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src/HOL/Library/Quotient_Type.thy

author | wenzelm |

Sun, 02 Nov 2014 17:20:45 +0100 | |

changeset 58881 | b9556a055632 |

parent 49834 | b27bbb021df1 |

child 59192 | a1d6d6db781b |

permissions | -rw-r--r-- |

modernized header;

(* Title: HOL/Library/Quotient_Type.thy Author: Markus Wenzel, TU Muenchen *) section {* 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}. *} definition "quot = {{x. a \<sim> x} | a::'a::eqv. True}" typedef 'a quot = "quot :: 'a::eqv set set" unfolding quot_def 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