(*  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