src/HOL/Library/Quotient.thy
 author wenzelm Wed Nov 15 19:43:42 2000 +0100 (2000-11-15) changeset 10473 4f15b844fea6 parent 10459 df3cd3e76046 child 10477 c21bee84cefe permissions -rw-r--r--
separate rules for function/operation definitions;
     1 (*  Title:      HOL/Library/Quotient.thy

     2     ID:         $Id$

     3     Author:     Gertrud Bauer and Markus Wenzel, TU Muenchen

     4 *)

     5

     6 header {*

     7   \title{Quotient types}

     8   \author{Gertrud Bauer and Markus Wenzel}

     9 *}

    10

    11 theory Quotient = Main:

    12

    13 text {*

    14  We introduce the notion of quotient types over equivalence relations

    15  via axiomatic type classes.

    16 *}

    17

    18 subsection {* Equivalence relations and quotient types *}

    19

    20 text {*

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

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

    23 *}

    24

    25 axclass eqv < "term"

    26 consts

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

    28

    29 axclass equiv < eqv

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

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

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

    33

    34 text {*

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

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

    37 *}

    38

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

    40   by blast

    41

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

    43   by (unfold quot_def) blast

    44

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

    46   by (unfold quot_def) blast

    47

    48 text {*

    49  \medskip Abstracted equivalence classes are the canonical

    50  representation of elements of a quotient type.

    51 *}

    52

    53 constdefs

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

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

    56

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

    58 proof (cases A)

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

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

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

    62   thus ?thesis by (unfold equivalence_class_def)

    63 qed

    64

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

    66   by (insert quot_exhaust) blast

    67

    68

    69 subsection {* Equality on quotients *}

    70

    71 text {*

    72  Equality of canonical quotient elements coincides with the original

    73  relation.

    74 *}

    75

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

    77 proof

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

    79   show "a \<sim> b"

    80   proof -

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

    82       by (simp only: equivalence_class_def Abs_quot_inject quotI)

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

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

    85     hence "b \<sim> a" by blast

    86     thus ?thesis ..

    87   qed

    88 next

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

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

    91   proof -

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

    93     proof (rule Collect_cong)

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

    95       proof

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

    97         also assume "a \<sim> x"

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

    99       next

   100         note ab

   101         also assume "b \<sim> x"

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

   103       qed

   104     qed

   105     thus ?thesis by (simp only: equivalence_class_def)

   106   qed

   107 qed

   108

   109

   110 subsection {* Picking representing elements *}

   111

   112 constdefs

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

   114   "pick A == SOME a. A = \<lfloor>a\<rfloor>"

   115

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

   117 proof (unfold pick_def)

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

   119   proof (rule someI2)

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

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

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

   123   qed

   124 qed

   125

   126 theorem pick_inverse: "\<lfloor>pick A\<rfloor> = A"

   127 proof (cases A)

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

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

   130   hence "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" ..

   131   with a show ?thesis by simp

   132 qed

   133

   134 text {*

   135  \medskip The following rules support canonical function definitions

   136  on quotient types.

   137 *}

   138

   139 theorem quot_cond_function1:

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

   141     (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x') ==>

   142     (!!x x'. x \<sim> x' ==> P x = P x') ==>

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

   144 proof -

   145   assume cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x'"

   146   assume cong_P: "!!x x'. x \<sim> x' ==> P x = P x'"

   147   assume P: "P a"

   148   assume "!!X. f X == g (pick X)"

   149   hence "f \<lfloor>a\<rfloor> = g (pick \<lfloor>a\<rfloor>)" by (simp only:)

   150   also have "\<dots> = g a"

   151   proof (rule cong_g)

   152     show "pick \<lfloor>a\<rfloor> \<sim> a" ..

   153     hence "P (pick \<lfloor>a\<rfloor>) = P a" by (rule cong_P)

   154     also note P

   155     finally show "P (pick \<lfloor>a\<rfloor>)" .

   156   qed

   157   finally show ?thesis .

   158 qed

   159

   160 theorem quot_function1:

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

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

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

   164 proof -

   165   case antecedent from this refl TrueI

   166   show ?thesis by (rule quot_cond_function1)

   167 qed

   168

   169 theorem quot_cond_operation1:

   170   "(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>

   171     (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x') ==>

   172     (!!x x'. x \<sim> x' ==> P x = P x') ==>

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

   174 proof -

   175   assume defn: "!!X. f X == \<lfloor>g (pick X)\<rfloor>"

   176   assume "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x'"

   177   hence cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> \<lfloor>g x\<rfloor> = \<lfloor>g x'\<rfloor>" ..

   178   assume "!!x x'. x \<sim> x' ==> P x = P x'" and "P a"

   179   with defn cong_g show ?thesis by (rule quot_cond_function1)

   180 qed

   181

   182 theorem quot_operation1:

   183   "(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>

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

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

   186 proof -

   187   case antecedent from this refl TrueI

   188   show ?thesis by (rule quot_cond_operation1)

   189 qed

   190

   191 theorem quot_cond_function2:

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

   193     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'

   194       ==> g x y = g x' y') ==>

   195     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y') ==>

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

   197 proof -

   198   assume cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'

   199     ==> g x y = g x' y'"

   200   assume cong_P: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'"

   201   assume P: "P a b"

   202   assume "!!X Y. f X Y == g (pick X) (pick Y)"

   203   hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)

   204   also have "\<dots> = g a b"

   205   proof (rule cong_g)

   206     show "pick \<lfloor>a\<rfloor> \<sim> a" ..

   207     moreover show "pick \<lfloor>b\<rfloor> \<sim> b" ..

   208     ultimately have "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) = P a b" by (rule cong_P)

   209     also show "P a b" .

   210     finally show "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" .

   211   qed

   212   finally show ?thesis .

   213 qed

   214

   215 theorem quot_function2:

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

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

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

   219 proof -

   220   case antecedent from this refl TrueI

   221   show ?thesis by (rule quot_cond_function2)

   222 qed

   223

   224 theorem quot_cond_operation2:

   225   "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>

   226     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'

   227       ==> g x y \<sim> g x' y') ==>

   228     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y') ==>

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

   230 proof -

   231   assume defn: "!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>"

   232   assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'

   233     ==> g x y \<sim> g x' y'"

   234   hence cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'

   235     ==> \<lfloor>g x y\<rfloor> = \<lfloor>g x' y'\<rfloor>" ..

   236   assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'" and "P a b"

   237   with defn cong_g show ?thesis by (rule quot_cond_function2)

   238 qed

   239

   240 theorem quot_operation2:

   241   "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>

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

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

   244 proof -

   245   case antecedent from this refl TrueI

   246   show ?thesis by (rule quot_cond_operation2)

   247 qed

   248

   249 text {*

   250  \medskip HOL's collection of overloaded standard operations is lifted

   251  to quotient types in the canonical manner.

   252 *}

   253

   254 instance quot :: (zero) zero ..

   255 instance quot :: (plus) plus ..

   256 instance quot :: (minus) minus ..

   257 instance quot :: (times) times ..

   258 instance quot :: (inverse) inverse ..

   259 instance quot :: (power) power ..

   260 instance quot :: (number) number ..

   261 instance quot :: (ord) ord ..

   262

   263 defs (overloaded)

   264   zero_quot_def: "0 == \<lfloor>0\<rfloor>"

   265   add_quot_def: "X + Y == \<lfloor>pick X + pick Y\<rfloor>"

   266   diff_quot_def: "X - Y == \<lfloor>pick X - pick Y\<rfloor>"

   267   minus_quot_def: "- X == \<lfloor>- pick X\<rfloor>"

   268   abs_quot_def: "abs X == \<lfloor>abs (pick X)\<rfloor>"

   269   mult_quot_def: "X * Y == \<lfloor>pick X * pick Y\<rfloor>"

   270   inverse_quot_def: "inverse X == \<lfloor>inverse (pick X)\<rfloor>"

   271   divide_quot_def: "X / Y == \<lfloor>pick X / pick Y\<rfloor>"

   272   power_quot_def: "X^n == \<lfloor>(pick X)^n\<rfloor>"

   273   number_of_quot_def: "number_of b == \<lfloor>number_of b\<rfloor>"

   274   le_quot_def: "X \<le> Y == pick X \<le> pick Y"

   275   less_quot_def: "X < Y == pick X < pick Y"

   276

   277 end