(* 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 \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"
and equiv_trans [trans]: "x \ y \ y \ z \ x \ z"
and equiv_sym [sym]: "x \ y \ y \ x"
begin
lemma equiv_not_sym [sym]: "\ x \ y \ \ y \ x"
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"
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"
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
end
text \The quotient type @{text "'a quot"} consists of all \emph{equivalence
classes} over elements of the base type @{typ 'a}.\
definition (in eqv) "quot = {{x. a \ x} | a. True}"
typedef 'a quot = "quot :: 'a::eqv set set"
unfolding quot_def by blast
lemma quotI [intro]: "{x. a \ x} \ quot"
unfolding quot_def by blast
lemma quotE [elim]:
assumes "R \ quot"
obtains a where "R = {x. a \ x}"
using assms unfolding quot_def by blast
text \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]:
obtains a where "A = \a\"
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 \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 \