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 
   116 setup {*
   117   TypecopyPackage.setup
   118   #> TypedefCodegen.setup
   119 *}
   120 
   121 end