src/HOL/Library/Quotient.thy
author wenzelm
Sat Nov 18 19:46:48 2000 +0100 (2000-11-18)
changeset 10491 e4a408728012
parent 10483 eb93ace45a6e
child 10499 2f33d0fd242e
permissions -rw-r--r--
quot_cond_function: simplified, support conditional definition;
     1 (*  Title:      HOL/Library/Quotient.thy
     2     ID:         $Id$
     3     Author:     Markus Wenzel, TU Muenchen
     4 *)
     5 
     6 header {*
     7   \title{Quotient types}
     8   \author{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 lemma not_equiv_sym [elim?]: "\<not> (x \<sim> y) ==> \<not> (y \<sim> (x::'a::equiv))"
    35 proof -
    36   assume "\<not> (x \<sim> y)" thus "\<not> (y \<sim> x)"
    37     by (rule contrapos_nn) (rule equiv_sym)
    38 qed
    39 
    40 lemma not_equiv_trans1 [trans]: "\<not> (x \<sim> y) ==> y \<sim> z ==> \<not> (x \<sim> (z::'a::equiv))"
    41 proof -
    42   assume "\<not> (x \<sim> y)" and yz: "y \<sim> z"
    43   show "\<not> (x \<sim> z)"
    44   proof
    45     assume "x \<sim> z"
    46     also from yz have "z \<sim> y" ..
    47     finally have "x \<sim> y" .
    48     thus False by contradiction
    49   qed
    50 qed
    51 
    52 lemma not_equiv_trans2 [trans]: "x \<sim> y ==> \<not> (y \<sim> z) ==> \<not> (x \<sim> (z::'a::equiv))"
    53 proof -
    54   assume "\<not> (y \<sim> z)" hence "\<not> (z \<sim> y)" ..
    55   also assume "x \<sim> y" hence "y \<sim> x" ..
    56   finally have "\<not> (z \<sim> x)" . thus "(\<not> x \<sim> z)" ..
    57 qed
    58 
    59 text {*
    60  \medskip The quotient type @{text "'a quot"} consists of all
    61  \emph{equivalence classes} over elements of the base type @{typ 'a}.
    62 *}
    63 
    64 typedef 'a quot = "{{x. a \<sim> x} | a::'a::eqv. True}"
    65   by blast
    66 
    67 lemma quotI [intro]: "{x. a \<sim> x} \<in> quot"
    68   by (unfold quot_def) blast
    69 
    70 lemma quotE [elim]: "R \<in> quot ==> (!!a. R = {x. a \<sim> x} ==> C) ==> C"
    71   by (unfold quot_def) blast
    72 
    73 text {*
    74  \medskip Abstracted equivalence classes are the canonical
    75  representation of elements of a quotient type.
    76 *}
    77 
    78 constdefs
    79   equivalence_class :: "'a::equiv => 'a quot"    ("\<lfloor>_\<rfloor>")
    80   "\<lfloor>a\<rfloor> == Abs_quot {x. a \<sim> x}"
    81 
    82 theorem quot_exhaust: "\<exists>a. A = \<lfloor>a\<rfloor>"
    83 proof (cases A)
    84   fix R assume R: "A = Abs_quot R"
    85   assume "R \<in> quot" hence "\<exists>a. R = {x. a \<sim> x}" by blast
    86   with R have "\<exists>a. A = Abs_quot {x. a \<sim> x}" by blast
    87   thus ?thesis by (unfold equivalence_class_def)
    88 qed
    89 
    90 lemma quot_cases [cases type: quot]: "(!!a. A = \<lfloor>a\<rfloor> ==> C) ==> C"
    91   by (insert quot_exhaust) blast
    92 
    93 
    94 subsection {* Equality on quotients *}
    95 
    96 text {*
    97  Equality of canonical quotient elements coincides with the original
    98  relation.
    99 *}
   100 
   101 theorem quot_equality: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
   102 proof
   103   assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   104   show "a \<sim> b"
   105   proof -
   106     from eq have "{x. a \<sim> x} = {x. b \<sim> x}"
   107       by (simp only: equivalence_class_def Abs_quot_inject quotI)
   108     moreover have "a \<sim> a" ..
   109     ultimately have "a \<in> {x. b \<sim> x}" by blast
   110     hence "b \<sim> a" by blast
   111     thus ?thesis ..
   112   qed
   113 next
   114   assume ab: "a \<sim> b"
   115   show "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   116   proof -
   117     have "{x. a \<sim> x} = {x. b \<sim> x}"
   118     proof (rule Collect_cong)
   119       fix x show "(a \<sim> x) = (b \<sim> x)"
   120       proof
   121         from ab have "b \<sim> a" ..
   122         also assume "a \<sim> x"
   123         finally show "b \<sim> x" .
   124       next
   125         note ab
   126         also assume "b \<sim> x"
   127         finally show "a \<sim> x" .
   128       qed
   129     qed
   130     thus ?thesis by (simp only: equivalence_class_def)
   131   qed
   132 qed
   133 
   134 lemma quot_equalI [intro?]: "a \<sim> b ==> \<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
   135   by (simp only: quot_equality)
   136 
   137 lemma quot_equalD [dest?]: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor> ==> a \<sim> b"
   138   by (simp only: quot_equality)
   139 
   140 lemma quot_not_equalI [intro?]: "\<not> (a \<sim> b) ==> \<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor>"
   141   by (simp add: quot_equality)
   142 
   143 lemma quot_not_equalD [dest?]: "\<lfloor>a\<rfloor> \<noteq> \<lfloor>b\<rfloor> ==> \<not> (a \<sim> b)"
   144   by (simp add: quot_equality)
   145 
   146 
   147 subsection {* Picking representing elements *}
   148 
   149 constdefs
   150   pick :: "'a::equiv quot => 'a"
   151   "pick A == SOME a. A = \<lfloor>a\<rfloor>"
   152 
   153 theorem pick_equiv [intro]: "pick \<lfloor>a\<rfloor> \<sim> a"
   154 proof (unfold pick_def)
   155   show "(SOME x. \<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>) \<sim> a"
   156   proof (rule someI2)
   157     show "\<lfloor>a\<rfloor> = \<lfloor>a\<rfloor>" ..
   158     fix x assume "\<lfloor>a\<rfloor> = \<lfloor>x\<rfloor>"
   159     hence "a \<sim> x" .. thus "x \<sim> a" ..
   160   qed
   161 qed
   162 
   163 theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"
   164 proof (cases A)
   165   fix a assume a: "A = \<lfloor>a\<rfloor>"
   166   hence "pick A \<sim> a" by (simp only: pick_equiv)
   167   hence "\<lfloor>pick A\<rfloor> = \<lfloor>a\<rfloor>" ..
   168   with a show ?thesis by simp
   169 qed
   170 
   171 text {*
   172  \medskip The following rules support canonical function definitions
   173  on quotient types (with up to two arguments).  Note that the
   174  stripped-down version without additional conditions is sufficient
   175  most of the time.
   176 *}
   177 
   178 theorem quot_cond_function:
   179   "(!!X Y. P X Y ==> f X Y == g (pick X) (pick Y)) ==>
   180     (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>
   181       ==> P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor> ==> P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
   182     P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
   183   (is "PROP ?eq ==> PROP ?cong ==> _ ==> _")
   184 proof -
   185   assume cong: "PROP ?cong"
   186   assume "PROP ?eq" and "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>"
   187   hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)
   188   also have "\<dots> = g a b"
   189   proof (rule cong)
   190     show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..
   191     moreover
   192     show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..
   193     moreover
   194     show "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>" .
   195     ultimately show "P \<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> \<lfloor>pick \<lfloor>b\<rfloor>\<rfloor>" by (simp only:)
   196   qed
   197   finally show ?thesis .
   198 qed
   199 
   200 theorem quot_function:
   201   "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
   202     (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
   203     f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
   204 proof -
   205   case antecedent from this TrueI
   206   show ?thesis by (rule quot_cond_function)
   207 qed
   208 
   209 end