src/HOL/Cardinals/Cardinal_Order_Relation_FP.thy
author traytel
Mon Nov 25 13:48:00 2013 +0100 (2013-11-25)
changeset 54581 1502a1f707d9
parent 54578 9387251b6a46
child 54794 e279c2ceb54c
permissions -rw-r--r--
eliminated dependence of Cardinals_FP on Set_Intervals, more precise imports
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(*  Title:      HOL/Cardinals/Cardinal_Order_Relation_FP.thy
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   2012
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Cardinal-order relations (FP).
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*)
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header {* Cardinal-Order Relations (FP) *}
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theory Cardinal_Order_Relation_FP
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imports Constructions_on_Wellorders_FP
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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 rel.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 rel.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 rel.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 rel.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: rel.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 rel.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
blanchet@48975
   313
  also have "... = (r \<le>o |Field r| \<and> |Field r| \<le>o r)"
blanchet@48975
   314
  unfolding ordIso_iff_ordLeq by simp
blanchet@48975
   315
  also have "... = (r \<le>o |Field r| )"
blanchet@48975
   316
  using card_of_Field_ordLess
blanchet@48975
   317
  by (auto simp: card_of_Field_ordLess ordLeq_Well_order_simp)
blanchet@48975
   318
  finally show ?thesis .
blanchet@48975
   319
qed
blanchet@48975
   320
blanchet@48975
   321
blanchet@48975
   322
lemma Card_order_iff_Restr_underS:
blanchet@48975
   323
assumes "Well_order r"
blanchet@48975
   324
shows "Card_order r = (\<forall>a \<in> Field r. Restr r (rel.underS r a) <o |Field r| )"
blanchet@48975
   325
using assms unfolding Card_order_iff_ordLeq_card_of
blanchet@48975
   326
using ordLeq_iff_ordLess_Restr card_of_Well_order by blast
blanchet@48975
   327
blanchet@48975
   328
blanchet@48975
   329
lemma card_of_underS:
blanchet@48975
   330
assumes r: "Card_order r" and a: "a : Field r"
blanchet@48975
   331
shows "|rel.underS r a| <o r"
blanchet@48975
   332
proof-
blanchet@48975
   333
  let ?A = "rel.underS r a"  let ?r' = "Restr r ?A"
blanchet@48975
   334
  have 1: "Well_order r"
blanchet@48975
   335
  using r unfolding card_order_on_def by simp
blanchet@48975
   336
  have "Well_order ?r'" using 1 Well_order_Restr by auto
blanchet@48975
   337
  moreover have "card_order_on (Field ?r') |Field ?r'|"
blanchet@48975
   338
  using card_of_card_order_on .
blanchet@48975
   339
  ultimately have "|Field ?r'| \<le>o ?r'"
blanchet@48975
   340
  unfolding card_order_on_def by simp
blanchet@48975
   341
  moreover have "Field ?r' = ?A"
blanchet@48975
   342
  using 1 wo_rel.underS_ofilter Field_Restr_ofilter
blanchet@48975
   343
  unfolding wo_rel_def by fastforce
blanchet@48975
   344
  ultimately have "|?A| \<le>o ?r'" by simp
blanchet@48975
   345
  also have "?r' <o |Field r|"
blanchet@48975
   346
  using 1 a r Card_order_iff_Restr_underS by blast
blanchet@48975
   347
  also have "|Field r| =o r"
blanchet@48975
   348
  using r ordIso_symmetric unfolding Card_order_iff_ordIso_card_of by auto
blanchet@48975
   349
  finally show ?thesis .
blanchet@48975
   350
qed
blanchet@48975
   351
blanchet@48975
   352
blanchet@48975
   353
lemma ordLess_Field:
blanchet@48975
   354
assumes "r <o r'"
blanchet@48975
   355
shows "|Field r| <o r'"
blanchet@48975
   356
proof-
blanchet@48975
   357
  have "well_order_on (Field r) r" using assms unfolding ordLess_def
blanchet@48975
   358
  by (auto simp add: rel.well_order_on_Well_order)
blanchet@48975
   359
  hence "|Field r| \<le>o r" using card_of_least by blast
blanchet@48975
   360
  thus ?thesis using assms ordLeq_ordLess_trans by blast
blanchet@48975
   361
qed
blanchet@48975
   362
blanchet@48975
   363
blanchet@48975
   364
lemma internalize_card_of_ordLeq:
blanchet@48975
   365
"( |A| \<le>o r) = (\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r)"
blanchet@48975
   366
proof
blanchet@48975
   367
  assume "|A| \<le>o r"
blanchet@48975
   368
  then obtain p where 1: "Field p \<le> Field r \<and> |A| =o p \<and> p \<le>o r"
blanchet@48975
   369
  using internalize_ordLeq[of "|A|" r] by blast
blanchet@48975
   370
  hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast
blanchet@48975
   371
  hence "|Field p| =o p" using card_of_Field_ordIso by blast
blanchet@48975
   372
  hence "|A| =o |Field p| \<and> |Field p| \<le>o r"
blanchet@48975
   373
  using 1 ordIso_equivalence ordIso_ordLeq_trans by blast
blanchet@48975
   374
  thus "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r" using 1 by blast
blanchet@48975
   375
next
blanchet@48975
   376
  assume "\<exists>B \<le> Field r. |A| =o |B| \<and> |B| \<le>o r"
blanchet@48975
   377
  thus "|A| \<le>o r" using ordIso_ordLeq_trans by blast
blanchet@48975
   378
qed
blanchet@48975
   379
blanchet@48975
   380
blanchet@48975
   381
lemma internalize_card_of_ordLeq2:
blanchet@48975
   382
"( |A| \<le>o |C| ) = (\<exists>B \<le> C. |A| =o |B| \<and> |B| \<le>o |C| )"
blanchet@48975
   383
using internalize_card_of_ordLeq[of "A" "|C|"] Field_card_of[of C] by auto
blanchet@48975
   384
blanchet@48975
   385
blanchet@48975
   386
blanchet@48975
   387
subsection {* Cardinals versus set operations on arbitrary sets *}
blanchet@48975
   388
blanchet@48975
   389
blanchet@48975
   390
text{* Here we embark in a long journey of simple results showing
blanchet@48975
   391
that the standard set-theoretic operations are well-behaved w.r.t. the notion of
blanchet@48975
   392
cardinal -- essentially, this means that they preserve the ``cardinal identity"
blanchet@48975
   393
@{text "=o"} and are monotonic w.r.t. @{text "\<le>o"}.
blanchet@48975
   394
*}
blanchet@48975
   395
blanchet@48975
   396
blanchet@48975
   397
lemma card_of_empty: "|{}| \<le>o |A|"
blanchet@48975
   398
using card_of_ordLeq inj_on_id by blast
blanchet@48975
   399
blanchet@48975
   400
blanchet@48975
   401
lemma card_of_empty1:
blanchet@48975
   402
assumes "Well_order r \<or> Card_order r"
blanchet@48975
   403
shows "|{}| \<le>o r"
blanchet@48975
   404
proof-
blanchet@48975
   405
  have "Well_order r" using assms unfolding card_order_on_def by auto
blanchet@48975
   406
  hence "|Field r| <=o r"
blanchet@48975
   407
  using assms card_of_Field_ordLess by blast
blanchet@48975
   408
  moreover have "|{}| \<le>o |Field r|" by (simp add: card_of_empty)
blanchet@48975
   409
  ultimately show ?thesis using ordLeq_transitive by blast
blanchet@48975
   410
qed
blanchet@48975
   411
blanchet@48975
   412
blanchet@48975
   413
corollary Card_order_empty:
blanchet@48975
   414
"Card_order r \<Longrightarrow> |{}| \<le>o r" by (simp add: card_of_empty1)
blanchet@48975
   415
blanchet@48975
   416
blanchet@48975
   417
lemma card_of_empty2:
blanchet@48975
   418
assumes LEQ: "|A| =o |{}|"
blanchet@48975
   419
shows "A = {}"
blanchet@48975
   420
using assms card_of_ordIso[of A] bij_betw_empty2 by blast
blanchet@48975
   421
blanchet@48975
   422
blanchet@48975
   423
lemma card_of_empty3:
blanchet@48975
   424
assumes LEQ: "|A| \<le>o |{}|"
blanchet@48975
   425
shows "A = {}"
blanchet@48975
   426
using assms
blanchet@48975
   427
by (simp add: ordIso_iff_ordLeq card_of_empty1 card_of_empty2
blanchet@48975
   428
              ordLeq_Well_order_simp)
blanchet@48975
   429
blanchet@48975
   430
blanchet@48975
   431
lemma card_of_empty_ordIso:
blanchet@48975
   432
"|{}::'a set| =o |{}::'b set|"
blanchet@48975
   433
using card_of_ordIso unfolding bij_betw_def inj_on_def by blast
blanchet@48975
   434
blanchet@48975
   435
blanchet@48975
   436
lemma card_of_image:
blanchet@48975
   437
"|f ` A| <=o |A|"
blanchet@48975
   438
proof(cases "A = {}", simp add: card_of_empty)
blanchet@48975
   439
  assume "A ~= {}"
blanchet@48975
   440
  hence "f ` A ~= {}" by auto
blanchet@48975
   441
  thus "|f ` A| \<le>o |A|"
blanchet@48975
   442
  using card_of_ordLeq2[of "f ` A" A] by auto
blanchet@48975
   443
qed
blanchet@48975
   444
blanchet@48975
   445
blanchet@48975
   446
lemma surj_imp_ordLeq:
blanchet@48975
   447
assumes "B <= f ` A"
blanchet@48975
   448
shows "|B| <=o |A|"
blanchet@48975
   449
proof-
blanchet@48975
   450
  have "|B| <=o |f ` A|" using assms card_of_mono1 by auto
blanchet@48975
   451
  thus ?thesis using card_of_image ordLeq_transitive by blast
blanchet@48975
   452
qed
blanchet@48975
   453
blanchet@48975
   454
blanchet@48975
   455
lemma card_of_ordLeqI2:
blanchet@48975
   456
assumes "B \<subseteq> f ` A"
blanchet@48975
   457
shows "|B| \<le>o |A|"
blanchet@48975
   458
using assms by (metis surj_imp_ordLeq)
blanchet@48975
   459
blanchet@48975
   460
blanchet@48975
   461
lemma card_of_singl_ordLeq:
blanchet@48975
   462
assumes "A \<noteq> {}"
blanchet@48975
   463
shows "|{b}| \<le>o |A|"
blanchet@48975
   464
proof-
blanchet@48975
   465
  obtain a where *: "a \<in> A" using assms by auto
blanchet@48975
   466
  let ?h = "\<lambda> b'::'b. if b' = b then a else undefined"
blanchet@48975
   467
  have "inj_on ?h {b} \<and> ?h ` {b} \<le> A"
blanchet@48975
   468
  using * unfolding inj_on_def by auto
blanchet@54482
   469
  thus ?thesis using card_of_ordLeq by fast
blanchet@48975
   470
qed
blanchet@48975
   471
blanchet@48975
   472
blanchet@48975
   473
corollary Card_order_singl_ordLeq:
blanchet@48975
   474
"\<lbrakk>Card_order r; Field r \<noteq> {}\<rbrakk> \<Longrightarrow> |{b}| \<le>o r"
blanchet@48975
   475
using card_of_singl_ordLeq[of "Field r" b]
blanchet@48975
   476
      card_of_Field_ordIso[of r] ordLeq_ordIso_trans by blast
blanchet@48975
   477
blanchet@48975
   478
blanchet@48975
   479
lemma card_of_Pow: "|A| <o |Pow A|"
blanchet@48975
   480
using card_of_ordLess2[of "Pow A" A]  Cantors_paradox[of A]
blanchet@48975
   481
      Pow_not_empty[of A] by auto
blanchet@48975
   482
blanchet@48975
   483
blanchet@48975
   484
corollary Card_order_Pow:
blanchet@48975
   485
"Card_order r \<Longrightarrow> r <o |Pow(Field r)|"
blanchet@48975
   486
using card_of_Pow card_of_Field_ordIso ordIso_ordLess_trans ordIso_symmetric by blast
blanchet@48975
   487
blanchet@48975
   488
blanchet@54481
   489
lemma infinite_Pow:
traytel@54578
   490
assumes "\<not> finite A"
traytel@54578
   491
shows "\<not> finite (Pow A)"
blanchet@54481
   492
proof-
blanchet@54481
   493
  have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq)
blanchet@54481
   494
  thus ?thesis by (metis assms finite_Pow_iff)
blanchet@54481
   495
qed
blanchet@54481
   496
blanchet@54481
   497
blanchet@48975
   498
lemma card_of_Plus1: "|A| \<le>o |A <+> B|"
blanchet@48975
   499
proof-
blanchet@48975
   500
  have "Inl ` A \<le> A <+> B" by auto
blanchet@48975
   501
  thus ?thesis using inj_Inl[of A] card_of_ordLeq by blast
blanchet@48975
   502
qed
blanchet@48975
   503
blanchet@48975
   504
blanchet@48975
   505
corollary Card_order_Plus1:
blanchet@48975
   506
"Card_order r \<Longrightarrow> r \<le>o |(Field r) <+> B|"
blanchet@48975
   507
using card_of_Plus1 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   508
blanchet@48975
   509
blanchet@48975
   510
lemma card_of_Plus2: "|B| \<le>o |A <+> B|"
blanchet@48975
   511
proof-
blanchet@48975
   512
  have "Inr ` B \<le> A <+> B" by auto
blanchet@48975
   513
  thus ?thesis using inj_Inr[of B] card_of_ordLeq by blast
blanchet@48975
   514
qed
blanchet@48975
   515
blanchet@48975
   516
blanchet@48975
   517
corollary Card_order_Plus2:
blanchet@48975
   518
"Card_order r \<Longrightarrow> r \<le>o |A <+> (Field r)|"
blanchet@48975
   519
using card_of_Plus2 card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   520
blanchet@48975
   521
blanchet@48975
   522
lemma card_of_Plus_empty1: "|A| =o |A <+> {}|"
blanchet@48975
   523
proof-
blanchet@48975
   524
  have "bij_betw Inl A (A <+> {})" unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   525
  thus ?thesis using card_of_ordIso by auto
blanchet@48975
   526
qed
blanchet@48975
   527
blanchet@48975
   528
blanchet@48975
   529
lemma card_of_Plus_empty2: "|A| =o |{} <+> A|"
blanchet@48975
   530
proof-
blanchet@48975
   531
  have "bij_betw Inr A ({} <+> A)" unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   532
  thus ?thesis using card_of_ordIso by auto
blanchet@48975
   533
qed
blanchet@48975
   534
blanchet@48975
   535
blanchet@48975
   536
lemma card_of_Plus_commute: "|A <+> B| =o |B <+> A|"
blanchet@48975
   537
proof-
blanchet@48975
   538
  let ?f = "\<lambda>(c::'a + 'b). case c of Inl a \<Rightarrow> Inr a
blanchet@48975
   539
                                   | Inr b \<Rightarrow> Inl b"
blanchet@48975
   540
  have "bij_betw ?f (A <+> B) (B <+> A)"
blanchet@48975
   541
  unfolding bij_betw_def inj_on_def by force
blanchet@48975
   542
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   543
qed
blanchet@48975
   544
blanchet@48975
   545
blanchet@48975
   546
lemma card_of_Plus_assoc:
blanchet@48975
   547
fixes A :: "'a set" and B :: "'b set" and C :: "'c set"
blanchet@48975
   548
shows "|(A <+> B) <+> C| =o |A <+> B <+> C|"
blanchet@48975
   549
proof -
blanchet@48975
   550
  def f \<equiv> "\<lambda>(k::('a + 'b) + 'c).
blanchet@48975
   551
  case k of Inl ab \<Rightarrow> (case ab of Inl a \<Rightarrow> Inl a
blanchet@48975
   552
                                 |Inr b \<Rightarrow> Inr (Inl b))
blanchet@48975
   553
           |Inr c \<Rightarrow> Inr (Inr c)"
blanchet@48975
   554
  have "A <+> B <+> C \<subseteq> f ` ((A <+> B) <+> C)"
blanchet@48975
   555
  proof
blanchet@48975
   556
    fix x assume x: "x \<in> A <+> B <+> C"
blanchet@48975
   557
    show "x \<in> f ` ((A <+> B) <+> C)"
blanchet@48975
   558
    proof(cases x)
blanchet@48975
   559
      case (Inl a)
blanchet@48975
   560
      hence "a \<in> A" "x = f (Inl (Inl a))"
blanchet@48975
   561
      using x unfolding f_def by auto
blanchet@48975
   562
      thus ?thesis by auto
blanchet@48975
   563
    next
blanchet@48975
   564
      case (Inr bc) note 1 = Inr show ?thesis
blanchet@48975
   565
      proof(cases bc)
blanchet@48975
   566
        case (Inl b)
blanchet@48975
   567
        hence "b \<in> B" "x = f (Inl (Inr b))"
blanchet@48975
   568
        using x 1 unfolding f_def by auto
blanchet@48975
   569
        thus ?thesis by auto
blanchet@48975
   570
      next
blanchet@48975
   571
        case (Inr c)
blanchet@48975
   572
        hence "c \<in> C" "x = f (Inr c)"
blanchet@48975
   573
        using x 1 unfolding f_def by auto
blanchet@48975
   574
        thus ?thesis by auto
blanchet@48975
   575
      qed
blanchet@48975
   576
    qed
blanchet@48975
   577
  qed
blanchet@48975
   578
  hence "bij_betw f ((A <+> B) <+> C) (A <+> B <+> C)"
blanchet@54482
   579
  unfolding bij_betw_def inj_on_def f_def by fastforce
blanchet@48975
   580
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   581
qed
blanchet@48975
   582
blanchet@48975
   583
blanchet@48975
   584
lemma card_of_Plus_mono1:
blanchet@48975
   585
assumes "|A| \<le>o |B|"
blanchet@48975
   586
shows "|A <+> C| \<le>o |B <+> C|"
blanchet@48975
   587
proof-
blanchet@48975
   588
  obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   589
  using assms card_of_ordLeq[of A] by fastforce
blanchet@48975
   590
  obtain g where g_def:
blanchet@48975
   591
  "g = (\<lambda>d. case d of Inl a \<Rightarrow> Inl(f a) | Inr (c::'c) \<Rightarrow> Inr c)" by blast
blanchet@48975
   592
  have "inj_on g (A <+> C) \<and> g ` (A <+> C) \<le> (B <+> C)"
blanchet@48975
   593
  proof-
blanchet@48975
   594
    {fix d1 and d2 assume "d1 \<in> A <+> C \<and> d2 \<in> A <+> C" and
blanchet@48975
   595
                          "g d1 = g d2"
blanchet@54482
   596
     hence "d1 = d2" using 1 unfolding inj_on_def g_def by force
blanchet@48975
   597
    }
blanchet@48975
   598
    moreover
blanchet@48975
   599
    {fix d assume "d \<in> A <+> C"
blanchet@48975
   600
     hence "g d \<in> B <+> C"  using 1
blanchet@48975
   601
     by(case_tac d, auto simp add: g_def)
blanchet@48975
   602
    }
blanchet@48975
   603
    ultimately show ?thesis unfolding inj_on_def by auto
blanchet@48975
   604
  qed
blanchet@48975
   605
  thus ?thesis using card_of_ordLeq by metis
blanchet@48975
   606
qed
blanchet@48975
   607
blanchet@48975
   608
blanchet@48975
   609
corollary ordLeq_Plus_mono1:
blanchet@48975
   610
assumes "r \<le>o r'"
blanchet@48975
   611
shows "|(Field r) <+> C| \<le>o |(Field r') <+> C|"
blanchet@48975
   612
using assms card_of_mono2 card_of_Plus_mono1 by blast
blanchet@48975
   613
blanchet@48975
   614
blanchet@48975
   615
lemma card_of_Plus_mono2:
blanchet@48975
   616
assumes "|A| \<le>o |B|"
blanchet@48975
   617
shows "|C <+> A| \<le>o |C <+> B|"
blanchet@48975
   618
using assms card_of_Plus_mono1[of A B C]
blanchet@48975
   619
      card_of_Plus_commute[of C A]  card_of_Plus_commute[of B C]
blanchet@48975
   620
      ordIso_ordLeq_trans[of "|C <+> A|"] ordLeq_ordIso_trans[of "|C <+> A|"]
blanchet@48975
   621
by blast
blanchet@48975
   622
blanchet@48975
   623
blanchet@48975
   624
corollary ordLeq_Plus_mono2:
blanchet@48975
   625
assumes "r \<le>o r'"
blanchet@48975
   626
shows "|A <+> (Field r)| \<le>o |A <+> (Field r')|"
blanchet@48975
   627
using assms card_of_mono2 card_of_Plus_mono2 by blast
blanchet@48975
   628
blanchet@48975
   629
blanchet@48975
   630
lemma card_of_Plus_mono:
blanchet@48975
   631
assumes "|A| \<le>o |B|" and "|C| \<le>o |D|"
blanchet@48975
   632
shows "|A <+> C| \<le>o |B <+> D|"
blanchet@48975
   633
using assms card_of_Plus_mono1[of A B C] card_of_Plus_mono2[of C D B]
blanchet@48975
   634
      ordLeq_transitive[of "|A <+> C|"] by blast
blanchet@48975
   635
blanchet@48975
   636
blanchet@48975
   637
corollary ordLeq_Plus_mono:
blanchet@48975
   638
assumes "r \<le>o r'" and "p \<le>o p'"
blanchet@48975
   639
shows "|(Field r) <+> (Field p)| \<le>o |(Field r') <+> (Field p')|"
blanchet@48975
   640
using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Plus_mono by blast
blanchet@48975
   641
blanchet@48975
   642
blanchet@48975
   643
lemma card_of_Plus_cong1:
blanchet@48975
   644
assumes "|A| =o |B|"
blanchet@48975
   645
shows "|A <+> C| =o |B <+> C|"
blanchet@48975
   646
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono1)
blanchet@48975
   647
blanchet@48975
   648
blanchet@48975
   649
corollary ordIso_Plus_cong1:
blanchet@48975
   650
assumes "r =o r'"
blanchet@48975
   651
shows "|(Field r) <+> C| =o |(Field r') <+> C|"
blanchet@48975
   652
using assms card_of_cong card_of_Plus_cong1 by blast
blanchet@48975
   653
blanchet@48975
   654
blanchet@48975
   655
lemma card_of_Plus_cong2:
blanchet@48975
   656
assumes "|A| =o |B|"
blanchet@48975
   657
shows "|C <+> A| =o |C <+> B|"
blanchet@48975
   658
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono2)
blanchet@48975
   659
blanchet@48975
   660
blanchet@48975
   661
corollary ordIso_Plus_cong2:
blanchet@48975
   662
assumes "r =o r'"
blanchet@48975
   663
shows "|A <+> (Field r)| =o |A <+> (Field r')|"
blanchet@48975
   664
using assms card_of_cong card_of_Plus_cong2 by blast
blanchet@48975
   665
blanchet@48975
   666
blanchet@48975
   667
lemma card_of_Plus_cong:
blanchet@48975
   668
assumes "|A| =o |B|" and "|C| =o |D|"
blanchet@48975
   669
shows "|A <+> C| =o |B <+> D|"
blanchet@48975
   670
using assms by (simp add: ordIso_iff_ordLeq card_of_Plus_mono)
blanchet@48975
   671
blanchet@48975
   672
blanchet@48975
   673
corollary ordIso_Plus_cong:
blanchet@48975
   674
assumes "r =o r'" and "p =o p'"
blanchet@48975
   675
shows "|(Field r) <+> (Field p)| =o |(Field r') <+> (Field p')|"
blanchet@48975
   676
using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Plus_cong by blast
blanchet@48975
   677
blanchet@48975
   678
blanchet@48975
   679
lemma card_of_Un_Plus_ordLeq:
blanchet@48975
   680
"|A \<union> B| \<le>o |A <+> B|"
blanchet@48975
   681
proof-
blanchet@48975
   682
   let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c"
blanchet@48975
   683
   have "inj_on ?f (A \<union> B) \<and> ?f ` (A \<union> B) \<le> A <+> B"
blanchet@48975
   684
   unfolding inj_on_def by auto
blanchet@48975
   685
   thus ?thesis using card_of_ordLeq by blast
blanchet@48975
   686
qed
blanchet@48975
   687
blanchet@48975
   688
blanchet@48975
   689
lemma card_of_Times1:
blanchet@48975
   690
assumes "A \<noteq> {}"
blanchet@48975
   691
shows "|B| \<le>o |B \<times> A|"
blanchet@48975
   692
proof(cases "B = {}", simp add: card_of_empty)
blanchet@48975
   693
  assume *: "B \<noteq> {}"
blanchet@48975
   694
  have "fst `(B \<times> A) = B" unfolding image_def using assms by auto
blanchet@48975
   695
  thus ?thesis using inj_on_iff_surj[of B "B \<times> A"]
blanchet@48975
   696
                     card_of_ordLeq[of B "B \<times> A"] * by blast
blanchet@48975
   697
qed
blanchet@48975
   698
blanchet@48975
   699
blanchet@48975
   700
lemma card_of_Times_commute: "|A \<times> B| =o |B \<times> A|"
blanchet@48975
   701
proof-
blanchet@48975
   702
  let ?f = "\<lambda>(a::'a,b::'b). (b,a)"
blanchet@48975
   703
  have "bij_betw ?f (A \<times> B) (B \<times> A)"
blanchet@48975
   704
  unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   705
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   706
qed
blanchet@48975
   707
blanchet@48975
   708
blanchet@48975
   709
lemma card_of_Times2:
blanchet@48975
   710
assumes "A \<noteq> {}"   shows "|B| \<le>o |A \<times> B|"
blanchet@48975
   711
using assms card_of_Times1[of A B] card_of_Times_commute[of B A]
blanchet@48975
   712
      ordLeq_ordIso_trans by blast
blanchet@48975
   713
blanchet@48975
   714
blanchet@54475
   715
corollary Card_order_Times1:
blanchet@54475
   716
"\<lbrakk>Card_order r; B \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |(Field r) \<times> B|"
blanchet@54475
   717
using card_of_Times1[of B] card_of_Field_ordIso
blanchet@54475
   718
      ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@54475
   719
blanchet@54475
   720
blanchet@48975
   721
corollary Card_order_Times2:
blanchet@48975
   722
"\<lbrakk>Card_order r; A \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o |A \<times> (Field r)|"
blanchet@48975
   723
using card_of_Times2[of A] card_of_Field_ordIso
blanchet@48975
   724
      ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
   725
blanchet@48975
   726
blanchet@48975
   727
lemma card_of_Times3: "|A| \<le>o |A \<times> A|"
blanchet@48975
   728
using card_of_Times1[of A]
blanchet@48975
   729
by(cases "A = {}", simp add: card_of_empty, blast)
blanchet@48975
   730
blanchet@48975
   731
blanchet@48975
   732
lemma card_of_Plus_Times_bool: "|A <+> A| =o |A \<times> (UNIV::bool set)|"
blanchet@48975
   733
proof-
blanchet@48975
   734
  let ?f = "\<lambda>c::'a + 'a. case c of Inl a \<Rightarrow> (a,True)
blanchet@48975
   735
                                  |Inr a \<Rightarrow> (a,False)"
blanchet@48975
   736
  have "bij_betw ?f (A <+> A) (A \<times> (UNIV::bool set))"
blanchet@48975
   737
  proof-
blanchet@48975
   738
    {fix  c1 and c2 assume "?f c1 = ?f c2"
blanchet@48975
   739
     hence "c1 = c2"
blanchet@48975
   740
     by(case_tac "c1", case_tac "c2", auto, case_tac "c2", auto)
blanchet@48975
   741
    }
blanchet@48975
   742
    moreover
blanchet@48975
   743
    {fix c assume "c \<in> A <+> A"
blanchet@48975
   744
     hence "?f c \<in> A \<times> (UNIV::bool set)"
blanchet@48975
   745
     by(case_tac c, auto)
blanchet@48975
   746
    }
blanchet@48975
   747
    moreover
blanchet@48975
   748
    {fix a bl assume *: "(a,bl) \<in> A \<times> (UNIV::bool set)"
blanchet@48975
   749
     have "(a,bl) \<in> ?f ` ( A <+> A)"
blanchet@48975
   750
     proof(cases bl)
blanchet@48975
   751
       assume bl hence "?f(Inl a) = (a,bl)" by auto
blanchet@48975
   752
       thus ?thesis using * by force
blanchet@48975
   753
     next
blanchet@48975
   754
       assume "\<not> bl" hence "?f(Inr a) = (a,bl)" by auto
blanchet@48975
   755
       thus ?thesis using * by force
blanchet@48975
   756
     qed
blanchet@48975
   757
    }
blanchet@48975
   758
    ultimately show ?thesis unfolding bij_betw_def inj_on_def by auto
blanchet@48975
   759
  qed
blanchet@48975
   760
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   761
qed
blanchet@48975
   762
blanchet@48975
   763
blanchet@48975
   764
lemma card_of_Times_mono1:
blanchet@48975
   765
assumes "|A| \<le>o |B|"
blanchet@48975
   766
shows "|A \<times> C| \<le>o |B \<times> C|"
blanchet@48975
   767
proof-
blanchet@48975
   768
  obtain f where 1: "inj_on f A \<and> f ` A \<le> B"
blanchet@48975
   769
  using assms card_of_ordLeq[of A] by fastforce
blanchet@48975
   770
  obtain g where g_def:
blanchet@48975
   771
  "g = (\<lambda>(a,c::'c). (f a,c))" by blast
blanchet@48975
   772
  have "inj_on g (A \<times> C) \<and> g ` (A \<times> C) \<le> (B \<times> C)"
blanchet@48975
   773
  using 1 unfolding inj_on_def using g_def by auto
blanchet@48975
   774
  thus ?thesis using card_of_ordLeq by metis
blanchet@48975
   775
qed
blanchet@48975
   776
blanchet@48975
   777
blanchet@48975
   778
corollary ordLeq_Times_mono1:
blanchet@48975
   779
assumes "r \<le>o r'"
blanchet@48975
   780
shows "|(Field r) \<times> C| \<le>o |(Field r') \<times> C|"
blanchet@48975
   781
using assms card_of_mono2 card_of_Times_mono1 by blast
blanchet@48975
   782
blanchet@48975
   783
blanchet@48975
   784
lemma card_of_Times_mono2:
blanchet@48975
   785
assumes "|A| \<le>o |B|"
blanchet@48975
   786
shows "|C \<times> A| \<le>o |C \<times> B|"
blanchet@48975
   787
using assms card_of_Times_mono1[of A B C]
blanchet@48975
   788
      card_of_Times_commute[of C A]  card_of_Times_commute[of B C]
blanchet@48975
   789
      ordIso_ordLeq_trans[of "|C \<times> A|"] ordLeq_ordIso_trans[of "|C \<times> A|"]
blanchet@48975
   790
by blast
blanchet@48975
   791
blanchet@48975
   792
blanchet@48975
   793
corollary ordLeq_Times_mono2:
blanchet@48975
   794
assumes "r \<le>o r'"
blanchet@48975
   795
shows "|A \<times> (Field r)| \<le>o |A \<times> (Field r')|"
blanchet@48975
   796
using assms card_of_mono2 card_of_Times_mono2 by blast
blanchet@48975
   797
blanchet@48975
   798
blanchet@48975
   799
lemma card_of_Sigma_mono1:
blanchet@48975
   800
assumes "\<forall>i \<in> I. |A i| \<le>o |B i|"
blanchet@48975
   801
shows "|SIGMA i : I. A i| \<le>o |SIGMA i : I. B i|"
blanchet@48975
   802
proof-
blanchet@48975
   803
  have "\<forall>i. i \<in> I \<longrightarrow> (\<exists>f. inj_on f (A i) \<and> f ` (A i) \<le> B i)"
blanchet@48975
   804
  using assms by (auto simp add: card_of_ordLeq)
blanchet@48975
   805
  with choice[of "\<lambda> i f. i \<in> I \<longrightarrow> inj_on f (A i) \<and> f ` (A i) \<le> B i"]
traytel@51764
   806
  obtain F where 1: "\<forall>i \<in> I. inj_on (F i) (A i) \<and> (F i) ` (A i) \<le> B i" by metis
blanchet@48975
   807
  obtain g where g_def: "g = (\<lambda>(i,a::'b). (i,F i a))" by blast
blanchet@48975
   808
  have "inj_on g (Sigma I A) \<and> g ` (Sigma I A) \<le> (Sigma I B)"
blanchet@48975
   809
  using 1 unfolding inj_on_def using g_def by force
blanchet@48975
   810
  thus ?thesis using card_of_ordLeq by metis
blanchet@48975
   811
qed
blanchet@48975
   812
blanchet@48975
   813
blanchet@48975
   814
corollary card_of_Sigma_Times:
blanchet@48975
   815
"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> |SIGMA i : I. A i| \<le>o |I \<times> B|"
blanchet@48975
   816
using card_of_Sigma_mono1[of I A "\<lambda>i. B"] .
blanchet@48975
   817
blanchet@48975
   818
blanchet@48975
   819
lemma card_of_UNION_Sigma:
blanchet@48975
   820
"|\<Union>i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
blanchet@48975
   821
using Ex_inj_on_UNION_Sigma[of I A] card_of_ordLeq by metis
blanchet@48975
   822
blanchet@48975
   823
blanchet@48975
   824
lemma card_of_bool:
blanchet@48975
   825
assumes "a1 \<noteq> a2"
blanchet@48975
   826
shows "|UNIV::bool set| =o |{a1,a2}|"
blanchet@48975
   827
proof-
blanchet@48975
   828
  let ?f = "\<lambda> bl. case bl of True \<Rightarrow> a1 | False \<Rightarrow> a2"
blanchet@48975
   829
  have "bij_betw ?f UNIV {a1,a2}"
blanchet@48975
   830
  proof-
blanchet@48975
   831
    {fix bl1 and bl2 assume "?f  bl1 = ?f bl2"
blanchet@48975
   832
     hence "bl1 = bl2" using assms by (case_tac bl1, case_tac bl2, auto)
blanchet@48975
   833
    }
blanchet@48975
   834
    moreover
blanchet@48975
   835
    {fix bl have "?f bl \<in> {a1,a2}" by (case_tac bl, auto)
blanchet@48975
   836
    }
blanchet@48975
   837
    moreover
blanchet@48975
   838
    {fix a assume *: "a \<in> {a1,a2}"
blanchet@48975
   839
     have "a \<in> ?f ` UNIV"
blanchet@48975
   840
     proof(cases "a = a1")
blanchet@48975
   841
       assume "a = a1"
blanchet@48975
   842
       hence "?f True = a" by auto  thus ?thesis by blast
blanchet@48975
   843
     next
blanchet@48975
   844
       assume "a \<noteq> a1" hence "a = a2" using * by auto
blanchet@48975
   845
       hence "?f False = a" by auto  thus ?thesis by blast
blanchet@48975
   846
     qed
blanchet@48975
   847
    }
blanchet@48975
   848
    ultimately show ?thesis unfolding bij_betw_def inj_on_def
blanchet@48975
   849
    by (metis image_subsetI order_eq_iff subsetI)
blanchet@48975
   850
  qed
blanchet@48975
   851
  thus ?thesis using card_of_ordIso by blast
blanchet@48975
   852
qed
blanchet@48975
   853
blanchet@48975
   854
blanchet@48975
   855
lemma card_of_Plus_Times_aux:
blanchet@48975
   856
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
blanchet@48975
   857
        LEQ: "|A| \<le>o |B|"
blanchet@48975
   858
shows "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   859
proof-
blanchet@48975
   860
  have 1: "|UNIV::bool set| \<le>o |A|"
blanchet@48975
   861
  using A2 card_of_mono1[of "{a1,a2}"] card_of_bool[of a1 a2]
blanchet@48975
   862
        ordIso_ordLeq_trans[of "|UNIV::bool set|"] by metis
blanchet@48975
   863
  (*  *)
blanchet@48975
   864
  have "|A <+> B| \<le>o |B <+> B|"
blanchet@48975
   865
  using LEQ card_of_Plus_mono1 by blast
blanchet@48975
   866
  moreover have "|B <+> B| =o |B \<times> (UNIV::bool set)|"
blanchet@48975
   867
  using card_of_Plus_Times_bool by blast
blanchet@48975
   868
  moreover have "|B \<times> (UNIV::bool set)| \<le>o |B \<times> A|"
blanchet@48975
   869
  using 1 by (simp add: card_of_Times_mono2)
blanchet@48975
   870
  moreover have " |B \<times> A| =o |A \<times> B|"
blanchet@48975
   871
  using card_of_Times_commute by blast
blanchet@48975
   872
  ultimately show "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   873
  using ordLeq_ordIso_trans[of "|A <+> B|" "|B <+> B|" "|B \<times> (UNIV::bool set)|"]
blanchet@48975
   874
        ordLeq_transitive[of "|A <+> B|" "|B \<times> (UNIV::bool set)|" "|B \<times> A|"]
blanchet@48975
   875
        ordLeq_ordIso_trans[of "|A <+> B|" "|B \<times> A|" "|A \<times> B|"]
blanchet@48975
   876
  by blast
blanchet@48975
   877
qed
blanchet@48975
   878
blanchet@48975
   879
blanchet@48975
   880
lemma card_of_Plus_Times:
blanchet@48975
   881
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and
blanchet@48975
   882
        B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B"
blanchet@48975
   883
shows "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
   884
proof-
blanchet@48975
   885
  {assume "|A| \<le>o |B|"
blanchet@48975
   886
   hence ?thesis using assms by (auto simp add: card_of_Plus_Times_aux)
blanchet@48975
   887
  }
blanchet@48975
   888
  moreover
blanchet@48975
   889
  {assume "|B| \<le>o |A|"
blanchet@48975
   890
   hence "|B <+> A| \<le>o |B \<times> A|"
blanchet@48975
   891
   using assms by (auto simp add: card_of_Plus_Times_aux)
blanchet@48975
   892
   hence ?thesis
blanchet@48975
   893
   using card_of_Plus_commute card_of_Times_commute
blanchet@48975
   894
         ordIso_ordLeq_trans ordLeq_ordIso_trans by metis
blanchet@48975
   895
  }
blanchet@48975
   896
  ultimately show ?thesis
blanchet@48975
   897
  using card_of_Well_order[of A] card_of_Well_order[of B]
blanchet@48975
   898
        ordLeq_total[of "|A|"] by metis
blanchet@48975
   899
qed
blanchet@48975
   900
blanchet@48975
   901
blanchet@48975
   902
lemma card_of_ordLeq_finite:
blanchet@48975
   903
assumes "|A| \<le>o |B|" and "finite B"
blanchet@48975
   904
shows "finite A"
blanchet@48975
   905
using assms unfolding ordLeq_def
blanchet@48975
   906
using embed_inj_on[of "|A|" "|B|"]  embed_Field[of "|A|" "|B|"]
blanchet@48975
   907
      Field_card_of[of "A"] Field_card_of[of "B"] inj_on_finite[of _ "A" "B"] by fastforce
blanchet@48975
   908
blanchet@48975
   909
blanchet@48975
   910
lemma card_of_ordLeq_infinite:
traytel@54578
   911
assumes "|A| \<le>o |B|" and "\<not> finite A"
traytel@54578
   912
shows "\<not> finite B"
blanchet@48975
   913
using assms card_of_ordLeq_finite by auto
blanchet@48975
   914
blanchet@48975
   915
blanchet@48975
   916
lemma card_of_ordIso_finite:
blanchet@48975
   917
assumes "|A| =o |B|"
blanchet@48975
   918
shows "finite A = finite B"
blanchet@48975
   919
using assms unfolding ordIso_def iso_def[abs_def]
blanchet@48975
   920
by (auto simp: bij_betw_finite Field_card_of)
blanchet@48975
   921
blanchet@48975
   922
blanchet@48975
   923
lemma card_of_ordIso_finite_Field:
blanchet@48975
   924
assumes "Card_order r" and "r =o |A|"
blanchet@48975
   925
shows "finite(Field r) = finite A"
blanchet@48975
   926
using assms card_of_Field_ordIso card_of_ordIso_finite ordIso_equivalence by blast
blanchet@48975
   927
blanchet@48975
   928
blanchet@48975
   929
subsection {* Cardinals versus set operations involving infinite sets *}
blanchet@48975
   930
blanchet@48975
   931
blanchet@48975
   932
text{* Here we show that, for infinite sets, most set-theoretic constructions
blanchet@48975
   933
do not increase the cardinality.  The cornerstone for this is
blanchet@48975
   934
theorem @{text "Card_order_Times_same_infinite"}, which states that self-product
blanchet@48975
   935
does not increase cardinality -- the proof of this fact adapts a standard
blanchet@48975
   936
set-theoretic argument, as presented, e.g., in the proof of theorem 1.5.11
blanchet@48975
   937
at page 47 in \cite{card-book}. Then everything else follows fairly easily.  *}
blanchet@48975
   938
blanchet@48975
   939
blanchet@48975
   940
lemma infinite_iff_card_of_nat:
traytel@54578
   941
"\<not> finite A \<longleftrightarrow> ( |UNIV::nat set| \<le>o |A| )"
traytel@54578
   942
unfolding infinite_iff_countable_subset card_of_ordLeq ..
blanchet@48975
   943
blanchet@48975
   944
text{* The next two results correspond to the ZF fact that all infinite cardinals are
blanchet@48975
   945
limit ordinals: *}
blanchet@48975
   946
blanchet@48975
   947
lemma Card_order_infinite_not_under:
traytel@54578
   948
assumes CARD: "Card_order r" and INF: "\<not>finite (Field r)"
blanchet@48975
   949
shows "\<not> (\<exists>a. Field r = rel.under r a)"
blanchet@48975
   950
proof(auto)
blanchet@48975
   951
  have 0: "Well_order r \<and> wo_rel r \<and> Refl r"
blanchet@48975
   952
  using CARD unfolding wo_rel_def card_order_on_def order_on_defs by auto
blanchet@48975
   953
  fix a assume *: "Field r = rel.under r a"
blanchet@48975
   954
  show False
blanchet@48975
   955
  proof(cases "a \<in> Field r")
blanchet@48975
   956
    assume Case1: "a \<notin> Field r"
blanchet@48975
   957
    hence "rel.under r a = {}" unfolding Field_def rel.under_def by auto
blanchet@48975
   958
    thus False using INF *  by auto
blanchet@48975
   959
  next
blanchet@48975
   960
    let ?r' = "Restr r (rel.underS r a)"
blanchet@48975
   961
    assume Case2: "a \<in> Field r"
blanchet@48975
   962
    hence 1: "rel.under r a = rel.underS r a \<union> {a} \<and> a \<notin> rel.underS r a"
blanchet@54482
   963
    using 0 rel.Refl_under_underS rel.underS_notIn by metis
blanchet@48975
   964
    have 2: "wo_rel.ofilter r (rel.underS r a) \<and> rel.underS r a < Field r"
blanchet@54482
   965
    using 0 wo_rel.underS_ofilter * 1 Case2 by fast
blanchet@48975
   966
    hence "?r' <o r" using 0 using ofilter_ordLess by blast
blanchet@48975
   967
    moreover
blanchet@48975
   968
    have "Field ?r' = rel.underS r a \<and> Well_order ?r'"
blanchet@48975
   969
    using  2 0 Field_Restr_ofilter[of r] Well_order_Restr[of r] by blast
blanchet@48975
   970
    ultimately have "|rel.underS r a| <o r" using ordLess_Field[of ?r'] by auto
blanchet@48975
   971
    moreover have "|rel.under r a| =o r" using * CARD card_of_Field_ordIso[of r] by auto
blanchet@48975
   972
    ultimately have "|rel.underS r a| <o |rel.under r a|"
blanchet@48975
   973
    using ordIso_symmetric ordLess_ordIso_trans by blast
blanchet@48975
   974
    moreover
blanchet@48975
   975
    {have "\<exists>f. bij_betw f (rel.under r a) (rel.underS r a)"
blanchet@48975
   976
     using infinite_imp_bij_betw[of "Field r" a] INF * 1 by auto
blanchet@48975
   977
     hence "|rel.under r a| =o |rel.underS r a|" using card_of_ordIso by blast
blanchet@48975
   978
    }
blanchet@48975
   979
    ultimately show False using not_ordLess_ordIso ordIso_symmetric by blast
blanchet@48975
   980
  qed
blanchet@48975
   981
qed
blanchet@48975
   982
blanchet@48975
   983
blanchet@48975
   984
lemma infinite_Card_order_limit:
traytel@54578
   985
assumes r: "Card_order r" and "\<not>finite (Field r)"
blanchet@48975
   986
and a: "a : Field r"
blanchet@48975
   987
shows "EX b : Field r. a \<noteq> b \<and> (a,b) : r"
blanchet@48975
   988
proof-
blanchet@48975
   989
  have "Field r \<noteq> rel.under r a"
blanchet@48975
   990
  using assms Card_order_infinite_not_under by blast
blanchet@48975
   991
  moreover have "rel.under r a \<le> Field r"
blanchet@48975
   992
  using rel.under_Field .
blanchet@48975
   993
  ultimately have "rel.under r a < Field r" by blast
blanchet@48975
   994
  then obtain b where 1: "b : Field r \<and> ~ (b,a) : r"
blanchet@48975
   995
  unfolding rel.under_def by blast
blanchet@48975
   996
  moreover have ba: "b \<noteq> a"
blanchet@48975
   997
  using 1 r unfolding card_order_on_def well_order_on_def
blanchet@48975
   998
  linear_order_on_def partial_order_on_def preorder_on_def refl_on_def by auto
blanchet@48975
   999
  ultimately have "(a,b) : r"
blanchet@48975
  1000
  using a r unfolding card_order_on_def well_order_on_def linear_order_on_def
blanchet@48975
  1001
  total_on_def by blast
blanchet@48975
  1002
  thus ?thesis using 1 ba by auto
blanchet@48975
  1003
qed
blanchet@48975
  1004
blanchet@48975
  1005
blanchet@48975
  1006
theorem Card_order_Times_same_infinite:
traytel@54578
  1007
assumes CO: "Card_order r" and INF: "\<not>finite(Field r)"
blanchet@48975
  1008
shows "|Field r \<times> Field r| \<le>o r"
blanchet@48975
  1009
proof-
blanchet@48975
  1010
  obtain phi where phi_def:
traytel@54578
  1011
  "phi = (\<lambda>r::'a rel. Card_order r \<and> \<not>finite(Field r) \<and>
blanchet@48975
  1012
                      \<not> |Field r \<times> Field r| \<le>o r )" by blast
blanchet@48975
  1013
  have temp1: "\<forall>r. phi r \<longrightarrow> Well_order r"
blanchet@48975
  1014
  unfolding phi_def card_order_on_def by auto
blanchet@48975
  1015
  have Ft: "\<not>(\<exists>r. phi r)"
blanchet@48975
  1016
  proof
blanchet@48975
  1017
    assume "\<exists>r. phi r"
blanchet@48975
  1018
    hence "{r. phi r} \<noteq> {} \<and> {r. phi r} \<le> {r. Well_order r}"
blanchet@48975
  1019
    using temp1 by auto
blanchet@48975
  1020
    then obtain r where 1: "phi r" and 2: "\<forall>r'. phi r' \<longrightarrow> r \<le>o r'" and
blanchet@48975
  1021
                   3: "Card_order r \<and> Well_order r"
blanchet@48975
  1022
    using exists_minim_Well_order[of "{r. phi r}"] temp1 phi_def by blast
blanchet@48975
  1023
    let ?A = "Field r"  let ?r' = "bsqr r"
blanchet@48975
  1024
    have 4: "Well_order ?r' \<and> Field ?r' = ?A \<times> ?A \<and> |?A| =o r"
blanchet@48975
  1025
    using 3 bsqr_Well_order Field_bsqr card_of_Field_ordIso by blast
blanchet@48975
  1026
    have 5: "Card_order |?A \<times> ?A| \<and> Well_order |?A \<times> ?A|"
blanchet@48975
  1027
    using card_of_Card_order card_of_Well_order by blast
blanchet@48975
  1028
    (*  *)
blanchet@48975
  1029
    have "r <o |?A \<times> ?A|"
blanchet@48975
  1030
    using 1 3 5 ordLess_or_ordLeq unfolding phi_def by blast
blanchet@48975
  1031
    moreover have "|?A \<times> ?A| \<le>o ?r'"
blanchet@48975
  1032
    using card_of_least[of "?A \<times> ?A"] 4 by auto
blanchet@48975
  1033
    ultimately have "r <o ?r'" using ordLess_ordLeq_trans by auto
blanchet@48975
  1034
    then obtain f where 6: "embed r ?r' f" and 7: "\<not> bij_betw f ?A (?A \<times> ?A)"
blanchet@48975
  1035
    unfolding ordLess_def embedS_def[abs_def]
blanchet@48975
  1036
    by (auto simp add: Field_bsqr)
blanchet@48975
  1037
    let ?B = "f ` ?A"
blanchet@48975
  1038
    have "|?A| =o |?B|"
blanchet@48975
  1039
    using 3 6 embed_inj_on inj_on_imp_bij_betw card_of_ordIso by blast
blanchet@48975
  1040
    hence 8: "r =o |?B|" using 4 ordIso_transitive ordIso_symmetric by blast
blanchet@48975
  1041
    (*  *)
blanchet@48975
  1042
    have "wo_rel.ofilter ?r' ?B"
blanchet@48975
  1043
    using 6 embed_Field_ofilter 3 4 by blast
blanchet@48975
  1044
    hence "wo_rel.ofilter ?r' ?B \<and> ?B \<noteq> ?A \<times> ?A \<and> ?B \<noteq> Field ?r'"
blanchet@48975
  1045
    using 7 unfolding bij_betw_def using 6 3 embed_inj_on 4 by auto
blanchet@48975
  1046
    hence temp2: "wo_rel.ofilter ?r' ?B \<and> ?B < ?A \<times> ?A"
blanchet@48975
  1047
    using 4 wo_rel_def[of ?r'] wo_rel.ofilter_def[of ?r' ?B] by blast
blanchet@48975
  1048
    have "\<not> (\<exists>a. Field r = rel.under r a)"
blanchet@48975
  1049
    using 1 unfolding phi_def using Card_order_infinite_not_under[of r] by auto
blanchet@48975
  1050
    then obtain A1 where temp3: "wo_rel.ofilter r A1 \<and> A1 < ?A" and 9: "?B \<le> A1 \<times> A1"
blanchet@48975
  1051
    using temp2 3 bsqr_ofilter[of r ?B] by blast
blanchet@48975
  1052
    hence "|?B| \<le>o |A1 \<times> A1|" using card_of_mono1 by blast
blanchet@48975
  1053
    hence 10: "r \<le>o |A1 \<times> A1|" using 8 ordIso_ordLeq_trans by blast
blanchet@48975
  1054
    let ?r1 = "Restr r A1"
blanchet@48975
  1055
    have "?r1 <o r" using temp3 ofilter_ordLess 3 by blast
blanchet@48975
  1056
    moreover
blanchet@48975
  1057
    {have "well_order_on A1 ?r1" using 3 temp3 well_order_on_Restr by blast
blanchet@48975
  1058
     hence "|A1| \<le>o ?r1" using 3 Well_order_Restr card_of_least by blast
blanchet@48975
  1059
    }
blanchet@48975
  1060
    ultimately have 11: "|A1| <o r" using ordLeq_ordLess_trans by blast
blanchet@48975
  1061
    (*  *)
traytel@54578
  1062
    have "\<not> finite (Field r)" using 1 unfolding phi_def by simp
traytel@54578
  1063
    hence "\<not> finite ?B" using 8 3 card_of_ordIso_finite_Field[of r ?B] by blast
traytel@54578
  1064
    hence "\<not> finite A1" using 9 finite_cartesian_product finite_subset by metis
blanchet@48975
  1065
    moreover have temp4: "Field |A1| = A1 \<and> Well_order |A1| \<and> Card_order |A1|"
blanchet@48975
  1066
    using card_of_Card_order[of A1] card_of_Well_order[of A1]
blanchet@48975
  1067
    by (simp add: Field_card_of)
blanchet@48975
  1068
    moreover have "\<not> r \<le>o | A1 |"
blanchet@48975
  1069
    using temp4 11 3 using not_ordLeq_iff_ordLess by blast
traytel@54578
  1070
    ultimately have "\<not> finite(Field |A1| ) \<and> Card_order |A1| \<and> \<not> r \<le>o | A1 |"
blanchet@48975
  1071
    by (simp add: card_of_card_order_on)
blanchet@48975
  1072
    hence "|Field |A1| \<times> Field |A1| | \<le>o |A1|"
blanchet@48975
  1073
    using 2 unfolding phi_def by blast
blanchet@48975
  1074
    hence "|A1 \<times> A1 | \<le>o |A1|" using temp4 by auto
blanchet@48975
  1075
    hence "r \<le>o |A1|" using 10 ordLeq_transitive by blast
blanchet@48975
  1076
    thus False using 11 not_ordLess_ordLeq by auto
blanchet@48975
  1077
  qed
blanchet@48975
  1078
  thus ?thesis using assms unfolding phi_def by blast
blanchet@48975
  1079
qed
blanchet@48975
  1080
blanchet@48975
  1081
blanchet@48975
  1082
corollary card_of_Times_same_infinite:
traytel@54578
  1083
assumes "\<not>finite A"
blanchet@48975
  1084
shows "|A \<times> A| =o |A|"
blanchet@48975
  1085
proof-
blanchet@48975
  1086
  let ?r = "|A|"
blanchet@48975
  1087
  have "Field ?r = A \<and> Card_order ?r"
blanchet@48975
  1088
  using Field_card_of card_of_Card_order[of A] by fastforce
blanchet@48975
  1089
  hence "|A \<times> A| \<le>o |A|"
blanchet@48975
  1090
  using Card_order_Times_same_infinite[of ?r] assms by auto
blanchet@48975
  1091
  thus ?thesis using card_of_Times3 ordIso_iff_ordLeq by blast
blanchet@48975
  1092
qed
blanchet@48975
  1093
blanchet@48975
  1094
blanchet@48975
  1095
lemma card_of_Times_infinite:
traytel@54578
  1096
assumes INF: "\<not>finite A" and NE: "B \<noteq> {}" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1097
shows "|A \<times> B| =o |A| \<and> |B \<times> A| =o |A|"
blanchet@48975
  1098
proof-
blanchet@48975
  1099
  have "|A| \<le>o |A \<times> B| \<and> |A| \<le>o |B \<times> A|"
blanchet@48975
  1100
  using assms by (simp add: card_of_Times1 card_of_Times2)
blanchet@48975
  1101
  moreover
blanchet@48975
  1102
  {have "|A \<times> B| \<le>o |A \<times> A| \<and> |B \<times> A| \<le>o |A \<times> A|"
blanchet@48975
  1103
   using LEQ card_of_Times_mono1 card_of_Times_mono2 by blast
blanchet@48975
  1104
   moreover have "|A \<times> A| =o |A|" using INF card_of_Times_same_infinite by blast
blanchet@48975
  1105
   ultimately have "|A \<times> B| \<le>o |A| \<and> |B \<times> A| \<le>o |A|"
blanchet@48975
  1106
   using ordLeq_ordIso_trans[of "|A \<times> B|"] ordLeq_ordIso_trans[of "|B \<times> A|"] by auto
blanchet@48975
  1107
  }
blanchet@48975
  1108
  ultimately show ?thesis by (simp add: ordIso_iff_ordLeq)
blanchet@48975
  1109
qed
blanchet@48975
  1110
blanchet@48975
  1111
blanchet@48975
  1112
corollary Card_order_Times_infinite:
traytel@54578
  1113
assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
blanchet@48975
  1114
        NE: "Field p \<noteq> {}" and LEQ: "p \<le>o r"
blanchet@48975
  1115
shows "| (Field r) \<times> (Field p) | =o r \<and> | (Field p) \<times> (Field r) | =o r"
blanchet@48975
  1116
proof-
blanchet@48975
  1117
  have "|Field r \<times> Field p| =o |Field r| \<and> |Field p \<times> Field r| =o |Field r|"
blanchet@48975
  1118
  using assms by (simp add: card_of_Times_infinite card_of_mono2)
blanchet@48975
  1119
  thus ?thesis
blanchet@48975
  1120
  using assms card_of_Field_ordIso[of r]
blanchet@48975
  1121
        ordIso_transitive[of "|Field r \<times> Field p|"]
blanchet@48975
  1122
        ordIso_transitive[of _ "|Field r|"] by blast
blanchet@48975
  1123
qed
blanchet@48975
  1124
blanchet@48975
  1125
blanchet@48975
  1126
lemma card_of_Sigma_ordLeq_infinite:
traytel@54578
  1127
assumes INF: "\<not>finite B" and
blanchet@48975
  1128
        LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
blanchet@48975
  1129
shows "|SIGMA i : I. A i| \<le>o |B|"
blanchet@48975
  1130
proof(cases "I = {}", simp add: card_of_empty)
blanchet@48975
  1131
  assume *: "I \<noteq> {}"
blanchet@48975
  1132
  have "|SIGMA i : I. A i| \<le>o |I \<times> B|"
blanchet@48975
  1133
  using LEQ card_of_Sigma_Times by blast
blanchet@48975
  1134
  moreover have "|I \<times> B| =o |B|"
blanchet@48975
  1135
  using INF * LEQ_I by (auto simp add: card_of_Times_infinite)
blanchet@48975
  1136
  ultimately show ?thesis using ordLeq_ordIso_trans by blast
blanchet@48975
  1137
qed
blanchet@48975
  1138
blanchet@48975
  1139
blanchet@48975
  1140
lemma card_of_Sigma_ordLeq_infinite_Field:
traytel@54578
  1141
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@48975
  1142
        LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@48975
  1143
shows "|SIGMA i : I. A i| \<le>o r"
blanchet@48975
  1144
proof-
blanchet@48975
  1145
  let ?B  = "Field r"
blanchet@48975
  1146
  have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
blanchet@48975
  1147
  ordIso_symmetric by blast
blanchet@48975
  1148
  hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"
blanchet@48975
  1149
  using LEQ_I LEQ ordLeq_ordIso_trans by blast+
blanchet@48975
  1150
  hence  "|SIGMA i : I. A i| \<le>o |?B|" using INF LEQ
blanchet@48975
  1151
  card_of_Sigma_ordLeq_infinite by blast
blanchet@48975
  1152
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
blanchet@48975
  1153
qed
blanchet@48975
  1154
blanchet@48975
  1155
blanchet@48975
  1156
lemma card_of_Times_ordLeq_infinite_Field:
traytel@54578
  1157
"\<lbrakk>\<not>finite (Field r); |A| \<le>o r; |B| \<le>o r; Card_order r\<rbrakk>
blanchet@48975
  1158
 \<Longrightarrow> |A <*> B| \<le>o r"
blanchet@48975
  1159
by(simp add: card_of_Sigma_ordLeq_infinite_Field)
blanchet@48975
  1160
blanchet@48975
  1161
blanchet@54475
  1162
lemma card_of_Times_infinite_simps:
traytel@54578
  1163
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A \<times> B| =o |A|"
traytel@54578
  1164
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |A \<times> B|"
traytel@54578
  1165
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |B \<times> A| =o |A|"
traytel@54578
  1166
"\<lbrakk>\<not>finite A; B \<noteq> {}; |B| \<le>o |A|\<rbrakk> \<Longrightarrow> |A| =o |B \<times> A|"
blanchet@54475
  1167
by (auto simp add: card_of_Times_infinite ordIso_symmetric)
blanchet@54475
  1168
blanchet@54475
  1169
blanchet@48975
  1170
lemma card_of_UNION_ordLeq_infinite:
traytel@54578
  1171
assumes INF: "\<not>finite B" and
blanchet@48975
  1172
        LEQ_I: "|I| \<le>o |B|" and LEQ: "\<forall>i \<in> I. |A i| \<le>o |B|"
blanchet@48975
  1173
shows "|\<Union> i \<in> I. A i| \<le>o |B|"
blanchet@48975
  1174
proof(cases "I = {}", simp add: card_of_empty)
blanchet@48975
  1175
  assume *: "I \<noteq> {}"
blanchet@48975
  1176
  have "|\<Union> i \<in> I. A i| \<le>o |SIGMA i : I. A i|"
blanchet@48975
  1177
  using card_of_UNION_Sigma by blast
blanchet@48975
  1178
  moreover have "|SIGMA i : I. A i| \<le>o |B|"
blanchet@48975
  1179
  using assms card_of_Sigma_ordLeq_infinite by blast
blanchet@48975
  1180
  ultimately show ?thesis using ordLeq_transitive by blast
blanchet@48975
  1181
qed
blanchet@48975
  1182
blanchet@48975
  1183
blanchet@48975
  1184
corollary card_of_UNION_ordLeq_infinite_Field:
traytel@54578
  1185
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@48975
  1186
        LEQ_I: "|I| \<le>o r" and LEQ: "\<forall>i \<in> I. |A i| \<le>o r"
blanchet@48975
  1187
shows "|\<Union> i \<in> I. A i| \<le>o r"
blanchet@48975
  1188
proof-
blanchet@48975
  1189
  let ?B  = "Field r"
blanchet@48975
  1190
  have 1: "r =o |?B| \<and> |?B| =o r" using r card_of_Field_ordIso
blanchet@48975
  1191
  ordIso_symmetric by blast
blanchet@48975
  1192
  hence "|I| \<le>o |?B|"  "\<forall>i \<in> I. |A i| \<le>o |?B|"
blanchet@48975
  1193
  using LEQ_I LEQ ordLeq_ordIso_trans by blast+
blanchet@48975
  1194
  hence  "|\<Union> i \<in> I. A i| \<le>o |?B|" using INF LEQ
blanchet@48975
  1195
  card_of_UNION_ordLeq_infinite by blast
blanchet@48975
  1196
  thus ?thesis using 1 ordLeq_ordIso_trans by blast
blanchet@48975
  1197
qed
blanchet@48975
  1198
blanchet@48975
  1199
blanchet@48975
  1200
lemma card_of_Plus_infinite1:
traytel@54578
  1201
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1202
shows "|A <+> B| =o |A|"
blanchet@48975
  1203
proof(cases "B = {}", simp add: card_of_Plus_empty1 card_of_Plus_empty2 ordIso_symmetric)
blanchet@48975
  1204
  let ?Inl = "Inl::'a \<Rightarrow> 'a + 'b"  let ?Inr = "Inr::'b \<Rightarrow> 'a + 'b"
blanchet@48975
  1205
  assume *: "B \<noteq> {}"
blanchet@48975
  1206
  then obtain b1 where 1: "b1 \<in> B" by blast
blanchet@48975
  1207
  show ?thesis
blanchet@48975
  1208
  proof(cases "B = {b1}")
blanchet@48975
  1209
    assume Case1: "B = {b1}"
blanchet@48975
  1210
    have 2: "bij_betw ?Inl A ((?Inl ` A))"
blanchet@48975
  1211
    unfolding bij_betw_def inj_on_def by auto
traytel@54578
  1212
    hence 3: "\<not>finite (?Inl ` A)"
blanchet@48975
  1213
    using INF bij_betw_finite[of ?Inl A] by blast
blanchet@48975
  1214
    let ?A' = "?Inl ` A \<union> {?Inr b1}"
blanchet@48975
  1215
    obtain g where "bij_betw g (?Inl ` A) ?A'"
blanchet@48975
  1216
    using 3 infinite_imp_bij_betw2[of "?Inl ` A"] by auto
blanchet@48975
  1217
    moreover have "?A' = A <+> B" using Case1 by blast
blanchet@48975
  1218
    ultimately have "bij_betw g (?Inl ` A) (A <+> B)" by simp
blanchet@48975
  1219
    hence "bij_betw (g o ?Inl) A (A <+> B)"
blanchet@48975
  1220
    using 2 by (auto simp add: bij_betw_trans)
blanchet@48975
  1221
    thus ?thesis using card_of_ordIso ordIso_symmetric by blast
blanchet@48975
  1222
  next
blanchet@48975
  1223
    assume Case2: "B \<noteq> {b1}"
blanchet@48975
  1224
    with * 1 obtain b2 where 3: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" by fastforce
blanchet@48975
  1225
    obtain f where "inj_on f B \<and> f ` B \<le> A"
blanchet@48975
  1226
    using LEQ card_of_ordLeq[of B] by fastforce
blanchet@48975
  1227
    with 3 have "f b1 \<noteq> f b2 \<and> {f b1, f b2} \<le> A"
blanchet@48975
  1228
    unfolding inj_on_def by auto
blanchet@48975
  1229
    with 3 have "|A <+> B| \<le>o |A \<times> B|"
blanchet@48975
  1230
    by (auto simp add: card_of_Plus_Times)
blanchet@48975
  1231
    moreover have "|A \<times> B| =o |A|"
blanchet@48975
  1232
    using assms * by (simp add: card_of_Times_infinite_simps)
blanchet@48975
  1233
    ultimately have "|A <+> B| \<le>o |A|" using ordLeq_ordIso_trans by metis
blanchet@48975
  1234
    thus ?thesis using card_of_Plus1 ordIso_iff_ordLeq by blast
blanchet@48975
  1235
  qed
blanchet@48975
  1236
qed
blanchet@48975
  1237
blanchet@48975
  1238
blanchet@48975
  1239
lemma card_of_Plus_infinite2:
traytel@54578
  1240
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1241
shows "|B <+> A| =o |A|"
blanchet@48975
  1242
using assms card_of_Plus_commute card_of_Plus_infinite1
blanchet@48975
  1243
ordIso_equivalence by blast
blanchet@48975
  1244
blanchet@48975
  1245
blanchet@48975
  1246
lemma card_of_Plus_infinite:
traytel@54578
  1247
assumes INF: "\<not>finite A" and LEQ: "|B| \<le>o |A|"
blanchet@48975
  1248
shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
blanchet@48975
  1249
using assms by (auto simp: card_of_Plus_infinite1 card_of_Plus_infinite2)
blanchet@48975
  1250
blanchet@48975
  1251
blanchet@48975
  1252
corollary Card_order_Plus_infinite:
traytel@54578
  1253
assumes INF: "\<not>finite(Field r)" and CARD: "Card_order r" and
blanchet@48975
  1254
        LEQ: "p \<le>o r"
blanchet@48975
  1255
shows "| (Field r) <+> (Field p) | =o r \<and> | (Field p) <+> (Field r) | =o r"
blanchet@48975
  1256
proof-
blanchet@48975
  1257
  have "| Field r <+> Field p | =o | Field r | \<and>
blanchet@48975
  1258
        | Field p <+> Field r | =o | Field r |"
blanchet@48975
  1259
  using assms by (simp add: card_of_Plus_infinite card_of_mono2)
blanchet@48975
  1260
  thus ?thesis
blanchet@48975
  1261
  using assms card_of_Field_ordIso[of r]
blanchet@48975
  1262
        ordIso_transitive[of "|Field r <+> Field p|"]
blanchet@48975
  1263
        ordIso_transitive[of _ "|Field r|"] by blast
blanchet@48975
  1264
qed
blanchet@48975
  1265
blanchet@48975
  1266
blanchet@48975
  1267
subsection {* The cardinal $\omega$ and the finite cardinals  *}
blanchet@48975
  1268
blanchet@48975
  1269
blanchet@48975
  1270
text{* The cardinal $\omega$, of natural numbers, shall be the standard non-strict
blanchet@48975
  1271
order relation on
blanchet@48975
  1272
@{text "nat"}, that we abbreviate by @{text "natLeq"}.  The finite cardinals
blanchet@48975
  1273
shall be the restrictions of these relations to the numbers smaller than
blanchet@48975
  1274
fixed numbers @{text "n"}, that we abbreviate by @{text "natLeq_on n"}.  *}
blanchet@48975
  1275
blanchet@48975
  1276
abbreviation "(natLeq::(nat * nat) set) \<equiv> {(x,y). x \<le> y}"
blanchet@48975
  1277
abbreviation "(natLess::(nat * nat) set) \<equiv> {(x,y). x < y}"
blanchet@48975
  1278
blanchet@48975
  1279
abbreviation natLeq_on :: "nat \<Rightarrow> (nat * nat) set"
blanchet@48975
  1280
where "natLeq_on n \<equiv> {(x,y). x < n \<and> y < n \<and> x \<le> y}"
blanchet@48975
  1281
blanchet@48975
  1282
lemma infinite_cartesian_product:
traytel@54578
  1283
assumes "\<not>finite A" "\<not>finite B"
traytel@54578
  1284
shows "\<not>finite (A \<times> B)"
blanchet@48975
  1285
proof
blanchet@48975
  1286
  assume "finite (A \<times> B)"
blanchet@48975
  1287
  from assms(1) have "A \<noteq> {}" by auto
blanchet@48975
  1288
  with `finite (A \<times> B)` have "finite B" using finite_cartesian_productD2 by auto
blanchet@48975
  1289
  with assms(2) show False by simp
blanchet@48975
  1290
qed
blanchet@48975
  1291
blanchet@48975
  1292
blanchet@48975
  1293
subsubsection {* First as well-orders *}
blanchet@48975
  1294
blanchet@48975
  1295
blanchet@48975
  1296
lemma Field_natLeq: "Field natLeq = (UNIV::nat set)"
blanchet@48975
  1297
by(unfold Field_def, auto)
blanchet@48975
  1298
blanchet@48975
  1299
blanchet@48975
  1300
lemma natLeq_Refl: "Refl natLeq"
blanchet@48975
  1301
unfolding refl_on_def Field_def by auto
blanchet@48975
  1302
blanchet@48975
  1303
blanchet@48975
  1304
lemma natLeq_trans: "trans natLeq"
blanchet@48975
  1305
unfolding trans_def by auto
blanchet@48975
  1306
blanchet@48975
  1307
blanchet@48975
  1308
lemma natLeq_Preorder: "Preorder natLeq"
blanchet@48975
  1309
unfolding preorder_on_def
blanchet@48975
  1310
by (auto simp add: natLeq_Refl natLeq_trans)
blanchet@48975
  1311
blanchet@48975
  1312
blanchet@48975
  1313
lemma natLeq_antisym: "antisym natLeq"
blanchet@48975
  1314
unfolding antisym_def by auto
blanchet@48975
  1315
blanchet@48975
  1316
blanchet@48975
  1317
lemma natLeq_Partial_order: "Partial_order natLeq"
blanchet@48975
  1318
unfolding partial_order_on_def
blanchet@48975
  1319
by (auto simp add: natLeq_Preorder natLeq_antisym)
blanchet@48975
  1320
blanchet@48975
  1321
blanchet@48975
  1322
lemma natLeq_Total: "Total natLeq"
blanchet@48975
  1323
unfolding total_on_def by auto
blanchet@48975
  1324
blanchet@48975
  1325
blanchet@48975
  1326
lemma natLeq_Linear_order: "Linear_order natLeq"
blanchet@48975
  1327
unfolding linear_order_on_def
blanchet@48975
  1328
by (auto simp add: natLeq_Partial_order natLeq_Total)
blanchet@48975
  1329
blanchet@48975
  1330
blanchet@48975
  1331
lemma natLeq_natLess_Id: "natLess = natLeq - Id"
blanchet@48975
  1332
by auto
blanchet@48975
  1333
blanchet@48975
  1334
blanchet@48975
  1335
lemma natLeq_Well_order: "Well_order natLeq"
blanchet@48975
  1336
unfolding well_order_on_def
blanchet@48975
  1337
using natLeq_Linear_order wf_less natLeq_natLess_Id by auto
blanchet@48975
  1338
blanchet@48975
  1339
traytel@54581
  1340
lemma Field_natLeq_on: "Field (natLeq_on n) = {x. x < n}"
blanchet@48975
  1341
unfolding Field_def by auto
blanchet@48975
  1342
blanchet@48975
  1343
traytel@54581
  1344
lemma natLeq_underS_less: "rel.underS natLeq n = {x. x < n}"
blanchet@48975
  1345
unfolding rel.underS_def by auto
blanchet@48975
  1346
blanchet@48975
  1347
traytel@54581
  1348
lemma Restr_natLeq: "Restr natLeq {x. x < n} = natLeq_on n"
blanchet@54482
  1349
by force
blanchet@48975
  1350
blanchet@48975
  1351
blanchet@48975
  1352
lemma Restr_natLeq2:
blanchet@48975
  1353
"Restr natLeq (rel.underS natLeq n) = natLeq_on n"
blanchet@48975
  1354
by (auto simp add: Restr_natLeq natLeq_underS_less)
blanchet@48975
  1355
blanchet@48975
  1356
blanchet@48975
  1357
lemma natLeq_on_Well_order: "Well_order(natLeq_on n)"
blanchet@48975
  1358
using Restr_natLeq[of n] natLeq_Well_order
traytel@54581
  1359
      Well_order_Restr[of natLeq "{x. x < n}"] by auto
blanchet@48975
  1360
blanchet@48975
  1361
traytel@54581
  1362
corollary natLeq_on_well_order_on: "well_order_on {x. x < n} (natLeq_on n)"
blanchet@48975
  1363
using natLeq_on_Well_order Field_natLeq_on by auto
blanchet@48975
  1364
blanchet@48975
  1365
blanchet@48975
  1366
lemma natLeq_on_wo_rel: "wo_rel(natLeq_on n)"
blanchet@48975
  1367
unfolding wo_rel_def using natLeq_on_Well_order .
blanchet@48975
  1368
blanchet@48975
  1369
blanchet@48975
  1370
blanchet@48975
  1371
subsubsection {* Then as cardinals *}
blanchet@48975
  1372
blanchet@48975
  1373
blanchet@48975
  1374
lemma natLeq_Card_order: "Card_order natLeq"
blanchet@48975
  1375
proof(auto simp add: natLeq_Well_order
blanchet@48975
  1376
      Card_order_iff_Restr_underS Restr_natLeq2, simp add:  Field_natLeq)
traytel@54581
  1377
  fix n have "finite(Field (natLeq_on n))" by (auto simp: Field_def)
traytel@54578
  1378
  moreover have "\<not>finite(UNIV::nat set)" by auto
blanchet@48975
  1379
  ultimately show "natLeq_on n <o |UNIV::nat set|"
blanchet@48975
  1380
  using finite_ordLess_infinite[of "natLeq_on n" "|UNIV::nat set|"]
blanchet@48975
  1381
        Field_card_of[of "UNIV::nat set"]
blanchet@48975
  1382
        card_of_Well_order[of "UNIV::nat set"] natLeq_on_Well_order[of n] by auto
blanchet@48975
  1383
qed
blanchet@48975
  1384
blanchet@48975
  1385
blanchet@48975
  1386
corollary card_of_Field_natLeq:
blanchet@48975
  1387
"|Field natLeq| =o natLeq"
blanchet@48975
  1388
using Field_natLeq natLeq_Card_order Card_order_iff_ordIso_card_of[of natLeq]
blanchet@48975
  1389
      ordIso_symmetric[of natLeq] by blast
blanchet@48975
  1390
blanchet@48975
  1391
blanchet@48975
  1392
corollary card_of_nat:
blanchet@48975
  1393
"|UNIV::nat set| =o natLeq"
blanchet@48975
  1394
using Field_natLeq card_of_Field_natLeq by auto
blanchet@48975
  1395
blanchet@48975
  1396
blanchet@48975
  1397
corollary infinite_iff_natLeq_ordLeq:
traytel@54578
  1398
"\<not>finite A = ( natLeq \<le>o |A| )"
blanchet@48975
  1399
using infinite_iff_card_of_nat[of A] card_of_nat
blanchet@48975
  1400
      ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric by blast
blanchet@48975
  1401
blanchet@48975
  1402
corollary finite_iff_ordLess_natLeq:
blanchet@48975
  1403
"finite A = ( |A| <o natLeq)"
blanchet@48975
  1404
using infinite_iff_natLeq_ordLeq not_ordLeq_iff_ordLess
traytel@54581
  1405
      card_of_Well_order natLeq_Well_order by metis
blanchet@48975
  1406
blanchet@48975
  1407
blanchet@48975
  1408
subsection {* The successor of a cardinal *}
blanchet@48975
  1409
blanchet@48975
  1410
blanchet@48975
  1411
text{* First we define @{text "isCardSuc r r'"}, the notion of @{text "r'"}
blanchet@48975
  1412
being a successor cardinal of @{text "r"}. Although the definition does
blanchet@48975
  1413
not require @{text "r"} to be a cardinal, only this case will be meaningful.  *}
blanchet@48975
  1414
blanchet@48975
  1415
blanchet@48975
  1416
definition isCardSuc :: "'a rel \<Rightarrow> 'a set rel \<Rightarrow> bool"
blanchet@48975
  1417
where
blanchet@48975
  1418
"isCardSuc r r' \<equiv>
blanchet@48975
  1419
 Card_order r' \<and> r <o r' \<and>
blanchet@48975
  1420
 (\<forall>(r''::'a set rel). Card_order r'' \<and> r <o r'' \<longrightarrow> r' \<le>o r'')"
blanchet@48975
  1421
blanchet@48975
  1422
blanchet@48975
  1423
text{* Now we introduce the cardinal-successor operator @{text "cardSuc"},
blanchet@48975
  1424
by picking {\em some} cardinal-order relation fulfilling @{text "isCardSuc"}.
blanchet@48975
  1425
Again, the picked item shall be proved unique up to order-isomorphism. *}
blanchet@48975
  1426
blanchet@48975
  1427
blanchet@48975
  1428
definition cardSuc :: "'a rel \<Rightarrow> 'a set rel"
blanchet@48975
  1429
where
blanchet@48975
  1430
"cardSuc r \<equiv> SOME r'. isCardSuc r r'"
blanchet@48975
  1431
blanchet@48975
  1432
blanchet@48975
  1433
lemma exists_minim_Card_order:
blanchet@48975
  1434
"\<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
  1435
unfolding card_order_on_def using exists_minim_Well_order by blast
blanchet@48975
  1436
blanchet@48975
  1437
blanchet@48975
  1438
lemma exists_isCardSuc:
blanchet@48975
  1439
assumes "Card_order r"
blanchet@48975
  1440
shows "\<exists>r'. isCardSuc r r'"
blanchet@48975
  1441
proof-
blanchet@48975
  1442
  let ?R = "{(r'::'a set rel). Card_order r' \<and> r <o r'}"
blanchet@48975
  1443
  have "|Pow(Field r)| \<in> ?R \<and> (\<forall>r \<in> ?R. Card_order r)" using assms
blanchet@48975
  1444
  by (simp add: card_of_Card_order Card_order_Pow)
blanchet@48975
  1445
  then obtain r where "r \<in> ?R \<and> (\<forall>r' \<in> ?R. r \<le>o r')"
blanchet@48975
  1446
  using exists_minim_Card_order[of ?R] by blast
blanchet@48975
  1447
  thus ?thesis unfolding isCardSuc_def by auto
blanchet@48975
  1448
qed
blanchet@48975
  1449
blanchet@48975
  1450
blanchet@48975
  1451
lemma cardSuc_isCardSuc:
blanchet@48975
  1452
assumes "Card_order r"
blanchet@48975
  1453
shows "isCardSuc r (cardSuc r)"
blanchet@48975
  1454
unfolding cardSuc_def using assms
blanchet@48975
  1455
by (simp add: exists_isCardSuc someI_ex)
blanchet@48975
  1456
blanchet@48975
  1457
blanchet@48975
  1458
lemma cardSuc_Card_order:
blanchet@48975
  1459
"Card_order r \<Longrightarrow> Card_order(cardSuc r)"
blanchet@48975
  1460
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1461
blanchet@48975
  1462
blanchet@48975
  1463
lemma cardSuc_greater:
blanchet@48975
  1464
"Card_order r \<Longrightarrow> r <o cardSuc r"
blanchet@48975
  1465
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1466
blanchet@48975
  1467
blanchet@48975
  1468
lemma cardSuc_ordLeq:
blanchet@48975
  1469
"Card_order r \<Longrightarrow> r \<le>o cardSuc r"
blanchet@48975
  1470
using cardSuc_greater ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
  1471
blanchet@48975
  1472
blanchet@48975
  1473
text{* The minimality property of @{text "cardSuc"} originally present in its definition
blanchet@48975
  1474
is local to the type @{text "'a set rel"}, i.e., that of @{text "cardSuc r"}:  *}
blanchet@48975
  1475
blanchet@48975
  1476
lemma cardSuc_least_aux:
blanchet@48975
  1477
"\<lbrakk>Card_order (r::'a rel); Card_order (r'::'a set rel); r <o r'\<rbrakk> \<Longrightarrow> cardSuc r \<le>o r'"
blanchet@48975
  1478
using cardSuc_isCardSuc unfolding isCardSuc_def by blast
blanchet@48975
  1479
blanchet@48975
  1480
blanchet@48975
  1481
text{* But from this we can infer general minimality: *}
blanchet@48975
  1482
blanchet@48975
  1483
lemma cardSuc_least:
blanchet@48975
  1484
assumes CARD: "Card_order r" and CARD': "Card_order r'" and LESS: "r <o r'"
blanchet@48975
  1485
shows "cardSuc r \<le>o r'"
blanchet@48975
  1486
proof-
blanchet@48975
  1487
  let ?p = "cardSuc r"
blanchet@48975
  1488
  have 0: "Well_order ?p \<and> Well_order r'"
blanchet@48975
  1489
  using assms cardSuc_Card_order unfolding card_order_on_def by blast
blanchet@48975
  1490
  {assume "r' <o ?p"
blanchet@48975
  1491
   then obtain r'' where 1: "Field r'' < Field ?p" and 2: "r' =o r'' \<and> r'' <o ?p"
blanchet@48975
  1492
   using internalize_ordLess[of r' ?p] by blast
blanchet@48975
  1493
   (*  *)
blanchet@48975
  1494
   have "Card_order r''" using CARD' Card_order_ordIso2 2 by blast
blanchet@48975
  1495
   moreover have "r <o r''" using LESS 2 ordLess_ordIso_trans by blast
blanchet@48975
  1496
   ultimately have "?p \<le>o r''" using cardSuc_least_aux CARD by blast
blanchet@48975
  1497
   hence False using 2 not_ordLess_ordLeq by blast
blanchet@48975
  1498
  }
blanchet@48975
  1499
  thus ?thesis using 0 ordLess_or_ordLeq by blast
blanchet@48975
  1500
qed
blanchet@48975
  1501
blanchet@48975
  1502
blanchet@48975
  1503
lemma cardSuc_ordLess_ordLeq:
blanchet@48975
  1504
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1505
shows "(r <o r') = (cardSuc r \<le>o r')"
blanchet@48975
  1506
proof(auto simp add: assms cardSuc_least)
blanchet@48975
  1507
  assume "cardSuc r \<le>o r'"
blanchet@48975
  1508
  thus "r <o r'" using assms cardSuc_greater ordLess_ordLeq_trans by blast
blanchet@48975
  1509
qed
blanchet@48975
  1510
blanchet@48975
  1511
blanchet@48975
  1512
lemma cardSuc_ordLeq_ordLess:
blanchet@48975
  1513
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1514
shows "(r' <o cardSuc r) = (r' \<le>o r)"
blanchet@48975
  1515
proof-
blanchet@48975
  1516
  have "Well_order r \<and> Well_order r'"
blanchet@48975
  1517
  using assms unfolding card_order_on_def by auto
blanchet@48975
  1518
  moreover have "Well_order(cardSuc r)"
blanchet@48975
  1519
  using assms cardSuc_Card_order card_order_on_def by blast
blanchet@48975
  1520
  ultimately show ?thesis
blanchet@48975
  1521
  using assms cardSuc_ordLess_ordLeq[of r r']
blanchet@48975
  1522
  not_ordLeq_iff_ordLess[of r r'] not_ordLeq_iff_ordLess[of r' "cardSuc r"] by blast
blanchet@48975
  1523
qed
blanchet@48975
  1524
blanchet@48975
  1525
blanchet@48975
  1526
lemma cardSuc_mono_ordLeq:
blanchet@48975
  1527
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1528
shows "(cardSuc r \<le>o cardSuc r') = (r \<le>o r')"
blanchet@48975
  1529
using assms cardSuc_ordLeq_ordLess cardSuc_ordLess_ordLeq cardSuc_Card_order by blast
blanchet@48975
  1530
blanchet@48975
  1531
blanchet@48975
  1532
lemma cardSuc_invar_ordIso:
blanchet@48975
  1533
assumes CARD: "Card_order r" and CARD': "Card_order r'"
blanchet@48975
  1534
shows "(cardSuc r =o cardSuc r') = (r =o r')"
blanchet@48975
  1535
proof-
blanchet@48975
  1536
  have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')"
blanchet@48975
  1537
  using assms by (simp add: card_order_on_well_order_on cardSuc_Card_order)
blanchet@48975
  1538
  thus ?thesis
blanchet@48975
  1539
  using ordIso_iff_ordLeq[of r r'] ordIso_iff_ordLeq
blanchet@48975
  1540
  using cardSuc_mono_ordLeq[of r r'] cardSuc_mono_ordLeq[of r' r] assms by blast
blanchet@48975
  1541
qed
blanchet@48975
  1542
blanchet@48975
  1543
blanchet@48975
  1544
lemma card_of_cardSuc_finite:
blanchet@48975
  1545
"finite(Field(cardSuc |A| )) = finite A"
blanchet@48975
  1546
proof
blanchet@48975
  1547
  assume *: "finite (Field (cardSuc |A| ))"
blanchet@48975
  1548
  have 0: "|Field(cardSuc |A| )| =o cardSuc |A|"
blanchet@48975
  1549
  using card_of_Card_order cardSuc_Card_order card_of_Field_ordIso by blast
blanchet@48975
  1550
  hence "|A| \<le>o |Field(cardSuc |A| )|"
blanchet@48975
  1551
  using card_of_Card_order[of A] cardSuc_ordLeq[of "|A|"] ordIso_symmetric
blanchet@48975
  1552
  ordLeq_ordIso_trans by blast
blanchet@48975
  1553
  thus "finite A" using * card_of_ordLeq_finite by blast
blanchet@48975
  1554
next
blanchet@48975
  1555
  assume "finite A"
traytel@54581
  1556
  then have "finite ( Field |Pow A| )" unfolding Field_card_of by simp
traytel@54581
  1557
  then show "finite (Field (cardSuc |A| ))"
traytel@54581
  1558
  proof (rule card_of_ordLeq_finite[OF card_of_mono2, rotated])
traytel@54581
  1559
    show "cardSuc |A| \<le>o |Pow A|"
traytel@54581
  1560
      by (metis cardSuc_ordLess_ordLeq card_of_Card_order card_of_Pow)
traytel@54581
  1561
  qed
blanchet@48975
  1562
qed
blanchet@48975
  1563
blanchet@48975
  1564
blanchet@48975
  1565
lemma cardSuc_finite:
blanchet@48975
  1566
assumes "Card_order r"
blanchet@48975
  1567
shows "finite (Field (cardSuc r)) = finite (Field r)"
blanchet@48975
  1568
proof-
blanchet@48975
  1569
  let ?A = "Field r"
blanchet@48975
  1570
  have "|?A| =o r" using assms by (simp add: card_of_Field_ordIso)
blanchet@48975
  1571
  hence "cardSuc |?A| =o cardSuc r" using assms
blanchet@48975
  1572
  by (simp add: card_of_Card_order cardSuc_invar_ordIso)
blanchet@48975
  1573
  moreover have "|Field (cardSuc |?A| ) | =o cardSuc |?A|"
blanchet@48975
  1574
  by (simp add: card_of_card_order_on Field_card_of card_of_Field_ordIso cardSuc_Card_order)
blanchet@48975
  1575
  moreover
blanchet@48975
  1576
  {have "|Field (cardSuc r) | =o cardSuc r"
blanchet@48975
  1577
   using assms by (simp add: card_of_Field_ordIso cardSuc_Card_order)
blanchet@48975
  1578
   hence "cardSuc r =o |Field (cardSuc r) |"
blanchet@48975
  1579
   using ordIso_symmetric by blast
blanchet@48975
  1580
  }
blanchet@48975
  1581
  ultimately have "|Field (cardSuc |?A| ) | =o |Field (cardSuc r) |"
blanchet@48975
  1582
  using ordIso_transitive by blast
blanchet@48975
  1583
  hence "finite (Field (cardSuc |?A| )) = finite (Field (cardSuc r))"
blanchet@48975
  1584
  using card_of_ordIso_finite by blast
blanchet@48975
  1585
  thus ?thesis by (simp only: card_of_cardSuc_finite)
blanchet@48975
  1586
qed
blanchet@48975
  1587
blanchet@48975
  1588
blanchet@54475
  1589
lemma card_of_Plus_ordLess_infinite:
traytel@54578
  1590
assumes INF: "\<not>finite C" and
blanchet@54475
  1591
        LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|"
blanchet@54475
  1592
shows "|A <+> B| <o |C|"
blanchet@54475
  1593
proof(cases "A = {} \<or> B = {}")
blanchet@54475
  1594
  assume Case1: "A = {} \<or> B = {}"
blanchet@54475
  1595
  hence "|A| =o |A <+> B| \<or> |B| =o |A <+> B|"
blanchet@54475
  1596
  using card_of_Plus_empty1 card_of_Plus_empty2 by blast
blanchet@54475
  1597
  hence "|A <+> B| =o |A| \<or> |A <+> B| =o |B|"
blanchet@54475
  1598
  using ordIso_symmetric[of "|A|"] ordIso_symmetric[of "|B|"] by blast
blanchet@54475
  1599
  thus ?thesis using LESS1 LESS2
blanchet@54475
  1600
       ordIso_ordLess_trans[of "|A <+> B|" "|A|"]
blanchet@54475
  1601
       ordIso_ordLess_trans[of "|A <+> B|" "|B|"] by blast
blanchet@54475
  1602
next
blanchet@54475
  1603
  assume Case2: "\<not>(A = {} \<or> B = {})"
blanchet@54475
  1604
  {assume *: "|C| \<le>o |A <+> B|"
traytel@54578
  1605
   hence "\<not>finite (A <+> B)" using INF card_of_ordLeq_finite by blast
traytel@54578
  1606
   hence 1: "\<not>finite A \<or> \<not>finite B" using finite_Plus by blast
blanchet@54475
  1607
   {assume Case21: "|A| \<le>o |B|"
traytel@54578
  1608
    hence "\<not>finite B" using 1 card_of_ordLeq_finite by blast
blanchet@54475
  1609
    hence "|A <+> B| =o |B|" using Case2 Case21
blanchet@54475
  1610
    by (auto simp add: card_of_Plus_infinite)
blanchet@54475
  1611
    hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@54475
  1612
   }
blanchet@54475
  1613
   moreover
blanchet@54475
  1614
   {assume Case22: "|B| \<le>o |A|"
traytel@54578
  1615
    hence "\<not>finite A" using 1 card_of_ordLeq_finite by blast
blanchet@54475
  1616
    hence "|A <+> B| =o |A|" using Case2 Case22
blanchet@54475
  1617
    by (auto simp add: card_of_Plus_infinite)
blanchet@54475
  1618
    hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast
blanchet@54475
  1619
   }
blanchet@54475
  1620
   ultimately have False using ordLeq_total card_of_Well_order[of A]
blanchet@54475
  1621
   card_of_Well_order[of B] by blast
blanchet@54475
  1622
  }
blanchet@54475
  1623
  thus ?thesis using ordLess_or_ordLeq[of "|A <+> B|" "|C|"]
blanchet@54475
  1624
  card_of_Well_order[of "A <+> B"] card_of_Well_order[of "C"] by auto
blanchet@54475
  1625
qed
blanchet@54475
  1626
blanchet@54475
  1627
blanchet@54475
  1628
lemma card_of_Plus_ordLess_infinite_Field:
traytel@54578
  1629
assumes INF: "\<not>finite (Field r)" and r: "Card_order r" and
blanchet@54475
  1630
        LESS1: "|A| <o r" and LESS2: "|B| <o r"
blanchet@54475
  1631
shows "|A <+> B| <o r"
blanchet@54475
  1632
proof-
blanchet@54475
  1633
  let ?C  = "Field r"
blanchet@54475
  1634
  have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso
blanchet@54475
  1635
  ordIso_symmetric by blast
blanchet@54475
  1636
  hence "|A| <o |?C|"  "|B| <o |?C|"
blanchet@54475
  1637
  using LESS1 LESS2 ordLess_ordIso_trans by blast+
blanchet@54475
  1638
  hence  "|A <+> B| <o |?C|" using INF
blanchet@54475
  1639
  card_of_Plus_ordLess_infinite by blast
blanchet@54475
  1640
  thus ?thesis using 1 ordLess_ordIso_trans by blast
blanchet@54475
  1641
qed
blanchet@54475
  1642
blanchet@54475
  1643
blanchet@48975
  1644
lemma card_of_Plus_ordLeq_infinite_Field:
traytel@54578
  1645
assumes r: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
blanchet@48975
  1646
and c: "Card_order r"
blanchet@48975
  1647
shows "|A <+> B| \<le>o r"
blanchet@48975
  1648
proof-
blanchet@48975
  1649
  let ?r' = "cardSuc r"
traytel@54578
  1650
  have "Card_order ?r' \<and> \<not>finite (Field ?r')" using assms
blanchet@48975
  1651
  by (simp add: cardSuc_Card_order cardSuc_finite)
blanchet@48975
  1652
  moreover have "|A| <o ?r'" and "|B| <o ?r'" using A B c
blanchet@48975
  1653
  by (auto simp: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
blanchet@48975
  1654
  ultimately have "|A <+> B| <o ?r'"
blanchet@48975
  1655
  using card_of_Plus_ordLess_infinite_Field by blast
blanchet@48975
  1656
  thus ?thesis using c r
blanchet@48975
  1657
  by (simp add: card_of_card_order_on Field_card_of cardSuc_ordLeq_ordLess)
blanchet@48975
  1658
qed
blanchet@48975
  1659
blanchet@48975
  1660
blanchet@48975
  1661
lemma card_of_Un_ordLeq_infinite_Field:
traytel@54578
  1662
assumes C: "\<not>finite (Field r)" and A: "|A| \<le>o r" and B: "|B| \<le>o r"
blanchet@48975
  1663
and "Card_order r"
blanchet@48975
  1664
shows "|A Un B| \<le>o r"
blanchet@48975
  1665
using assms card_of_Plus_ordLeq_infinite_Field card_of_Un_Plus_ordLeq
blanchet@54482
  1666
ordLeq_transitive by fast
blanchet@48975
  1667
blanchet@48975
  1668
blanchet@48975
  1669
blanchet@48975
  1670
subsection {* Regular cardinals *}
blanchet@48975
  1671
blanchet@48975
  1672
blanchet@48975
  1673
definition cofinal where
blanchet@48975
  1674
"cofinal A r \<equiv>
blanchet@48975
  1675
 ALL a : Field r. EX b : A. a \<noteq> b \<and> (a,b) : r"
blanchet@48975
  1676
blanchet@48975
  1677
blanchet@48975
  1678
definition regular where
blanchet@48975
  1679
"regular r \<equiv>
blanchet@48975
  1680
 ALL K. K \<le> Field r \<and> cofinal K r \<longrightarrow> |K| =o r"
blanchet@48975
  1681
blanchet@48975
  1682
blanchet@48975
  1683
definition relChain where
blanchet@48975
  1684
"relChain r As \<equiv>
blanchet@48975
  1685
 ALL i j. (i,j) \<in> r \<longrightarrow> As i \<le> As j"
blanchet@48975
  1686
blanchet@48975
  1687
lemma regular_UNION:
blanchet@48975
  1688
assumes r: "Card_order r"   "regular r"
blanchet@48975
  1689
and As: "relChain r As"
blanchet@48975
  1690
and Bsub: "B \<le> (UN i : Field r. As i)"
blanchet@48975
  1691
and cardB: "|B| <o r"
blanchet@48975
  1692
shows "EX i : Field r. B \<le> As i"
blanchet@48975
  1693
proof-
blanchet@48975
  1694
  let ?phi = "%b j. j : Field r \<and> b : As j"
blanchet@48975
  1695
  have "ALL b : B. EX j. ?phi b j" using Bsub by blast
blanchet@48975
  1696
  then obtain f where f: "!! b. b : B \<Longrightarrow> ?phi b (f b)"
blanchet@48975
  1697
  using bchoice[of B ?phi] by blast
blanchet@48975
  1698
  let ?K = "f ` B"
blanchet@48975
  1699
  {assume 1: "!! i. i : Field r \<Longrightarrow> ~ B \<le> As i"
blanchet@48975
  1700
   have 2: "cofinal ?K r"
blanchet@48975
  1701
   unfolding cofinal_def proof auto
blanchet@48975
  1702
     fix i assume i: "i : Field r"
blanchet@48975
  1703
     with 1 obtain b where b: "b : B \<and> b \<notin> As i" by blast
blanchet@48975
  1704
     hence "i \<noteq> f b \<and> ~ (f b,i) : r"
blanchet@48975
  1705
     using As f unfolding relChain_def by auto
blanchet@48975
  1706
     hence "i \<noteq> f b \<and> (i, f b) : r" using r
blanchet@48975
  1707
     unfolding card_order_on_def well_order_on_def linear_order_on_def
blanchet@48975
  1708
     total_on_def using i f b by auto
blanchet@48975
  1709
     with b show "\<exists>b\<in>B. i \<noteq> f b \<and> (i, f b) \<in> r" by blast
blanchet@48975
  1710
   qed
blanchet@48975
  1711
   moreover have "?K \<le> Field r" using f by blast
blanchet@48975
  1712
   ultimately have "|?K| =o r" using 2 r unfolding regular_def by blast
blanchet@48975
  1713
   moreover
blanchet@48975
  1714
   {
blanchet@48975
  1715
    have "|?K| <=o |B|" using card_of_image .
blanchet@48975
  1716
    hence "|?K| <o r" using cardB ordLeq_ordLess_trans by blast
blanchet@48975
  1717
   }
blanchet@48975
  1718
   ultimately have False using not_ordLess_ordIso by blast
blanchet@48975
  1719
  }
blanchet@48975
  1720
  thus ?thesis by blast
blanchet@48975
  1721
qed
blanchet@48975
  1722
blanchet@48975
  1723
blanchet@48975
  1724
lemma infinite_cardSuc_regular:
traytel@54578
  1725
assumes r_inf: "\<not>finite (Field r)" and r_card: "Card_order r"
blanchet@48975
  1726
shows "regular (cardSuc r)"
blanchet@48975
  1727
proof-
blanchet@48975
  1728
  let ?r' = "cardSuc r"
blanchet@48975
  1729
  have r': "Card_order ?r'"
blanchet@48975
  1730
  "!! p. Card_order p \<longrightarrow> (p \<le>o r) = (p <o ?r')"
blanchet@48975
  1731
  using r_card by (auto simp: cardSuc_Card_order cardSuc_ordLeq_ordLess)
blanchet@48975
  1732
  show ?thesis
blanchet@48975
  1733
  unfolding regular_def proof auto
blanchet@48975
  1734
    fix K assume 1: "K \<le> Field ?r'" and 2: "cofinal K ?r'"
blanchet@48975
  1735
    hence "|K| \<le>o |Field ?r'|" by (simp only: card_of_mono1)
blanchet@48975
  1736
    also have 22: "|Field ?r'| =o ?r'"
blanchet@48975
  1737
    using r' by (simp add: card_of_Field_ordIso[of ?r'])
blanchet@48975
  1738
    finally have "|K| \<le>o ?r'" .
blanchet@48975
  1739
    moreover
blanchet@48975
  1740
    {let ?L = "UN j : K. rel.underS ?r' j"
blanchet@48975
  1741
     let ?J = "Field r"
blanchet@48975
  1742
     have rJ: "r =o |?J|"
blanchet@48975
  1743
     using r_card card_of_Field_ordIso ordIso_symmetric by blast
blanchet@48975
  1744
     assume "|K| <o ?r'"
blanchet@48975
  1745
     hence "|K| <=o r" using r' card_of_Card_order[of K] by blast
blanchet@48975
  1746
     hence "|K| \<le>o |?J|" using rJ ordLeq_ordIso_trans by blast
blanchet@48975
  1747
     moreover
blanchet@48975
  1748
     {have "ALL j : K. |rel.underS ?r' j| <o ?r'"
blanchet@48975
  1749
      using r' 1 by (auto simp: card_of_underS)
blanchet@48975
  1750
      hence "ALL j : K. |rel.underS ?r' j| \<le>o r"
blanchet@48975
  1751
      using r' card_of_Card_order by blast
blanchet@48975
  1752
      hence "ALL j : K. |rel.underS ?r' j| \<le>o |?J|"
blanchet@48975
  1753
      using rJ ordLeq_ordIso_trans by blast
blanchet@48975
  1754
     }
blanchet@48975
  1755
     ultimately have "|?L| \<le>o |?J|"
blanchet@48975
  1756
     using r_inf card_of_UNION_ordLeq_infinite by blast
blanchet@48975
  1757
     hence "|?L| \<le>o r" using rJ ordIso_symmetric ordLeq_ordIso_trans by blast
blanchet@48975
  1758
     hence "|?L| <o ?r'" using r' card_of_Card_order by blast
blanchet@48975
  1759
     moreover
blanchet@48975
  1760
     {
blanchet@48975
  1761
      have "Field ?r' \<le> ?L"
blanchet@48975
  1762
      using 2 unfolding rel.underS_def cofinal_def by auto
blanchet@48975
  1763
      hence "|Field ?r'| \<le>o |?L|" by (simp add: card_of_mono1)
blanchet@48975
  1764
      hence "?r' \<le>o |?L|"
blanchet@48975
  1765
      using 22 ordIso_ordLeq_trans ordIso_symmetric by blast
blanchet@48975
  1766
     }
blanchet@48975
  1767
     ultimately have "|?L| <o |?L|" using ordLess_ordLeq_trans by blast
blanchet@48975
  1768
     hence False using ordLess_irreflexive by blast
blanchet@48975
  1769
    }
blanchet@48975
  1770
    ultimately show "|K| =o ?r'"
blanchet@48975
  1771
    unfolding ordLeq_iff_ordLess_or_ordIso by blast
blanchet@48975
  1772
  qed
blanchet@48975
  1773
qed
blanchet@48975
  1774
blanchet@48975
  1775
lemma cardSuc_UNION:
traytel@54578
  1776
assumes r: "Card_order r" and "\<not>finite (Field r)"
blanchet@48975
  1777
and As: "relChain (cardSuc r) As"
blanchet@48975
  1778
and Bsub: "B \<le> (UN i : Field (cardSuc r). As i)"
blanchet@48975
  1779
and cardB: "|B| <=o r"
blanchet@48975
  1780
shows "EX i : Field (cardSuc r). B \<le> As i"
blanchet@48975
  1781
proof-
blanchet@48975
  1782
  let ?r' = "cardSuc r"
blanchet@48975
  1783
  have "Card_order ?r' \<and> |B| <o ?r'"
blanchet@48975
  1784
  using r cardB cardSuc_ordLeq_ordLess cardSuc_Card_order
blanchet@48975
  1785
  card_of_Card_order by blast
blanchet@48975
  1786
  moreover have "regular ?r'"
blanchet@48975
  1787
  using assms by(simp add: infinite_cardSuc_regular)
blanchet@48975
  1788
  ultimately show ?thesis
blanchet@48975
  1789
  using As Bsub cardB regular_UNION by blast
blanchet@48975
  1790
qed
blanchet@48975
  1791
blanchet@48975
  1792
blanchet@48975
  1793
subsection {* Others *}
blanchet@48975
  1794
blanchet@48975
  1795
(* function space *)
blanchet@48975
  1796
definition Func where
traytel@52545
  1797
"Func A B = {f . (\<forall> a \<in> A. f a \<in> B) \<and> (\<forall> a. a \<notin> A \<longrightarrow> f a = undefined)}"
blanchet@48975
  1798
blanchet@48975
  1799
lemma Func_empty:
traytel@52545
  1800
"Func {} B = {\<lambda>x. undefined}"
blanchet@48975
  1801
unfolding Func_def by auto
blanchet@48975
  1802
blanchet@48975
  1803
lemma Func_elim:
blanchet@48975
  1804
assumes "g \<in> Func A B" and "a \<in> A"
traytel@52545
  1805
shows "\<exists> b. b \<in> B \<and> g a = b"
traytel@52545
  1806
using assms unfolding Func_def by (cases "g a = undefined") auto
blanchet@48975
  1807
blanchet@48975
  1808
definition curr where
traytel@52545
  1809
"curr A f \<equiv> \<lambda> a. if a \<in> A then \<lambda>b. f (a,b) else undefined"
blanchet@48975
  1810
blanchet@48975
  1811
lemma curr_in:
blanchet@48975
  1812
assumes f: "f \<in> Func (A <*> B) C"
blanchet@48975
  1813
shows "curr A f \<in> Func A (Func B C)"
blanchet@48975
  1814
using assms unfolding curr_def Func_def by auto
blanchet@48975
  1815
blanchet@48975
  1816
lemma curr_inj:
blanchet@48975
  1817
assumes "f1 \<in> Func (A <*> B) C" and "f2 \<in> Func (A <*> B) C"
blanchet@48975
  1818
shows "curr A f1 = curr A f2 \<longleftrightarrow> f1 = f2"
blanchet@48975
  1819
proof safe
blanchet@48975
  1820
  assume c: "curr A f1 = curr A f2"
blanchet@48975
  1821
  show "f1 = f2"
blanchet@48975
  1822
  proof (rule ext, clarify)
blanchet@48975
  1823
    fix a b show "f1 (a, b) = f2 (a, b)"
blanchet@48975
  1824
    proof (cases "(a,b) \<in> A <*> B")
blanchet@48975
  1825
      case False
traytel@52545
  1826
      thus ?thesis using assms unfolding Func_def by auto
blanchet@48975
  1827
    next
blanchet@48975
  1828
      case True hence a: "a \<in> A" and b: "b \<in> B" by auto
blanchet@48975
  1829
      thus ?thesis
traytel@52545
  1830
      using c unfolding curr_def fun_eq_iff by(elim allE[of _ a]) simp
blanchet@48975
  1831
    qed
blanchet@48975
  1832
  qed
blanchet@48975
  1833
qed
blanchet@48975
  1834
blanchet@48975
  1835
lemma curr_surj:
blanchet@48975
  1836
assumes "g \<in> Func A (Func B C)"
blanchet@48975
  1837
shows "\<exists> f \<in> Func (A <*> B) C. curr A f = g"
blanchet@48975
  1838
proof
traytel@52545
  1839
  let ?f = "\<lambda> ab. if fst ab \<in> A \<and> snd ab \<in> B then g (fst ab) (snd ab) else undefined"
blanchet@48975
  1840
  show "curr A ?f = g"
blanchet@48975
  1841
  proof (rule ext)
blanchet@48975
  1842
    fix a show "curr A ?f a = g a"
blanchet@48975
  1843
    proof (cases "a \<in> A")
blanchet@48975
  1844
      case False
traytel@52545
  1845
      hence "g a = undefined" using assms unfolding Func_def by auto
blanchet@48975
  1846
      thus ?thesis unfolding curr_def using False by simp
blanchet@48975
  1847
    next
blanchet@48975
  1848
      case True
traytel@52545
  1849
      obtain g1 where "g1 \<in> Func B C" and "g a = g1"
blanchet@48975
  1850
      using assms using Func_elim[OF assms True] by blast
traytel@52545
  1851
      thus ?thesis using True unfolding Func_def curr_def by auto
blanchet@48975
  1852
    qed
blanchet@48975
  1853
  qed
traytel@52545
  1854
  show "?f \<in> Func (A <*> B) C" using assms unfolding Func_def mem_Collect_eq by auto
blanchet@48975
  1855
qed
blanchet@48975
  1856
traytel@52544
  1857
lemma bij_betw_curr:
blanchet@48975
  1858
"bij_betw (curr A) (Func (A <*> B) C) (Func A (Func B C))"
blanchet@48975
  1859
unfolding bij_betw_def inj_on_def image_def
blanchet@54482
  1860
apply (intro impI conjI ballI)
blanchet@54482
  1861
apply (erule curr_inj[THEN iffD1], assumption+)
blanchet@54482
  1862
apply auto
blanchet@54482
  1863
apply (erule curr_in)
blanchet@54482
  1864
using curr_surj by blast
blanchet@48975
  1865
blanchet@48975
  1866
lemma card_of_Func_Times:
blanchet@48975
  1867
"|Func (A <*> B) C| =o |Func A (Func B C)|"
blanchet@48975
  1868
unfolding card_of_ordIso[symmetric]
traytel@52544
  1869
using bij_betw_curr by blast
blanchet@48975
  1870
blanchet@48975
  1871
definition Func_map where
traytel@52545
  1872
"Func_map B2 f1 f2 g b2 \<equiv> if b2 \<in> B2 then f1 (g (f2 b2)) else undefined"
blanchet@48975
  1873
blanchet@48975
  1874
lemma Func_map:
blanchet@48975
  1875
assumes g: "g \<in> Func A2 A1" and f1: "f1 ` A1 \<subseteq> B1" and f2: "f2 ` B2 \<subseteq> A2"
blanchet@48975
  1876
shows "Func_map B2 f1 f2 g \<in> Func B2 B1"
traytel@52545
  1877
using assms unfolding Func_def Func_map_def mem_Collect_eq by auto
blanchet@48975
  1878
blanchet@48975
  1879
lemma Func_non_emp:
blanchet@48975
  1880
assumes "B \<noteq> {}"
blanchet@48975
  1881
shows "Func A B \<noteq> {}"
blanchet@48975
  1882
proof-
blanchet@48975
  1883
  obtain b where b: "b \<in> B" using assms by auto
traytel@52545
  1884
  hence "(\<lambda> a. if a \<in> A then b else undefined) \<in> Func A B" unfolding Func_def by auto
blanchet@48975
  1885
  thus ?thesis by blast
blanchet@48975
  1886
qed
blanchet@48975
  1887
blanchet@48975
  1888
lemma Func_is_emp:
blanchet@48975
  1889
"Func A B = {} \<longleftrightarrow> A \<noteq> {} \<and> B = {}" (is "?L \<longleftrightarrow> ?R")
blanchet@48975
  1890
proof
blanchet@48975
  1891
  assume L: ?L
blanchet@48975
  1892
  moreover {assume "A = {}" hence False using L Func_empty by auto}
blanchet@48975
  1893
  moreover {assume "B \<noteq> {}" hence False using L Func_non_emp by metis}
blanchet@48975
  1894
  ultimately show ?R by blast
blanchet@48975
  1895
next
blanchet@48975
  1896
  assume R: ?R
blanchet@48975
  1897
  moreover
blanchet@48975
  1898
  {fix f assume "f \<in> Func A B"
blanchet@48975
  1899
   moreover obtain a where "a \<in> A" using R by blast
blanchet@54482
  1900
   ultimately obtain b where "b \<in> B" unfolding Func_def by blast
blanchet@54482
  1901
   with R have False by blast
blanchet@48975
  1902
  }
blanchet@48975
  1903
  thus ?L by blast
blanchet@48975
  1904
qed
blanchet@48975
  1905
blanchet@48975
  1906
lemma Func_map_surj:
blanchet@48975
  1907
assumes B1: "f1 ` A1 = B1" and A2: "inj_on f2 B2" "f2 ` B2 \<subseteq> A2"
blanchet@48975
  1908
and B2A2: "B2 = {} \<Longrightarrow> A2 = {}"
blanchet@48975
  1909
shows "Func B2 B1 = Func_map B2 f1 f2 ` Func A2 A1"
blanchet@48975
  1910
proof(cases "B2 = {}")
blanchet@48975
  1911
  case True
traytel@52545
  1912
  thus ?thesis using B2A2 by (auto simp: Func_empty Func_map_def)
blanchet@48975
  1913
next
blanchet@48975
  1914
  case False note B2 = False
blanchet@48975
  1915
  show ?thesis
traytel@52545
  1916
  proof safe
traytel@52545
  1917
    fix h assume h: "h \<in> Func B2 B1"
traytel@52545
  1918
    def j1 \<equiv> "inv_into A1 f1"
traytel@52545
  1919
    have "\<forall> a2 \<in> f2 ` B2. \<exists> b2. b2 \<in> B2 \<and> f2 b2 = a2" by blast
traytel@52545
  1920
    then obtain k where k: "\<forall> a2 \<in> f2 ` B2. k a2 \<in> B2 \<and> f2 (k a2) = a2" by metis
traytel@52545
  1921
    {fix b2 assume b2: "b2 \<in> B2"
traytel@52545
  1922
     hence "f2 (k (f2 b2)) = f2 b2" using k A2(2) by auto
traytel@52545
  1923
     moreover have "k (f2 b2) \<in> B2" using b2 A2(2) k by auto
traytel@52545
  1924
     ultimately have "k (f2 b2) = b2" using b2 A2(1) unfolding inj_on_def by blast
traytel@52545
  1925
    } note kk = this
traytel@52545
  1926
    obtain b22 where b22: "b22 \<in> B2" using B2 by auto
traytel@52545
  1927
    def j2 \<equiv> "\<lambda> a2. if a2 \<in> f2 ` B2 then k a2 else b22"
traytel@52545
  1928
    have j2A2: "j2 ` A2 \<subseteq> B2" unfolding j2_def using k b22 by auto
traytel@52545
  1929
    have j2: "\<And> b2. b2 \<in> B2 \<Longrightarrow> j2 (f2 b2) = b2"
traytel@52545
  1930
    using kk unfolding j2_def by auto
traytel@52545
  1931
    def g \<equiv> "Func_map A2 j1 j2 h"
traytel@52545
  1932
    have "Func_map B2 f1 f2 g = h"
traytel@52545
  1933
    proof (rule ext)
traytel@52545
  1934
      fix b2 show "Func_map B2 f1 f2 g b2 = h b2"
traytel@52545
  1935
      proof(cases "b2 \<in> B2")
traytel@52545
  1936
        case True
blanchet@48975
  1937
        show ?thesis
traytel@52545
  1938
        proof (cases "h b2 = undefined")
traytel@52545
  1939
          case True
traytel@52545
  1940
          hence b1: "h b2 \<in> f1 ` A1" using h `b2 \<in> B2` unfolding B1 Func_def by auto
traytel@52545
  1941
          show ?thesis using A2 f_inv_into_f[OF b1]
traytel@52545
  1942
            unfolding True g_def Func_map_def j1_def j2[OF `b2 \<in> B2`] by auto
traytel@52545
  1943
        qed(insert A2 True j2[OF True] h B1, unfold j1_def g_def Func_def Func_map_def,
traytel@52545
  1944
          auto intro: f_inv_into_f)
traytel@52545
  1945
      qed(insert h, unfold Func_def Func_map_def, auto)
traytel@52545
  1946
    qed
traytel@52545
  1947
    moreover have "g \<in> Func A2 A1" unfolding g_def apply(rule Func_map[OF h])
traytel@52545
  1948
    using inv_into_into j2A2 B1 A2 inv_into_into
traytel@52545
  1949
    unfolding j1_def image_def by fast+
traytel@52545
  1950
    ultimately show "h \<in> Func_map B2 f1 f2 ` Func A2 A1"
traytel@52545
  1951
    unfolding Func_map_def[abs_def] unfolding image_def by auto
traytel@52545
  1952
  qed(insert B1 Func_map[OF _ _ A2(2)], auto)
blanchet@48975
  1953
qed
blanchet@48975
  1954
blanchet@48975
  1955
lemma card_of_Pow_Func:
blanchet@48975
  1956
"|Pow A| =o |Func A (UNIV::bool set)|"
blanchet@48975
  1957
proof-
traytel@52545
  1958
  def F \<equiv> "\<lambda> A' a. if a \<in> A then (if a \<in> A' then True else False)
traytel@52545
  1959
                            else undefined"
blanchet@48975
  1960
  have "bij_betw F (Pow A) (Func A (UNIV::bool set))"
blanchet@48975
  1961
  unfolding bij_betw_def inj_on_def proof (intro ballI impI conjI)
traytel@52545
  1962
    fix A1 A2 assume "A1 \<in> Pow A" "A2 \<in> Pow A" "F A1 = F A2"
traytel@52545
  1963
    thus "A1 = A2" unfolding F_def Pow_def fun_eq_iff by (auto split: split_if_asm)
blanchet@48975
  1964
  next
blanchet@48975
  1965
    show "F ` Pow A = Func A UNIV"
blanchet@48975
  1966
    proof safe
blanchet@48975
  1967
      fix f assume f: "f \<in> Func A (UNIV::bool set)"
blanchet@48975
  1968
      show "f \<in> F ` Pow A" unfolding image_def mem_Collect_eq proof(intro bexI)
traytel@52545
  1969
        let ?A1 = "{a \<in> A. f a = True}"
blanchet@48975
  1970
        show "f = F ?A1" unfolding F_def apply(rule ext)
traytel@52545
  1971
        using f unfolding Func_def mem_Collect_eq by auto
blanchet@48975
  1972
      qed auto
blanchet@48975
  1973
    qed(unfold Func_def mem_Collect_eq F_def, auto)
blanchet@48975
  1974
  qed
blanchet@48975
  1975
  thus ?thesis unfolding card_of_ordIso[symmetric] by blast
blanchet@48975
  1976
qed
blanchet@48975
  1977
blanchet@48975
  1978
lemma card_of_Func_UNIV:
blanchet@48975
  1979
"|Func (UNIV::'a set) (B::'b set)| =o |{f::'a \<Rightarrow> 'b. range f \<subseteq> B}|"
blanchet@48975
  1980
apply(rule ordIso_symmetric) proof(intro card_of_ordIsoI)
traytel@52545
  1981
  let ?F = "\<lambda> f (a::'a). ((f a)::'b)"
blanchet@48975
  1982
  show "bij_betw ?F {f. range f \<subseteq> B} (Func UNIV B)"
blanchet@48975
  1983
  unfolding bij_betw_def inj_on_def proof safe
traytel@52545
  1984
    fix h :: "'a \<Rightarrow> 'b" assume h: "h \<in> Func UNIV B"
traytel@52545
  1985
    hence "\<forall> a. \<exists> b. h a = b" unfolding Func_def by auto
traytel@52545
  1986
    then obtain f where f: "\<forall> a. h a = f a" by metis
blanchet@48975
  1987
    hence "range f \<subseteq> B" using h unfolding Func_def by auto
traytel@52545
  1988
    thus "h \<in> (\<lambda>f a. f a) ` {f. range f \<subseteq> B}" using f unfolding image_def by auto
blanchet@48975
  1989
  qed(unfold Func_def fun_eq_iff, auto)
blanchet@48975
  1990
qed
blanchet@48975
  1991
blanchet@48975
  1992
end