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