summary |
shortlog |
changelog |
graph |
tags |
bookmarks |
branches |
files |
changeset |
raw | gz |
help

author | wenzelm |

Wed, 10 Feb 2010 19:37:34 +0100 | |

changeset 35100 | 53754ec7360b |

parent 35099 | 7722bcb5c37c |

child 35101 | 6ce9177d6b38 |

child 35103 | d74fe18f01e9 |

renamed Library/Quotient.thy to Library/Quotient_Type.thy to avoid clash with new theory Quotient in Main HOL;

NEWS | file | annotate | diff | comparison | revisions | |

src/HOL/IsaMakefile | file | annotate | diff | comparison | revisions | |

src/HOL/Library/Library.thy | file | annotate | diff | comparison | revisions | |

src/HOL/Library/Quotient.thy | file | annotate | diff | comparison | revisions | |

src/HOL/Library/Quotient_Type.thy | file | annotate | diff | comparison | revisions |

--- 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 ***

--- 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 \

--- 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

--- 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

--- /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