header {* Bounds *}
theory Bounds
imports Main 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 lub_compat: "lub A x = isLub UNIV A x"
proof -
have "isUb UNIV A = (λx. A *<= x ∧ x ∈ UNIV)"
by (rule ext) (simp only: isUb_def)
then show ?thesis
by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
qed
lemma real_complete:
fixes A :: "real set"
assumes nonempty: "∃a. a ∈ A"
and ex_upper: "∃y. ∀a ∈ A. a ≤ y"
shows "∃x. lub A x"
proof -
from ex_upper have "∃y. isUb UNIV A y"
unfolding isUb_def setle_def by blast
with nonempty have "∃x. isLub UNIV A x"
by (rule reals_complete)
then show ?thesis by (simp only: lub_compat)
qed
end