(*  Title:      HOL/Typedef.thy

     Author:     Markus Wenzel, TU Munich

 *)

 header {* HOL type definitions *}

 theory Typedef

 imports Set

 uses ("Tools/typedef.ML")

 begin

 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.ML" setup Typedef.setup

 end