(* 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 \ A"
and Rep_inverse: "Abs (Rep x) = x"
and Abs_inverse: "y \ 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 \ A" and y: "y \ 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 \ 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 \ A ==> P"
shows P
proof (rule r)
have "Abs (Rep x) = x" by (rule Rep_inverse)
thus "x = Abs (Rep x)" ..
show "Rep x \ A" by (rule Rep)
qed
lemma Rep_induct [induct set]:
assumes y: "y \ 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 \ A ==> P (Abs y)"
shows "P x"
proof -
have "Rep x \ 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:
shows "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
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, (("", []), eq)))
|> snd
|> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
|> LocalTheory.exit
|> ProofContext.theory_of
end
in TypedefPackage.interpretation add_itself end
*}
instantiation bool :: itself
begin
definition "itself = TYPE(bool)"
instance ..
end
instantiation "fun" :: ("type", "type") itself
begin
definition "itself = TYPE('a \ 'b)"
instance ..
end
instantiation "set" :: ("type") itself
begin
definition "itself = TYPE('a set)"
instance ..
end
hide (open) const itself
end