--- a/src/HOL/Typedef.thy Thu Sep 03 16:41:43 2015 +0200
+++ b/src/HOL/Typedef.thy Thu Sep 03 17:14:57 2015 +0200
@@ -13,12 +13,11 @@
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"
+ 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) = (x = y)"
+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:)
@@ -27,44 +26,44 @@
ultimately show "x = y" by simp
next
assume "x = y"
- thus "Rep x = Rep y" by (simp only:)
+ then show "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)"
+ 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 x have "Rep (Abs x) = x" by (rule Abs_inverse)
- moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse)
+ 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"
- thus "Abs x = Abs y" by (simp only:)
+ then show "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"
+ assumes "y \<in> A"
+ and hyp: "\<And>x. y = Rep x \<Longrightarrow> P"
shows P
proof (rule hyp)
- from y have "Rep (Abs y) = y" by (rule Abs_inverse)
- thus "y = Rep (Abs y)" ..
+ 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: "!!y. x = Abs y ==> y \<in> A ==> P"
+ 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)
- thus "x = Abs (Rep x)" ..
+ 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: "!!x. P (Rep x)"
+ and hyp: "\<And>x. P (Rep x)"
shows "P y"
proof -
have "P (Rep (Abs y))" by (rule hyp)
@@ -73,7 +72,7 @@
qed
lemma Abs_induct [induct type]:
- assumes r: "!!y. y \<in> A ==> P (Abs y)"
+ assumes r: "\<And>y. y \<in> A \<Longrightarrow> P (Abs y)"
shows "P x"
proof -
have "Rep x \<in> A" by (rule Rep)
@@ -84,25 +83,24 @@
lemma Rep_range: "range Rep = A"
proof
- show "range Rep <= A" using Rep by (auto simp add: image_def)
- show "A <= range Rep"
+ show "range Rep \<subseteq> A" using Rep by (auto simp add: image_def)
+ show "A \<subseteq> 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)
+ 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 <= UNIV" by (rule subset_UNIV)
-next
- show "UNIV <= Abs ` A"
+ 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 : A" by (rule Rep)
- ultimately show "x : Abs ` A" by (rule image_eqI)
+ moreover have "Rep x \<in> A" by (rule Rep)
+ ultimately show "x \<in> Abs ` A" by (rule image_eqI)
qed
qed