src/HOL/Typedef.thy
 author obua Mon Apr 10 16:00:34 2006 +0200 (2006-04-10) changeset 19404 9bf2cdc9e8e8 parent 16417 9bc16273c2d4 child 19459 2041d472fc17 permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Munich
```
```     4 *)
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```     5
```
```     6 header {* HOL type definitions *}
```
```     7
```
```     8 theory Typedef
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```     9 imports Set
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```    10 uses ("Tools/typedef_package.ML")
```
```    11 begin
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```    12
```
```    13 locale type_definition =
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```    14   fixes Rep and Abs and A
```
```    15   assumes Rep: "Rep x \<in> A"
```
```    16     and Rep_inverse: "Abs (Rep x) = x"
```
```    17     and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
```
```    18   -- {* This will be axiomatized for each typedef! *}
```
```    19
```
```    20 lemma (in type_definition) Rep_inject:
```
```    21   "(Rep x = Rep y) = (x = y)"
```
```    22 proof
```
```    23   assume "Rep x = Rep y"
```
```    24   hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
```
```    25   also have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    26   also have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    27   finally show "x = y" .
```
```    28 next
```
```    29   assume "x = y"
```
```    30   thus "Rep x = Rep y" by (simp only:)
```
```    31 qed
```
```    32
```
```    33 lemma (in type_definition) Abs_inject:
```
```    34   assumes x: "x \<in> A" and y: "y \<in> A"
```
```    35   shows "(Abs x = Abs y) = (x = y)"
```
```    36 proof
```
```    37   assume "Abs x = Abs y"
```
```    38   hence "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    39   also from x have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    40   also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    41   finally show "x = y" .
```
```    42 next
```
```    43   assume "x = y"
```
```    44   thus "Abs x = Abs y" by (simp only:)
```
```    45 qed
```
```    46
```
```    47 lemma (in type_definition) Rep_cases [cases set]:
```
```    48   assumes y: "y \<in> A"
```
```    49     and hyp: "!!x. y = Rep x ==> P"
```
```    50   shows P
```
```    51 proof (rule hyp)
```
```    52   from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    53   thus "y = Rep (Abs y)" ..
```
```    54 qed
```
```    55
```
```    56 lemma (in type_definition) Abs_cases [cases type]:
```
```    57   assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
```
```    58   shows P
```
```    59 proof (rule r)
```
```    60   have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    61   thus "x = Abs (Rep x)" ..
```
```    62   show "Rep x \<in> A" by (rule Rep)
```
```    63 qed
```
```    64
```
```    65 lemma (in type_definition) Rep_induct [induct set]:
```
```    66   assumes y: "y \<in> A"
```
```    67     and hyp: "!!x. P (Rep x)"
```
```    68   shows "P y"
```
```    69 proof -
```
```    70   have "P (Rep (Abs y))" by (rule hyp)
```
```    71   also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    72   finally show "P y" .
```
```    73 qed
```
```    74
```
```    75 lemma (in type_definition) Abs_induct [induct type]:
```
```    76   assumes r: "!!y. y \<in> A ==> P (Abs y)"
```
```    77   shows "P x"
```
```    78 proof -
```
```    79   have "Rep x \<in> A" by (rule Rep)
```
```    80   hence "P (Abs (Rep x))" by (rule r)
```
```    81   also have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    82   finally show "P x" .
```
```    83 qed
```
```    84
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```    85 use "Tools/typedef_package.ML"
```
```    86
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```    87 setup TypedefPackage.setup
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```    88
```
```    89 end
```