(* 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?]: "(\a. a \ A \ a \ b) \ x \ b"
and upper [intro?]: "a \ A \ a \ x"
lemmas [elim?] = lub.least lub.upper
definition the_lub :: "'a::order set \ 'a"
where "the_lub A = The (lub A)"
notation (xsymbols)
the_lub ("\_" [90] 90)
lemma the_lub_equality [elim?]:
assumes "lub A x"
shows "\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' \ x"
proof
fix a assume "a \ A"
then show "a \ x" ..
qed
show "x \ x'"
proof
fix a assume "a \ A"
with lub' show "a \ x'" ..
qed
qed
qed
qed
qed
lemma the_lubI_ex:
assumes ex: "\x. lub A x"
shows "lub A (\A)"
proof -
from ex obtain x where x: "lub A x" ..
also from x have [symmetric]: "\A = x" ..
finally show ?thesis .
qed
lemma real_complete: "\a::real. a \ A \ \y. \a \ A. a \ y \ \x. lub A x"
by (intro exI[of _ "Sup A"]) (auto intro!: cSup_upper cSup_least simp: lub_def)
end