(*  Title:      HOL/Real/HahnBanach/Bounds.thy

     ID:         $Id$

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Bounds *}

 theory Bounds

 imports Main Real

 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"

     by (unfold isUb_def setle_def) blast

   with nonempty have "\<exists>x. isLub UNIV A x"

     by (rule reals_complete)

   then show ?thesis by (simp only: lub_compat)

 qed

 end