--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HahnBanach/ZornLemma.thy Tue Dec 30 11:10:01 2008 +0100
@@ -0,0 +1,57 @@
+(* 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