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

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Zorn's Lemma *}

 theory ZornLemma

 imports Zorn

 begin

 text {*

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

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

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

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

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

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

   suffices to show that for every non-empty chain @{text c} in @{text

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

 *}

 theorem Zorn's_Lemma:

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

     and aS: "a \<in> S"

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

 proof (rule Zorn_Lemma2)

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

   proof

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

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

     proof cases

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

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

       assume "c = {}"

       with aS show ?thesis by fast

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

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

     next

       assume "c \<noteq> {}"

       show ?thesis

       proof

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

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

         proof (rule r)

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

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

         qed

       qed

     qed

   qed

 qed

 end