src/HOL/Hahn_Banach/Bounds.thy
 author wenzelm Tue Oct 21 10:58:19 2014 +0200 (2014-10-21) changeset 58744 c434e37f290e parent 54263 c4159fe6fa46 child 58745 5d452cf4bce7 permissions -rw-r--r--
update_cartouches;
```     1 (*  Title:      HOL/Hahn_Banach/Bounds.thy
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```     2     Author:     Gertrud Bauer, TU Munich
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```     3 *)
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```     4
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```     5 header \<open>Bounds\<close>
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```     6
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```     7 theory Bounds
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```     8 imports Main "~~/src/HOL/Library/ContNotDenum"
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```     9 begin
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```    10
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```    11 locale lub =
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```    12   fixes A and x
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```    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
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```    16 lemmas [elim?] = lub.least lub.upper
```
```    17
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```    18 definition the_lub :: "'a::order set \<Rightarrow> 'a"
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```    19   where "the_lub A = The (lub A)"
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```    20
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```    21 notation (xsymbols)
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```    22   the_lub  ("\<Squnion>_" [90] 90)
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```    23
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```    24 lemma the_lub_equality [elim?]:
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```    25   assumes "lub A x"
```
```    26   shows "\<Squnion>A = (x::'a::order)"
```
```    27 proof -
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```    28   interpret lub A x by fact
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```    29   show ?thesis
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```    30   proof (unfold the_lub_def)
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```    31     from \<open>lub A x\<close> show "The (lub A) = x"
```
```    32     proof
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```    33       fix x' assume lub': "lub A x'"
```
```    34       show "x' = x"
```
```    35       proof (rule order_antisym)
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```    36         from lub' show "x' \<le> x"
```
```    37         proof
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```    38           fix a assume "a \<in> A"
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```    39           then show "a \<le> x" ..
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```    40         qed
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```    41         show "x \<le> x'"
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```    42         proof
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```    43           fix a assume "a \<in> A"
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```    44           with lub' show "a \<le> x'" ..
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```    45         qed
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```    46       qed
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```    47     qed
```
```    48   qed
```
```    49 qed
```
```    50
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```    51 lemma the_lubI_ex:
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```    52   assumes ex: "\<exists>x. lub A x"
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```    53   shows "lub A (\<Squnion>A)"
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```    54 proof -
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```    55   from ex obtain x where x: "lub A x" ..
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```    56   also from x have [symmetric]: "\<Squnion>A = x" ..
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```    57   finally show ?thesis .
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```    58 qed
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```    59
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```    60 lemma real_complete: "\<exists>a::real. a \<in> A \<Longrightarrow> \<exists>y. \<forall>a \<in> A. a \<le> y \<Longrightarrow> \<exists>x. lub A x"
```
```    61   by (intro exI[of _ "Sup A"]) (auto intro!: cSup_upper cSup_least simp: lub_def)
```
```    62
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```    63 end
```