src/HOL/BNF_Cardinal_Arithmetic.thy
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(*  Title:      HOL/BNF_Cardinal_Arithmetic.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Copyright   2012
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Cardinal arithmetic as needed by bounded natural functors.
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*)
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header {* Cardinal Arithmetic as Needed by Bounded Natural Functors *}
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theory BNF_Cardinal_Arithmetic
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imports BNF_Cardinal_Order_Relation
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begin
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lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
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by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
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(*should supersede a weaker lemma from the library*)
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lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
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unfolding dir_image_def Field_def Range_def Domain_def by fast
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lemma card_order_dir_image:
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  assumes bij: "bij f" and co: "card_order r"
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  shows "card_order (dir_image r f)"
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proof -
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  from assms have "Field (dir_image r f) = UNIV"
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    using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
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  moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
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  with co have "Card_order (dir_image r f)"
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    using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
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  ultimately show ?thesis by auto
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qed
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lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
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by (rule card_order_on_ordIso)
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lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
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by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
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lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
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by (simp only: ordIso_refl card_of_Card_order)
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lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
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using card_order_on_Card_order[of UNIV r] by simp
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lemma card_of_Times_Plus_distrib:
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  "|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
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proof -
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  let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
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  have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
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  thus ?thesis using card_of_ordIso by blast
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qed
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lemma Func_Times_Range:
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  "|Func A (B <*> C)| =o |Func A B <*> Func A C|" (is "|?LHS| =o |?RHS|")
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proof -
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  let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,
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                  \<lambda>x. if x \<in> A then snd (fg x) else undefined)"
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  let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"
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  have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def
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  apply safe
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     apply (simp add: Func_def fun_eq_iff)
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     apply (metis (no_types) pair_collapse)
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    apply (auto simp: Func_def fun_eq_iff)[2]
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  proof -
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    fix f g assume "f \<in> Func A B" "g \<in> Func A C"
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    thus "(f, g) \<in> ?F ` Func A (B \<times> C)"
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      by (intro image_eqI[of _ _ "?G (f, g)"]) (auto simp: Func_def)
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  qed
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  thus ?thesis using card_of_ordIso by blast
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qed
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subsection {* Zero *}
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definition czero where
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  "czero = card_of {}"
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lemma czero_ordIso:
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  "czero =o czero"
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using card_of_empty_ordIso by (simp add: czero_def)
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lemma card_of_ordIso_czero_iff_empty:
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  "|A| =o (czero :: 'b rel) \<longleftrightarrow> A = ({} :: 'a set)"
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unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl card_of_empty_ordIso)
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(* A "not czero" Cardinal predicate *)
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abbreviation Cnotzero where
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  "Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
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(*helper*)
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lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
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by (metis Card_order_iff_ordIso_card_of czero_def)
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lemma czeroI:
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  "\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
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using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
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lemma czeroE:
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  "r =o czero \<Longrightarrow> Field r = {}"
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unfolding czero_def
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by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
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lemma Cnotzero_mono:
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  "\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
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apply (rule ccontr)
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apply auto
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apply (drule czeroE)
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apply (erule notE)
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apply (erule czeroI)
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apply (drule card_of_mono2)
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apply (simp only: card_of_empty3)
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done
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subsection {* (In)finite cardinals *}
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definition cinfinite where
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  "cinfinite r = (\<not> finite (Field r))"
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abbreviation Cinfinite where
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  "Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
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definition cfinite where
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  "cfinite r = finite (Field r)"
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abbreviation Cfinite where
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  "Cfinite r \<equiv> cfinite r \<and> Card_order r"
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lemma Cfinite_ordLess_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r <o s"
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  unfolding cfinite_def cinfinite_def
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  by (metis card_order_on_well_order_on finite_ordLess_infinite)
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lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
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lemma natLeq_cinfinite: "cinfinite natLeq"
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unfolding cinfinite_def Field_natLeq by (metis infinite_UNIV_nat)
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lemma natLeq_ordLeq_cinfinite:
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  assumes inf: "Cinfinite r"
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  shows "natLeq \<le>o r"
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proof -
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  from inf have "natLeq \<le>o |Field r|" by (metis cinfinite_def infinite_iff_natLeq_ordLeq)
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  also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
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  finally show ?thesis .
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qed
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lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
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unfolding cinfinite_def by (metis czeroE finite.emptyI)
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lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
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by (metis cinfinite_not_czero)
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lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
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by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
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lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
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by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
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subsection {* Binary sum *}
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definition csum (infixr "+c" 65) where
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  "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
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lemma Field_csum: "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
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  unfolding csum_def Field_card_of by auto
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lemma Card_order_csum:
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  "Card_order (r1 +c r2)"
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unfolding csum_def by (simp add: card_of_Card_order)
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lemma csum_Cnotzero1:
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  "Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
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unfolding csum_def
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by (metis Cnotzero_imp_not_empty Plus_eq_empty_conv card_of_Card_order card_of_ordIso_czero_iff_empty)
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lemma card_order_csum:
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  assumes "card_order r1" "card_order r2"
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  shows "card_order (r1 +c r2)"
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proof -
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  have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
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  thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
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qed
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lemma cinfinite_csum:
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  "cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
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unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
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lemma Cinfinite_csum1:
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  "Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
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unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
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lemma Cinfinite_csum:
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  "Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
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unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
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lemma Cinfinite_csum_strong:
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  "\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
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by (metis Cinfinite_csum)
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lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
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by (simp only: csum_def ordIso_Plus_cong)
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lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
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by (simp only: csum_def ordIso_Plus_cong1)
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lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
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by (simp only: csum_def ordIso_Plus_cong2)
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lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
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by (simp only: csum_def ordLeq_Plus_mono)
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lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
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by (simp only: csum_def ordLeq_Plus_mono1)
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lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
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by (simp only: csum_def ordLeq_Plus_mono2)
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lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
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by (simp only: csum_def Card_order_Plus1)
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lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
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by (simp only: csum_def Card_order_Plus2)
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lemma csum_com: "p1 +c p2 =o p2 +c p1"
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by (simp only: csum_def card_of_Plus_commute)
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lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
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by (simp only: csum_def Field_card_of card_of_Plus_assoc)
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lemma Cfinite_csum: "\<lbrakk>Cfinite r; Cfinite s\<rbrakk> \<Longrightarrow> Cfinite (r +c s)"
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  unfolding cfinite_def csum_def Field_card_of using card_of_card_order_on by simp
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lemma csum_csum: "(r1 +c r2) +c (r3 +c r4) =o (r1 +c r3) +c (r2 +c r4)"
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proof -
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  have "(r1 +c r2) +c (r3 +c r4) =o r1 +c r2 +c (r3 +c r4)"
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    by (metis csum_assoc)
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  also have "r1 +c r2 +c (r3 +c r4) =o r1 +c (r2 +c r3) +c r4"
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    by (metis csum_assoc csum_cong2 ordIso_symmetric)
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  also have "r1 +c (r2 +c r3) +c r4 =o r1 +c (r3 +c r2) +c r4"
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    by (metis csum_com csum_cong1 csum_cong2)
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  also have "r1 +c (r3 +c r2) +c r4 =o r1 +c r3 +c r2 +c r4"
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    by (metis csum_assoc csum_cong2 ordIso_symmetric)
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  also have "r1 +c r3 +c r2 +c r4 =o (r1 +c r3) +c (r2 +c r4)"
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    by (metis csum_assoc ordIso_symmetric)
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  finally show ?thesis .
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qed
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lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
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by (simp only: csum_def Field_card_of card_of_refl)
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lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
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using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
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subsection {* One *}
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definition cone where
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  "cone = card_of {()}"
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lemma Card_order_cone: "Card_order cone"
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unfolding cone_def by (rule card_of_Card_order)
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lemma Cfinite_cone: "Cfinite cone"
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  unfolding cfinite_def by (simp add: Card_order_cone)
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lemma cone_not_czero: "\<not> (cone =o czero)"
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unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
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lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
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unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
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subsection {* Two *}
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definition ctwo where
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  "ctwo = |UNIV :: bool set|"
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lemma Card_order_ctwo: "Card_order ctwo"
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unfolding ctwo_def by (rule card_of_Card_order)
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lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
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using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
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unfolding czero_def ctwo_def by (metis UNIV_not_empty)
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lemma ctwo_Cnotzero: "Cnotzero ctwo"
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by (simp add: ctwo_not_czero Card_order_ctwo)
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subsection {* Family sum *}
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definition Csum where
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  "Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
blanchet@54474
   293
blanchet@54474
   294
(* Similar setup to the one for SIGMA from theory Big_Operators: *)
blanchet@54474
   295
syntax "_Csum" ::
blanchet@54474
   296
  "pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
blanchet@54474
   297
  ("(3CSUM _:_. _)" [0, 51, 10] 10)
blanchet@54474
   298
blanchet@54474
   299
translations
blanchet@54474
   300
  "CSUM i:r. rs" == "CONST Csum r (%i. rs)"
blanchet@54474
   301
blanchet@54474
   302
lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
blanchet@54474
   303
by (auto simp: Csum_def Field_card_of)
blanchet@54474
   304
blanchet@54474
   305
(* NB: Always, under the cardinal operator,
blanchet@54474
   306
operations on sets are reduced automatically to operations on cardinals.
blanchet@54474
   307
This should make cardinal reasoning more direct and natural.  *)
blanchet@54474
   308
blanchet@54474
   309
blanchet@54474
   310
subsection {* Product *}
blanchet@54474
   311
blanchet@54474
   312
definition cprod (infixr "*c" 80) where
blanchet@54474
   313
  "r1 *c r2 = |Field r1 <*> Field r2|"
blanchet@54474
   314
blanchet@54474
   315
lemma card_order_cprod:
blanchet@54474
   316
  assumes "card_order r1" "card_order r2"
blanchet@54474
   317
  shows "card_order (r1 *c r2)"
blanchet@54474
   318
proof -
blanchet@54474
   319
  have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
blanchet@54474
   320
  thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
blanchet@54474
   321
qed
blanchet@54474
   322
blanchet@54474
   323
lemma Card_order_cprod: "Card_order (r1 *c r2)"
blanchet@54474
   324
by (simp only: cprod_def Field_card_of card_of_card_order_on)
blanchet@54474
   325
blanchet@54474
   326
lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
blanchet@54474
   327
by (simp only: cprod_def ordLeq_Times_mono1)
blanchet@54474
   328
blanchet@54474
   329
lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
blanchet@54474
   330
by (simp only: cprod_def ordLeq_Times_mono2)
blanchet@54474
   331
blanchet@54474
   332
lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
blanchet@54474
   333
unfolding cprod_def by (metis Card_order_Times2 czeroI)
blanchet@54474
   334
blanchet@54474
   335
lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
blanchet@54474
   336
by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
blanchet@54474
   337
blanchet@54474
   338
lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
blanchet@54474
   339
by (metis cinfinite_mono ordLeq_cprod2)
blanchet@54474
   340
blanchet@54474
   341
lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
blanchet@54474
   342
by (blast intro: cinfinite_cprod2 Card_order_cprod)
blanchet@54474
   343
blanchet@54474
   344
lemma cprod_com: "p1 *c p2 =o p2 *c p1"
blanchet@54474
   345
by (simp only: cprod_def card_of_Times_commute)
blanchet@54474
   346
blanchet@54474
   347
lemma card_of_Csum_Times:
blanchet@54474
   348
  "\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
blanchet@54474
   349
by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
blanchet@54474
   350
blanchet@54474
   351
lemma card_of_Csum_Times':
blanchet@54474
   352
  assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@54474
   353
  shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
blanchet@54474
   354
proof -
blanchet@54474
   355
  from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
blanchet@54474
   356
  with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
blanchet@54474
   357
  hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
blanchet@54474
   358
  also from * have "|I| *c |Field r| \<le>o |I| *c r"
blanchet@54474
   359
    by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
blanchet@54474
   360
  finally show ?thesis .
blanchet@54474
   361
qed
blanchet@54474
   362
blanchet@54474
   363
lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
blanchet@54474
   364
unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
blanchet@54474
   365
blanchet@54474
   366
lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
blanchet@54474
   367
unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
blanchet@54474
   368
blanchet@54474
   369
lemma csum_absorb1':
blanchet@54474
   370
  assumes card: "Card_order r2"
blanchet@54474
   371
  and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
blanchet@54474
   372
  shows "r2 +c r1 =o r2"
blanchet@54474
   373
by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
blanchet@54474
   374
blanchet@54474
   375
lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
blanchet@54474
   376
by (rule csum_absorb1') auto
blanchet@54474
   377
blanchet@54474
   378
blanchet@54474
   379
subsection {* Exponentiation *}
blanchet@54474
   380
blanchet@54474
   381
definition cexp (infixr "^c" 90) where
blanchet@54474
   382
  "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
blanchet@54474
   383
blanchet@54474
   384
lemma Card_order_cexp: "Card_order (r1 ^c r2)"
blanchet@54474
   385
unfolding cexp_def by (rule card_of_Card_order)
blanchet@54474
   386
blanchet@54474
   387
lemma cexp_mono':
blanchet@54474
   388
  assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
blanchet@54474
   389
  and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
blanchet@54474
   390
  shows "p1 ^c p2 \<le>o r1 ^c r2"
blanchet@54474
   391
proof(cases "Field p1 = {}")
blanchet@54474
   392
  case True
blanchet@54474
   393
  hence "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
blanchet@54474
   394
    unfolding cone_def Field_card_of
blanchet@54474
   395
    by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
blanchet@54474
   396
       (metis Func_is_emp card_of_empty ex_in_conv)
blanchet@54474
   397
  hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
blanchet@54474
   398
  hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
blanchet@54474
   399
  thus ?thesis
blanchet@54474
   400
  proof (cases "Field p2 = {}")
blanchet@54474
   401
    case True
blanchet@54474
   402
    with n have "Field r2 = {}" .
noschinl@55604
   403
    hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def
noschinl@55604
   404
      by (auto intro: card_of_ordLeqI[where f="\<lambda>_ _. undefined"])
blanchet@54474
   405
    thus ?thesis using `p1 ^c p2 \<le>o cone` ordLeq_transitive by auto
blanchet@54474
   406
  next
blanchet@54474
   407
    case False with True have "|Field (p1 ^c p2)| =o czero"
blanchet@54474
   408
      unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
blanchet@54474
   409
    thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
blanchet@54474
   410
      by (simp add: card_of_empty)
blanchet@54474
   411
  qed
blanchet@54474
   412
next
blanchet@54474
   413
  case False
blanchet@54474
   414
  have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
blanchet@54474
   415
    using 1 2 by (auto simp: card_of_mono2)
blanchet@54474
   416
  obtain f1 where f1: "f1 ` Field r1 = Field p1"
blanchet@54474
   417
    using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
blanchet@54474
   418
  obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
blanchet@54474
   419
    using 2 unfolding card_of_ordLeq[symmetric] by blast
blanchet@54474
   420
  have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
blanchet@54474
   421
    unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
blanchet@54474
   422
  have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
blanchet@54474
   423
    using False by simp
blanchet@54474
   424
  show ?thesis
blanchet@54474
   425
    using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
blanchet@54474
   426
qed
blanchet@54474
   427
blanchet@54474
   428
lemma cexp_mono:
blanchet@54474
   429
  assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
blanchet@54474
   430
  and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
blanchet@54474
   431
  shows "p1 ^c p2 \<le>o r1 ^c r2"
blanchet@54474
   432
  by (metis (full_types) "1" "2" card cexp_mono' czeroE czeroI n)
blanchet@54474
   433
blanchet@54474
   434
lemma cexp_mono1:
blanchet@54474
   435
  assumes 1: "p1 \<le>o r1" and q: "Card_order q"
blanchet@54474
   436
  shows "p1 ^c q \<le>o r1 ^c q"
blanchet@54474
   437
using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
blanchet@54474
   438
blanchet@54474
   439
lemma cexp_mono2':
blanchet@54474
   440
  assumes 2: "p2 \<le>o r2" and q: "Card_order q"
blanchet@54474
   441
  and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
blanchet@54474
   442
  shows "q ^c p2 \<le>o q ^c r2"
blanchet@54474
   443
using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
blanchet@54474
   444
blanchet@54474
   445
lemma cexp_mono2:
blanchet@54474
   446
  assumes 2: "p2 \<le>o r2" and q: "Card_order q"
blanchet@54474
   447
  and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
blanchet@54474
   448
  shows "q ^c p2 \<le>o q ^c r2"
blanchet@54474
   449
using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
blanchet@54474
   450
blanchet@54474
   451
lemma cexp_mono2_Cnotzero:
blanchet@54474
   452
  assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
blanchet@54474
   453
  shows "q ^c p2 \<le>o q ^c r2"
blanchet@54474
   454
by (metis assms cexp_mono2' czeroI)
blanchet@54474
   455
blanchet@54474
   456
lemma cexp_cong:
blanchet@54474
   457
  assumes 1: "p1 =o r1" and 2: "p2 =o r2"
blanchet@54474
   458
  and Cr: "Card_order r2"
blanchet@54474
   459
  and Cp: "Card_order p2"
blanchet@54474
   460
  shows "p1 ^c p2 =o r1 ^c r2"
blanchet@54474
   461
proof -
blanchet@54474
   462
  obtain f where "bij_betw f (Field p2) (Field r2)"
blanchet@54474
   463
    using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
blanchet@54474
   464
  hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
blanchet@54474
   465
  have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
blanchet@54474
   466
    and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
blanchet@54474
   467
     using 0 Cr Cp czeroE czeroI by auto
blanchet@54474
   468
  show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
blanchet@54482
   469
    using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by metis
blanchet@54474
   470
qed
blanchet@54474
   471
blanchet@54474
   472
lemma cexp_cong1:
blanchet@54474
   473
  assumes 1: "p1 =o r1" and q: "Card_order q"
blanchet@54474
   474
  shows "p1 ^c q =o r1 ^c q"
blanchet@54474
   475
by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
blanchet@54474
   476
blanchet@54474
   477
lemma cexp_cong2:
blanchet@54474
   478
  assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
blanchet@54474
   479
  shows "q ^c p2 =o q ^c r2"
blanchet@54474
   480
by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
blanchet@54474
   481
blanchet@54474
   482
lemma cexp_cone:
blanchet@54474
   483
  assumes "Card_order r"
blanchet@54474
   484
  shows "r ^c cone =o r"
blanchet@54474
   485
proof -
blanchet@54474
   486
  have "r ^c cone =o |Field r|"
blanchet@54474
   487
    unfolding cexp_def cone_def Field_card_of Func_empty
blanchet@54474
   488
      card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
blanchet@54474
   489
    by (rule exI[of _ "\<lambda>f. f ()"]) auto
blanchet@54474
   490
  also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
blanchet@54474
   491
  finally show ?thesis .
blanchet@54474
   492
qed
blanchet@54474
   493
blanchet@54474
   494
lemma cexp_cprod:
blanchet@54474
   495
  assumes r1: "Card_order r1"
blanchet@54474
   496
  shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
blanchet@54474
   497
proof -
blanchet@54474
   498
  have "?L =o r1 ^c (r3 *c r2)"
blanchet@54474
   499
    unfolding cprod_def cexp_def Field_card_of
blanchet@54474
   500
    using card_of_Func_Times by(rule ordIso_symmetric)
blanchet@54474
   501
  also have "r1 ^c (r3 *c r2) =o ?R"
blanchet@54474
   502
    apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
blanchet@54474
   503
  finally show ?thesis .
blanchet@54474
   504
qed
blanchet@54474
   505
blanchet@54474
   506
lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
blanchet@54474
   507
unfolding cinfinite_def cprod_def
blanchet@54474
   508
by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
blanchet@54474
   509
blanchet@54474
   510
lemma cexp_cprod_ordLeq:
blanchet@54474
   511
  assumes r1: "Card_order r1" and r2: "Cinfinite r2"
blanchet@54474
   512
  and r3: "Cnotzero r3" "r3 \<le>o r2"
blanchet@54474
   513
  shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
blanchet@54474
   514
proof-
blanchet@54474
   515
  have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
blanchet@54474
   516
  also have "r1 ^c (r2 *c r3) =o ?R"
blanchet@54474
   517
  apply(rule cexp_cong2)
blanchet@54474
   518
  apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
blanchet@54474
   519
  finally show ?thesis .
blanchet@54474
   520
qed
blanchet@54474
   521
blanchet@54474
   522
lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
blanchet@54474
   523
by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
blanchet@54474
   524
blanchet@54474
   525
lemma ordLess_ctwo_cexp:
blanchet@54474
   526
  assumes "Card_order r"
blanchet@54474
   527
  shows "r <o ctwo ^c r"
blanchet@54474
   528
proof -
blanchet@54474
   529
  have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
blanchet@54474
   530
  also have "|Pow (Field r)| =o ctwo ^c r"
blanchet@54474
   531
    unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
blanchet@54474
   532
  finally show ?thesis .
blanchet@54474
   533
qed
blanchet@54474
   534
blanchet@54474
   535
lemma ordLeq_cexp1:
blanchet@54474
   536
  assumes "Cnotzero r" "Card_order q"
blanchet@54474
   537
  shows "q \<le>o q ^c r"
blanchet@54474
   538
proof (cases "q =o (czero :: 'a rel)")
blanchet@54474
   539
  case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
blanchet@54474
   540
next
blanchet@54474
   541
  case False
blanchet@54474
   542
  thus ?thesis
blanchet@54474
   543
    apply -
blanchet@54474
   544
    apply (rule ordIso_ordLeq_trans)
blanchet@54474
   545
    apply (rule ordIso_symmetric)
blanchet@54474
   546
    apply (rule cexp_cone)
blanchet@54474
   547
    apply (rule assms(2))
blanchet@54474
   548
    apply (rule cexp_mono2)
blanchet@54474
   549
    apply (rule cone_ordLeq_Cnotzero)
blanchet@54474
   550
    apply (rule assms(1))
blanchet@54474
   551
    apply (rule assms(2))
blanchet@54474
   552
    apply (rule notE)
blanchet@54474
   553
    apply (rule cone_not_czero)
blanchet@54474
   554
    apply assumption
blanchet@54474
   555
    apply (rule Card_order_cone)
blanchet@54474
   556
  done
blanchet@54474
   557
qed
blanchet@54474
   558
blanchet@54474
   559
lemma ordLeq_cexp2:
blanchet@54474
   560
  assumes "ctwo \<le>o q" "Card_order r"
blanchet@54474
   561
  shows "r \<le>o q ^c r"
blanchet@54474
   562
proof (cases "r =o (czero :: 'a rel)")
blanchet@54474
   563
  case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
blanchet@54474
   564
next
blanchet@54474
   565
  case False thus ?thesis
blanchet@54474
   566
    apply -
blanchet@54474
   567
    apply (rule ordLess_imp_ordLeq)
blanchet@54474
   568
    apply (rule ordLess_ordLeq_trans)
blanchet@54474
   569
    apply (rule ordLess_ctwo_cexp)
blanchet@54474
   570
    apply (rule assms(2))
blanchet@54474
   571
    apply (rule cexp_mono1)
blanchet@54474
   572
    apply (rule assms(1))
blanchet@54474
   573
    apply (rule assms(2))
blanchet@54474
   574
  done
blanchet@54474
   575
qed
blanchet@54474
   576
blanchet@54474
   577
lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
blanchet@54474
   578
by (metis assms cinfinite_mono ordLeq_cexp2)
blanchet@54474
   579
blanchet@54474
   580
lemma Cinfinite_cexp:
blanchet@54474
   581
  "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
blanchet@54474
   582
by (simp add: cinfinite_cexp Card_order_cexp)
blanchet@54474
   583
blanchet@54474
   584
lemma ctwo_ordLess_natLeq: "ctwo <o natLeq"
traytel@54581
   585
unfolding ctwo_def using finite_UNIV natLeq_cinfinite natLeq_Card_order
traytel@54581
   586
by (intro Cfinite_ordLess_Cinfinite) (auto simp: cfinite_def card_of_Card_order)
blanchet@54474
   587
blanchet@54474
   588
lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
blanchet@54474
   589
by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
blanchet@54474
   590
blanchet@54474
   591
lemma ctwo_ordLeq_Cinfinite:
blanchet@54474
   592
  assumes "Cinfinite r"
blanchet@54474
   593
  shows "ctwo \<le>o r"
blanchet@54474
   594
by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
blanchet@54474
   595
blanchet@54474
   596
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
   597
by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
blanchet@54474
   598
blanchet@54474
   599
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
   600
by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
blanchet@54474
   601
blanchet@54474
   602
lemma csum_cinfinite_bound:
blanchet@54474
   603
  assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
blanchet@54474
   604
  shows "p +c q \<le>o r"
blanchet@54474
   605
proof -
blanchet@54474
   606
  from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
blanchet@54474
   607
    unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
blanchet@54474
   608
  with assms show ?thesis unfolding cinfinite_def csum_def
blanchet@54474
   609
    by (blast intro: card_of_Plus_ordLeq_infinite_Field)
blanchet@54474
   610
qed
blanchet@54474
   611
blanchet@54474
   612
lemma cprod_cinfinite_bound:
blanchet@54474
   613
  assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
blanchet@54474
   614
  shows "p *c q \<le>o r"
blanchet@54474
   615
proof -
blanchet@54474
   616
  from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
blanchet@54474
   617
    unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
blanchet@54474
   618
  with assms show ?thesis unfolding cinfinite_def cprod_def
blanchet@54474
   619
    by (blast intro: card_of_Times_ordLeq_infinite_Field)
blanchet@54474
   620
qed
blanchet@54474
   621
blanchet@54474
   622
lemma cprod_csum_cexp:
blanchet@54474
   623
  "r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
blanchet@54474
   624
unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
blanchet@54474
   625
proof -
blanchet@54474
   626
  let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
blanchet@54474
   627
  have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
blanchet@54474
   628
    by (auto simp: inj_on_def fun_eq_iff split: bool.split)
blanchet@54474
   629
  moreover
blanchet@54474
   630
  have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
blanchet@54474
   631
    by (auto simp: Func_def)
blanchet@54474
   632
  ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
blanchet@54474
   633
qed
blanchet@54474
   634
blanchet@54474
   635
lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
blanchet@54474
   636
by (intro cprod_cinfinite_bound)
blanchet@54474
   637
  (auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
blanchet@54474
   638
blanchet@54474
   639
lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
blanchet@54474
   640
  unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
blanchet@54474
   641
blanchet@54474
   642
lemma cprod_cexp_csum_cexp_Cinfinite:
blanchet@54474
   643
  assumes t: "Cinfinite t"
blanchet@54474
   644
  shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
blanchet@54474
   645
proof -
blanchet@54474
   646
  have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
blanchet@54474
   647
    by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
blanchet@54474
   648
  also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
blanchet@54474
   649
    by (rule cexp_cprod[OF Card_order_csum])
blanchet@54474
   650
  also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
blanchet@54474
   651
    by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
blanchet@54474
   652
  also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
blanchet@54474
   653
    by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
blanchet@54474
   654
  also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
blanchet@54474
   655
    by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
blanchet@54474
   656
  finally show ?thesis .
blanchet@54474
   657
qed
blanchet@54474
   658
blanchet@54474
   659
lemma Cfinite_cexp_Cinfinite:
blanchet@54474
   660
  assumes s: "Cfinite s" and t: "Cinfinite t"
blanchet@54474
   661
  shows "s ^c t \<le>o ctwo ^c t"
blanchet@54474
   662
proof (cases "s \<le>o ctwo")
blanchet@54474
   663
  case True thus ?thesis using t by (blast intro: cexp_mono1)
blanchet@54474
   664
next
blanchet@54474
   665
  case False
blanchet@54474
   666
  hence "ctwo \<le>o s" by (metis card_order_on_well_order_on ctwo_Cnotzero ordLeq_total s)
blanchet@54474
   667
  hence "Cnotzero s" by (metis Cnotzero_mono ctwo_Cnotzero s)
blanchet@54474
   668
  hence st: "Cnotzero (s *c t)" by (metis Cinfinite_cprod2 cinfinite_not_czero t)
blanchet@54474
   669
  have "s ^c t \<le>o (ctwo ^c s) ^c t"
blanchet@54474
   670
    using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
blanchet@54474
   671
  also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
blanchet@54474
   672
    by (blast intro: Card_order_ctwo cexp_cprod)
blanchet@54474
   673
  also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
blanchet@54474
   674
    using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
blanchet@54474
   675
  finally show ?thesis .
blanchet@54474
   676
qed
blanchet@54474
   677
blanchet@54474
   678
lemma csum_Cfinite_cexp_Cinfinite:
blanchet@54474
   679
  assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
blanchet@54474
   680
  shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
blanchet@54474
   681
proof (cases "Cinfinite r")
blanchet@54474
   682
  case True
blanchet@54474
   683
  hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
blanchet@54474
   684
  hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
blanchet@54474
   685
  also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
blanchet@54474
   686
  finally show ?thesis .
blanchet@54474
   687
next
blanchet@54474
   688
  case False
blanchet@54474
   689
  with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
blanchet@54474
   690
  hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
blanchet@54474
   691
  hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
blanchet@54474
   692
  also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
blanchet@54474
   693
    by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
blanchet@54474
   694
  finally show ?thesis .
blanchet@54474
   695
qed
blanchet@54474
   696
blanchet@54474
   697
(* cardSuc *)
blanchet@54474
   698
blanchet@54474
   699
lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
blanchet@54474
   700
by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
blanchet@54474
   701
blanchet@54474
   702
lemma cardSuc_UNION_Cinfinite:
blanchet@54474
   703
  assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
blanchet@54474
   704
  shows "EX i : Field (cardSuc r). B \<le> As i"
blanchet@54474
   705
using cardSuc_UNION assms unfolding cinfinite_def by blast
blanchet@54474
   706
blanchet@54474
   707
end