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 ("\<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' ≤ 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 (\<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: "∃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