--- a/src/HOL/Library/Quotient_Type.thy Sun Dec 28 12:18:01 2014 +0100
+++ b/src/HOL/Library/Quotient_Type.thy Sun Dec 28 12:37:03 2014 +0100
@@ -2,63 +2,65 @@
Author: Markus Wenzel, TU Muenchen
*)
-section {* Quotient types *}
+section \<open>Quotient types\<close>
theory Quotient_Type
imports Main
begin
-text {*
- We introduce the notion of quotient types over equivalence relations
- via type classes.
-*}
+text \<open>We introduce the notion of quotient types over equivalence relations
+ via type classes.\<close>
+
-subsection {* Equivalence relations and quotient types *}
+subsection \<open>Equivalence relations and quotient types\<close>
-text {*
- \medskip Type class @{text equiv} models equivalence relations @{text
- "\<sim> :: 'a => 'a => bool"}.
-*}
+text \<open>Type class @{text equiv} models equivalence relations
+ @{text "\<sim> :: 'a \<Rightarrow> 'a \<Rightarrow> bool"}.\<close>
class eqv =
- fixes eqv :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sim>" 50)
+ fixes eqv :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sim>" 50)
class equiv = eqv +
assumes equiv_refl [intro]: "x \<sim> x"
- assumes equiv_trans [trans]: "x \<sim> y \<Longrightarrow> y \<sim> z \<Longrightarrow> x \<sim> z"
- assumes equiv_sym [sym]: "x \<sim> y \<Longrightarrow> y \<sim> x"
+ and equiv_trans [trans]: "x \<sim> y \<Longrightarrow> y \<sim> z \<Longrightarrow> x \<sim> z"
+ and equiv_sym [sym]: "x \<sim> y \<Longrightarrow> y \<sim> x"
+begin
-lemma equiv_not_sym [sym]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"
+lemma equiv_not_sym [sym]: "\<not> x \<sim> y \<Longrightarrow> \<not> y \<sim> x"
proof -
- assume "\<not> (x \<sim> y)" then show "\<not> (y \<sim> x)"
- by (rule contrapos_nn) (rule equiv_sym)
+ assume "\<not> x \<sim> y"
+ then show "\<not> y \<sim> x" by (rule contrapos_nn) (rule equiv_sym)
qed
-lemma not_equiv_trans1 [trans]: "\<not> (x \<sim> y) ==> y \<sim> z ==> \<not> (x \<sim> (z::'a::equiv))"
+lemma not_equiv_trans1 [trans]: "\<not> x \<sim> y \<Longrightarrow> y \<sim> z \<Longrightarrow> \<not> x \<sim> z"
proof -
- assume "\<not> (x \<sim> y)" and "y \<sim> z"
- show "\<not> (x \<sim> z)"
+ assume "\<not> x \<sim> y" and "y \<sim> z"
+ show "\<not> x \<sim> z"
proof
assume "x \<sim> z"
- also from `y \<sim> z` have "z \<sim> y" ..
+ also from \<open>y \<sim> z\<close> have "z \<sim> y" ..
finally have "x \<sim> y" .
- with `\<not> (x \<sim> y)` show False by contradiction
+ with \<open>\<not> x \<sim> y\<close> show False by contradiction
qed
qed
-lemma not_equiv_trans2 [trans]: "x \<sim> y ==> \<not> (y \<sim> z) ==> \<not> (x \<sim> (z::'a::equiv))"
+lemma not_equiv_trans2 [trans]: "x \<sim> y \<Longrightarrow> \<not> y \<sim> z \<Longrightarrow> \<not> x \<sim> z"
proof -
- assume "\<not> (y \<sim> z)" then have "\<not> (z \<sim> y)" ..
- also assume "x \<sim> y" then have "y \<sim> x" ..
- finally have "\<not> (z \<sim> x)" . then show "(\<not> x \<sim> z)" ..
+ assume "\<not> y \<sim> z"
+ then have "\<not> z \<sim> y" ..
+ also
+ assume "x \<sim> y"
+ then have "y \<sim> x" ..
+ finally have "\<not> z \<sim> x" .
+ then show "\<not> x \<sim> z" ..
qed
-text {*
- \medskip The quotient type @{text "'a quot"} consists of all
- \emph{equivalence classes} over elements of the base type @{typ 'a}.
-*}
+end
-definition "quot = {{x. a \<sim> x} | a::'a::eqv. True}"
+text \<open>The quotient type @{text "'a quot"} consists of all \emph{equivalence
+ classes} over elements of the base type @{typ 'a}.\<close>
+
+definition (in eqv) "quot = {{x. a \<sim> x} | a. True}"
typedef 'a quot = "quot :: 'a::eqv set set"
unfolding quot_def by blast
@@ -66,38 +68,38 @@
lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"
unfolding quot_def by blast
-lemma quotE [elim]: "R \<in> quot ==> (!!a. R = {x. a \<sim> x} ==> C) ==> C"
- unfolding quot_def by blast
+lemma quotE [elim]:
+ assumes "R \<in> quot"
+ obtains a where "R = {x. a \<sim> x}"
+ using assms unfolding quot_def by blast
-text {*
- \medskip Abstracted equivalence classes are the canonical
- representation of elements of a quotient type.
-*}
+text \<open>Abstracted equivalence classes are the canonical representation of
+ elements of a quotient type.\<close>
-definition
- "class" :: "'a::equiv => 'a quot" ("\<lfloor>_\<rfloor>") where
- "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"
+definition "class" :: "'a::equiv \<Rightarrow> 'a quot" ("\<lfloor>_\<rfloor>")
+ where "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"
theorem quot_exhaust: "\<exists>a. A = \<lfloor>a\<rfloor>"
proof (cases A)
- fix R assume R: "A = Abs_quot R"
- assume "R \<in> quot" then have "\<exists>a. R = {x. a \<sim> x}" by blast
+ fix R
+ assume R: "A = Abs_quot R"
+ assume "R \<in> quot"
+ then have "\<exists>a. R = {x. a \<sim> x}" by blast
with R have "\<exists>a. A = Abs_quot {x. a \<sim> x}" by blast
then show ?thesis unfolding class_def .
qed
-lemma quot_cases [cases type: quot]: "(!!a. A = \<lfloor>a\<rfloor> ==> C) ==> C"
+lemma quot_cases [cases type: quot]:
+ obtains a where "A = \<lfloor>a\<rfloor>"
using quot_exhaust by blast
-subsection {* Equality on quotients *}
+subsection \<open>Equality on quotients\<close>
-text {*
- Equality of canonical quotient elements coincides with the original
- relation.
-*}
+text \<open>Equality of canonical quotient elements coincides with the original
+ relation.\<close>
-theorem quot_equality [iff?]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
+theorem quot_equality [iff?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> \<longleftrightarrow> a \<sim> b"
proof
assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
show "a \<sim> b"
@@ -131,11 +133,10 @@
qed
-subsection {* Picking representing elements *}
+subsection \<open>Picking representing elements\<close>
-definition
- pick :: "'a::equiv quot => 'a" where
- "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"
+definition pick :: "'a::equiv quot \<Rightarrow> 'a"
+ where "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"
theorem pick_equiv [intro]: "pick \<lfloor>a\<rfloor> \<sim> a"
proof (unfold pick_def)
@@ -143,7 +144,8 @@
proof (rule someI2)
show "\<lfloor>a\<rfloor> = \<lfloor>a\<rfloor>" ..
fix x assume "\<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>"
- then have "a \<sim> x" .. then show "x \<sim> a" ..
+ then have "a \<sim> x" ..
+ then show "x \<sim> a" ..
qed
qed
@@ -155,17 +157,14 @@
with a show ?thesis by simp
qed
-text {*
- \medskip The following rules support canonical function definitions
- on quotient types (with up to two arguments). Note that the
- stripped-down version without additional conditions is sufficient
- most of the time.
-*}
+text \<open>The following rules support canonical function definitions on quotient
+ types (with up to two arguments). Note that the stripped-down version
+ without additional conditions is sufficient most of the time.\<close>
theorem quot_cond_function:
- assumes eq: "!!X Y. P X Y ==> f X Y == g (pick X) (pick Y)"
- and cong: "!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>
- ==> P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor> ==> P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor> ==> g x y = g x' y'"
+ assumes eq: "\<And>X Y. P X Y \<Longrightarrow> f X Y \<equiv> g (pick X) (pick Y)"
+ and cong: "\<And>x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> \<Longrightarrow> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>
+ \<Longrightarrow> P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor> \<Longrightarrow> P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor> \<Longrightarrow> g x y = g x' y'"
and P: "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"
shows "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
proof -
@@ -183,15 +182,15 @@
qed
theorem quot_function:
- assumes "!!X Y. f X Y == g (pick X) (pick Y)"
- and "!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y'"
+ assumes "\<And>X Y. f X Y \<equiv> g (pick X) (pick Y)"
+ and "\<And>x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> \<Longrightarrow> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> \<Longrightarrow> g x y = g x' y'"
shows "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
using assms and TrueI
by (rule quot_cond_function)
theorem quot_function':
- "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
- (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y = g x' y') ==>
+ "(\<And>X Y. f X Y \<equiv> g (pick X) (pick Y)) \<Longrightarrow>
+ (\<And>x x' y y'. x \<sim> x' \<Longrightarrow> y \<sim> y' \<Longrightarrow> g x y = g x' y') \<Longrightarrow>
f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
by (rule quot_function) (simp_all only: quot_equality)