# Theory Bounds

theory Bounds
imports Continuum_Not_Denumerable
```(*  Title:      HOL/Hahn_Banach/Bounds.thy
Author:     Gertrud Bauer, TU Munich
*)

section ‹Bounds›

theory Bounds
imports Main "HOL-Analysis.Continuum_Not_Denumerable"
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"  ("⨆_" [90] 90)
where "the_lub A = The (lub A)"

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
```