src/HOL/BNF_Cardinal_Order_Relation.thy
author haftmann
Thu Mar 13 08:56:08 2014 +0100 (2014-03-13)
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tuned
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(*  Title:      HOL/BNF_Cardinal_Order_Relation.thy
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Cardinal-order relations as needed by bounded natural functors.
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*)
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header {* Cardinal-Order Relations as Needed by Bounded Natural Functors *}
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theory BNF_Cardinal_Order_Relation
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imports Zorn BNF_Constructions_on_Wellorders
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begin
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text{* In this section, we define cardinal-order relations to be minim well-orders
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on their field.  Then we define the cardinal of a set to be {\em some} cardinal-order
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relation on that set, which will be unique up to order isomorphism.  Then we study
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the connection between cardinals and:
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\begin{itemize}
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\item standard set-theoretic constructions: products,
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sums, unions, lists, powersets, set-of finite sets operator;
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\item finiteness and infiniteness (in particular, with the numeric cardinal operator
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for finite sets, @{text "card"}, from the theory @{text "Finite_Sets.thy"}).
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\end{itemize}
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%
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On the way, we define the canonical $\omega$ cardinal and finite cardinals.  We also
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define (again, up to order isomorphism) the successor of a cardinal, and show that
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any cardinal admits a successor.
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Main results of this section are the existence of cardinal relations and the
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facts that, in the presence of infiniteness,
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most of the standard set-theoretic constructions (except for the powerset)
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{\em do not increase cardinality}.  In particular, e.g., the set of words/lists over
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any infinite set has the same cardinality (hence, is in bijection) with that set.
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*}
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subsection {* Cardinal orders *}
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text{* A cardinal order in our setting shall be a well-order {\em minim} w.r.t. the
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order-embedding relation, @{text "\<le>o"} (which is the same as being {\em minimal} w.r.t. the
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strict order-embedding relation, @{text "<o"}), among all the well-orders on its field. *}
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definition card_order_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
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where
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"card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
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abbreviation "Card_order r \<equiv> card_order_on (Field r) r"
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abbreviation "card_order r \<equiv> card_order_on UNIV r"
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lemma card_order_on_well_order_on:
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assumes "card_order_on A r"
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shows "well_order_on A r"
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using assms unfolding card_order_on_def by simp
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lemma card_order_on_Card_order:
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"card_order_on A r \<Longrightarrow> A = Field r \<and> Card_order r"
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unfolding card_order_on_def using well_order_on_Field by blast
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text{* The existence of a cardinal relation on any given set (which will mean
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that any set has a cardinal) follows from two facts:
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\begin{itemize}
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\item Zermelo's theorem (proved in @{text "Zorn.thy"} as theorem @{text "well_order_on"}),
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which states that on any given set there exists a well-order;
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\item The well-founded-ness of @{text "<o"}, ensuring that then there exists a minimal
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such well-order, i.e., a cardinal order.
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\end{itemize}
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*}
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theorem card_order_on: "\<exists>r. card_order_on A r"
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proof-
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  obtain R where R_def: "R = {r. well_order_on A r}" by blast
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  have 1: "R \<noteq> {} \<and> (\<forall>r \<in> R. Well_order r)"
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  using well_order_on[of A] R_def well_order_on_Well_order by blast
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  hence "\<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
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  using  exists_minim_Well_order[of R] by auto
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  thus ?thesis using R_def unfolding card_order_on_def by auto
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qed
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lemma card_order_on_ordIso:
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assumes CO: "card_order_on A r" and CO': "card_order_on A r'"
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shows "r =o r'"
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using assms unfolding card_order_on_def
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using ordIso_iff_ordLeq by blast
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lemma Card_order_ordIso:
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assumes CO: "Card_order r" and ISO: "r' =o r"
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shows "Card_order r'"
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using ISO unfolding ordIso_def
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proof(unfold card_order_on_def, auto)
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  fix p' assume "well_order_on (Field r') p'"
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  hence 0: "Well_order p' \<and> Field p' = Field r'"
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  using well_order_on_Well_order by blast
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  obtain f where 1: "iso r' r f" and 2: "Well_order r \<and> Well_order r'"
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  using ISO unfolding ordIso_def by auto
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  hence 3: "inj_on f (Field r') \<and> f ` (Field r') = Field r"
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  by (auto simp add: iso_iff embed_inj_on)
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  let ?p = "dir_image p' f"
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  have 4: "p' =o ?p \<and> Well_order ?p"
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  using 0 2 3 by (auto simp add: dir_image_ordIso Well_order_dir_image)
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  moreover have "Field ?p =  Field r"
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  using 0 3 by (auto simp add: dir_image_Field2 order_on_defs)
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  ultimately have "well_order_on (Field r) ?p" by auto
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  hence "r \<le>o ?p" using CO unfolding card_order_on_def by auto
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  thus "r' \<le>o p'"
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  using ISO 4 ordLeq_ordIso_trans ordIso_ordLeq_trans ordIso_symmetric by blast
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qed
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lemma Card_order_ordIso2:
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assumes CO: "Card_order r" and ISO: "r =o r'"
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shows "Card_order r'"
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using assms Card_order_ordIso ordIso_symmetric by blast
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subsection {* Cardinal of a set *}
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text{* We define the cardinal of set to be {\em some} cardinal order on that set.
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We shall prove that this notion is unique up to order isomorphism, meaning
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that order isomorphism shall be the true identity of cardinals. *}
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definition card_of :: "'a set \<Rightarrow> 'a rel" ("|_|" )
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where "card_of A = (SOME r. card_order_on A r)"
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lemma card_of_card_order_on: "card_order_on A |A|"
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unfolding card_of_def by (auto simp add: card_order_on someI_ex)
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lemma card_of_well_order_on: "well_order_on A |A|"
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using card_of_card_order_on card_order_on_def by blast
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lemma Field_card_of: "Field |A| = A"
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using card_of_card_order_on[of A] unfolding card_order_on_def
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using well_order_on_Field by blast
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lemma card_of_Card_order: "Card_order |A|"
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by (simp only: card_of_card_order_on Field_card_of)
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corollary ordIso_card_of_imp_Card_order:
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"r =o |A| \<Longrightarrow> Card_order r"
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using card_of_Card_order Card_order_ordIso by blast
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lemma card_of_Well_order: "Well_order |A|"
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using card_of_Card_order unfolding card_order_on_def by auto
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lemma card_of_refl: "|A| =o |A|"
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using card_of_Well_order ordIso_reflexive by blast
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lemma card_of_least: "well_order_on A r \<Longrightarrow> |A| \<le>o r"
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using card_of_card_order_on unfolding card_order_on_def by blast
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lemma card_of_ordIso:
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"(\<exists>f. bij_betw f A B) = ( |A| =o |B| )"
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proof(auto)
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  fix f assume *: "bij_betw f A B"
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  then obtain r where "well_order_on B r \<and> |A| =o r"
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  using Well_order_iso_copy card_of_well_order_on by blast
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  hence "|B| \<le>o |A|" using card_of_least
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  ordLeq_ordIso_trans ordIso_symmetric by blast
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  moreover
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  {let ?g = "inv_into A f"
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   have "bij_betw ?g B A" using * bij_betw_inv_into by blast
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   then obtain r where "well_order_on A r \<and> |B| =o r"
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   using Well_order_iso_copy card_of_well_order_on by blast
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   hence "|A| \<le>o |B|" using card_of_least
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   ordLeq_ordIso_trans ordIso_symmetric by blast
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  }
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  ultimately show "|A| =o |B|" using ordIso_iff_ordLeq by blast
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next
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  assume "|A| =o |B|"
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  then obtain f where "iso ( |A| ) ( |B| ) f"
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  unfolding ordIso_def by auto
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  hence "bij_betw f A B" unfolding iso_def Field_card_of by simp
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  thus "\<exists>f. bij_betw f A B" by auto
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qed
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lemma card_of_ordLeq:
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"(\<exists>f. inj_on f A \<and> f ` A \<le> B) = ( |A| \<le>o |B| )"
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proof(auto)
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  fix f assume *: "inj_on f A" and **: "f ` A \<le> B"
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  {assume "|B| <o |A|"
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   hence "|B| \<le>o |A|" using ordLeq_iff_ordLess_or_ordIso by blast
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   then obtain g where "embed ( |B| ) ( |A| ) g"
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   unfolding ordLeq_def by auto
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   hence 1: "inj_on g B \<and> g ` B \<le> A" using embed_inj_on[of "|B|" "|A|" "g"]
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   card_of_Well_order[of "B"] Field_card_of[of "B"] Field_card_of[of "A"]
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   embed_Field[of "|B|" "|A|" g] by auto
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   obtain h where "bij_betw h A B"
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   using * ** 1 Cantor_Bernstein[of f] by fastforce
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   hence "|A| =o |B|" using card_of_ordIso by blast
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   hence "|A| \<le>o |B|" using ordIso_iff_ordLeq by auto
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  }
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  thus "|A| \<le>o |B|" using ordLess_or_ordLeq[of "|B|" "|A|"]
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  by (auto simp: card_of_Well_order)
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next
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  assume *: "|A| \<le>o |B|"
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  obtain f where "embed ( |A| ) ( |B| ) f"
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  using * unfolding ordLeq_def by auto
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  hence "inj_on f A \<and> f ` A \<le> B" using embed_inj_on[of "|A|" "|B|" f]
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  card_of_Well_order[of "A"] Field_card_of[of "A"] Field_card_of[of "B"]
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  embed_Field[of "|A|" "|B|" f] by auto
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  thus "\<exists>f. inj_on f A \<and> f ` A \<le> B" by auto
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qed
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lemma card_of_ordLeq2:
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"A \<noteq> {} \<Longrightarrow> (\<exists>g. g ` B = A) = ( |A| \<le>o |B| )"
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using card_of_ordLeq[of A B] inj_on_iff_surj[of A B] by auto
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lemma card_of_ordLess:
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"(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = ( |B| <o |A| )"
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proof-
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  have "(\<not>(\<exists>f. inj_on f A \<and> f ` A \<le> B)) = (\<not> |A| \<le>o |B| )"
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  using card_of_ordLeq by blast
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  also have "\<dots> = ( |B| <o |A| )"
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  using card_of_Well_order[of A] card_of_Well_order[of B]
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        not_ordLeq_iff_ordLess by blast
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  finally show ?thesis .
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qed
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lemma card_of_ordLess2:
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"B \<noteq> {} \<Longrightarrow> (\<not>(\<exists>f. f ` A = B)) = ( |A| <o |B| )"
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using card_of_ordLess[of B A] inj_on_iff_surj[of B A] by auto
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lemma card_of_ordIsoI:
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assumes "bij_betw f A B"
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shows "|A| =o |B|"
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using assms unfolding card_of_ordIso[symmetric] by auto
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lemma card_of_ordLeqI:
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assumes "inj_on f A" and "\<And> a. a \<in> A \<Longrightarrow> f a \<in> B"
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shows "|A| \<le>o |B|"
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using assms unfolding card_of_ordLeq[symmetric] by auto
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lemma card_of_unique:
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"card_order_on A r \<Longrightarrow> r =o |A|"
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by (simp only: card_order_on_ordIso card_of_card_order_on)
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lemma card_of_mono1:
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"A \<le> B \<Longrightarrow> |A| \<le>o |B|"
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using inj_on_id[of A] card_of_ordLeq[of A B] by fastforce
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lemma card_of_mono2:
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assumes "r \<le>o r'"
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shows "|Field r| \<le>o |Field r'|"
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proof-
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  obtain f where
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  1: "well_order_on (Field r) r \<and> well_order_on (Field r) r \<and> embed r r' f"
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  using assms unfolding ordLeq_def
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  by (auto simp add: well_order_on_Well_order)
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  hence "inj_on f (Field r) \<and> f ` (Field r) \<le> Field r'"
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  by (auto simp add: embed_inj_on embed_Field)
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  thus "|Field r| \<le>o |Field r'|" using card_of_ordLeq by blast
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qed
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lemma card_of_cong: "r =o r' \<Longrightarrow> |Field r| =o |Field r'|"
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by (simp add: ordIso_iff_ordLeq card_of_mono2)
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lemma card_of_Field_ordLess: "Well_order r \<Longrightarrow> |Field r| \<le>o r"
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using card_of_least card_of_well_order_on well_order_on_Well_order by blast
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lemma card_of_Field_ordIso:
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assumes "Card_order r"
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shows "|Field r| =o r"
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proof-
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  have "card_order_on (Field r) r"
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  using assms card_order_on_Card_order by blast
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  moreover have "card_order_on (Field r) |Field r|"
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  using card_of_card_order_on by blast
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  ultimately show ?thesis using card_order_on_ordIso by blast
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qed
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lemma Card_order_iff_ordIso_card_of:
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"Card_order r = (r =o |Field r| )"
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using ordIso_card_of_imp_Card_order card_of_Field_ordIso ordIso_symmetric by blast
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lemma Card_order_iff_ordLeq_card_of:
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"Card_order r = (r \<le>o |Field r| )"
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proof-
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  have "Card_order r = (r =o |Field r| )"
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  unfolding Card_order_iff_ordIso_card_of by simp
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  also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
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  unfolding ordIso_iff_ordLeq by simp
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  also have "... = (r \<le>o |Field r| )"
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  using card_of_Field_ordLess
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  by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
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  finally show ?thesis .
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qed
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lemma Card_order_iff_Restr_underS:
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assumes "Well_order r"
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shows "Card_order r = (\<forall>a \<in> Field r. Restr r (underS r a) <o |Field r| )"
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using assms unfolding Card_order_iff_ordLeq_card_of
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using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
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lemma card_of_underS:
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   293
assumes r: "Card_order r" and a: "a : Field r"
blanchet@55023
   294
shows "|underS r a| <o r"
blanchet@48975
   295
proof-
blanchet@55023
   296
  let ?A = "underS r a"  let ?r' = "Restr r ?A"
blanchet@48975
   297
  have 1: "Well_order r"
blanchet@48975
   298
  using r unfolding card_order_on_def by simp
blanchet@48975
   299
  have "Well_order ?r'" using 1 Well_order_Restr by auto
blanchet@48975
   300
  moreover have "card_order_on (Field ?r') |Field ?r'|"
blanchet@48975
   301
  using card_of_card_order_on .
blanchet@48975
   302
  ultimately have "|Field ?r'| \<le>o ?r'"
blanchet@48975
   303
  unfolding card_order_on_def by simp
blanchet@48975
   304
  moreover have "Field ?r' = ?A"
blanchet@48975
   305
  using 1 wo_rel.underS_ofilter Field_Restr_ofilter
blanchet@48975
   306
  unfolding wo_rel_def by fastforce
blanchet@48975
   307
  ultimately have "|?A| \<le>o ?r'" by simp
blanchet@48975
   308
  also have "?r' <o |Field r|"
blanchet@48975
   309
  using 1 a r Card_order_iff_Restr_underS by blast
blanchet@48975
   310
  also have "|Field r| =o r"
blanchet@48975
   311
  using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
blanchet@48975
   312
  finally show ?thesis .
blanchet@48975
   313
qed
blanchet@48975
   314
blanchet@48975
   315
lemma ordLess_Field:
blanchet@48975
   316
assumes "r <o r'"
blanchet@48975
   317
shows "|Field r| <o r'"
blanchet@48975
   318
proof-
blanchet@48975
   319
  have "well_order_on (Field r) r" using assms unfolding ordLess_def
blanchet@55023
   320
  by (auto simp add: well_order_on_Well_order)
blanchet@48975
   321
  hence "|Field r| \<le>o r" using card_of_least by blast
blanchet@48975
   322
  thus ?thesis using assms ordLeq_ordLess_trans by blast
blanchet@48975
   323
qed
blanchet@48975
   324
blanchet@48975
   325
lemma internalize_card_of_ordLeq:
blanchet@48975
   326
"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
blanchet@48975
   327
proof
blanchet@48975
   328
  assume "|A| \<le>o r"
blanchet@48975
   329
  then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
blanchet@48975
   330
  using internalize_ordLeq[of "|A|" r] by blast
blanchet@48975
   331
  hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
blanchet@48975
   332
  hence "|Field p| =o p" using card_of_Field_ordIso by blast
blanchet@48975
   333
  hence "|A| =o |Field p| \<and> |Field p| \<le>o r"
blanchet@48975
   334
  using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
blanchet@48975
   335
  thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
blanchet@48975
   336
next
blanchet@48975
   337
  assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
blanchet@48975
   338
  thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
blanchet@48975
   339
qed
blanchet@48975
   340
blanchet@48975
   341
lemma internalize_card_of_ordLeq2:
blanchet@48975
   342
"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
blanchet@48975
   343
using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
blanchet@48975
   344
blanchet@48975
   345
blanchet@48975
   346
subsection {* Cardinals versus set operations on arbitrary sets *}
blanchet@48975
   347
blanchet@48975
   348
text{* Here we embark in a long journey of simple results showing
blanchet@48975
   349
that the standard set-theoretic operations are well-behaved w.r.t. the notion of
blanchet@48975
   350
cardinal -- essentially, this means that they preserve the ``cardinal identity"
blanchet@48975
   351
@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
blanchet@48975
   352
*}
blanchet@48975
   353
blanchet@48975
   354
lemma card_of_empty: "|{}| \<le>o |A|"
blanchet@48975
   355
using card_of_ordLeq inj_on_id by blast
blanchet@48975
   356
blanchet@48975
   357
lemma card_of_empty1:
blanchet@48975
   358
assumes "Well_order r \<or> Card_order r"
blanchet@48975
   359
shows "|{}| \<le>o r"
blanchet@48975
   360
proof-
blanchet@48975
   361
  have "Well_order r" using assms unfolding card_order_on_def by auto
blanchet@48975
   362
  hence "|Field r| <=o r"
blanchet@48975
   363
  using assms card_of_Field_ordLess by blast
blanchet@48975
   364
  moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
blanchet@48975
   365
  ultimately show ?thesis using ordLeq_transitive by blast
blanchet@48975
   366
qed
blanchet@48975
   367
blanchet@48975
   368
corollary Card_order_empty:
blanchet@48975
   369
"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
blanchet@48975
   370
blanchet@48975
   371
lemma card_of_empty2:
blanchet@48975
   372
assumes LEQ: "|A| =o |{}|"
blanchet@48975
   373
shows "A = {}"
blanchet@48975
   374
using assms card_of_ordIso[of A] bij_betw_empty2 by blast
blanchet@48975
   375
blanchet@48975
   376
lemma card_of_empty3:
blanchet@48975
   377
assumes LEQ: "|A| \<le>o |{}|"
blanchet@48975
   378
shows "A = {}"
blanchet@48975
   379
using assms
blanchet@48975
   380
by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
blanchet@48975
   381
              ordLeq_Well_order_simp)
blanchet@48975
   382
blanchet@48975
   383
lemma card_of_empty_ordIso:
blanchet@48975
   384
"|{}::'a set| =o |{}::'b set|"
blanchet@48975
   385
using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
blanchet@48975
   386
blanchet@48975
   387
lemma card_of_image:
blanchet@48975
   388
"|f ` A| <=o |A|"
blanchet@48975
   389
proof(cases "A = {}", simp add: card_of_empty)
blanchet@48975
   390
  assume "A ~= {}"
blanchet@48975
   391
  hence "f ` A ~= {}" by auto
blanchet@48975
   392
  thus "|f ` A| \<le>o |A|"
blanchet@48975
   393
  using card_of_ordLeq2[of "f ` A" A] by auto
blanchet@48975
   394
qed
blanchet@48975
   395
blanchet@48975
   396
lemma surj_imp_ordLeq:
traytel@55811
   397
assumes "B \<subseteq> f ` A"
traytel@55811
   398
shows "|B| \<le>o |A|"
blanchet@48975
   399
proof-
blanchet@48975
   400
  have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
blanchet@48975
   401
  thus ?thesis using card_of_image ordLeq_transitive by blast
blanchet@48975
   402
qed
blanchet@48975
   403
blanchet@48975
   404
lemma card_of_singl_ordLeq:
blanchet@48975
   405
assumes "A \<noteq> {}"
blanchet@48975
   406
shows "|{b}| \<le>o |A|"
blanchet@48975
   407
proof-
blanchet@48975
   408
  obtain a where *: "a \<in> A" using assms by auto
blanchet@48975
   409
  let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
blanchet@48975
   410
  have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
blanchet@48975
   411
  using * unfolding inj_on_def by auto
traytel@55603
   412
  thus ?thesis unfolding card_of_ordLeq[symmetric] by (intro exI)
blanchet@48975
   413
qed
blanchet@48975
   414
blanchet@48975
   415
corollary Card_order_singl_ordLeq:
blanchet@48975
   416
"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
blanchet@48975
   417
using card_of_singl_ordLeq[of "Field r" b]
blanchet@48975
   418
      card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
blanchet@48975
   419
blanchet@48975
   420
lemma card_of_Pow: "|A| <o |Pow A|"
blanchet@48975
   421
using card_of_ordLess2[of "Pow A" A]  Cantors_paradox[of A]
blanchet@48975
   422
      Pow_not_empty[of A] by auto
blanchet@48975
   423
blanchet@48975
   424
corollary Card_order_Pow:
blanchet@48975
   425
"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
blanchet@48975
   426
using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
blanchet@48975
   427
blanchet@48975
   428
lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
blanchet@48975
   429
proof-
blanchet@48975
   430
  have "Inl ` A \<le> A <+> B" by auto
blanchet@48975
   431
  thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
blanchet@48975
   432
qed
blanchet@48975
   433
blanchet@48975
   434
corollary Card_order_Plus1:
blanchet@48975
   435
"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
blanchet@48975
   436
using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   437
blanchet@48975
   438
lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
blanchet@48975
   439
proof-
blanchet@48975
   440
  have "Inr ` B \<le> A <+> B" by auto
blanchet@48975
   441
  thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
blanchet@48975
   442
qed
blanchet@48975
   443
blanchet@48975
   444
corollary Card_order_Plus2:
blanchet@48975
   445
"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
blanchet@48975
   446
using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   447
blanchet@48975
   448
lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
blanchet@48975
   449
proof-
blanchet@48975
   450
  have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   451
  thus ?thesis using card_of_ordIso by auto
blanchet@48975
   452
qed
blanchet@48975
   453
blanchet@48975
   454
lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
blanchet@48975
   455
proof-
blanchet@48975
   456
  have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   457
  thus ?thesis using card_of_ordIso by auto
blanchet@48975
   458
qed
blanchet@48975
   459
blanchet@48975
   460
lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
blanchet@48975
   461
proof-
blanchet@48975
   462
  let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
blanchet@48975
   463
                                   | Inr b \<Rightarrow> Inl b"
blanchet@48975
   464
  have "bij_betw ?f (A <+> B) (B <+> A)"
blanchet@48975
   465
  unfolding bij_betw_def inj_on_def by force
blanchet@48975
   466
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   467
qed
blanchet@48975
   468
blanchet@48975
   469
lemma card_of_Plus_assoc:
blanchet@48975
   470
fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
blanchet@48975
   471
shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
blanchet@48975
   472
proof -
blanchet@48975
   473
  def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
blanchet@48975
   474
  case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
blanchet@48975
   475
                                 |Inr b \<Rightarrow> Inr (Inl b))
blanchet@48975
   476
           |Inr c \<Rightarrow> Inr (Inr c)"
blanchet@48975
   477
  have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
blanchet@48975
   478
  proof
blanchet@48975
   479
    fix x assume x: "x \<in> A <+> B <+> C"
blanchet@48975
   480
    show "x \<in> f ` ((A <+> B) <+> C)"
blanchet@48975
   481
    proof(cases x)
blanchet@48975
   482
      case (Inl a)
blanchet@48975
   483
      hence "a \<in> A" "x = f (Inl (Inl a))"
blanchet@48975
   484
      using x unfolding f_def by auto
blanchet@48975
   485
      thus ?thesis by auto
blanchet@48975
   486
    next
blanchet@48975
   487
      case (Inr bc) note 1 = Inr show ?thesis
blanchet@48975
   488
      proof(cases bc)
blanchet@48975
   489
        case (Inl b)
blanchet@48975
   490
        hence "b \<in> B" "x = f (Inl (Inr b))"
blanchet@48975
   491
        using x 1 unfolding f_def by auto
blanchet@48975
   492
        thus ?thesis by auto
blanchet@48975
   493
      next
blanchet@48975
   494
        case (Inr c)
blanchet@48975
   495
        hence "c \<in> C" "x = f (Inr c)"
blanchet@48975
   496
        using x 1 unfolding f_def by auto
blanchet@48975
   497
        thus ?thesis by auto
blanchet@48975
   498
      qed
blanchet@48975
   499
    qed
blanchet@48975
   500
  qed
blanchet@48975
   501
  hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
blanchet@54482
   502
  unfolding bij_betw_def inj_on_def f_def by fastforce
blanchet@48975
   503
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   504
qed
blanchet@48975
   505
blanchet@48975
   506
lemma card_of_Plus_mono1:
blanchet@48975
   507
assumes "|A| \<le>o |B|"
blanchet@48975
   508
shows "|A <+> C| \<le>o |B <+> C|"
blanchet@48975
   509
proof-
blanchet@48975
   510
  obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   511
  using assms card_of_ordLeq[of A] by fastforce
blanchet@48975
   512
  obtain g where g_def:
blanchet@48975
   513
  "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
blanchet@48975
   514
  have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
blanchet@48975
   515
  proof-
blanchet@48975
   516
    {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
blanchet@48975
   517
                          "g d1 = g d2"
blanchet@54482
   518
     hence "d1 = d2" using 1 unfolding inj_on_def g_def by force
blanchet@48975
   519
    }
blanchet@48975
   520
    moreover
blanchet@48975
   521
    {fix d assume "d \<in> A <+> C"
blanchet@48975
   522
     hence "g d \<in> B <+> C"  using 1
blanchet@48975
   523
     by(case_tac d, auto simp add: g_def)
blanchet@48975
   524
    }
blanchet@48975
   525
    ultimately show ?thesis unfolding inj_on_def by auto
blanchet@48975
   526
  qed
traytel@55811
   527
  thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   528
qed
blanchet@48975
   529
blanchet@48975
   530
corollary ordLeq_Plus_mono1:
blanchet@48975
   531
assumes "r \<le>o r'"
blanchet@48975
   532
shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
blanchet@48975
   533
using assms card_of_mono2 card_of_Plus_mono1 by blast
blanchet@48975
   534
blanchet@48975
   535
lemma card_of_Plus_mono2:
blanchet@48975
   536
assumes "|A| \<le>o |B|"
blanchet@48975
   537
shows "|C <+> A| \<le>o |C <+> B|"
blanchet@48975
   538
using assms card_of_Plus_mono1[of A B C]
blanchet@48975
   539
      card_of_Plus_commute[of C A]  card_of_Plus_commute[of B C]
blanchet@48975
   540
      ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
blanchet@48975
   541
by blast
blanchet@48975
   542
blanchet@48975
   543
corollary ordLeq_Plus_mono2:
blanchet@48975
   544
assumes "r \<le>o r'"
blanchet@48975
   545
shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
blanchet@48975
   546
using assms card_of_mono2 card_of_Plus_mono2 by blast
blanchet@48975
   547
blanchet@48975
   548
lemma card_of_Plus_mono:
blanchet@48975
   549
assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
blanchet@48975
   550
shows "|A <+> C| \<le>o |B <+> D|"
blanchet@48975
   551
using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
blanchet@48975
   552
      ordLeq_transitive[of "|A <+> C|"] by blast
blanchet@48975
   553
blanchet@48975
   554
corollary ordLeq_Plus_mono:
blanchet@48975
   555
assumes "r \<le>o r'" and "p \<le>o p'"
blanchet@48975
   556
shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
blanchet@48975
   557
using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
blanchet@48975
   558
blanchet@48975
   559
lemma card_of_Plus_cong1:
blanchet@48975
   560
assumes "|A| =o |B|"
blanchet@48975
   561
shows "|A <+> C| =o |B <+> C|"
blanchet@48975
   562
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
blanchet@48975
   563
blanchet@48975
   564
corollary ordIso_Plus_cong1:
blanchet@48975
   565
assumes "r =o r'"
blanchet@48975
   566
shows "|(Field r) <+> C| =o |(Field r') <+> C|"
blanchet@48975
   567
using assms card_of_cong card_of_Plus_cong1 by blast
blanchet@48975
   568
blanchet@48975
   569
lemma card_of_Plus_cong2:
blanchet@48975
   570
assumes "|A| =o |B|"
blanchet@48975
   571
shows "|C <+> A| =o |C <+> B|"
blanchet@48975
   572
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
blanchet@48975
   573
blanchet@48975
   574
corollary ordIso_Plus_cong2:
blanchet@48975
   575
assumes "r =o r'"
blanchet@48975
   576
shows "|A <+> (Field r)| =o |A <+> (Field r')|"
blanchet@48975
   577
using assms card_of_cong card_of_Plus_cong2 by blast
blanchet@48975
   578
blanchet@48975
   579
lemma card_of_Plus_cong:
blanchet@48975
   580
assumes "|A| =o |B|" and "|C| =o |D|"
blanchet@48975
   581
shows "|A <+> C| =o |B <+> D|"
blanchet@48975
   582
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
blanchet@48975
   583
blanchet@48975
   584
corollary ordIso_Plus_cong:
blanchet@48975
   585
assumes "r =o r'" and "p =o p'"
blanchet@48975
   586
shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
blanchet@48975
   587
using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
blanchet@48975
   588
blanchet@48975
   589
lemma card_of_Un_Plus_ordLeq:
blanchet@48975
   590
"|A \<union> B| \<le>o |A <+> B|"
blanchet@48975
   591
proof-
blanchet@48975
   592
   let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
blanchet@48975
   593
   have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
blanchet@48975
   594
   unfolding inj_on_def by auto
blanchet@48975
   595
   thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   596
qed
blanchet@48975
   597
blanchet@48975
   598
lemma card_of_Times1:
blanchet@48975
   599
assumes "A \<noteq> {}"
blanchet@48975
   600
shows "|B| \<le>o |B \<times> A|"
blanchet@48975
   601
proof(cases "B = {}", simp add: card_of_empty)
blanchet@48975
   602
  assume *: "B \<noteq> {}"
blanchet@48975
   603
  have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
blanchet@48975
   604
  thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
blanchet@48975
   605
                     card_of_ordLeq[of B "B \<times> A"] * by blast
blanchet@48975
   606
qed
blanchet@48975
   607
blanchet@48975
   608
lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
blanchet@48975
   609
proof-
blanchet@48975
   610
  let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
blanchet@48975
   611
  have "bij_betw ?f (A \<times> B) (B \<times> A)"
blanchet@48975
   612
  unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   613
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   614
qed
blanchet@48975
   615
blanchet@48975
   616
lemma card_of_Times2:
blanchet@48975
   617
assumes "A \<noteq> {}"   shows "|B| \<le>o |A \<times> B|"
blanchet@48975
   618
using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
blanchet@48975
   619
      ordLeq_ordIso_trans by blast
blanchet@48975
   620
blanchet@54475
   621
corollary Card_order_Times1:
blanchet@54475
   622
"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
blanchet@54475
   623
using card_of_Times1[of B] card_of_Field_ordIso
blanchet@54475
   624
      ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@54475
   625
blanchet@48975
   626
corollary Card_order_Times2:
blanchet@48975
   627
"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
blanchet@48975
   628
using card_of_Times2[of A] card_of_Field_ordIso
blanchet@48975
   629
      ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   630
blanchet@48975
   631
lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
blanchet@48975
   632
using card_of_Times1[of A]
blanchet@48975
   633
by(cases "A = {}", simp add: card_of_empty, blast)
blanchet@48975
   634
blanchet@48975
   635
lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
blanchet@48975
   636
proof-
blanchet@48975
   637
  let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
blanchet@48975
   638
                                  |Inr a \<Rightarrow> (a,False)"
blanchet@48975
   639
  have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
blanchet@48975
   640
  proof-
blanchet@48975
   641
    {fix  c1 and c2 assume "?f c1 = ?f c2"
blanchet@48975
   642
     hence "c1 = c2"
blanchet@48975
   643
     by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
blanchet@48975
   644
    }
blanchet@48975
   645
    moreover
blanchet@48975
   646
    {fix c assume "c \<in> A <+> A"
blanchet@48975
   647
     hence "?f c \<in> A \<times> (UNIV::bool set)"
blanchet@48975
   648
     by(case_tac c, auto)
blanchet@48975
   649
    }
blanchet@48975
   650
    moreover
blanchet@48975
   651
    {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
blanchet@48975
   652
     have "(a,bl) \<in> ?f ` ( A <+> A)"
blanchet@48975
   653
     proof(cases bl)
blanchet@48975
   654
       assume bl hence "?f(Inl a) = (a,bl)" by auto
blanchet@48975
   655
       thus ?thesis using * by force
blanchet@48975
   656
     next
blanchet@48975
   657
       assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
blanchet@48975
   658
       thus ?thesis using * by force
blanchet@48975
   659
     qed
blanchet@48975
   660
    }
blanchet@48975
   661
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   662
  qed
blanchet@48975
   663
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   664
qed
blanchet@48975
   665
blanchet@48975
   666
lemma card_of_Times_mono1:
blanchet@48975
   667
assumes "|A| \<le>o |B|"
blanchet@48975
   668
shows "|A \<times> C| \<le>o |B \<times> C|"
blanchet@48975
   669
proof-
blanchet@48975
   670
  obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   671
  using assms card_of_ordLeq[of A] by fastforce
blanchet@48975
   672
  obtain g where g_def:
blanchet@48975
   673
  "g = (\<lambda>(a,c::'c). (f a,c))" by blast
blanchet@48975
   674
  have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
blanchet@48975
   675
  using 1 unfolding inj_on_def using g_def by auto
traytel@55811
   676
  thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   677
qed
blanchet@48975
   678
blanchet@48975
   679
corollary ordLeq_Times_mono1:
blanchet@48975
   680
assumes "r \<le>o r'"
blanchet@48975
   681
shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
blanchet@48975
   682
using assms card_of_mono2 card_of_Times_mono1 by blast
blanchet@48975
   683
blanchet@48975
   684
lemma card_of_Times_mono2:
blanchet@48975
   685
assumes "|A| \<le>o |B|"
blanchet@48975
   686
shows "|C \<times> A| \<le>o |C \<times> B|"
blanchet@48975
   687
using assms card_of_Times_mono1[of A B C]
blanchet@48975
   688
      card_of_Times_commute[of C A]  card_of_Times_commute[of B C]
blanchet@48975
   689
      ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
blanchet@48975
   690
by blast
blanchet@48975
   691
blanchet@48975
   692
corollary ordLeq_Times_mono2:
blanchet@48975
   693
assumes "r \<le>o r'"
blanchet@48975
   694
shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
blanchet@48975
   695
using assms card_of_mono2 card_of_Times_mono2 by blast
blanchet@48975
   696
blanchet@48975
   697
lemma card_of_Sigma_mono1:
blanchet@48975
   698
assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
blanchet@48975
   699
shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
blanchet@48975
   700
proof-
blanchet@48975
   701
  have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
blanchet@48975
   702
  using assms by (auto simp add: card_of_ordLeq)
blanchet@48975
   703
  with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
traytel@55811
   704
  obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i"
traytel@55811
   705
    by atomize_elim (auto intro: bchoice)
blanchet@48975
   706
  obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
blanchet@48975
   707
  have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
blanchet@48975
   708
  using 1 unfolding inj_on_def using g_def by force
traytel@55811
   709
  thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   710
qed
blanchet@48975
   711
blanchet@48975
   712
corollary card_of_Sigma_Times:
blanchet@48975
   713
"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
haftmann@56075
   714
by (fact card_of_Sigma_mono1)
blanchet@48975
   715
blanchet@48975
   716
lemma card_of_UNION_Sigma:
blanchet@48975
   717
"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
traytel@55811
   718
using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by blast
blanchet@48975
   719
blanchet@48975
   720
lemma card_of_bool:
blanchet@48975
   721
assumes "a1 \<noteq> a2"
blanchet@48975
   722
shows "|UNIV::bool set| =o |{a1,a2}|"
blanchet@48975
   723
proof-
blanchet@48975
   724
  let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
blanchet@48975
   725
  have "bij_betw ?f UNIV {a1,a2}"
blanchet@48975
   726
  proof-
blanchet@48975
   727
    {fix bl1 and bl2 assume "?f  bl1 = ?f bl2"
blanchet@48975
   728
     hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
blanchet@48975
   729
    }
blanchet@48975
   730
    moreover
blanchet@48975
   731
    {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)
blanchet@48975
   732
    }
blanchet@48975
   733
    moreover
blanchet@48975
   734
    {fix a assume *: "a \<in> {a1,a2}"
blanchet@48975
   735
     have "a \<in> ?f ` UNIV"
blanchet@48975
   736
     proof(cases "a = a1")
blanchet@48975
   737
       assume "a = a1"
blanchet@48975
   738
       hence "?f True = a" by auto  thus ?thesis by blast
blanchet@48975
   739
     next
blanchet@48975
   740
       assume "a \<noteq> a1" hence "a = a2" using * by auto
blanchet@48975
   741
       hence "?f False = a" by auto  thus ?thesis by blast
blanchet@48975
   742
     qed
blanchet@48975
   743
    }
traytel@55811
   744
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by blast
blanchet@48975
   745
  qed
blanchet@48975
   746
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   747
qed
blanchet@48975
   748
blanchet@48975
   749
lemma card_of_Plus_Times_aux:
blanchet@48975
   750
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
blanchet@48975
   751
        LEQ: "|A| \<le>o |B|"
blanchet@48975
   752
shows "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   753
proof-
blanchet@48975
   754
  have 1: "|UNIV::bool set| \<le>o |A|"
blanchet@48975
   755
  using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
traytel@55811
   756
        ordIso_ordLeq_trans[of "|UNIV::bool set|"] by blast
blanchet@48975
   757
  (*  *)
blanchet@48975
   758
  have "|A <+> B| \<le>o |B <+> B|"
blanchet@48975
   759
  using LEQ card_of_Plus_mono1 by blast
blanchet@48975
   760
  moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
blanchet@48975
   761
  using card_of_Plus_Times_bool by blast
blanchet@48975
   762
  moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
blanchet@48975
   763
  using 1 by (simp add: card_of_Times_mono2)
blanchet@48975
   764
  moreover have " |B \<times> A| =o |A \<times> B|"
blanchet@48975
   765
  using card_of_Times_commute by blast
blanchet@48975
   766
  ultimately show "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   767
  using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
blanchet@48975
   768
        ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
blanchet@48975
   769
        ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
blanchet@48975
   770
  by blast
blanchet@48975
   771
qed
blanchet@48975
   772
blanchet@48975
   773
lemma card_of_Plus_Times:
blanchet@48975
   774
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
blanchet@48975
   775
        B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
blanchet@48975
   776
shows "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   777
proof-
blanchet@48975
   778
  {assume "|A| \<le>o |B|"
blanchet@48975
   779
   hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
blanchet@48975
   780
  }
blanchet@48975
   781
  moreover
blanchet@48975
   782
  {assume "|B| \<le>o |A|"
blanchet@48975
   783
   hence "|B <+> A| \<le>o |B \<times> A|"
blanchet@48975
   784
   using assms by (auto simp add: card_of_Plus_Times_aux)
blanchet@48975
   785
   hence ?thesis
blanchet@48975
   786
   using card_of_Plus_commute card_of_Times_commute
traytel@55811
   787
         ordIso_ordLeq_trans ordLeq_ordIso_trans by blast
blanchet@48975
   788
  }
blanchet@48975
   789
  ultimately show ?thesis
blanchet@48975
   790
  using card_of_Well_order[of A] card_of_Well_order[of B]
traytel@55811
   791
        ordLeq_total[of "|A|"] by blast
blanchet@48975
   792
qed
blanchet@48975
   793
blanchet@48975
   794
lemma card_of_ordLeq_finite:
blanchet@48975
   795
assumes "|A| \<le>o |B|" and "finite B"
blanchet@48975
   796
shows "finite A"
blanchet@48975
   797
using assms unfolding ordLeq_def
blanchet@48975
   798
using embed_inj_on[of "|A|" "|B|"]  embed_Field[of "|A|" "|B|"]
blanchet@48975
   799
      Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
blanchet@48975
   800
blanchet@48975
   801
lemma card_of_ordLeq_infinite:
traytel@54578
   802
assumes "|A| \<le>o |B|" and "\<not> finite A"
traytel@54578
   803
shows "\<not> finite B"
blanchet@48975
   804
using assms card_of_ordLeq_finite by auto
blanchet@48975
   805
blanchet@48975
   806
lemma card_of_ordIso_finite:
blanchet@48975
   807
assumes "|A| =o |B|"
blanchet@48975
   808
shows "finite A = finite B"
blanchet@48975
   809
using assms unfolding ordIso_def iso_def[abs_def]
blanchet@48975
   810
by (auto simp: bij_betw_finite Field_card_of)
blanchet@48975
   811
blanchet@48975
   812
lemma card_of_ordIso_finite_Field:
blanchet@48975
   813
assumes "Card_order r" and "r =o |A|"
blanchet@48975
   814
shows "finite(Field r) = finite A"
blanchet@48975
   815
using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
blanchet@48975
   816
blanchet@48975
   817
blanchet@48975
   818
subsection {* Cardinals versus set operations involving infinite sets *}
blanchet@48975
   819
blanchet@48975
   820
text{* Here we show that, for infinite sets, most set-theoretic constructions
blanchet@48975
   821
do not increase the cardinality.  The cornerstone for this is
blanchet@48975
   822
theorem @{text "Card_order_Times_same_infinite"}, which states that self-product
blanchet@48975
   823
does not increase cardinality -- the proof of this fact adapts a standard
blanchet@48975
   824
set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
blanchet@55101
   825
at page 47 in \cite{card-book}. Then everything else follows fairly easily. *}
blanchet@48975
   826
blanchet@48975
   827
lemma infinite_iff_card_of_nat:
traytel@54578
   828
"\<not> finite A \<longleftrightarrow> ( |UNIV::nat set| \<le>o |A| )"
traytel@54578
   829
unfolding infinite_iff_countable_subset card_of_ordLeq ..
blanchet@48975
   830
blanchet@48975
   831
text{* The next two results correspond to the ZF fact that all infinite cardinals are
blanchet@48975
   832
limit ordinals: *}
blanchet@48975
   833
blanchet@48975
   834
lemma Card_order_infinite_not_under:
traytel@54578
   835
assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"
blanchet@55023
   836
shows "\<not> (\<exists>a. Field r = under r a)"
blanchet@48975
   837
proof(auto)
blanchet@48975
   838
  have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
blanchet@48975
   839
  using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
blanchet@55023
   840
  fix a assume *: "Field r = under r a"
blanchet@48975
   841
  show False
blanchet@48975
   842
  proof(cases "a \<in> Field r")
blanchet@48975
   843
    assume Case1: "a \<notin> Field r"
blanchet@55023
   844
    hence "under r a = {}" unfolding Field_def under_def by auto
blanchet@48975
   845
    thus False using INF *  by auto
blanchet@48975
   846
  next
blanchet@55023
   847
    let ?r' = "Restr r (underS r a)"
blanchet@48975
   848
    assume Case2: "a \<in> Field r"
blanchet@55023
   849
    hence 1: "under r a = underS r a \<union> {a} \<and> a \<notin> underS r a"
traytel@55811
   850
    using 0 Refl_under_underS[of r a] underS_notIn[of a r] by blast
blanchet@55023
   851
    have 2: "wo_rel.ofilter r (underS r a) \<and> underS r a < Field r"
blanchet@54482
   852
    using 0 wo_rel.underS_ofilter * 1 Case2 by fast
blanchet@48975
   853
    hence "?r' <o r" using 0 using ofilter_ordLess by blast
blanchet@48975
   854
    moreover
blanchet@55023
   855
    have "Field ?r' = underS r a \<and> Well_order ?r'"
blanchet@48975
   856
    using  2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
blanchet@55023
   857
    ultimately have "|underS r a| <o r" using ordLess_Field[of ?r'] by auto
blanchet@55023
   858
    moreover have "|under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
blanchet@55023
   859
    ultimately have "|underS r a| <o |under r a|"
blanchet@48975
   860
    using ordIso_symmetric ordLess_ordIso_trans by blast
blanchet@48975
   861
    moreover
blanchet@55023
   862
    {have "\<exists>f. bij_betw f (under r a) (underS r a)"
blanchet@48975
   863
     using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
blanchet@55023
   864
     hence "|under r a| =o |underS r a|" using card_of_ordIso by blast
blanchet@48975
   865
    }
blanchet@48975
   866
    ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
blanchet@48975
   867
  qed
blanchet@48975
   868
qed
blanchet@48975
   869
blanchet@48975
   870
lemma infinite_Card_order_limit:
traytel@54578
   871
assumes r: "Card_order r" and "\<not>finite (Field r)"
blanchet@48975
   872
and a: "a : Field r"
blanchet@48975
   873
shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
blanchet@48975
   874
proof-
blanchet@55023
   875
  have "Field r \<noteq> under r a"
blanchet@48975
   876
  using assms Card_order_infinite_not_under by blast
blanchet@55023
   877
  moreover have "under r a \<le> Field r"
blanchet@55023
   878
  using under_Field .
blanchet@55023
   879
  ultimately have "under r a < Field r" by blast
blanchet@48975
   880
  then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
blanchet@55023
   881
  unfolding under_def by blast
blanchet@48975
   882
  moreover have ba: "b \<noteq> a"
blanchet@48975
   883
  using 1 r unfolding card_order_on_def well_order_on_def
blanchet@48975
   884
  linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
blanchet@48975
   885
  ultimately have "(a,b) : r"
blanchet@48975
   886
  using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
blanchet@48975
   887
  total_on_def by blast
blanchet@48975
   888
  thus ?thesis using 1 ba by auto
blanchet@48975
   889
qed
blanchet@48975
   890
blanchet@48975
   891
theorem Card_order_Times_same_infinite:
traytel@54578
   892
assumes CO: "Card_order r" and INF: "\<not>finite(Field r)"
blanchet@48975
   893
shows "|Field r \<times> Field r| \<le>o r"
blanchet@48975
   894
proof-
blanchet@48975
   895
  obtain phi where phi_def:
traytel@54578
   896
  "phi = (\<lambda>r::'a rel. Card_order r \<and> \<not>finite(Field r) \<and>
blanchet@48975
   897
                      \<not> |Field r \<times> Field r| \<le>o r )" by blast
blanchet@48975
   898
  have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
blanchet@48975
   899
  unfolding phi_def card_order_on_def by auto
blanchet@48975
   900
  have Ft: "\<not>(\<exists>r. phi r)"
blanchet@48975
   901
  proof
blanchet@48975
   902
    assume "\<exists>r. phi r"
blanchet@48975
   903
    hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
blanchet@48975
   904
    using temp1 by auto
blanchet@48975
   905
    then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
blanchet@48975
   906
                   3: "Card_order r \<and> Well_order r"
blanchet@48975
   907
    using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
blanchet@48975
   908
    let ?A = "Field r"  let ?r' = "bsqr r"
blanchet@48975
   909
    have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
blanchet@48975
   910
    using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
blanchet@48975
   911
    have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
blanchet@48975
   912
    using card_of_Card_order card_of_Well_order by blast
blanchet@48975
   913
    (*  *)
blanchet@48975
   914
    have "r <o |?A \<times> ?A|"
blanchet@48975
   915
    using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
blanchet@48975
   916
    moreover have "|?A \<times> ?A| \<le>o ?r'"
blanchet@48975
   917
    using card_of_least[of "?A \<times> ?A"] 4 by auto
blanchet@48975
   918
    ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
blanchet@48975
   919
    then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
blanchet@48975
   920
    unfolding ordLess_def embedS_def[abs_def]
blanchet@48975
   921
    by (auto simp add: Field_bsqr)
blanchet@48975
   922
    let ?B = "f ` ?A"
blanchet@48975
   923
    have "|?A| =o |?B|"
blanchet@48975
   924
    using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
blanchet@48975
   925
    hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
blanchet@48975
   926
    (*  *)
blanchet@48975
   927
    have "wo_rel.ofilter ?r' ?B"
blanchet@48975
   928
    using 6 embed_Field_ofilter 3 4 by blast
blanchet@48975
   929
    hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
blanchet@48975
   930
    using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
blanchet@48975
   931
    hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
blanchet@48975
   932
    using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
blanchet@55023
   933
    have "\<not> (\<exists>a. Field r = under r a)"
blanchet@48975
   934
    using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
blanchet@48975
   935
    then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
blanchet@48975
   936
    using temp2 3 bsqr_ofilter[of r ?B] by blast
blanchet@48975
   937
    hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
blanchet@48975
   938
    hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
blanchet@48975
   939
    let ?r1 = "Restr r A1"
blanchet@48975
   940
    have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
blanchet@48975
   941
    moreover
blanchet@48975
   942
    {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
blanchet@48975
   943
     hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
blanchet@48975
   944
    }
blanchet@48975
   945
    ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
blanchet@48975
   946
    (*  *)
traytel@54578
   947
    have "\<not> finite (Field r)" using 1 unfolding phi_def by simp
traytel@54578
   948
    hence "\<not> finite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
traytel@55811
   949
    hence "\<not> finite A1" using 9 finite_cartesian_product finite_subset by blast
blanchet@48975
   950
    moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
blanchet@48975
   951
    using card_of_Card_order[of A1] card_of_Well_order[of A1]
blanchet@48975
   952
    by (simp add: Field_card_of)
blanchet@48975
   953
    moreover have "\<not> r \<le>o | A1 |"
blanchet@48975
   954
    using temp4 11 3 using not_ordLeq_iff_ordLess by blast
traytel@54578
   955
    ultimately have "\<not> finite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
blanchet@48975
   956
    by (simp add: card_of_card_order_on)
blanchet@48975
   957
    hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
blanchet@48975
   958
    using 2 unfolding phi_def by blast
blanchet@48975
   959
    hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
blanchet@48975
   960
    hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
blanchet@48975
   961
    thus False using 11 not_ordLess_ordLeq by auto
blanchet@48975
   962
  qed
blanchet@48975
   963
  thus ?thesis using assms unfolding phi_def by blast
blanchet@48975
   964
qed
blanchet@48975
   965
blanchet@48975
   966
corollary card_of_Times_same_infinite:
traytel@54578
   967
assumes "\<not>finite A"
blanchet@48975
   968
shows "|A \<times> A| =o |A|"
blanchet@48975
   969
proof-
blanchet@48975
   970
  let ?r = "|A|"
blanchet@48975
   971
  have "Field ?r = A \<and> Card_order ?r"
blanchet@48975
   972
  using Field_card_of card_of_Card_order[of A] by fastforce
blanchet@48975
   973
  hence "|A \<times> A| \<le>o |A|"
blanchet@48975
   974
  using Card_order_Times_same_infinite[of ?r] assms by auto
blanchet@48975
   975
  thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
blanchet@48975
   976
qed
blanchet@48975
   977
blanchet@48975
   978
lemma card_of_Times_infinite:
traytel@54578
   979
assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
blanchet@48975
   980
shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
blanchet@48975
   981
proof-
blanchet@48975
   982
  have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
blanchet@48975
   983
  using assms by (simp add: card_of_Times1 card_of_Times2)
blanchet@48975
   984
  moreover
blanchet@48975
   985
  {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
blanchet@48975
   986
   using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
blanchet@48975
   987
   moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
blanchet@48975
   988
   ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
blanchet@48975
   989
   using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
blanchet@48975
   990
  }
blanchet@48975
   991
  ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
blanchet@48975
   992
qed
blanchet@48975
   993
blanchet@48975
   994
corollary Card_order_Times_infinite:
traytel@54578
   995
assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
blanchet@48975
   996
        NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
blanchet@48975
   997
shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
blanchet@48975
   998
proof-
blanchet@48975
   999
  have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
blanchet@48975
  1000
  using assms by (simp add: card_of_Times_infinite card_of_mono2)
blanchet@48975
  1001
  thus ?thesis
blanchet@48975
  1002
  using assms card_of_Field_ordIso[of r]
blanchet@48975
  1003
        ordIso_transitive[of "|Field r \<times> Field p|"]
blanchet@48975
  1004
        ordIso_transitive[of _ "|Field r|"] by blast
blanchet@48975
  1005
qed
blanchet@48975
  1006
blanchet@48975
  1007
lemma card_of_Sigma_ordLeq_infinite:
traytel@54578
  1008
assumes INF: "\<not>finite B" and
blanchet@48975
  1009
        LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
blanchet@48975
  1010
shows "|SIGMA i : I. A i| \<le>o |B|"
blanchet@48975
  1011
proof(cases "I = {}", simp add: card_of_empty)
blanchet@48975
  1012
  assume *: "I \<noteq> {}"
blanchet@48975
  1013
  have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
blanchet@48975
  1014
  using LEQ card_of_Sigma_Times by blast
blanchet@48975
  1015
  moreover have "|I \<times> B| =o |B|"
blanchet@48975
  1016
  using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
blanchet@48975
  1017
  ultimately show ?thesis using ordLeq_ordIso_trans by blast
blanchet@48975
  1018
qed
blanchet@48975
  1019
blanchet@48975
  1020
lemma card_of_Sigma_ordLeq_infinite_Field:
traytel@54578
  1021
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@48975
  1022
        LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@48975
  1023
shows "|SIGMA i : I. A i| \<le>o r"
blanchet@48975
  1024
proof-
blanchet@48975
  1025
  let ?B  = "Field r"
blanchet@48975
  1026
  have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
blanchet@48975
  1027
  ordIso_symmetric by blast
blanchet@48975
  1028
  hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"
blanchet@48975
  1029
  using LEQ_I LEQ ordLeq_ordIso_trans by blast+
blanchet@48975
  1030
  hence  "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
blanchet@48975
  1031
  card_of_Sigma_ordLeq_infinite by blast
blanchet@48975
  1032
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
blanchet@48975
  1033
qed
blanchet@48975
  1034
blanchet@48975
  1035
lemma card_of_Times_ordLeq_infinite_Field:
traytel@54578
  1036
"\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
blanchet@48975
  1037
 \<Longrightarrow> |A <*> B| \<le>o r"
blanchet@48975
  1038
by(simp add: card_of_Sigma_ordLeq_infinite_Field)
blanchet@48975
  1039
blanchet@54475
  1040
lemma card_of_Times_infinite_simps:
traytel@54578
  1041
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
traytel@54578
  1042
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
traytel@54578
  1043
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
traytel@54578
  1044
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
blanchet@54475
  1045
by (auto simp add: card_of_Times_infinite ordIso_symmetric)
blanchet@54475
  1046
blanchet@48975
  1047
lemma card_of_UNION_ordLeq_infinite:
traytel@54578
  1048
assumes INF: "\<not>finite B" and
blanchet@48975
  1049
        LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
blanchet@48975
  1050
shows "|\<Union> i \<in> I. A i| \<le>o |B|"
blanchet@48975
  1051
proof(cases "I = {}", simp add: card_of_empty)
blanchet@48975
  1052
  assume *: "I \<noteq> {}"
blanchet@48975
  1053
  have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
blanchet@48975
  1054
  using card_of_UNION_Sigma by blast
blanchet@48975
  1055
  moreover have "|SIGMA i : I. A i| \<le>o |B|"
blanchet@48975
  1056
  using assms card_of_Sigma_ordLeq_infinite by blast
blanchet@48975
  1057
  ultimately show ?thesis using ordLeq_transitive by blast
blanchet@48975
  1058
qed
blanchet@48975
  1059
blanchet@48975
  1060
corollary card_of_UNION_ordLeq_infinite_Field:
traytel@54578
  1061
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@48975
  1062
        LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@48975
  1063
shows "|\<Union> i \<in> I. A i| \<le>o r"
blanchet@48975
  1064
proof-
blanchet@48975
  1065
  let ?B  = "Field r"
blanchet@48975
  1066
  have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
blanchet@48975
  1067
  ordIso_symmetric by blast
blanchet@48975
  1068
  hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"
blanchet@48975
  1069
  using LEQ_I LEQ ordLeq_ordIso_trans by blast+
blanchet@48975
  1070
  hence  "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ
blanchet@48975
  1071
  card_of_UNION_ordLeq_infinite by blast
blanchet@48975
  1072
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
blanchet@48975
  1073
qed
blanchet@48975
  1074
blanchet@48975
  1075
lemma card_of_Plus_infinite1:
traytel@54578
  1076
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1077
shows "|A <+> B| =o |A|"
blanchet@48975
  1078
proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
blanchet@48975
  1079
  let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b"  let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
blanchet@48975
  1080
  assume *: "B \<noteq> {}"
blanchet@48975
  1081
  then obtain b1 where 1: "b1 \<in> B" by blast
blanchet@48975
  1082
  show ?thesis
blanchet@48975
  1083
  proof(cases "B = {b1}")
blanchet@48975
  1084
    assume Case1: "B = {b1}"
blanchet@48975
  1085
    have 2: "bij_betw ?Inl A ((?Inl ` A))"
blanchet@48975
  1086
    unfolding bij_betw_def inj_on_def by auto
traytel@54578
  1087
    hence 3: "\<not>finite (?Inl ` A)"
blanchet@48975
  1088
    using INF bij_betw_finite[of ?Inl A] by blast
blanchet@48975
  1089
    let ?A' = "?Inl ` A \<union> {?Inr b1}"
blanchet@48975
  1090
    obtain g where "bij_betw g (?Inl ` A) ?A'"
blanchet@48975
  1091
    using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
blanchet@48975
  1092
    moreover have "?A' = A <+> B" using Case1 by blast
blanchet@48975
  1093
    ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
blanchet@48975
  1094
    hence "bij_betw (g o ?Inl) A (A <+> B)"
blanchet@48975
  1095
    using 2 by (auto simp add: bij_betw_trans)
blanchet@48975
  1096
    thus ?thesis using card_of_ordIso ordIso_symmetric by blast
blanchet@48975
  1097
  next
blanchet@48975
  1098
    assume Case2: "B \<noteq> {b1}"
blanchet@48975
  1099
    with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
blanchet@48975
  1100
    obtain f where "inj_on f B \<and> f ` B \<le> A"
blanchet@48975
  1101
    using LEQ card_of_ordLeq[of B] by fastforce
blanchet@48975
  1102
    with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
blanchet@48975
  1103
    unfolding inj_on_def by auto
blanchet@48975
  1104
    with 3 have "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
  1105
    by (auto simp add: card_of_Plus_Times)
blanchet@48975
  1106
    moreover have "|A \<times> B| =o |A|"
blanchet@48975
  1107
    using assms * by (simp add: card_of_Times_infinite_simps)
traytel@55811
  1108
    ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by blast
blanchet@48975
  1109
    thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
blanchet@48975
  1110
  qed
blanchet@48975
  1111
qed
blanchet@48975
  1112
blanchet@48975
  1113
lemma card_of_Plus_infinite2:
traytel@54578
  1114
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1115
shows "|B <+> A| =o |A|"
blanchet@48975
  1116
using assms card_of_Plus_commute card_of_Plus_infinite1
blanchet@48975
  1117
ordIso_equivalence by blast
blanchet@48975
  1118
blanchet@48975
  1119
lemma card_of_Plus_infinite:
traytel@54578
  1120
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1121
shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
blanchet@48975
  1122
using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
blanchet@48975
  1123
blanchet@48975
  1124
corollary Card_order_Plus_infinite:
traytel@54578
  1125
assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
blanchet@48975
  1126
        LEQ: "p \<le>o r"
blanchet@48975
  1127
shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
blanchet@48975
  1128
proof-
blanchet@48975
  1129
  have "| Field r <+> Field p | =o | Field r | \<and>
blanchet@48975
  1130
        | Field p <+> Field r | =o | Field r |"
blanchet@48975
  1131
  using assms by (simp add: card_of_Plus_infinite card_of_mono2)
blanchet@48975
  1132
  thus ?thesis
blanchet@48975
  1133
  using assms card_of_Field_ordIso[of r]
blanchet@48975
  1134
        ordIso_transitive[of "|Field r <+> Field p|"]
blanchet@48975
  1135
        ordIso_transitive[of _ "|Field r|"] by blast
blanchet@48975
  1136
qed
blanchet@48975
  1137
blanchet@48975
  1138
blanchet@55101
  1139
subsection {* The cardinal $\omega$ and the finite cardinals *}
blanchet@48975
  1140
blanchet@48975
  1141
text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
blanchet@48975
  1142
order relation on
blanchet@48975
  1143
@{text "nat"}, that we abbreviate by @{text "natLeq"}.  The finite cardinals
blanchet@48975
  1144
shall be the restrictions of these relations to the numbers smaller than
blanchet@55101
  1145
fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}. *}
blanchet@48975
  1146
traytel@56011
  1147
definition "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
traytel@56011
  1148
definition "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
blanchet@48975
  1149
blanchet@48975
  1150
abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
blanchet@48975
  1151
where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
blanchet@48975
  1152
blanchet@48975
  1153
lemma infinite_cartesian_product:
traytel@54578
  1154
assumes "\<not>finite A" "\<not>finite B"
traytel@54578
  1155
shows "\<not>finite (A \<times> B)"
blanchet@48975
  1156
proof
blanchet@48975
  1157
  assume "finite (A \<times> B)"
blanchet@48975
  1158
  from assms(1) have "A \<noteq> {}" by auto
blanchet@48975
  1159
  with `finite (A \<times> B)` have "finite B" using finite_cartesian_productD2 by auto
blanchet@48975
  1160
  with assms(2) show False by simp
blanchet@48975
  1161
qed
blanchet@48975
  1162
blanchet@48975
  1163
blanchet@48975
  1164
subsubsection {* First as well-orders *}
blanchet@48975
  1165
blanchet@48975
  1166
lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
traytel@56011
  1167
by(unfold Field_def natLeq_def, auto)
blanchet@48975
  1168
blanchet@48975
  1169
lemma natLeq_Refl: "Refl natLeq"
traytel@56011
  1170
unfolding refl_on_def Field_def natLeq_def by auto
blanchet@48975
  1171
blanchet@48975
  1172
lemma natLeq_trans: "trans natLeq"
traytel@56011
  1173
unfolding trans_def natLeq_def by auto
blanchet@48975
  1174
blanchet@48975
  1175
lemma natLeq_Preorder: "Preorder natLeq"
blanchet@48975
  1176
unfolding preorder_on_def
blanchet@48975
  1177
by (auto simp add: natLeq_Refl natLeq_trans)
blanchet@48975
  1178
blanchet@48975
  1179
lemma natLeq_antisym: "antisym natLeq"
traytel@56011
  1180
unfolding antisym_def natLeq_def by auto
blanchet@48975
  1181
blanchet@48975
  1182
lemma natLeq_Partial_order: "Partial_order natLeq"
blanchet@48975
  1183
unfolding partial_order_on_def
blanchet@48975
  1184
by (auto simp add: natLeq_Preorder natLeq_antisym)
blanchet@48975
  1185
blanchet@48975
  1186
lemma natLeq_Total: "Total natLeq"
traytel@56011
  1187
unfolding total_on_def natLeq_def by auto
blanchet@48975
  1188
blanchet@48975
  1189
lemma natLeq_Linear_order: "Linear_order natLeq"
blanchet@48975
  1190
unfolding linear_order_on_def
blanchet@48975
  1191
by (auto simp add: natLeq_Partial_order natLeq_Total)
blanchet@48975
  1192
blanchet@48975
  1193
lemma natLeq_natLess_Id: "natLess = natLeq - Id"
traytel@56011
  1194
unfolding natLeq_def natLess_def by auto
blanchet@48975
  1195
blanchet@48975
  1196
lemma natLeq_Well_order: "Well_order natLeq"
blanchet@48975
  1197
unfolding well_order_on_def
traytel@56011
  1198
using natLeq_Linear_order wf_less natLeq_natLess_Id natLeq_def natLess_def by auto
blanchet@48975
  1199
traytel@54581
  1200
lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"
blanchet@48975
  1201
unfolding Field_def by auto
blanchet@48975
  1202
blanchet@55023
  1203
lemma natLeq_underS_less: "underS natLeq n = {x. x < n}"
traytel@56011
  1204
unfolding underS_def natLeq_def by auto
blanchet@48975
  1205
traytel@54581
  1206
lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"
traytel@56011
  1207
unfolding natLeq_def by force
blanchet@48975
  1208
blanchet@48975
  1209
lemma Restr_natLeq2:
blanchet@55023
  1210
"Restr natLeq (underS natLeq n) = natLeq_on n"
blanchet@48975
  1211
by (auto simp add: Restr_natLeq natLeq_underS_less)
blanchet@48975
  1212
blanchet@48975
  1213
lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
blanchet@48975
  1214
using Restr_natLeq[of n] natLeq_Well_order
traytel@54581
  1215
      Well_order_Restr[of natLeq "{x. x < n}"] by auto
blanchet@48975
  1216
traytel@54581
  1217
corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"
blanchet@48975
  1218
using natLeq_on_Well_order Field_natLeq_on by auto
blanchet@48975
  1219
blanchet@48975
  1220
lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
blanchet@48975
  1221
unfolding wo_rel_def using natLeq_on_Well_order .
blanchet@48975
  1222
blanchet@48975
  1223
blanchet@48975
  1224
subsubsection {* Then as cardinals *}
blanchet@48975
  1225
blanchet@48975
  1226
lemma natLeq_Card_order: "Card_order natLeq"
blanchet@48975
  1227
proof(auto simp add: natLeq_Well_order
blanchet@48975
  1228
      Card_order_iff_Restr_underS Restr_natLeq2, simp add:  Field_natLeq)
traytel@54581
  1229
  fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
traytel@54578
  1230
  moreover have "\<not>finite(UNIV::nat set)" by auto
blanchet@48975
  1231
  ultimately show "natLeq_on n <o |UNIV::nat set|"
blanchet@48975
  1232
  using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
blanchet@48975
  1233
        Field_card_of[of "UNIV::nat set"]
blanchet@48975
  1234
        card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
blanchet@48975
  1235
qed
blanchet@48975
  1236
blanchet@48975
  1237
corollary card_of_Field_natLeq:
blanchet@48975
  1238
"|Field natLeq| =o natLeq"
blanchet@48975
  1239
using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
blanchet@48975
  1240
      ordIso_symmetric[of natLeq] by blast
blanchet@48975
  1241
blanchet@48975
  1242
corollary card_of_nat:
blanchet@48975
  1243
"|UNIV::nat set| =o natLeq"
blanchet@48975
  1244
using Field_natLeq card_of_Field_natLeq by auto
blanchet@48975
  1245
blanchet@48975
  1246
corollary infinite_iff_natLeq_ordLeq:
traytel@54578
  1247
"\<not>finite A = ( natLeq \<le>o |A| )"
blanchet@48975
  1248
using infinite_iff_card_of_nat[of A] card_of_nat
blanchet@48975
  1249
      ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
blanchet@48975
  1250
blanchet@48975
  1251
corollary finite_iff_ordLess_natLeq:
blanchet@48975
  1252
"finite A = ( |A| <o natLeq)"
blanchet@48975
  1253
using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
traytel@55811
  1254
      card_of_Well_order natLeq_Well_order by blast
blanchet@48975
  1255
blanchet@48975
  1256
blanchet@48975
  1257
subsection {* The successor of a cardinal *}
blanchet@48975
  1258
blanchet@48975
  1259
text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
blanchet@48975
  1260
being a successor cardinal of @{text "r"}. Although the definition does
blanchet@55101
  1261
not require @{text "r"} to be a cardinal, only this case will be meaningful. *}
blanchet@48975
  1262
blanchet@48975
  1263
definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
blanchet@48975
  1264
where
blanchet@48975
  1265
"isCardSuc r r' \<equiv>
blanchet@48975
  1266
 Card_order r' \<and> r <o r' \<and>
blanchet@48975
  1267
 (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
blanchet@48975
  1268
blanchet@48975
  1269
text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
blanchet@48975
  1270
by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
blanchet@48975
  1271
Again, the picked item shall be proved unique up to order-isomorphism. *}
blanchet@48975
  1272
blanchet@48975
  1273
definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
blanchet@48975
  1274
where
blanchet@48975
  1275
"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
blanchet@48975
  1276
blanchet@48975
  1277
lemma exists_minim_Card_order:
blanchet@48975
  1278
"\<lbrakk>R \<noteq> {}; \<forall>r \<in> R. Card_order r\<rbrakk> \<Longrightarrow> \<exists>r \<in> R. \<forall>r' \<in> R. r \<le>o r'"
blanchet@48975
  1279
unfolding card_order_on_def using exists_minim_Well_order by blast
blanchet@48975
  1280
blanchet@48975
  1281
lemma exists_isCardSuc:
blanchet@48975
  1282
assumes "Card_order r"
blanchet@48975
  1283
shows "\<exists>r'. isCardSuc r r'"
blanchet@48975
  1284
proof-
blanchet@48975
  1285
  let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
blanchet@48975
  1286
  have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
blanchet@48975
  1287
  by (simp add: card_of_Card_order Card_order_Pow)
blanchet@48975
  1288
  then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
blanchet@48975
  1289
  using exists_minim_Card_order[of ?R] by blast
blanchet@48975
  1290
  thus ?thesis unfolding isCardSuc_def by auto
blanchet@48975
  1291
qed
blanchet@48975
  1292
blanchet@48975
  1293
lemma cardSuc_isCardSuc:
blanchet@48975
  1294
assumes "Card_order r"
blanchet@48975
  1295
shows "isCardSuc r (cardSuc r)"
blanchet@48975
  1296
unfolding cardSuc_def using assms
blanchet@48975
  1297
by (simp add: exists_isCardSuc someI_ex)
blanchet@48975
  1298
blanchet@48975
  1299
lemma cardSuc_Card_order:
blanchet@48975
  1300
"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
blanchet@48975
  1301
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1302
blanchet@48975
  1303
lemma cardSuc_greater:
blanchet@48975
  1304
"Card_order r \<Longrightarrow> r <o cardSuc r"
blanchet@48975
  1305
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1306
blanchet@48975
  1307
lemma cardSuc_ordLeq:
blanchet@48975
  1308
"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
blanchet@48975
  1309
using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
  1310
blanchet@48975
  1311
text{* The minimality property of @{text "cardSuc"} originally present in its definition
blanchet@55101
  1312
is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}: *}
blanchet@48975
  1313
blanchet@48975
  1314
lemma cardSuc_least_aux:
blanchet@48975
  1315
"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
blanchet@48975
  1316
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1317
blanchet@48975
  1318
text{* But from this we can infer general minimality: *}
blanchet@48975
  1319
blanchet@48975
  1320
lemma cardSuc_least:
blanchet@48975
  1321
assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
blanchet@48975
  1322
shows "cardSuc r \<le>o r'"
blanchet@48975
  1323
proof-
blanchet@48975
  1324
  let ?p = "cardSuc r"
blanchet@48975
  1325
  have 0: "Well_order ?p \<and> Well_order r'"
blanchet@48975
  1326
  using assms cardSuc_Card_order unfolding card_order_on_def by blast
blanchet@48975
  1327
  {assume "r' <o ?p"
blanchet@48975
  1328
   then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
blanchet@48975
  1329
   using internalize_ordLess[of r' ?p] by blast
blanchet@48975
  1330
   (*  *)
blanchet@48975
  1331
   have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
blanchet@48975
  1332
   moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
blanchet@48975
  1333
   ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
blanchet@48975
  1334
   hence False using 2 not_ordLess_ordLeq by blast
blanchet@48975
  1335
  }
blanchet@48975
  1336
  thus ?thesis using 0 ordLess_or_ordLeq by blast
blanchet@48975
  1337
qed
blanchet@48975
  1338
blanchet@48975
  1339
lemma cardSuc_ordLess_ordLeq:
blanchet@48975
  1340
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1341
shows "(r <o r') = (cardSuc r \<le>o r')"
blanchet@48975
  1342
proof(auto simp add: assms cardSuc_least)
blanchet@48975
  1343
  assume "cardSuc r \<le>o r'"
blanchet@48975
  1344
  thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
blanchet@48975
  1345
qed
blanchet@48975
  1346
blanchet@48975
  1347
lemma cardSuc_ordLeq_ordLess:
blanchet@48975
  1348
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1349
shows "(r' <o cardSuc r) = (r' \<le>o r)"
blanchet@48975
  1350
proof-
blanchet@48975
  1351
  have "Well_order r \<and> Well_order r'"
blanchet@48975
  1352
  using assms unfolding card_order_on_def by auto
blanchet@48975
  1353
  moreover have "Well_order(cardSuc r)"
blanchet@48975
  1354
  using assms cardSuc_Card_order card_order_on_def by blast
blanchet@48975
  1355
  ultimately show ?thesis
blanchet@48975
  1356
  using assms cardSuc_ordLess_ordLeq[of r r']
blanchet@48975
  1357
  not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
blanchet@48975
  1358
qed
blanchet@48975
  1359
blanchet@48975
  1360
lemma cardSuc_mono_ordLeq:
blanchet@48975
  1361
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1362
shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
blanchet@48975
  1363
using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
blanchet@48975
  1364
blanchet@48975
  1365
lemma cardSuc_invar_ordIso:
blanchet@48975
  1366
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1367
shows "(cardSuc r =o cardSuc r') = (r =o r')"
blanchet@48975
  1368
proof-
blanchet@48975
  1369
  have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
blanchet@48975
  1370
  using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
blanchet@48975
  1371
  thus ?thesis
blanchet@48975
  1372
  using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
blanchet@48975
  1373
  using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
blanchet@48975
  1374
qed
blanchet@48975
  1375
blanchet@48975
  1376
lemma card_of_cardSuc_finite:
blanchet@48975
  1377
"finite(Field(cardSuc |A| )) = finite A"
blanchet@48975
  1378
proof
blanchet@48975
  1379
  assume *: "finite (Field (cardSuc |A| ))"
blanchet@48975
  1380
  have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
blanchet@48975
  1381
  using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
blanchet@48975
  1382
  hence "|A| \<le>o |Field(cardSuc |A| )|"
blanchet@48975
  1383
  using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
blanchet@48975
  1384
  ordLeq_ordIso_trans by blast
blanchet@48975
  1385
  thus "finite A" using * card_of_ordLeq_finite by blast
blanchet@48975
  1386
next
blanchet@48975
  1387
  assume "finite A"
traytel@54581
  1388
  then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp
traytel@54581
  1389
  then show "finite (Field (cardSuc |A| ))"
traytel@54581
  1390
  proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])
traytel@54581
  1391
    show "cardSuc |A| \<le>o |Pow A|"
traytel@55811
  1392
      by (rule iffD1[OF cardSuc_ordLess_ordLeq card_of_Pow]) (simp_all add: card_of_Card_order)
traytel@54581
  1393
  qed
blanchet@48975
  1394
qed
blanchet@48975
  1395
blanchet@48975
  1396
lemma cardSuc_finite:
blanchet@48975
  1397
assumes "Card_order r"
blanchet@48975
  1398
shows "finite (Field (cardSuc r)) = finite (Field r)"
blanchet@48975
  1399
proof-
blanchet@48975
  1400
  let ?A = "Field r"
blanchet@48975
  1401
  have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
blanchet@48975
  1402
  hence "cardSuc |?A| =o cardSuc r" using assms
blanchet@48975
  1403
  by (simp add: card_of_Card_order cardSuc_invar_ordIso)
blanchet@48975
  1404
  moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
blanchet@48975
  1405
  by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
blanchet@48975
  1406
  moreover
blanchet@48975
  1407
  {have "|Field (cardSuc r) | =o cardSuc r"
blanchet@48975
  1408
   using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
blanchet@48975
  1409
   hence "cardSuc r =o |Field (cardSuc r) |"
blanchet@48975
  1410
   using ordIso_symmetric by blast
blanchet@48975
  1411
  }
blanchet@48975
  1412
  ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
blanchet@48975
  1413
  using ordIso_transitive by blast
blanchet@48975
  1414
  hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
blanchet@48975
  1415
  using card_of_ordIso_finite by blast
blanchet@48975
  1416
  thus ?thesis by (simp only: card_of_cardSuc_finite)
blanchet@48975
  1417
qed
blanchet@48975
  1418
blanchet@54475
  1419
lemma card_of_Plus_ordLess_infinite:
traytel@54578
  1420
assumes INF: "\<not>finite C" and
blanchet@54475
  1421
        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
blanchet@54475
  1422
shows "|A <+> B| <o |C|"
blanchet@54475
  1423
proof(cases "A = {} \<or> B = {}")
blanchet@54475
  1424
  assume Case1: "A = {} \<or> B = {}"
blanchet@54475
  1425
  hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
blanchet@54475
  1426
  using card_of_Plus_empty1 card_of_Plus_empty2 by blast
blanchet@54475
  1427
  hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
blanchet@54475
  1428
  using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
blanchet@54475
  1429
  thus ?thesis using LESS1 LESS2
blanchet@54475
  1430
       ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
blanchet@54475
  1431
       ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
blanchet@54475
  1432
next
blanchet@54475
  1433
  assume Case2: "\<not>(A = {} \<or> B = {})"
blanchet@54475
  1434
  {assume *: "|C| \<le>o |A <+> B|"
traytel@54578
  1435
   hence "\<not>finite (A <+> B)" using INF card_of_ordLeq_finite by blast
traytel@54578
  1436
   hence 1: "\<not>finite A \<or> \<not>finite B" using finite_Plus by blast
blanchet@54475
  1437
   {assume Case21: "|A| \<le>o |B|"
traytel@54578
  1438
    hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast
blanchet@54475
  1439
    hence "|A <+> B| =o |B|" using Case2 Case21
blanchet@54475
  1440
    by (auto simp add: card_of_Plus_infinite)
blanchet@54475
  1441
    hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@54475
  1442
   }
blanchet@54475
  1443
   moreover
blanchet@54475
  1444
   {assume Case22: "|B| \<le>o |A|"
traytel@54578
  1445
    hence "\<not>finite A" using 1 card_of_ordLeq_finite by blast
blanchet@54475
  1446
    hence "|A <+> B| =o |A|" using Case2 Case22
blanchet@54475
  1447
    by (auto simp add: card_of_Plus_infinite)
blanchet@54475
  1448
    hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@54475
  1449
   }
blanchet@54475
  1450
   ultimately have False using ordLeq_total card_of_Well_order[of A]
blanchet@54475
  1451
   card_of_Well_order[of B] by blast
blanchet@54475
  1452
  }
blanchet@54475
  1453
  thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
blanchet@54475
  1454
  card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
blanchet@54475
  1455
qed
blanchet@54475
  1456
blanchet@54475
  1457
lemma card_of_Plus_ordLess_infinite_Field:
traytel@54578
  1458
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@54475
  1459
        LESS1: "|A| <o r" and LESS2: "|B| <o r"
blanchet@54475
  1460
shows "|A <+> B| <o r"
blanchet@54475
  1461
proof-
blanchet@54475
  1462
  let ?C  = "Field r"
blanchet@54475
  1463
  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
blanchet@54475
  1464
  ordIso_symmetric by blast
blanchet@54475
  1465
  hence "|A| <o |?C|"  "|B| <o |?C|"
blanchet@54475
  1466
  using LESS1 LESS2 ordLess_ordIso_trans by blast+
blanchet@54475
  1467
  hence  "|A <+> B| <o |?C|" using INF
blanchet@54475
  1468
  card_of_Plus_ordLess_infinite by blast
blanchet@54475
  1469
  thus ?thesis using 1 ordLess_ordIso_trans by blast
blanchet@54475
  1470
qed
blanchet@54475
  1471
blanchet@48975
  1472
lemma card_of_Plus_ordLeq_infinite_Field:
traytel@54578
  1473
assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
blanchet@48975
  1474
and c: "Card_order r"
blanchet@48975
  1475
shows "|A <+> B| \<le>o r"
blanchet@48975
  1476
proof-
blanchet@48975
  1477
  let ?r' = "cardSuc r"
traytel@54578
  1478
  have "Card_order ?r' \<and> \<not>finite (Field ?r')" using assms
blanchet@48975
  1479
  by (simp add: cardSuc_Card_order cardSuc_finite)
blanchet@48975
  1480
  moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
blanchet@48975
  1481
  by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
blanchet@48975
  1482
  ultimately have "|A <+> B| <o ?r'"
blanchet@48975
  1483
  using card_of_Plus_ordLess_infinite_Field by blast
blanchet@48975
  1484
  thus ?thesis using c r
blanchet@48975
  1485
  by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
blanchet@48975
  1486
qed
blanchet@48975
  1487
blanchet@48975
  1488
lemma card_of_Un_ordLeq_infinite_Field:
traytel@54578
  1489
assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
blanchet@48975
  1490
and "Card_order r"
blanchet@48975
  1491
shows "|A Un B| \<le>o r"
blanchet@48975
  1492
using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
blanchet@54482
  1493
ordLeq_transitive by fast
blanchet@48975
  1494
blanchet@48975
  1495
blanchet@48975
  1496
subsection {* Regular cardinals *}
blanchet@48975
  1497
blanchet@48975
  1498
definition cofinal where
blanchet@48975
  1499
"cofinal A r \<equiv>
blanchet@48975
  1500
 ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
blanchet@48975
  1501
blanchet@55087
  1502
definition regularCard where
blanchet@55087
  1503
"regularCard r \<equiv>
blanchet@48975
  1504
 ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
blanchet@48975
  1505
blanchet@48975
  1506
definition relChain where
blanchet@48975
  1507
"relChain r As \<equiv>
blanchet@48975
  1508
 ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
blanchet@48975
  1509
blanchet@55087
  1510
lemma regularCard_UNION:
blanchet@55087
  1511
assumes r: "Card_order r"   "regularCard r"
blanchet@48975
  1512
and As: "relChain r As"
blanchet@48975
  1513
and Bsub: "B \<le> (UN i : Field r. As i)"
blanchet@48975
  1514
and cardB: "|B| <o r"
blanchet@48975
  1515
shows "EX i : Field r. B \<le> As i"
blanchet@48975
  1516
proof-
blanchet@48975
  1517
  let ?phi = "%b j. j : Field r \<and> b : As j"
blanchet@48975
  1518
  have "ALL b : B. EX j. ?phi b j" using Bsub by blast
blanchet@48975
  1519
  then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
blanchet@48975
  1520
  using bchoice[of B ?phi] by blast
blanchet@48975
  1521
  let ?K = "f ` B"
blanchet@48975
  1522
  {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"
blanchet@48975
  1523
   have 2: "cofinal ?K r"
blanchet@48975
  1524
   unfolding cofinal_def proof auto
blanchet@48975
  1525
     fix i assume i: "i : Field r"
blanchet@48975
  1526
     with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast
blanchet@48975
  1527
     hence "i \<noteq> f b \<and> ~ (f b,i) : r"
blanchet@48975
  1528
     using As f unfolding relChain_def by auto
blanchet@48975
  1529
     hence "i \<noteq> f b \<and> (i, f b) : r" using r
blanchet@48975
  1530
     unfolding card_order_on_def well_order_on_def linear_order_on_def
blanchet@48975
  1531
     total_on_def using i f b by auto
blanchet@48975
  1532
     with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
blanchet@48975
  1533
   qed
blanchet@48975
  1534
   moreover have "?K \<le> Field r" using f by blast
blanchet@55087
  1535
   ultimately have "|?K| =o r" using 2 r unfolding regularCard_def by blast
blanchet@48975
  1536
   moreover
blanchet@48975
  1537
   {
blanchet@48975
  1538
    have "|?K| <=o |B|" using card_of_image .
blanchet@48975
  1539
    hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
blanchet@48975
  1540
   }
blanchet@48975
  1541
   ultimately have False using not_ordLess_ordIso by blast
blanchet@48975
  1542
  }
blanchet@48975
  1543
  thus ?thesis by blast
blanchet@48975
  1544
qed
blanchet@48975
  1545
blanchet@55087
  1546
lemma infinite_cardSuc_regularCard:
traytel@54578
  1547
assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"
blanchet@55087
  1548
shows "regularCard (cardSuc r)"
blanchet@48975
  1549
proof-
blanchet@48975
  1550
  let ?r' = "cardSuc r"
blanchet@48975
  1551
  have r': "Card_order ?r'"
blanchet@48975
  1552
  "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
blanchet@48975
  1553
  using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
blanchet@48975
  1554
  show ?thesis
blanchet@55087
  1555
  unfolding regularCard_def proof auto
blanchet@48975
  1556
    fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
blanchet@48975
  1557
    hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
blanchet@48975
  1558
    also have 22: "|Field ?r'| =o ?r'"
blanchet@48975
  1559
    using r' by (simp add: card_of_Field_ordIso[of ?r'])
blanchet@48975
  1560
    finally have "|K| \<le>o ?r'" .
blanchet@48975
  1561
    moreover
blanchet@55023
  1562
    {let ?L = "UN j : K. underS ?r' j"
blanchet@48975
  1563
     let ?J = "Field r"
blanchet@48975
  1564
     have rJ: "r =o |?J|"
blanchet@48975
  1565
     using r_card card_of_Field_ordIso ordIso_symmetric by blast
blanchet@48975
  1566
     assume "|K| <o ?r'"
blanchet@48975
  1567
     hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
blanchet@48975
  1568
     hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
blanchet@48975
  1569
     moreover
blanchet@55023
  1570
     {have "ALL j : K. |underS ?r' j| <o ?r'"
blanchet@48975
  1571
      using r' 1 by (auto simp: card_of_underS)
blanchet@55023
  1572
      hence "ALL j : K. |underS ?r' j| \<le>o r"
blanchet@48975
  1573
      using r' card_of_Card_order by blast
blanchet@55023
  1574
      hence "ALL j : K. |underS ?r' j| \<le>o |?J|"
blanchet@48975
  1575
      using rJ ordLeq_ordIso_trans by blast
blanchet@48975
  1576
     }
blanchet@48975
  1577
     ultimately have "|?L| \<le>o |?J|"
blanchet@48975
  1578
     using r_inf card_of_UNION_ordLeq_infinite by blast
blanchet@48975
  1579
     hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
blanchet@48975
  1580
     hence "|?L| <o ?r'" using r' card_of_Card_order by blast
blanchet@48975
  1581
     moreover
blanchet@48975
  1582
     {
blanchet@48975
  1583
      have "Field ?r' \<le> ?L"
blanchet@55023
  1584
      using 2 unfolding underS_def cofinal_def by auto
blanchet@48975
  1585
      hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
blanchet@48975
  1586
      hence "?r' \<le>o |?L|"
blanchet@48975
  1587
      using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
  1588
     }
blanchet@48975
  1589
     ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
blanchet@48975
  1590
     hence False using ordLess_irreflexive by blast
blanchet@48975
  1591
    }
blanchet@48975
  1592
    ultimately show "|K| =o ?r'"
blanchet@48975
  1593
    unfolding ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
  1594
  qed
blanchet@48975
  1595
qed
blanchet@48975
  1596
blanchet@48975
  1597
lemma cardSuc_UNION:
traytel@54578
  1598
assumes r: "Card_order r" and "\<not>finite (Field r)"
blanchet@48975
  1599
and As: "relChain (cardSuc r) As"
blanchet@48975
  1600
and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
blanchet@48975
  1601
and cardB: "|B| <=o r"
blanchet@48975
  1602
shows "EX i : Field (cardSuc r). B \<le> As i"
blanchet@48975
  1603
proof-
blanchet@48975
  1604
  let ?r' = "cardSuc r"
blanchet@48975
  1605
  have "Card_order ?r' \<and> |B| <o ?r'"
blanchet@48975
  1606
  using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
blanchet@48975
  1607
  card_of_Card_order by blast
blanchet@55087
  1608
  moreover have "regularCard ?r'"
blanchet@55087
  1609
  using assms by(simp add: infinite_cardSuc_regularCard)
blanchet@48975
  1610
  ultimately show ?thesis
blanchet@55087
  1611
  using As Bsub cardB regularCard_UNION by blast
blanchet@48975
  1612
qed
blanchet@48975
  1613
blanchet@48975
  1614
blanchet@48975
  1615
subsection {* Others *}
blanchet@48975
  1616
blanchet@48975
  1617
lemma card_of_Func_Times:
blanchet@48975
  1618
"|Func (A <*> B) C| =o |Func A (Func B C)|"
blanchet@48975
  1619
unfolding card_of_ordIso[symmetric]
traytel@52544
  1620
using bij_betw_curr by blast
blanchet@48975
  1621
blanchet@48975
  1622
lemma card_of_Pow_Func:
blanchet@48975
  1623
"|Pow A| =o |Func A (UNIV::bool set)|"
blanchet@48975
  1624
proof-
traytel@52545
  1625
  def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
traytel@52545
  1626
                            else undefined"
blanchet@48975
  1627
  have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
blanchet@48975
  1628
  unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
traytel@52545
  1629
    fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
traytel@52545
  1630
    thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
blanchet@48975
  1631
  next
blanchet@48975
  1632
    show "F ` Pow A = Func A UNIV"
blanchet@48975
  1633
    proof safe
blanchet@48975
  1634
      fix f assume f: "f \<in> Func A (UNIV::bool set)"
blanchet@48975
  1635
      show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
traytel@52545
  1636
        let ?A1 = "{a \<in> A. f a = True}"
blanchet@48975
  1637
        show "f = F ?A1" unfolding F_def apply(rule ext)
traytel@52545
  1638
        using f unfolding Func_def mem_Collect_eq by auto
blanchet@48975
  1639
      qed auto
blanchet@48975
  1640
    qed(unfold Func_def mem_Collect_eq F_def, auto)
blanchet@48975
  1641
  qed
blanchet@48975
  1642
  thus ?thesis unfolding card_of_ordIso[symmetric] by blast
blanchet@48975
  1643
qed
blanchet@48975
  1644
blanchet@48975
  1645
lemma card_of_Func_UNIV:
blanchet@48975
  1646
"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
blanchet@48975
  1647
apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
traytel@52545
  1648
  let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
blanchet@48975
  1649
  show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
blanchet@48975
  1650
  unfolding bij_betw_def inj_on_def proof safe
traytel@52545
  1651
    fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
traytel@52545
  1652
    hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
traytel@55811
  1653
    then obtain f where f: "\<forall> a. h a = f a" by blast
blanchet@48975
  1654
    hence "range f \<subseteq> B" using h unfolding Func_def by auto
traytel@52545
  1655
    thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
blanchet@48975
  1656
  qed(unfold Func_def fun_eq_iff, auto)
blanchet@48975
  1657
qed
blanchet@48975
  1658
blanchet@48975
  1659
end