(* Title: HOL/Hahn_Banach/Bounds.thy
Author: Gertrud Bauer, TU Munich
*)
section \<open>Bounds\<close>
theory Bounds
imports Main "HOL-Analysis.Continuum_Not_Denumerable"
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" ("\<Squnion>_" [90] 90)
where "the_lub A = The (lub A)"
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 \<open>lub A x\<close> 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