author | wenzelm |
Fri, 17 Nov 2006 02:20:03 +0100 | |
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parent 21210 | c17fd2df4e9e |
child 25596 | ad9e3594f3f3 |
permissions | -rw-r--r-- |
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(* Title: HOL/Real/HahnBanach/Bounds.thy |
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ID: $Id$ |
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Author: Gertrud Bauer, TU Munich |
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*) |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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header {* Bounds *} |
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theory Bounds imports Main Real begin |
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The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
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locale lub = |
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fixes A and x |
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assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b" |
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and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x" |
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lemmas [elim?] = lub.least lub.upper |
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definition |
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more robust syntax for definition/abbreviation/notation;
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the_lub :: "'a::order set \<Rightarrow> 'a" where |
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"the_lub A = The (lub A)" |
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notation (xsymbols) |
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the_lub ("\<Squnion>_" [90] 90) |
7535
599d3414b51d
The Hahn-Banach theorem for real vectorspaces (Isabelle/Isar)
wenzelm
parents:
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changeset
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lemma the_lub_equality [elim?]: |
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includes lub |
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shows "\<Squnion>A = (x::'a::order)" |
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proof (unfold the_lub_def) |
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from lub_axioms show "The (lub A) = x" |
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proof |
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fix x' assume lub': "lub A x'" |
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show "x' = x" |
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proof (rule order_antisym) |
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from lub' show "x' \<le> x" |
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proof |
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fix a assume "a \<in> A" |
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then show "a \<le> x" .. |
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qed |
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show "x \<le> x'" |
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proof |
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fix a assume "a \<in> A" |
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with lub' show "a \<le> x'" .. |
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qed |
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qed |
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qed |
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qed |
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lemma the_lubI_ex: |
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assumes ex: "\<exists>x. lub A x" |
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shows "lub A (\<Squnion>A)" |
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proof - |
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from ex obtain x where x: "lub A x" .. |
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also from x have [symmetric]: "\<Squnion>A = x" .. |
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finally show ?thesis . |
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qed |
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lemma lub_compat: "lub A x = isLub UNIV A x" |
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proof - |
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have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)" |
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by (rule ext) (simp only: isUb_def) |
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then show ?thesis |
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by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast |
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qed |
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lemma real_complete: |
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fixes A :: "real set" |
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assumes nonempty: "\<exists>a. a \<in> A" |
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and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y" |
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shows "\<exists>x. lub A x" |
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proof - |
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from ex_upper have "\<exists>y. isUb UNIV A y" |
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by (unfold isUb_def setle_def) blast |
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with nonempty have "\<exists>x. isLub UNIV A x" |
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by (rule reals_complete) |
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then show ?thesis by (simp only: lub_compat) |
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qed |
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end |