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(* Title: HOL/Real/HahnBanach/ZornLemma.thy


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ID: $Id$


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Author: Gertrud Bauer, TU Munich


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*)


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header {* Zorn's Lemma *}

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theory ZornLemma


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imports Zorn


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begin

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text {*


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Zorn's Lemmas states: if every linear ordered subset of an ordered


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set @{text S} has an upper bound in @{text S}, then there exists a


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maximal element in @{text S}. In our application, @{text S} is a


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set of sets ordered by set inclusion. Since the union of a chain of


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sets is an upper bound for all elements of the chain, the conditions


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of Zorn's lemma can be modified: if @{text S} is nonempty, it


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suffices to show that for every nonempty chain @{text c} in @{text


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S} the union of @{text c} also lies in @{text S}.


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*}

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theorem Zorn's_Lemma:

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assumes r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"


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and aS: "a \<in> S"


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shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"

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proof (rule Zorn_Lemma2)

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show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"

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proof

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fix c assume "c \<in> chain S"


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show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"

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proof cases

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txt {* If @{text c} is an empty chain, then every element in

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@{text S} is an upper bound of @{text c}. *}

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assume "c = {}"

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with aS show ?thesis by fast

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txt {* If @{text c} is nonempty, then @{text "\<Union>c"} is an upper

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bound of @{text c}, lying in @{text S}. *}

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next

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assume "c \<noteq> {}"

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show ?thesis


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proof

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show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast

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show "\<Union>c \<in> S"

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proof (rule r)

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from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast

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show "c \<in> chain S" by fact

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qed


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qed


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qed


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qed


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qed

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end
