src/HOL/Library/Quotient.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30738 0842e906300c permissions -rw-r--r--
power operation defined generic
     1 (*  Title:      HOL/Library/Quotient.thy

     2     Author:     Markus Wenzel, TU Muenchen

     3 *)

     4

     5 header {* Quotient types *}

     6

     7 theory Quotient

     8 imports Main

     9 begin

    10

    11 text {*

    12  We introduce the notion of quotient types over equivalence relations

    13  via type classes.

    14 *}

    15

    16 subsection {* Equivalence relations and quotient types *}

    17

    18 text {*

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

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

    21 *}

    22

    23 class eqv =

    24   fixes eqv :: "'a \<Rightarrow> 'a \<Rightarrow> bool"    (infixl "\<sim>" 50)

    25

    26 class equiv = eqv +

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

    28   assumes equiv_trans [trans]: "x \<sim> y \<Longrightarrow> y \<sim> z \<Longrightarrow> x \<sim> z"

    29   assumes equiv_sym [sym]: "x \<sim> y \<Longrightarrow> y \<sim> x"

    30

    31 lemma equiv_not_sym [sym]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"

    32 proof -

    33   assume "\<not> (x \<sim> y)" then show "\<not> (y \<sim> x)"

    34     by (rule contrapos_nn) (rule equiv_sym)

    35 qed

    36

    37 lemma not_equiv_trans1 [trans]: "\<not> (x \<sim> y) ==> y \<sim> z ==> \<not> (x \<sim> (z::'a::equiv))"

    38 proof -

    39   assume "\<not> (x \<sim> y)" and "y \<sim> z"

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

    41   proof

    42     assume "x \<sim> z"

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

    44     finally have "x \<sim> y" .

    45     with \<not> (x \<sim> y) show False by contradiction

    46   qed

    47 qed

    48

    49 lemma not_equiv_trans2 [trans]: "x \<sim> y ==> \<not> (y \<sim> z) ==> \<not> (x \<sim> (z::'a::equiv))"

    50 proof -

    51   assume "\<not> (y \<sim> z)" then have "\<not> (z \<sim> y)" ..

    52   also assume "x \<sim> y" then have "y \<sim> x" ..

    53   finally have "\<not> (z \<sim> x)" . then show "(\<not> x \<sim> z)" ..

    54 qed

    55

    56 text {*

    57  \medskip The quotient type @{text "'a quot"} consists of all

    58  \emph{equivalence classes} over elements of the base type @{typ 'a}.

    59 *}

    60

    61 typedef 'a quot = "{{x. a \<sim> x} | a::'a::eqv. True}"

    62   by blast

    63

    64 lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"

    65   unfolding quot_def by blast

    66

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

    68   unfolding quot_def by blast

    69

    70 text {*

    71  \medskip Abstracted equivalence classes are the canonical

    72  representation of elements of a quotient type.

    73 *}

    74

    75 definition

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

    77   "\<lfloor>a\<rfloor> = Abs_quot {x. a \<sim> x}"

    78

    79 theorem quot_exhaust: "\<exists>a. A = \<lfloor>a\<rfloor>"

    80 proof (cases A)

    81   fix R assume R: "A = Abs_quot R"

    82   assume "R \<in> quot" then have "\<exists>a. R = {x. a \<sim> x}" by blast

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

    84   then show ?thesis unfolding class_def .

    85 qed

    86

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

    88   using quot_exhaust by blast

    89

    90

    91 subsection {* Equality on quotients *}

    92

    93 text {*

    94  Equality of canonical quotient elements coincides with the original

    95  relation.

    96 *}

    97

    98 theorem quot_equality [iff?]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"

    99 proof

   100   assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"

   101   show "a \<sim> b"

   102   proof -

   103     from eq have "{x. a \<sim> x} = {x. b \<sim> x}"

   104       by (simp only: class_def Abs_quot_inject quotI)

   105     moreover have "a \<sim> a" ..

   106     ultimately have "a \<in> {x. b \<sim> x}" by blast

   107     then have "b \<sim> a" by blast

   108     then show ?thesis ..

   109   qed

   110 next

   111   assume ab: "a \<sim> b"

   112   show "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"

   113   proof -

   114     have "{x. a \<sim> x} = {x. b \<sim> x}"

   115     proof (rule Collect_cong)

   116       fix x show "(a \<sim> x) = (b \<sim> x)"

   117       proof

   118         from ab have "b \<sim> a" ..

   119         also assume "a \<sim> x"

   120         finally show "b \<sim> x" .

   121       next

   122         note ab

   123         also assume "b \<sim> x"

   124         finally show "a \<sim> x" .

   125       qed

   126     qed

   127     then show ?thesis by (simp only: class_def)

   128   qed

   129 qed

   130

   131

   132 subsection {* Picking representing elements *}

   133

   134 definition

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

   136   "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"

   137

   138 theorem pick_equiv [intro]: "pick \<lfloor>a\<rfloor> \<sim> a"

   139 proof (unfold pick_def)

   140   show "(SOME x. \<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>) \<sim> a"

   141   proof (rule someI2)

   142     show "\<lfloor>a\<rfloor> = \<lfloor>a\<rfloor>" ..

   143     fix x assume "\<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>"

   144     then have "a \<sim> x" .. then show "x \<sim> a" ..

   145   qed

   146 qed

   147

   148 theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"

   149 proof (cases A)

   150   fix a assume a: "A = \<lfloor>a\<rfloor>"

   151   then have "pick A \<sim> a" by (simp only: pick_equiv)

   152   then have "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" ..

   153   with a show ?thesis by simp

   154 qed

   155

   156 text {*

   157  \medskip The following rules support canonical function definitions

   158  on quotient types (with up to two arguments).  Note that the

   159  stripped-down version without additional conditions is sufficient

   160  most of the time.

   161 *}

   162

   163 theorem quot_cond_function:

   164   assumes eq: "!!X Y. P X Y ==> f X Y == g (pick X) (pick Y)"

   165     and cong: "!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>

   166       ==> P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor> ==> P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor> ==> g x y = g x' y'"

   167     and P: "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"

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

   169 proof -

   170   from eq and P have "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)

   171   also have "... = g a b"

   172   proof (rule cong)

   173     show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..

   174     moreover

   175     show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..

   176     moreover

   177     show "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>" by (rule P)

   178     ultimately show "P \<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> \<lfloor>pick \<lfloor>b\<rfloor>\<rfloor>" by (simp only:)

   179   qed

   180   finally show ?thesis .

   181 qed

   182

   183 theorem quot_function:

   184   assumes "!!X Y. f X Y == g (pick X) (pick Y)"

   185     and "!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y'"

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

   187   using assms and TrueI

   188   by (rule quot_cond_function)

   189

   190 theorem quot_function':

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

   192     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y = g x' y') ==>

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

   194   by (rule quot_function) (simp_all only: quot_equality)

   195

   196 end