src/HOL/Typedef.thy
 author wenzelm Thu, 03 Sep 2015 17:14:57 +0200 changeset 61102 0ec9fd8d8119 parent 60758 d8d85a8172b5 child 61799 4cf66f21b764 permissions -rw-r--r--
misc tuning and modernization;
```
(*  Title:      HOL/Typedef.thy
Author:     Markus Wenzel, TU Munich
*)

section \<open>HOL type definitions\<close>

theory Typedef
imports Set
keywords "typedef" :: thy_goal and "morphisms"
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 \<Longrightarrow> Rep (Abs y) = y"
-- \<open>This will be axiomatized for each typedef!\<close>
begin

lemma Rep_inject: "Rep x = Rep y \<longleftrightarrow> 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 \<in> A" and "y \<in> A"
shows "Abs x = Abs y \<longleftrightarrow> x = y"
proof
assume "Abs x = Abs y"
then have "Rep (Abs x) = Rep (Abs y)" by (simp only:)
moreover from \<open>x \<in> A\<close> have "Rep (Abs x) = x" by (rule Abs_inverse)
moreover from \<open>y \<in> A\<close> 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 \<in> A"
and hyp: "\<And>x. y = Rep x \<Longrightarrow> P"
shows P
proof (rule hyp)
from \<open>y \<in> A\<close> have "Rep (Abs y) = y" by (rule Abs_inverse)
then show "y = Rep (Abs y)" ..
qed

lemma Abs_cases [cases type]:
assumes r: "\<And>y. x = Abs y \<Longrightarrow> y \<in> A \<Longrightarrow> P"
shows P
proof (rule r)
have "Abs (Rep x) = x" by (rule Rep_inverse)
then show "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: "\<And>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: "\<And>y. y \<in> A \<Longrightarrow> 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 \<subseteq> A" using Rep by (auto simp add: image_def)
show "A \<subseteq> range Rep"
proof
fix x assume "x \<in> A"
then have "x = Rep (Abs x)" by (rule Abs_inverse [symmetric])
then show "x \<in> range Rep" by (rule range_eqI)
qed
qed

lemma Abs_image: "Abs ` A = UNIV"
proof
show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
show "UNIV \<subseteq> Abs ` A"
proof
fix x
have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
moreover have "Rep x \<in> A" by (rule Rep)
ultimately show "x \<in> Abs ` A" by (rule image_eqI)
qed
qed

end

ML_file "Tools/typedef.ML"

end
```