src/HOL/Typedef.thy
 author huffman Sat Jun 06 09:11:12 2009 -0700 (2009-06-06) changeset 31488 5691ccb8d6b5 parent 29797 08ef36ed2f8a child 31723 f5cafe803b55 permissions -rw-r--r--
generalize tendsto to class topological_space
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     Author:     Markus Wenzel, TU Munich
```
```     3 *)
```
```     4
```
```     5 header {* HOL type definitions *}
```
```     6
```
```     7 theory Typedef
```
```     8 imports Set
```
```     9 uses
```
```    10   ("Tools/typedef_package.ML")
```
```    11   ("Tools/typecopy_package.ML")
```
```    12   ("Tools/typedef_codegen.ML")
```
```    13 begin
```
```    14
```
```    15 ML {*
```
```    16 structure HOL = struct val thy = theory "HOL" end;
```
```    17 *}  -- "belongs to theory HOL"
```
```    18
```
```    19 locale type_definition =
```
```    20   fixes Rep and Abs and A
```
```    21   assumes Rep: "Rep x \<in> A"
```
```    22     and Rep_inverse: "Abs (Rep x) = x"
```
```    23     and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
```
```    24   -- {* This will be axiomatized for each typedef! *}
```
```    25 begin
```
```    26
```
```    27 lemma Rep_inject:
```
```    28   "(Rep x = Rep y) = (x = y)"
```
```    29 proof
```
```    30   assume "Rep x = Rep y"
```
```    31   then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
```
```    32   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    33   moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
```
```    34   ultimately show "x = y" by simp
```
```    35 next
```
```    36   assume "x = y"
```
```    37   thus "Rep x = Rep y" by (simp only:)
```
```    38 qed
```
```    39
```
```    40 lemma Abs_inject:
```
```    41   assumes x: "x \<in> A" and y: "y \<in> A"
```
```    42   shows "(Abs x = Abs y) = (x = y)"
```
```    43 proof
```
```    44   assume "Abs x = Abs y"
```
```    45   then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
```
```    46   moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)
```
```    47   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    48   ultimately show "x = y" by simp
```
```    49 next
```
```    50   assume "x = y"
```
```    51   thus "Abs x = Abs y" by (simp only:)
```
```    52 qed
```
```    53
```
```    54 lemma Rep_cases [cases set]:
```
```    55   assumes y: "y \<in> A"
```
```    56     and hyp: "!!x. y = Rep x ==> P"
```
```    57   shows P
```
```    58 proof (rule hyp)
```
```    59   from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    60   thus "y = Rep (Abs y)" ..
```
```    61 qed
```
```    62
```
```    63 lemma Abs_cases [cases type]:
```
```    64   assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
```
```    65   shows P
```
```    66 proof (rule r)
```
```    67   have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    68   thus "x = Abs (Rep x)" ..
```
```    69   show "Rep x \<in> A" by (rule Rep)
```
```    70 qed
```
```    71
```
```    72 lemma Rep_induct [induct set]:
```
```    73   assumes y: "y \<in> A"
```
```    74     and hyp: "!!x. P (Rep x)"
```
```    75   shows "P y"
```
```    76 proof -
```
```    77   have "P (Rep (Abs y))" by (rule hyp)
```
```    78   moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
```
```    79   ultimately show "P y" by simp
```
```    80 qed
```
```    81
```
```    82 lemma Abs_induct [induct type]:
```
```    83   assumes r: "!!y. y \<in> A ==> P (Abs y)"
```
```    84   shows "P x"
```
```    85 proof -
```
```    86   have "Rep x \<in> A" by (rule Rep)
```
```    87   then have "P (Abs (Rep x))" by (rule r)
```
```    88   moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
```
```    89   ultimately show "P x" by simp
```
```    90 qed
```
```    91
```
```    92 lemma Rep_range: "range Rep = A"
```
```    93 proof
```
```    94   show "range Rep <= A" using Rep by (auto simp add: image_def)
```
```    95   show "A <= range Rep"
```
```    96   proof
```
```    97     fix x assume "x : A"
```
```    98     hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
```
```    99     thus "x : range Rep" by (rule range_eqI)
```
```   100   qed
```
```   101 qed
```
```   102
```
```   103 lemma Abs_image: "Abs ` A = UNIV"
```
```   104 proof
```
```   105   show "Abs ` A <= UNIV" by (rule subset_UNIV)
```
```   106 next
```
```   107   show "UNIV <= Abs ` A"
```
```   108   proof
```
```   109     fix x
```
```   110     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
```
```   111     moreover have "Rep x : A" by (rule Rep)
```
```   112     ultimately show "x : Abs ` A" by (rule image_eqI)
```
```   113   qed
```
```   114 qed
```
```   115
```
```   116 end
```
```   117
```
```   118 use "Tools/typedef_package.ML" setup TypedefPackage.setup
```
```   119 use "Tools/typecopy_package.ML" setup TypecopyPackage.setup
```
```   120 use "Tools/typedef_codegen.ML" setup TypedefCodegen.setup
```
```   121
```
```   122 end
```