1.1 --- a/src/HOL/Real/HahnBanach/ZornLemma.thy	Tue Dec 30 08:18:54 2008 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,57 +0,0 @@
     1.4 -(*  Title:      HOL/Real/HahnBanach/ZornLemma.thy
     1.5 -    Author:     Gertrud Bauer, TU Munich
     1.6 -*)
     1.7 -
     1.8 -header {* Zorn's Lemma *}
     1.9 -
    1.10 -theory ZornLemma
    1.11 -imports Zorn
    1.12 -begin
    1.13 -
    1.14 -text {*
    1.15 -  Zorn's Lemmas states: if every linear ordered subset of an ordered
    1.16 -  set @{text S} has an upper bound in @{text S}, then there exists a
    1.17 -  maximal element in @{text S}.  In our application, @{text S} is a
    1.18 -  set of sets ordered by set inclusion. Since the union of a chain of
    1.19 -  sets is an upper bound for all elements of the chain, the conditions
    1.20 -  of Zorn's lemma can be modified: if @{text S} is non-empty, it
    1.21 -  suffices to show that for every non-empty chain @{text c} in @{text
    1.22 -  S} the union of @{text c} also lies in @{text S}.
    1.23 -*}
    1.24 -
    1.25 -theorem Zorn's_Lemma:
    1.26 -  assumes r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"
    1.27 -    and aS: "a \<in> S"
    1.28 -  shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"
    1.29 -proof (rule Zorn_Lemma2)
    1.30 -  show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
    1.31 -  proof
    1.32 -    fix c assume "c \<in> chain S"
    1.33 -    show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
    1.34 -    proof cases
    1.35 -
    1.36 -      txt {* If @{text c} is an empty chain, then every element in
    1.37 -	@{text S} is an upper bound of @{text c}. *}
    1.38 -
    1.39 -      assume "c = {}"
    1.40 -      with aS show ?thesis by fast
    1.41 -
    1.42 -      txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper
    1.43 -	bound of @{text c}, lying in @{text S}. *}
    1.44 -
    1.45 -    next
    1.46 -      assume "c \<noteq> {}"
    1.47 -      show ?thesis
    1.48 -      proof
    1.49 -        show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast
    1.50 -        show "\<Union>c \<in> S"
    1.51 -        proof (rule r)
    1.52 -          from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast
    1.53 -	  show "c \<in> chain S" by fact
    1.54 -        qed
    1.55 -      qed
    1.56 -    qed
    1.57 -  qed
    1.58 -qed
    1.59 -
    1.60 -end