# HG changeset patch # User wenzelm # Date 1265827054 -3600 # Node ID 53754ec7360b7b70dc7c81a46809d026f7b1c783 # Parent 7722bcb5c37c799bf486466ce5060a7b81f8c8e2 renamed Library/Quotient.thy to Library/Quotient_Type.thy to avoid clash with new theory Quotient in Main HOL; diff -r 7722bcb5c37c -r 53754ec7360b NEWS --- a/NEWS Wed Feb 10 17:05:40 2010 +0100 +++ b/NEWS Wed Feb 10 19:37:34 2010 +0100 @@ -126,6 +126,9 @@ * Theory List: added transpose. +* Renamed Library/Quotient.thy to Library/Quotient_Type.thy to avoid +clash with new theory Quotient in Main HOL. + *** ML *** diff -r 7722bcb5c37c -r 53754ec7360b src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Wed Feb 10 17:05:40 2010 +0100 +++ b/src/HOL/IsaMakefile Wed Feb 10 19:37:34 2010 +0100 @@ -386,12 +386,12 @@ Library/Permutations.thy Library/Bit.thy Library/FrechetDeriv.thy \ Library/Fraction_Field.thy Library/Fundamental_Theorem_Algebra.thy \ Library/Inner_Product.thy Library/Kleene_Algebra.thy \ - Library/Lattice_Algebras.thy \ - Library/Lattice_Syntax.thy Library/Library.thy \ - Library/List_Prefix.thy Library/List_Set.thy Library/State_Monad.thy \ - Library/Nat_Int_Bij.thy Library/Multiset.thy Library/Permutation.thy \ - Library/Quotient.thy Library/Quicksort.thy Library/Nat_Infinity.thy \ - Library/Word.thy Library/README.html Library/Continuity.thy \ + Library/Lattice_Algebras.thy Library/Lattice_Syntax.thy \ + Library/Library.thy Library/List_Prefix.thy Library/List_Set.thy \ + Library/State_Monad.thy Library/Nat_Int_Bij.thy Library/Multiset.thy \ + Library/Permutation.thy Library/Quotient_Type.thy \ + Library/Quicksort.thy Library/Nat_Infinity.thy Library/Word.thy \ + Library/README.html Library/Continuity.thy \ Library/Order_Relation.thy Library/Nested_Environment.thy \ Library/Ramsey.thy Library/Zorn.thy Library/Library/ROOT.ML \ Library/Library/document/root.tex Library/Library/document/root.bib \ diff -r 7722bcb5c37c -r 53754ec7360b src/HOL/Library/Library.thy --- a/src/HOL/Library/Library.thy Wed Feb 10 17:05:40 2010 +0100 +++ b/src/HOL/Library/Library.thy Wed Feb 10 19:37:34 2010 +0100 @@ -45,7 +45,7 @@ Preorder Product_Vector Quicksort - Quotient + Quotient_Type Ramsey Reflection RBT diff -r 7722bcb5c37c -r 53754ec7360b src/HOL/Library/Quotient.thy --- 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 - "\ :: 'a => 'a => bool"}. -*} - -class eqv = - fixes eqv :: "'a \ 'a \ bool" (infixl "\" 50) - -class equiv = eqv + - assumes equiv_refl [intro]: "x \ x" - assumes equiv_trans [trans]: "x \ y \ y \ z \ x \ z" - assumes equiv_sym [sym]: "x \ y \ y \ x" - -lemma equiv_not_sym [sym]: "\ (x \ y) ==> \ (y \ (x::'a::equiv))" -proof - - assume "\ (x \ y)" then show "\ (y \ x)" - by (rule contrapos_nn) (rule equiv_sym) -qed - -lemma not_equiv_trans1 [trans]: "\ (x \ y) ==> y \ z ==> \ (x \ (z::'a::equiv))" -proof - - assume "\ (x \ y)" and "y \ z" - show "\ (x \ z)" - proof - assume "x \ z" - also from `y \ z` have "z \ y" .. - finally have "x \ y" . - with `\ (x \ y)` show False by contradiction - qed -qed - -lemma not_equiv_trans2 [trans]: "x \ y ==> \ (y \ z) ==> \ (x \ (z::'a::equiv))" -proof - - assume "\ (y \ z)" then have "\ (z \ y)" .. - also assume "x \ y" then have "y \ x" .. - finally have "\ (z \ x)" . then show "(\ x \ 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 \ x} | a::'a::eqv. True}" - by blast - -lemma quotI [intro]: "{x. a \ x} \ quot" - unfolding quot_def by blast - -lemma quotE [elim]: "R \ quot ==> (!!a. R = {x. a \ 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" ("\_\") where - "\a\ = Abs_quot {x. a \ x}" - -theorem quot_exhaust: "\a. A = \a\" -proof (cases A) - fix R assume R: "A = Abs_quot R" - assume "R \ quot" then have "\a. R = {x. a \ x}" by blast - with R have "\a. A = Abs_quot {x. a \ x}" by blast - then show ?thesis unfolding class_def . -qed - -lemma quot_cases [cases type: quot]: "(!!a. A = \a\ ==> 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?]: "(\a\ = \b\) = (a \ b)" -proof - assume eq: "\a\ = \b\" - show "a \ b" - proof - - from eq have "{x. a \ x} = {x. b \ x}" - by (simp only: class_def Abs_quot_inject quotI) - moreover have "a \ a" .. - ultimately have "a \ {x. b \ x}" by blast - then have "b \ a" by blast - then show ?thesis .. - qed -next - assume ab: "a \ b" - show "\a\ = \b\" - proof - - have "{x. a \ x} = {x. b \ x}" - proof (rule Collect_cong) - fix x show "(a \ x) = (b \ x)" - proof - from ab have "b \ a" .. - also assume "a \ x" - finally show "b \ x" . - next - note ab - also assume "b \ x" - finally show "a \ 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 = \a\)" - -theorem pick_equiv [intro]: "pick \a\ \ a" -proof (unfold pick_def) - show "(SOME x. \a\ = \x\) \ a" - proof (rule someI2) - show "\a\ = \a\" .. - fix x assume "\a\ = \x\" - then have "a \ x" .. then show "x \ a" .. - qed -qed - -theorem pick_inverse [intro]: "\pick A\ = A" -proof (cases A) - fix a assume a: "A = \a\" - then have "pick A \ a" by (simp only: pick_equiv) - then have "\pick A\ = \a\" .. - 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'. \x\ = \x'\ ==> \y\ = \y'\ - ==> P \x\ \y\ ==> P \x'\ \y'\ ==> g x y = g x' y'" - and P: "P \a\ \b\" - shows "f \a\ \b\ = g a b" -proof - - from eq and P have "f \a\ \b\ = g (pick \a\) (pick \b\)" by (simp only:) - also have "... = g a b" - proof (rule cong) - show "\pick \a\\ = \a\" .. - moreover - show "\pick \b\\ = \b\" .. - moreover - show "P \a\ \b\" by (rule P) - ultimately show "P \pick \a\\ \pick \b\\" 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'. \x\ = \x'\ ==> \y\ = \y'\ ==> g x y = g x' y'" - shows "f \a\ \b\ = 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 \ x' ==> y \ y' ==> g x y = g x' y') ==> - f \a\ \b\ = g a b" - by (rule quot_function) (simp_all only: quot_equality) - -end diff -r 7722bcb5c37c -r 53754ec7360b src/HOL/Library/Quotient_Type.thy --- /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 + "\ :: 'a => 'a => bool"}. +*} + +class eqv = + fixes eqv :: "'a \ 'a \ bool" (infixl "\" 50) + +class equiv = eqv + + assumes equiv_refl [intro]: "x \ x" + assumes equiv_trans [trans]: "x \ y \ y \ z \ x \ z" + assumes equiv_sym [sym]: "x \ y \ y \ x" + +lemma equiv_not_sym [sym]: "\ (x \ y) ==> \ (y \ (x::'a::equiv))" +proof - + assume "\ (x \ y)" then show "\ (y \ x)" + by (rule contrapos_nn) (rule equiv_sym) +qed + +lemma not_equiv_trans1 [trans]: "\ (x \ y) ==> y \ z ==> \ (x \ (z::'a::equiv))" +proof - + assume "\ (x \ y)" and "y \ z" + show "\ (x \ z)" + proof + assume "x \ z" + also from `y \ z` have "z \ y" .. + finally have "x \ y" . + with `\ (x \ y)` show False by contradiction + qed +qed + +lemma not_equiv_trans2 [trans]: "x \ y ==> \ (y \ z) ==> \ (x \ (z::'a::equiv))" +proof - + assume "\ (y \ z)" then have "\ (z \ y)" .. + also assume "x \ y" then have "y \ x" .. + finally have "\ (z \ x)" . then show "(\ x \ 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 \ x} | a::'a::eqv. True}" + by blast + +lemma quotI [intro]: "{x. a \ x} \ quot" + unfolding quot_def by blast + +lemma quotE [elim]: "R \ quot ==> (!!a. R = {x. a \ 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" ("\_\") where + "\a\ = Abs_quot {x. a \ x}" + +theorem quot_exhaust: "\a. A = \a\" +proof (cases A) + fix R assume R: "A = Abs_quot R" + assume "R \ quot" then have "\a. R = {x. a \ x}" by blast + with R have "\a. A = Abs_quot {x. a \ x}" by blast + then show ?thesis unfolding class_def . +qed + +lemma quot_cases [cases type: quot]: "(!!a. A = \a\ ==> 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?]: "(\a\ = \b\) = (a \ b)" +proof + assume eq: "\a\ = \b\" + show "a \ b" + proof - + from eq have "{x. a \ x} = {x. b \ x}" + by (simp only: class_def Abs_quot_inject quotI) + moreover have "a \ a" .. + ultimately have "a \ {x. b \ x}" by blast + then have "b \ a" by blast + then show ?thesis .. + qed +next + assume ab: "a \ b" + show "\a\ = \b\" + proof - + have "{x. a \ x} = {x. b \ x}" + proof (rule Collect_cong) + fix x show "(a \ x) = (b \ x)" + proof + from ab have "b \ a" .. + also assume "a \ x" + finally show "b \ x" . + next + note ab + also assume "b \ x" + finally show "a \ 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 = \a\)" + +theorem pick_equiv [intro]: "pick \a\ \ a" +proof (unfold pick_def) + show "(SOME x. \a\ = \x\) \ a" + proof (rule someI2) + show "\a\ = \a\" .. + fix x assume "\a\ = \x\" + then have "a \ x" .. then show "x \ a" .. + qed +qed + +theorem pick_inverse [intro]: "\pick A\ = A" +proof (cases A) + fix a assume a: "A = \a\" + then have "pick A \ a" by (simp only: pick_equiv) + then have "\pick A\ = \a\" .. + 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'. \x\ = \x'\ ==> \y\ = \y'\ + ==> P \x\ \y\ ==> P \x'\ \y'\ ==> g x y = g x' y'" + and P: "P \a\ \b\" + shows "f \a\ \b\ = g a b" +proof - + from eq and P have "f \a\ \b\ = g (pick \a\) (pick \b\)" by (simp only:) + also have "... = g a b" + proof (rule cong) + show "\pick \a\\ = \a\" .. + moreover + show "\pick \b\\ = \b\" .. + moreover + show "P \a\ \b\" by (rule P) + ultimately show "P \pick \a\\ \pick \b\\" 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'. \x\ = \x'\ ==> \y\ = \y'\ ==> g x y = g x' y'" + shows "f \