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(* Title: HOL/Cardinals/Cardinal_Arithmetic_LFP.thy
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Author: Dmitriy Traytel, TU Muenchen
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Copyright 2012
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Cardinal arithmetic.
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*)
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header {* Cardinal Arithmetic *}
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theory Cardinal_Arithmetic_LFP
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imports Cardinal_Order_Relation_LFP
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begin
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(*library candidate*)
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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 fastforce
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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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(*library candidate*)
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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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(*library candidate*)
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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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(*library candidate*)
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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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(*library candidate*)
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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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(*library candidate*)
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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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(*library candidate*)
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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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proof safe
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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 (auto simp: Func_def fun_eq_iff, metis pair_collapse)
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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 = infinite (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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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 (simp add: 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 Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)
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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 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)|"
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(* Similar setup to the one for SIGMA from theory Big_Operators: *)
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syntax "_Csum" ::
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"pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
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("(3CSUM _:_. _)" [0, 51, 10] 10)
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translations
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"CSUM i:r. rs" == "CONST Csum r (%i. rs)"
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lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
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by (auto simp: Csum_def Field_card_of)
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(* NB: Always, under the cardinal operator,
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operations on sets are reduced automatically to operations on cardinals.
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This should make cardinal reasoning more direct and natural. *)
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subsection {* Product *}
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definition cprod (infixr "*c" 80) where
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"r1 *c r2 = |Field r1 <*> Field r2|"
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lemma card_order_cprod:
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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 by (auto simp: cprod_def card_of_card_order_on)
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qed
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lemma Card_order_cprod: "Card_order (r1 *c r2)"
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by (simp only: cprod_def Field_card_of card_of_card_order_on)
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lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
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by (simp only: cprod_def ordLeq_Times_mono1)
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lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
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by (simp only: cprod_def ordLeq_Times_mono2)
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lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
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unfolding cprod_def by (metis Card_order_Times2 czeroI)
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lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
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326 |
by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
|
|
327 |
|
|
328 |
lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
|
|
329 |
by (metis cinfinite_mono ordLeq_cprod2)
|
|
330 |
|
|
331 |
lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
|
|
332 |
by (blast intro: cinfinite_cprod2 Card_order_cprod)
|
|
333 |
|
|
334 |
lemma cprod_com: "p1 *c p2 =o p2 *c p1"
|
|
335 |
by (simp only: cprod_def card_of_Times_commute)
|
|
336 |
|
|
337 |
lemma card_of_Csum_Times:
|
|
338 |
"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
|
|
339 |
by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
|
|
340 |
|
|
341 |
lemma card_of_Csum_Times':
|
|
342 |
assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
|
|
343 |
shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
|
|
344 |
proof -
|
|
345 |
from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
|
|
346 |
with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
|
|
347 |
hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
|
|
348 |
also from * have "|I| *c |Field r| \<le>o |I| *c r"
|
|
349 |
by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
|
|
350 |
finally show ?thesis .
|
|
351 |
qed
|
|
352 |
|
|
353 |
lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
|
|
354 |
unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
|
|
355 |
|
|
356 |
lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
|
|
357 |
unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
|
|
358 |
|
|
359 |
lemma csum_absorb1':
|
|
360 |
assumes card: "Card_order r2"
|
|
361 |
and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
|
|
362 |
shows "r2 +c r1 =o r2"
|
|
363 |
by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
|
|
364 |
|
|
365 |
lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
|
|
366 |
by (rule csum_absorb1') auto
|
|
367 |
|
|
368 |
|
|
369 |
subsection {* Exponentiation *}
|
|
370 |
|
|
371 |
definition cexp (infixr "^c" 90) where
|
|
372 |
"r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
|
|
373 |
|
|
374 |
lemma Card_order_cexp: "Card_order (r1 ^c r2)"
|
|
375 |
unfolding cexp_def by (rule card_of_Card_order)
|
|
376 |
|
|
377 |
lemma cexp_mono':
|
|
378 |
assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
|
|
379 |
and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
|
|
380 |
shows "p1 ^c p2 \<le>o r1 ^c r2"
|
|
381 |
proof(cases "Field p1 = {}")
|
|
382 |
case True
|
|
383 |
hence "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
|
|
384 |
unfolding cone_def Field_card_of
|
|
385 |
by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
|
|
386 |
(metis Func_is_emp card_of_empty ex_in_conv)
|
|
387 |
hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
|
|
388 |
hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
|
|
389 |
thus ?thesis
|
|
390 |
proof (cases "Field p2 = {}")
|
|
391 |
case True
|
|
392 |
with n have "Field r2 = {}" .
|
|
393 |
hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def by (auto intro: card_of_ordLeqI)
|
|
394 |
thus ?thesis using `p1 ^c p2 \<le>o cone` ordLeq_transitive by auto
|
|
395 |
next
|
|
396 |
case False with True have "|Field (p1 ^c p2)| =o czero"
|
|
397 |
unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
|
|
398 |
thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
|
|
399 |
by (simp add: card_of_empty)
|
|
400 |
qed
|
|
401 |
next
|
|
402 |
case False
|
|
403 |
have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
|
|
404 |
using 1 2 by (auto simp: card_of_mono2)
|
|
405 |
obtain f1 where f1: "f1 ` Field r1 = Field p1"
|
|
406 |
using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
|
|
407 |
obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
|
|
408 |
using 2 unfolding card_of_ordLeq[symmetric] by blast
|
|
409 |
have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
|
|
410 |
unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
|
|
411 |
have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
|
|
412 |
using False by simp
|
|
413 |
show ?thesis
|
|
414 |
using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
|
|
415 |
qed
|
|
416 |
|
|
417 |
lemma cexp_mono:
|
|
418 |
assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
|
|
419 |
and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
|
|
420 |
shows "p1 ^c p2 \<le>o r1 ^c r2"
|
|
421 |
by (metis (full_types) "1" "2" card cexp_mono' czeroE czeroI n)
|
|
422 |
|
|
423 |
lemma cexp_mono1:
|
|
424 |
assumes 1: "p1 \<le>o r1" and q: "Card_order q"
|
|
425 |
shows "p1 ^c q \<le>o r1 ^c q"
|
|
426 |
using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
|
|
427 |
|
|
428 |
lemma cexp_mono2':
|
|
429 |
assumes 2: "p2 \<le>o r2" and q: "Card_order q"
|
|
430 |
and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
|
|
431 |
shows "q ^c p2 \<le>o q ^c r2"
|
|
432 |
using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
|
|
433 |
|
|
434 |
lemma cexp_mono2:
|
|
435 |
assumes 2: "p2 \<le>o r2" and q: "Card_order q"
|
|
436 |
and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
|
|
437 |
shows "q ^c p2 \<le>o q ^c r2"
|
|
438 |
using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
|
|
439 |
|
|
440 |
lemma cexp_mono2_Cnotzero:
|
|
441 |
assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
|
|
442 |
shows "q ^c p2 \<le>o q ^c r2"
|
|
443 |
by (metis assms cexp_mono2' czeroI)
|
|
444 |
|
|
445 |
lemma cexp_cong:
|
|
446 |
assumes 1: "p1 =o r1" and 2: "p2 =o r2"
|
|
447 |
and Cr: "Card_order r2"
|
|
448 |
and Cp: "Card_order p2"
|
|
449 |
shows "p1 ^c p2 =o r1 ^c r2"
|
|
450 |
proof -
|
|
451 |
obtain f where "bij_betw f (Field p2) (Field r2)"
|
|
452 |
using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
|
|
453 |
hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
|
|
454 |
have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
|
|
455 |
and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
|
|
456 |
using 0 Cr Cp czeroE czeroI by auto
|
|
457 |
show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
|
|
458 |
using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by blast
|
|
459 |
qed
|
|
460 |
|
|
461 |
lemma cexp_cong1:
|
|
462 |
assumes 1: "p1 =o r1" and q: "Card_order q"
|
|
463 |
shows "p1 ^c q =o r1 ^c q"
|
|
464 |
by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
|
|
465 |
|
|
466 |
lemma cexp_cong2:
|
|
467 |
assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
|
|
468 |
shows "q ^c p2 =o q ^c r2"
|
|
469 |
by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
|
|
470 |
|
|
471 |
lemma cexp_cone:
|
|
472 |
assumes "Card_order r"
|
|
473 |
shows "r ^c cone =o r"
|
|
474 |
proof -
|
|
475 |
have "r ^c cone =o |Field r|"
|
|
476 |
unfolding cexp_def cone_def Field_card_of Func_empty
|
|
477 |
card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
|
|
478 |
by (rule exI[of _ "\<lambda>f. f ()"]) auto
|
|
479 |
also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
|
|
480 |
finally show ?thesis .
|
|
481 |
qed
|
|
482 |
|
|
483 |
lemma cexp_cprod:
|
|
484 |
assumes r1: "Card_order r1"
|
|
485 |
shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
|
|
486 |
proof -
|
|
487 |
have "?L =o r1 ^c (r3 *c r2)"
|
|
488 |
unfolding cprod_def cexp_def Field_card_of
|
|
489 |
using card_of_Func_Times by(rule ordIso_symmetric)
|
|
490 |
also have "r1 ^c (r3 *c r2) =o ?R"
|
|
491 |
apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
|
|
492 |
finally show ?thesis .
|
|
493 |
qed
|
|
494 |
|
|
495 |
lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
|
|
496 |
unfolding cinfinite_def cprod_def
|
|
497 |
by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
|
|
498 |
|
|
499 |
lemma cexp_cprod_ordLeq:
|
|
500 |
assumes r1: "Card_order r1" and r2: "Cinfinite r2"
|
|
501 |
and r3: "Cnotzero r3" "r3 \<le>o r2"
|
|
502 |
shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
|
|
503 |
proof-
|
|
504 |
have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
|
|
505 |
also have "r1 ^c (r2 *c r3) =o ?R"
|
|
506 |
apply(rule cexp_cong2)
|
|
507 |
apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
|
|
508 |
finally show ?thesis .
|
|
509 |
qed
|
|
510 |
|
|
511 |
lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
|
|
512 |
by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
|
|
513 |
|
|
514 |
lemma ordLess_ctwo_cexp:
|
|
515 |
assumes "Card_order r"
|
|
516 |
shows "r <o ctwo ^c r"
|
|
517 |
proof -
|
|
518 |
have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
|
|
519 |
also have "|Pow (Field r)| =o ctwo ^c r"
|
|
520 |
unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
|
|
521 |
finally show ?thesis .
|
|
522 |
qed
|
|
523 |
|
|
524 |
lemma ordLeq_cexp1:
|
|
525 |
assumes "Cnotzero r" "Card_order q"
|
|
526 |
shows "q \<le>o q ^c r"
|
|
527 |
proof (cases "q =o (czero :: 'a rel)")
|
|
528 |
case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
|
|
529 |
next
|
|
530 |
case False
|
|
531 |
thus ?thesis
|
|
532 |
apply -
|
|
533 |
apply (rule ordIso_ordLeq_trans)
|
|
534 |
apply (rule ordIso_symmetric)
|
|
535 |
apply (rule cexp_cone)
|
|
536 |
apply (rule assms(2))
|
|
537 |
apply (rule cexp_mono2)
|
|
538 |
apply (rule cone_ordLeq_Cnotzero)
|
|
539 |
apply (rule assms(1))
|
|
540 |
apply (rule assms(2))
|
|
541 |
apply (rule notE)
|
|
542 |
apply (rule cone_not_czero)
|
|
543 |
apply assumption
|
|
544 |
apply (rule Card_order_cone)
|
|
545 |
done
|
|
546 |
qed
|
|
547 |
|
|
548 |
lemma ordLeq_cexp2:
|
|
549 |
assumes "ctwo \<le>o q" "Card_order r"
|
|
550 |
shows "r \<le>o q ^c r"
|
|
551 |
proof (cases "r =o (czero :: 'a rel)")
|
|
552 |
case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
|
|
553 |
next
|
|
554 |
case False thus ?thesis
|
|
555 |
apply -
|
|
556 |
apply (rule ordLess_imp_ordLeq)
|
|
557 |
apply (rule ordLess_ordLeq_trans)
|
|
558 |
apply (rule ordLess_ctwo_cexp)
|
|
559 |
apply (rule assms(2))
|
|
560 |
apply (rule cexp_mono1)
|
|
561 |
apply (rule assms(1))
|
|
562 |
apply (rule assms(2))
|
|
563 |
done
|
|
564 |
qed
|
|
565 |
|
|
566 |
lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
|
|
567 |
by (metis assms cinfinite_mono ordLeq_cexp2)
|
|
568 |
|
|
569 |
lemma Cinfinite_cexp:
|
|
570 |
"\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
|
|
571 |
by (simp add: cinfinite_cexp Card_order_cexp)
|
|
572 |
|
|
573 |
lemma ctwo_ordLess_natLeq: "ctwo <o natLeq"
|
|
574 |
unfolding ctwo_def using finite_iff_ordLess_natLeq finite_UNIV by fast
|
|
575 |
|
|
576 |
lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
|
|
577 |
by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
|
|
578 |
|
|
579 |
lemma ctwo_ordLeq_Cinfinite:
|
|
580 |
assumes "Cinfinite r"
|
|
581 |
shows "ctwo \<le>o r"
|
|
582 |
by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
|
|
583 |
|
|
584 |
lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
|
|
585 |
by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
|
|
586 |
|
|
587 |
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"
|
|
588 |
by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
|
|
589 |
|
|
590 |
lemma csum_cinfinite_bound:
|
|
591 |
assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
|
|
592 |
shows "p +c q \<le>o r"
|
|
593 |
proof -
|
|
594 |
from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
|
|
595 |
unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
|
|
596 |
with assms show ?thesis unfolding cinfinite_def csum_def
|
|
597 |
by (blast intro: card_of_Plus_ordLeq_infinite_Field)
|
|
598 |
qed
|
|
599 |
|
|
600 |
lemma cprod_cinfinite_bound:
|
|
601 |
assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
|
|
602 |
shows "p *c q \<le>o r"
|
|
603 |
proof -
|
|
604 |
from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
|
|
605 |
unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
|
|
606 |
with assms show ?thesis unfolding cinfinite_def cprod_def
|
|
607 |
by (blast intro: card_of_Times_ordLeq_infinite_Field)
|
|
608 |
qed
|
|
609 |
|
|
610 |
lemma cprod_csum_cexp:
|
|
611 |
"r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
|
|
612 |
unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
|
|
613 |
proof -
|
|
614 |
let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
|
|
615 |
have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
|
|
616 |
by (auto simp: inj_on_def fun_eq_iff split: bool.split)
|
|
617 |
moreover
|
|
618 |
have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
|
|
619 |
by (auto simp: Func_def)
|
|
620 |
ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
|
|
621 |
qed
|
|
622 |
|
|
623 |
lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
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|
624 |
by (intro cprod_cinfinite_bound)
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625 |
(auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
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|
626 |
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|
627 |
lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
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|
628 |
unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
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|
629 |
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|
630 |
lemma cprod_cexp_csum_cexp_Cinfinite:
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|
631 |
assumes t: "Cinfinite t"
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|
632 |
shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
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|
633 |
proof -
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|
634 |
have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
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|
635 |
by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
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|
636 |
also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
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|
637 |
by (rule cexp_cprod[OF Card_order_csum])
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|
638 |
also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
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|
639 |
by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
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|
640 |
also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
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|
641 |
by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
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|
642 |
also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
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|
643 |
by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
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|
644 |
finally show ?thesis .
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|
645 |
qed
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|
646 |
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|
647 |
lemma Cfinite_cexp_Cinfinite:
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|
648 |
assumes s: "Cfinite s" and t: "Cinfinite t"
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|
649 |
shows "s ^c t \<le>o ctwo ^c t"
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|
650 |
proof (cases "s \<le>o ctwo")
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|
651 |
case True thus ?thesis using t by (blast intro: cexp_mono1)
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|
652 |
next
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|
653 |
case False
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|
654 |
hence "ctwo \<le>o s" by (metis card_order_on_well_order_on ctwo_Cnotzero ordLeq_total s)
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|
655 |
hence "Cnotzero s" by (metis Cnotzero_mono ctwo_Cnotzero s)
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|
656 |
hence st: "Cnotzero (s *c t)" by (metis Cinfinite_cprod2 cinfinite_not_czero t)
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|
657 |
have "s ^c t \<le>o (ctwo ^c s) ^c t"
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|
658 |
using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
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|
659 |
also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
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|
660 |
by (blast intro: Card_order_ctwo cexp_cprod)
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|
661 |
also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
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|
662 |
using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
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|
663 |
finally show ?thesis .
|
|
664 |
qed
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|
665 |
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|
666 |
lemma csum_Cfinite_cexp_Cinfinite:
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|
667 |
assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
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|
668 |
shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
|
|
669 |
proof (cases "Cinfinite r")
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|
670 |
case True
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|
671 |
hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
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|
672 |
hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
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|
673 |
also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
|
|
674 |
finally show ?thesis .
|
|
675 |
next
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|
676 |
case False
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|
677 |
with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
|
|
678 |
hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
|
|
679 |
hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
|
|
680 |
also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
|
|
681 |
by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
|
|
682 |
finally show ?thesis .
|
|
683 |
qed
|
|
684 |
|
|
685 |
|
|
686 |
(* cardSuc *)
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|
687 |
|
|
688 |
lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
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|
689 |
by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
|
|
690 |
|
|
691 |
lemma cardSuc_UNION_Cinfinite:
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|
692 |
assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
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|
693 |
shows "EX i : Field (cardSuc r). B \<le> As i"
|
|
694 |
using cardSuc_UNION assms unfolding cinfinite_def by blast
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|
695 |
|
|
696 |
subsection {* Powerset *}
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|
697 |
|
|
698 |
definition cpow where "cpow r = |Pow (Field r)|"
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|
699 |
|
|
700 |
lemma card_order_cpow: "card_order r \<Longrightarrow> card_order (cpow r)"
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|
701 |
by (simp only: cpow_def Field_card_order Pow_UNIV card_of_card_order_on)
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|
702 |
|
|
703 |
lemma cpow_greater_eq: "Card_order r \<Longrightarrow> r \<le>o cpow r"
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|
704 |
by (rule ordLess_imp_ordLeq) (simp only: cpow_def Card_order_Pow)
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|
705 |
|
|
706 |
subsection {* Lists *}
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|
707 |
|
|
708 |
definition clists where "clists r = |lists (Field r)|"
|
|
709 |
|
|
710 |
end
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