(*  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, ((Name.no_binding, []), 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 \<Rightarrow> 'b)"

 instance ..

 end

 hide (open) const itself

 end