1 (* Title: HOL/Real/HahnBanach/Bounds.thy |
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2 ID: $Id$ |
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3 Author: Gertrud Bauer, TU Munich |
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4 *) |
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5 |
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6 header {* Bounds *} |
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7 |
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8 theory Bounds |
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9 imports Main ContNotDenum |
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10 begin |
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11 |
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12 locale lub = |
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13 fixes A and x |
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14 assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b" |
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15 and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x" |
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16 |
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17 lemmas [elim?] = lub.least lub.upper |
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18 |
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19 definition |
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20 the_lub :: "'a::order set \<Rightarrow> 'a" where |
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21 "the_lub A = The (lub A)" |
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22 |
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23 notation (xsymbols) |
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24 the_lub ("\<Squnion>_" [90] 90) |
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25 |
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26 lemma the_lub_equality [elim?]: |
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27 assumes "lub A x" |
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28 shows "\<Squnion>A = (x::'a::order)" |
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29 proof - |
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30 interpret lub [A x] by fact |
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31 show ?thesis |
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32 proof (unfold the_lub_def) |
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33 from `lub A x` show "The (lub A) = x" |
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34 proof |
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35 fix x' assume lub': "lub A x'" |
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36 show "x' = x" |
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37 proof (rule order_antisym) |
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38 from lub' show "x' \<le> x" |
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39 proof |
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40 fix a assume "a \<in> A" |
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41 then show "a \<le> x" .. |
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42 qed |
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43 show "x \<le> x'" |
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44 proof |
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45 fix a assume "a \<in> A" |
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46 with lub' show "a \<le> x'" .. |
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47 qed |
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48 qed |
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49 qed |
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50 qed |
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51 qed |
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52 |
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53 lemma the_lubI_ex: |
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54 assumes ex: "\<exists>x. lub A x" |
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55 shows "lub A (\<Squnion>A)" |
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56 proof - |
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57 from ex obtain x where x: "lub A x" .. |
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58 also from x have [symmetric]: "\<Squnion>A = x" .. |
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59 finally show ?thesis . |
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60 qed |
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61 |
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62 lemma lub_compat: "lub A x = isLub UNIV A x" |
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63 proof - |
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64 have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)" |
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65 by (rule ext) (simp only: isUb_def) |
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66 then show ?thesis |
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67 by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast |
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68 qed |
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69 |
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70 lemma real_complete: |
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71 fixes A :: "real set" |
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72 assumes nonempty: "\<exists>a. a \<in> A" |
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73 and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y" |
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74 shows "\<exists>x. lub A x" |
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75 proof - |
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76 from ex_upper have "\<exists>y. isUb UNIV A y" |
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77 unfolding isUb_def setle_def by blast |
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78 with nonempty have "\<exists>x. isLub UNIV A x" |
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79 by (rule reals_complete) |
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80 then show ?thesis by (simp only: lub_compat) |
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81 qed |
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82 |
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83 end |
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