src/HOL/Equiv_Relations.thy
 author haftmann Wed Jan 28 11:03:42 2009 +0100 (2009-01-28) changeset 29655 ac31940cfb69 parent 28562 4e74209f113e child 30198 922f944f03b2 permissions -rw-r--r--
Plain, Main form meeting points in import hierarchy
     1 (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory

     2     Copyright   1996  University of Cambridge

     3 *)

     4

     5 header {* Equivalence Relations in Higher-Order Set Theory *}

     6

     7 theory Equiv_Relations

     8 imports Finite_Set Relation Plain

     9 begin

    10

    11 subsection {* Equivalence relations *}

    12

    13 locale equiv =

    14   fixes A and r

    15   assumes refl: "refl A r"

    16     and sym: "sym r"

    17     and trans: "trans r"

    18

    19 text {*

    20   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O

    21   r = r"}.

    22

    23   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.

    24 *}

    25

    26 lemma sym_trans_comp_subset:

    27     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"

    28   by (unfold trans_def sym_def converse_def) blast

    29

    30 lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"

    31   by (unfold refl_def) blast

    32

    33 lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"

    34   apply (unfold equiv_def)

    35   apply clarify

    36   apply (rule equalityI)

    37    apply (iprover intro: sym_trans_comp_subset refl_comp_subset)+

    38   done

    39

    40 text {* Second half. *}

    41

    42 lemma comp_equivI:

    43     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"

    44   apply (unfold equiv_def refl_def sym_def trans_def)

    45   apply (erule equalityE)

    46   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")

    47    apply fast

    48   apply fast

    49   done

    50

    51

    52 subsection {* Equivalence classes *}

    53

    54 lemma equiv_class_subset:

    55   "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"

    56   -- {* lemma for the next result *}

    57   by (unfold equiv_def trans_def sym_def) blast

    58

    59 theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r{a} = r{b}"

    60   apply (assumption | rule equalityI equiv_class_subset)+

    61   apply (unfold equiv_def sym_def)

    62   apply blast

    63   done

    64

    65 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"

    66   by (unfold equiv_def refl_def) blast

    67

    68 lemma subset_equiv_class:

    69     "equiv A r ==> r{b} \<subseteq> r{a} ==> b \<in> A ==> (a,b) \<in> r"

    70   -- {* lemma for the next result *}

    71   by (unfold equiv_def refl_def) blast

    72

    73 lemma eq_equiv_class:

    74     "r{a} = r{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"

    75   by (iprover intro: equalityD2 subset_equiv_class)

    76

    77 lemma equiv_class_nondisjoint:

    78     "equiv A r ==> x \<in> (r{a} \<inter> r{b}) ==> (a, b) \<in> r"

    79   by (unfold equiv_def trans_def sym_def) blast

    80

    81 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"

    82   by (unfold equiv_def refl_def) blast

    83

    84 theorem equiv_class_eq_iff:

    85   "equiv A r ==> ((x, y) \<in> r) = (r{x} = r{y} & x \<in> A & y \<in> A)"

    86   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)

    87

    88 theorem eq_equiv_class_iff:

    89   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r{x} = r{y}) = ((x, y) \<in> r)"

    90   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)

    91

    92

    93 subsection {* Quotients *}

    94

    95 definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90) where

    96   [code del]: "A//r = (\<Union>x \<in> A. {r{x}})"  -- {* set of equiv classes *}

    97

    98 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"

    99   by (unfold quotient_def) blast

   100

   101 lemma quotientE:

   102   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"

   103   by (unfold quotient_def) blast

   104

   105 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"

   106   by (unfold equiv_def refl_def quotient_def) blast

   107

   108 lemma quotient_disj:

   109   "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"

   110   apply (unfold quotient_def)

   111   apply clarify

   112   apply (rule equiv_class_eq)

   113    apply assumption

   114   apply (unfold equiv_def trans_def sym_def)

   115   apply blast

   116   done

   117

   118 lemma quotient_eqI:

   119   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y"

   120   apply (clarify elim!: quotientE)

   121   apply (rule equiv_class_eq, assumption)

   122   apply (unfold equiv_def sym_def trans_def, blast)

   123   done

   124

   125 lemma quotient_eq_iff:

   126   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)"

   127   apply (rule iffI)

   128    prefer 2 apply (blast del: equalityI intro: quotient_eqI)

   129   apply (clarify elim!: quotientE)

   130   apply (unfold equiv_def sym_def trans_def, blast)

   131   done

   132

   133 lemma eq_equiv_class_iff2:

   134   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"

   135 by(simp add:quotient_def eq_equiv_class_iff)

   136

   137

   138 lemma quotient_empty [simp]: "{}//r = {}"

   139 by(simp add: quotient_def)

   140

   141 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"

   142 by(simp add: quotient_def)

   143

   144 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"

   145 by(simp add: quotient_def)

   146

   147

   148 lemma singleton_quotient: "{x}//r = {r  {x}}"

   149 by(simp add:quotient_def)

   150

   151 lemma quotient_diff1:

   152   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"

   153 apply(simp add:quotient_def inj_on_def)

   154 apply blast

   155 done

   156

   157 subsection {* Defining unary operations upon equivalence classes *}

   158

   159 text{*A congruence-preserving function*}

   160 locale congruent =

   161   fixes r and f

   162   assumes congruent: "(y,z) \<in> r ==> f y = f z"

   163

   164 abbreviation

   165   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"

   166     (infixr "respects" 80) where

   167   "f respects r == congruent r f"

   168

   169

   170 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"

   171   -- {* lemma required to prove @{text UN_equiv_class} *}

   172   by auto

   173

   174 lemma UN_equiv_class:

   175   "equiv A r ==> f respects r ==> a \<in> A

   176     ==> (\<Union>x \<in> r{a}. f x) = f a"

   177   -- {* Conversion rule *}

   178   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)

   179   apply (unfold equiv_def congruent_def sym_def)

   180   apply (blast del: equalityI)

   181   done

   182

   183 lemma UN_equiv_class_type:

   184   "equiv A r ==> f respects r ==> X \<in> A//r ==>

   185     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"

   186   apply (unfold quotient_def)

   187   apply clarify

   188   apply (subst UN_equiv_class)

   189      apply auto

   190   done

   191

   192 text {*

   193   Sufficient conditions for injectiveness.  Could weaken premises!

   194   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>

   195   A ==> f y \<in> B"}.

   196 *}

   197

   198 lemma UN_equiv_class_inject:

   199   "equiv A r ==> f respects r ==>

   200     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r

   201     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)

   202     ==> X = Y"

   203   apply (unfold quotient_def)

   204   apply clarify

   205   apply (rule equiv_class_eq)

   206    apply assumption

   207   apply (subgoal_tac "f x = f xa")

   208    apply blast

   209   apply (erule box_equals)

   210    apply (assumption | rule UN_equiv_class)+

   211   done

   212

   213

   214 subsection {* Defining binary operations upon equivalence classes *}

   215

   216 text{*A congruence-preserving function of two arguments*}

   217 locale congruent2 =

   218   fixes r1 and r2 and f

   219   assumes congruent2:

   220     "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"

   221

   222 text{*Abbreviation for the common case where the relations are identical*}

   223 abbreviation

   224   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"

   225     (infixr "respects2" 80) where

   226   "f respects2 r == congruent2 r r f"

   227

   228

   229 lemma congruent2_implies_congruent:

   230     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"

   231   by (unfold congruent_def congruent2_def equiv_def refl_def) blast

   232

   233 lemma congruent2_implies_congruent_UN:

   234   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>

   235     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2{a}. f x1 x2)"

   236   apply (unfold congruent_def)

   237   apply clarify

   238   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)

   239   apply (simp add: UN_equiv_class congruent2_implies_congruent)

   240   apply (unfold congruent2_def equiv_def refl_def)

   241   apply (blast del: equalityI)

   242   done

   243

   244 lemma UN_equiv_class2:

   245   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2

   246     ==> (\<Union>x1 \<in> r1{a1}. \<Union>x2 \<in> r2{a2}. f x1 x2) = f a1 a2"

   247   by (simp add: UN_equiv_class congruent2_implies_congruent

   248     congruent2_implies_congruent_UN)

   249

   250 lemma UN_equiv_class_type2:

   251   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f

   252     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2

   253     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)

   254     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"

   255   apply (unfold quotient_def)

   256   apply clarify

   257   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN

   258     congruent2_implies_congruent quotientI)

   259   done

   260

   261 lemma UN_UN_split_split_eq:

   262   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =

   263     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"

   264   -- {* Allows a natural expression of binary operators, *}

   265   -- {* without explicit calls to @{text split} *}

   266   by auto

   267

   268 lemma congruent2I:

   269   "equiv A1 r1 ==> equiv A2 r2

   270     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)

   271     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)

   272     ==> congruent2 r1 r2 f"

   273   -- {* Suggested by John Harrison -- the two subproofs may be *}

   274   -- {* \emph{much} simpler than the direct proof. *}

   275   apply (unfold congruent2_def equiv_def refl_def)

   276   apply clarify

   277   apply (blast intro: trans)

   278   done

   279

   280 lemma congruent2_commuteI:

   281   assumes equivA: "equiv A r"

   282     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"

   283     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"

   284   shows "f respects2 r"

   285   apply (rule congruent2I [OF equivA equivA])

   286    apply (rule commute [THEN trans])

   287      apply (rule_tac  commute [THEN trans, symmetric])

   288        apply (rule_tac  sym)

   289        apply (rule congt | assumption |

   290          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+

   291   done

   292

   293

   294 subsection {* Quotients and finiteness *}

   295

   296 text {*Suggested by Florian Kamm�ller*}

   297

   298 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"

   299   -- {* recall @{thm equiv_type} *}

   300   apply (rule finite_subset)

   301    apply (erule_tac  finite_Pow_iff [THEN iffD2])

   302   apply (unfold quotient_def)

   303   apply blast

   304   done

   305

   306 lemma finite_equiv_class:

   307   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"

   308   apply (unfold quotient_def)

   309   apply (rule finite_subset)

   310    prefer 2 apply assumption

   311   apply blast

   312   done

   313

   314 lemma equiv_imp_dvd_card:

   315   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X

   316     ==> k dvd card A"

   317   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])

   318    apply assumption

   319   apply (rule dvd_partition)

   320      prefer 3 apply (blast dest: quotient_disj)

   321     apply (simp_all add: Union_quotient equiv_type)

   322   done

   323

   324 lemma card_quotient_disjoint:

   325  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"

   326 apply(simp add:quotient_def)

   327 apply(subst card_UN_disjoint)

   328    apply assumption

   329   apply simp

   330  apply(fastsimp simp add:inj_on_def)

   331 apply (simp add:setsum_constant)

   332 done

   333

   334 end