--- a/src/HOL/Real/HahnBanach/ZornLemma.thy Mon Dec 29 11:04:27 2008 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,58 +0,0 @@
-(* Title: HOL/Real/HahnBanach/ZornLemma.thy
- ID: $Id$
- 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