src/HOL/Equiv_Relations.thy
 author haftmann Mon Nov 29 22:32:06 2010 +0100 (2010-11-29) changeset 40816 19c492929756 parent 40815 6e2d17cc0d1d child 40817 781da1e8808c permissions -rw-r--r--
replaced slightly odd locale congruent by plain definition
     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 Big_Operators Relation Plain

     9 begin

    10

    11 subsection {* Equivalence relations -- set version *}

    12

    13 definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where

    14   "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"

    15

    16 lemma equivI:

    17   "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"

    18   by (simp add: equiv_def)

    19

    20 lemma equivE:

    21   assumes "equiv A r"

    22   obtains "refl_on A r" and "sym r" and "trans r"

    23   using assms by (simp add: equiv_def)

    24

    25 text {*

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

    27   r = r"}.

    28

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

    30 *}

    31

    32 lemma sym_trans_comp_subset:

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

    34   by (unfold trans_def sym_def converse_def) blast

    35

    36 lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"

    37   by (unfold refl_on_def) blast

    38

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

    40   apply (unfold equiv_def)

    41   apply clarify

    42   apply (rule equalityI)

    43    apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+

    44   done

    45

    46 text {* Second half. *}

    47

    48 lemma comp_equivI:

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

    50   apply (unfold equiv_def refl_on_def sym_def trans_def)

    51   apply (erule equalityE)

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

    53    apply fast

    54   apply fast

    55   done

    56

    57

    58 subsection {* Equivalence classes *}

    59

    60 lemma equiv_class_subset:

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

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

    63   by (unfold equiv_def trans_def sym_def) blast

    64

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

    66   apply (assumption | rule equalityI equiv_class_subset)+

    67   apply (unfold equiv_def sym_def)

    68   apply blast

    69   done

    70

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

    72   by (unfold equiv_def refl_on_def) blast

    73

    74 lemma subset_equiv_class:

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

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

    77   by (unfold equiv_def refl_on_def) blast

    78

    79 lemma eq_equiv_class:

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

    81   by (iprover intro: equalityD2 subset_equiv_class)

    82

    83 lemma equiv_class_nondisjoint:

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

    85   by (unfold equiv_def trans_def sym_def) blast

    86

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

    88   by (unfold equiv_def refl_on_def) blast

    89

    90 theorem equiv_class_eq_iff:

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

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

    93

    94 theorem eq_equiv_class_iff:

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

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

    97

    98

    99 subsection {* Quotients *}

   100

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

   102   "A//r = (\<Union>x \<in> A. {r{x}})"  -- {* set of equiv classes *}

   103

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

   105   by (unfold quotient_def) blast

   106

   107 lemma quotientE:

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

   109   by (unfold quotient_def) blast

   110

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

   112   by (unfold equiv_def refl_on_def quotient_def) blast

   113

   114 lemma quotient_disj:

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

   116   apply (unfold quotient_def)

   117   apply clarify

   118   apply (rule equiv_class_eq)

   119    apply assumption

   120   apply (unfold equiv_def trans_def sym_def)

   121   apply blast

   122   done

   123

   124 lemma quotient_eqI:

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

   126   apply (clarify elim!: quotientE)

   127   apply (rule equiv_class_eq, assumption)

   128   apply (unfold equiv_def sym_def trans_def, blast)

   129   done

   130

   131 lemma quotient_eq_iff:

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

   133   apply (rule iffI)

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

   135   apply (clarify elim!: quotientE)

   136   apply (unfold equiv_def sym_def trans_def, blast)

   137   done

   138

   139 lemma eq_equiv_class_iff2:

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

   141 by(simp add:quotient_def eq_equiv_class_iff)

   142

   143

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

   145 by(simp add: quotient_def)

   146

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

   148 by(simp add: quotient_def)

   149

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

   151 by(simp add: quotient_def)

   152

   153

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

   155 by(simp add:quotient_def)

   156

   157 lemma quotient_diff1:

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

   159 apply(simp add:quotient_def inj_on_def)

   160 apply blast

   161 done

   162

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

   164

   165 text{*A congruence-preserving function*}

   166

   167 definition congruent where

   168   "congruent r f \<longleftrightarrow> (\<forall>y z. (y, z) \<in> r \<longrightarrow> f y = f z)"

   169

   170 lemma congruentI:

   171   "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"

   172   by (simp add: congruent_def)

   173

   174 lemma congruentD:

   175   "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"

   176   by (simp add: congruent_def)

   177

   178 abbreviation

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

   180     (infixr "respects" 80) where

   181   "f respects r == congruent r f"

   182

   183

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

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

   186   by auto

   187

   188 lemma UN_equiv_class:

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

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

   191   -- {* Conversion rule *}

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

   193   apply (unfold equiv_def congruent_def sym_def)

   194   apply (blast del: equalityI)

   195   done

   196

   197 lemma UN_equiv_class_type:

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

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

   200   apply (unfold quotient_def)

   201   apply clarify

   202   apply (subst UN_equiv_class)

   203      apply auto

   204   done

   205

   206 text {*

   207   Sufficient conditions for injectiveness.  Could weaken premises!

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

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

   210 *}

   211

   212 lemma UN_equiv_class_inject:

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

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

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

   216     ==> X = Y"

   217   apply (unfold quotient_def)

   218   apply clarify

   219   apply (rule equiv_class_eq)

   220    apply assumption

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

   222    apply blast

   223   apply (erule box_equals)

   224    apply (assumption | rule UN_equiv_class)+

   225   done

   226

   227

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

   229

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

   231 locale congruent2 =

   232   fixes r1 and r2 and f

   233   assumes congruent2:

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

   235

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

   237 abbreviation

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

   239     (infixr "respects2" 80) where

   240   "f respects2 r == congruent2 r r f"

   241

   242

   243 lemma congruent2_implies_congruent:

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

   245   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast

   246

   247 lemma congruent2_implies_congruent_UN:

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

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

   250   apply (unfold congruent_def)

   251   apply clarify

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

   253   apply (simp add: UN_equiv_class congruent2_implies_congruent)

   254   apply (unfold congruent2_def equiv_def refl_on_def)

   255   apply (blast del: equalityI)

   256   done

   257

   258 lemma UN_equiv_class2:

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

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

   261   by (simp add: UN_equiv_class congruent2_implies_congruent

   262     congruent2_implies_congruent_UN)

   263

   264 lemma UN_equiv_class_type2:

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

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

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

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

   269   apply (unfold quotient_def)

   270   apply clarify

   271   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN

   272     congruent2_implies_congruent quotientI)

   273   done

   274

   275 lemma UN_UN_split_split_eq:

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

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

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

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

   280   by auto

   281

   282 lemma congruent2I:

   283   "equiv A1 r1 ==> equiv A2 r2

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

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

   286     ==> congruent2 r1 r2 f"

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

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

   289   apply (unfold congruent2_def equiv_def refl_on_def)

   290   apply clarify

   291   apply (blast intro: trans)

   292   done

   293

   294 lemma congruent2_commuteI:

   295   assumes equivA: "equiv A r"

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

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

   298   shows "f respects2 r"

   299   apply (rule congruent2I [OF equivA equivA])

   300    apply (rule commute [THEN trans])

   301      apply (rule_tac [3] commute [THEN trans, symmetric])

   302        apply (rule_tac [5] sym)

   303        apply (rule congt | assumption |

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

   305   done

   306

   307

   308 subsection {* Quotients and finiteness *}

   309

   310 text {*Suggested by Florian Kammüller*}

   311

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

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

   314   apply (rule finite_subset)

   315    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])

   316   apply (unfold quotient_def)

   317   apply blast

   318   done

   319

   320 lemma finite_equiv_class:

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

   322   apply (unfold quotient_def)

   323   apply (rule finite_subset)

   324    prefer 2 apply assumption

   325   apply blast

   326   done

   327

   328 lemma equiv_imp_dvd_card:

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

   330     ==> k dvd card A"

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

   332    apply assumption

   333   apply (rule dvd_partition)

   334      prefer 3 apply (blast dest: quotient_disj)

   335     apply (simp_all add: Union_quotient equiv_type)

   336   done

   337

   338 lemma card_quotient_disjoint:

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

   340 apply(simp add:quotient_def)

   341 apply(subst card_UN_disjoint)

   342    apply assumption

   343   apply simp

   344  apply(fastsimp simp add:inj_on_def)

   345 apply simp

   346 done

   347

   348

   349 subsection {* Equivalence relations -- predicate version *}

   350

   351 text {* Partial equivalences *}

   352

   353 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   354   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"

   355     -- {* John-Harrison-style characterization *}

   356

   357 lemma part_equivpI:

   358   "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"

   359   by (auto simp add: part_equivp_def mem_def) (auto elim: sympE transpE)

   360

   361 lemma part_equivpE:

   362   assumes "part_equivp R"

   363   obtains x where "R x x" and "symp R" and "transp R"

   364 proof -

   365   from assms have 1: "\<exists>x. R x x"

   366     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"

   367     by (unfold part_equivp_def) blast+

   368   from 1 obtain x where "R x x" ..

   369   moreover have "symp R"

   370   proof (rule sympI)

   371     fix x y

   372     assume "R x y"

   373     with 2 [of x y] show "R y x" by auto

   374   qed

   375   moreover have "transp R"

   376   proof (rule transpI)

   377     fix x y z

   378     assume "R x y" and "R y z"

   379     with 2 [of x y] 2 [of y z] show "R x z" by auto

   380   qed

   381   ultimately show thesis by (rule that)

   382 qed

   383

   384 lemma part_equivp_refl_symp_transp:

   385   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"

   386   by (auto intro: part_equivpI elim: part_equivpE)

   387

   388 lemma part_equivp_symp:

   389   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"

   390   by (erule part_equivpE, erule sympE)

   391

   392 lemma part_equivp_transp:

   393   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"

   394   by (erule part_equivpE, erule transpE)

   395

   396 lemma part_equivp_typedef:

   397   "part_equivp R \<Longrightarrow> \<exists>d. d \<in> (\<lambda>c. \<exists>x. R x x \<and> c = R x)"

   398   by (auto elim: part_equivpE simp add: mem_def)

   399

   400

   401 text {* Total equivalences *}

   402

   403 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   404   "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}

   405

   406 lemma equivpI:

   407   "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"

   408   by (auto elim: reflpE sympE transpE simp add: equivp_def mem_def)

   409

   410 lemma equivpE:

   411   assumes "equivp R"

   412   obtains "reflp R" and "symp R" and "transp R"

   413   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)

   414

   415 lemma equivp_implies_part_equivp:

   416   "equivp R \<Longrightarrow> part_equivp R"

   417   by (auto intro: part_equivpI elim: equivpE reflpE)

   418

   419 lemma equivp_equiv:

   420   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"

   421   by (auto intro: equivpI elim: equivpE simp add: equiv_def reflp_def symp_def transp_def)

   422

   423 lemma equivp_reflp_symp_transp:

   424   shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"

   425   by (auto intro: equivpI elim: equivpE)

   426

   427 lemma identity_equivp:

   428   "equivp (op =)"

   429   by (auto intro: equivpI reflpI sympI transpI)

   430

   431 lemma equivp_reflp:

   432   "equivp R \<Longrightarrow> R x x"

   433   by (erule equivpE, erule reflpE)

   434

   435 lemma equivp_symp:

   436   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"

   437   by (erule equivpE, erule sympE)

   438

   439 lemma equivp_transp:

   440   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"

   441   by (erule equivpE, erule transpE)

   442

   443 end