Theory Typedef

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

(*  Title:      HOL/Typedef.thy
    ID:         $Id: Typedef.thy,v 1.21 2007/08/14 21:05:55 huffman Exp $
    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 {*
  TypecopyPackage.setup
  #> TypedefCodegen.setup
*}

end

lemma Rep_inject:

  (Rep x = Rep y) = (x = y)

lemma Abs_inject:

  [| xA; yA |] ==> (Abs x = Abs y) = (x = y)

lemma Rep_cases:

  [| yA; !!x. y = Rep x ==> P |] ==> P

lemma Abs_cases:

  (!!y. [| x = Abs y; yA |] ==> P) ==> P

lemma Rep_induct:

  [| yA; !!x. P (Rep x) |] ==> P y

lemma Abs_induct:

  (!!y. yA ==> P (Abs y)) ==> P x

lemma Rep_range:

  range Rep = A