(*  Title:      HOL/Library/Quotient.thy

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

*)

header {*

  \title{Quotient types}

  \author{Markus Wenzel}

*}

theory Quotient = Main:

text {*

 We introduce the notion of quotient types over equivalence relations

 via axiomatic type classes.

*}

subsection {* Equivalence relations and quotient types *}

text {*

 \medskip Type class @{text equiv} models equivalence relations @{text

 "\<sim> :: 'a => 'a => bool"}.

*}

axclass eqv < "term"

consts

  eqv :: "('a::eqv) => 'a => bool"    (infixl "\<sim>" 50)

axclass equiv < eqv

  equiv_refl [intro]: "x \<sim> x"

  equiv_trans [trans]: "x \<sim> y ==> y \<sim> z ==> x \<sim> z"

  equiv_sym [elim?]: "x \<sim> y ==> y \<sim> x"

lemma not_equiv_sym [elim?]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"

proof -

  assume "\<not> (x \<sim> y)" thus "\<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 yz: "y \<sim> z"

  show "\<not> (x \<sim> z)"

  proof

    assume "x \<sim> z"

    also from yz have "z \<sim> y" ..

    finally have "x \<sim> y" .

    thus 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)" hence "\<not> (z \<sim> y)" ..

  also assume "x \<sim> y" hence "y \<sim> x" ..

  finally have "\<not> (z \<sim> x)" . thus "(\<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"

  by (unfold quot_def) blast

lemma quotE [elim]: "R \<in> quot ==> (!!a. R = {x. a \<sim> x} ==> C) ==> C"

  by (unfold quot_def) blast

text {*

 \medskip Abstracted equivalence classes are the canonical

 representation of elements of a quotient type.

*}

constdefs

  class :: "'a::equiv => 'a quot"    ("\<lfloor>_\<rfloor>")

  "\<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" hence "\<exists>a. R = {x. a \<sim> x}" by blast

  with R have "\<exists>a. A = Abs_quot {x. a \<sim> x}" by blast

  thus ?thesis by (unfold class_def)

qed

lemma quot_cases [cases type: quot]: "(!!a. A = \<lfloor>a\<rfloor> ==> C) ==> C"

  by (insert quot_exhaust) blast

subsection {* Equality on quotients *}

text {*

 Equality of canonical quotient elements coincides with the original

 relation.

*}

theorem quot_equality: "(\<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

    hence "b \<sim> a" by blast

    thus ?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

    thus ?thesis by (simp only: class_def)

  qed

qed

lemma quot_equalI [intro?]: "a \<sim> b ==> \<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"

  by (simp only: quot_equality)

lemma quot_equalD [dest?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> ==> a \<sim> b"

  by (simp only: quot_equality)

lemma quot_not_equalI [intro?]: "\<not> (a \<sim> b) ==> \<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor>"

  by (simp add: quot_equality)

lemma quot_not_equalD [dest?]: "\<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor> ==> \<not> (a \<sim> b)"

  by (simp add: quot_equality)

subsection {* Picking representing elements *}

constdefs

  pick :: "'a::equiv quot => 'a"

  "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>"

    hence "a \<sim> x" .. thus "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>"

  hence "pick A \<sim> a" by (simp only: pick_equiv)

  hence "\<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:

  "(!!X Y. P X Y ==> f X Y == g (pick X) (pick Y)) ==>

    (!!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') ==>

    P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"

  (is "PROP ?eq ==> PROP ?cong ==> _ ==> _")

proof -

  assume cong: "PROP ?cong"

  assume "PROP ?eq" and "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"

  hence "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>" .

    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:

  "(!!X Y. f X Y == g (pick X) (pick Y)) ==>

    (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>

    f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"

proof -

  case antecedent from this TrueI

  show ?thesis by (rule quot_cond_function)

qed

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 only: quot_equality)+

end