src/HOL/Equiv_Relations.thy
author nipkow
Thu Dec 22 13:00:53 2005 +0100 (2005-12-22)
changeset 18490 434e34392c40
parent 17589 58eeffd73be1
child 18493 343da052b961
permissions -rw-r--r--
new lemmas
     1 (*  ID:         $Id$
     2     Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1996  University of Cambridge
     4 *)
     5 
     6 header {* Equivalence Relations in Higher-Order Set Theory *}
     7 
     8 theory Equiv_Relations
     9 imports Relation Finite_Set
    10 begin
    11 
    12 subsection {* Equivalence relations *}
    13 
    14 locale equiv =
    15   fixes A and r
    16   assumes refl: "refl A r"
    17     and sym: "sym r"
    18     and trans: "trans r"
    19 
    20 text {*
    21   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
    22   r = r"}.
    23 
    24   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
    25 *}
    26 
    27 lemma sym_trans_comp_subset:
    28     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
    29   by (unfold trans_def sym_def converse_def) blast
    30 
    31 lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
    32   by (unfold refl_def) blast
    33 
    34 lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
    35   apply (unfold equiv_def)
    36   apply clarify
    37   apply (rule equalityI)
    38    apply (iprover intro: sym_trans_comp_subset refl_comp_subset)+
    39   done
    40 
    41 text {* Second half. *}
    42 
    43 lemma comp_equivI:
    44     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
    45   apply (unfold equiv_def refl_def sym_def trans_def)
    46   apply (erule equalityE)
    47   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
    48    apply fast
    49   apply fast
    50   done
    51 
    52 
    53 subsection {* Equivalence classes *}
    54 
    55 lemma equiv_class_subset:
    56   "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
    57   -- {* lemma for the next result *}
    58   by (unfold equiv_def trans_def sym_def) blast
    59 
    60 theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
    61   apply (assumption | rule equalityI equiv_class_subset)+
    62   apply (unfold equiv_def sym_def)
    63   apply blast
    64   done
    65 
    66 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
    67   by (unfold equiv_def refl_def) blast
    68 
    69 lemma subset_equiv_class:
    70     "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
    71   -- {* lemma for the next result *}
    72   by (unfold equiv_def refl_def) blast
    73 
    74 lemma eq_equiv_class:
    75     "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
    76   by (iprover intro: equalityD2 subset_equiv_class)
    77 
    78 lemma equiv_class_nondisjoint:
    79     "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
    80   by (unfold equiv_def trans_def sym_def) blast
    81 
    82 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
    83   by (unfold equiv_def refl_def) blast
    84 
    85 theorem equiv_class_eq_iff:
    86   "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
    87   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
    88 
    89 theorem eq_equiv_class_iff:
    90   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
    91   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
    92 
    93 lemma eq_equiv_class_iff2:
    94   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
    95 by(simp add:quotient_def eq_equiv_class_iff)
    96 
    97 
    98 subsection {* Quotients *}
    99 
   100 constdefs
   101   quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
   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_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 
   140 lemma quotient_empty [simp]: "{}//r = {}"
   141 by(simp add: quotient_def)
   142 
   143 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
   144 by(simp add: quotient_def)
   145 
   146 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
   147 by(simp add: quotient_def)
   148 
   149 
   150 lemma singleton_quotient: "{x}//r = {r `` {x}}"
   151 by(simp add:quotient_def)
   152 
   153 lemma quotient_diff1:
   154   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
   155 apply(simp add:quotient_def inj_on_def)
   156 apply blast
   157 done
   158 
   159 subsection {* Defining unary operations upon equivalence classes *}
   160 
   161 text{*A congruence-preserving function*}
   162 locale congruent =
   163   fixes r and f
   164   assumes congruent: "(y,z) \<in> r ==> f y = f z"
   165 
   166 syntax
   167   RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)
   168 
   169 translations
   170   "f respects r" == "congruent r f"
   171 
   172 
   173 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
   174   -- {* lemma required to prove @{text UN_equiv_class} *}
   175   by auto
   176 
   177 lemma UN_equiv_class:
   178   "equiv A r ==> f respects r ==> a \<in> A
   179     ==> (\<Union>x \<in> r``{a}. f x) = f a"
   180   -- {* Conversion rule *}
   181   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
   182   apply (unfold equiv_def congruent_def sym_def)
   183   apply (blast del: equalityI)
   184   done
   185 
   186 lemma UN_equiv_class_type:
   187   "equiv A r ==> f respects r ==> X \<in> A//r ==>
   188     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
   189   apply (unfold quotient_def)
   190   apply clarify
   191   apply (subst UN_equiv_class)
   192      apply auto
   193   done
   194 
   195 text {*
   196   Sufficient conditions for injectiveness.  Could weaken premises!
   197   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
   198   A ==> f y \<in> B"}.
   199 *}
   200 
   201 lemma UN_equiv_class_inject:
   202   "equiv A r ==> f respects r ==>
   203     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
   204     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
   205     ==> X = Y"
   206   apply (unfold quotient_def)
   207   apply clarify
   208   apply (rule equiv_class_eq)
   209    apply assumption
   210   apply (subgoal_tac "f x = f xa")
   211    apply blast
   212   apply (erule box_equals)
   213    apply (assumption | rule UN_equiv_class)+
   214   done
   215 
   216 
   217 subsection {* Defining binary operations upon equivalence classes *}
   218 
   219 text{*A congruence-preserving function of two arguments*}
   220 locale congruent2 =
   221   fixes r1 and r2 and f
   222   assumes congruent2:
   223     "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
   224 
   225 text{*Abbreviation for the common case where the relations are identical*}
   226 syntax
   227   RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects2 " 80)
   228 
   229 translations
   230   "f respects2 r" => "congruent2 r r f"
   231 
   232 lemma congruent2_implies_congruent:
   233     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
   234   by (unfold congruent_def congruent2_def equiv_def refl_def) blast
   235 
   236 lemma congruent2_implies_congruent_UN:
   237   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
   238     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
   239   apply (unfold congruent_def)
   240   apply clarify
   241   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
   242   apply (simp add: UN_equiv_class congruent2_implies_congruent)
   243   apply (unfold congruent2_def equiv_def refl_def)
   244   apply (blast del: equalityI)
   245   done
   246 
   247 lemma UN_equiv_class2:
   248   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
   249     ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
   250   by (simp add: UN_equiv_class congruent2_implies_congruent
   251     congruent2_implies_congruent_UN)
   252 
   253 lemma UN_equiv_class_type2:
   254   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
   255     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
   256     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
   257     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
   258   apply (unfold quotient_def)
   259   apply clarify
   260   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
   261     congruent2_implies_congruent quotientI)
   262   done
   263 
   264 lemma UN_UN_split_split_eq:
   265   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
   266     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
   267   -- {* Allows a natural expression of binary operators, *}
   268   -- {* without explicit calls to @{text split} *}
   269   by auto
   270 
   271 lemma congruent2I:
   272   "equiv A1 r1 ==> equiv A2 r2
   273     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
   274     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
   275     ==> congruent2 r1 r2 f"
   276   -- {* Suggested by John Harrison -- the two subproofs may be *}
   277   -- {* \emph{much} simpler than the direct proof. *}
   278   apply (unfold congruent2_def equiv_def refl_def)
   279   apply clarify
   280   apply (blast intro: trans)
   281   done
   282 
   283 lemma congruent2_commuteI:
   284   assumes equivA: "equiv A r"
   285     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
   286     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
   287   shows "f respects2 r"
   288   apply (rule congruent2I [OF equivA equivA])
   289    apply (rule commute [THEN trans])
   290      apply (rule_tac [3] commute [THEN trans, symmetric])
   291        apply (rule_tac [5] sym)
   292        apply (assumption | rule congt |
   293          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
   294   done
   295 
   296 
   297 subsection {* Cardinality results *}
   298 
   299 text {*Suggested by Florian Kammüller*}
   300 
   301 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
   302   -- {* recall @{thm equiv_type} *}
   303   apply (rule finite_subset)
   304    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
   305   apply (unfold quotient_def)
   306   apply blast
   307   done
   308 
   309 lemma finite_equiv_class:
   310   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
   311   apply (unfold quotient_def)
   312   apply (rule finite_subset)
   313    prefer 2 apply assumption
   314   apply blast
   315   done
   316 
   317 lemma equiv_imp_dvd_card:
   318   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
   319     ==> k dvd card A"
   320   apply (rule Union_quotient [THEN subst])
   321    apply assumption
   322   apply (rule dvd_partition)
   323      prefer 3 apply (blast dest: quotient_disj)
   324     apply (simp_all add: Union_quotient equiv_type)
   325   done
   326 
   327 lemma card_quotient_disjoint:
   328  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
   329 apply(simp add:quotient_def)
   330 apply(subst card_UN_disjoint)
   331    apply assumption
   332   apply simp
   333  apply(fastsimp simp add:inj_on_def)
   334 apply (simp add:setsum_constant)
   335 done
   336 
   337 ML
   338 {*
   339 val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
   340 val UN_constant_eq = thm "UN_constant_eq";
   341 val UN_equiv_class = thm "UN_equiv_class";
   342 val UN_equiv_class2 = thm "UN_equiv_class2";
   343 val UN_equiv_class_inject = thm "UN_equiv_class_inject";
   344 val UN_equiv_class_type = thm "UN_equiv_class_type";
   345 val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
   346 val Union_quotient = thm "Union_quotient";
   347 val comp_equivI = thm "comp_equivI";
   348 val congruent2I = thm "congruent2I";
   349 val congruent2_commuteI = thm "congruent2_commuteI";
   350 val congruent2_def = thm "congruent2_def";
   351 val congruent2_implies_congruent = thm "congruent2_implies_congruent";
   352 val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
   353 val congruent_def = thm "congruent_def";
   354 val eq_equiv_class = thm "eq_equiv_class";
   355 val eq_equiv_class_iff = thm "eq_equiv_class_iff";
   356 val equiv_class_eq = thm "equiv_class_eq";
   357 val equiv_class_eq_iff = thm "equiv_class_eq_iff";
   358 val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
   359 val equiv_class_self = thm "equiv_class_self";
   360 val equiv_comp_eq = thm "equiv_comp_eq";
   361 val equiv_def = thm "equiv_def";
   362 val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
   363 val equiv_type = thm "equiv_type";
   364 val finite_equiv_class = thm "finite_equiv_class";
   365 val finite_quotient = thm "finite_quotient";
   366 val quotientE = thm "quotientE";
   367 val quotientI = thm "quotientI";
   368 val quotient_def = thm "quotient_def";
   369 val quotient_disj = thm "quotient_disj";
   370 val refl_comp_subset = thm "refl_comp_subset";
   371 val subset_equiv_class = thm "subset_equiv_class";
   372 val sym_trans_comp_subset = thm "sym_trans_comp_subset";
   373 *}
   374 
   375 end