(*  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