src/HOL/Hahn_Banach/Bounds.thy
 author wenzelm Sun Sep 11 22:55:26 2011 +0200 (2011-09-11) changeset 44887 7ca82df6e951 parent 41413 64cd30d6b0b8 child 54263 c4159fe6fa46 permissions -rw-r--r--
misc tuning and clarification;
```     1 (*  Title:      HOL/Hahn_Banach/Bounds.thy
```
```     2     Author:     Gertrud Bauer, TU Munich
```
```     3 *)
```
```     4
```
```     5 header {* Bounds *}
```
```     6
```
```     7 theory Bounds
```
```     8 imports Main "~~/src/HOL/Library/ContNotDenum"
```
```     9 begin
```
```    10
```
```    11 locale lub =
```
```    12   fixes A and x
```
```    13   assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b"
```
```    14     and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
```
```    15
```
```    16 lemmas [elim?] = lub.least lub.upper
```
```    17
```
```    18 definition the_lub :: "'a::order set \<Rightarrow> 'a"
```
```    19   where "the_lub A = The (lub A)"
```
```    20
```
```    21 notation (xsymbols)
```
```    22   the_lub  ("\<Squnion>_"  90)
```
```    23
```
```    24 lemma the_lub_equality [elim?]:
```
```    25   assumes "lub A x"
```
```    26   shows "\<Squnion>A = (x::'a::order)"
```
```    27 proof -
```
```    28   interpret lub A x by fact
```
```    29   show ?thesis
```
```    30   proof (unfold the_lub_def)
```
```    31     from `lub A x` show "The (lub A) = x"
```
```    32     proof
```
```    33       fix x' assume lub': "lub A x'"
```
```    34       show "x' = x"
```
```    35       proof (rule order_antisym)
```
```    36         from lub' show "x' \<le> x"
```
```    37         proof
```
```    38           fix a assume "a \<in> A"
```
```    39           then show "a \<le> x" ..
```
```    40         qed
```
```    41         show "x \<le> x'"
```
```    42         proof
```
```    43           fix a assume "a \<in> A"
```
```    44           with lub' show "a \<le> x'" ..
```
```    45         qed
```
```    46       qed
```
```    47     qed
```
```    48   qed
```
```    49 qed
```
```    50
```
```    51 lemma the_lubI_ex:
```
```    52   assumes ex: "\<exists>x. lub A x"
```
```    53   shows "lub A (\<Squnion>A)"
```
```    54 proof -
```
```    55   from ex obtain x where x: "lub A x" ..
```
```    56   also from x have [symmetric]: "\<Squnion>A = x" ..
```
```    57   finally show ?thesis .
```
```    58 qed
```
```    59
```
```    60 lemma lub_compat: "lub A x = isLub UNIV A x"
```
```    61 proof -
```
```    62   have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
```
```    63     by (rule ext) (simp only: isUb_def)
```
```    64   then show ?thesis
```
```    65     by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
```
```    66 qed
```
```    67
```
```    68 lemma real_complete:
```
```    69   fixes A :: "real set"
```
```    70   assumes nonempty: "\<exists>a. a \<in> A"
```
```    71     and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
```
```    72   shows "\<exists>x. lub A x"
```
```    73 proof -
```
```    74   from ex_upper have "\<exists>y. isUb UNIV A y"
```
```    75     unfolding isUb_def setle_def by blast
```
```    76   with nonempty have "\<exists>x. isLub UNIV A x"
```
```    77     by (rule reals_complete)
```
```    78   then show ?thesis by (simp only: lub_compat)
```
```    79 qed
```
```    80
```
```    81 end
```