src/HOL/subset.thy
 author oheimb Wed Jan 31 10:15:55 2001 +0100 (2001-01-31) changeset 11008 f7333f055ef6 parent 10291 a88d347db404 child 11083 d8fda557e476 permissions -rw-r--r--
improved theory reference in comment
```     1 (*  Title:      HOL/subset.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1994  University of Cambridge
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```     5
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```     6 Subset lemmas and HOL type definitions.
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```     7 *)
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```     8
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```     9 theory subset = Set
```
```    10 files "Tools/induct_attrib.ML" ("Tools/typedef_package.ML"):
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```    11
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```    12 (*belongs to theory Ord*)
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```    13 theorems linorder_cases [case_names less equal greater] =
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```    14   linorder_less_split
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```    15
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```    16 (*belongs to theory Set*)
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```    17 setup Rulify.setup
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```    18
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```    19
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```    20 section {* HOL type definitions *}
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```    21
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```    22 constdefs
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```    23   type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool"
```
```    24   "type_definition Rep Abs A ==
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```    25     (\<forall>x. Rep x \<in> A) \<and>
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```    26     (\<forall>x. Abs (Rep x) = x) \<and>
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```    27     (\<forall>y \<in> A. Rep (Abs y) = y)"
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```    28   -- {* This will be stated as an axiom for each typedef! *}
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```    29
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```    30 lemma type_definitionI [intro]:
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```    31   "(!!x. Rep x \<in> A) ==>
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```    32     (!!x. Abs (Rep x) = x) ==>
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```    33     (!!y. y \<in> A ==> Rep (Abs y) = y) ==>
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```    34     type_definition Rep Abs A"
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```    35   by (unfold type_definition_def) blast
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```    36
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```    37 theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A"
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```    38   by (unfold type_definition_def) blast
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```    39
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```    40 theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x"
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```    41   by (unfold type_definition_def) blast
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```    42
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```    43 theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y"
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```    44   by (unfold type_definition_def) blast
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```    45
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```    46 theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)"
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```    47 proof -
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```    48   assume tydef: "type_definition Rep Abs A"
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```    49   show ?thesis
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```    50   proof
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```    51     assume "Rep x = Rep y"
```
```    52     hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
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```    53     thus "x = y" by (simp only: Rep_inverse [OF tydef])
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```    54   next
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```    55     assume "x = y"
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```    56     thus "Rep x = Rep y" by simp
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```    57   qed
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```    58 qed
```
```    59
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```    60 theorem Abs_inject:
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```    61   "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)"
```
```    62 proof -
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```    63   assume tydef: "type_definition Rep Abs A"
```
```    64   assume x: "x \<in> A" and y: "y \<in> A"
```
```    65   show ?thesis
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```    66   proof
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```    67     assume "Abs x = Abs y"
```
```    68     hence "Rep (Abs x) = Rep (Abs y)" by simp
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```    69     moreover note x hence "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])
```
```    70     moreover note y hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
```
```    71     ultimately show "x = y" by (simp only:)
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```    72   next
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```    73     assume "x = y"
```
```    74     thus "Abs x = Abs y" by simp
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```    75   qed
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```    76 qed
```
```    77
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```    78 theorem Rep_cases:
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```    79   "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P"
```
```    80 proof -
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```    81   assume tydef: "type_definition Rep Abs A"
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```    82   assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)"
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```    83   show P
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```    84   proof (rule r)
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```    85     from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
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```    86     thus "y = Rep (Abs y)" ..
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```    87   qed
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```    88 qed
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```    89
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```    90 theorem Abs_cases:
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```    91   "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P"
```
```    92 proof -
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```    93   assume tydef: "type_definition Rep Abs A"
```
```    94   assume r: "!!y. x = Abs y ==> y \<in> A ==> P"
```
```    95   show P
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```    96   proof (rule r)
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```    97     have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
```
```    98     thus "x = Abs (Rep x)" ..
```
```    99     show "Rep x \<in> A" by (rule Rep [OF tydef])
```
```   100   qed
```
```   101 qed
```
```   102
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```   103 theorem Rep_induct:
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```   104   "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y"
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```   105 proof -
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```   106   assume tydef: "type_definition Rep Abs A"
```
```   107   assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" .
```
```   108   moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
```
```   109   ultimately show "P y" by (simp only:)
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```   110 qed
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```   111
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```   112 theorem Abs_induct:
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```   113   "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x"
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```   114 proof -
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```   115   assume tydef: "type_definition Rep Abs A"
```
```   116   assume r: "!!y. y \<in> A ==> P (Abs y)"
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```   117   have "Rep x \<in> A" by (rule Rep [OF tydef])
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```   118   hence "P (Abs (Rep x))" by (rule r)
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```   119   moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
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```   120   ultimately show "P x" by (simp only:)
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```   121 qed
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```   122
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```   123 setup InductAttrib.setup
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```   124 use "Tools/typedef_package.ML"
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```   125
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```   126 end
```