--- a/src/HOL/Library/Quotient.thy Wed Feb 10 17:05:40 2010 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,196 +0,0 @@
-(* Title: HOL/Library/Quotient.thy
- Author: Markus Wenzel, TU Muenchen
-*)
-
-header {* Quotient types *}
-
-theory Quotient
-imports Main
-begin
-
-text {*
- We introduce the notion of quotient types over equivalence relations
- via type classes.
-*}
-
-subsection {* Equivalence relations and quotient types *}
-
-text {*
- \medskip Type class @{text equiv} models equivalence relations @{text
- "\<sim> :: 'a => 'a => bool"}.
-*}
-
-class eqv =
- 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"
-
-lemma equiv_not_sym [sym]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"
-proof -
- 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))"
-proof -
- 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" ..
- finally have "x \<sim> y" .
- with `\<not> (x \<sim> y)` show False by contradiction
- qed
-qed
-
-lemma not_equiv_trans2 [trans]: "x \<sim> y ==> \<not> (y \<sim> z) ==> \<not> (x \<sim> (z::'a::equiv))"
-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)" ..
-qed
-
-text {*
- \medskip The quotient type @{text "'a quot"} consists of all
- \emph{equivalence classes} over elements of the base type @{typ 'a}.
-*}
-
-typedef 'a quot = "{{x. a \<sim> x} | a::'a::eqv. True}"
- by blast
-
-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
-
-text {*
- \medskip Abstracted equivalence classes are the canonical
- representation of elements of a quotient type.
-*}
-
-definition
- "class" :: "'a::equiv => '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
- 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"
- using quot_exhaust by blast
-
-
-subsection {* Equality on quotients *}
-
-text {*
- Equality of canonical quotient elements coincides with the original
- relation.
-*}
-
-theorem quot_equality [iff?]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
-proof
- assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
- show "a \<sim> b"
- proof -
- from eq have "{x. a \<sim> x} = {x. b \<sim> x}"
- by (simp only: class_def Abs_quot_inject quotI)
- moreover have "a \<sim> a" ..
- ultimately have "a \<in> {x. b \<sim> x}" by blast
- then have "b \<sim> a" by blast
- then show ?thesis ..
- qed
-next
- assume ab: "a \<sim> b"
- show "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
- proof -
- have "{x. a \<sim> x} = {x. b \<sim> x}"
- proof (rule Collect_cong)
- fix x show "(a \<sim> x) = (b \<sim> x)"
- proof
- from ab have "b \<sim> a" ..
- also assume "a \<sim> x"
- finally show "b \<sim> x" .
- next
- note ab
- also assume "b \<sim> x"
- finally show "a \<sim> x" .
- qed
- qed
- then show ?thesis by (simp only: class_def)
- qed
-qed
-
-
-subsection {* Picking representing elements *}
-
-definition
- pick :: "'a::equiv quot => 'a" where
- "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"
-
-theorem pick_equiv [intro]: "pick \<lfloor>a\<rfloor> \<sim> a"
-proof (unfold pick_def)
- show "(SOME x. \<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>) \<sim> a"
- 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" ..
- qed
-qed
-
-theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"
-proof (cases A)
- fix a assume a: "A = \<lfloor>a\<rfloor>"
- then have "pick A \<sim> a" by (simp only: pick_equiv)
- then have "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" ..
- 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.
-*}
-
-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'"
- and P: "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"
- shows "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
-proof -
- from eq and P have "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)
- also have "... = g a b"
- proof (rule cong)
- show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..
- moreover
- show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..
- moreover
- show "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>" by (rule P)
- ultimately show "P \<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> \<lfloor>pick \<lfloor>b\<rfloor>\<rfloor>" by (simp only:)
- qed
- finally show ?thesis .
-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'"
- 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') ==>
- f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
- by (rule quot_function) (simp_all only: quot_equality)
-
-end