# Theory Typedef

```(*  Title:      HOL/Typedef.thy
Author:     Markus Wenzel, TU Munich
*)

section ‹HOL type definitions›

theory Typedef
imports Set
keywords
"typedef" :: thy_goal_defn and
"morphisms" :: quasi_command
begin

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"
then show "Rep x = Rep y" by (simp only:)
qed

lemma Abs_inject:
assumes "x ∈ A" and "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 ∈ A› have "Rep (Abs x) = x" by (rule Abs_inverse)
moreover from ‹y ∈ A› have "Rep (Abs y) = y" by (rule Abs_inverse)
ultimately show "x = y" by simp
next
assume "x = y"
then show "Abs x = Abs y" by (simp only:)
qed

lemma Rep_cases [cases set]:
assumes "y ∈ A"
and hyp: "⋀x. y = Rep x ⟹ P"
shows P
proof (rule hyp)
from ‹y ∈ A› have "Rep (Abs y) = y" by (rule Abs_inverse)
then show "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)
then show "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: "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"
then have "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
then show "x ∈ range Rep" by (rule range_eqI)
qed
qed

lemma Abs_image: "Abs ` A = UNIV"
proof
show "Abs ` A ⊆ UNIV" by (rule subset_UNIV)
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

ML_file ‹Tools/typedef.ML›

end
```