src/HOL/Hahn_Banach/Bounds.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (20 months ago) changeset 67003 49850a679c2c parent 66453 cc19f7ca2ed6 permissions -rw-r--r--
more robust sorted_entries;
```     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 section \<open>Bounds\<close>
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```     6
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```     7 theory Bounds
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```     8 imports Main "HOL-Analysis.Continuum_Not_Denumerable"
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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"
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```    14     and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
```
```    15
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```    16 lemmas [elim?] = lub.least lub.upper
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```    17
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```    18 definition the_lub :: "'a::order set \<Rightarrow> 'a"  ("\<Squnion>_"  90)
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```    19   where "the_lub A = The (lub A)"
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```    20
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```    21 lemma the_lub_equality [elim?]:
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```    22   assumes "lub A x"
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```    23   shows "\<Squnion>A = (x::'a::order)"
```
```    24 proof -
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```    25   interpret lub A x by fact
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```    26   show ?thesis
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```    27   proof (unfold the_lub_def)
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```    28     from \<open>lub A x\<close> show "The (lub A) = x"
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```    29     proof
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```    30       fix x' assume lub': "lub A x'"
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```    31       show "x' = x"
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```    32       proof (rule order_antisym)
```
```    33         from lub' show "x' \<le> x"
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```    34         proof
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```    35           fix a assume "a \<in> A"
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```    36           then show "a \<le> x" ..
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```    37         qed
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```    38         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           with lub' show "a \<le> x'" ..
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```    42         qed
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```    43       qed
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```    44     qed
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```    45   qed
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```    46 qed
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```    47
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```    48 lemma the_lubI_ex:
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```    49   assumes ex: "\<exists>x. lub A x"
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```    50   shows "lub A (\<Squnion>A)"
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```    51 proof -
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```    52   from ex obtain x where x: "lub A x" ..
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```    53   also from x have [symmetric]: "\<Squnion>A = x" ..
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```    54   finally show ?thesis .
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```    55 qed
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```    56
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```    57 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"
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```    58   by (intro exI[of _ "Sup A"]) (auto intro!: cSup_upper cSup_least simp: lub_def)
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```    59
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```    60 end
```