src/HOL/Hahn_Banach/Bounds.thy
 author nipkow Fri Aug 28 18:52:41 2009 +0200 (2009-08-28) changeset 32436 10cd49e0c067 parent 31795 be3e1cc5005c child 32960 69916a850301 permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
```     1 (*  Title:      HOL/Hahn_Banach/Bounds.thy
```
```     2     Author:     Gertrud Bauer, TU Munich
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```     3 *)
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```     4
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```     5 header {* Bounds *}
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```     6
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```     7 theory Bounds
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```     8 imports Main 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
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```    17
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```    18 definition
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```    19   the_lub :: "'a::order set \<Rightarrow> 'a" where
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```    20   "the_lub A = The (lub A)"
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```    21
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```    22 notation (xsymbols)
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```    23   the_lub  ("\<Squnion>_"  90)
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```    24
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```    25 lemma the_lub_equality [elim?]:
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```    26   assumes "lub A x"
```
```    27   shows "\<Squnion>A = (x::'a::order)"
```
```    28 proof -
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```    29   interpret lub A x by fact
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```    30   show ?thesis
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```    31   proof (unfold the_lub_def)
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```    32     from `lub A x` show "The (lub A) = x"
```
```    33     proof
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```    34       fix x' assume lub': "lub A x'"
```
```    35       show "x' = x"
```
```    36       proof (rule order_antisym)
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```    37 	from lub' show "x' \<le> x"
```
```    38 	proof
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```    39           fix a assume "a \<in> A"
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```    40           then show "a \<le> x" ..
```
```    41 	qed
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```    42 	show "x \<le> x'"
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```    43 	proof
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```    44           fix a assume "a \<in> A"
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```    45           with lub' show "a \<le> x'" ..
```
```    46 	qed
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```    47       qed
```
```    48     qed
```
```    49   qed
```
```    50 qed
```
```    51
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```    52 lemma the_lubI_ex:
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```    53   assumes ex: "\<exists>x. lub A x"
```
```    54   shows "lub A (\<Squnion>A)"
```
```    55 proof -
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```    56   from ex obtain x where x: "lub A x" ..
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```    57   also from x have [symmetric]: "\<Squnion>A = x" ..
```
```    58   finally show ?thesis .
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```    59 qed
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```    60
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```    61 lemma lub_compat: "lub A x = isLub UNIV A x"
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```    62 proof -
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```    63   have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
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```    64     by (rule ext) (simp only: isUb_def)
```
```    65   then show ?thesis
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```    66     by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
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```    67 qed
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```    68
```
```    69 lemma real_complete:
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```    70   fixes A :: "real set"
```
```    71   assumes nonempty: "\<exists>a. a \<in> A"
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```    72     and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
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```    73   shows "\<exists>x. lub A x"
```
```    74 proof -
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```    75   from ex_upper have "\<exists>y. isUb UNIV A y"
```
```    76     unfolding isUb_def setle_def by blast
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```    77   with nonempty have "\<exists>x. isLub UNIV A x"
```
```    78     by (rule reals_complete)
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```    79   then show ?thesis by (simp only: lub_compat)
```
```    80 qed
```
```    81
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```    82 end
```