(* 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}.
*}
definition "quot = {{x. a \<sim> x} | a::'a::eqv. True}"
typedef (open) '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