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src/HOL/Library/Quotient_Type.thy

changeset 35100 | 53754ec7360b |

parent 30738 | 0842e906300c |

child 45694 | 4a8743618257 |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Quotient_Type.thy Wed Feb 10 19:37:34 2010 +0100 @@ -0,0 +1,196 @@ +(* Title: HOL/Library/Quotient_Type.thy + Author: Markus Wenzel, TU Muenchen +*) + +header {* Quotient types *} + +theory Quotient_Type +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