src/HOL/Typedef.thy
 author haftmann Tue Jul 10 17:30:51 2007 +0200 (2007-07-10) changeset 23710 a8ac2305eaf2 parent 23433 c2c10abd2a1e child 24269 4b2aac7669b3 permissions -rw-r--r--
removed proof dependency on transitivity theorems
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Munich
```
```     4 *)
```
```     5
```
```     6 header {* HOL type definitions *}
```
```     7
```
```     8 theory Typedef
```
```     9 imports Set
```
```    10 uses
```
```    11   ("Tools/typedef_package.ML")
```
```    12   ("Tools/typecopy_package.ML")
```
```    13   ("Tools/typedef_codegen.ML")
```
```    14 begin
```
```    15
```
```    16 ML {*
```
```    17 structure HOL = struct val thy = theory "HOL" end;
```
```    18 *}  -- "belongs to theory HOL"
```
```    19
```
```    20 locale type_definition =
```
```    21   fixes Rep and Abs and A
```
```    22   assumes Rep: "Rep x \<in> A"
```
```    23     and Rep_inverse: "Abs (Rep x) = x"
```
```    24     and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
```
```    25   -- {* This will be axiomatized for each typedef! *}
```
```    26 begin
```
```    27
```
```    28 lemma Rep_inject:
```
```    29   "(Rep x = Rep y) = (x = y)"
```
```    30 proof
```
```    31   assume "Rep x = Rep y"
```
```    32   then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
```
```    33   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    34   moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    35   ultimately show "x = y" by simp
```
```    36 next
```
```    37   assume "x = y"
```
```    38   thus "Rep x = Rep y" by (simp only:)
```
```    39 qed
```
```    40
```
```    41 lemma Abs_inject:
```
```    42   assumes x: "x \<in> A" and y: "y \<in> A"
```
```    43   shows "(Abs x = Abs y) = (x = y)"
```
```    44 proof
```
```    45   assume "Abs x = Abs y"
```
```    46   then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    47   moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    48   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    49   ultimately show "x = y" by simp
```
```    50 next
```
```    51   assume "x = y"
```
```    52   thus "Abs x = Abs y" by (simp only:)
```
```    53 qed
```
```    54
```
```    55 lemma Rep_cases [cases set]:
```
```    56   assumes y: "y \<in> A"
```
```    57     and hyp: "!!x. y = Rep x ==> P"
```
```    58   shows P
```
```    59 proof (rule hyp)
```
```    60   from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    61   thus "y = Rep (Abs y)" ..
```
```    62 qed
```
```    63
```
```    64 lemma Abs_cases [cases type]:
```
```    65   assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
```
```    66   shows P
```
```    67 proof (rule r)
```
```    68   have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    69   thus "x = Abs (Rep x)" ..
```
```    70   show "Rep x \<in> A" by (rule Rep)
```
```    71 qed
```
```    72
```
```    73 lemma Rep_induct [induct set]:
```
```    74   assumes y: "y \<in> A"
```
```    75     and hyp: "!!x. P (Rep x)"
```
```    76   shows "P y"
```
```    77 proof -
```
```    78   have "P (Rep (Abs y))" by (rule hyp)
```
```    79   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    80   ultimately show "P y" by simp
```
```    81 qed
```
```    82
```
```    83 lemma Abs_induct [induct type]:
```
```    84   assumes r: "!!y. y \<in> A ==> P (Abs y)"
```
```    85   shows "P x"
```
```    86 proof -
```
```    87   have "Rep x \<in> A" by (rule Rep)
```
```    88   then have "P (Abs (Rep x))" by (rule r)
```
```    89   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    90   ultimately show "P x" by simp
```
```    91 qed
```
```    92
```
```    93 lemma Rep_range:
```
```    94 assumes "type_definition Rep Abs A"
```
```    95 shows "range Rep = A"
```
```    96 proof -
```
```    97   from assms have A1: "!!x. Rep x : A"
```
```    98               and A2: "!!y. y : A ==> y = Rep(Abs y)"
```
```    99      by (auto simp add: type_definition_def)
```
```   100   have "range Rep <= A" using A1 by (auto simp add: image_def)
```
```   101   moreover have "A <= range Rep"
```
```   102   proof
```
```   103     fix x assume "x : A"
```
```   104     hence "x = Rep (Abs x)" by (rule A2)
```
```   105     thus "x : range Rep" by (auto simp add: image_def)
```
```   106   qed
```
```   107   ultimately show ?thesis ..
```
```   108 qed
```
```   109
```
```   110 end
```
```   111
```
```   112 use "Tools/typedef_package.ML"
```
```   113 use "Tools/typecopy_package.ML"
```
```   114 use "Tools/typedef_codegen.ML"
```
```   115
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```   116 setup {*
```
```   117   TypecopyPackage.setup
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```   118   #> TypedefCodegen.setup
```
```   119 *}
```
```   120
```
```   121 end
```