src/HOL/Typedef.thy
 author wenzelm Mon, 29 Sep 2008 10:58:01 +0200 changeset 28394 b9c8e3a12a98 parent 28084 a05ca48ef263 child 28965 1de908189869 permissions -rw-r--r--
LocalTheory.exit_global;
```
(*  Title:      HOL/Typedef.thy
ID:         \$Id\$
Author:     Markus Wenzel, TU Munich
*)

header {* HOL type definitions *}

theory Typedef
imports Set
uses
("Tools/typedef_package.ML")
("Tools/typecopy_package.ML")
("Tools/typedef_codegen.ML")
begin

ML {*
structure HOL = struct val thy = theory "HOL" end;
*}  -- "belongs to theory HOL"

locale type_definition =
fixes Rep and Abs and A
assumes Rep: "Rep x \<in> A"
and Rep_inverse: "Abs (Rep x) = x"
and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
-- {* This will be axiomatized for each typedef! *}
begin

lemma Rep_inject:
"(Rep x = Rep y) = (x = y)"
proof
assume "Rep x = Rep y"
then have "Abs (Rep x) = Abs (Rep y)" by (simp only:)
moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
moreover have "Abs (Rep y) = y" by (rule Rep_inverse)
ultimately show "x = y" by simp
next
assume "x = y"
thus "Rep x = Rep y" by (simp only:)
qed

lemma Abs_inject:
assumes x: "x \<in> A" and y: "y \<in> A"
shows "(Abs x = Abs y) = (x = y)"
proof
assume "Abs x = Abs y"
then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse)
moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
ultimately show "x = y" by simp
next
assume "x = y"
thus "Abs x = Abs y" by (simp only:)
qed

lemma Rep_cases [cases set]:
assumes y: "y \<in> A"
and hyp: "!!x. y = Rep x ==> P"
shows P
proof (rule hyp)
from y have "Rep (Abs y) = y" by (rule Abs_inverse)
thus "y = Rep (Abs y)" ..
qed

lemma Abs_cases [cases type]:
assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
shows P
proof (rule r)
have "Abs (Rep x) = x" by (rule Rep_inverse)
thus "x = Abs (Rep x)" ..
show "Rep x \<in> A" by (rule Rep)
qed

lemma Rep_induct [induct set]:
assumes y: "y \<in> A"
and hyp: "!!x. P (Rep x)"
shows "P y"
proof -
have "P (Rep (Abs y))" by (rule hyp)
moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
ultimately show "P y" by simp
qed

lemma Abs_induct [induct type]:
assumes r: "!!y. y \<in> A ==> P (Abs y)"
shows "P x"
proof -
have "Rep x \<in> A" by (rule Rep)
then have "P (Abs (Rep x))" by (rule r)
moreover have "Abs (Rep x) = x" by (rule Rep_inverse)
ultimately show "P x" by simp
qed

lemma Rep_range: "range Rep = A"
proof
show "range Rep <= A" using Rep by (auto simp add: image_def)
show "A <= range Rep"
proof
fix x assume "x : A"
hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
thus "x : range Rep" by (rule range_eqI)
qed
qed

lemma Abs_image: "Abs ` A = UNIV"
proof
show "Abs ` A <= UNIV" by (rule subset_UNIV)
next
show "UNIV <= Abs ` A"
proof
fix x
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x : A" by (rule Rep)
ultimately show "x : Abs ` A" by (rule image_eqI)
qed
qed

end

use "Tools/typedef_package.ML"
use "Tools/typecopy_package.ML"
use "Tools/typedef_codegen.ML"

setup {*
TypedefPackage.setup
#> TypecopyPackage.setup
#> TypedefCodegen.setup
*}

text {* This class is just a workaround for classes without parameters;
it shall disappear as soon as possible. *}

class itself = type +
fixes itself :: "'a itself"

setup {*
let fun add_itself tyco thy =
let
val vs = Name.names Name.context "'a"
(replicate (Sign.arity_number thy tyco) @{sort type});
val ty = Type (tyco, map TFree vs);
val lhs = Const (@{const_name itself}, Term.itselfT ty);
val rhs = Logic.mk_type ty;
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
in
thy
|> TheoryTarget.instantiation ([tyco], vs, @{sort itself})
|> `(fn lthy => Syntax.check_term lthy eq)
|-> (fn eq => Specification.definition (NONE, (Attrib.no_binding, eq)))
|> snd
|> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
|> LocalTheory.exit_global
end
*}

instantiation bool :: itself
begin

definition "itself = TYPE(bool)"

instance ..

end

instantiation "fun" :: ("type", "type") itself
begin

definition "itself = TYPE('a \<Rightarrow> 'b)"

instance ..

end

hide (open) const itself

end
```