src/HOL/Typedef.thy
 author haftmann Tue Aug 29 14:31:13 2006 +0200 (2006-08-29) changeset 20426 9ffea7a8b31c parent 19459 2041d472fc17 child 22846 fb79144af9a3 permissions -rw-r--r--
```     1 (*  Title:      HOL/Typedef.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Markus Wenzel, TU Munich
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```     4 *)
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```     5
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```     6 header {* HOL type definitions *}
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```     7
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```     8 theory Typedef
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```     9 imports Set
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```    10 uses
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```    11   ("Tools/typedef_package.ML")
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```    12   ("Tools/typecopy_package.ML")
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```    13   ("Tools/typedef_codegen.ML")
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```    14 begin
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```    15
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```    16 locale type_definition =
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```    17   fixes Rep and Abs and A
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```    18   assumes Rep: "Rep x \<in> A"
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```    19     and Rep_inverse: "Abs (Rep x) = x"
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```    20     and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
```
```    21   -- {* This will be axiomatized for each typedef! *}
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```    22
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```    23 lemma (in type_definition) Rep_inject:
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```    24   "(Rep x = Rep y) = (x = y)"
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```    25 proof
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```    26   assume "Rep x = Rep y"
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```    27   hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
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```    28   also have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    29   also have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    30   finally show "x = y" .
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```    31 next
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```    32   assume "x = y"
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```    33   thus "Rep x = Rep y" by (simp only:)
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```    34 qed
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```    35
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```    36 lemma (in type_definition) Abs_inject:
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```    37   assumes x: "x \<in> A" and y: "y \<in> A"
```
```    38   shows "(Abs x = Abs y) = (x = y)"
```
```    39 proof
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```    40   assume "Abs x = Abs y"
```
```    41   hence "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    42   also from x have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    43   also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    44   finally show "x = y" .
```
```    45 next
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```    46   assume "x = y"
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```    47   thus "Abs x = Abs y" by (simp only:)
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```    48 qed
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```    49
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```    50 lemma (in type_definition) Rep_cases [cases set]:
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```    51   assumes y: "y \<in> A"
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```    52     and hyp: "!!x. y = Rep x ==> P"
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```    53   shows P
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```    54 proof (rule hyp)
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```    55   from y have "Rep (Abs y) = y" by (rule Abs_inverse)
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```    56   thus "y = Rep (Abs y)" ..
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```    57 qed
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```    58
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```    59 lemma (in type_definition) Abs_cases [cases type]:
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```    60   assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
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```    61   shows P
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```    62 proof (rule r)
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```    63   have "Abs (Rep x) = x" by (rule Rep_inverse)
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```    64   thus "x = Abs (Rep x)" ..
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```    65   show "Rep x \<in> A" by (rule Rep)
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```    66 qed
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```    67
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```    68 lemma (in type_definition) Rep_induct [induct set]:
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```    69   assumes y: "y \<in> A"
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```    70     and hyp: "!!x. P (Rep x)"
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```    71   shows "P y"
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```    72 proof -
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```    73   have "P (Rep (Abs y))" by (rule hyp)
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```    74   also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    75   finally show "P y" .
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```    76 qed
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```    77
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```    78 lemma (in type_definition) Abs_induct [induct type]:
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```    79   assumes r: "!!y. y \<in> A ==> P (Abs y)"
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```    80   shows "P x"
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```    81 proof -
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```    82   have "Rep x \<in> A" by (rule Rep)
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```    83   hence "P (Abs (Rep x))" by (rule r)
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```    84   also have "Abs (Rep x) = x" by (rule Rep_inverse)
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```    85   finally show "P x" .
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```    86 qed
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```    87
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```    88 use "Tools/typedef_package.ML"
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```    89 use "Tools/typecopy_package.ML"
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```    90 use "Tools/typedef_codegen.ML"
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```    91
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```    92 setup {*
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```    93   TypedefPackage.setup
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```    94   #> TypecopyPackage.setup
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```    95   #> TypedefCodegen.setup
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```    96 *}
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```    97
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```    98 end
```