(*  Title:      HOL/Hahn_Banach/Bounds.thy

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Bounds *}

 theory Bounds

 imports Main "~~/src/HOL/Library/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 real_complete: "\<exists>a::real. a \<in> A \<Longrightarrow> \<exists>y. \<forall>a \<in> A. a \<le> y \<Longrightarrow> \<exists>x. lub A x"

   by (intro exI[of _ "Sup A"]) (auto intro!: cSup_upper cSup_least simp: lub_def)

 end