--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Ordinals_and_Cardinals/Cardinal_Arithmetic.thy Tue Aug 28 17:16:00 2012 +0200
@@ -0,0 +1,878 @@
+(* Title: HOL/Ordinals_and_Cardinals/Cardinal_Arithmetic.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2012
+
+Cardinal arithmetic.
+*)
+
+header {* Cardinal Arithmetic *}
+
+theory Cardinal_Arithmetic
+imports Cardinal_Order_Relation_Base
+begin
+
+text {*
+ The following collection of lemmas should be seen as an user interface to the HOL Theory
+ of cardinals. It is not expected to be complete in any sense, since its
+ development was driven by demand arising from the development of the (co)datatype package.
+*}
+
+(*library candidate*)
+lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
+by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
+
+(*should supersede a weaker lemma from the library*)
+lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
+unfolding dir_image_def Field_def Range_def Domain_def by fastforce
+
+lemma card_order_dir_image:
+ assumes bij: "bij f" and co: "card_order r"
+ shows "card_order (dir_image r f)"
+proof -
+ from assms have "Field (dir_image r f) = UNIV"
+ using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
+ moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
+ with co have "Card_order (dir_image r f)"
+ using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
+ ultimately show ?thesis by auto
+qed
+
+(*library candidate*)
+lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
+by (rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
+by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
+
+(*library candidate*)
+lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
+by (simp only: ordIso_refl card_of_Card_order)
+
+(*library candidate*)
+lemma card_of_Times_Plus_distrib:
+ "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
+proof -
+ let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
+ have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
+ thus ?thesis using card_of_ordIso by blast
+qed
+
+(*library candidate*)
+lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
+using card_order_on_Card_order[of UNIV r] by simp
+
+subsection {* Zero *}
+
+definition czero where
+ "czero = card_of {}"
+
+lemma czero_ordIso:
+ "czero =o czero"
+using card_of_empty_ordIso by (simp add: czero_def)
+
+lemma card_of_ordIso_czero_iff_empty:
+ "|A| =o (czero :: 'a rel) \<longleftrightarrow> A = ({} :: 'a set)"
+unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl)
+
+(* A "not czero" Cardinal predicate *)
+abbreviation Cnotzero where
+ "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
+
+(*helper*)
+lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
+by (metis Card_order_iff_ordIso_card_of czero_def)
+
+lemma czeroI:
+ "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
+using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
+
+lemma czeroE:
+ "r =o czero \<Longrightarrow> Field r = {}"
+unfolding czero_def
+by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
+
+lemma Cnotzero_mono:
+ "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
+apply (rule ccontr)
+apply auto
+apply (drule czeroE)
+apply (erule notE)
+apply (erule czeroI)
+apply (drule card_of_mono2)
+apply (simp only: card_of_empty3)
+done
+
+subsection {* Infinite cardinals *}
+
+definition cinfinite where
+ "cinfinite r = infinite (Field r)"
+
+abbreviation Cinfinite where
+ "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
+
+lemma natLeq_ordLeq_cinfinite:
+ assumes inf: "Cinfinite r"
+ shows "natLeq \<le>o r"
+proof -
+ from inf have "natLeq \<le>o |Field r|" by (simp add: cinfinite_def infinite_iff_natLeq_ordLeq)
+ also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
+ finally show ?thesis .
+qed
+
+lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
+unfolding cinfinite_def by (metis czeroE finite.emptyI)
+
+lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
+by (metis cinfinite_not_czero)
+
+lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
+by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
+
+lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
+by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
+
+
+subsection {* Binary sum *}
+
+definition csum (infixr "+c" 65) where
+ "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
+
+lemma Card_order_csum:
+ "Card_order (r1 +c r2)"
+unfolding csum_def by (simp add: card_of_Card_order)
+
+lemma csum_Cnotzero1:
+ "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
+unfolding csum_def
+by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
+
+lemma csum_Cnotzero2:
+ "Cnotzero r2 \<Longrightarrow> Cnotzero (r1 +c r2)"
+unfolding csum_def
+by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
+
+lemma card_order_csum:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 +c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
+qed
+
+lemma cinfinite_csum:
+ "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
+
+lemma Cinfinite_csum:
+ "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma Cinfinite_csum_strong:
+ "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
+by (metis Cinfinite_csum)
+
+lemma Cinfinite_csum1:
+ "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
+unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
+
+lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
+by (simp only: csum_def ordIso_Plus_cong)
+
+lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
+by (simp only: csum_def ordIso_Plus_cong1)
+
+lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
+by (simp only: csum_def ordIso_Plus_cong2)
+
+lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
+by (simp only: csum_def ordLeq_Plus_mono)
+
+lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
+by (simp only: csum_def ordLeq_Plus_mono1)
+
+lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
+by (simp only: csum_def ordLeq_Plus_mono2)
+
+lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus1)
+
+lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
+by (simp only: csum_def Card_order_Plus2)
+
+lemma csum_com: "p1 +c p2 =o p2 +c p1"
+by (simp only: csum_def card_of_Plus_commute)
+
+lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
+by (simp only: csum_def Field_card_of card_of_Plus_assoc)
+
+lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
+by (simp only: csum_def Field_card_of card_of_refl)
+
+lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
+using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
+
+
+subsection {* One *}
+
+definition cone where
+ "cone = card_of {()}"
+
+lemma Card_order_cone: "Card_order cone"
+unfolding cone_def by (rule card_of_Card_order)
+
+lemma single_cone:
+ "|{x}| =o cone"
+proof -
+ let ?f = "\<lambda>x. ()"
+ have "bij_betw ?f {x} {()}" unfolding bij_betw_def by auto
+ thus ?thesis unfolding cone_def using card_of_ordIso by blast
+qed
+
+lemma cone_not_czero: "\<not> (cone =o czero)"
+unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
+
+lemma cone_Cnotzero: "Cnotzero cone"
+by (simp add: cone_not_czero Card_order_cone)
+
+lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
+unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
+
+
+subsection{* Two *}
+
+definition ctwo where
+ "ctwo = |UNIV :: bool set|"
+
+lemma Card_order_ctwo: "Card_order ctwo"
+unfolding ctwo_def by (rule card_of_Card_order)
+
+lemma cone_ordLeq_ctwo: "cone \<le>o ctwo"
+unfolding cone_def ctwo_def card_of_ordLeq[symmetric] by auto
+
+lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
+using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
+unfolding czero_def ctwo_def by (metis UNIV_not_empty)
+
+lemma ctwo_Cnotzero: "Cnotzero ctwo"
+by (simp add: ctwo_not_czero Card_order_ctwo)
+
+
+subsection {* Family sum *}
+
+definition Csum where
+ "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
+
+(* Similar setup to the one for SIGMA from theory Big_Operators: *)
+syntax "_Csum" ::
+ "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
+ ("(3CSUM _:_. _)" [0, 51, 10] 10)
+
+translations
+ "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
+
+lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
+by (auto simp: Csum_def Field_card_of)
+
+(* NB: Always, under the cardinal operator,
+operations on sets are reduced automatically to operations on cardinals.
+This should make cardinal reasoning more direct and natural. *)
+
+
+subsection {* Product *}
+
+definition cprod (infixr "*c" 80) where
+ "r1 *c r2 = |Field r1 <*> Field r2|"
+
+lemma Times_cprod: "|A \<times> B| =o |A| *c |B|"
+by (simp only: cprod_def Field_card_of card_of_refl)
+
+lemma card_order_cprod:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 *c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
+qed
+
+lemma Card_order_cprod: "Card_order (r1 *c r2)"
+by (simp only: cprod_def Field_card_of card_of_card_order_on)
+
+lemma cprod_cong2: "p2 =o r2 \<Longrightarrow> q *c p2 =o q *c r2"
+by (simp only: cprod_def ordIso_Times_cong2)
+
+lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
+by (simp only: cprod_def ordLeq_Times_mono1)
+
+lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
+by (simp only: cprod_def ordLeq_Times_mono2)
+
+lemma ordLeq_cprod1: "\<lbrakk>Card_order p1; Cnotzero p2\<rbrakk> \<Longrightarrow> p1 \<le>o p1 *c p2"
+unfolding cprod_def by (metis Card_order_Times1 czeroI)
+
+lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
+unfolding cprod_def by (metis Card_order_Times2 czeroI)
+
+lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
+
+lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
+by (metis cinfinite_mono ordLeq_cprod2)
+
+lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
+by (blast intro: cinfinite_cprod2 Card_order_cprod)
+
+lemma cprod_com: "p1 *c p2 =o p2 *c p1"
+by (simp only: cprod_def card_of_Times_commute)
+
+lemma card_of_Csum_Times:
+ "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
+by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
+
+lemma card_of_Csum_Times':
+ assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
+ shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
+proof -
+ from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
+ with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
+ hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
+ also from * have "|I| *c |Field r| \<le>o |I| *c r"
+ by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
+ finally show ?thesis .
+qed
+
+lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
+unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
+
+lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
+unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
+
+lemma csum_absorb1':
+ assumes card: "Card_order r2"
+ and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
+ shows "r2 +c r1 =o r2"
+by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
+
+lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
+by (rule csum_absorb1') auto
+
+lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
+unfolding cinfinite_def cprod_def
+by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
+
+lemma cprod_infinite: "Cinfinite r \<Longrightarrow> r *c r =o r"
+using cprod_infinite1' Cinfinite_Cnotzero ordLeq_refl by blast
+
+
+subsection {* Exponentiation *}
+
+definition cexp (infixr "^c" 80) where
+ "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
+
+definition ccexp (infixr "^^c" 80) where
+ "r1 ^^c r2 \<equiv> |Pfunc (Field r2) (Field r1)|"
+
+lemma cexp_ordLeq_ccexp: "r1 ^c r2 \<le>o r1 ^^c r2"
+unfolding cexp_def ccexp_def by (rule card_of_mono1) (rule Func_Pfunc)
+
+lemma card_order_ccexp:
+ assumes "card_order r1" "card_order r2"
+ shows "card_order (r1 ^^c r2)"
+proof -
+ have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
+ thus ?thesis unfolding ccexp_def Pfunc_def
+ by (auto simp: card_of_card_order_on split: option.split)
+qed
+
+lemma Card_order_cexp: "Card_order (r1 ^c r2)"
+unfolding cexp_def by (rule card_of_Card_order)
+
+lemma cexp_mono':
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n1: "Field p1 \<noteq> {} \<or> cone \<le>o r1 ^c r2"
+ and n2: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+proof(cases "Field p1 = {}")
+ case True
+ hence "|Field (p1 ^c p2)| \<le>o cone"
+ unfolding czero_def cone_def cexp_def Field_card_of
+ by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
+ (metis Func_is_emp card_of_empty ex_in_conv)
+ hence "p1 ^c p2 \<le>o cone" by (simp add: Field_card_of cexp_def)
+ thus ?thesis using True n1 ordLeq_transitive by auto
+next
+ case False
+ have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
+ using 1 2 by (auto simp: card_of_mono2)
+ obtain f1 where f1: "f1 ` Field r1 = Field p1"
+ using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
+ obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
+ using 2 unfolding card_of_ordLeq[symmetric] by blast
+ have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
+ unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n2, symmetric] .
+ have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
+ using False by simp
+ show ?thesis
+ using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
+qed
+
+lemma cexp_mono:
+ assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
+ and n1: "Cnotzero p1 \<or> cone \<le>o r1 ^c r2"
+ and n2: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "p1 ^c p2 \<le>o r1 ^c r2"
+proof (rule cexp_mono'[OF 1 2])
+ show "Field p1 \<noteq> {} \<or> cone \<le>o r1 ^c r2"
+ proof (cases "Cnotzero p1")
+ case True show ?thesis using Cnotzero_imp_not_empty[OF True] by (rule disjI1)
+ next
+ case False with n1 show ?thesis by blast
+ qed
+qed (rule czeroI[OF card, THEN n2, THEN czeroE])
+
+lemma cexp_mono1:
+ assumes 1: "p1 \<le>o r1"
+ and n1: "Cnotzero p1 \<or> cone \<le>o r1 ^c q" and q: "Card_order q"
+ shows "p1 ^c q \<le>o r1 ^c q"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF 1 _ n1]) (auto simp: q)
+
+lemma cexp_mono1_Cnotzero: "\<lbrakk>p1 \<le>o r1; Cnotzero p1; Card_order q\<rbrakk> \<Longrightarrow> p1 ^c q \<le>o r1 ^c q"
+by (simp add: cexp_mono1)
+
+lemma cexp_mono1_cone_ordLeq: "\<lbrakk>p1 \<le>o r1; cone \<le>o r1 ^c q; Card_order q\<rbrakk> \<Longrightarrow> p1 ^c q \<le>o r1 ^c q"
+using assms by (simp add: cexp_mono1)
+
+lemma cexp_mono2':
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n1: "Field q \<noteq> {} \<or> cone \<le>o q ^c r2"
+ and n2: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n1 n2]) auto
+
+lemma cexp_mono2:
+ assumes 2: "p2 \<le>o r2" and q: "Card_order q"
+ and n1: "Cnotzero q \<or> cone \<le>o q ^c r2"
+ and n2: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n1 n2 card]) auto
+
+lemma cexp_mono2_Cnotzero:
+ assumes "p2 \<le>o r2" "Cnotzero q" and n2: "Cnotzero p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+proof (rule cexp_mono)
+ assume *: "p2 =o czero"
+ have False using n2 czeroI czeroE[OF *] by blast
+ thus "r2 =o czero" by blast
+qed (auto simp add: assms ordLeq_refl)
+
+lemma cexp_cong:
+ assumes 1: "p1 =o r1" and 2: "p2 =o r2"
+ and p1: "Cnotzero p1 \<or> cone \<le>o r1 ^c r2" and Cr: "Card_order r2"
+ and r1: "Cnotzero r1 \<or> cone \<le>o p1 ^c p2" and Cp: "Card_order p2"
+ shows "p1 ^c p2 =o r1 ^c r2"
+proof -
+ obtain f where "bij_betw f (Field p2) (Field r2)"
+ using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
+ hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
+ have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
+ and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
+ using 0 Cr Cp czeroE czeroI by auto
+ show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
+ using r p cexp_mono[OF _ _ p1 _ Cp] cexp_mono[OF _ _ r1 _ Cr]
+ by blast
+qed
+
+lemma cexp_cong1:
+ assumes 1: "p1 =o r1" and q: "Card_order q"
+ and p1: "Cnotzero p1 \<or> cone \<le>o r1 ^c q"
+ and r1: "Cnotzero r1 \<or> cone \<le>o p1 ^c q"
+ shows "p1 ^c q =o r1 ^c q"
+by (rule cexp_cong[OF 1 _ p1 q r1 q]) (rule ordIso_refl[OF q])
+
+lemma cexp_cong1_Cnotzero:
+ assumes "p1 =o r1" "Card_order q" "Cnotzero p1" "Cnotzero r1"
+ shows "p1 ^c q =o r1 ^c q"
+by (rule cexp_cong1, auto simp add: assms)
+
+lemma cexp_cong2:
+ assumes 2: "p2 =o r2" and q: "Card_order q"
+ and p: "Card_order p2" and r: "Card_order r2"
+ shows "Cnotzero q \<or> (cone \<le>o q ^c p2 \<and> cone \<le>o q ^c r2) \<Longrightarrow>
+ q ^c p2 =o q ^c r2"
+by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl q p r)
+
+lemma cexp_cong2_Cnotzero:
+ assumes 2: "p2 =o r2" and q: "Cnotzero q"
+ and p: "Card_order p2"
+ shows "q ^c p2 =o q ^c r2"
+by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
+
+lemma cexp_czero: "r ^c czero =o cone"
+unfolding cexp_def czero_def Field_card_of Func_empty by (rule single_cone)
+
+lemma cexp_cone:
+ assumes "Card_order r"
+ shows "r ^c cone =o r"
+proof -
+ have "r ^c cone =o |Field r|"
+ unfolding cexp_def cone_def Field_card_of Func_empty
+ card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
+ by (rule exI[of _ "\<lambda>f. case f () of Some a \<Rightarrow> a"]) auto
+ also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
+ finally show ?thesis .
+qed
+
+lemma cexp_cprod:
+ assumes r1: "Cnotzero r1"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
+proof -
+ have "?L =o r1 ^c (r3 *c r2)"
+ unfolding cprod_def cexp_def Field_card_of
+ using card_of_Func_Times by(rule ordIso_symmetric)
+ also have "r1 ^c (r3 *c r2) =o ?R"
+ apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
+ finally show ?thesis .
+qed
+
+lemma cexp_cprod_ordLeq:
+ assumes r1: "Cnotzero r1" and r2: "Cinfinite r2"
+ and r3: "Cnotzero r3" "r3 \<le>o r2"
+ shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
+proof-
+ have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
+ also have "r1 ^c (r2 *c r3) =o ?R"
+ apply(rule cexp_cong2)
+ apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
+ finally show ?thesis .
+qed
+
+lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
+by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
+
+lemma Pow_cexp_ctwo:
+ "|Pow A| =o ctwo ^c |A|"
+unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+
+lemma ordLess_ctwo_cexp:
+ assumes "Card_order r"
+ shows "r <o ctwo ^c r"
+proof -
+ have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
+ also have "|Pow (Field r)| =o ctwo ^c r"
+ unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+ finally show ?thesis .
+qed
+
+lemma ordLeq_cexp1:
+ assumes "Cnotzero r" "Card_order q"
+ shows "q \<le>o q ^c r"
+proof (cases "q =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False
+ thus ?thesis
+ apply -
+ apply (rule ordIso_ordLeq_trans)
+ apply (rule ordIso_symmetric)
+ apply (rule cexp_cone)
+ apply (rule assms(2))
+ apply (rule cexp_mono2)
+ apply (rule cone_ordLeq_Cnotzero)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ apply (rule disjI1)
+ apply (rule conjI)
+ apply (rule notI)
+ apply (erule notE)
+ apply (rule ordIso_transitive)
+ apply assumption
+ apply (rule czero_ordIso)
+ apply (rule assms(2))
+ apply (rule notE)
+ apply (rule cone_not_czero)
+ apply assumption
+ apply (rule Card_order_cone)
+ done
+qed
+
+lemma ordLeq_cexp2:
+ assumes "ctwo \<le>o q" "Card_order r"
+ shows "r \<le>o q ^c r"
+proof (cases "r =o (czero :: 'a rel)")
+ case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
+next
+ case False thus ?thesis
+ apply -
+ apply (rule ordLess_imp_ordLeq)
+ apply (rule ordLess_ordLeq_trans)
+ apply (rule ordLess_ctwo_cexp)
+ apply (rule assms(2))
+ apply (rule cexp_mono1)
+ apply (rule assms(1))
+ apply (rule disjI1)
+ apply (rule ctwo_Cnotzero)
+ apply (rule assms(2))
+ done
+qed
+
+lemma Cnotzero_cexp:
+ assumes "Cnotzero q" "Card_order r"
+ shows "Cnotzero (q ^c r)"
+proof (cases "r =o czero")
+ case False thus ?thesis
+ apply -
+ apply (rule Cnotzero_mono)
+ apply (rule assms(1))
+ apply (rule Card_order_cexp)
+ apply (rule ordLeq_cexp1)
+ apply (rule conjI)
+ apply (rule notI)
+ apply (erule notE)
+ apply (rule ordIso_transitive)
+ apply assumption
+ apply (rule czero_ordIso)
+ apply (rule assms(2))
+ apply (rule conjunct2)
+ apply (rule assms(1))
+ done
+next
+ case True thus ?thesis
+ apply -
+ apply (rule Cnotzero_mono)
+ apply (rule cone_Cnotzero)
+ apply (rule Card_order_cexp)
+ apply (rule ordIso_imp_ordLeq)
+ apply (rule ordIso_symmetric)
+ apply (rule ordIso_transitive)
+ apply (rule cexp_cong2)
+ apply assumption
+ apply (rule conjunct2)
+ apply (rule assms(1))
+ apply (rule assms(2))
+ apply (simp only: card_of_Card_order czero_def)
+ apply (rule disjI1)
+ apply (rule assms(1))
+ apply (rule cexp_czero)
+ done
+qed
+
+lemma Cinfinite_ctwo_cexp:
+ "Cinfinite r \<Longrightarrow> Cinfinite (ctwo ^c r)"
+unfolding ctwo_def cexp_def cinfinite_def Field_card_of
+by (rule conjI, rule infinite_Func, auto, rule card_of_card_order_on)
+
+lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
+by (metis assms cinfinite_mono ordLeq_cexp2)
+
+lemma cinfinite_ccexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^^c r)"
+by (rule cinfinite_mono[OF cexp_ordLeq_ccexp cinfinite_cexp])
+
+lemma Cinfinite_cexp:
+ "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
+by (simp add: cinfinite_cexp Card_order_cexp)
+
+lemma ctwo_ordLess_natLeq:
+ "ctwo <o natLeq"
+unfolding ctwo_def using finite_iff_ordLess_natLeq finite_UNIV by fast
+
+lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
+by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
+
+lemma ctwo_ordLeq_Cinfinite:
+ assumes "Cinfinite r"
+ shows "ctwo \<le>o r"
+by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
+
+lemma Cinfinite_ordLess_cexp:
+ assumes r: "Cinfinite r"
+ shows "r <o r ^c r"
+proof -
+ have "r <o ctwo ^c r" using r by (simp only: ordLess_ctwo_cexp)
+ also have "ctwo ^c r \<le>o r ^c r"
+ by (rule cexp_mono1[OF ctwo_ordLeq_Cinfinite]) (auto simp: r ctwo_not_czero Card_order_ctwo)
+ finally show ?thesis .
+qed
+
+lemma infinite_ordLeq_cexp:
+ assumes "Cinfinite r"
+ shows "r \<le>o r ^c r"
+by (rule ordLess_imp_ordLeq[OF Cinfinite_ordLess_cexp[OF assms]])
+
+lemma cone_ordLeq_iff_Field:
+ assumes "cone \<le>o r"
+ shows "Field r \<noteq> {}"
+proof (rule ccontr)
+ assume "\<not> Field r \<noteq> {}"
+ hence "Field r = {}" by simp
+ thus False using card_of_empty3
+ card_of_mono2[OF assms] Cnotzero_imp_not_empty[OF cone_Cnotzero] by auto
+qed
+
+lemma cone_ordLeq_cexp: "cone \<le>o r1 \<Longrightarrow> cone \<le>o r1 ^c r2"
+by (simp add: cexp_def cone_def Func_non_emp card_of_singl_ordLeq cone_ordLeq_iff_Field)
+
+lemma Card_order_czero: "Card_order czero"
+by (simp only: card_of_Card_order czero_def)
+
+lemma cexp_mono2'':
+ assumes 2: "p2 \<le>o r2"
+ and n1: "Cnotzero q"
+ and n2: "Card_order p2"
+ shows "q ^c p2 \<le>o q ^c r2"
+proof (cases "p2 =o (czero :: 'a rel)")
+ case True
+ hence "q ^c p2 =o q ^c (czero :: 'a rel)" using n1 n2 cexp_cong2 Card_order_czero by blast
+ also have "q ^c (czero :: 'a rel) =o cone" using cexp_czero by blast
+ also have "cone \<le>o q ^c r2" using cone_ordLeq_cexp cone_ordLeq_Cnotzero n1 by blast
+ finally show ?thesis .
+next
+ case False thus ?thesis using assms cexp_mono2' czeroI by metis
+qed
+
+lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
+by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
+
+lemma UNION_Cinfinite_bound: "\<lbrakk>|I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |\<Union>i \<in> I. A i| \<le>o r"
+by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
+
+lemma csum_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p +c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def csum_def
+ by (blast intro: card_of_Plus_ordLeq_infinite_Field)
+qed
+
+lemma csum_cexp: "\<lbrakk>Cinfinite r1; Cinfinite r2; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow>
+ q ^c r1 +c q ^c r2 \<le>o q ^c (r1 +c r2)"
+apply (rule csum_cinfinite_bound)
+apply (rule cexp_mono2)
+apply (rule ordLeq_csum1)
+apply (erule conjunct2)
+apply assumption
+apply (rule disjI2)
+apply (rule ordLeq_transitive)
+apply (rule cone_ordLeq_ctwo)
+apply (rule ordLeq_transitive)
+apply assumption
+apply (rule ordLeq_cexp1)
+apply (rule Cinfinite_Cnotzero)
+apply (rule Cinfinite_csum)
+apply (rule disjI1)
+apply assumption
+apply assumption
+apply (rule notE)
+apply (rule cinfinite_not_czero[of r1])
+apply (erule conjunct1)
+apply assumption
+apply (erule conjunct2)
+apply (rule cexp_mono2)
+apply (rule ordLeq_csum2)
+apply (erule conjunct2)
+apply assumption
+apply (rule disjI2)
+apply (rule ordLeq_transitive)
+apply (rule cone_ordLeq_ctwo)
+apply (rule ordLeq_transitive)
+apply assumption
+apply (rule ordLeq_cexp1)
+apply (rule Cinfinite_Cnotzero)
+apply (rule Cinfinite_csum)
+apply (rule disjI1)
+apply assumption
+apply assumption
+apply (rule notE)
+apply (rule cinfinite_not_czero[of r2])
+apply (erule conjunct1)
+apply assumption
+apply (erule conjunct2)
+apply (rule Card_order_cexp)
+apply (rule Card_order_cexp)
+apply (rule Cinfinite_cexp)
+apply assumption
+apply (rule Cinfinite_csum)
+by (rule disjI1)
+
+lemma csum_cexp': "\<lbrakk>Cinfinite r; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow> q +c r \<le>o q ^c r"
+apply (rule csum_cinfinite_bound)
+ apply (metis Cinfinite_Cnotzero ordLeq_cexp1)
+ apply (metis ordLeq_cexp2)
+ apply blast+
+by (metis Cinfinite_cexp)
+
+lemma cprod_cinfinite_bound:
+ assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
+ shows "p *c q \<le>o r"
+proof -
+ from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
+ unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
+ with assms show ?thesis unfolding cinfinite_def cprod_def
+ by (blast intro: card_of_Times_ordLeq_infinite_Field)
+qed
+
+lemma cprod_csum_cexp:
+ "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
+unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
+proof -
+ let ?f = "\<lambda>(a, b). %x. if x then Some (Inl a) else Some (Inr b)"
+ have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
+ by (auto simp: inj_on_def fun_eq_iff split: bool.split)
+ moreover
+ have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
+ by (auto simp: Func_def)
+ ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
+qed
+
+lemma card_of_Sigma_ordLeq_Cinfinite:
+ "\<lbrakk>Cinfinite r; |I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r\<rbrakk> \<Longrightarrow> |SIGMA i : I. A i| \<le>o r"
+unfolding cinfinite_def by (blast intro: card_of_Sigma_ordLeq_infinite_Field)
+
+
+(* cardSuc *)
+
+lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
+by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
+
+lemma cardSuc_UNION_Cinfinite:
+ assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
+ shows "EX i : Field (cardSuc r). B \<le> As i"
+using cardSuc_UNION assms unfolding cinfinite_def by blast
+
+subsection {* Powerset *}
+
+definition cpow where "cpow r = |Pow (Field r)|"
+
+lemma card_order_cpow: "card_order r \<Longrightarrow> card_order (cpow r)"
+by (simp only: cpow_def Field_card_order Pow_UNIV card_of_card_order_on)
+
+lemma cpow_greater_eq: "Card_order r \<Longrightarrow> r \<le>o cpow r"
+by (rule ordLess_imp_ordLeq) (simp only: cpow_def Card_order_Pow)
+
+lemma Card_order_cpow: "Card_order (cpow r)"
+unfolding cpow_def by (rule card_of_Card_order)
+
+lemma Cinfinite_cpow: "Cinfinite r \<Longrightarrow> Cinfinite (cpow r)"
+unfolding cpow_def cinfinite_def by (metis Field_card_of card_of_Card_order infinite_Pow)
+
+lemma cardSuc_ordLeq_cpow: "Card_order r \<Longrightarrow> cardSuc r \<le>o cpow r"
+unfolding cpow_def by (metis Card_order_Pow cardSuc_ordLess_ordLeq card_of_Card_order)
+
+lemma cpow_cexp_ctwo: "cpow r =o ctwo ^c r"
+unfolding cpow_def ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
+
+subsection {* Lists *}
+
+definition clists where "clists r = |lists (Field r)|"
+
+lemma clists_Cinfinite: "Cinfinite r \<Longrightarrow> clists r =o r"
+unfolding cinfinite_def clists_def by (blast intro: Card_order_lists_infinite)
+
+lemma Card_order_clists: "Card_order (clists r)"
+unfolding clists_def by (rule card_of_Card_order)
+
+lemma Cnotzero_clists: "Cnotzero (clists r)"
+by (simp add: clists_def card_of_ordIso_czero_iff_empty lists_not_empty) (rule card_of_Card_order)
+
+end