--- a/src/HOL/Real/HahnBanach/Bounds.thy Mon Dec 29 13:23:53 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,83 +0,0 @@
-(* Title: HOL/Real/HahnBanach/Bounds.thy
- ID: $Id$
- Author: Gertrud Bauer, TU Munich
-*)
-
-header {* Bounds *}
-
-theory Bounds
-imports Main ContNotDenum
-begin
-
-locale lub =
- fixes A and x
- assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b"
- and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
-
-lemmas [elim?] = lub.least lub.upper
-
-definition
- the_lub :: "'a::order set \<Rightarrow> 'a" where
- "the_lub A = The (lub A)"
-
-notation (xsymbols)
- the_lub ("\<Squnion>_" [90] 90)
-
-lemma the_lub_equality [elim?]:
- assumes "lub A x"
- shows "\<Squnion>A = (x::'a::order)"
-proof -
- interpret lub [A x] by fact
- show ?thesis
- proof (unfold the_lub_def)
- from `lub A x` show "The (lub A) = x"
- proof
- fix x' assume lub': "lub A x'"
- show "x' = x"
- proof (rule order_antisym)
- from lub' show "x' \<le> x"
- proof
- fix a assume "a \<in> A"
- then show "a \<le> x" ..
- qed
- show "x \<le> x'"
- proof
- fix a assume "a \<in> A"
- with lub' show "a \<le> x'" ..
- qed
- qed
- qed
- qed
-qed
-
-lemma the_lubI_ex:
- assumes ex: "\<exists>x. lub A x"
- shows "lub A (\<Squnion>A)"
-proof -
- from ex obtain x where x: "lub A x" ..
- also from x have [symmetric]: "\<Squnion>A = x" ..
- finally show ?thesis .
-qed
-
-lemma lub_compat: "lub A x = isLub UNIV A x"
-proof -
- have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
- by (rule ext) (simp only: isUb_def)
- then show ?thesis
- by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
-qed
-
-lemma real_complete:
- fixes A :: "real set"
- assumes nonempty: "\<exists>a. a \<in> A"
- and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
- shows "\<exists>x. lub A x"
-proof -
- from ex_upper have "\<exists>y. isUb UNIV A y"
- unfolding isUb_def setle_def by blast
- with nonempty have "\<exists>x. isLub UNIV A x"
- by (rule reals_complete)
- then show ?thesis by (simp only: lub_compat)
-qed
-
-end