src/HOL/Typedef.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 48891 c0eafbd55de3 child 58239 1c5bc387bd4c permissions -rw-r--r--
introduce order topology
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     Author:     Markus Wenzel, TU Munich
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```     3 *)
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```     4
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```     5 header {* HOL type definitions *}
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```     6
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```     7 theory Typedef
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```     8 imports Set
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```     9 keywords "typedef" :: thy_goal and "morphisms"
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```    10 begin
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```    11
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```    12 locale type_definition =
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```    13   fixes Rep and Abs and A
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```    14   assumes Rep: "Rep x \<in> A"
```
```    15     and Rep_inverse: "Abs (Rep x) = x"
```
```    16     and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
```
```    17   -- {* This will be axiomatized for each typedef! *}
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```    18 begin
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```    19
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```    20 lemma Rep_inject:
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```    21   "(Rep x = Rep y) = (x = y)"
```
```    22 proof
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```    23   assume "Rep x = Rep y"
```
```    24   then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
```
```    25   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    26   moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    27   ultimately show "x = y" by simp
```
```    28 next
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```    29   assume "x = y"
```
```    30   thus "Rep x = Rep y" by (simp only:)
```
```    31 qed
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```    32
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```    33 lemma Abs_inject:
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```    34   assumes x: "x \<in> A" and y: "y \<in> A"
```
```    35   shows "(Abs x = Abs y) = (x = y)"
```
```    36 proof
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```    37   assume "Abs x = Abs y"
```
```    38   then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    39   moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    40   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    41   ultimately show "x = y" by simp
```
```    42 next
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```    43   assume "x = y"
```
```    44   thus "Abs x = Abs y" by (simp only:)
```
```    45 qed
```
```    46
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```    47 lemma 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
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```    56 lemma Abs_cases [cases type]:
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```    57   assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
```
```    58   shows P
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```    59 proof (rule r)
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```    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
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```    65 lemma Rep_induct [induct set]:
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```    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   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    72   ultimately show "P y" by simp
```
```    73 qed
```
```    74
```
```    75 lemma 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   then have "P (Abs (Rep x))" by (rule r)
```
```    81   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    82   ultimately show "P x" by simp
```
```    83 qed
```
```    84
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```    85 lemma Rep_range: "range Rep = A"
```
```    86 proof
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```    87   show "range Rep <= A" using Rep by (auto simp add: image_def)
```
```    88   show "A <= range Rep"
```
```    89   proof
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```    90     fix x assume "x : A"
```
```    91     hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
```
```    92     thus "x : range Rep" by (rule range_eqI)
```
```    93   qed
```
```    94 qed
```
```    95
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```    96 lemma Abs_image: "Abs ` A = UNIV"
```
```    97 proof
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```    98   show "Abs ` A <= UNIV" by (rule subset_UNIV)
```
```    99 next
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```   100   show "UNIV <= Abs ` A"
```
```   101   proof
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```   102     fix x
```
```   103     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
```
```   104     moreover have "Rep x : A" by (rule Rep)
```
```   105     ultimately show "x : Abs ` A" by (rule image_eqI)
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```   106   qed
```
```   107 qed
```
```   108
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```   109 end
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```   110
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```   111 ML_file "Tools/typedef.ML" setup Typedef.setup
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```   112
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```   113 end
```