src/HOL/Real/HahnBanach/ZornLemma.thy
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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 = Aux + Zorn:;
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text {* Zorn's Lemmas states: if every linear ordered subset of an
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ordered set $S$ has an upper bound in $S$, then there exists a maximal
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element in $S$.  In our application, $S$ is a set of sets ordered by
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set inclusion. Since the union of a chain of sets is an upper bound
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for all elements of the chain, the conditions of Zorn's lemma can be
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modified: if $S$ is non-empty, it suffices to show that for every
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non-empty chain $c$ in $S$ the union of $c$ also lies in $S$. *};
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theorem Zorn's_Lemma: 
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  "(!!c. c: chain S ==> EX x. x:c ==> Union c : S) ==> a:S
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  ==>  EX y: S. ALL z: S. y <= z --> y = z";
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proof (rule Zorn_Lemma2);
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  txt_raw {* \footnote{See
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  \url{http://isabelle.in.tum.de/library/HOL/HOL-Real/Zorn.html}}*};
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  assume r: "!!c. c: chain S ==> EX x. x:c ==> Union c : S";
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  assume aS: "a:S";
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  show "ALL c:chain S. EX y:S. ALL z:c. z <= y";
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  proof;
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    fix c; assume "c:chain S"; 
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    show "EX y:S. ALL z:c. z <= y";
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    proof (rule case_split);
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      txt{* If $c$ is an empty chain, then every element
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      in $S$ is an upper bound of $c$. *};
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      assume "c={}"; 
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      with aS; show ?thesis; by fast;
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      txt{* If $c$ is non-empty, then $\Union c$ 
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      is an upper bound of $c$, lying in $S$. *};
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    next;
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      assume c: "c~={}";
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      show ?thesis; 
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      proof; 
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        show "ALL z:c. z <= Union c"; by fast;
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        show "Union c : S"; 
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        proof (rule r);
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          from c; show "EX x. x:c"; by fast;  
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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;