src/HOL/Library/Quotient.thy
author wenzelm
Mon Feb 12 20:43:12 2001 +0100 (2001-02-12)
changeset 11099 b301d1f72552
parent 10681 ec76e17f73c5
child 11549 e7265e70fd7c
permissions -rw-r--r--
\<subseteq>;
     1 (*  Title:      HOL/Library/Quotient.thy
     2     ID:         $Id$
     3     Author:     Markus Wenzel, TU Muenchen
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 *)
     6 
     7 header {*
     8   \title{Quotient types}
     9   \author{Markus Wenzel}
    10 *}
    11 
    12 theory Quotient = Main:
    13 
    14 text {*
    15  We introduce the notion of quotient types over equivalence relations
    16  via axiomatic type classes.
    17 *}
    18 
    19 subsection {* Equivalence relations and quotient types *}
    20 
    21 text {*
    22  \medskip Type class @{text equiv} models equivalence relations @{text
    23  "\<sim> :: 'a => 'a => bool"}.
    24 *}
    25 
    26 axclass eqv \<subseteq> "term"
    27 consts
    28   eqv :: "('a::eqv) => 'a => bool"    (infixl "\<sim>" 50)
    29 
    30 axclass equiv \<subseteq> eqv
    31   equiv_refl [intro]: "x \<sim> x"
    32   equiv_trans [trans]: "x \<sim> y ==> y \<sim> z ==> x \<sim> z"
    33   equiv_sym [elim?]: "x \<sim> y ==> y \<sim> x"
    34 
    35 lemma not_equiv_sym [elim?]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"
    36 proof -
    37   assume "\<not> (x \<sim> y)" thus "\<not> (y \<sim> x)"
    38     by (rule contrapos_nn) (rule equiv_sym)
    39 qed
    40 
    41 lemma not_equiv_trans1 [trans]: "\<not> (x \<sim> y) ==> y \<sim> z ==> \<not> (x \<sim> (z::'a::equiv))"
    42 proof -
    43   assume "\<not> (x \<sim> y)" and yz: "y \<sim> z"
    44   show "\<not> (x \<sim> z)"
    45   proof
    46     assume "x \<sim> z"
    47     also from yz have "z \<sim> y" ..
    48     finally have "x \<sim> y" .
    49     thus False by contradiction
    50   qed
    51 qed
    52 
    53 lemma not_equiv_trans2 [trans]: "x \<sim> y ==> \<not> (y \<sim> z) ==> \<not> (x \<sim> (z::'a::equiv))"
    54 proof -
    55   assume "\<not> (y \<sim> z)" hence "\<not> (z \<sim> y)" ..
    56   also assume "x \<sim> y" hence "y \<sim> x" ..
    57   finally have "\<not> (z \<sim> x)" . thus "(\<not> x \<sim> z)" ..
    58 qed
    59 
    60 text {*
    61  \medskip The quotient type @{text "'a quot"} consists of all
    62  \emph{equivalence classes} over elements of the base type @{typ 'a}.
    63 *}
    64 
    65 typedef 'a quot = "{{x. a \<sim> x} | a::'a::eqv. True}"
    66   by blast
    67 
    68 lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"
    69   by (unfold quot_def) blast
    70 
    71 lemma quotE [elim]: "R \<in> quot ==> (!!a. R = {x. a \<sim> x} ==> C) ==> C"
    72   by (unfold quot_def) blast
    73 
    74 text {*
    75  \medskip Abstracted equivalence classes are the canonical
    76  representation of elements of a quotient type.
    77 *}
    78 
    79 constdefs
    80   class :: "'a::equiv => 'a quot"    ("\<lfloor>_\<rfloor>")
    81   "\<lfloor>a\<rfloor> == Abs_quot {x. a \<sim> x}"
    82 
    83 theorem quot_exhaust: "\<exists>a. A = \<lfloor>a\<rfloor>"
    84 proof (cases A)
    85   fix R assume R: "A = Abs_quot R"
    86   assume "R \<in> quot" hence "\<exists>a. R = {x. a \<sim> x}" by blast
    87   with R have "\<exists>a. A = Abs_quot {x. a \<sim> x}" by blast
    88   thus ?thesis by (unfold class_def)
    89 qed
    90 
    91 lemma quot_cases [cases type: quot]: "(!!a. A = \<lfloor>a\<rfloor> ==> C) ==> C"
    92   by (insert quot_exhaust) blast
    93 
    94 
    95 subsection {* Equality on quotients *}
    96 
    97 text {*
    98  Equality of canonical quotient elements coincides with the original
    99  relation.
   100 *}
   101 
   102 theorem quot_equality: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
   103 proof
   104   assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   105   show "a \<sim> b"
   106   proof -
   107     from eq have "{x. a \<sim> x} = {x. b \<sim> x}"
   108       by (simp only: class_def Abs_quot_inject quotI)
   109     moreover have "a \<sim> a" ..
   110     ultimately have "a \<in> {x. b \<sim> x}" by blast
   111     hence "b \<sim> a" by blast
   112     thus ?thesis ..
   113   qed
   114 next
   115   assume ab: "a \<sim> b"
   116   show "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   117   proof -
   118     have "{x. a \<sim> x} = {x. b \<sim> x}"
   119     proof (rule Collect_cong)
   120       fix x show "(a \<sim> x) = (b \<sim> x)"
   121       proof
   122         from ab have "b \<sim> a" ..
   123         also assume "a \<sim> x"
   124         finally show "b \<sim> x" .
   125       next
   126         note ab
   127         also assume "b \<sim> x"
   128         finally show "a \<sim> x" .
   129       qed
   130     qed
   131     thus ?thesis by (simp only: class_def)
   132   qed
   133 qed
   134 
   135 lemma quot_equalI [intro?]: "a \<sim> b ==> \<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   136   by (simp only: quot_equality)
   137 
   138 lemma quot_equalD [dest?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> ==> a \<sim> b"
   139   by (simp only: quot_equality)
   140 
   141 lemma quot_not_equalI [intro?]: "\<not> (a \<sim> b) ==> \<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor>"
   142   by (simp add: quot_equality)
   143 
   144 lemma quot_not_equalD [dest?]: "\<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor> ==> \<not> (a \<sim> b)"
   145   by (simp add: quot_equality)
   146 
   147 
   148 subsection {* Picking representing elements *}
   149 
   150 constdefs
   151   pick :: "'a::equiv quot => 'a"
   152   "pick A == SOME a. A = \<lfloor>a\<rfloor>"
   153 
   154 theorem pick_equiv [intro]: "pick \<lfloor>a\<rfloor> \<sim> a"
   155 proof (unfold pick_def)
   156   show "(SOME x. \<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>) \<sim> a"
   157   proof (rule someI2)
   158     show "\<lfloor>a\<rfloor> = \<lfloor>a\<rfloor>" ..
   159     fix x assume "\<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>"
   160     hence "a \<sim> x" .. thus "x \<sim> a" ..
   161   qed
   162 qed
   163 
   164 theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"
   165 proof (cases A)
   166   fix a assume a: "A = \<lfloor>a\<rfloor>"
   167   hence "pick A \<sim> a" by (simp only: pick_equiv)
   168   hence "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" ..
   169   with a show ?thesis by simp
   170 qed
   171 
   172 text {*
   173  \medskip The following rules support canonical function definitions
   174  on quotient types (with up to two arguments).  Note that the
   175  stripped-down version without additional conditions is sufficient
   176  most of the time.
   177 *}
   178 
   179 theorem quot_cond_function:
   180   "(!!X Y. P X Y ==> f X Y == g (pick X) (pick Y)) ==>
   181     (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>
   182       ==> P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor> ==> P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
   183     P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
   184   (is "PROP ?eq ==> PROP ?cong ==> _ ==> _")
   185 proof -
   186   assume cong: "PROP ?cong"
   187   assume "PROP ?eq" and "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"
   188   hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)
   189   also have "... = g a b"
   190   proof (rule cong)
   191     show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..
   192     moreover
   193     show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..
   194     moreover
   195     show "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>" .
   196     ultimately show "P \<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> \<lfloor>pick \<lfloor>b\<rfloor>\<rfloor>" by (simp only:)
   197   qed
   198   finally show ?thesis .
   199 qed
   200 
   201 theorem quot_function:
   202   "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
   203     (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
   204     f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
   205 proof -
   206   case antecedent from this TrueI
   207   show ?thesis by (rule quot_cond_function)
   208 qed
   209 
   210 theorem quot_function':
   211   "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
   212     (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y = g x' y') ==>
   213     f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
   214   by  (rule quot_function) (simp only: quot_equality)+
   215 
   216 end