src/HOL/Typedef.thy
 author wenzelm Wed Sep 17 21:27:14 2008 +0200 (2008-09-17) changeset 28263 69eaa97e7e96 parent 28084 a05ca48ef263 child 28394 b9c8e3a12a98 permissions -rw-r--r--
moved global ML bindings to global place;
```     1 (*  Title:      HOL/Typedef.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Markus Wenzel, TU Munich
```
```     4 *)
```
```     5
```
```     6 header {* HOL type definitions *}
```
```     7
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```     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
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```    15
```
```    16 ML {*
```
```    17 structure HOL = struct val thy = theory "HOL" end;
```
```    18 *}  -- "belongs to theory HOL"
```
```    19
```
```    20 locale type_definition =
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```    21   fixes Rep and Abs and A
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```    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:
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```    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
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```    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: "range Rep = A"
```
```    94 proof
```
```    95   show "range Rep <= A" using Rep by (auto simp add: image_def)
```
```    96   show "A <= range Rep"
```
```    97   proof
```
```    98     fix x assume "x : A"
```
```    99     hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
```
```   100     thus "x : range Rep" by (rule range_eqI)
```
```   101   qed
```
```   102 qed
```
```   103
```
```   104 lemma Abs_image: "Abs ` A = UNIV"
```
```   105 proof
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```   106   show "Abs ` A <= UNIV" by (rule subset_UNIV)
```
```   107 next
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```   108   show "UNIV <= Abs ` A"
```
```   109   proof
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```   110     fix x
```
```   111     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
```
```   112     moreover have "Rep x : A" by (rule Rep)
```
```   113     ultimately show "x : Abs ` A" by (rule image_eqI)
```
```   114   qed
```
```   115 qed
```
```   116
```
```   117 end
```
```   118
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```   119 use "Tools/typedef_package.ML"
```
```   120 use "Tools/typecopy_package.ML"
```
```   121 use "Tools/typedef_codegen.ML"
```
```   122
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```   123 setup {*
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```   124   TypedefPackage.setup
```
```   125   #> TypecopyPackage.setup
```
```   126   #> TypedefCodegen.setup
```
```   127 *}
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```   128
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```   129 text {* This class is just a workaround for classes without parameters;
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```   130   it shall disappear as soon as possible. *}
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```   131
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```   132 class itself = type +
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```   133   fixes itself :: "'a itself"
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```   134
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```   135 setup {*
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```   136 let fun add_itself tyco thy =
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```   137   let
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```   138     val vs = Name.names Name.context "'a"
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```   139       (replicate (Sign.arity_number thy tyco) @{sort type});
```
```   140     val ty = Type (tyco, map TFree vs);
```
```   141     val lhs = Const (@{const_name itself}, Term.itselfT ty);
```
```   142     val rhs = Logic.mk_type ty;
```
```   143     val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
```
```   144   in
```
```   145     thy
```
```   146     |> TheoryTarget.instantiation ([tyco], vs, @{sort itself})
```
```   147     |> `(fn lthy => Syntax.check_term lthy eq)
```
```   148     |-> (fn eq => Specification.definition (NONE, (Attrib.no_binding, eq)))
```
```   149     |> snd
```
```   150     |> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
```
```   151     |> LocalTheory.exit
```
```   152     |> ProofContext.theory_of
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```   153   end
```
```   154 in TypedefPackage.interpretation add_itself end
```
```   155 *}
```
```   156
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```   157 instantiation bool :: itself
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```   158 begin
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```   159
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```   160 definition "itself = TYPE(bool)"
```
```   161
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```   162 instance ..
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```   163
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```   164 end
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```   165
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```   166 instantiation "fun" :: ("type", "type") itself
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```   167 begin
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```   168
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```   169 definition "itself = TYPE('a \<Rightarrow> 'b)"
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```   170
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```   171 instance ..
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```   172
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```   173 end
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```   174
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```   175 hide (open) const itself
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```   176
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```   177 end
```