wenzelm@31795: (*  Title:      HOL/Hahn_Banach/Bounds.thy
wenzelm@7566:     Author:     Gertrud Bauer, TU Munich
wenzelm@7566: *)
wenzelm@7535: 
wenzelm@9035: header {* Bounds *}
wenzelm@7808: 
haftmann@25596: theory Bounds
wenzelm@41413: imports Main "~~/src/HOL/Library/ContNotDenum"
haftmann@25596: begin
wenzelm@7535: 
wenzelm@13515: locale lub =
wenzelm@13515:   fixes A and x
wenzelm@13515:   assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b"
wenzelm@13515:     and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
wenzelm@13515: 
wenzelm@13515: lemmas [elim?] = lub.least lub.upper
wenzelm@13515: 
wenzelm@44887: definition the_lub :: "'a::order set \<Rightarrow> 'a"
wenzelm@44887:   where "the_lub A = The (lub A)"
wenzelm@14653: 
wenzelm@21210: notation (xsymbols)
wenzelm@19736:   the_lub  ("\<Squnion>_" [90] 90)
wenzelm@7535: 
wenzelm@13515: lemma the_lub_equality [elim?]:
ballarin@27611:   assumes "lub A x"
wenzelm@13515:   shows "\<Squnion>A = (x::'a::order)"
ballarin@27611: proof -
ballarin@29234:   interpret lub A x by fact
wenzelm@27612:   show ?thesis
wenzelm@27612:   proof (unfold the_lub_def)
ballarin@27611:     from `lub A x` show "The (lub A) = x"
ballarin@27611:     proof
ballarin@27611:       fix x' assume lub': "lub A x'"
ballarin@27611:       show "x' = x"
ballarin@27611:       proof (rule order_antisym)
wenzelm@32960:         from lub' show "x' \<le> x"
wenzelm@32960:         proof
ballarin@27611:           fix a assume "a \<in> A"
ballarin@27611:           then show "a \<le> x" ..
wenzelm@32960:         qed
wenzelm@32960:         show "x \<le> x'"
wenzelm@32960:         proof
ballarin@27611:           fix a assume "a \<in> A"
ballarin@27611:           with lub' show "a \<le> x'" ..
wenzelm@32960:         qed
wenzelm@13515:       qed
wenzelm@13515:     qed
wenzelm@13515:   qed
wenzelm@13515: qed
wenzelm@7917: 
wenzelm@13515: lemma the_lubI_ex:
wenzelm@13515:   assumes ex: "\<exists>x. lub A x"
wenzelm@13515:   shows "lub A (\<Squnion>A)"
wenzelm@13515: proof -
wenzelm@13515:   from ex obtain x where x: "lub A x" ..
wenzelm@13515:   also from x have [symmetric]: "\<Squnion>A = x" ..
wenzelm@13515:   finally show ?thesis .
wenzelm@13515: qed
wenzelm@7917: 
wenzelm@13515: lemma lub_compat: "lub A x = isLub UNIV A x"
wenzelm@13515: proof -
wenzelm@13515:   have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
wenzelm@13515:     by (rule ext) (simp only: isUb_def)
wenzelm@13515:   then show ?thesis
wenzelm@13515:     by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
wenzelm@9035: qed
wenzelm@13515: 
wenzelm@13515: lemma real_complete:
wenzelm@13515:   fixes A :: "real set"
wenzelm@13515:   assumes nonempty: "\<exists>a. a \<in> A"
wenzelm@13515:     and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
wenzelm@13515:   shows "\<exists>x. lub A x"
wenzelm@13515: proof -
wenzelm@13515:   from ex_upper have "\<exists>y. isUb UNIV A y"
wenzelm@27612:     unfolding isUb_def setle_def by blast
wenzelm@13515:   with nonempty have "\<exists>x. isLub UNIV A x"
wenzelm@13515:     by (rule reals_complete)
wenzelm@13515:   then show ?thesis by (simp only: lub_compat)
wenzelm@13515: qed
wenzelm@13515: 
wenzelm@10687: end