: "infinite A" and LESS: "|B| <o |A|"
-shows "|A - B| =o |A|"
-proof-
- obtain C where C_def: "C = A - B" by blast
- have "|A \<union> B| =o |A|"
- using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
- moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
- ultimately have 1: "|C \<union> B| =o |A|" by auto
- (* *)
- {assume *: "|C| \<le>o |B|"
- moreover
- {assume **: "finite B"
- hence "finite C"
- using card_of_ordLeq_finite * by blast
- hence False using ** INF card_of_ordIso_finite 1 by blast
- }
- hence "infinite B" by auto
- ultimately have False
- using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
- }
- hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
- {assume *: "finite C"
- hence "finite B" using card_of_ordLeq_finite 2 by blast
- hence False using * INF card_of_ordIso_finite 1 by blast
- }
- hence "infinite C" by auto
- hence "|C| =o |A|"
- using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
- thus ?thesis unfolding C_def .
-qed
-
-
-lemma card_of_Plus_ordLess_infinite:
-assumes INF: "infinite C" and
- LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
-shows "|A <+> B| <o |C|"
-proof(cases "A = {} \<or> B = {}")
- assume Case1: "A = {} \<or> B = {}"
- hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
- using card_of_Plus_empty1 card_of_Plus_empty2 by blast
- hence "|A <+> B| =o |A| \<or> |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: "\<not>(A = {} \<or> B = {})"
- {assume *: "|C| \<le>o |A <+> B|"
- hence "infinite (A <+> B)" using INF card_of_ordLeq_finite by blast
- hence 1: "infinite A \<or> infinite B" using finite_Plus by blast
- {assume Case21: "|A| \<le>o |B|"
- hence "infinite 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| \<le>o |A|"
- hence "infinite 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: "infinite (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| \<and> |?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 infinite_card_of_insert:
-assumes "infinite A"
-shows "|insert a A| =o |A|"
-proof-
- have iA: "insert a A = A \<union> {a}" by simp
- show ?thesis
- using infinite_imp_bij_betw2[OF assms] unfolding iA
- by (metis bij_betw_inv card_of_ordIso)
-qed
-
-
-subsection {* Cardinals versus lists *}
-
-
-text{* The next is an auxiliary operator, which shall be used for inductive
-proofs of facts concerning the cardinality of @{text "List"} : *}
-
-definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
-where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
-
-
-lemma lists_def2: "lists A = {l. set l \<le> A}"
-using in_listsI by blast
-
-
-lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
-unfolding lists_def2 nlists_def by blast
-
-
-lemma card_of_lists: "|A| \<le>o |lists A|"
-proof-
- let ?h = "\<lambda> a. [a]"
- have "inj_on ?h A \<and> ?h ` A \<le> lists A"
- unfolding inj_on_def lists_def2 by auto
- thus ?thesis by (metis card_of_ordLeq)
-qed
-
-
-lemma nlists_0: "nlists A 0 = {[]}"
-unfolding nlists_def by auto
-
-
-lemma nlists_not_empty:
-assumes "A \<noteq> {}"
-shows "nlists A n \<noteq> {}"
-proof(induct n, simp add: nlists_0)
- fix n assume "nlists A n \<noteq> {}"
- then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
- hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
- thus "nlists A (Suc n) \<noteq> {}" by auto
-qed
-
-
-lemma Nil_in_lists: "[] \<in> lists A"
-unfolding lists_def2 by auto
-
-
-lemma lists_not_empty: "lists A \<noteq> {}"
-using Nil_in_lists by blast
-
-
-lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
-proof-
- let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l"
- have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
- unfolding inj_on_def nlists_def by auto
- moreover have "nlists A (Suc n) \<le> ?h ` ?B"
- proof(auto)
- fix l assume "l \<in> nlists A (Suc n)"
- hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
- then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
- hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
- thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto
- qed
- ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
- unfolding bij_betw_def by auto
- thus ?thesis using card_of_ordIso ordIso_symmetric by blast
-qed
-
-
-lemma card_of_nlists_infinite:
-assumes "infinite A"
-shows "|nlists A n| \<le>o |A|"
-proof(induct n)
- have "A \<noteq> {}" using assms by auto
- thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
-next
- fix n assume IH: "|nlists A n| \<le>o |A|"
- have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
- using card_of_nlists_Succ by blast
- moreover
- {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
- hence "|A \<times> (nlists A n)| =o |A|"
- using assms IH by (auto simp add: card_of_Times_infinite)
- }
- ultimately show "|nlists A (Suc n)| \<le>o |A|"
- using ordIso_transitive ordIso_iff_ordLeq by blast
-qed
-
-
-lemma card_of_lists_infinite:
-assumes "infinite A"
-shows "|lists A| =o |A|"
-proof-
- have "|lists A| \<le>o |A|"
- using assms
- by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
- infinite_iff_card_of_nat card_of_nlists_infinite)
- thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
-qed
-
-
-lemma Card_order_lists_infinite:
-assumes "Card_order r" and "infinite(Field r)"
-shows "|lists(Field r)| =o r"
-using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
-
-
-
-subsection {* The cardinal $\omega$ and the finite cardinals *}
-
-
-text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
-order relation on
-@{text "nat"}, that we abbreviate by @{text "natLeq"}. The finite cardinals
-shall be the restrictions of these relations to the numbers smaller than
-fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
-
-abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
-abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
-
-abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
-where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
-
-lemma infinite_cartesian_product:
-assumes "infinite A" "infinite B"
-shows "infinite (A \<times> B)"
-proof
- assume "finite (A \<times> B)"
- from assms(1) have "A \<noteq> {}" by auto
- with `finite (A \<times> 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, auto)
-
-
-lemma natLeq_Refl: "Refl natLeq"
-unfolding refl_on_def Field_def by auto
-
-
-lemma natLeq_trans: "trans natLeq"
-unfolding trans_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 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 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"
-by auto
-
-
-lemma natLeq_Well_order: "Well_order natLeq"
-unfolding well_order_on_def
-using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
-
-
-corollary natLeq_well_order_on: "well_order_on UNIV natLeq"
-using natLeq_Well_order Field_natLeq by auto
-
-
-lemma natLeq_wo_rel: "wo_rel natLeq"
-unfolding wo_rel_def using natLeq_Well_order .
-
-
-lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
-using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
-
-
-lemma closed_nat_set_iff:
-assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
-shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
-proof-
- {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
- moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
- ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
- using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
- have "A = {0 ..< n}"
- proof(auto simp add: 1)
- fix m assume *: "m \<in> A"
- {assume "n \<le> m" with assms * have "n \<in> A" by blast
- hence False using 1 by auto
- }
- thus "m < n" by fastforce
- qed
- hence "\<exists>n. A = {0 ..< n}" by blast
- }
- thus ?thesis by blast
-qed
-
-
-lemma Field_natLeq_on: "Field (natLeq_on n) = {0 ..< n}"
-unfolding Field_def by auto
-
-
-lemma natLeq_underS_less: "rel.underS natLeq n = {0 ..< n}"
-unfolding rel.underS_def by auto
-
-
-lemma Restr_natLeq: "Restr natLeq {0 ..< n} = natLeq_on n"
-by auto
-
-
-lemma Restr_natLeq2:
-"Restr natLeq (rel.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 "{0..<n}"] by auto
-
-
-corollary natLeq_on_well_order_on: "well_order_on {0 ..< 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 .
-
-
-lemma natLeq_on_ofilter_less_eq:
-"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
-by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
- simp add: Field_natLeq_on, unfold rel.under_def, auto)
-
-
-lemma natLeq_on_ofilter_iff:
-"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
-proof(rule iffI)
- assume *: "wo_rel.ofilter (natLeq_on m) A"
- hence 1: "A \<le> {0..<m}"
- by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def Field_natLeq_on)
- hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
- using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def)
- hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
- thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
-next
- assume "(\<exists>n\<le>m. A = {0 ..< n})"
- thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
-qed
-
-
-
-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))"
- unfolding Field_natLeq_on by auto
- moreover have "infinite(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:
-"infinite A = ( natLeq \<le>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
-
-
-lemma ordIso_natLeq_on_imp_finite:
-"|A| =o natLeq_on n \<Longrightarrow> finite A"
-unfolding ordIso_def iso_def[abs_def]
-by (auto simp: Field_natLeq_on bij_betw_finite Field_card_of)
-
-
-lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
-proof(unfold card_order_on_def,
- auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
- fix r assume "well_order_on {0..<n} r"
- thus "natLeq_on n \<le>o r"
- using finite_atLeastLessThan natLeq_on_well_order_on
- finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
-qed
-
-
-corollary card_of_Field_natLeq_on:
-"|Field (natLeq_on n)| =o natLeq_on n"
-using Field_natLeq_on natLeq_on_Card_order
- Card_order_iff_ordIso_card_of[of "natLeq_on n"]
- ordIso_symmetric[of "natLeq_on n"] by blast
-
-
-corollary card_of_less:
-"|{0 ..< n}| =o natLeq_on n"
-using Field_natLeq_on card_of_Field_natLeq_on by auto
-
-
-lemma natLeq_on_ordLeq_less_eq:
-"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
-proof
- assume "natLeq_on m \<le>o natLeq_on n"
- then obtain f where "inj_on f {0..<m} \<and> f ` {0..<m} \<le> {0..<n}"
- unfolding ordLeq_def using
- embed_inj_on[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on]
- embed_Field[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on] by blast
- thus "m \<le> n" using atLeastLessThan_less_eq2 by blast
-next
- assume "m \<le> n"
- hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
- hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
- thus "natLeq_on m \<le>o natLeq_on n"
- using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
-qed
-
-
-lemma natLeq_on_ordLeq_less:
-"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
-using not_ordLeq_iff_ordLess[of "natLeq_on m" "natLeq_on n"]
-natLeq_on_Well_order natLeq_on_ordLeq_less_eq by auto
-
-
-
-subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
-
-
-lemma finite_card_of_iff_card2:
-assumes FIN: "finite A" and FIN': "finite B"
-shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
-using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
-
-
-lemma finite_imp_card_of_natLeq_on:
-assumes "finite A"
-shows "|A| =o natLeq_on (card A)"
-proof-
- obtain h where "bij_betw h A {0 ..< card A}"
- using assms ex_bij_betw_finite_nat by blast
- thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
-qed
-
-
-lemma finite_iff_card_of_natLeq_on:
-"finite A = (\<exists>n. |A| =o natLeq_on n)"
-using finite_imp_card_of_natLeq_on[of A]
-by(auto simp add: ordIso_natLeq_on_imp_finite)
-
-
-
-subsection {* The successor of a cardinal *}
-
-
-text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
-being a successor cardinal of @{text "r"}. Although the definition does
-not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
-
-
-definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
-where
-"isCardSuc r r' \<equiv>
- Card_order r' \<and> r <o r' \<and>
- (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
-
-
-text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
-by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
-Again, the picked item shall be proved unique up to order-isomorphism. *}
-
-
-definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
-where
-"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
-
-
-lemma exists_minim_Card_order:
-"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
-unfolding card_order_on_def using exists_minim_Well_order by blast
-
-
-lemma exists_isCardSuc:
-assumes "Card_order r"
-shows "\<exists>r'. isCardSuc r r'"
-proof-
- let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
- have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
- by (simp add: card_of_Card_order Card_order_Pow)
- then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>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 \<Longrightarrow> Card_order(cardSuc r)"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
-
-
-lemma cardSuc_greater:
-"Card_order r \<Longrightarrow> r <o cardSuc r"
-using cardSuc_isCardSuc unfolding isCardSuc_def by blast
-
-
-lemma cardSuc_ordLeq:
-"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
-using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
-
-
-text{* The minimality property of @{text "cardSuc"} originally present in its definition
-is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
-
-lemma cardSuc_least_aux:
-"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>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 \<le>o r'"
-proof-
- let ?p = "cardSuc r"
- have 0: "Well_order ?p \<and> 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'' \<and> 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 \<le>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 \<le>o r')"
-proof(auto simp add: assms cardSuc_least)
- assume "cardSuc r \<le>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' \<le>o r)"
-proof-
- have "Well_order r \<and> 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 \<le>o cardSuc r') = (r \<le>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 \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> 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 cardSuc_natLeq_on_Suc:
-"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
-proof-
- obtain r r' p where r_def: "r = natLeq_on n" and
- r'_def: "r' = cardSuc(natLeq_on n)" and
- p_def: "p = natLeq_on(Suc n)" by blast
- (* Preliminary facts: *)
- have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
- using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
- hence WELL: "Well_order r \<and> Well_order r' \<and> Well_order p"
- unfolding card_order_on_def by force
- have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
- unfolding r_def p_def Field_natLeq_on by simp
- hence FIN: "finite (Field r)" by force
- have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
- hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
- hence LESS: "|Field r| <o |Field r'|"
- using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
- (* Main proof: *)
- have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
- using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
- moreover have "p \<le>o r'"
- proof-
- {assume "r' <o p"
- then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
- let ?q = "Restr p (f ` Field r')"
- have 1: "embed r' p f" using 0 unfolding embedS_def by force
- hence 2: "f ` Field r' < {0..<(Suc n)}"
- using WELL FIELD 0 by (auto simp add: embedS_iff)
- have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
- then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
- unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
- hence 4: "m \<le> n" using 2 by force
- (* *)
- have "bij_betw f (Field r') (f ` (Field r'))"
- using 1 WELL embed_inj_on unfolding bij_betw_def by force
- moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
- ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
- using bij_betw_same_card bij_betw_finite by metis
- hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
- hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
- hence False using LESS not_ordLess_ordLeq by auto
- }
- thus ?thesis using WELL CARD by (fastforce simp: not_ordLess_iff_ordLeq)
- qed
- ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def 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| \<le>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 obtain n where "|A| =o natLeq_on n" using finite_iff_card_of_natLeq_on by blast
- hence "cardSuc |A| =o cardSuc(natLeq_on n)"
- using card_of_Card_order cardSuc_invar_ordIso natLeq_on_Card_order by blast
- hence "cardSuc |A| =o natLeq_on(Suc n)"
- using cardSuc_natLeq_on_Suc ordIso_transitive by blast
- hence "cardSuc |A| =o |{0..<(Suc n)}|" using card_of_less ordIso_equivalence by blast
- moreover have "|Field (cardSuc |A| ) | =o cardSuc |A|"
- using card_of_Field_ordIso cardSuc_Card_order card_of_Card_order by blast
- ultimately have "|Field (cardSuc |A| ) | =o |{0..<(Suc n)}|"
- using ordIso_equivalence by blast
- thus "finite (Field (cardSuc |A| ))"
- using card_of_ordIso_finite finite_atLeastLessThan by blast
-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_ordLeq_infinite_Field:
-assumes r: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and c: "Card_order r"
-shows "|A <+> B| \<le>o r"
-proof-
- let ?r' = "cardSuc r"
- have "Card_order ?r' \<and> infinite (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: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
-and "Card_order r"
-shows "|A Un B| \<le>o r"
-using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
-ordLeq_transitive by blast
-
-
-
-subsection {* Regular cardinals *}
-
-
-definition cofinal where
-"cofinal A r \<equiv>
- ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
-
-
-definition regular where
-"regular r \<equiv>
- ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
-
-
-definition relChain where
-"relChain r As \<equiv>
- ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
-
-lemma regular_UNION:
-assumes r: "Card_order r" "regular r"
-and As: "relChain r As"
-and Bsub: "B \<le> (UN i : Field r. As i)"
-and cardB: "|B| <o r"
-shows "EX i : Field r. B \<le> As i"
-proof-
- let ?phi = "%b j. j : Field r \<and> b : As j"
- have "ALL b : B. EX j. ?phi b j" using Bsub by blast
- then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
- using bchoice[of B ?phi] by blast
- let ?K = "f ` B"
- {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> 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 \<and> b \<notin> As i" by blast
- hence "i \<noteq> f b \<and> ~ (f b,i) : r"
- using As f unfolding relChain_def by auto
- hence "i \<noteq> f b \<and> (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 "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
- qed
- moreover have "?K \<le> Field r" using f by blast
- ultimately have "|?K| =o r" using 2 r unfolding regular_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_regular:
-assumes r_inf: "infinite (Field r)" and r_card: "Card_order r"
-shows "regular (cardSuc r)"
-proof-
- let ?r' = "cardSuc r"
- have r': "Card_order ?r'"
- "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
- using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
- show ?thesis
- unfolding regular_def proof auto
- fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
- hence "|K| \<le>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| \<le>o ?r'" .
- moreover
- {let ?L = "UN j : K. rel.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| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
- moreover
- {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
- using r' 1 by (auto simp: card_of_underS)
- hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
- using r' card_of_Card_order by blast
- hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
- using rJ ordLeq_ordIso_trans by blast
- }
- ultimately have "|?L| \<le>o |?J|"
- using r_inf card_of_UNION_ordLeq_infinite by blast
- hence "|?L| \<le>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' \<le> ?L"
- using 2 unfolding rel.underS_def cofinal_def by auto
- hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
- hence "?r' \<le>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 "infinite (Field r)"
-and As: "relChain (cardSuc r) As"
-and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
-and cardB: "|B| <=o r"
-shows "EX i : Field (cardSuc r). B \<le> As i"
-proof-
- let ?r' = "cardSuc r"
- have "Card_order ?r' \<and> |B| <o ?r'"
- using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
- card_of_Card_order by blast
- moreover have "regular ?r'"
- using assms by(simp add: infinite_cardSuc_regular)
- ultimately show ?thesis
- using As Bsub cardB regular_UNION by blast
-qed
-
-
-subsection {* Others *}
-
-lemma card_of_infinite_diff_finite:
-assumes "infinite A" and "finite B"
-shows "|A - B| =o |A|"
-by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
-
-(* function space *)
-definition Func where
-"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
-
-lemma Func_empty:
-"Func {} B = {\<lambda>x. undefined}"
-unfolding Func_def by auto
-
-lemma Func_elim:
-assumes "g \<in> Func A B" and "a \<in> A"
-shows "\<exists> b. b \<in> B \<and> g a = b"
-using assms unfolding Func_def by (cases "g a = undefined") auto
-
-definition curr where
-"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
-
-lemma curr_in:
-assumes f: "f \<in> Func (A <*> B) C"
-shows "curr A f \<in> Func A (Func B C)"
-using assms unfolding curr_def Func_def by auto
-
-lemma curr_inj:
-assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
-shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
-proof safe
- assume c: "curr A f1 = curr A f2"
- show "f1 = f2"
- proof (rule ext, clarify)
- fix a b show "f1 (a, b) = f2 (a, b)"
- proof (cases "(a,b) \<in> A <*> B")
- case False
- thus ?thesis using assms unfolding Func_def by auto
- next
- case True hence a: "a \<in> A" and b: "b \<in> B" by auto
- thus ?thesis
- using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
- qed
- qed
-qed
-
-lemma curr_surj:
-assumes "g \<in> Func A (Func B C)"
-shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
-proof
- let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
- show "curr A ?f = g"
- proof (rule ext)
- fix a show "curr A ?f a = g a"
- proof (cases "a \<in> A")
- case False
- hence "g a = undefined" using assms unfolding Func_def by auto
- thus ?thesis unfolding curr_def using False by simp
- next
- case True
- obtain g1 where "g1 \<in> Func B C" and "g a = g1"
- using assms using Func_elim[OF assms True] by blast
- thus ?thesis using True unfolding Func_def curr_def by auto
- qed
- qed
- show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
-qed
-
-lemma bij_betw_curr:
-"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
-unfolding bij_betw_def inj_on_def image_def
-using curr_in curr_inj curr_surj by blast
-
-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
-
-definition Func_map where
-"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
-
-lemma Func_map:
-assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
-shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
-using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
-
-lemma Func_non_emp:
-assumes "B \<noteq> {}"
-shows "Func A B \<noteq> {}"
-proof-
- obtain b where b: "b \<in> B" using assms by auto
- hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
- thus ?thesis by blast
-qed
-
-lemma Func_is_emp:
-"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
-proof
- assume L: ?L
- moreover {assume "A = {}" hence False using L Func_empty by auto}
- moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
- ultimately show ?R by blast
-next
- assume R: ?R
- moreover
- {fix f assume "f \<in> Func A B"
- moreover obtain a where "a \<in> A" using R by blast
- ultimately obtain b where "b \<in> B" unfolding Func_def by(cases "f a = undefined", force+)
- with R have False by auto
- }
- thus ?L by blast
-qed
-
-lemma Func_map_surj:
-assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
-and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
-shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
-proof(cases "B2 = {}")
- case True
- thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
-next
- case False note B2 = False
- show ?thesis
- proof safe
- fix h assume h: "h \<in> Func B2 B1"
- def j1 \<equiv> "inv_into A1 f1"
- have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
- then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
- {fix b2 assume b2: "b2 \<in> B2"
- hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
- moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
- ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
- } note kk = this
- obtain b22 where b22: "b22 \<in> B2" using B2 by auto
- def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
- have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
- have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
- using kk unfolding j2_def by auto
- def g \<equiv> "Func_map A2 j1 j2 h"
- have "Func_map B2 f1 f2 g = h"
- proof (rule ext)
- fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
- proof(cases "b2 \<in> B2")
- case True
- show ?thesis
- proof (cases "h b2 = undefined")
- case True
- hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
- show ?thesis using A2 f_inv_into_f[OF b1]
- unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
- qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
- auto intro: f_inv_into_f)
- qed(insert h, unfold Func_def Func_map_def, auto)
- qed
- moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
- using inv_into_into j2A2 B1 A2 inv_into_into
- unfolding j1_def image_def by fast+
- ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
- unfolding Func_map_def[abs_def] unfolding image_def by auto
- qed(insert B1 Func_map[OF _ _ A2(2)], auto)
-qed
-
-lemma card_of_Pow_Func:
-"|Pow A| =o |Func A (UNIV::bool set)|"
-proof-
- def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
- else undefined"
- 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 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
- thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
- next
- show "F ` Pow A = Func A UNIV"
- proof safe
- fix f assume f: "f \<in> Func A (UNIV::bool set)"
- show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
- let ?A1 = "{a \<in> 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_mono:
-fixes A1 A2 :: "'a set" and B :: "'b set"
-assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
-shows "|Func A1 B| \<le>o |Func A2 B|"
-proof-
- obtain bb where bb: "bb \<in> B" using B by auto
- def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb)
- else undefined"
- show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
- show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
- fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
- show "f = g"
- proof(rule ext)
- fix a show "f a = g a"
- proof(cases "a \<in> A1")
- case True
- thus ?thesis using eq A12 unfolding F_def fun_eq_iff
- by (elim allE[of _ a]) auto
- qed(insert f g, unfold Func_def, fastforce)
- qed
- qed
- qed(insert bb, unfold Func_def F_def, force)
-qed
-
-lemma ordLeq_Func:
-assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
-shows "|A| \<le>o |Func A B|"
-unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
- let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
- show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
- show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
-qed
-
-lemma infinite_Func:
-assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
-shows "infinite (Func A B)"
-using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
-
-lemma card_of_Func_UNIV:
-"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
-apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
- let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
- show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
- unfolding bij_betw_def inj_on_def proof safe
- fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
- hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
- then obtain f where f: "\<forall> a. h a = f a" by metis
- hence "range f \<subseteq> B" using h unfolding Func_def by auto
- thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
- qed(unfold Func_def fun_eq_iff, auto)
-qed
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Cardinal_Order_Relation_LFP.thy Mon Nov 18 18:04:44 2013 +0100
@@ -0,0 +1,2438 @@
+(* Title: HOL/Cardinals/Cardinal_Order_Relation_LFP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Cardinal-order relations (LFP).
+*)
+
+header {* Cardinal-Order Relations (LFP) *}
+
+theory Cardinal_Order_Relation_LFP
+imports Constructions_on_Wellorders_LFP
+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, @{text "card"}, from the theory @{text "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, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
+strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
+
+definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
+
+
+abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
+abbreviation "card_order r \<equiv> 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 \<Longrightarrow> A = Field r \<and> Card_order r"
+unfolding card_order_on_def using rel.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 @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
+which states that on any given set there exists a well-order;
+\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
+such well-order, i.e., a cardinal order.
+\end{itemize}
+*}
+
+
+theorem card_order_on: "\<exists>r. card_order_on A r"
+proof-
+ obtain R where R_def: "R = {r. well_order_on A r}" by blast
+ have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
+ using well_order_on[of A] R_def rel.well_order_on_Well_order by blast
+ hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>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' \<and> Field p' = Field r'"
+ using rel.well_order_on_Well_order by blast
+ obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
+ using ISO unfolding ordIso_def by auto
+ hence 3: "inj_on f (Field r') \<and> 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 \<and> 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_Field2 order_on_defs)
+ ultimately have "well_order_on (Field r) ?p" by auto
+ hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
+ thus "r' \<le>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 \<Rightarrow> '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 rel.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| \<Longrightarrow> 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 \<Longrightarrow> |A| \<le>o r"
+using card_of_card_order_on unfolding card_order_on_def by blast
+
+
+lemma card_of_ordIso:
+"(\<exists>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 \<and> |A| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|B| \<le>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 \<and> |B| =o r"
+ using Well_order_iso_copy card_of_well_order_on by blast
+ hence "|A| \<le>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 "\<exists>f. bij_betw f A B" by auto
+qed
+
+
+lemma card_of_ordLeq:
+"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
+proof(auto)
+ fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
+ {assume "|B| <o |A|"
+ hence "|B| \<le>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 \<and> g ` B \<le> 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 Cantor_Bernstein[of f] by fastforce
+ hence "|A| =o |B|" using card_of_ordIso by blast
+ hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
+ }
+ thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
+ by (auto simp: card_of_Well_order)
+next
+ assume *: "|A| \<le>o |B|"
+ obtain f where "embed ( |A| ) ( |B| ) f"
+ using * unfolding ordLeq_def by auto
+ hence "inj_on f A \<and> f ` A \<le> 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 "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
+qed
+
+
+lemma card_of_ordLeq2:
+"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
+using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
+
+
+lemma card_of_ordLess:
+"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
+proof-
+ have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
+ using card_of_ordLeq by blast
+ also have "\<dots> = ( |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 \<noteq> {} \<Longrightarrow> (\<not>(\<exists>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 "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
+shows "|A| \<le>o |B|"
+using assms unfolding card_of_ordLeq[symmetric] by auto
+
+
+lemma card_of_unique:
+"card_order_on A r \<Longrightarrow> r =o |A|"
+by (simp only: card_order_on_ordIso card_of_card_order_on)
+
+
+lemma card_of_mono1:
+"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
+using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
+
+
+lemma card_of_mono2:
+assumes "r \<le>o r'"
+shows "|Field r| \<le>o |Field r'|"
+proof-
+ obtain f where
+ 1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
+ using assms unfolding ordLeq_def
+ by (auto simp add: rel.well_order_on_Well_order)
+ hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
+ by (auto simp add: embed_inj_on embed_Field)
+ thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
+by (simp add: ordIso_iff_ordLeq card_of_mono2)
+
+
+lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
+using card_of_least card_of_well_order_on rel.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 \<le>o |Field r| )"
+proof-
+ have "Card_order r = (r =o |Field r| )"
+ unfolding Card_order_iff_ordIso_card_of by simp
+ also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
+ unfolding ordIso_iff_ordLeq by simp
+ also have "... = (r \<le>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 = (\<forall>a \<in> Field r. Restr r (rel.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 "|rel.underS r a| <o r"
+proof-
+ let ?A = "rel.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'| \<le>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| \<le>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: rel.well_order_on_Well_order)
+ hence "|Field r| \<le>o r" using card_of_least by blast
+ thus ?thesis using assms ordLeq_ordLess_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq:
+"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
+proof
+ assume "|A| \<le>o r"
+ then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>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| \<and> |Field p| \<le>o r"
+ using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
+ thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
+next
+ assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
+ thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma internalize_card_of_ordLeq2:
+"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>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"
+@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
+*}
+
+
+lemma card_of_empty: "|{}| \<le>o |A|"
+using card_of_ordLeq inj_on_id by blast
+
+
+lemma card_of_empty1:
+assumes "Well_order r \<or> Card_order r"
+shows "|{}| \<le>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 "|{}| \<le>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 \<Longrightarrow> |{}| \<le>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| \<le>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| \<le>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_ordLeqI2:
+assumes "B \<subseteq> f ` A"
+shows "|B| \<le>o |A|"
+using assms by (metis surj_imp_ordLeq)
+
+
+lemma card_of_singl_ordLeq:
+assumes "A \<noteq> {}"
+shows "|{b}| \<le>o |A|"
+proof-
+ obtain a where *: "a \<in> A" using assms by auto
+ let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
+ have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
+ using * unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_singl_ordLeq:
+"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>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
+
+
+lemma infinite_Pow:
+assumes "infinite A"
+shows "infinite (Pow A)"
+proof-
+ have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
+ thus ?thesis by (metis assms finite_Pow_iff)
+qed
+
+
+corollary Card_order_Pow:
+"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
+using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
+
+
+corollary card_of_set_type: "|UNIV::'a set| <o |UNIV::'a set set|"
+using card_of_Pow[of "UNIV::'a set"] by simp
+
+
+lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
+proof-
+ have "Inl ` A \<le> A <+> B" by auto
+ thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus1:
+"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
+using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
+proof-
+ have "Inr ` B \<le> A <+> B" by auto
+ thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
+qed
+
+
+corollary Card_order_Plus2:
+"Card_order r \<Longrightarrow> r \<le>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 = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
+ | Inr b \<Rightarrow> 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 -
+ def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
+ case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
+ |Inr b \<Rightarrow> Inr (Inl b))
+ |Inr c \<Rightarrow> Inr (Inr c)"
+ have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
+ proof
+ fix x assume x: "x \<in> A <+> B <+> C"
+ show "x \<in> f ` ((A <+> B) <+> C)"
+ proof(cases x)
+ case (Inl a)
+ hence "a \<in> 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 \<in> B" "x = f (Inl (Inr b))"
+ using x 1 unfolding f_def by auto
+ thus ?thesis by auto
+ next
+ case (Inr c)
+ hence "c \<in> 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 force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_mono1:
+assumes "|A| \<le>o |B|"
+shows "|A <+> C| \<le>o |B <+> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
+ have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
+ proof-
+ {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
+ "g d1 = g d2"
+ hence "d1 = d2" using 1 unfolding inj_on_def
+ by(case_tac d1, case_tac d2, auto simp add: g_def)
+ }
+ moreover
+ {fix d assume "d \<in> A <+> C"
+ hence "g d \<in> 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 metis
+qed
+
+
+corollary ordLeq_Plus_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
+using assms card_of_mono2 card_of_Plus_mono1 by blast
+
+
+lemma card_of_Plus_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C <+> A| \<le>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 \<le>o r'"
+shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
+using assms card_of_mono2 card_of_Plus_mono2 by blast
+
+
+lemma card_of_Plus_mono:
+assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
+shows "|A <+> C| \<le>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 \<le>o r'" and "p \<le>o p'"
+shows "|(Field r) <+> (Field p)| \<le>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_Un1:
+shows "|A| \<le>o |A \<union> B| "
+using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce
+
+
+lemma card_of_diff:
+shows "|A - B| \<le>o |A|"
+using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce
+
+
+lemma card_of_Un_Plus_ordLeq:
+"|A \<union> B| \<le>o |A <+> B|"
+proof-
+ let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
+ have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
+ unfolding inj_on_def by auto
+ thus ?thesis using card_of_ordLeq by blast
+qed
+
+
+lemma card_of_Times1:
+assumes "A \<noteq> {}"
+shows "|B| \<le>o |B \<times> A|"
+proof(cases "B = {}", simp add: card_of_empty)
+ assume *: "B \<noteq> {}"
+ have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
+ thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
+ card_of_ordLeq[of B "B \<times> A"] * by blast
+qed
+
+
+corollary Card_order_Times1:
+"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
+using card_of_Times1[of B] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
+proof-
+ let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
+ have "bij_betw ?f (A \<times> B) (B \<times> 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 \<noteq> {}" shows "|B| \<le>o |A \<times> B|"
+using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
+ ordLeq_ordIso_trans by blast
+
+
+corollary Card_order_Times2:
+"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
+using card_of_Times2[of A] card_of_Field_ordIso
+ ordIso_ordLeq_trans ordIso_symmetric by blast
+
+
+lemma card_of_Times3: "|A| \<le>o |A \<times> 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 \<times> (UNIV::bool set)|"
+proof-
+ let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
+ |Inr a \<Rightarrow> (a,False)"
+ have "bij_betw ?f (A <+> A) (A \<times> (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 \<in> A <+> A"
+ hence "?f c \<in> A \<times> (UNIV::bool set)"
+ by(case_tac c, auto)
+ }
+ moreover
+ {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
+ have "(a,bl) \<in> ?f ` ( A <+> A)"
+ proof(cases bl)
+ assume bl hence "?f(Inl a) = (a,bl)" by auto
+ thus ?thesis using * by force
+ next
+ assume "\<not> 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| \<le>o |B|"
+shows "|A \<times> C| \<le>o |B \<times> C|"
+proof-
+ obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
+ using assms card_of_ordLeq[of A] by fastforce
+ obtain g where g_def:
+ "g = (\<lambda>(a,c::'c). (f a,c))" by blast
+ have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
+ using 1 unfolding inj_on_def using g_def by auto
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary ordLeq_Times_mono1:
+assumes "r \<le>o r'"
+shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
+using assms card_of_mono2 card_of_Times_mono1 by blast
+
+
+lemma card_of_Times_mono2:
+assumes "|A| \<le>o |B|"
+shows "|C \<times> A| \<le>o |C \<times> 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 \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
+by blast
+
+
+corollary ordLeq_Times_mono2:
+assumes "r \<le>o r'"
+shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
+using assms card_of_mono2 card_of_Times_mono2 by blast
+
+
+lemma card_of_Times_cong1:
+assumes "|A| =o |B|"
+shows "|A \<times> C| =o |B \<times> C|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1)
+
+
+lemma card_of_Times_cong2:
+assumes "|A| =o |B|"
+shows "|C \<times> A| =o |C \<times> B|"
+using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2)
+
+
+corollary ordIso_Times_cong2:
+assumes "r =o r'"
+shows "|A \<times> (Field r)| =o |A \<times> (Field r')|"
+using assms card_of_cong card_of_Times_cong2 by blast
+
+
+lemma card_of_Sigma_mono1:
+assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
+shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
+proof-
+ have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
+ using assms by (auto simp add: card_of_ordLeq)
+ with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
+ obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by metis
+ obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
+ have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
+ using 1 unfolding inj_on_def using g_def by force
+ thus ?thesis using card_of_ordLeq by metis
+qed
+
+
+corollary card_of_Sigma_Times:
+"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
+using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
+
+
+lemma card_of_UNION_Sigma:
+"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
+
+
+lemma card_of_bool:
+assumes "a1 \<noteq> a2"
+shows "|UNIV::bool set| =o |{a1,a2}|"
+proof-
+ let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> 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 \<in> {a1,a2}" by (case_tac bl, auto)
+ }
+ moreover
+ {fix a assume *: "a \<in> {a1,a2}"
+ have "a \<in> ?f ` UNIV"
+ proof(cases "a = a1")
+ assume "a = a1"
+ hence "?f True = a" by auto thus ?thesis by blast
+ next
+ assume "a \<noteq> 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 (metis image_subsetI order_eq_iff subsetI)
+ qed
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+
+lemma card_of_Plus_Times_aux:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ LEQ: "|A| \<le>o |B|"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ have 1: "|UNIV::bool set| \<le>o |A|"
+ using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
+ ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
+ (* *)
+ have "|A <+> B| \<le>o |B <+> B|"
+ using LEQ card_of_Plus_mono1 by blast
+ moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
+ using card_of_Plus_Times_bool by blast
+ moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
+ using 1 by (simp add: card_of_Times_mono2)
+ moreover have " |B \<times> A| =o |A \<times> B|"
+ using card_of_Times_commute by blast
+ ultimately show "|A <+> B| \<le>o |A \<times> B|"
+ using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
+ ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
+ ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
+ by blast
+qed
+
+
+lemma card_of_Plus_Times:
+assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
+ B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
+shows "|A <+> B| \<le>o |A \<times> B|"
+proof-
+ {assume "|A| \<le>o |B|"
+ hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
+ }
+ moreover
+ {assume "|B| \<le>o |A|"
+ hence "|B <+> A| \<le>o |B \<times> 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 metis
+ }
+ ultimately show ?thesis
+ using card_of_Well_order[of A] card_of_Well_order[of B]
+ ordLeq_total[of "|A|"] by metis
+qed
+
+
+lemma card_of_ordLeq_finite:
+assumes "|A| \<le>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| \<le>o |B|" and "infinite A"
+shows "infinite 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 @{text "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:
+"infinite A = ( |UNIV::nat set| \<le>o |A| )"
+by (auto simp add: infinite_iff_countable_subset card_of_ordLeq)
+
+
+lemma finite_iff_cardOf_nat:
+"finite A = ( |A| <o |UNIV :: nat set| )"
+using infinite_iff_card_of_nat[of A]
+not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"]
+by (fastforce simp: card_of_Well_order)
+
+lemma finite_ordLess_infinite2:
+assumes "finite A" and "infinite B"
+shows "|A| <o |B|"
+using assms
+finite_ordLess_infinite[of "|A|" "|B|"]
+card_of_Well_order[of A] card_of_Well_order[of B]
+Field_card_of[of A] Field_card_of[of B] by auto
+
+
+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: "infinite (Field r)"
+shows "\<not> (\<exists>a. Field r = rel.under r a)"
+proof(auto)
+ have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
+ using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
+ fix a assume *: "Field r = rel.under r a"
+ show False
+ proof(cases "a \<in> Field r")
+ assume Case1: "a \<notin> Field r"
+ hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
+ thus False using INF * by auto
+ next
+ let ?r' = "Restr r (rel.underS r a)"
+ assume Case2: "a \<in> Field r"
+ hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
+ using 0 rel.Refl_under_underS rel.underS_notIn by fastforce
+ have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
+ using 0 wo_rel.underS_ofilter * 1 Case2 by auto
+ hence "?r' <o r" using 0 using ofilter_ordLess by blast
+ moreover
+ have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
+ using 2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
+ ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
+ moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
+ ultimately have "|rel.underS r a| <o |rel.under r a|"
+ using ordIso_symmetric ordLess_ordIso_trans by blast
+ moreover
+ {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
+ using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
+ hence "|rel.under r a| =o |rel.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 "infinite (Field r)"
+and a: "a : Field r"
+shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
+proof-
+ have "Field r \<noteq> rel.under r a"
+ using assms Card_order_infinite_not_under by blast
+ moreover have "rel.under r a \<le> Field r"
+ using rel.under_Field .
+ ultimately have "rel.under r a < Field r" by blast
+ then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
+ unfolding rel.under_def by blast
+ moreover have ba: "b \<noteq> 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: "infinite(Field r)"
+shows "|Field r \<times> Field r| \<le>o r"
+proof-
+ obtain phi where phi_def:
+ "phi = (\<lambda>r::'a rel. Card_order r \<and> infinite(Field r) \<and>
+ \<not> |Field r \<times> Field r| \<le>o r )" by blast
+ have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
+ unfolding phi_def card_order_on_def by auto
+ have Ft: "\<not>(\<exists>r. phi r)"
+ proof
+ assume "\<exists>r. phi r"
+ hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
+ using temp1 by auto
+ then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
+ 3: "Card_order r \<and> 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' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
+ using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
+ have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
+ using card_of_Card_order card_of_Well_order by blast
+ (* *)
+ have "r <o |?A \<times> ?A|"
+ using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
+ moreover have "|?A \<times> ?A| \<le>o ?r'"
+ using card_of_least[of "?A \<times> ?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: "\<not> bij_betw f ?A (?A \<times> ?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 \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
+ using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
+ hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
+ using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
+ have "\<not> (\<exists>a. Field r = rel.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 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
+ using temp2 3 bsqr_ofilter[of r ?B] by blast
+ hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
+ hence 10: "r \<le>o |A1 \<times> 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| \<le>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 "infinite (Field r)" using 1 unfolding phi_def by simp
+ hence "infinite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
+ hence "infinite A1" using 9 infinite_super finite_cartesian_product by blast
+ moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
+ using card_of_Card_order[of A1] card_of_Well_order[of A1]
+ by (simp add: Field_card_of)
+ moreover have "\<not> r \<le>o | A1 |"
+ using temp4 11 3 using not_ordLeq_iff_ordLess by blast
+ ultimately have "infinite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
+ by (simp add: card_of_card_order_on)
+ hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
+ using 2 unfolding phi_def by blast
+ hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
+ hence "r \<le>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 "infinite A"
+shows "|A \<times> A| =o |A|"
+proof-
+ let ?r = "|A|"
+ have "Field ?r = A \<and> Card_order ?r"
+ using Field_card_of card_of_Card_order[of A] by fastforce
+ hence "|A \<times> A| \<le>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: "infinite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
+shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
+proof-
+ have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
+ using assms by (simp add: card_of_Times1 card_of_Times2)
+ moreover
+ {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
+ using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
+ moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
+ ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
+ using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
+ }
+ ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
+qed
+
+
+corollary card_of_Times_infinite_simps:
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
+"\<lbrakk>infinite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
+by (auto simp add: card_of_Times_infinite ordIso_symmetric)
+
+
+corollary Card_order_Times_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
+shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
+proof-
+ have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> 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 \<times> Field p|"]
+ ordIso_transitive[of _ "|Field r|"] by blast
+qed
+
+
+lemma card_of_Sigma_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|SIGMA i : I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
+ using LEQ card_of_Sigma_Times by blast
+ moreover have "|I \<times> 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: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|SIGMA i : I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|SIGMA i : I. A i| \<le>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:
+"\<lbrakk>infinite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
+ \<Longrightarrow> |A <*> B| \<le>o r"
+by(simp add: card_of_Sigma_ordLeq_infinite_Field)
+
+
+lemma card_of_UNION_ordLeq_infinite:
+assumes INF: "infinite B" and
+ LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
+shows "|\<Union> i \<in> I. A i| \<le>o |B|"
+proof(cases "I = {}", simp add: card_of_empty)
+ assume *: "I \<noteq> {}"
+ have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
+ using card_of_UNION_Sigma by blast
+ moreover have "|SIGMA i : I. A i| \<le>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: "infinite (Field r)" and r: "Card_order r" and
+ LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
+shows "|\<Union> i \<in> I. A i| \<le>o r"
+proof-
+ let ?B = "Field r"
+ have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
+ ordIso_symmetric by blast
+ hence "|I| \<le>o |?B|" "\<forall>i \<in> I. |A i| \<le>o |?B|"
+ using LEQ_I LEQ ordLeq_ordIso_trans by blast+
+ hence "|\<Union> i \<in> I. A i| \<le>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: "infinite A" and LEQ: "|B| \<le>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 \<Rightarrow> 'a + 'b" let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
+ assume *: "B \<noteq> {}"
+ then obtain b1 where 1: "b1 \<in> 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: "infinite (?Inl ` A)"
+ using INF bij_betw_finite[of ?Inl A] by blast
+ let ?A' = "?Inl ` A \<union> {?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 \<noteq> {b1}"
+ with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
+ obtain f where "inj_on f B \<and> f ` B \<le> A"
+ using LEQ card_of_ordLeq[of B] by fastforce
+ with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
+ unfolding inj_on_def by auto
+ with 3 have "|A <+> B| \<le>o |A \<times> B|"
+ by (auto simp add: card_of_Plus_Times)
+ moreover have "|A \<times> B| =o |A|"
+ using assms * by (simp add: card_of_Times_infinite_simps)
+ ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
+ thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
+ qed
+qed
+
+
+lemma card_of_Plus_infinite2:
+assumes INF: "infinite A" and LEQ: "|B| \<le>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: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
+using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
+
+
+corollary Card_order_Plus_infinite:
+assumes INF: "infinite(Field r)" and CARD: "Card_order r" and
+ LEQ: "p \<le>o r"
+shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
+proof-
+ have "| Field r <+> Field p | =o | Field r | \<and>
+ | 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
+
+
+lemma card_of_Un_infinite:
+assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|"
+shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
+proof-
+ have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq)
+ moreover have "|A <+> B| =o |A|"
+ using assms by (metis card_of_Plus_infinite)
+ ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
+ hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast
+ thus ?thesis using Un_commute[of B A] by auto
+qed
+
+
+lemma card_of_Un_diff_infinite:
+assumes INF: "infinite A" and LESS: "|B| <o |A|"
+shows "|A - B| =o |A|"
+proof-
+ obtain C where C_def: "C = A - B" by blast
+ have "|A \<union> B| =o |A|"
+ using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast
+ moreover have "C \<union> B = A \<union> B" unfolding C_def by auto
+ ultimately have 1: "|C \<union> B| =o |A|" by auto
+ (* *)
+ {assume *: "|C| \<le>o |B|"
+ moreover
+ {assume **: "finite B"
+ hence "finite C"
+ using card_of_ordLeq_finite * by blast
+ hence False using ** INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite B" by auto
+ ultimately have False
+ using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis
+ }
+ hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast
+ {assume *: "finite C"
+ hence "finite B" using card_of_ordLeq_finite 2 by blast
+ hence False using * INF card_of_ordIso_finite 1 by blast
+ }
+ hence "infinite C" by auto
+ hence "|C| =o |A|"
+ using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis
+ thus ?thesis unfolding C_def .
+qed
+
+
+lemma card_of_Plus_ordLess_infinite:
+assumes INF: "infinite C" and
+ LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
+shows "|A <+> B| <o |C|"
+proof(cases "A = {} \<or> B = {}")
+ assume Case1: "A = {} \<or> B = {}"
+ hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
+ using card_of_Plus_empty1 card_of_Plus_empty2 by blast
+ hence "|A <+> B| =o |A| \<or> |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: "\<not>(A = {} \<or> B = {})"
+ {assume *: "|C| \<le>o |A <+> B|"
+ hence "infinite (A <+> B)" using INF card_of_ordLeq_finite by blast
+ hence 1: "infinite A \<or> infinite B" using finite_Plus by blast
+ {assume Case21: "|A| \<le>o |B|"
+ hence "infinite 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| \<le>o |A|"
+ hence "infinite 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: "infinite (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| \<and> |?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 infinite_card_of_insert:
+assumes "infinite A"
+shows "|insert a A| =o |A|"
+proof-
+ have iA: "insert a A = A \<union> {a}" by simp
+ show ?thesis
+ using infinite_imp_bij_betw2[OF assms] unfolding iA
+ by (metis bij_betw_inv card_of_ordIso)
+qed
+
+
+subsection {* Cardinals versus lists *}
+
+
+text{* The next is an auxiliary operator, which shall be used for inductive
+proofs of facts concerning the cardinality of @{text "List"} : *}
+
+definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set"
+where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}"
+
+
+lemma lists_def2: "lists A = {l. set l \<le> A}"
+using in_listsI by blast
+
+
+lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)"
+unfolding lists_def2 nlists_def by blast
+
+
+lemma card_of_lists: "|A| \<le>o |lists A|"
+proof-
+ let ?h = "\<lambda> a. [a]"
+ have "inj_on ?h A \<and> ?h ` A \<le> lists A"
+ unfolding inj_on_def lists_def2 by auto
+ thus ?thesis by (metis card_of_ordLeq)
+qed
+
+
+lemma nlists_0: "nlists A 0 = {[]}"
+unfolding nlists_def by auto
+
+
+lemma nlists_not_empty:
+assumes "A \<noteq> {}"
+shows "nlists A n \<noteq> {}"
+proof(induct n, simp add: nlists_0)
+ fix n assume "nlists A n \<noteq> {}"
+ then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto
+ hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto
+ thus "nlists A (Suc n) \<noteq> {}" by auto
+qed
+
+
+lemma Nil_in_lists: "[] \<in> lists A"
+unfolding lists_def2 by auto
+
+
+lemma lists_not_empty: "lists A \<noteq> {}"
+using Nil_in_lists by blast
+
+
+lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+proof-
+ let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l"
+ have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)"
+ unfolding inj_on_def nlists_def by auto
+ moreover have "nlists A (Suc n) \<le> ?h ` ?B"
+ proof(auto)
+ fix l assume "l \<in> nlists A (Suc n)"
+ hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto
+ then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv)
+ hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto
+ thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto
+ qed
+ ultimately have "bij_betw ?h ?B (nlists A (Suc n))"
+ unfolding bij_betw_def by auto
+ thus ?thesis using card_of_ordIso ordIso_symmetric by blast
+qed
+
+
+lemma card_of_nlists_infinite:
+assumes "infinite A"
+shows "|nlists A n| \<le>o |A|"
+proof(induct n)
+ have "A \<noteq> {}" using assms by auto
+ thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq)
+next
+ fix n assume IH: "|nlists A n| \<le>o |A|"
+ have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|"
+ using card_of_nlists_Succ by blast
+ moreover
+ {have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast
+ hence "|A \<times> (nlists A n)| =o |A|"
+ using assms IH by (auto simp add: card_of_Times_infinite)
+ }
+ ultimately show "|nlists A (Suc n)| \<le>o |A|"
+ using ordIso_transitive ordIso_iff_ordLeq by blast
+qed
+
+
+lemma card_of_lists_infinite:
+assumes "infinite A"
+shows "|lists A| =o |A|"
+proof-
+ have "|lists A| \<le>o |A|"
+ using assms
+ by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite
+ infinite_iff_card_of_nat card_of_nlists_infinite)
+ thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast
+qed
+
+
+lemma Card_order_lists_infinite:
+assumes "Card_order r" and "infinite(Field r)"
+shows "|lists(Field r)| =o r"
+using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast
+
+
+
+subsection {* The cardinal $\omega$ and the finite cardinals *}
+
+
+text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
+order relation on
+@{text "nat"}, that we abbreviate by @{text "natLeq"}. The finite cardinals
+shall be the restrictions of these relations to the numbers smaller than
+fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
+
+abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
+abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
+
+abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
+where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
+
+lemma infinite_cartesian_product:
+assumes "infinite A" "infinite B"
+shows "infinite (A \<times> B)"
+proof
+ assume "finite (A \<times> B)"
+ from assms(1) have "A \<noteq> {}" by auto
+ with `finite (A \<times> 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, auto)
+
+
+lemma natLeq_Refl: "Refl natLeq"
+unfolding refl_on_def Field_def by auto
+
+
+lemma natLeq_trans: "trans natLeq"
+unfolding trans_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 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 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"
+by auto
+
+
+lemma natLeq_Well_order: "Well_order natLeq"
+unfolding well_order_on_def
+using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
+
+
+corollary natLeq_well_order_on: "well_order_on UNIV natLeq"
+using natLeq_Well_order Field_natLeq by auto
+
+
+lemma natLeq_wo_rel: "wo_rel natLeq"
+unfolding wo_rel_def using natLeq_Well_order .
+
+
+lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV"
+using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto
+
+
+lemma closed_nat_set_iff:
+assumes "\<forall>(m::nat) n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A"
+shows "A = UNIV \<or> (\<exists>n. A = {0 ..< n})"
+proof-
+ {assume "A \<noteq> UNIV" hence "\<exists>n. n \<notin> A" by blast
+ moreover obtain n where n_def: "n = (LEAST n. n \<notin> A)" by blast
+ ultimately have 1: "n \<notin> A \<and> (\<forall>m. m < n \<longrightarrow> m \<in> A)"
+ using LeastI_ex[of "\<lambda> n. n \<notin> A"] n_def Least_le[of "\<lambda> n. n \<notin> A"] by fastforce
+ have "A = {0 ..< n}"
+ proof(auto simp add: 1)
+ fix m assume *: "m \<in> A"
+ {assume "n \<le> m" with assms * have "n \<in> A" by blast
+ hence False using 1 by auto
+ }
+ thus "m < n" by fastforce
+ qed
+ hence "\<exists>n. A = {0 ..< n}" by blast
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma Field_natLeq_on: "Field (natLeq_on n) = {0 ..< n}"
+unfolding Field_def by auto
+
+
+lemma natLeq_underS_less: "rel.underS natLeq n = {0 ..< n}"
+unfolding rel.underS_def by auto
+
+
+lemma Restr_natLeq: "Restr natLeq {0 ..< n} = natLeq_on n"
+by auto
+
+
+lemma Restr_natLeq2:
+"Restr natLeq (rel.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 "{0..<n}"] by auto
+
+
+corollary natLeq_on_well_order_on: "well_order_on {0 ..< 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 .
+
+
+lemma natLeq_on_ofilter_less_eq:
+"n \<le> m \<Longrightarrow> wo_rel.ofilter (natLeq_on m) {0 ..< n}"
+by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def,
+ simp add: Field_natLeq_on, unfold rel.under_def, auto)
+
+
+lemma natLeq_on_ofilter_iff:
+"wo_rel.ofilter (natLeq_on m) A = (\<exists>n \<le> m. A = {0 ..< n})"
+proof(rule iffI)
+ assume *: "wo_rel.ofilter (natLeq_on m) A"
+ hence 1: "A \<le> {0..<m}"
+ by (auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def Field_natLeq_on)
+ hence "\<forall>n1 n2. n2 \<in> A \<and> n1 \<le> n2 \<longrightarrow> n1 \<in> A"
+ using * by(fastforce simp add: natLeq_on_wo_rel wo_rel.ofilter_def rel.under_def)
+ hence "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast
+ thus "\<exists>n \<le> m. A = {0 ..< n}" using 1 atLeastLessThan_less_eq by blast
+next
+ assume "(\<exists>n\<le>m. A = {0 ..< n})"
+ thus "wo_rel.ofilter (natLeq_on m) A" by (auto simp add: natLeq_on_ofilter_less_eq)
+qed
+
+
+
+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))"
+ unfolding Field_natLeq_on by auto
+ moreover have "infinite(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:
+"infinite A = ( natLeq \<le>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
+
+
+lemma ordIso_natLeq_on_imp_finite:
+"|A| =o natLeq_on n \<Longrightarrow> finite A"
+unfolding ordIso_def iso_def[abs_def]
+by (auto simp: Field_natLeq_on bij_betw_finite Field_card_of)
+
+
+lemma natLeq_on_Card_order: "Card_order (natLeq_on n)"
+proof(unfold card_order_on_def,
+ auto simp add: natLeq_on_Well_order, simp add: Field_natLeq_on)
+ fix r assume "well_order_on {0..<n} r"
+ thus "natLeq_on n \<le>o r"
+ using finite_atLeastLessThan natLeq_on_well_order_on
+ finite_well_order_on_ordIso ordIso_iff_ordLeq by blast
+qed
+
+
+corollary card_of_Field_natLeq_on:
+"|Field (natLeq_on n)| =o natLeq_on n"
+using Field_natLeq_on natLeq_on_Card_order
+ Card_order_iff_ordIso_card_of[of "natLeq_on n"]
+ ordIso_symmetric[of "natLeq_on n"] by blast
+
+
+corollary card_of_less:
+"|{0 ..< n}| =o natLeq_on n"
+using Field_natLeq_on card_of_Field_natLeq_on by auto
+
+
+lemma natLeq_on_ordLeq_less_eq:
+"((natLeq_on m) \<le>o (natLeq_on n)) = (m \<le> n)"
+proof
+ assume "natLeq_on m \<le>o natLeq_on n"
+ then obtain f where "inj_on f {0..<m} \<and> f ` {0..<m} \<le> {0..<n}"
+ unfolding ordLeq_def using
+ embed_inj_on[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on]
+ embed_Field[OF natLeq_on_Well_order[of m], of "natLeq_on n", unfolded Field_natLeq_on] by blast
+ thus "m \<le> n" using atLeastLessThan_less_eq2 by blast
+next
+ assume "m \<le> n"
+ hence "inj_on id {0..<m} \<and> id ` {0..<m} \<le> {0..<n}" unfolding inj_on_def by auto
+ hence "|{0..<m}| \<le>o |{0..<n}|" using card_of_ordLeq by blast
+ thus "natLeq_on m \<le>o natLeq_on n"
+ using card_of_less ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
+qed
+
+
+lemma natLeq_on_ordLeq_less:
+"((natLeq_on m) <o (natLeq_on n)) = (m < n)"
+using not_ordLeq_iff_ordLess[of "natLeq_on m" "natLeq_on n"]
+natLeq_on_Well_order natLeq_on_ordLeq_less_eq by auto
+
+
+
+subsubsection {* "Backwards compatibility" with the numeric cardinal operator for finite sets *}
+
+
+lemma finite_card_of_iff_card2:
+assumes FIN: "finite A" and FIN': "finite B"
+shows "( |A| \<le>o |B| ) = (card A \<le> card B)"
+using assms card_of_ordLeq[of A B] inj_on_iff_card_le[of A B] by blast
+
+
+lemma finite_imp_card_of_natLeq_on:
+assumes "finite A"
+shows "|A| =o natLeq_on (card A)"
+proof-
+ obtain h where "bij_betw h A {0 ..< card A}"
+ using assms ex_bij_betw_finite_nat by blast
+ thus ?thesis using card_of_ordIso card_of_less ordIso_equivalence by blast
+qed
+
+
+lemma finite_iff_card_of_natLeq_on:
+"finite A = (\<exists>n. |A| =o natLeq_on n)"
+using finite_imp_card_of_natLeq_on[of A]
+by(auto simp add: ordIso_natLeq_on_imp_finite)
+
+
+
+subsection {* The successor of a cardinal *}
+
+
+text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
+being a successor cardinal of @{text "r"}. Although the definition does
+not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
+
+
+definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
+where
+"isCardSuc r r' \<equiv>
+ Card_order r' \<and> r <o r' \<and>
+ (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
+
+
+text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
+by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
+Again, the picked item shall be proved unique up to order-isomorphism. *}
+
+
+definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
+where
+"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
+
+
+lemma exists_minim_Card_order:
+"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+unfolding card_order_on_def using exists_minim_Well_order by blast
+
+
+lemma exists_isCardSuc:
+assumes "Card_order r"
+shows "\<exists>r'. isCardSuc r r'"
+proof-
+ let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
+ have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
+ by (simp add: card_of_Card_order Card_order_Pow)
+ then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>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 \<Longrightarrow> Card_order(cardSuc r)"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_greater:
+"Card_order r \<Longrightarrow> r <o cardSuc r"
+using cardSuc_isCardSuc unfolding isCardSuc_def by blast
+
+
+lemma cardSuc_ordLeq:
+"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
+using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
+
+
+text{* The minimality property of @{text "cardSuc"} originally present in its definition
+is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
+
+lemma cardSuc_least_aux:
+"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>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 \<le>o r'"
+proof-
+ let ?p = "cardSuc r"
+ have 0: "Well_order ?p \<and> 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'' \<and> 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 \<le>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 \<le>o r')"
+proof(auto simp add: assms cardSuc_least)
+ assume "cardSuc r \<le>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' \<le>o r)"
+proof-
+ have "Well_order r \<and> 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 \<le>o cardSuc r') = (r \<le>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 \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> 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 cardSuc_natLeq_on_Suc:
+"cardSuc(natLeq_on n) =o natLeq_on(Suc n)"
+proof-
+ obtain r r' p where r_def: "r = natLeq_on n" and
+ r'_def: "r' = cardSuc(natLeq_on n)" and
+ p_def: "p = natLeq_on(Suc n)" by blast
+ (* Preliminary facts: *)
+ have CARD: "Card_order r \<and> Card_order r' \<and> Card_order p" unfolding r_def r'_def p_def
+ using cardSuc_ordLess_ordLeq natLeq_on_Card_order cardSuc_Card_order by blast
+ hence WELL: "Well_order r \<and> Well_order r' \<and> Well_order p"
+ unfolding card_order_on_def by force
+ have FIELD: "Field r = {0..<n} \<and> Field p = {0..<(Suc n)}"
+ unfolding r_def p_def Field_natLeq_on by simp
+ hence FIN: "finite (Field r)" by force
+ have "r <o r'" using CARD unfolding r_def r'_def using cardSuc_greater by blast
+ hence "|Field r| <o r'" using CARD card_of_Field_ordIso ordIso_ordLess_trans by blast
+ hence LESS: "|Field r| <o |Field r'|"
+ using CARD card_of_Field_ordIso ordLess_ordIso_trans ordIso_symmetric by blast
+ (* Main proof: *)
+ have "r' \<le>o p" using CARD unfolding r_def r'_def p_def
+ using natLeq_on_ordLeq_less cardSuc_ordLess_ordLeq by blast
+ moreover have "p \<le>o r'"
+ proof-
+ {assume "r' <o p"
+ then obtain f where 0: "embedS r' p f" unfolding ordLess_def by force
+ let ?q = "Restr p (f ` Field r')"
+ have 1: "embed r' p f" using 0 unfolding embedS_def by force
+ hence 2: "f ` Field r' < {0..<(Suc n)}"
+ using WELL FIELD 0 by (auto simp add: embedS_iff)
+ have "wo_rel.ofilter p (f ` Field r')" using embed_Field_ofilter 1 WELL by blast
+ then obtain m where "m \<le> Suc n" and 3: "f ` (Field r') = {0..<m}"
+ unfolding p_def by (auto simp add: natLeq_on_ofilter_iff)
+ hence 4: "m \<le> n" using 2 by force
+ (* *)
+ have "bij_betw f (Field r') (f ` (Field r'))"
+ using 1 WELL embed_inj_on unfolding bij_betw_def by force
+ moreover have "finite(f ` (Field r'))" using 3 finite_atLeastLessThan[of 0 m] by force
+ ultimately have 5: "finite (Field r') \<and> card(Field r') = card (f ` (Field r'))"
+ using bij_betw_same_card bij_betw_finite by metis
+ hence "card(Field r') \<le> card(Field r)" using 3 4 FIELD by force
+ hence "|Field r'| \<le>o |Field r|" using FIN 5 finite_card_of_iff_card2 by blast
+ hence False using LESS not_ordLess_ordLeq by auto
+ }
+ thus ?thesis using WELL CARD by (fastforce simp: not_ordLess_iff_ordLeq)
+ qed
+ ultimately show ?thesis using ordIso_iff_ordLeq unfolding r'_def p_def 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| \<le>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 obtain n where "|A| =o natLeq_on n" using finite_iff_card_of_natLeq_on by blast
+ hence "cardSuc |A| =o cardSuc(natLeq_on n)"
+ using card_of_Card_order cardSuc_invar_ordIso natLeq_on_Card_order by blast
+ hence "cardSuc |A| =o natLeq_on(Suc n)"
+ using cardSuc_natLeq_on_Suc ordIso_transitive by blast
+ hence "cardSuc |A| =o |{0..<(Suc n)}|" using card_of_less ordIso_equivalence by blast
+ moreover have "|Field (cardSuc |A| ) | =o cardSuc |A|"
+ using card_of_Field_ordIso cardSuc_Card_order card_of_Card_order by blast
+ ultimately have "|Field (cardSuc |A| ) | =o |{0..<(Suc n)}|"
+ using ordIso_equivalence by blast
+ thus "finite (Field (cardSuc |A| ))"
+ using card_of_ordIso_finite finite_atLeastLessThan by blast
+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_ordLeq_infinite_Field:
+assumes r: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and c: "Card_order r"
+shows "|A <+> B| \<le>o r"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> infinite (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: "infinite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
+and "Card_order r"
+shows "|A Un B| \<le>o r"
+using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
+ordLeq_transitive by blast
+
+
+
+subsection {* Regular cardinals *}
+
+
+definition cofinal where
+"cofinal A r \<equiv>
+ ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
+
+
+definition regular where
+"regular r \<equiv>
+ ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
+
+
+definition relChain where
+"relChain r As \<equiv>
+ ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
+
+lemma regular_UNION:
+assumes r: "Card_order r" "regular r"
+and As: "relChain r As"
+and Bsub: "B \<le> (UN i : Field r. As i)"
+and cardB: "|B| <o r"
+shows "EX i : Field r. B \<le> As i"
+proof-
+ let ?phi = "%b j. j : Field r \<and> b : As j"
+ have "ALL b : B. EX j. ?phi b j" using Bsub by blast
+ then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
+ using bchoice[of B ?phi] by blast
+ let ?K = "f ` B"
+ {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> 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 \<and> b \<notin> As i" by blast
+ hence "i \<noteq> f b \<and> ~ (f b,i) : r"
+ using As f unfolding relChain_def by auto
+ hence "i \<noteq> f b \<and> (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 "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
+ qed
+ moreover have "?K \<le> Field r" using f by blast
+ ultimately have "|?K| =o r" using 2 r unfolding regular_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_regular:
+assumes r_inf: "infinite (Field r)" and r_card: "Card_order r"
+shows "regular (cardSuc r)"
+proof-
+ let ?r' = "cardSuc r"
+ have r': "Card_order ?r'"
+ "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
+ using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
+ show ?thesis
+ unfolding regular_def proof auto
+ fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
+ hence "|K| \<le>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| \<le>o ?r'" .
+ moreover
+ {let ?L = "UN j : K. rel.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| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
+ moreover
+ {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
+ using r' 1 by (auto simp: card_of_underS)
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
+ using r' card_of_Card_order by blast
+ hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
+ using rJ ordLeq_ordIso_trans by blast
+ }
+ ultimately have "|?L| \<le>o |?J|"
+ using r_inf card_of_UNION_ordLeq_infinite by blast
+ hence "|?L| \<le>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' \<le> ?L"
+ using 2 unfolding rel.underS_def cofinal_def by auto
+ hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
+ hence "?r' \<le>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 "infinite (Field r)"
+and As: "relChain (cardSuc r) As"
+and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
+and cardB: "|B| <=o r"
+shows "EX i : Field (cardSuc r). B \<le> As i"
+proof-
+ let ?r' = "cardSuc r"
+ have "Card_order ?r' \<and> |B| <o ?r'"
+ using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
+ card_of_Card_order by blast
+ moreover have "regular ?r'"
+ using assms by(simp add: infinite_cardSuc_regular)
+ ultimately show ?thesis
+ using As Bsub cardB regular_UNION by blast
+qed
+
+
+subsection {* Others *}
+
+lemma card_of_infinite_diff_finite:
+assumes "infinite A" and "finite B"
+shows "|A - B| =o |A|"
+by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2)
+
+(* function space *)
+definition Func where
+"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
+
+lemma Func_empty:
+"Func {} B = {\<lambda>x. undefined}"
+unfolding Func_def by auto
+
+lemma Func_elim:
+assumes "g \<in> Func A B" and "a \<in> A"
+shows "\<exists> b. b \<in> B \<and> g a = b"
+using assms unfolding Func_def by (cases "g a = undefined") auto
+
+definition curr where
+"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
+
+lemma curr_in:
+assumes f: "f \<in> Func (A <*> B) C"
+shows "curr A f \<in> Func A (Func B C)"
+using assms unfolding curr_def Func_def by auto
+
+lemma curr_inj:
+assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
+shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
+proof safe
+ assume c: "curr A f1 = curr A f2"
+ show "f1 = f2"
+ proof (rule ext, clarify)
+ fix a b show "f1 (a, b) = f2 (a, b)"
+ proof (cases "(a,b) \<in> A <*> B")
+ case False
+ thus ?thesis using assms unfolding Func_def by auto
+ next
+ case True hence a: "a \<in> A" and b: "b \<in> B" by auto
+ thus ?thesis
+ using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
+ qed
+ qed
+qed
+
+lemma curr_surj:
+assumes "g \<in> Func A (Func B C)"
+shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
+proof
+ let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
+ show "curr A ?f = g"
+ proof (rule ext)
+ fix a show "curr A ?f a = g a"
+ proof (cases "a \<in> A")
+ case False
+ hence "g a = undefined" using assms unfolding Func_def by auto
+ thus ?thesis unfolding curr_def using False by simp
+ next
+ case True
+ obtain g1 where "g1 \<in> Func B C" and "g a = g1"
+ using assms using Func_elim[OF assms True] by blast
+ thus ?thesis using True unfolding Func_def curr_def by auto
+ qed
+ qed
+ show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
+qed
+
+lemma bij_betw_curr:
+"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
+unfolding bij_betw_def inj_on_def image_def
+using curr_in curr_inj curr_surj by blast
+
+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
+
+definition Func_map where
+"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
+
+lemma Func_map:
+assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
+shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
+using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
+
+lemma Func_non_emp:
+assumes "B \<noteq> {}"
+shows "Func A B \<noteq> {}"
+proof-
+ obtain b where b: "b \<in> B" using assms by auto
+ hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
+ thus ?thesis by blast
+qed
+
+lemma Func_is_emp:
+"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
+proof
+ assume L: ?L
+ moreover {assume "A = {}" hence False using L Func_empty by auto}
+ moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
+ ultimately show ?R by blast
+next
+ assume R: ?R
+ moreover
+ {fix f assume "f \<in> Func A B"
+ moreover obtain a where "a \<in> A" using R by blast
+ ultimately obtain b where "b \<in> B" unfolding Func_def by(cases "f a = undefined", force+)
+ with R have False by auto
+ }
+ thus ?L by blast
+qed
+
+lemma Func_map_surj:
+assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
+and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
+shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
+proof(cases "B2 = {}")
+ case True
+ thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
+next
+ case False note B2 = False
+ show ?thesis
+ proof safe
+ fix h assume h: "h \<in> Func B2 B1"
+ def j1 \<equiv> "inv_into A1 f1"
+ have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
+ then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
+ {fix b2 assume b2: "b2 \<in> B2"
+ hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
+ moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
+ ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
+ } note kk = this
+ obtain b22 where b22: "b22 \<in> B2" using B2 by auto
+ def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
+ have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
+ have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
+ using kk unfolding j2_def by auto
+ def g \<equiv> "Func_map A2 j1 j2 h"
+ have "Func_map B2 f1 f2 g = h"
+ proof (rule ext)
+ fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
+ proof(cases "b2 \<in> B2")
+ case True
+ show ?thesis
+ proof (cases "h b2 = undefined")
+ case True
+ hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
+ show ?thesis using A2 f_inv_into_f[OF b1]
+ unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
+ qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
+ auto intro: f_inv_into_f)
+ qed(insert h, unfold Func_def Func_map_def, auto)
+ qed
+ moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
+ using inv_into_into j2A2 B1 A2 inv_into_into
+ unfolding j1_def image_def by fast+
+ ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
+ unfolding Func_map_def[abs_def] unfolding image_def by auto
+ qed(insert B1 Func_map[OF _ _ A2(2)], auto)
+qed
+
+lemma card_of_Pow_Func:
+"|Pow A| =o |Func A (UNIV::bool set)|"
+proof-
+ def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
+ else undefined"
+ 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 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
+ thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
+ next
+ show "F ` Pow A = Func A UNIV"
+ proof safe
+ fix f assume f: "f \<in> Func A (UNIV::bool set)"
+ show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
+ let ?A1 = "{a \<in> 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_mono:
+fixes A1 A2 :: "'a set" and B :: "'b set"
+assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}"
+shows "|Func A1 B| \<le>o |Func A2 B|"
+proof-
+ obtain bb where bb: "bb \<in> B" using B by auto
+ def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb)
+ else undefined"
+ show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI)
+ show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe
+ fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g"
+ show "f = g"
+ proof(rule ext)
+ fix a show "f a = g a"
+ proof(cases "a \<in> A1")
+ case True
+ thus ?thesis using eq A12 unfolding F_def fun_eq_iff
+ by (elim allE[of _ a]) auto
+ qed(insert f g, unfold Func_def, fastforce)
+ qed
+ qed
+ qed(insert bb, unfold Func_def F_def, force)
+qed
+
+lemma ordLeq_Func:
+assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "|A| \<le>o |Func A B|"
+unfolding card_of_ordLeq[symmetric] proof(intro exI conjI)
+ let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined"
+ show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto
+ show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto
+qed
+
+lemma infinite_Func:
+assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2"
+shows "infinite (Func A B)"
+using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite)
+
+lemma card_of_Func_UNIV:
+"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
+apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
+ let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
+ show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
+ unfolding bij_betw_def inj_on_def proof safe
+ fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
+ hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
+ then obtain f where f: "\<forall> a. h a = f a" by metis
+ hence "range f \<subseteq> B" using h unfolding Func_def by auto
+ thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
+ qed(unfold Func_def fun_eq_iff, auto)
+qed
+
+end
--- a/src/HOL/Cardinals/Constructions_on_Wellorders.thy Mon Nov 18 18:04:44 2013 +0100
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders.thy Mon Nov 18 18:04:44 2013 +0100
@@ -8,7 +8,7 @@
header {* Constructions on Wellorders *}
theory Constructions_on_Wellorders
-imports Constructions_on_Wellorders_Base Wellorder_Embedding
+imports Constructions_on_Wellorders_LFP Wellorder_Embedding
begin
declare
--- a/src/HOL/Cardinals/Constructions_on_Wellorders_Base.thy Mon Nov 18 18:04:44 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1633 +0,0 @@
-(* Title: HOL/Cardinals/Constructions_on_Wellorders_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Constructions on wellorders (base).
-*)
-
-header {* Constructions on Wellorders (Base) *}
-
-theory Constructions_on_Wellorders_Base
-imports Wellorder_Embedding_Base
-begin
-
-
-text {* In this section, we study basic constructions on well-orders, such as restriction to
-a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
-and bounded square. We also define between well-orders
-the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
-@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
-@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}). We study the
-connections between these relations, order filters, and the aforementioned constructions.
-A main result of this section is that @{text "<o"} is well-founded. *}
-
-
-subsection {* Restriction to a set *}
-
-
-abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
-where "Restr r A \<equiv> r Int (A \<times> A)"
-
-
-lemma Restr_subset:
-"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
-by blast
-
-
-lemma Restr_Field: "Restr r (Field r) = r"
-unfolding Field_def by auto
-
-
-lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
-unfolding refl_on_def Field_def by auto
-
-
-lemma antisym_Restr:
-"antisym r \<Longrightarrow> antisym(Restr r A)"
-unfolding antisym_def Field_def by auto
-
-
-lemma Total_Restr:
-"Total r \<Longrightarrow> Total(Restr r A)"
-unfolding total_on_def Field_def by auto
-
-
-lemma trans_Restr:
-"trans r \<Longrightarrow> trans(Restr r A)"
-unfolding trans_def Field_def by blast
-
-
-lemma Preorder_Restr:
-"Preorder r \<Longrightarrow> Preorder(Restr r A)"
-unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
-
-
-lemma Partial_order_Restr:
-"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
-unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
-
-
-lemma Linear_order_Restr:
-"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
-unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
-
-
-lemma Well_order_Restr:
-assumes "Well_order r"
-shows "Well_order(Restr r A)"
-proof-
- have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
- hence "wf(Restr r A - Id)" using assms
- using well_order_on_def wf_subset by blast
- thus ?thesis using assms unfolding well_order_on_def
- by (simp add: Linear_order_Restr)
-qed
-
-
-lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
-by (auto simp add: Field_def)
-
-
-lemma Refl_Field_Restr:
-"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
-by (auto simp add: refl_on_def Field_def)
-
-
-lemma Refl_Field_Restr2:
-"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
-by (auto simp add: Refl_Field_Restr)
-
-
-lemma well_order_on_Restr:
-assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
-shows "well_order_on A (Restr r A)"
-using assms
-using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
- order_on_defs[of "Field r" r] by auto
-
-
-subsection {* Order filters versus restrictions and embeddings *}
-
-
-lemma Field_Restr_ofilter:
-"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
-by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
-
-
-lemma ofilter_Restr_under:
-assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
-shows "rel.under (Restr r A) a = rel.under r a"
-using assms wo_rel_def
-proof(auto simp add: wo_rel.ofilter_def rel.under_def)
- fix b assume *: "a \<in> A" and "(b,a) \<in> r"
- hence "b \<in> rel.under r a \<and> a \<in> Field r"
- unfolding rel.under_def using Field_def by fastforce
- thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
-qed
-
-
-lemma ofilter_embed:
-assumes "Well_order r"
-shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
-proof
- assume *: "wo_rel.ofilter r A"
- show "A \<le> Field r \<and> embed (Restr r A) r id"
- proof(unfold embed_def, auto)
- fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
- by (auto simp add: wo_rel_def wo_rel.ofilter_def)
- next
- fix a assume "a \<in> Field (Restr r A)"
- thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
- by (simp add: ofilter_Restr_under Field_Restr_ofilter)
- qed
-next
- assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
- hence "Field(Restr r A) \<le> Field r"
- using assms embed_Field[of "Restr r A" r id] id_def
- Well_order_Restr[of r] by auto
- {fix a assume "a \<in> A"
- hence "a \<in> Field(Restr r A)" using * assms
- by (simp add: order_on_defs Refl_Field_Restr2)
- hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
- using * unfolding embed_def by auto
- hence "rel.under r a \<le> rel.under (Restr r A) a"
- unfolding bij_betw_def by auto
- also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
- also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
- finally have "rel.under r a \<le> A" .
- }
- thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
-qed
-
-
-lemma ofilter_Restr_Int:
-assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
-shows "wo_rel.ofilter (Restr r B) (A Int B)"
-proof-
- let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence Field: "Field ?rB = Field r Int B"
- using Refl_Field_Restr by blast
- have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- (* Main proof *)
- show ?thesis using WellB assms
- proof(auto simp add: wo_rel.ofilter_def rel.under_def)
- fix a assume "a \<in> A" and *: "a \<in> B"
- hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
- with * show "a \<in> Field ?rB" using Field by auto
- next
- fix a b assume "a \<in> A" and "(b,a) \<in> r"
- thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
- qed
-qed
-
-
-lemma ofilter_Restr_subset:
-assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
-shows "wo_rel.ofilter (Restr r B) A"
-proof-
- have "A Int B = A" using SUB by blast
- thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
-qed
-
-
-lemma ofilter_subset_embed:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence FieldA: "Field ?rA = Field r Int A"
- using Refl_Field_Restr by blast
- have FieldB: "Field ?rB = Field r Int B"
- using Refl Refl_Field_Restr by blast
- have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
- by (simp add: Well_order_Restr wo_rel_def)
- (* Main proof *)
- show ?thesis
- proof
- assume *: "A \<le> B"
- hence "wo_rel.ofilter (Restr r B) A" using assms
- by (simp add: ofilter_Restr_subset)
- hence "embed (Restr ?rB A) (Restr r B) id"
- using WellB ofilter_embed[of "?rB" A] by auto
- thus "embed (Restr r A) (Restr r B) id"
- using * by (simp add: Restr_subset)
- next
- assume *: "embed (Restr r A) (Restr r B) id"
- {fix a assume **: "a \<in> A"
- hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
- with ** FieldA have "a \<in> Field ?rA" by auto
- hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
- hence "a \<in> B" using FieldB by auto
- }
- thus "A \<le> B" by blast
- qed
-qed
-
-
-lemma ofilter_subset_embedS_iso:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
- ((A = B) = (iso (Restr r A) (Restr r B) id))"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have Well: "wo_rel r" unfolding wo_rel_def using WELL .
- hence Refl: "Refl r" by (simp add: wo_rel.REFL)
- hence "Field ?rA = Field r Int A"
- using Refl_Field_Restr by blast
- hence FieldA: "Field ?rA = A" using OFA Well
- by (auto simp add: wo_rel.ofilter_def)
- have "Field ?rB = Field r Int B"
- using Refl Refl_Field_Restr by blast
- hence FieldB: "Field ?rB = B" using OFB Well
- by (auto simp add: wo_rel.ofilter_def)
- (* Main proof *)
- show ?thesis unfolding embedS_def iso_def
- using assms ofilter_subset_embed[of r A B]
- FieldA FieldB bij_betw_id_iff[of A B] by auto
-qed
-
-
-lemma ofilter_subset_embedS:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A < B) = embedS (Restr r A) (Restr r B) id"
-using assms by (simp add: ofilter_subset_embedS_iso)
-
-
-lemma embed_implies_iso_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- EMB: "embed r' r f"
-shows "iso r' (Restr r (f ` (Field r'))) f"
-proof-
- let ?A' = "Field r'"
- let ?r'' = "Restr r (f ` ?A')"
- have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
- have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter by blast
- hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
- hence "bij_betw f ?A' (Field ?r'')"
- using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
- moreover
- {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
- unfolding Field_def by auto
- hence "compat r' ?r'' f"
- using assms embed_iff_compat_inj_on_ofilter
- unfolding compat_def by blast
- }
- ultimately show ?thesis using WELL' 0 iso_iff3 by blast
-qed
-
-
-subsection {* The strict inclusion on proper ofilters is well-founded *}
-
-
-definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
-where
-"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
- wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
-
-
-lemma wf_ofilterIncl:
-assumes WELL: "Well_order r"
-shows "wf(ofilterIncl r)"
-proof-
- have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
- hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
- let ?h = "(\<lambda> A. wo_rel.suc r A)"
- let ?rS = "r - Id"
- have "wf ?rS" using WELL by (simp add: order_on_defs)
- moreover
- have "compat (ofilterIncl r) ?rS ?h"
- proof(unfold compat_def ofilterIncl_def,
- intro allI impI, simp, elim conjE)
- fix A B
- assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
- **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
- then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
- 1: "A = rel.underS r a \<and> B = rel.underS r b"
- using Well by (auto simp add: wo_rel.ofilter_underS_Field)
- hence "a \<noteq> b" using *** by auto
- moreover
- have "(a,b) \<in> r" using 0 1 Lo ***
- by (auto simp add: rel.underS_incl_iff)
- moreover
- have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
- using Well 0 1 by (simp add: wo_rel.suc_underS)
- ultimately
- show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
- by simp
- qed
- ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
-qed
-
-
-
-subsection {* Ordering the well-orders by existence of embeddings *}
-
-
-text {* We define three relations between well-orders:
-\begin{itemize}
-\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
-\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
-\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
-\end{itemize}
-%
-The prefix "ord" and the index "o" in these names stand for "ordinal-like".
-These relations shall be proved to be inter-connected in a similar fashion as the trio
-@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
-*}
-
-
-definition ordLeq :: "('a rel * 'a' rel) set"
-where
-"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
-
-
-abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
-where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
-
-
-abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
-where "r \<le>o r' \<equiv> r <=o r'"
-
-
-definition ordLess :: "('a rel * 'a' rel) set"
-where
-"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
-
-abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
-where "r <o r' \<equiv> (r,r') \<in> ordLess"
-
-
-definition ordIso :: "('a rel * 'a' rel) set"
-where
-"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
-
-abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
-where "r =o r' \<equiv> (r,r') \<in> ordIso"
-
-
-lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
-
-lemma ordLeq_Well_order_simp:
-assumes "r \<le>o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordLeq_def by simp
-
-
-lemma ordLess_Well_order_simp:
-assumes "r <o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordLess_def by simp
-
-
-lemma ordIso_Well_order_simp:
-assumes "r =o r'"
-shows "Well_order r \<and> Well_order r'"
-using assms unfolding ordIso_def by simp
-
-
-text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
-on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
-restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
-to @{text "'a rel rel"}. *}
-
-
-lemma ordLeq_reflexive:
-"Well_order r \<Longrightarrow> r \<le>o r"
-unfolding ordLeq_def using id_embed[of r] by blast
-
-
-lemma ordLeq_transitive[trans]:
-assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
-shows "r \<le>o r''"
-proof-
- obtain f and f'
- where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
- "embed r r' f" and "embed r' r'' f'"
- using * ** unfolding ordLeq_def by blast
- hence "embed r r'' (f' o f)"
- using comp_embed[of r r' f r'' f'] by auto
- thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
-qed
-
-
-lemma ordLeq_total:
-"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
-unfolding ordLeq_def using wellorders_totally_ordered by blast
-
-
-lemma ordIso_reflexive:
-"Well_order r \<Longrightarrow> r =o r"
-unfolding ordIso_def using id_iso[of r] by blast
-
-
-lemma ordIso_transitive[trans]:
-assumes *: "r =o r'" and **: "r' =o r''"
-shows "r =o r''"
-proof-
- obtain f and f'
- where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
- "iso r r' f" and 3: "iso r' r'' f'"
- using * ** unfolding ordIso_def by auto
- hence "iso r r'' (f' o f)"
- using comp_iso[of r r' f r'' f'] by auto
- thus "r =o r''" unfolding ordIso_def using 1 by auto
-qed
-
-
-lemma ordIso_symmetric:
-assumes *: "r =o r'"
-shows "r' =o r"
-proof-
- obtain f where 1: "Well_order r \<and> Well_order r'" and
- 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
- using * by (auto simp add: ordIso_def iso_def)
- let ?f' = "inv_into (Field r) f"
- have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
- using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
- thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
-qed
-
-
-lemma ordLeq_ordLess_trans[trans]:
-assumes "r \<le>o r'" and " r' <o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordLess_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordLess_def
- using embed_comp_embedS by blast
-qed
-
-
-lemma ordLess_ordLeq_trans[trans]:
-assumes "r <o r'" and " r' \<le>o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordLess_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordLess_def
- using embedS_comp_embed by blast
-qed
-
-
-lemma ordLeq_ordIso_trans[trans]:
-assumes "r \<le>o r'" and " r' =o r''"
-shows "r \<le>o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordIso_def
- using embed_comp_iso by blast
-qed
-
-
-lemma ordIso_ordLeq_trans[trans]:
-assumes "r =o r'" and " r' \<le>o r''"
-shows "r \<le>o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLeq_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLeq_def ordIso_def
- using iso_comp_embed by blast
-qed
-
-
-lemma ordLess_ordIso_trans[trans]:
-assumes "r <o r'" and " r' =o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLess_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLess_def ordIso_def
- using embedS_comp_iso by blast
-qed
-
-
-lemma ordIso_ordLess_trans[trans]:
-assumes "r =o r'" and " r' <o r''"
-shows "r <o r''"
-proof-
- have "Well_order r \<and> Well_order r''"
- using assms unfolding ordLess_def ordIso_def by auto
- thus ?thesis using assms unfolding ordLess_def ordIso_def
- using iso_comp_embedS by blast
-qed
-
-
-lemma ordLess_not_embed:
-assumes "r <o r'"
-shows "\<not>(\<exists>f'. embed r' r f')"
-proof-
- obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
- 3: " \<not> bij_betw f (Field r) (Field r')"
- using assms unfolding ordLess_def by (auto simp add: embedS_def)
- {fix f' assume *: "embed r' r f'"
- hence "bij_betw f (Field r) (Field r')" using 1 2
- by (simp add: embed_bothWays_Field_bij_betw)
- with 3 have False by contradiction
- }
- thus ?thesis by blast
-qed
-
-
-lemma ordLess_Field:
-assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
-shows "\<not> (f`(Field r1) = Field r2)"
-proof-
- let ?A1 = "Field r1" let ?A2 = "Field r2"
- obtain g where
- 0: "Well_order r1 \<and> Well_order r2" and
- 1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
- using OL unfolding ordLess_def by (auto simp add: embedS_def)
- hence "\<forall>a \<in> ?A1. f a = g a"
- using 0 EMB embed_unique[of r1] by auto
- hence "\<not>(bij_betw f ?A1 ?A2)"
- using 1 bij_betw_cong[of ?A1] by blast
- moreover
- have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
- ultimately show ?thesis by (simp add: bij_betw_def)
-qed
-
-
-lemma ordLess_iff:
-"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
-proof
- assume *: "r <o r'"
- hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
- with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
- unfolding ordLess_def by auto
-next
- assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
- then obtain f where 1: "embed r r' f"
- using wellorders_totally_ordered[of r r'] by blast
- moreover
- {assume "bij_betw f (Field r) (Field r')"
- with * 1 have "embed r' r (inv_into (Field r) f) "
- using inv_into_Field_embed_bij_betw[of r r' f] by auto
- with * have False by blast
- }
- ultimately show "(r,r') \<in> ordLess"
- unfolding ordLess_def using * by (fastforce simp add: embedS_def)
-qed
-
-
-lemma ordLess_irreflexive: "\<not> r <o r"
-proof
- assume "r <o r"
- hence "Well_order r \<and> \<not>(\<exists>f. embed r r f)"
- unfolding ordLess_iff ..
- moreover have "embed r r id" using id_embed[of r] .
- ultimately show False by blast
-qed
-
-
-lemma ordLeq_iff_ordLess_or_ordIso:
-"r \<le>o r' = (r <o r' \<or> r =o r')"
-unfolding ordRels_def embedS_defs iso_defs by blast
-
-
-lemma ordIso_iff_ordLeq:
-"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
-proof
- assume "r =o r'"
- then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
- embed r r' f \<and> bij_betw f (Field r) (Field r')"
- unfolding ordIso_def iso_defs by auto
- hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
- by (simp add: inv_into_Field_embed_bij_betw)
- thus "r \<le>o r' \<and> r' \<le>o r"
- unfolding ordLeq_def using 1 by auto
-next
- assume "r \<le>o r' \<and> r' \<le>o r"
- then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
- embed r r' f \<and> embed r' r g"
- unfolding ordLeq_def by auto
- hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
- thus "r =o r'" unfolding ordIso_def using 1 by auto
-qed
-
-
-lemma not_ordLess_ordLeq:
-"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
-using ordLess_ordLeq_trans ordLess_irreflexive by blast
-
-
-lemma ordLess_or_ordLeq:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "r <o r' \<or> r' \<le>o r"
-proof-
- have "r \<le>o r' \<or> r' \<le>o r"
- using assms by (simp add: ordLeq_total)
- moreover
- {assume "\<not> r <o r' \<and> r \<le>o r'"
- hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
- hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
- }
- ultimately show ?thesis by blast
-qed
-
-
-lemma not_ordLess_ordIso:
-"r <o r' \<Longrightarrow> \<not> r =o r'"
-using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
-
-
-lemma not_ordLeq_iff_ordLess:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(\<not> r' \<le>o r) = (r <o r')"
-using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
-
-
-lemma not_ordLess_iff_ordLeq:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(\<not> r' <o r) = (r \<le>o r')"
-using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
-
-
-lemma ordLess_transitive[trans]:
-"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
-using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
-
-
-corollary ordLess_trans: "trans ordLess"
-unfolding trans_def using ordLess_transitive by blast
-
-
-lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
-
-
-lemma ordIso_imp_ordLeq:
-"r =o r' \<Longrightarrow> r \<le>o r'"
-using ordIso_iff_ordLeq by blast
-
-
-lemma ordLess_imp_ordLeq:
-"r <o r' \<Longrightarrow> r \<le>o r'"
-using ordLeq_iff_ordLess_or_ordIso by blast
-
-
-lemma ofilter_subset_ordLeq:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
-proof
- assume "A \<le> B"
- thus "Restr r A \<le>o Restr r B"
- unfolding ordLeq_def using assms
- Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
-next
- assume *: "Restr r A \<le>o Restr r B"
- then obtain f where "embed (Restr r A) (Restr r B) f"
- unfolding ordLeq_def by blast
- {assume "B < A"
- hence "Restr r B <o Restr r A"
- unfolding ordLess_def using assms
- Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
- hence False using * not_ordLess_ordLeq by blast
- }
- thus "A \<le> B" using OFA OFB WELL
- wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
-qed
-
-
-lemma ofilter_subset_ordLess:
-assumes WELL: "Well_order r" and
- OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
-shows "(A < B) = (Restr r A <o Restr r B)"
-proof-
- let ?rA = "Restr r A" let ?rB = "Restr r B"
- have 1: "Well_order ?rA \<and> Well_order ?rB"
- using WELL Well_order_Restr by blast
- have "(A < B) = (\<not> B \<le> A)" using assms
- wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
- also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
- using assms ofilter_subset_ordLeq by blast
- also have "\<dots> = (Restr r A <o Restr r B)"
- using 1 not_ordLeq_iff_ordLess by blast
- finally show ?thesis .
-qed
-
-
-lemma ofilter_ordLess:
-"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
-by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
- wo_rel_def Restr_Field)
-
-
-corollary underS_Restr_ordLess:
-assumes "Well_order r" and "Field r \<noteq> {}"
-shows "Restr r (rel.underS r a) <o r"
-proof-
- have "rel.underS r a < Field r" using assms
- by (simp add: rel.underS_Field3)
- thus ?thesis using assms
- by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
-qed
-
-
-lemma embed_ordLess_ofilterIncl:
-assumes
- OL12: "r1 <o r2" and OL23: "r2 <o r3" and
- EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
-shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
-proof-
- have OL13: "r1 <o r3"
- using OL12 OL23 using ordLess_transitive by auto
- let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A3 ="Field r3"
- obtain f12 g23 where
- 0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
- 1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
- 2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
- using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
- hence "\<forall>a \<in> ?A2. f23 a = g23 a"
- using EMB23 embed_unique[of r2 r3] by blast
- hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
- using 2 bij_betw_cong[of ?A2 f23 g23] by blast
- (* *)
- have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
- using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
- have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
- using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
- have 6: "wo_rel.ofilter r3 (f13 ` ?A1) \<and> f13 ` ?A1 \<noteq> ?A3"
- using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
- (* *)
- have "f12 ` ?A1 < ?A2"
- using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
- moreover have "inj_on f23 ?A2"
- using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
- ultimately
- have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
- moreover
- {have "embed r1 r3 (f23 o f12)"
- using 1 EMB23 0 by (auto simp add: comp_embed)
- hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
- using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
- hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
- }
- ultimately
- have "f13 ` ?A1 < f23 ` ?A2" by simp
- (* *)
- with 5 6 show ?thesis
- unfolding ofilterIncl_def by auto
-qed
-
-
-lemma ordLess_iff_ordIso_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
-proof(auto)
- fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
- hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
- thus "r' <o r" using ** ordIso_ordLess_trans by blast
-next
- assume "r' <o r"
- then obtain f where 1: "Well_order r \<and> Well_order r'" and
- 2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
- unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
- hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
- then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
- using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
- have "iso r' (Restr r (f ` (Field r'))) f"
- using embed_implies_iso_Restr 2 assms by blast
- moreover have "Well_order (Restr r (f ` (Field r')))"
- using WELL Well_order_Restr by blast
- ultimately have "r' =o Restr r (f ` (Field r'))"
- using WELL' unfolding ordIso_def by auto
- hence "r' =o Restr r (rel.underS r a)" using 4 by auto
- thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
-qed
-
-
-lemma internalize_ordLess:
-"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
-proof
- assume *: "r' <o r"
- hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
- with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
- using ordLess_iff_ordIso_Restr by blast
- let ?p = "Restr r (rel.underS r a)"
- have "wo_rel.ofilter r (rel.underS r a)" using 0
- by (simp add: wo_rel_def wo_rel.underS_ofilter)
- hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
- hence "Field ?p < Field r" using rel.underS_Field2 1 by fastforce
- moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
- ultimately
- show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
-next
- assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
- thus "r' <o r" using ordIso_ordLess_trans by blast
-qed
-
-
-lemma internalize_ordLeq:
-"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
-proof
- assume *: "r' \<le>o r"
- moreover
- {assume "r' <o r"
- then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
- using internalize_ordLess[of r' r] by blast
- hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- using ordLeq_iff_ordLess_or_ordIso by blast
- }
- moreover
- have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
- ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- using ordLeq_iff_ordLess_or_ordIso by blast
-next
- assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
- thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
-qed
-
-
-lemma ordLeq_iff_ordLess_Restr:
-assumes WELL: "Well_order r" and WELL': "Well_order r'"
-shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
-proof(auto)
- assume *: "r \<le>o r'"
- fix a assume "a \<in> Field r"
- hence "Restr r (rel.underS r a) <o r"
- using WELL underS_Restr_ordLess[of r] by blast
- thus "Restr r (rel.underS r a) <o r'"
- using * ordLess_ordLeq_trans by blast
-next
- assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
- {assume "r' <o r"
- then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
- using assms ordLess_iff_ordIso_Restr by blast
- hence False using * not_ordLess_ordIso ordIso_symmetric by blast
- }
- thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
-qed
-
-
-lemma finite_ordLess_infinite:
-assumes WELL: "Well_order r" and WELL': "Well_order r'" and
- FIN: "finite(Field r)" and INF: "infinite(Field r')"
-shows "r <o r'"
-proof-
- {assume "r' \<le>o r"
- then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
- unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
- hence False using finite_imageD finite_subset FIN INF by metis
- }
- thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
-qed
-
-
-lemma finite_well_order_on_ordIso:
-assumes FIN: "finite A" and
- WELL: "well_order_on A r" and WELL': "well_order_on A r'"
-shows "r =o r'"
-proof-
- have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
- using assms rel.well_order_on_Well_order by blast
- moreover
- have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
- \<longrightarrow> r =o r'"
- proof(clarify)
- fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
- have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
- using * ** rel.well_order_on_Well_order by blast
- assume "r \<le>o r'"
- then obtain f where 1: "embed r r' f" and
- "inj_on f A \<and> f ` A \<le> A"
- unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
- hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
- thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
- qed
- ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
-qed
-
-
-subsection{* @{text "<o"} is well-founded *}
-
-
-text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
-on the restricted type @{text "'a rel rel"}. We prove this by first showing that, for any set
-of well-orders all embedded in a fixed well-order, the function mapping each well-order
-in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
-{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
-
-
-definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
-where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
-
-
-lemma ord_to_filter_compat:
-"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
- (ofilterIncl r0)
- (ord_to_filter r0)"
-proof(unfold compat_def ord_to_filter_def, clarify)
- fix r1::"'a rel" and r2::"'a rel"
- let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A0 ="Field r0"
- let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
- let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
- assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
- hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
- by (auto simp add: ordLess_def embedS_def)
- hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
- thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
- using * ** by (simp add: embed_ordLess_ofilterIncl)
-qed
-
-
-theorem wf_ordLess: "wf ordLess"
-proof-
- {fix r0 :: "('a \<times> 'a) set"
- (* need to annotate here!*)
- let ?ordLess = "ordLess::('d rel * 'd rel) set"
- let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
- {assume Case1: "Well_order r0"
- hence "wf ?R"
- using wf_ofilterIncl[of r0]
- compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
- ord_to_filter_compat[of r0] by auto
- }
- moreover
- {assume Case2: "\<not> Well_order r0"
- hence "?R = {}" unfolding ordLess_def by auto
- hence "wf ?R" using wf_empty by simp
- }
- ultimately have "wf ?R" by blast
- }
- thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
-qed
-
-corollary exists_minim_Well_order:
-assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
-shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
-proof-
- obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
- using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
- equals0I[of R] by blast
- with not_ordLeq_iff_ordLess WELL show ?thesis by blast
-qed
-
-
-
-subsection {* Copy via direct images *}
-
-
-text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
-from @{text "Relation.thy"}. It is useful for transporting a well-order between
-different types. *}
-
-
-definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
-where
-"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
-
-
-lemma dir_image_Field:
-"Field(dir_image r f) \<le> f ` (Field r)"
-unfolding dir_image_def Field_def by auto
-
-
-lemma dir_image_minus_Id:
-"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
-unfolding inj_on_def Field_def dir_image_def by auto
-
-
-lemma Refl_dir_image:
-assumes "Refl r"
-shows "Refl(dir_image r f)"
-proof-
- {fix a' b'
- assume "(a',b') \<in> dir_image r f"
- then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
- unfolding dir_image_def by blast
- hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
- hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
- with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
- unfolding dir_image_def by auto
- }
- thus ?thesis
- by(unfold refl_on_def Field_def Domain_def Range_def, auto)
-qed
-
-
-lemma trans_dir_image:
-assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
-shows "trans(dir_image r f)"
-proof(unfold trans_def, auto)
- fix a' b' c'
- assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
- then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
- 2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
- unfolding dir_image_def by blast
- hence "b1 \<in> Field r \<and> b2 \<in> Field r"
- unfolding Field_def by auto
- hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
- hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
- thus "(a',c') \<in> dir_image r f"
- unfolding dir_image_def using 1 by auto
-qed
-
-
-lemma Preorder_dir_image:
-"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
-by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
-
-
-lemma antisym_dir_image:
-assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
-shows "antisym(dir_image r f)"
-proof(unfold antisym_def, auto)
- fix a' b'
- assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
- then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
- 2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
- 3: "{a1,a2,b1,b2} \<le> Field r"
- unfolding dir_image_def Field_def by blast
- hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
- hence "a1 = b2" using 2 AN unfolding antisym_def by auto
- thus "a' = b'" using 1 by auto
-qed
-
-
-lemma Partial_order_dir_image:
-"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
-by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
-
-
-lemma Total_dir_image:
-assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
-shows "Total(dir_image r f)"
-proof(unfold total_on_def, intro ballI impI)
- fix a' b'
- assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
- then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
- using dir_image_Field[of r f] by blast
- moreover assume "a' \<noteq> b'"
- ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
- hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
- thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
- using 1 unfolding dir_image_def by auto
-qed
-
-
-lemma Linear_order_dir_image:
-"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
-by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
-
-
-lemma wf_dir_image:
-assumes WF: "wf r" and INJ: "inj_on f (Field r)"
-shows "wf(dir_image r f)"
-proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
- fix A'::"'b set"
- assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
- obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
- have "A \<noteq> {} \<and> A \<le> Field r"
- using A_def dir_image_Field[of r f] SUB NE by blast
- then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
- using WF unfolding wf_eq_minimal2 by metis
- have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
- proof(clarify)
- fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
- obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
- 3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
- using ** unfolding dir_image_def Field_def by blast
- hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
- hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
- with 1 show False by auto
- qed
- thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
- using A_def 1 by blast
-qed
-
-
-lemma Well_order_dir_image:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
-using assms unfolding well_order_on_def
-using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
- dir_image_minus_Id[of f r]
- subset_inj_on[of f "Field r" "Field(r - Id)"]
- mono_Field[of "r - Id" r] by auto
-
-
-lemma dir_image_Field2:
-"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
-unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
-
-
-lemma dir_image_bij_betw:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
-unfolding bij_betw_def
-by (simp add: dir_image_Field2 order_on_defs)
-
-
-lemma dir_image_compat:
-"compat r (dir_image r f) f"
-unfolding compat_def dir_image_def by auto
-
-
-lemma dir_image_iso:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> iso r (dir_image r f) f"
-using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
-
-
-lemma dir_image_ordIso:
-"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> r =o dir_image r f"
-unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
-
-
-lemma Well_order_iso_copy:
-assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
-shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
-proof-
- let ?r' = "dir_image r f"
- have 1: "A = Field r \<and> Well_order r"
- using WELL rel.well_order_on_Well_order by blast
- hence 2: "iso r ?r' f"
- using dir_image_iso using BIJ unfolding bij_betw_def by auto
- hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
- hence "Field ?r' = A'"
- using 1 BIJ unfolding bij_betw_def by auto
- moreover have "Well_order ?r'"
- using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
- ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
-qed
-
-
-
-subsection {* Bounded square *}
-
-
-text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
-order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
-following criteria (in this order):
-\begin{itemize}
-\item compare the maximums;
-\item compare the first components;
-\item compare the second components.
-\end{itemize}
-%
-The only application of this construction that we are aware of is
-at proving that the square of an infinite set has the same cardinal
-as that set. The essential property required there (and which is ensured by this
-construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
-in a product of proper filters on the original relation (assumed to be a well-order). *}
-
-
-definition bsqr :: "'a rel => ('a * 'a)rel"
-where
-"bsqr r = {((a1,a2),(b1,b2)).
- {a1,a2,b1,b2} \<le> Field r \<and>
- (a1 = b1 \<and> a2 = b2 \<or>
- (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id
- )}"
-
-
-lemma Field_bsqr:
-"Field (bsqr r) = Field r \<times> Field r"
-proof
- show "Field (bsqr r) \<le> Field r \<times> Field r"
- proof-
- {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
- moreover
- have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
- a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
- ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
- }
- thus ?thesis unfolding Field_def by force
- qed
-next
- show "Field r \<times> Field r \<le> Field (bsqr r)"
- proof(auto)
- fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
- hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
- thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
- qed
-qed
-
-
-lemma bsqr_Refl: "Refl(bsqr r)"
-by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
-
-
-lemma bsqr_Trans:
-assumes "Well_order r"
-shows "trans (bsqr r)"
-proof(unfold trans_def, auto)
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Trans: "trans r" using wo_rel.TRANS by auto
- have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
- hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
- (* Main proof *)
- fix a1 a2 b1 b2 c1 c2
- assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
- hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
- have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- using * unfolding bsqr_def by auto
- have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
- using ** unfolding bsqr_def by auto
- show "((a1,a2),(c1,c2)) \<in> bsqr r"
- proof-
- {assume Case1: "a1 = b1 \<and> a2 = b2"
- hence ?thesis using ** by simp
- }
- moreover
- {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
- {assume Case21: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
- using Case2 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
- hence ?thesis using Case2 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
- {assume Case31: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence ?thesis using Case3 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
- hence "(a1,c1) \<in> r - Id"
- using Case3 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
- hence ?thesis using Case3 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- {assume Case41: "b1 = c1 \<and> b2 = c2"
- hence ?thesis using * by simp
- }
- moreover
- {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
- hence ?thesis using Case4 0 unfolding bsqr_def by force
- }
- moreover
- {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
- hence ?thesis using Case4 0 unfolding bsqr_def by auto
- }
- moreover
- {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
- hence "(a2,c2) \<in> r - Id"
- using Case4 TransS trans_def[of "r - Id"] by blast
- hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- ultimately show ?thesis using 0 1 by auto
- qed
-qed
-
-
-lemma bsqr_antisym:
-assumes "Well_order r"
-shows "antisym (bsqr r)"
-proof(unfold antisym_def, clarify)
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Trans: "trans r" using wo_rel.TRANS by auto
- have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
- hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
- hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
- using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
- (* Main proof *)
- fix a1 a2 b1 b2
- assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
- hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
- have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
- wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- using * unfolding bsqr_def by auto
- have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
- wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
- using ** unfolding bsqr_def by auto
- show "a1 = b1 \<and> a2 = b2"
- proof-
- {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
- {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case1 IrrS by blast
- }
- moreover
- {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
- hence False using Case1 by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
- {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case2 by auto
- }
- moreover
- {assume Case22: "(b1,a1) \<in> r - Id"
- hence False using Case2 IrrS by blast
- }
- moreover
- {assume Case23: "b1 = a1"
- hence False using Case2 by auto
- }
- ultimately have ?thesis using 0 2 by auto
- }
- moreover
- {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
- moreover
- {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
- hence False using Case3 by auto
- }
- moreover
- {assume Case32: "(b1,a1) \<in> r - Id"
- hence False using Case3 by auto
- }
- moreover
- {assume Case33: "(b2,a2) \<in> r - Id"
- hence False using Case3 IrrS by blast
- }
- ultimately have ?thesis using 0 2 by auto
- }
- ultimately show ?thesis using 0 1 by blast
- qed
-qed
-
-
-lemma bsqr_Total:
-assumes "Well_order r"
-shows "Total(bsqr r)"
-proof-
- (* Preliminary facts *)
- have Well: "wo_rel r" using assms wo_rel_def by auto
- hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
- using wo_rel.TOTALS by auto
- (* Main proof *)
- {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
- hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
- using Field_bsqr by blast
- have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
- proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
- (* Why didn't clarsimp simp add: Well 0 do the same job? *)
- assume Case1: "(a1,a2) \<in> r"
- hence 1: "wo_rel.max2 r a1 a2 = a2"
- using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case11: "(b1,b2) \<in> r"
- hence 2: "wo_rel.max2 r b1 b2 = b2"
- using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 2 unfolding bsqr_def by auto
- next
- assume Case112: "a2 = b2"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
- next
- assume Case1122: "a1 = b1"
- thus ?thesis using Case112 by auto
- qed
- qed
- next
- assume Case12: "(b2,b1) \<in> r"
- hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
- assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 unfolding bsqr_def by auto
- next
- assume Case122: "a2 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
- next
- assume Case1222: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
- next
- assume Case12222: "a2 = b2"
- thus ?thesis using Case122 Case1222 by auto
- qed
- qed
- qed
- qed
- next
- assume Case2: "(a2,a1) \<in> r"
- hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case21: "(b1,b2) \<in> r"
- hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
- assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 unfolding bsqr_def by auto
- next
- assume Case212: "a1 = b2"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
- next
- assume Case2122: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
- next
- assume Case21222: "a2 = b2"
- thus ?thesis using Case2122 Case212 by auto
- qed
- qed
- qed
- next
- assume Case22: "(b2,b1) \<in> r"
- hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
- assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
- thus ?thesis using 0 1 3 unfolding bsqr_def by auto
- next
- assume Case222: "a1 = b1"
- show ?thesis
- proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
- assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
- thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
- next
- assume Case2222: "a2 = b2"
- thus ?thesis using Case222 by auto
- qed
- qed
- qed
- qed
- }
- thus ?thesis unfolding total_on_def by fast
-qed
-
-
-lemma bsqr_Linear_order:
-assumes "Well_order r"
-shows "Linear_order(bsqr r)"
-unfolding order_on_defs
-using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
-
-
-lemma bsqr_Well_order:
-assumes "Well_order r"
-shows "Well_order(bsqr r)"
-using assms
-proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
- have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
- using assms well_order_on_def Linear_order_Well_order_iff by blast
- fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
- hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
- (* *)
- obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
- have "M \<noteq> {}" using 1 M_def ** by auto
- moreover
- have "M \<le> Field r" unfolding M_def
- using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
- ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
- using 0 by blast
- (* *)
- obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
- have "A1 \<le> Field r" unfolding A1_def using 1 by auto
- moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
- ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
- using 0 by blast
- (* *)
- obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
- have "A2 \<le> Field r" unfolding A2_def using 1 by auto
- moreover have "A2 \<noteq> {}" unfolding A2_def
- using m_min a1_min unfolding A1_def M_def by blast
- ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
- using 0 by blast
- (* *)
- have 2: "wo_rel.max2 r a1 a2 = m"
- using a1_min a2_min unfolding A1_def A2_def by auto
- have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
- (* *)
- moreover
- {fix b1 b2 assume ***: "(b1,b2) \<in> D"
- hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
- have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
- using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
- have "((a1,a2),(b1,b2)) \<in> bsqr r"
- proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
- assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
- thus ?thesis unfolding bsqr_def using 4 5 by auto
- next
- assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
- hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
- hence 6: "(a1,b1) \<in> r" using a1_min by auto
- show ?thesis
- proof(cases "a1 = b1")
- assume Case21: "a1 \<noteq> b1"
- thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
- next
- assume Case22: "a1 = b1"
- hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
- hence 7: "(a2,b2) \<in> r" using a2_min by auto
- thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
- qed
- qed
- }
- (* *)
- ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
-qed
-
-
-lemma bsqr_max2:
-assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
-shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
-proof-
- have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
- using LEQ unfolding Field_def by auto
- hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
- hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
- using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
- moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
- using LEQ unfolding bsqr_def by auto
- ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
-qed
-
-
-lemma bsqr_ofilter:
-assumes WELL: "Well_order r" and
- OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
- NE: "\<not> (\<exists>a. Field r = rel.under r a)"
-shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
-proof-
- let ?r' = "bsqr r"
- have Well: "wo_rel r" using WELL wo_rel_def by blast
- hence Trans: "trans r" using wo_rel.TRANS by blast
- have Well': "Well_order ?r' \<and> wo_rel ?r'"
- using WELL bsqr_Well_order wo_rel_def by blast
- (* *)
- have "D < Field ?r'" unfolding Field_bsqr using SUB .
- with OF obtain a1 and a2 where
- "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
- using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
- hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
- let ?m = "wo_rel.max2 r a1 a2"
- have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
- proof(unfold 1)
- {fix b1 b2
- let ?n = "wo_rel.max2 r b1 b2"
- assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
- hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
- unfolding rel.underS_def by blast
- hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
- moreover
- {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
- hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
- hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
- using Well by (simp add: wo_rel.max2_greater)
- }
- ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
- using Trans trans_def[of r] by blast
- hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
- thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
- qed
- moreover have "wo_rel.ofilter r (rel.under r ?m)"
- using Well by (simp add: wo_rel.under_ofilter)
- moreover have "rel.under r ?m < Field r"
- using NE rel.under_Field[of r ?m] by blast
- ultimately show ?thesis by blast
-qed
-
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders_LFP.thy Mon Nov 18 18:04:44 2013 +0100
@@ -0,0 +1,1633 @@
+(* Title: HOL/Cardinals/Constructions_on_Wellorders_LFP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Constructions on wellorders (LFP).
+*)
+
+header {* Constructions on Wellorders (LFP) *}
+
+theory Constructions_on_Wellorders_LFP
+imports Wellorder_Embedding_LFP
+begin
+
+
+text {* In this section, we study basic constructions on well-orders, such as restriction to
+a set/order filter, copy via direct images, ordinal-like sum of disjoint well-orders,
+and bounded square. We also define between well-orders
+the relations @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"}),
+@{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"}), and
+@{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}). We study the
+connections between these relations, order filters, and the aforementioned constructions.
+A main result of this section is that @{text "<o"} is well-founded. *}
+
+
+subsection {* Restriction to a set *}
+
+
+abbreviation Restr :: "'a rel \<Rightarrow> 'a set \<Rightarrow> 'a rel"
+where "Restr r A \<equiv> r Int (A \<times> A)"
+
+
+lemma Restr_subset:
+"A \<le> B \<Longrightarrow> Restr (Restr r B) A = Restr r A"
+by blast
+
+
+lemma Restr_Field: "Restr r (Field r) = r"
+unfolding Field_def by auto
+
+
+lemma Refl_Restr: "Refl r \<Longrightarrow> Refl(Restr r A)"
+unfolding refl_on_def Field_def by auto
+
+
+lemma antisym_Restr:
+"antisym r \<Longrightarrow> antisym(Restr r A)"
+unfolding antisym_def Field_def by auto
+
+
+lemma Total_Restr:
+"Total r \<Longrightarrow> Total(Restr r A)"
+unfolding total_on_def Field_def by auto
+
+
+lemma trans_Restr:
+"trans r \<Longrightarrow> trans(Restr r A)"
+unfolding trans_def Field_def by blast
+
+
+lemma Preorder_Restr:
+"Preorder r \<Longrightarrow> Preorder(Restr r A)"
+unfolding preorder_on_def by (simp add: Refl_Restr trans_Restr)
+
+
+lemma Partial_order_Restr:
+"Partial_order r \<Longrightarrow> Partial_order(Restr r A)"
+unfolding partial_order_on_def by (simp add: Preorder_Restr antisym_Restr)
+
+
+lemma Linear_order_Restr:
+"Linear_order r \<Longrightarrow> Linear_order(Restr r A)"
+unfolding linear_order_on_def by (simp add: Partial_order_Restr Total_Restr)
+
+
+lemma Well_order_Restr:
+assumes "Well_order r"
+shows "Well_order(Restr r A)"
+proof-
+ have "Restr r A - Id \<le> r - Id" using Restr_subset by blast
+ hence "wf(Restr r A - Id)" using assms
+ using well_order_on_def wf_subset by blast
+ thus ?thesis using assms unfolding well_order_on_def
+ by (simp add: Linear_order_Restr)
+qed
+
+
+lemma Field_Restr_subset: "Field(Restr r A) \<le> A"
+by (auto simp add: Field_def)
+
+
+lemma Refl_Field_Restr:
+"Refl r \<Longrightarrow> Field(Restr r A) = (Field r) Int A"
+by (auto simp add: refl_on_def Field_def)
+
+
+lemma Refl_Field_Restr2:
+"\<lbrakk>Refl r; A \<le> Field r\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: Refl_Field_Restr)
+
+
+lemma well_order_on_Restr:
+assumes WELL: "Well_order r" and SUB: "A \<le> Field r"
+shows "well_order_on A (Restr r A)"
+using assms
+using Well_order_Restr[of r A] Refl_Field_Restr2[of r A]
+ order_on_defs[of "Field r" r] by auto
+
+
+subsection {* Order filters versus restrictions and embeddings *}
+
+
+lemma Field_Restr_ofilter:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> Field(Restr r A) = A"
+by (auto simp add: wo_rel_def wo_rel.ofilter_def wo_rel.REFL Refl_Field_Restr2)
+
+
+lemma ofilter_Restr_under:
+assumes WELL: "Well_order r" and OF: "wo_rel.ofilter r A" and IN: "a \<in> A"
+shows "rel.under (Restr r A) a = rel.under r a"
+using assms wo_rel_def
+proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix b assume *: "a \<in> A" and "(b,a) \<in> r"
+ hence "b \<in> rel.under r a \<and> a \<in> Field r"
+ unfolding rel.under_def using Field_def by fastforce
+ thus "b \<in> A" using * assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_embed:
+assumes "Well_order r"
+shows "wo_rel.ofilter r A = (A \<le> Field r \<and> embed (Restr r A) r id)"
+proof
+ assume *: "wo_rel.ofilter r A"
+ show "A \<le> Field r \<and> embed (Restr r A) r id"
+ proof(unfold embed_def, auto)
+ fix a assume "a \<in> A" thus "a \<in> Field r" using assms *
+ by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ next
+ fix a assume "a \<in> Field (Restr r A)"
+ thus "bij_betw id (rel.under (Restr r A) a) (rel.under r a)" using assms *
+ by (simp add: ofilter_Restr_under Field_Restr_ofilter)
+ qed
+next
+ assume *: "A \<le> Field r \<and> embed (Restr r A) r id"
+ hence "Field(Restr r A) \<le> Field r"
+ using assms embed_Field[of "Restr r A" r id] id_def
+ Well_order_Restr[of r] by auto
+ {fix a assume "a \<in> A"
+ hence "a \<in> Field(Restr r A)" using * assms
+ by (simp add: order_on_defs Refl_Field_Restr2)
+ hence "bij_betw id (rel.under (Restr r A) a) (rel.under r a)"
+ using * unfolding embed_def by auto
+ hence "rel.under r a \<le> rel.under (Restr r A) a"
+ unfolding bij_betw_def by auto
+ also have "\<dots> \<le> Field(Restr r A)" by (simp add: rel.under_Field)
+ also have "\<dots> \<le> A" by (simp add: Field_Restr_subset)
+ finally have "rel.under r a \<le> A" .
+ }
+ thus "wo_rel.ofilter r A" using assms * by (simp add: wo_rel_def wo_rel.ofilter_def)
+qed
+
+
+lemma ofilter_Restr_Int:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A"
+shows "wo_rel.ofilter (Restr r B) (A Int B)"
+proof-
+ let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence Field: "Field ?rB = Field r Int B"
+ using Refl_Field_Restr by blast
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis using WellB assms
+ proof(auto simp add: wo_rel.ofilter_def rel.under_def)
+ fix a assume "a \<in> A" and *: "a \<in> B"
+ hence "a \<in> Field r" using OFA Well by (auto simp add: wo_rel.ofilter_def)
+ with * show "a \<in> Field ?rB" using Field by auto
+ next
+ fix a b assume "a \<in> A" and "(b,a) \<in> r"
+ thus "b \<in> A" using Well OFA by (auto simp add: wo_rel.ofilter_def rel.under_def)
+ qed
+qed
+
+
+lemma ofilter_Restr_subset:
+assumes WELL: "Well_order r" and OFA: "wo_rel.ofilter r A" and SUB: "A \<le> B"
+shows "wo_rel.ofilter (Restr r B) A"
+proof-
+ have "A Int B = A" using SUB by blast
+ thus ?thesis using assms ofilter_Restr_Int[of r A B] by auto
+qed
+
+
+lemma ofilter_subset_embed:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (embed (Restr r A) (Restr r B) id)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence FieldA: "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ have FieldB: "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ have WellA: "wo_rel ?rA \<and> Well_order ?rA" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
+ by (simp add: Well_order_Restr wo_rel_def)
+ (* Main proof *)
+ show ?thesis
+ proof
+ assume *: "A \<le> B"
+ hence "wo_rel.ofilter (Restr r B) A" using assms
+ by (simp add: ofilter_Restr_subset)
+ hence "embed (Restr ?rB A) (Restr r B) id"
+ using WellB ofilter_embed[of "?rB" A] by auto
+ thus "embed (Restr r A) (Restr r B) id"
+ using * by (simp add: Restr_subset)
+ next
+ assume *: "embed (Restr r A) (Restr r B) id"
+ {fix a assume **: "a \<in> A"
+ hence "a \<in> Field r" using Well OFA by (auto simp add: wo_rel.ofilter_def)
+ with ** FieldA have "a \<in> Field ?rA" by auto
+ hence "a \<in> Field ?rB" using * WellA embed_Field[of ?rA ?rB id] by auto
+ hence "a \<in> B" using FieldB by auto
+ }
+ thus "A \<le> B" by blast
+ qed
+qed
+
+
+lemma ofilter_subset_embedS_iso:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "((A < B) = (embedS (Restr r A) (Restr r B) id)) \<and>
+ ((A = B) = (iso (Restr r A) (Restr r B) id))"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have Well: "wo_rel r" unfolding wo_rel_def using WELL .
+ hence Refl: "Refl r" by (simp add: wo_rel.REFL)
+ hence "Field ?rA = Field r Int A"
+ using Refl_Field_Restr by blast
+ hence FieldA: "Field ?rA = A" using OFA Well
+ by (auto simp add: wo_rel.ofilter_def)
+ have "Field ?rB = Field r Int B"
+ using Refl Refl_Field_Restr by blast
+ hence FieldB: "Field ?rB = B" using OFB Well
+ by (auto simp add: wo_rel.ofilter_def)
+ (* Main proof *)
+ show ?thesis unfolding embedS_def iso_def
+ using assms ofilter_subset_embed[of r A B]
+ FieldA FieldB bij_betw_id_iff[of A B] by auto
+qed
+
+
+lemma ofilter_subset_embedS:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = embedS (Restr r A) (Restr r B) id"
+using assms by (simp add: ofilter_subset_embedS_iso)
+
+
+lemma embed_implies_iso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ EMB: "embed r' r f"
+shows "iso r' (Restr r (f ` (Field r'))) f"
+proof-
+ let ?A' = "Field r'"
+ let ?r'' = "Restr r (f ` ?A')"
+ have 0: "Well_order ?r''" using WELL Well_order_Restr by blast
+ have 1: "wo_rel.ofilter r (f ` ?A')" using assms embed_Field_ofilter by blast
+ hence "Field ?r'' = f ` (Field r')" using WELL Field_Restr_ofilter by blast
+ hence "bij_betw f ?A' (Field ?r'')"
+ using EMB embed_inj_on WELL' unfolding bij_betw_def by blast
+ moreover
+ {have "\<forall>a b. (a,b) \<in> r' \<longrightarrow> a \<in> Field r' \<and> b \<in> Field r'"
+ unfolding Field_def by auto
+ hence "compat r' ?r'' f"
+ using assms embed_iff_compat_inj_on_ofilter
+ unfolding compat_def by blast
+ }
+ ultimately show ?thesis using WELL' 0 iso_iff3 by blast
+qed
+
+
+subsection {* The strict inclusion on proper ofilters is well-founded *}
+
+
+definition ofilterIncl :: "'a rel \<Rightarrow> 'a set rel"
+where
+"ofilterIncl r \<equiv> {(A,B). wo_rel.ofilter r A \<and> A \<noteq> Field r \<and>
+ wo_rel.ofilter r B \<and> B \<noteq> Field r \<and> A < B}"
+
+
+lemma wf_ofilterIncl:
+assumes WELL: "Well_order r"
+shows "wf(ofilterIncl r)"
+proof-
+ have Well: "wo_rel r" using WELL by (simp add: wo_rel_def)
+ hence Lo: "Linear_order r" by (simp add: wo_rel.LIN)
+ let ?h = "(\<lambda> A. wo_rel.suc r A)"
+ let ?rS = "r - Id"
+ have "wf ?rS" using WELL by (simp add: order_on_defs)
+ moreover
+ have "compat (ofilterIncl r) ?rS ?h"
+ proof(unfold compat_def ofilterIncl_def,
+ intro allI impI, simp, elim conjE)
+ fix A B
+ assume *: "wo_rel.ofilter r A" "A \<noteq> Field r" and
+ **: "wo_rel.ofilter r B" "B \<noteq> Field r" and ***: "A < B"
+ then obtain a and b where 0: "a \<in> Field r \<and> b \<in> Field r" and
+ 1: "A = rel.underS r a \<and> B = rel.underS r b"
+ using Well by (auto simp add: wo_rel.ofilter_underS_Field)
+ hence "a \<noteq> b" using *** by auto
+ moreover
+ have "(a,b) \<in> r" using 0 1 Lo ***
+ by (auto simp add: rel.underS_incl_iff)
+ moreover
+ have "a = wo_rel.suc r A \<and> b = wo_rel.suc r B"
+ using Well 0 1 by (simp add: wo_rel.suc_underS)
+ ultimately
+ show "(wo_rel.suc r A, wo_rel.suc r B) \<in> r \<and> wo_rel.suc r A \<noteq> wo_rel.suc r B"
+ by simp
+ qed
+ ultimately show "wf (ofilterIncl r)" by (simp add: compat_wf)
+qed
+
+
+
+subsection {* Ordering the well-orders by existence of embeddings *}
+
+
+text {* We define three relations between well-orders:
+\begin{itemize}
+\item @{text "ordLeq"}, of being embedded (abbreviated @{text "\<le>o"});
+\item @{text "ordLess"}, of being strictly embedded (abbreviated @{text "<o"});
+\item @{text "ordIso"}, of being isomorphic (abbreviated @{text "=o"}).
+\end{itemize}
+%
+The prefix "ord" and the index "o" in these names stand for "ordinal-like".
+These relations shall be proved to be inter-connected in a similar fashion as the trio
+@{text "\<le>"}, @{text "<"}, @{text "="} associated to a total order on a set.
+*}
+
+
+definition ordLeq :: "('a rel * 'a' rel) set"
+where
+"ordLeq = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embed r r' f)}"
+
+
+abbreviation ordLeq2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<=o" 50)
+where "r <=o r' \<equiv> (r,r') \<in> ordLeq"
+
+
+abbreviation ordLeq3 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "\<le>o" 50)
+where "r \<le>o r' \<equiv> r <=o r'"
+
+
+definition ordLess :: "('a rel * 'a' rel) set"
+where
+"ordLess = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
+
+abbreviation ordLess2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "<o" 50)
+where "r <o r' \<equiv> (r,r') \<in> ordLess"
+
+
+definition ordIso :: "('a rel * 'a' rel) set"
+where
+"ordIso = {(r,r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. iso r r' f)}"
+
+abbreviation ordIso2 :: "'a rel \<Rightarrow> 'a' rel \<Rightarrow> bool" (infix "=o" 50)
+where "r =o r' \<equiv> (r,r') \<in> ordIso"
+
+
+lemmas ordRels_def = ordLeq_def ordLess_def ordIso_def
+
+lemma ordLeq_Well_order_simp:
+assumes "r \<le>o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLeq_def by simp
+
+
+lemma ordLess_Well_order_simp:
+assumes "r <o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordLess_def by simp
+
+
+lemma ordIso_Well_order_simp:
+assumes "r =o r'"
+shows "Well_order r \<and> Well_order r'"
+using assms unfolding ordIso_def by simp
+
+
+text{* Notice that the relations @{text "\<le>o"}, @{text "<o"}, @{text "=o"} connect well-orders
+on potentially {\em distinct} types. However, some of the lemmas below, including the next one,
+restrict implicitly the type of these relations to @{text "(('a rel) * ('a rel)) set"} , i.e.,
+to @{text "'a rel rel"}. *}
+
+
+lemma ordLeq_reflexive:
+"Well_order r \<Longrightarrow> r \<le>o r"
+unfolding ordLeq_def using id_embed[of r] by blast
+
+
+lemma ordLeq_transitive[trans]:
+assumes *: "r \<le>o r'" and **: "r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "embed r r' f" and "embed r' r'' f'"
+ using * ** unfolding ordLeq_def by blast
+ hence "embed r r'' (f' o f)"
+ using comp_embed[of r r' f r'' f'] by auto
+ thus "r \<le>o r''" unfolding ordLeq_def using 1 by auto
+qed
+
+
+lemma ordLeq_total:
+"\<lbrakk>Well_order r; Well_order r'\<rbrakk> \<Longrightarrow> r \<le>o r' \<or> r' \<le>o r"
+unfolding ordLeq_def using wellorders_totally_ordered by blast
+
+
+lemma ordIso_reflexive:
+"Well_order r \<Longrightarrow> r =o r"
+unfolding ordIso_def using id_iso[of r] by blast
+
+
+lemma ordIso_transitive[trans]:
+assumes *: "r =o r'" and **: "r' =o r''"
+shows "r =o r''"
+proof-
+ obtain f and f'
+ where 1: "Well_order r \<and> Well_order r' \<and> Well_order r''" and
+ "iso r r' f" and 3: "iso r' r'' f'"
+ using * ** unfolding ordIso_def by auto
+ hence "iso r r'' (f' o f)"
+ using comp_iso[of r r' f r'' f'] by auto
+ thus "r =o r''" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma ordIso_symmetric:
+assumes *: "r =o r'"
+shows "r' =o r"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ using * by (auto simp add: ordIso_def iso_def)
+ let ?f' = "inv_into (Field r) f"
+ have "embed r' r ?f' \<and> bij_betw ?f' (Field r') (Field r)"
+ using 1 2 by (simp add: bij_betw_inv_into inv_into_Field_embed_bij_betw)
+ thus "r' =o r" unfolding ordIso_def using 1 by (auto simp add: iso_def)
+qed
+
+
+lemma ordLeq_ordLess_trans[trans]:
+assumes "r \<le>o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embed_comp_embedS by blast
+qed
+
+
+lemma ordLess_ordLeq_trans[trans]:
+assumes "r <o r'" and " r' \<le>o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordLess_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordLess_def
+ using embedS_comp_embed by blast
+qed
+
+
+lemma ordLeq_ordIso_trans[trans]:
+assumes "r \<le>o r'" and " r' =o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using embed_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLeq_trans[trans]:
+assumes "r =o r'" and " r' \<le>o r''"
+shows "r \<le>o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLeq_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLeq_def ordIso_def
+ using iso_comp_embed by blast
+qed
+
+
+lemma ordLess_ordIso_trans[trans]:
+assumes "r <o r'" and " r' =o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using embedS_comp_iso by blast
+qed
+
+
+lemma ordIso_ordLess_trans[trans]:
+assumes "r =o r'" and " r' <o r''"
+shows "r <o r''"
+proof-
+ have "Well_order r \<and> Well_order r''"
+ using assms unfolding ordLess_def ordIso_def by auto
+ thus ?thesis using assms unfolding ordLess_def ordIso_def
+ using iso_comp_embedS by blast
+qed
+
+
+lemma ordLess_not_embed:
+assumes "r <o r'"
+shows "\<not>(\<exists>f'. embed r' r f')"
+proof-
+ obtain f where 1: "Well_order r \<and> Well_order r'" and 2: "embed r r' f" and
+ 3: " \<not> bij_betw f (Field r) (Field r')"
+ using assms unfolding ordLess_def by (auto simp add: embedS_def)
+ {fix f' assume *: "embed r' r f'"
+ hence "bij_betw f (Field r) (Field r')" using 1 2
+ by (simp add: embed_bothWays_Field_bij_betw)
+ with 3 have False by contradiction
+ }
+ thus ?thesis by blast
+qed
+
+
+lemma ordLess_Field:
+assumes OL: "r1 <o r2" and EMB: "embed r1 r2 f"
+shows "\<not> (f`(Field r1) = Field r2)"
+proof-
+ let ?A1 = "Field r1" let ?A2 = "Field r2"
+ obtain g where
+ 0: "Well_order r1 \<and> Well_order r2" and
+ 1: "embed r1 r2 g \<and> \<not>(bij_betw g ?A1 ?A2)"
+ using OL unfolding ordLess_def by (auto simp add: embedS_def)
+ hence "\<forall>a \<in> ?A1. f a = g a"
+ using 0 EMB embed_unique[of r1] by auto
+ hence "\<not>(bij_betw f ?A1 ?A2)"
+ using 1 bij_betw_cong[of ?A1] by blast
+ moreover
+ have "inj_on f ?A1" using EMB 0 by (simp add: embed_inj_on)
+ ultimately show ?thesis by (simp add: bij_betw_def)
+qed
+
+
+lemma ordLess_iff:
+"r <o r' = (Well_order r \<and> Well_order r' \<and> \<not>(\<exists>f'. embed r' r f'))"
+proof
+ assume *: "r <o r'"
+ hence "\<not>(\<exists>f'. embed r' r f')" using ordLess_not_embed[of r r'] by simp
+ with * show "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ unfolding ordLess_def by auto
+next
+ assume *: "Well_order r \<and> Well_order r' \<and> \<not> (\<exists>f'. embed r' r f')"
+ then obtain f where 1: "embed r r' f"
+ using wellorders_totally_ordered[of r r'] by blast
+ moreover
+ {assume "bij_betw f (Field r) (Field r')"
+ with * 1 have "embed r' r (inv_into (Field r) f) "
+ using inv_into_Field_embed_bij_betw[of r r' f] by auto
+ with * have False by blast
+ }
+ ultimately show "(r,r') \<in> ordLess"
+ unfolding ordLess_def using * by (fastforce simp add: embedS_def)
+qed
+
+
+lemma ordLess_irreflexive: "\<not> r <o r"
+proof
+ assume "r <o r"
+ hence "Well_order r \<and> \<not>(\<exists>f. embed r r f)"
+ unfolding ordLess_iff ..
+ moreover have "embed r r id" using id_embed[of r] .
+ ultimately show False by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_or_ordIso:
+"r \<le>o r' = (r <o r' \<or> r =o r')"
+unfolding ordRels_def embedS_defs iso_defs by blast
+
+
+lemma ordIso_iff_ordLeq:
+"(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
+proof
+ assume "r =o r'"
+ then obtain f where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> bij_betw f (Field r) (Field r')"
+ unfolding ordIso_def iso_defs by auto
+ hence "embed r r' f \<and> embed r' r (inv_into (Field r) f)"
+ by (simp add: inv_into_Field_embed_bij_betw)
+ thus "r \<le>o r' \<and> r' \<le>o r"
+ unfolding ordLeq_def using 1 by auto
+next
+ assume "r \<le>o r' \<and> r' \<le>o r"
+ then obtain f and g where 1: "Well_order r \<and> Well_order r' \<and>
+ embed r r' f \<and> embed r' r g"
+ unfolding ordLeq_def by auto
+ hence "iso r r' f" by (auto simp add: embed_bothWays_iso)
+ thus "r =o r'" unfolding ordIso_def using 1 by auto
+qed
+
+
+lemma not_ordLess_ordLeq:
+"r <o r' \<Longrightarrow> \<not> r' \<le>o r"
+using ordLess_ordLeq_trans ordLess_irreflexive by blast
+
+
+lemma ordLess_or_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "r <o r' \<or> r' \<le>o r"
+proof-
+ have "r \<le>o r' \<or> r' \<le>o r"
+ using assms by (simp add: ordLeq_total)
+ moreover
+ {assume "\<not> r <o r' \<and> r \<le>o r'"
+ hence "r =o r'" using ordLeq_iff_ordLess_or_ordIso by blast
+ hence "r' \<le>o r" using ordIso_symmetric ordIso_iff_ordLeq by blast
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma not_ordLess_ordIso:
+"r <o r' \<Longrightarrow> \<not> r =o r'"
+using assms ordLess_ordIso_trans ordIso_symmetric ordLess_irreflexive by blast
+
+
+lemma not_ordLeq_iff_ordLess:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' \<le>o r) = (r <o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma not_ordLess_iff_ordLeq:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<not> r' <o r) = (r \<le>o r')"
+using assms not_ordLess_ordLeq ordLess_or_ordLeq by blast
+
+
+lemma ordLess_transitive[trans]:
+"\<lbrakk>r <o r'; r' <o r''\<rbrakk> \<Longrightarrow> r <o r''"
+using assms ordLess_ordLeq_trans ordLeq_iff_ordLess_or_ordIso by blast
+
+
+corollary ordLess_trans: "trans ordLess"
+unfolding trans_def using ordLess_transitive by blast
+
+
+lemmas ordIso_equivalence = ordIso_transitive ordIso_reflexive ordIso_symmetric
+
+
+lemma ordIso_imp_ordLeq:
+"r =o r' \<Longrightarrow> r \<le>o r'"
+using ordIso_iff_ordLeq by blast
+
+
+lemma ordLess_imp_ordLeq:
+"r <o r' \<Longrightarrow> r \<le>o r'"
+using ordLeq_iff_ordLess_or_ordIso by blast
+
+
+lemma ofilter_subset_ordLeq:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A \<le> B) = (Restr r A \<le>o Restr r B)"
+proof
+ assume "A \<le> B"
+ thus "Restr r A \<le>o Restr r B"
+ unfolding ordLeq_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embed by blast
+next
+ assume *: "Restr r A \<le>o Restr r B"
+ then obtain f where "embed (Restr r A) (Restr r B) f"
+ unfolding ordLeq_def by blast
+ {assume "B < A"
+ hence "Restr r B <o Restr r A"
+ unfolding ordLess_def using assms
+ Well_order_Restr Well_order_Restr ofilter_subset_embedS by blast
+ hence False using * not_ordLess_ordLeq by blast
+ }
+ thus "A \<le> B" using OFA OFB WELL
+ wo_rel_def[of r] wo_rel.ofilter_linord[of r A B] by blast
+qed
+
+
+lemma ofilter_subset_ordLess:
+assumes WELL: "Well_order r" and
+ OFA: "wo_rel.ofilter r A" and OFB: "wo_rel.ofilter r B"
+shows "(A < B) = (Restr r A <o Restr r B)"
+proof-
+ let ?rA = "Restr r A" let ?rB = "Restr r B"
+ have 1: "Well_order ?rA \<and> Well_order ?rB"
+ using WELL Well_order_Restr by blast
+ have "(A < B) = (\<not> B \<le> A)" using assms
+ wo_rel_def wo_rel.ofilter_linord[of r A B] by blast
+ also have "\<dots> = (\<not> Restr r B \<le>o Restr r A)"
+ using assms ofilter_subset_ordLeq by blast
+ also have "\<dots> = (Restr r A <o Restr r B)"
+ using 1 not_ordLeq_iff_ordLess by blast
+ finally show ?thesis .
+qed
+
+
+lemma ofilter_ordLess:
+"\<lbrakk>Well_order r; wo_rel.ofilter r A\<rbrakk> \<Longrightarrow> (A < Field r) = (Restr r A <o r)"
+by (simp add: ofilter_subset_ordLess wo_rel.Field_ofilter
+ wo_rel_def Restr_Field)
+
+
+corollary underS_Restr_ordLess:
+assumes "Well_order r" and "Field r \<noteq> {}"
+shows "Restr r (rel.underS r a) <o r"
+proof-
+ have "rel.underS r a < Field r" using assms
+ by (simp add: rel.underS_Field3)
+ thus ?thesis using assms
+ by (simp add: ofilter_ordLess wo_rel.underS_ofilter wo_rel_def)
+qed
+
+
+lemma embed_ordLess_ofilterIncl:
+assumes
+ OL12: "r1 <o r2" and OL23: "r2 <o r3" and
+ EMB13: "embed r1 r3 f13" and EMB23: "embed r2 r3 f23"
+shows "(f13`(Field r1), f23`(Field r2)) \<in> (ofilterIncl r3)"
+proof-
+ have OL13: "r1 <o r3"
+ using OL12 OL23 using ordLess_transitive by auto
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A3 ="Field r3"
+ obtain f12 g23 where
+ 0: "Well_order r1 \<and> Well_order r2 \<and> Well_order r3" and
+ 1: "embed r1 r2 f12 \<and> \<not>(bij_betw f12 ?A1 ?A2)" and
+ 2: "embed r2 r3 g23 \<and> \<not>(bij_betw g23 ?A2 ?A3)"
+ using OL12 OL23 by (auto simp add: ordLess_def embedS_def)
+ hence "\<forall>a \<in> ?A2. f23 a = g23 a"
+ using EMB23 embed_unique[of r2 r3] by blast
+ hence 3: "\<not>(bij_betw f23 ?A2 ?A3)"
+ using 2 bij_betw_cong[of ?A2 f23 g23] by blast
+ (* *)
+ have 4: "wo_rel.ofilter r2 (f12 ` ?A1) \<and> f12 ` ?A1 \<noteq> ?A2"
+ using 0 1 OL12 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 5: "wo_rel.ofilter r3 (f23 ` ?A2) \<and> f23 ` ?A2 \<noteq> ?A3"
+ using 0 EMB23 OL23 by (simp add: embed_Field_ofilter ordLess_Field)
+ have 6: "wo_rel.ofilter r3 (f13 ` ?A1) \<and> f13 ` ?A1 \<noteq> ?A3"
+ using 0 EMB13 OL13 by (simp add: embed_Field_ofilter ordLess_Field)
+ (* *)
+ have "f12 ` ?A1 < ?A2"
+ using 0 4 by (auto simp add: wo_rel_def wo_rel.ofilter_def)
+ moreover have "inj_on f23 ?A2"
+ using EMB23 0 by (simp add: wo_rel_def embed_inj_on)
+ ultimately
+ have "f23 ` (f12 ` ?A1) < f23 ` ?A2" by (simp add: inj_on_strict_subset)
+ moreover
+ {have "embed r1 r3 (f23 o f12)"
+ using 1 EMB23 0 by (auto simp add: comp_embed)
+ hence "\<forall>a \<in> ?A1. f23(f12 a) = f13 a"
+ using EMB13 0 embed_unique[of r1 r3 "f23 o f12" f13] by auto
+ hence "f23 ` (f12 ` ?A1) = f13 ` ?A1" by force
+ }
+ ultimately
+ have "f13 ` ?A1 < f23 ` ?A2" by simp
+ (* *)
+ with 5 6 show ?thesis
+ unfolding ofilterIncl_def by auto
+qed
+
+
+lemma ordLess_iff_ordIso_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r' <o r) = (\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a))"
+proof(auto)
+ fix a assume *: "a \<in> Field r" and **: "r' =o Restr r (rel.underS r a)"
+ hence "Restr r (rel.underS r a) <o r" using WELL underS_Restr_ordLess[of r] by blast
+ thus "r' <o r" using ** ordIso_ordLess_trans by blast
+next
+ assume "r' <o r"
+ then obtain f where 1: "Well_order r \<and> Well_order r'" and
+ 2: "embed r' r f \<and> f ` (Field r') \<noteq> Field r"
+ unfolding ordLess_def embedS_def[abs_def] bij_betw_def using embed_inj_on by blast
+ hence "wo_rel.ofilter r (f ` (Field r'))" using embed_Field_ofilter by blast
+ then obtain a where 3: "a \<in> Field r" and 4: "rel.underS r a = f ` (Field r')"
+ using 1 2 by (auto simp add: wo_rel.ofilter_underS_Field wo_rel_def)
+ have "iso r' (Restr r (f ` (Field r'))) f"
+ using embed_implies_iso_Restr 2 assms by blast
+ moreover have "Well_order (Restr r (f ` (Field r')))"
+ using WELL Well_order_Restr by blast
+ ultimately have "r' =o Restr r (f ` (Field r'))"
+ using WELL' unfolding ordIso_def by auto
+ hence "r' =o Restr r (rel.underS r a)" using 4 by auto
+ thus "\<exists>a \<in> Field r. r' =o Restr r (rel.underS r a)" using 3 by auto
+qed
+
+
+lemma internalize_ordLess:
+"(r' <o r) = (\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r)"
+proof
+ assume *: "r' <o r"
+ hence 0: "Well_order r \<and> Well_order r'" unfolding ordLess_def by auto
+ with * obtain a where 1: "a \<in> Field r" and 2: "r' =o Restr r (rel.underS r a)"
+ using ordLess_iff_ordIso_Restr by blast
+ let ?p = "Restr r (rel.underS r a)"
+ have "wo_rel.ofilter r (rel.underS r a)" using 0
+ by (simp add: wo_rel_def wo_rel.underS_ofilter)
+ hence "Field ?p = rel.underS r a" using 0 Field_Restr_ofilter by blast
+ hence "Field ?p < Field r" using rel.underS_Field2 1 by fastforce
+ moreover have "?p <o r" using underS_Restr_ordLess[of r a] 0 1 by blast
+ ultimately
+ show "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r" using 2 by blast
+next
+ assume "\<exists>p. Field p < Field r \<and> r' =o p \<and> p <o r"
+ thus "r' <o r" using ordIso_ordLess_trans by blast
+qed
+
+
+lemma internalize_ordLeq:
+"(r' \<le>o r) = (\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r)"
+proof
+ assume *: "r' \<le>o r"
+ moreover
+ {assume "r' <o r"
+ then obtain p where "Field p < Field r \<and> r' =o p \<and> p <o r"
+ using internalize_ordLess[of r' r] by blast
+ hence "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+ }
+ moreover
+ have "r \<le>o r" using * ordLeq_def ordLeq_reflexive by blast
+ ultimately show "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ using ordLeq_iff_ordLess_or_ordIso by blast
+next
+ assume "\<exists>p. Field p \<le> Field r \<and> r' =o p \<and> p \<le>o r"
+ thus "r' \<le>o r" using ordIso_ordLeq_trans by blast
+qed
+
+
+lemma ordLeq_iff_ordLess_Restr:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(r \<le>o r') = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r')"
+proof(auto)
+ assume *: "r \<le>o r'"
+ fix a assume "a \<in> Field r"
+ hence "Restr r (rel.underS r a) <o r"
+ using WELL underS_Restr_ordLess[of r] by blast
+ thus "Restr r (rel.underS r a) <o r'"
+ using * ordLess_ordLeq_trans by blast
+next
+ assume *: "\<forall>a \<in> Field r. Restr r (rel.underS r a) <o r'"
+ {assume "r' <o r"
+ then obtain a where "a \<in> Field r \<and> r' =o Restr r (rel.underS r a)"
+ using assms ordLess_iff_ordIso_Restr by blast
+ hence False using * not_ordLess_ordIso ordIso_symmetric by blast
+ }
+ thus "r \<le>o r'" using ordLess_or_ordLeq assms by blast
+qed
+
+
+lemma finite_ordLess_infinite:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and
+ FIN: "finite(Field r)" and INF: "infinite(Field r')"
+shows "r <o r'"
+proof-
+ {assume "r' \<le>o r"
+ then obtain h where "inj_on h (Field r') \<and> h ` (Field r') \<le> Field r"
+ unfolding ordLeq_def using assms embed_inj_on embed_Field by blast
+ hence False using finite_imageD finite_subset FIN INF by metis
+ }
+ thus ?thesis using WELL WELL' ordLess_or_ordLeq by blast
+qed
+
+
+lemma finite_well_order_on_ordIso:
+assumes FIN: "finite A" and
+ WELL: "well_order_on A r" and WELL': "well_order_on A r'"
+shows "r =o r'"
+proof-
+ have 0: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using assms rel.well_order_on_Well_order by blast
+ moreover
+ have "\<forall>r r'. well_order_on A r \<and> well_order_on A r' \<and> r \<le>o r'
+ \<longrightarrow> r =o r'"
+ proof(clarify)
+ fix r r' assume *: "well_order_on A r" and **: "well_order_on A r'"
+ have 2: "Well_order r \<and> Well_order r' \<and> Field r = A \<and> Field r' = A"
+ using * ** rel.well_order_on_Well_order by blast
+ assume "r \<le>o r'"
+ then obtain f where 1: "embed r r' f" and
+ "inj_on f A \<and> f ` A \<le> A"
+ unfolding ordLeq_def using 2 embed_inj_on embed_Field by blast
+ hence "bij_betw f A A" unfolding bij_betw_def using FIN endo_inj_surj by blast
+ thus "r =o r'" unfolding ordIso_def iso_def[abs_def] using 1 2 by auto
+ qed
+ ultimately show ?thesis using assms ordLeq_total ordIso_symmetric by blast
+qed
+
+
+subsection{* @{text "<o"} is well-founded *}
+
+
+text {* Of course, it only makes sense to state that the @{text "<o"} is well-founded
+on the restricted type @{text "'a rel rel"}. We prove this by first showing that, for any set
+of well-orders all embedded in a fixed well-order, the function mapping each well-order
+in the set to an order filter of the fixed well-order is compatible w.r.t. to @{text "<o"} versus
+{\em strict inclusion}; and we already know that strict inclusion of order filters is well-founded. *}
+
+
+definition ord_to_filter :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a set"
+where "ord_to_filter r0 r \<equiv> (SOME f. embed r r0 f) ` (Field r)"
+
+
+lemma ord_to_filter_compat:
+"compat (ordLess Int (ordLess^-1``{r0} \<times> ordLess^-1``{r0}))
+ (ofilterIncl r0)
+ (ord_to_filter r0)"
+proof(unfold compat_def ord_to_filter_def, clarify)
+ fix r1::"'a rel" and r2::"'a rel"
+ let ?A1 = "Field r1" let ?A2 ="Field r2" let ?A0 ="Field r0"
+ let ?phi10 = "\<lambda> f10. embed r1 r0 f10" let ?f10 = "SOME f. ?phi10 f"
+ let ?phi20 = "\<lambda> f20. embed r2 r0 f20" let ?f20 = "SOME f. ?phi20 f"
+ assume *: "r1 <o r0" "r2 <o r0" and **: "r1 <o r2"
+ hence "(\<exists>f. ?phi10 f) \<and> (\<exists>f. ?phi20 f)"
+ by (auto simp add: ordLess_def embedS_def)
+ hence "?phi10 ?f10 \<and> ?phi20 ?f20" by (auto simp add: someI_ex)
+ thus "(?f10 ` ?A1, ?f20 ` ?A2) \<in> ofilterIncl r0"
+ using * ** by (simp add: embed_ordLess_ofilterIncl)
+qed
+
+
+theorem wf_ordLess: "wf ordLess"
+proof-
+ {fix r0 :: "('a \<times> 'a) set"
+ (* need to annotate here!*)
+ let ?ordLess = "ordLess::('d rel * 'd rel) set"
+ let ?R = "?ordLess Int (?ordLess^-1``{r0} \<times> ?ordLess^-1``{r0})"
+ {assume Case1: "Well_order r0"
+ hence "wf ?R"
+ using wf_ofilterIncl[of r0]
+ compat_wf[of ?R "ofilterIncl r0" "ord_to_filter r0"]
+ ord_to_filter_compat[of r0] by auto
+ }
+ moreover
+ {assume Case2: "\<not> Well_order r0"
+ hence "?R = {}" unfolding ordLess_def by auto
+ hence "wf ?R" using wf_empty by simp
+ }
+ ultimately have "wf ?R" by blast
+ }
+ thus ?thesis by (simp add: trans_wf_iff ordLess_trans)
+qed
+
+corollary exists_minim_Well_order:
+assumes NE: "R \<noteq> {}" and WELL: "\<forall>r \<in> R. Well_order r"
+shows "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
+proof-
+ obtain r where "r \<in> R \<and> (\<forall>r' \<in> R. \<not> r' <o r)"
+ using NE spec[OF spec[OF subst[OF wf_eq_minimal, of "%x. x", OF wf_ordLess]], of _ R]
+ equals0I[of R] by blast
+ with not_ordLeq_iff_ordLess WELL show ?thesis by blast
+qed
+
+
+
+subsection {* Copy via direct images *}
+
+
+text{* The direct image operator is the dual of the inverse image operator @{text "inv_image"}
+from @{text "Relation.thy"}. It is useful for transporting a well-order between
+different types. *}
+
+
+definition dir_image :: "'a rel \<Rightarrow> ('a \<Rightarrow> 'a') \<Rightarrow> 'a' rel"
+where
+"dir_image r f = {(f a, f b)| a b. (a,b) \<in> r}"
+
+
+lemma dir_image_Field:
+"Field(dir_image r f) \<le> f ` (Field r)"
+unfolding dir_image_def Field_def by auto
+
+
+lemma dir_image_minus_Id:
+"inj_on f (Field r) \<Longrightarrow> (dir_image r f) - Id = dir_image (r - Id) f"
+unfolding inj_on_def Field_def dir_image_def by auto
+
+
+lemma Refl_dir_image:
+assumes "Refl r"
+shows "Refl(dir_image r f)"
+proof-
+ {fix a' b'
+ assume "(a',b') \<in> dir_image r f"
+ then obtain a b where 1: "a' = f a \<and> b' = f b \<and> (a,b) \<in> r"
+ unfolding dir_image_def by blast
+ hence "a \<in> Field r \<and> b \<in> Field r" using Field_def by fastforce
+ hence "(a,a) \<in> r \<and> (b,b) \<in> r" using assms by (simp add: refl_on_def)
+ with 1 have "(a',a') \<in> dir_image r f \<and> (b',b') \<in> dir_image r f"
+ unfolding dir_image_def by auto
+ }
+ thus ?thesis
+ by(unfold refl_on_def Field_def Domain_def Range_def, auto)
+qed
+
+
+lemma trans_dir_image:
+assumes TRANS: "trans r" and INJ: "inj_on f (Field r)"
+shows "trans(dir_image r f)"
+proof(unfold trans_def, auto)
+ fix a' b' c'
+ assume "(a',b') \<in> dir_image r f" "(b',c') \<in> dir_image r f"
+ then obtain a b1 b2 c where 1: "a' = f a \<and> b' = f b1 \<and> b' = f b2 \<and> c' = f c" and
+ 2: "(a,b1) \<in> r \<and> (b2,c) \<in> r"
+ unfolding dir_image_def by blast
+ hence "b1 \<in> Field r \<and> b2 \<in> Field r"
+ unfolding Field_def by auto
+ hence "b1 = b2" using 1 INJ unfolding inj_on_def by auto
+ hence "(a,c): r" using 2 TRANS unfolding trans_def by blast
+ thus "(a',c') \<in> dir_image r f"
+ unfolding dir_image_def using 1 by auto
+qed
+
+
+lemma Preorder_dir_image:
+"\<lbrakk>Preorder r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Preorder (dir_image r f)"
+by (simp add: preorder_on_def Refl_dir_image trans_dir_image)
+
+
+lemma antisym_dir_image:
+assumes AN: "antisym r" and INJ: "inj_on f (Field r)"
+shows "antisym(dir_image r f)"
+proof(unfold antisym_def, auto)
+ fix a' b'
+ assume "(a',b') \<in> dir_image r f" "(b',a') \<in> dir_image r f"
+ then obtain a1 b1 a2 b2 where 1: "a' = f a1 \<and> a' = f a2 \<and> b' = f b1 \<and> b' = f b2" and
+ 2: "(a1,b1) \<in> r \<and> (b2,a2) \<in> r " and
+ 3: "{a1,a2,b1,b2} \<le> Field r"
+ unfolding dir_image_def Field_def by blast
+ hence "a1 = a2 \<and> b1 = b2" using INJ unfolding inj_on_def by auto
+ hence "a1 = b2" using 2 AN unfolding antisym_def by auto
+ thus "a' = b'" using 1 by auto
+qed
+
+
+lemma Partial_order_dir_image:
+"\<lbrakk>Partial_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Partial_order (dir_image r f)"
+by (simp add: partial_order_on_def Preorder_dir_image antisym_dir_image)
+
+
+lemma Total_dir_image:
+assumes TOT: "Total r" and INJ: "inj_on f (Field r)"
+shows "Total(dir_image r f)"
+proof(unfold total_on_def, intro ballI impI)
+ fix a' b'
+ assume "a' \<in> Field (dir_image r f)" "b' \<in> Field (dir_image r f)"
+ then obtain a and b where 1: "a \<in> Field r \<and> b \<in> Field r \<and> f a = a' \<and> f b = b'"
+ using dir_image_Field[of r f] by blast
+ moreover assume "a' \<noteq> b'"
+ ultimately have "a \<noteq> b" using INJ unfolding inj_on_def by auto
+ hence "(a,b) \<in> r \<or> (b,a) \<in> r" using 1 TOT unfolding total_on_def by auto
+ thus "(a',b') \<in> dir_image r f \<or> (b',a') \<in> dir_image r f"
+ using 1 unfolding dir_image_def by auto
+qed
+
+
+lemma Linear_order_dir_image:
+"\<lbrakk>Linear_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Linear_order (dir_image r f)"
+by (simp add: linear_order_on_def Partial_order_dir_image Total_dir_image)
+
+
+lemma wf_dir_image:
+assumes WF: "wf r" and INJ: "inj_on f (Field r)"
+shows "wf(dir_image r f)"
+proof(unfold wf_eq_minimal2, intro allI impI, elim conjE)
+ fix A'::"'b set"
+ assume SUB: "A' \<le> Field(dir_image r f)" and NE: "A' \<noteq> {}"
+ obtain A where A_def: "A = {a \<in> Field r. f a \<in> A'}" by blast
+ have "A \<noteq> {} \<and> A \<le> Field r"
+ using A_def dir_image_Field[of r f] SUB NE by blast
+ then obtain a where 1: "a \<in> A \<and> (\<forall>b \<in> A. (b,a) \<notin> r)"
+ using WF unfolding wf_eq_minimal2 by metis
+ have "\<forall>b' \<in> A'. (b',f a) \<notin> dir_image r f"
+ proof(clarify)
+ fix b' assume *: "b' \<in> A'" and **: "(b',f a) \<in> dir_image r f"
+ obtain b1 a1 where 2: "b' = f b1 \<and> f a = f a1" and
+ 3: "(b1,a1) \<in> r \<and> {a1,b1} \<le> Field r"
+ using ** unfolding dir_image_def Field_def by blast
+ hence "a = a1" using 1 A_def INJ unfolding inj_on_def by auto
+ hence "b1 \<in> A \<and> (b1,a) \<in> r" using 2 3 A_def * by auto
+ with 1 show False by auto
+ qed
+ thus "\<exists>a'\<in>A'. \<forall>b'\<in>A'. (b', a') \<notin> dir_image r f"
+ using A_def 1 by blast
+qed
+
+
+lemma Well_order_dir_image:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> Well_order (dir_image r f)"
+using assms unfolding well_order_on_def
+using Linear_order_dir_image[of r f] wf_dir_image[of "r - Id" f]
+ dir_image_minus_Id[of f r]
+ subset_inj_on[of f "Field r" "Field(r - Id)"]
+ mono_Field[of "r - Id" r] by auto
+
+
+lemma dir_image_Field2:
+"Refl r \<Longrightarrow> Field(dir_image r f) = f ` (Field r)"
+unfolding Field_def dir_image_def refl_on_def Domain_def Range_def by blast
+
+
+lemma dir_image_bij_betw:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> bij_betw f (Field r) (Field (dir_image r f))"
+unfolding bij_betw_def
+by (simp add: dir_image_Field2 order_on_defs)
+
+
+lemma dir_image_compat:
+"compat r (dir_image r f) f"
+unfolding compat_def dir_image_def by auto
+
+
+lemma dir_image_iso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> iso r (dir_image r f) f"
+using iso_iff3 dir_image_compat dir_image_bij_betw Well_order_dir_image by blast
+
+
+lemma dir_image_ordIso:
+"\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+unfolding ordIso_def using dir_image_iso Well_order_dir_image by blast
+
+
+lemma Well_order_iso_copy:
+assumes WELL: "well_order_on A r" and BIJ: "bij_betw f A A'"
+shows "\<exists>r'. well_order_on A' r' \<and> r =o r'"
+proof-
+ let ?r' = "dir_image r f"
+ have 1: "A = Field r \<and> Well_order r"
+ using WELL rel.well_order_on_Well_order by blast
+ hence 2: "iso r ?r' f"
+ using dir_image_iso using BIJ unfolding bij_betw_def by auto
+ hence "f ` (Field r) = Field ?r'" using 1 iso_iff[of r ?r'] by blast
+ hence "Field ?r' = A'"
+ using 1 BIJ unfolding bij_betw_def by auto
+ moreover have "Well_order ?r'"
+ using 1 Well_order_dir_image BIJ unfolding bij_betw_def by blast
+ ultimately show ?thesis unfolding ordIso_def using 1 2 by blast
+qed
+
+
+
+subsection {* Bounded square *}
+
+
+text{* This construction essentially defines, for an order relation @{text "r"}, a lexicographic
+order @{text "bsqr r"} on @{text "(Field r) \<times> (Field r)"}, applying the
+following criteria (in this order):
+\begin{itemize}
+\item compare the maximums;
+\item compare the first components;
+\item compare the second components.
+\end{itemize}
+%
+The only application of this construction that we are aware of is
+at proving that the square of an infinite set has the same cardinal
+as that set. The essential property required there (and which is ensured by this
+construction) is that any proper order filter of the product order is included in a rectangle, i.e.,
+in a product of proper filters on the original relation (assumed to be a well-order). *}
+
+
+definition bsqr :: "'a rel => ('a * 'a)rel"
+where
+"bsqr r = {((a1,a2),(b1,b2)).
+ {a1,a2,b1,b2} \<le> Field r \<and>
+ (a1 = b1 \<and> a2 = b2 \<or>
+ (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id
+ )}"
+
+
+lemma Field_bsqr:
+"Field (bsqr r) = Field r \<times> Field r"
+proof
+ show "Field (bsqr r) \<le> Field r \<times> Field r"
+ proof-
+ {fix a1 a2 assume "(a1,a2) \<in> Field (bsqr r)"
+ moreover
+ have "\<And> b1 b2. ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r \<Longrightarrow>
+ a1 \<in> Field r \<and> a2 \<in> Field r" unfolding bsqr_def by auto
+ ultimately have "a1 \<in> Field r \<and> a2 \<in> Field r" unfolding Field_def by auto
+ }
+ thus ?thesis unfolding Field_def by force
+ qed
+next
+ show "Field r \<times> Field r \<le> Field (bsqr r)"
+ proof(auto)
+ fix a1 a2 assume "a1 \<in> Field r" and "a2 \<in> Field r"
+ hence "((a1,a2),(a1,a2)) \<in> bsqr r" unfolding bsqr_def by blast
+ thus "(a1,a2) \<in> Field (bsqr r)" unfolding Field_def by auto
+ qed
+qed
+
+
+lemma bsqr_Refl: "Refl(bsqr r)"
+by(unfold refl_on_def Field_bsqr, auto simp add: bsqr_def)
+
+
+lemma bsqr_Trans:
+assumes "Well_order r"
+shows "trans (bsqr r)"
+proof(unfold trans_def, auto)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ (* Main proof *)
+ fix a1 a2 b1 b2 c1 c2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(c1,c2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2,c1,c2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = c1 \<and> b2 = c2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "((a1,a2),(c1,c2)) \<in> bsqr r"
+ proof-
+ {assume Case1: "a1 = b1 \<and> a2 = b2"
+ hence ?thesis using ** by simp
+ }
+ moreover
+ {assume Case2: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case21: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case22: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence "(wo_rel.max2 r a1 a2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ using Case2 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case23_4: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2"
+ hence ?thesis using Case2 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case31: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case32: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence "(a1,c1) \<in> r - Id"
+ using Case3 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case3 Case33 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case33: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1"
+ hence ?thesis using Case3 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case4: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ {assume Case41: "b1 = c1 \<and> b2 = c2"
+ hence ?thesis using * by simp
+ }
+ moreover
+ {assume Case42: "(wo_rel.max2 r b1 b2, wo_rel.max2 r c1 c2) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by force
+ }
+ moreover
+ {assume Case43: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> (b1,c1) \<in> r - Id"
+ hence ?thesis using Case4 0 unfolding bsqr_def by auto
+ }
+ moreover
+ {assume Case44: "wo_rel.max2 r b1 b2 = wo_rel.max2 r c1 c2 \<and> b1 = c1 \<and> (b2,c2) \<in> r - Id"
+ hence "(a2,c2) \<in> r - Id"
+ using Case4 TransS trans_def[of "r - Id"] by blast
+ hence ?thesis using Case4 Case44 0 unfolding bsqr_def by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by auto
+ qed
+qed
+
+
+lemma bsqr_antisym:
+assumes "Well_order r"
+shows "antisym (bsqr r)"
+proof(unfold antisym_def, clarify)
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Trans: "trans r" using wo_rel.TRANS by auto
+ have Anti: "antisym r" using wo_rel.ANTISYM Well by auto
+ hence TransS: "trans(r - Id)" using Trans by (simp add: trans_diff_Id)
+ hence IrrS: "\<forall>a b. \<not>((a,b) \<in> r - Id \<and> (b,a) \<in> r - Id)"
+ using Anti trans_def[of "r - Id"] antisym_def[of "r - Id"] by blast
+ (* Main proof *)
+ fix a1 a2 b1 b2
+ assume *: "((a1,a2),(b1,b2)) \<in> bsqr r" and **: "((b1,b2),(a1,a2)) \<in> bsqr r"
+ hence 0: "{a1,a2,b1,b2} \<le> Field r" unfolding bsqr_def by auto
+ have 1: "a1 = b1 \<and> a2 = b2 \<or> (wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id \<or>
+ wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ using * unfolding bsqr_def by auto
+ have 2: "b1 = a1 \<and> b2 = a2 \<or> (wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> (b1,a1) \<in> r - Id \<or>
+ wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2 \<and> b1 = a1 \<and> (b2,a2) \<in> r - Id"
+ using ** unfolding bsqr_def by auto
+ show "a1 = b1 \<and> a2 = b2"
+ proof-
+ {assume Case1: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r - Id"
+ {assume Case11: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case1 IrrS by blast
+ }
+ moreover
+ {assume Case12_3: "wo_rel.max2 r b1 b2 = wo_rel.max2 r a1 a2"
+ hence False using Case1 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> (a1,b1) \<in> r - Id"
+ {assume Case21: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case2 by auto
+ }
+ moreover
+ {assume Case22: "(b1,a1) \<in> r - Id"
+ hence False using Case2 IrrS by blast
+ }
+ moreover
+ {assume Case23: "b1 = a1"
+ hence False using Case2 by auto
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ moreover
+ {assume Case3: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2 \<and> a1 = b1 \<and> (a2,b2) \<in> r - Id"
+ moreover
+ {assume Case31: "(wo_rel.max2 r b1 b2, wo_rel.max2 r a1 a2) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case32: "(b1,a1) \<in> r - Id"
+ hence False using Case3 by auto
+ }
+ moreover
+ {assume Case33: "(b2,a2) \<in> r - Id"
+ hence False using Case3 IrrS by blast
+ }
+ ultimately have ?thesis using 0 2 by auto
+ }
+ ultimately show ?thesis using 0 1 by blast
+ qed
+qed
+
+
+lemma bsqr_Total:
+assumes "Well_order r"
+shows "Total(bsqr r)"
+proof-
+ (* Preliminary facts *)
+ have Well: "wo_rel r" using assms wo_rel_def by auto
+ hence Total: "\<forall>a \<in> Field r. \<forall>b \<in> Field r. (a,b) \<in> r \<or> (b,a) \<in> r"
+ using wo_rel.TOTALS by auto
+ (* Main proof *)
+ {fix a1 a2 b1 b2 assume "{(a1,a2), (b1,b2)} \<le> Field(bsqr r)"
+ hence 0: "a1 \<in> Field r \<and> a2 \<in> Field r \<and> b1 \<in> Field r \<and> b2 \<in> Field r"
+ using Field_bsqr by blast
+ have "((a1,a2) = (b1,b2) \<or> ((a1,a2),(b1,b2)) \<in> bsqr r \<or> ((b1,b2),(a1,a2)) \<in> bsqr r)"
+ proof(rule wo_rel.cases_Total[of r a1 a2], clarsimp simp add: Well, simp add: 0)
+ (* Why didn't clarsimp simp add: Well 0 do the same job? *)
+ assume Case1: "(a1,a2) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case11: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2"
+ using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case111: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case112: "a2 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case112 unfolding bsqr_def by auto
+ next
+ assume Case1122: "a1 = b1"
+ thus ?thesis using Case112 by auto
+ qed
+ qed
+ next
+ assume Case12: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case121: "(a2,b1) \<in> r - Id \<or> (b1,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case122: "a2 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case1221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 unfolding bsqr_def by auto
+ next
+ assume Case1222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case12221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case122 Case1222 unfolding bsqr_def by auto
+ next
+ assume Case12222: "a2 = b2"
+ thus ?thesis using Case122 Case1222 by auto
+ qed
+ qed
+ qed
+ qed
+ next
+ assume Case2: "(a2,a1) \<in> r"
+ hence 1: "wo_rel.max2 r a1 a2 = a1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total[of r b1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21: "(b1,b2) \<in> r"
+ hence 2: "wo_rel.max2 r b1 b2 = b2" using Well 0 by (simp add: wo_rel.max2_equals2)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case211: "(a1,b2) \<in> r - Id \<or> (b2,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 unfolding bsqr_def by auto
+ next
+ assume Case212: "a1 = b2"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case2121: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 unfolding bsqr_def by auto
+ next
+ assume Case2122: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case21221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 2 Case212 Case2122 unfolding bsqr_def by auto
+ next
+ assume Case21222: "a2 = b2"
+ thus ?thesis using Case2122 Case212 by auto
+ qed
+ qed
+ qed
+ next
+ assume Case22: "(b2,b1) \<in> r"
+ hence 3: "wo_rel.max2 r b1 b2 = b1" using Well 0 by (simp add: wo_rel.max2_equals1)
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a1 b1], clarsimp simp add: Well, simp add: 0)
+ assume Case221: "(a1,b1) \<in> r - Id \<or> (b1,a1) \<in> r - Id"
+ thus ?thesis using 0 1 3 unfolding bsqr_def by auto
+ next
+ assume Case222: "a1 = b1"
+ show ?thesis
+ proof(rule wo_rel.cases_Total3[of r a2 b2], clarsimp simp add: Well, simp add: 0)
+ assume Case2221: "(a2,b2) \<in> r - Id \<or> (b2,a2) \<in> r - Id"
+ thus ?thesis using 0 1 3 Case222 unfolding bsqr_def by auto
+ next
+ assume Case2222: "a2 = b2"
+ thus ?thesis using Case222 by auto
+ qed
+ qed
+ qed
+ qed
+ }
+ thus ?thesis unfolding total_on_def by fast
+qed
+
+
+lemma bsqr_Linear_order:
+assumes "Well_order r"
+shows "Linear_order(bsqr r)"
+unfolding order_on_defs
+using assms bsqr_Refl bsqr_Trans bsqr_antisym bsqr_Total by blast
+
+
+lemma bsqr_Well_order:
+assumes "Well_order r"
+shows "Well_order(bsqr r)"
+using assms
+proof(simp add: bsqr_Linear_order Linear_order_Well_order_iff, intro allI impI)
+ have 0: "\<forall>A \<le> Field r. A \<noteq> {} \<longrightarrow> (\<exists>a \<in> A. \<forall>a' \<in> A. (a,a') \<in> r)"
+ using assms well_order_on_def Linear_order_Well_order_iff by blast
+ fix D assume *: "D \<le> Field (bsqr r)" and **: "D \<noteq> {}"
+ hence 1: "D \<le> Field r \<times> Field r" unfolding Field_bsqr by simp
+ (* *)
+ obtain M where M_def: "M = {wo_rel.max2 r a1 a2| a1 a2. (a1,a2) \<in> D}" by blast
+ have "M \<noteq> {}" using 1 M_def ** by auto
+ moreover
+ have "M \<le> Field r" unfolding M_def
+ using 1 assms wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ ultimately obtain m where m_min: "m \<in> M \<and> (\<forall>a \<in> M. (m,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A1 where A1_def: "A1 = {a1. \<exists>a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A1 \<le> Field r" unfolding A1_def using 1 by auto
+ moreover have "A1 \<noteq> {}" unfolding A1_def using m_min unfolding M_def by blast
+ ultimately obtain a1 where a1_min: "a1 \<in> A1 \<and> (\<forall>a \<in> A1. (a1,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ obtain A2 where A2_def: "A2 = {a2. (a1,a2) \<in> D \<and> wo_rel.max2 r a1 a2 = m}" by blast
+ have "A2 \<le> Field r" unfolding A2_def using 1 by auto
+ moreover have "A2 \<noteq> {}" unfolding A2_def
+ using m_min a1_min unfolding A1_def M_def by blast
+ ultimately obtain a2 where a2_min: "a2 \<in> A2 \<and> (\<forall>a \<in> A2. (a2,a) \<in> r)"
+ using 0 by blast
+ (* *)
+ have 2: "wo_rel.max2 r a1 a2 = m"
+ using a1_min a2_min unfolding A1_def A2_def by auto
+ have 3: "(a1,a2) \<in> D" using a2_min unfolding A2_def by auto
+ (* *)
+ moreover
+ {fix b1 b2 assume ***: "(b1,b2) \<in> D"
+ hence 4: "{a1,a2,b1,b2} \<le> Field r" using 1 3 by blast
+ have 5: "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+ using *** a1_min a2_min m_min unfolding A1_def A2_def M_def by auto
+ have "((a1,a2),(b1,b2)) \<in> bsqr r"
+ proof(cases "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2")
+ assume Case1: "wo_rel.max2 r a1 a2 \<noteq> wo_rel.max2 r b1 b2"
+ thus ?thesis unfolding bsqr_def using 4 5 by auto
+ next
+ assume Case2: "wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ hence "b1 \<in> A1" unfolding A1_def using 2 *** by auto
+ hence 6: "(a1,b1) \<in> r" using a1_min by auto
+ show ?thesis
+ proof(cases "a1 = b1")
+ assume Case21: "a1 \<noteq> b1"
+ thus ?thesis unfolding bsqr_def using 4 Case2 6 by auto
+ next
+ assume Case22: "a1 = b1"
+ hence "b2 \<in> A2" unfolding A2_def using 2 *** Case2 by auto
+ hence 7: "(a2,b2) \<in> r" using a2_min by auto
+ thus ?thesis unfolding bsqr_def using 4 7 Case2 Case22 by auto
+ qed
+ qed
+ }
+ (* *)
+ ultimately show "\<exists>d \<in> D. \<forall>d' \<in> D. (d,d') \<in> bsqr r" by fastforce
+qed
+
+
+lemma bsqr_max2:
+assumes WELL: "Well_order r" and LEQ: "((a1,a2),(b1,b2)) \<in> bsqr r"
+shows "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r"
+proof-
+ have "{(a1,a2),(b1,b2)} \<le> Field(bsqr r)"
+ using LEQ unfolding Field_def by auto
+ hence "{a1,a2,b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "{wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2} \<le> Field r"
+ using WELL wo_rel_def[of r] wo_rel.max2_among[of r] by fastforce
+ moreover have "(wo_rel.max2 r a1 a2, wo_rel.max2 r b1 b2) \<in> r \<or> wo_rel.max2 r a1 a2 = wo_rel.max2 r b1 b2"
+ using LEQ unfolding bsqr_def by auto
+ ultimately show ?thesis using WELL unfolding order_on_defs refl_on_def by auto
+qed
+
+
+lemma bsqr_ofilter:
+assumes WELL: "Well_order r" and
+ OF: "wo_rel.ofilter (bsqr r) D" and SUB: "D < Field r \<times> Field r" and
+ NE: "\<not> (\<exists>a. Field r = rel.under r a)"
+shows "\<exists>A. wo_rel.ofilter r A \<and> A < Field r \<and> D \<le> A \<times> A"
+proof-
+ let ?r' = "bsqr r"
+ have Well: "wo_rel r" using WELL wo_rel_def by blast
+ hence Trans: "trans r" using wo_rel.TRANS by blast
+ have Well': "Well_order ?r' \<and> wo_rel ?r'"
+ using WELL bsqr_Well_order wo_rel_def by blast
+ (* *)
+ have "D < Field ?r'" unfolding Field_bsqr using SUB .
+ with OF obtain a1 and a2 where
+ "(a1,a2) \<in> Field ?r'" and 1: "D = rel.underS ?r' (a1,a2)"
+ using Well' wo_rel.ofilter_underS_Field[of ?r' D] by auto
+ hence 2: "{a1,a2} \<le> Field r" unfolding Field_bsqr by auto
+ let ?m = "wo_rel.max2 r a1 a2"
+ have "D \<le> (rel.under r ?m) \<times> (rel.under r ?m)"
+ proof(unfold 1)
+ {fix b1 b2
+ let ?n = "wo_rel.max2 r b1 b2"
+ assume "(b1,b2) \<in> rel.underS ?r' (a1,a2)"
+ hence 3: "((b1,b2),(a1,a2)) \<in> ?r'"
+ unfolding rel.underS_def by blast
+ hence "(?n,?m) \<in> r" using WELL by (simp add: bsqr_max2)
+ moreover
+ {have "(b1,b2) \<in> Field ?r'" using 3 unfolding Field_def by auto
+ hence "{b1,b2} \<le> Field r" unfolding Field_bsqr by auto
+ hence "(b1,?n) \<in> r \<and> (b2,?n) \<in> r"
+ using Well by (simp add: wo_rel.max2_greater)
+ }
+ ultimately have "(b1,?m) \<in> r \<and> (b2,?m) \<in> r"
+ using Trans trans_def[of r] by blast
+ hence "(b1,b2) \<in> (rel.under r ?m) \<times> (rel.under r ?m)" unfolding rel.under_def by simp}
+ thus "rel.underS ?r' (a1,a2) \<le> (rel.under r ?m) \<times> (rel.under r ?m)" by auto
+ qed
+ moreover have "wo_rel.ofilter r (rel.under r ?m)"
+ using Well by (simp add: wo_rel.under_ofilter)
+ moreover have "rel.under r ?m < Field r"
+ using NE rel.under_Field[of r ?m] by blast
+ ultimately show ?thesis by blast
+qed
+
+
+end
--- a/src/HOL/Cardinals/Fun_More.thy Mon Nov 18 18:04:44 2013 +0100
+++ b/src/HOL/Cardinals/Fun_More.thy Mon Nov 18 18:04:44 2013 +0100
@@ -8,7 +8,7 @@
header {* More on Injections, Bijections and Inverses *}
theory Fun_More
-imports Fun_More_Base
+imports Fun_More_LFP
begin
--- a/src/HOL/Cardinals/Fun_More_Base.thy Mon Nov 18 18:04:44 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,252 +0,0 @@
-(* Title: HOL/Cardinals/Fun_More_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-More on injections, bijections and inverses (base).
-*)
-
-header {* More on Injections, Bijections and Inverses (Base) *}
-
-theory Fun_More_Base
-imports "~~/src/HOL/Library/Infinite_Set"
-begin
-
-
-text {* This section proves more facts (additional to those in @{text "Fun.thy"},
-@{text "Hilbert_Choice.thy"}, @{text "Finite_Set.thy"} and @{text "Infinite_Set.thy"}),
-mainly concerning injections, bijections, inverses and (numeric) cardinals of
-finite sets. *}
-
-
-subsection {* Purely functional properties *}
-
-
-(*2*)lemma bij_betw_id_iff:
-"(A = B) = (bij_betw id A B)"
-by(simp add: bij_betw_def)
-
-
-(*2*)lemma bij_betw_byWitness:
-assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
- RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
- IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
-shows "bij_betw f A A'"
-using assms
-proof(unfold bij_betw_def inj_on_def, safe)
- fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
- have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
- with ** show "a = b" by simp
-next
- fix a' assume *: "a' \<in> A'"
- hence "f' a' \<in> A" using IM2 by blast
- moreover
- have "a' = f(f' a')" using * RIGHT by simp
- ultimately show "a' \<in> f ` A" by blast
-qed
-
-
-(*3*)corollary notIn_Un_bij_betw:
-assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
- BIJ: "bij_betw f A A'"
-shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
-proof-
- have "bij_betw f {b} {f b}"
- unfolding bij_betw_def inj_on_def by simp
- with assms show ?thesis
- using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
-qed
-
-
-(*1*)lemma notIn_Un_bij_betw3:
-assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
-shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
-proof
- assume "bij_betw f A A'"
- thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
- using assms notIn_Un_bij_betw[of b A f A'] by blast
-next
- assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
- have "f ` A = A'"
- proof(auto)
- fix a assume **: "a \<in> A"
- hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
- moreover
- {assume "f a = f b"
- hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
- with NIN ** have False by blast
- }
- ultimately show "f a \<in> A'" by blast
- next
- fix a' assume **: "a' \<in> A'"
- hence "a' \<in> f`(A \<union> {b})"
- using * by (auto simp add: bij_betw_def)
- then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
- moreover
- {assume "a = b" with 1 ** NIN' have False by blast
- }
- ultimately have "a \<in> A" by blast
- with 1 show "a' \<in> f ` A" by blast
- qed
- thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
-qed
-
-
-subsection {* Properties involving finite and infinite sets *}
-
-
-(*3*)lemma inj_on_finite:
-assumes "inj_on f A" "f ` A \<le> B" "finite B"
-shows "finite A"
-using assms infinite_super by (fast dest: finite_imageD)
-
-
-(*3*)lemma infinite_imp_bij_betw:
-assumes INF: "infinite A"
-shows "\<exists>h. bij_betw h A (A - {a})"
-proof(cases "a \<in> A")
- assume Case1: "a \<notin> A" hence "A - {a} = A" by blast
- thus ?thesis using bij_betw_id[of A] by auto
-next
- assume Case2: "a \<in> A"
- have "infinite (A - {a})" using INF infinite_remove by auto
- with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
- where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
- obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
- obtain A' where A'_def: "A' = g ` UNIV" by blast
- have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
- have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
- proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
- case_tac "x = 0", auto simp add: 2)
- fix y assume "a = (if y = 0 then a else f (Suc y))"
- thus "y = 0" using temp by (case_tac "y = 0", auto)
- next
- fix x y
- assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
- thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
- next
- fix n show "f (Suc n) \<in> A" using 2 by blast
- qed
- hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
- using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
- hence 5: "bij_betw (inv g) A' UNIV"
- by (auto simp add: bij_betw_inv_into)
- (* *)
- obtain n where "g n = a" using 3 by auto
- hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
- using 3 4 unfolding A'_def
- by clarify (rule bij_betw_subset, auto simp: image_set_diff)
- (* *)
- obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
- have 7: "bij_betw v UNIV (UNIV - {n})"
- proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
- fix m1 m2 assume "v m1 = v m2"
- thus "m1 = m2"
- by(case_tac "m1 < n", case_tac "m2 < n",
- auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
- next
- show "v ` UNIV = UNIV - {n}"
- proof(auto simp add: v_def)
- fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
- {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
- then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
- with 71 have "n \<le> m'" by auto
- with 72 ** have False by auto
- }
- thus "m < n" by force
- qed
- qed
- (* *)
- obtain h' where h'_def: "h' = g o v o (inv g)" by blast
- hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
- by (auto simp add: bij_betw_trans)
- (* *)
- obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
- have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
- hence "bij_betw h A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
- moreover
- {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
- hence "bij_betw h (A - A') (A - A')"
- using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
- }
- moreover
- have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
- ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
- using 4 by blast
- ultimately have "bij_betw h A (A - {a})"
- using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
- thus ?thesis by blast
-qed
-
-
-(*3*)lemma infinite_imp_bij_betw2:
-assumes INF: "infinite A"
-shows "\<exists>h. bij_betw h A (A \<union> {a})"
-proof(cases "a \<in> A")
- assume Case1: "a \<in> A" hence "A \<union> {a} = A" by blast
- thus ?thesis using bij_betw_id[of A] by auto
-next
- let ?A' = "A \<union> {a}"
- assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
- moreover have "infinite ?A'" using INF by auto
- ultimately obtain f where "bij_betw f ?A' A"
- using infinite_imp_bij_betw[of ?A' a] by auto
- hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
- thus ?thesis by auto
-qed
-
-
-subsection {* Properties involving Hilbert choice *}
-
-
-(*2*)lemma bij_betw_inv_into_left:
-assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
-shows "(inv_into A f) (f a) = a"
-using assms unfolding bij_betw_def
-by clarify (rule inv_into_f_f)
-
-(*2*)lemma bij_betw_inv_into_right:
-assumes "bij_betw f A A'" "a' \<in> A'"
-shows "f(inv_into A f a') = a'"
-using assms unfolding bij_betw_def using f_inv_into_f by force
-
-
-(*1*)lemma bij_betw_inv_into_LEFT:
-assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A"
-shows "(inv_into A f)`(f ` B) = B"
-using assms unfolding bij_betw_def using inv_into_image_cancel by force
-
-
-(*1*)lemma bij_betw_inv_into_LEFT_RIGHT:
-assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A" and
- IM: "f ` B = B'"
-shows "(inv_into A f) ` B' = B"
-using assms bij_betw_inv_into_LEFT[of f A A' B] by fast
-
-
-(*1*)lemma bij_betw_inv_into_subset:
-assumes BIJ: "bij_betw f A A'" and
- SUB: "B \<le> A" and IM: "f ` B = B'"
-shows "bij_betw (inv_into A f) B' B"
-using assms unfolding bij_betw_def
-by (auto intro: inj_on_inv_into)
-
-
-subsection {* Other facts *}
-
-
-(*2*)lemma atLeastLessThan_less_eq:
-"({0..<m} \<le> {0..<n}) = ((m::nat) \<le> n)"
-unfolding ivl_subset by arith
-
-
-(*2*)lemma atLeastLessThan_less_eq2:
-assumes "inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}"
-shows "m \<le> n"
-using assms
-using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
- card_atLeastLessThan[of m] card_atLeastLessThan[of n]
- card_inj_on_le[of f "{0 ..< m}" "{0 ..< n}"] by auto
-
-
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Fun_More_LFP.thy Mon Nov 18 18:04:44 2013 +0100
@@ -0,0 +1,252 @@
+(* Title: HOL/Cardinals/Fun_More_LFP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+More on injections, bijections and inverses (LFP).
+*)
+
+header {* More on Injections, Bijections and Inverses (LFP) *}
+
+theory Fun_More_LFP
+imports "~~/src/HOL/Library/Infinite_Set"
+begin
+
+
+text {* This section proves more facts (additional to those in @{text "Fun.thy"},
+@{text "Hilbert_Choice.thy"}, @{text "Finite_Set.thy"} and @{text "Infinite_Set.thy"}),
+mainly concerning injections, bijections, inverses and (numeric) cardinals of
+finite sets. *}
+
+
+subsection {* Purely functional properties *}
+
+
+(*2*)lemma bij_betw_id_iff:
+"(A = B) = (bij_betw id A B)"
+by(simp add: bij_betw_def)
+
+
+(*2*)lemma bij_betw_byWitness:
+assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
+ RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
+ IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
+shows "bij_betw f A A'"
+using assms
+proof(unfold bij_betw_def inj_on_def, safe)
+ fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
+ have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
+ with ** show "a = b" by simp
+next
+ fix a' assume *: "a' \<in> A'"
+ hence "f' a' \<in> A" using IM2 by blast
+ moreover
+ have "a' = f(f' a')" using * RIGHT by simp
+ ultimately show "a' \<in> f ` A" by blast
+qed
+
+
+(*3*)corollary notIn_Un_bij_betw:
+assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
+ BIJ: "bij_betw f A A'"
+shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+proof-
+ have "bij_betw f {b} {f b}"
+ unfolding bij_betw_def inj_on_def by simp
+ with assms show ?thesis
+ using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
+qed
+
+
+(*1*)lemma notIn_Un_bij_betw3:
+assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
+shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+proof
+ assume "bij_betw f A A'"
+ thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ using assms notIn_Un_bij_betw[of b A f A'] by blast
+next
+ assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
+ have "f ` A = A'"
+ proof(auto)
+ fix a assume **: "a \<in> A"
+ hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
+ moreover
+ {assume "f a = f b"
+ hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
+ with NIN ** have False by blast
+ }
+ ultimately show "f a \<in> A'" by blast
+ next
+ fix a' assume **: "a' \<in> A'"
+ hence "a' \<in> f`(A \<union> {b})"
+ using * by (auto simp add: bij_betw_def)
+ then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
+ moreover
+ {assume "a = b" with 1 ** NIN' have False by blast
+ }
+ ultimately have "a \<in> A" by blast
+ with 1 show "a' \<in> f ` A" by blast
+ qed
+ thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
+qed
+
+
+subsection {* Properties involving finite and infinite sets *}
+
+
+(*3*)lemma inj_on_finite:
+assumes "inj_on f A" "f ` A \<le> B" "finite B"
+shows "finite A"
+using assms infinite_super by (fast dest: finite_imageD)
+
+
+(*3*)lemma infinite_imp_bij_betw:
+assumes INF: "infinite A"
+shows "\<exists>h. bij_betw h A (A - {a})"
+proof(cases "a \<in> A")
+ assume Case1: "a \<notin> A" hence "A - {a} = A" by blast
+ thus ?thesis using bij_betw_id[of A] by auto
+next
+ assume Case2: "a \<in> A"
+ have "infinite (A - {a})" using INF infinite_remove by auto
+ with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
+ where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
+ obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
+ obtain A' where A'_def: "A' = g ` UNIV" by blast
+ have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
+ have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
+ proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
+ case_tac "x = 0", auto simp add: 2)
+ fix y assume "a = (if y = 0 then a else f (Suc y))"
+ thus "y = 0" using temp by (case_tac "y = 0", auto)
+ next
+ fix x y
+ assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
+ thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
+ next
+ fix n show "f (Suc n) \<in> A" using 2 by blast
+ qed
+ hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
+ using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
+ hence 5: "bij_betw (inv g) A' UNIV"
+ by (auto simp add: bij_betw_inv_into)
+ (* *)
+ obtain n where "g n = a" using 3 by auto
+ hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
+ using 3 4 unfolding A'_def
+ by clarify (rule bij_betw_subset, auto simp: image_set_diff)
+ (* *)
+ obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
+ have 7: "bij_betw v UNIV (UNIV - {n})"
+ proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
+ fix m1 m2 assume "v m1 = v m2"
+ thus "m1 = m2"
+ by(case_tac "m1 < n", case_tac "m2 < n",
+ auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
+ next
+ show "v ` UNIV = UNIV - {n}"
+ proof(auto simp add: v_def)
+ fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
+ {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
+ then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
+ with 71 have "n \<le> m'" by auto
+ with 72 ** have False by auto
+ }
+ thus "m < n" by force
+ qed
+ qed
+ (* *)
+ obtain h' where h'_def: "h' = g o v o (inv g)" by blast
+ hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
+ by (auto simp add: bij_betw_trans)
+ (* *)
+ obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
+ have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
+ hence "bij_betw h A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
+ moreover
+ {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
+ hence "bij_betw h (A - A') (A - A')"
+ using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
+ }
+ moreover
+ have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
+ ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
+ using 4 by blast
+ ultimately have "bij_betw h A (A - {a})"
+ using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
+ thus ?thesis by blast
+qed
+
+
+(*3*)lemma infinite_imp_bij_betw2:
+assumes INF: "infinite A"
+shows "\<exists>h. bij_betw h A (A \<union> {a})"
+proof(cases "a \<in> A")
+ assume Case1: "a \<in> A" hence "A \<union> {a} = A" by blast
+ thus ?thesis using bij_betw_id[of A] by auto
+next
+ let ?A' = "A \<union> {a}"
+ assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
+ moreover have "infinite ?A'" using INF by auto
+ ultimately obtain f where "bij_betw f ?A' A"
+ using infinite_imp_bij_betw[of ?A' a] by auto
+ hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
+ thus ?thesis by auto
+qed
+
+
+subsection {* Properties involving Hilbert choice *}
+
+
+(*2*)lemma bij_betw_inv_into_left:
+assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
+shows "(inv_into A f) (f a) = a"
+using assms unfolding bij_betw_def
+by clarify (rule inv_into_f_f)
+
+(*2*)lemma bij_betw_inv_into_right:
+assumes "bij_betw f A A'" "a' \<in> A'"
+shows "f(inv_into A f a') = a'"
+using assms unfolding bij_betw_def using f_inv_into_f by force
+
+
+(*1*)lemma bij_betw_inv_into_LEFT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A"
+shows "(inv_into A f)`(f ` B) = B"
+using assms unfolding bij_betw_def using inv_into_image_cancel by force
+
+
+(*1*)lemma bij_betw_inv_into_LEFT_RIGHT:
+assumes BIJ: "bij_betw f A A'" and SUB: "B \<le> A" and
+ IM: "f ` B = B'"
+shows "(inv_into A f) ` B' = B"
+using assms bij_betw_inv_into_LEFT[of f A A' B] by fast
+
+
+(*1*)lemma bij_betw_inv_into_subset:
+assumes BIJ: "bij_betw f A A'" and
+ SUB: "B \<le> A" and IM: "f ` B = B'"
+shows "bij_betw (inv_into A f) B' B"
+using assms unfolding bij_betw_def
+by (auto intro: inj_on_inv_into)
+
+
+subsection {* Other facts *}
+
+
+(*2*)lemma atLeastLessThan_less_eq:
+"({0..<m} \<le> {0..<n}) = ((m::nat) \<le> n)"
+unfolding ivl_subset by arith
+
+
+(*2*)lemma atLeastLessThan_less_eq2:
+assumes "inj_on f {0..<(m::nat)} \<and> f ` {0..<m} \<le> {0..<n}"
+shows "m \<le> n"
+using assms
+using finite_atLeastLessThan[of m] finite_atLeastLessThan[of n]
+ card_atLeastLessThan[of m] card_atLeastLessThan[of n]
+ card_inj_on_le[of f "{0 ..< m}" "{0 ..< n}"] by auto
+
+
+
+end
--- a/src/HOL/Cardinals/Order_Relation_More.thy Mon Nov 18 18:04:44 2013 +0100
+++ b/src/HOL/Cardinals/Order_Relation_More.thy Mon Nov 18 18:04:44 2013 +0100
@@ -8,7 +8,7 @@
header {* Basics on Order-Like Relations *}
theory Order_Relation_More
-imports Order_Relation_More_Base
+imports Order_Relation_More_LFP
begin
--- a/src/HOL/Cardinals/Order_Relation_More_Base.thy Mon Nov 18 18:04:44 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,286 +0,0 @@
-(* Title: HOL/Cardinals/Order_Relation_More_Base.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Basics on order-like relations (base).
-*)
-
-header {* Basics on Order-Like Relations (Base) *}
-
-theory Order_Relation_More_Base
-imports "~~/src/HOL/Library/Order_Relation"
-begin
-
-
-text{* In this section, we develop basic concepts and results pertaining
-to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
-total relations. The development is placed on top of the definitions
-from the theory @{text "Order_Relation"}. We also
-further define upper and lower bounds operators. *}
-
-
-locale rel = fixes r :: "'a rel"
-
-text{* The following context encompasses all this section, except
-for its last subsection. In other words, for the rest of this section except its last
-subsection, we consider a fixed relation @{text "r"}. *}
-
-context rel
-begin
-
-
-subsection {* Auxiliaries *}
-
-
-lemma refl_on_domain:
-"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(auto simp add: refl_on_def)
-
-
-corollary well_order_on_domain:
-"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(simp add: refl_on_domain order_on_defs)
-
-
-lemma well_order_on_Field:
-"well_order_on A r \<Longrightarrow> A = Field r"
-by(auto simp add: refl_on_def Field_def order_on_defs)
-
-
-lemma well_order_on_Well_order:
-"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
-using well_order_on_Field by simp
-
-
-lemma Total_subset_Id:
-assumes TOT: "Total r" and SUB: "r \<le> Id"
-shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
-proof-
- {assume "r \<noteq> {}"
- then obtain a b where 1: "(a,b) \<in> r" by fast
- hence "a = b" using SUB by blast
- hence 2: "(a,a) \<in> r" using 1 by simp
- {fix c d assume "(c,d) \<in> r"
- hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
- hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
- ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
- using TOT unfolding total_on_def by blast
- hence "a = c \<and> a = d" using SUB by blast
- }
- hence "r \<le> {(a,a)}" by auto
- with 2 have "\<exists>a. r = {(a,a)}" by blast
- }
- thus ?thesis by blast
-qed
-
-
-lemma Linear_order_in_diff_Id:
-assumes LI: "Linear_order r" and
- IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
-shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
-using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
-
-
-subsection {* The upper and lower bounds operators *}
-
-
-text{* Here we define upper (``above") and lower (``below") bounds operators.
-We think of @{text "r"} as a {\em non-strict} relation. The suffix ``S"
-at the names of some operators indicates that the bounds are strict -- e.g.,
-@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}).
-Capitalization of the first letter in the name reminds that the operator acts on sets, rather
-than on individual elements. *}
-
-definition under::"'a \<Rightarrow> 'a set"
-where "under a \<equiv> {b. (b,a) \<in> r}"
-
-definition underS::"'a \<Rightarrow> 'a set"
-where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition Under::"'a set \<Rightarrow> 'a set"
-where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
-
-definition UnderS::"'a set \<Rightarrow> 'a set"
-where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition above::"'a \<Rightarrow> 'a set"
-where "above a \<equiv> {b. (a,b) \<in> r}"
-
-definition aboveS::"'a \<Rightarrow> 'a set"
-where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
-
-definition Above::"'a set \<Rightarrow> 'a set"
-where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
-
-definition AboveS::"'a set \<Rightarrow> 'a set"
-where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
-(* *)
-
-text{* Note: In the definitions of @{text "Above[S]"} and @{text "Under[S]"},
- we bounded comprehension by @{text "Field r"} in order to properly cover
- the case of @{text "A"} being empty. *}
-
-
-lemma UnderS_subset_Under: "UnderS A \<le> Under A"
-by(auto simp add: UnderS_def Under_def)
-
-
-lemma underS_subset_under: "underS a \<le> under a"
-by(auto simp add: underS_def under_def)
-
-
-lemma underS_notIn: "a \<notin> underS a"
-by(simp add: underS_def)
-
-
-lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a"
-by(simp add: refl_on_def under_def)
-
-
-lemma AboveS_disjoint: "A Int (AboveS A) = {}"
-by(auto simp add: AboveS_def)
-
-
-lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)"
-by(auto simp add: AboveS_def underS_def)
-
-
-lemma Refl_under_underS:
-assumes "Refl r" "a \<in> Field r"
-shows "under a = underS a \<union> {a}"
-unfolding under_def underS_def
-using assms refl_on_def[of _ r] by fastforce
-
-
-lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}"
-by (auto simp: Field_def underS_def)
-
-
-lemma under_Field: "under a \<le> Field r"
-by(unfold under_def Field_def, auto)
-
-
-lemma underS_Field: "underS a \<le> Field r"
-by(unfold underS_def Field_def, auto)
-
-
-lemma underS_Field2:
-"a \<in> Field r \<Longrightarrow> underS a < Field r"
-using assms underS_notIn underS_Field by blast
-
-
-lemma underS_Field3:
-"Field r \<noteq> {} \<Longrightarrow> underS a < Field r"
-by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
-
-
-lemma Under_Field: "Under A \<le> Field r"
-by(unfold Under_def Field_def, auto)
-
-
-lemma UnderS_Field: "UnderS A \<le> Field r"
-by(unfold UnderS_def Field_def, auto)
-
-
-lemma AboveS_Field: "AboveS A \<le> Field r"
-by(unfold AboveS_def Field_def, auto)
-
-
-lemma under_incr:
-assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
-shows "under a \<le> under b"
-proof(unfold under_def, auto)
- fix x assume "(x,a) \<in> r"
- with REL TRANS trans_def[of r]
- show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incr:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
- REL: "(a,b) \<in> r"
-shows "underS a \<le> underS b"
-proof(unfold underS_def, auto)
- assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
- with ANTISYM antisym_def[of r] REL
- show False by blast
-next
- fix x assume "x \<noteq> a" "(x,a) \<in> r"
- with REL TRANS trans_def[of r]
- show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incl_iff:
-assumes LO: "Linear_order r" and
- INa: "a \<in> Field r" and INb: "b \<in> Field r"
-shows "(underS a \<le> underS b) = ((a,b) \<in> r)"
-proof
- assume "(a,b) \<in> r"
- thus "underS a \<le> underS b" using LO
- by (simp add: order_on_defs underS_incr)
-next
- assume *: "underS a \<le> underS b"
- {assume "a = b"
- hence "(a,b) \<in> r" using assms
- by (simp add: order_on_defs refl_on_def)
- }
- moreover
- {assume "a \<noteq> b \<and> (b,a) \<in> r"
- hence "b \<in> underS a" unfolding underS_def by blast
- hence "b \<in> underS b" using * by blast
- hence False by (simp add: underS_notIn)
- }
- ultimately
- show "(a,b) \<in> r" using assms
- order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
-qed
-
-
-lemma under_Under_trans:
-assumes TRANS: "trans r" and
- IN1: "a \<in> under b" and IN2: "b \<in> Under C"
-shows "a \<in> Under C"
-proof-
- have "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)"
- using IN1 IN2 under_def Under_def by blast
- hence "\<forall>c \<in> C. (a,c) \<in> r"
- using TRANS trans_def[of r] by blast
- moreover
- have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast
- ultimately
- show ?thesis unfolding Under_def by blast
-qed
-
-
-lemma under_UnderS_trans:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
- IN1: "a \<in> under b" and IN2: "b \<in> UnderS C"
-shows "a \<in> UnderS C"
-proof-
- from IN2 have "b \<in> Under C"
- using UnderS_subset_Under[of C] by blast
- with assms under_Under_trans
- have "a \<in> Under C" by blast
- (* *)
- moreover
- have "a \<notin> C"
- proof
- assume *: "a \<in> C"
- have 1: "(a,b) \<in> r"
- using IN1 under_def[of b] by auto
- have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r"
- using IN2 UnderS_def[of C] by blast
- with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast
- with 1 ANTISYM antisym_def[of r]
- show False by blast
- qed
- (* *)
- ultimately
- show ?thesis unfolding UnderS_def Under_def by fast
-qed
-
-
-end (* context rel *)
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Order_Relation_More_LFP.thy Mon Nov 18 18:04:44 2013 +0100
@@ -0,0 +1,286 @@
+(* Title: HOL/Cardinals/Order_Relation_More_LFP.thy
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Basics on order-like relations (LFP).
+*)
+
+header {* Basics on Order-Like Relations (LFP) *}
+
+theory Order_Relation_More_LFP
+imports "~~/src/HOL/Library/Order_Relation"
+begin
+
+
+text{* In this section, we develop basic concepts and results pertaining
+to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
+total relations. The development is placed on top of the definitions
+from the theory @{text "Order_Relation"}. We also
+further define upper and lower bounds operators. *}
+
+
+locale rel = fixes r :: "'a rel"
+
+text{* The following context encompasses all this section, except
+for its last subsection. In other words, for the rest of this section except its last
+subsection, we consider a fixed relation @{text "r"}. *}
+
+context rel
+begin
+
+
+subsection {* Auxiliaries *}
+
+
+lemma refl_on_domain:
+"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(auto simp add: refl_on_def)
+
+
+corollary well_order_on_domain:
+"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
+by(simp add: refl_on_domain order_on_defs)
+
+
+lemma well_order_on_Field:
+"well_order_on A r \<Longrightarrow> A = Field r"
+by(auto simp add: refl_on_def Field_def order_on_defs)
+
+
+lemma well_order_on_Well_order:
+"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
+using well_order_on_Field by simp
+
+
+lemma Total_subset_Id:
+assumes TOT: "Total r" and SUB: "r \<le> Id"
+shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
+proof-
+ {assume "r \<noteq> {}"
+ then obtain a b where 1: "(a,b) \<in> r" by fast
+ hence "a = b" using SUB by blast
+ hence 2: "(a,a) \<in> r" using 1 by simp
+ {fix c d assume "(c,d) \<in> r"
+ hence