blanchet@55056: (*  Title:      HOL/BNF_Cardinal_Arithmetic.thy
blanchet@54474:     Author:     Dmitriy Traytel, TU Muenchen
blanchet@54474:     Copyright   2012
blanchet@54474: 
blanchet@55059: Cardinal arithmetic as needed by bounded natural functors.
blanchet@54474: *)
blanchet@54474: 
wenzelm@60758: section \<open>Cardinal Arithmetic as Needed by Bounded Natural Functors\<close>
blanchet@54474: 
blanchet@55056: theory BNF_Cardinal_Arithmetic
blanchet@55056: imports BNF_Cardinal_Order_Relation
blanchet@54474: begin
blanchet@54474: 
blanchet@54474: lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
blanchet@54474: by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
blanchet@54474: 
blanchet@54474: lemma card_order_dir_image:
blanchet@54474:   assumes bij: "bij f" and co: "card_order r"
blanchet@54474:   shows "card_order (dir_image r f)"
blanchet@54474: proof -
blanchet@54474:   from assms have "Field (dir_image r f) = UNIV"
blanchet@54474:     using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
blanchet@54474:   moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
blanchet@54474:   with co have "Card_order (dir_image r f)"
blanchet@54474:     using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
blanchet@54474:   ultimately show ?thesis by auto
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
blanchet@54474: by (rule card_order_on_ordIso)
blanchet@54474: 
blanchet@54474: lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
blanchet@54474: by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
blanchet@54474: 
blanchet@54474: lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
blanchet@54474: by (simp only: ordIso_refl card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
blanchet@54474: using card_order_on_Card_order[of UNIV r] by simp
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Zero\<close>
blanchet@54474: 
blanchet@54474: definition czero where
blanchet@54474:   "czero = card_of {}"
blanchet@54474: 
blanchet@54474: lemma czero_ordIso:
blanchet@54474:   "czero =o czero"
blanchet@54474: using card_of_empty_ordIso by (simp add: czero_def)
blanchet@54474: 
blanchet@54474: lemma card_of_ordIso_czero_iff_empty:
blanchet@54474:   "|A| =o (czero :: 'b rel) \<longleftrightarrow> A = ({} :: 'a set)"
blanchet@54474: unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl card_of_empty_ordIso)
blanchet@54474: 
blanchet@54474: (* A "not czero" Cardinal predicate *)
blanchet@54474: abbreviation Cnotzero where
blanchet@54474:   "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
blanchet@54474: 
blanchet@54474: (*helper*)
blanchet@54474: lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
traytel@55811:   unfolding Card_order_iff_ordIso_card_of czero_def by force
blanchet@54474: 
blanchet@54474: lemma czeroI:
blanchet@54474:   "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
blanchet@54474: using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
blanchet@54474: 
blanchet@54474: lemma czeroE:
blanchet@54474:   "r =o czero \<Longrightarrow> Field r = {}"
blanchet@54474: unfolding czero_def
blanchet@54474: by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
blanchet@54474: 
blanchet@54474: lemma Cnotzero_mono:
blanchet@54474:   "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
blanchet@54474: apply (rule ccontr)
blanchet@54474: apply auto
blanchet@54474: apply (drule czeroE)
blanchet@54474: apply (erule notE)
blanchet@54474: apply (erule czeroI)
blanchet@54474: apply (drule card_of_mono2)
blanchet@54474: apply (simp only: card_of_empty3)
blanchet@54474: done
blanchet@54474: 
wenzelm@60758: subsection \<open>(In)finite cardinals\<close>
blanchet@54474: 
blanchet@54474: definition cinfinite where
traytel@54578:   "cinfinite r = (\<not> finite (Field r))"
blanchet@54474: 
blanchet@54474: abbreviation Cinfinite where
blanchet@54474:   "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
blanchet@54474: 
blanchet@54474: definition cfinite where
blanchet@54474:   "cfinite r = finite (Field r)"
blanchet@54474: 
blanchet@54474: abbreviation Cfinite where
blanchet@54474:   "Cfinite r \<equiv> cfinite r \<and> Card_order r"
blanchet@54474: 
blanchet@54474: lemma Cfinite_ordLess_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r <o s"
blanchet@54474:   unfolding cfinite_def cinfinite_def
traytel@55811:   by (blast intro: finite_ordLess_infinite card_order_on_well_order_on)
blanchet@54474: 
traytel@54581: lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
traytel@54581: 
traytel@54581: lemma natLeq_cinfinite: "cinfinite natLeq"
traytel@55811: unfolding cinfinite_def Field_natLeq by (rule infinite_UNIV_nat)
traytel@54581: 
blanchet@54474: lemma natLeq_ordLeq_cinfinite:
blanchet@54474:   assumes inf: "Cinfinite r"
blanchet@54474:   shows "natLeq \<le>o r"
blanchet@54474: proof -
traytel@55811:   from inf have "natLeq \<le>o |Field r|" unfolding cinfinite_def
traytel@55811:     using infinite_iff_natLeq_ordLeq by blast
blanchet@54474:   also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
traytel@55811: unfolding cinfinite_def by (cases "Field r = {}") (auto dest: czeroE)
blanchet@54474: 
blanchet@54474: lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
traytel@55811: by (rule conjI[OF cinfinite_not_czero]) simp_all
blanchet@54474: 
blanchet@54474: lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
traytel@55811: using Card_order_ordIso2[of r1 r2] unfolding cinfinite_def ordIso_iff_ordLeq
traytel@55811: by (auto dest: card_of_ordLeq_infinite[OF card_of_mono2])
blanchet@54474: 
blanchet@54474: lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
traytel@55811: unfolding cinfinite_def by (auto dest: card_of_ordLeq_infinite[OF card_of_mono2])
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Binary sum\<close>
blanchet@54474: 
blanchet@54474: definition csum (infixr "+c" 65) where
blanchet@54474:   "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
blanchet@54474: 
blanchet@54474: lemma Field_csum: "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
blanchet@54474:   unfolding csum_def Field_card_of by auto
blanchet@54474: 
blanchet@54474: lemma Card_order_csum:
blanchet@54474:   "Card_order (r1 +c r2)"
blanchet@54474: unfolding csum_def by (simp add: card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma csum_Cnotzero1:
blanchet@54474:   "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
traytel@55811: unfolding csum_def using Cnotzero_imp_not_empty[of r1] Plus_eq_empty_conv[of "Field r1" "Field r2"]
traytel@55811:    card_of_ordIso_czero_iff_empty[of "Field r1 <+> Field r2"] by (auto intro: card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma card_order_csum:
blanchet@54474:   assumes "card_order r1" "card_order r2"
blanchet@54474:   shows "card_order (r1 +c r2)"
blanchet@54474: proof -
blanchet@54474:   have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
blanchet@54474:   thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cinfinite_csum:
blanchet@54474:   "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
blanchet@54474: unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
blanchet@54474: 
blanchet@54474: lemma Cinfinite_csum1:
blanchet@54474:   "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
traytel@55811: unfolding cinfinite_def csum_def by (rule conjI[OF _ card_of_Card_order]) (auto simp: Field_card_of)
blanchet@54474: 
blanchet@54480: lemma Cinfinite_csum:
blanchet@54480:   "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
traytel@55811: unfolding cinfinite_def csum_def by (rule conjI[OF _ card_of_Card_order]) (auto simp: Field_card_of)
blanchet@54480: 
blanchet@55851: lemma Cinfinite_csum_weak:
blanchet@54480:   "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
traytel@55811: by (erule Cinfinite_csum1)
blanchet@54480: 
blanchet@54474: lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
blanchet@54474: by (simp only: csum_def ordIso_Plus_cong)
blanchet@54474: 
blanchet@54474: lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
blanchet@54474: by (simp only: csum_def ordIso_Plus_cong1)
blanchet@54474: 
blanchet@54474: lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
blanchet@54474: by (simp only: csum_def ordIso_Plus_cong2)
blanchet@54474: 
blanchet@54474: lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
blanchet@54474: by (simp only: csum_def ordLeq_Plus_mono)
blanchet@54474: 
blanchet@54474: lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
blanchet@54474: by (simp only: csum_def ordLeq_Plus_mono1)
blanchet@54474: 
blanchet@54474: lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
blanchet@54474: by (simp only: csum_def ordLeq_Plus_mono2)
blanchet@54474: 
blanchet@54474: lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
blanchet@54474: by (simp only: csum_def Card_order_Plus1)
blanchet@54474: 
blanchet@54474: lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
blanchet@54474: by (simp only: csum_def Card_order_Plus2)
blanchet@54474: 
blanchet@54474: lemma csum_com: "p1 +c p2 =o p2 +c p1"
blanchet@54474: by (simp only: csum_def card_of_Plus_commute)
blanchet@54474: 
blanchet@54474: lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
blanchet@54474: by (simp only: csum_def Field_card_of card_of_Plus_assoc)
blanchet@54474: 
blanchet@54474: lemma Cfinite_csum: "\<lbrakk>Cfinite r; Cfinite s\<rbrakk> \<Longrightarrow> Cfinite (r +c s)"
blanchet@54474:   unfolding cfinite_def csum_def Field_card_of using card_of_card_order_on by simp
blanchet@54474: 
blanchet@54474: lemma csum_csum: "(r1 +c r2) +c (r3 +c r4) =o (r1 +c r3) +c (r2 +c r4)"
blanchet@54474: proof -
blanchet@54474:   have "(r1 +c r2) +c (r3 +c r4) =o r1 +c r2 +c (r3 +c r4)"
traytel@55811:     by (rule csum_assoc)
blanchet@54474:   also have "r1 +c r2 +c (r3 +c r4) =o r1 +c (r2 +c r3) +c r4"
traytel@55811:     by (intro csum_assoc csum_cong2 ordIso_symmetric)
blanchet@54474:   also have "r1 +c (r2 +c r3) +c r4 =o r1 +c (r3 +c r2) +c r4"
traytel@55811:     by (intro csum_com csum_cong1 csum_cong2)
blanchet@54474:   also have "r1 +c (r3 +c r2) +c r4 =o r1 +c r3 +c r2 +c r4"
traytel@55811:     by (intro csum_assoc csum_cong2 ordIso_symmetric)
blanchet@54474:   also have "r1 +c r3 +c r2 +c r4 =o (r1 +c r3) +c (r2 +c r4)"
traytel@55811:     by (intro csum_assoc ordIso_symmetric)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
blanchet@54474: by (simp only: csum_def Field_card_of card_of_refl)
blanchet@54474: 
blanchet@54474: lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
blanchet@54474: using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>One\<close>
blanchet@54474: 
blanchet@54474: definition cone where
blanchet@54474:   "cone = card_of {()}"
blanchet@54474: 
blanchet@54474: lemma Card_order_cone: "Card_order cone"
blanchet@54474: unfolding cone_def by (rule card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma Cfinite_cone: "Cfinite cone"
blanchet@54474:   unfolding cfinite_def by (simp add: Card_order_cone)
blanchet@54474: 
blanchet@54474: lemma cone_not_czero: "\<not> (cone =o czero)"
traytel@55811: unfolding czero_def cone_def ordIso_iff_ordLeq using card_of_empty3 empty_not_insert by blast
blanchet@54474: 
blanchet@54474: lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
traytel@55811: unfolding cone_def by (rule Card_order_singl_ordLeq) (auto intro: czeroI)
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Two\<close>
blanchet@54474: 
blanchet@54474: definition ctwo where
blanchet@54474:   "ctwo = |UNIV :: bool set|"
blanchet@54474: 
blanchet@54474: lemma Card_order_ctwo: "Card_order ctwo"
blanchet@54474: unfolding ctwo_def by (rule card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
blanchet@54474: using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
traytel@55811: unfolding czero_def ctwo_def using UNIV_not_empty by auto
blanchet@54474: 
blanchet@54474: lemma ctwo_Cnotzero: "Cnotzero ctwo"
blanchet@54474: by (simp add: ctwo_not_czero Card_order_ctwo)
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Family sum\<close>
blanchet@54474: 
blanchet@54474: definition Csum where
blanchet@54474:   "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
blanchet@54474: 
blanchet@54474: (* Similar setup to the one for SIGMA from theory Big_Operators: *)
blanchet@54474: syntax "_Csum" ::
blanchet@54474:   "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
blanchet@54474:   ("(3CSUM _:_. _)" [0, 51, 10] 10)
blanchet@54474: 
blanchet@54474: translations
blanchet@54474:   "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
blanchet@54474: 
blanchet@54474: lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
blanchet@54474: by (auto simp: Csum_def Field_card_of)
blanchet@54474: 
blanchet@54474: (* NB: Always, under the cardinal operator,
blanchet@54474: operations on sets are reduced automatically to operations on cardinals.
blanchet@54474: This should make cardinal reasoning more direct and natural.  *)
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Product\<close>
blanchet@54474: 
blanchet@54474: definition cprod (infixr "*c" 80) where
wenzelm@61943:   "r1 *c r2 = |Field r1 \<times> Field r2|"
blanchet@54474: 
blanchet@54474: lemma card_order_cprod:
blanchet@54474:   assumes "card_order r1" "card_order r2"
blanchet@54474:   shows "card_order (r1 *c r2)"
blanchet@54474: proof -
blanchet@54474:   have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
blanchet@54474:   thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma Card_order_cprod: "Card_order (r1 *c r2)"
blanchet@54474: by (simp only: cprod_def Field_card_of card_of_card_order_on)
blanchet@54474: 
blanchet@54474: lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
blanchet@54474: by (simp only: cprod_def ordLeq_Times_mono1)
blanchet@54474: 
blanchet@54474: lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
blanchet@54474: by (simp only: cprod_def ordLeq_Times_mono2)
blanchet@54474: 
blanchet@55851: lemma cprod_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 *c p2 \<le>o r1 *c r2"
blanchet@55851: by (rule ordLeq_transitive[OF cprod_mono1 cprod_mono2])
blanchet@55851: 
blanchet@54474: lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
traytel@55811: unfolding cprod_def by (rule Card_order_Times2) (auto intro: czeroI)
blanchet@54474: 
blanchet@54474: lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
blanchet@54474: by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
blanchet@54474: 
blanchet@54474: lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
traytel@55811: by (rule cinfinite_mono) (auto intro: ordLeq_cprod2)
blanchet@54474: 
blanchet@54474: lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
blanchet@54474: by (blast intro: cinfinite_cprod2 Card_order_cprod)
blanchet@54474: 
blanchet@55851: lemma cprod_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 *c p2 =o r1 *c r2"
blanchet@55866: unfolding ordIso_iff_ordLeq by (blast intro: cprod_mono)
blanchet@55851: 
blanchet@55851: lemma cprod_cong1: "\<lbrakk>p1 =o r1\<rbrakk> \<Longrightarrow> p1 *c p2 =o r1 *c p2"
blanchet@55866: unfolding ordIso_iff_ordLeq by (blast intro: cprod_mono1)
blanchet@55851: 
blanchet@55851: lemma cprod_cong2: "p2 =o r2 \<Longrightarrow> q *c p2 =o q *c r2"
blanchet@55866: unfolding ordIso_iff_ordLeq by (blast intro: cprod_mono2)
blanchet@55851: 
blanchet@54474: lemma cprod_com: "p1 *c p2 =o p2 *c p1"
blanchet@54474: by (simp only: cprod_def card_of_Times_commute)
blanchet@54474: 
blanchet@54474: lemma card_of_Csum_Times:
blanchet@54474:   "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
traytel@56191: by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_mono1)
blanchet@54474: 
blanchet@54474: lemma card_of_Csum_Times':
blanchet@54474:   assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@54474:   shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
blanchet@54474: proof -
blanchet@54474:   from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
blanchet@54474:   with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
blanchet@54474:   hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
blanchet@54474:   also from * have "|I| *c |Field r| \<le>o |I| *c r"
blanchet@54474:     by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
blanchet@54474: unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
blanchet@54474: 
blanchet@54474: lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
traytel@55811: unfolding csum_def by (rule conjunct2[OF Card_order_Plus_infinite])
traytel@55811:   (auto simp: cinfinite_def dest: cinfinite_mono)
blanchet@54474: 
blanchet@54474: lemma csum_absorb1':
blanchet@54474:   assumes card: "Card_order r2"
blanchet@54474:   and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
blanchet@54474:   shows "r2 +c r1 =o r2"
blanchet@54474: by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
blanchet@54474: 
blanchet@54474: lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
blanchet@54474: by (rule csum_absorb1') auto
blanchet@54474: 
blanchet@54474: 
wenzelm@60758: subsection \<open>Exponentiation\<close>
blanchet@54474: 
blanchet@54474: definition cexp (infixr "^c" 90) where
blanchet@54474:   "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
blanchet@54474: 
blanchet@54474: lemma Card_order_cexp: "Card_order (r1 ^c r2)"
blanchet@54474: unfolding cexp_def by (rule card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma cexp_mono':
blanchet@54474:   assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
blanchet@54474:   and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
blanchet@54474:   shows "p1 ^c p2 \<le>o r1 ^c r2"
blanchet@54474: proof(cases "Field p1 = {}")
blanchet@54474:   case True
traytel@55811:   hence "Field p2 \<noteq> {} \<Longrightarrow> Func (Field p2) {} = {}" unfolding Func_is_emp by simp
traytel@55811:   with True have "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
blanchet@54474:     unfolding cone_def Field_card_of
traytel@55811:     by (cases "Field p2 = {}", auto intro: surj_imp_ordLeq simp: Func_empty)
blanchet@54474:   hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
blanchet@54474:   hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
blanchet@54474:   thus ?thesis
blanchet@54474:   proof (cases "Field p2 = {}")
blanchet@54474:     case True
blanchet@54474:     with n have "Field r2 = {}" .
noschinl@55604:     hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def
noschinl@55604:       by (auto intro: card_of_ordLeqI[where f="\<lambda>_ _. undefined"])
wenzelm@60758:     thus ?thesis using \<open>p1 ^c p2 \<le>o cone\<close> ordLeq_transitive by auto
blanchet@54474:   next
blanchet@54474:     case False with True have "|Field (p1 ^c p2)| =o czero"
blanchet@54474:       unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
blanchet@54474:     thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
blanchet@54474:       by (simp add: card_of_empty)
blanchet@54474:   qed
blanchet@54474: next
blanchet@54474:   case False
blanchet@54474:   have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
blanchet@54474:     using 1 2 by (auto simp: card_of_mono2)
blanchet@54474:   obtain f1 where f1: "f1 ` Field r1 = Field p1"
blanchet@54474:     using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
blanchet@54474:   obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
blanchet@54474:     using 2 unfolding card_of_ordLeq[symmetric] by blast
blanchet@54474:   have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
blanchet@54474:     unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
blanchet@54474:   have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
blanchet@54474:     using False by simp
blanchet@54474:   show ?thesis
blanchet@54474:     using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cexp_mono:
blanchet@54474:   assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
blanchet@54474:   and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
blanchet@54474:   shows "p1 ^c p2 \<le>o r1 ^c r2"
traytel@55811:   by (rule cexp_mono'[OF 1 2 czeroE[OF n[OF czeroI[OF card]]]])
blanchet@54474: 
blanchet@54474: lemma cexp_mono1:
blanchet@54474:   assumes 1: "p1 \<le>o r1" and q: "Card_order q"
blanchet@54474:   shows "p1 ^c q \<le>o r1 ^c q"
blanchet@54474: using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
blanchet@54474: 
blanchet@54474: lemma cexp_mono2':
blanchet@54474:   assumes 2: "p2 \<le>o r2" and q: "Card_order q"
blanchet@54474:   and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
blanchet@54474:   shows "q ^c p2 \<le>o q ^c r2"
blanchet@54474: using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
blanchet@54474: 
blanchet@54474: lemma cexp_mono2:
blanchet@54474:   assumes 2: "p2 \<le>o r2" and q: "Card_order q"
blanchet@54474:   and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
blanchet@54474:   shows "q ^c p2 \<le>o q ^c r2"
blanchet@54474: using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
blanchet@54474: 
blanchet@54474: lemma cexp_mono2_Cnotzero:
blanchet@54474:   assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
blanchet@54474:   shows "q ^c p2 \<le>o q ^c r2"
traytel@55811: using assms(3) czeroI by (blast intro: cexp_mono2'[OF assms(1,2)])
blanchet@54474: 
blanchet@54474: lemma cexp_cong:
blanchet@54474:   assumes 1: "p1 =o r1" and 2: "p2 =o r2"
blanchet@54474:   and Cr: "Card_order r2"
blanchet@54474:   and Cp: "Card_order p2"
blanchet@54474:   shows "p1 ^c p2 =o r1 ^c r2"
blanchet@54474: proof -
blanchet@54474:   obtain f where "bij_betw f (Field p2) (Field r2)"
blanchet@54474:     using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
blanchet@54474:   hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
blanchet@54474:   have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
blanchet@54474:     and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
blanchet@54474:      using 0 Cr Cp czeroE czeroI by auto
blanchet@54474:   show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
traytel@55811:     using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by blast
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cexp_cong1:
blanchet@54474:   assumes 1: "p1 =o r1" and q: "Card_order q"
blanchet@54474:   shows "p1 ^c q =o r1 ^c q"
blanchet@54474: by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
blanchet@54474: 
blanchet@54474: lemma cexp_cong2:
blanchet@54474:   assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
blanchet@54474:   shows "q ^c p2 =o q ^c r2"
blanchet@54474: by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
blanchet@54474: 
blanchet@54474: lemma cexp_cone:
blanchet@54474:   assumes "Card_order r"
blanchet@54474:   shows "r ^c cone =o r"
blanchet@54474: proof -
blanchet@54474:   have "r ^c cone =o |Field r|"
blanchet@54474:     unfolding cexp_def cone_def Field_card_of Func_empty
blanchet@54474:       card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
blanchet@54474:     by (rule exI[of _ "\<lambda>f. f ()"]) auto
blanchet@54474:   also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cexp_cprod:
blanchet@54474:   assumes r1: "Card_order r1"
blanchet@54474:   shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
blanchet@54474: proof -
blanchet@54474:   have "?L =o r1 ^c (r3 *c r2)"
blanchet@54474:     unfolding cprod_def cexp_def Field_card_of
blanchet@54474:     using card_of_Func_Times by(rule ordIso_symmetric)
blanchet@54474:   also have "r1 ^c (r3 *c r2) =o ?R"
blanchet@54474:     apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
blanchet@54474: unfolding cinfinite_def cprod_def
blanchet@54474: by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
blanchet@54474: 
blanchet@55851: lemma cprod_infinite: "Cinfinite r \<Longrightarrow> r *c r =o r"
blanchet@55851: using cprod_infinite1' Cinfinite_Cnotzero ordLeq_refl by blast
blanchet@55851: 
blanchet@54474: lemma cexp_cprod_ordLeq:
blanchet@54474:   assumes r1: "Card_order r1" and r2: "Cinfinite r2"
blanchet@54474:   and r3: "Cnotzero r3" "r3 \<le>o r2"
blanchet@54474:   shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
blanchet@54474: proof-
blanchet@54474:   have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
blanchet@54474:   also have "r1 ^c (r2 *c r3) =o ?R"
blanchet@54474:   apply(rule cexp_cong2)
blanchet@54474:   apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
blanchet@54474: by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
blanchet@54474: 
blanchet@54474: lemma ordLess_ctwo_cexp:
blanchet@54474:   assumes "Card_order r"
blanchet@54474:   shows "r <o ctwo ^c r"
blanchet@54474: proof -
blanchet@54474:   have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
blanchet@54474:   also have "|Pow (Field r)| =o ctwo ^c r"
blanchet@54474:     unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma ordLeq_cexp1:
blanchet@54474:   assumes "Cnotzero r" "Card_order q"
blanchet@54474:   shows "q \<le>o q ^c r"
blanchet@54474: proof (cases "q =o (czero :: 'a rel)")
blanchet@54474:   case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
blanchet@54474: next
blanchet@54474:   case False
blanchet@54474:   thus ?thesis
blanchet@54474:     apply -
blanchet@54474:     apply (rule ordIso_ordLeq_trans)
blanchet@54474:     apply (rule ordIso_symmetric)
blanchet@54474:     apply (rule cexp_cone)
blanchet@54474:     apply (rule assms(2))
blanchet@54474:     apply (rule cexp_mono2)
blanchet@54474:     apply (rule cone_ordLeq_Cnotzero)
blanchet@54474:     apply (rule assms(1))
blanchet@54474:     apply (rule assms(2))
blanchet@54474:     apply (rule notE)
blanchet@54474:     apply (rule cone_not_czero)
blanchet@54474:     apply assumption
blanchet@54474:     apply (rule Card_order_cone)
blanchet@54474:   done
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma ordLeq_cexp2:
blanchet@54474:   assumes "ctwo \<le>o q" "Card_order r"
blanchet@54474:   shows "r \<le>o q ^c r"
blanchet@54474: proof (cases "r =o (czero :: 'a rel)")
blanchet@54474:   case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
blanchet@54474: next
blanchet@54474:   case False thus ?thesis
blanchet@54474:     apply -
blanchet@54474:     apply (rule ordLess_imp_ordLeq)
blanchet@54474:     apply (rule ordLess_ordLeq_trans)
blanchet@54474:     apply (rule ordLess_ctwo_cexp)
blanchet@54474:     apply (rule assms(2))
blanchet@54474:     apply (rule cexp_mono1)
blanchet@54474:     apply (rule assms(1))
blanchet@54474:     apply (rule assms(2))
blanchet@54474:   done
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
traytel@55811: by (rule cinfinite_mono[OF ordLeq_cexp2]) simp_all
blanchet@54474: 
blanchet@54474: lemma Cinfinite_cexp:
blanchet@54474:   "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
blanchet@54474: by (simp add: cinfinite_cexp Card_order_cexp)
blanchet@54474: 
blanchet@54474: lemma ctwo_ordLess_natLeq: "ctwo <o natLeq"
traytel@54581: unfolding ctwo_def using finite_UNIV natLeq_cinfinite natLeq_Card_order
traytel@54581: by (intro Cfinite_ordLess_Cinfinite) (auto simp: cfinite_def card_of_Card_order)
blanchet@54474: 
blanchet@54474: lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
traytel@55811: by (rule ordLess_ordLeq_trans[OF ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite])
blanchet@54474: 
blanchet@54474: lemma ctwo_ordLeq_Cinfinite:
blanchet@54474:   assumes "Cinfinite r"
blanchet@54474:   shows "ctwo \<le>o r"
blanchet@54474: by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
blanchet@54474: 
blanchet@54474: lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
blanchet@54474: by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
blanchet@54474: 
blanchet@54474: 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"
blanchet@54474: by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
blanchet@54474: 
blanchet@54474: lemma csum_cinfinite_bound:
blanchet@54474:   assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
blanchet@54474:   shows "p +c q \<le>o r"
blanchet@54474: proof -
blanchet@54474:   from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
blanchet@54474:     unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
blanchet@54474:   with assms show ?thesis unfolding cinfinite_def csum_def
blanchet@54474:     by (blast intro: card_of_Plus_ordLeq_infinite_Field)
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cprod_cinfinite_bound:
blanchet@54474:   assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
blanchet@54474:   shows "p *c q \<le>o r"
blanchet@54474: proof -
blanchet@54474:   from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
blanchet@54474:     unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
blanchet@54474:   with assms show ?thesis unfolding cinfinite_def cprod_def
blanchet@54474:     by (blast intro: card_of_Times_ordLeq_infinite_Field)
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma cprod_csum_cexp:
blanchet@54474:   "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
blanchet@54474: unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
blanchet@54474: proof -
blanchet@54474:   let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
blanchet@54474:   have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
blanchet@54474:     by (auto simp: inj_on_def fun_eq_iff split: bool.split)
blanchet@54474:   moreover
blanchet@54474:   have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
blanchet@54474:     by (auto simp: Func_def)
blanchet@54474:   ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
blanchet@54474: by (intro cprod_cinfinite_bound)
blanchet@54474:   (auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
blanchet@54474: 
blanchet@54474: lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
blanchet@54474:   unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
blanchet@54474: 
blanchet@54474: lemma cprod_cexp_csum_cexp_Cinfinite:
blanchet@54474:   assumes t: "Cinfinite t"
blanchet@54474:   shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
blanchet@54474: proof -
blanchet@54474:   have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
blanchet@54474:     by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
blanchet@54474:   also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
blanchet@54474:     by (rule cexp_cprod[OF Card_order_csum])
blanchet@54474:   also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
blanchet@54474:     by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
blanchet@54474:   also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
blanchet@54474:     by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
blanchet@54474:   also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
blanchet@54474:     by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma Cfinite_cexp_Cinfinite:
blanchet@54474:   assumes s: "Cfinite s" and t: "Cinfinite t"
blanchet@54474:   shows "s ^c t \<le>o ctwo ^c t"
blanchet@54474: proof (cases "s \<le>o ctwo")
blanchet@54474:   case True thus ?thesis using t by (blast intro: cexp_mono1)
blanchet@54474: next
blanchet@54474:   case False
traytel@55811:   hence "ctwo \<le>o s" using ordLeq_total[of s ctwo] Card_order_ctwo s
traytel@55811:     by (auto intro: card_order_on_well_order_on)
traytel@55811:   hence "Cnotzero s" using Cnotzero_mono[OF ctwo_Cnotzero] s by blast
traytel@55811:   hence st: "Cnotzero (s *c t)" by (intro Cinfinite_Cnotzero[OF Cinfinite_cprod2]) (auto simp: t)
blanchet@54474:   have "s ^c t \<le>o (ctwo ^c s) ^c t"
blanchet@54474:     using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
blanchet@54474:   also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
blanchet@54474:     by (blast intro: Card_order_ctwo cexp_cprod)
blanchet@54474:   also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
blanchet@54474:     using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: lemma csum_Cfinite_cexp_Cinfinite:
blanchet@54474:   assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
blanchet@54474:   shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
blanchet@54474: proof (cases "Cinfinite r")
blanchet@54474:   case True
blanchet@54474:   hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
blanchet@54474:   hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
blanchet@54474:   also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
blanchet@54474:   finally show ?thesis .
blanchet@54474: next
blanchet@54474:   case False
blanchet@54474:   with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
blanchet@54474:   hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
blanchet@54474:   hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
blanchet@54474:   also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
blanchet@54474:     by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
blanchet@54474:   finally show ?thesis .
blanchet@54474: qed
blanchet@54474: 
blanchet@54474: (* cardSuc *)
blanchet@54474: 
blanchet@54474: lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
blanchet@54474: by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
blanchet@54474: 
blanchet@54474: lemma cardSuc_UNION_Cinfinite:
blanchet@54474:   assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
blanchet@54474:   shows "EX i : Field (cardSuc r). B \<le> As i"
blanchet@54474: using cardSuc_UNION assms unfolding cinfinite_def by blast
blanchet@54474: 
blanchet@54474: end