src/HOL/Equiv_Relations.thy
author nipkow
Mon Feb 21 15:04:10 2005 +0100 (2005-02-21)
changeset 15539 333a88244569
parent 15392 290bc97038c7
child 17589 58eeffd73be1
permissions -rw-r--r--
comprehensive cleanup, replacing sumr by setsum
     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 (rules 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 (rules 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 
    94 subsection {* Quotients *}
    95 
    96 constdefs
    97   quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
    98   "A//r == \<Union>x \<in> A. {r``{x}}"  -- {* set of equiv classes *}
    99 
   100 lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
   101   by (unfold quotient_def) blast
   102 
   103 lemma quotientE:
   104   "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
   105   by (unfold quotient_def) blast
   106 
   107 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
   108   by (unfold equiv_def refl_def quotient_def) blast
   109 
   110 lemma quotient_disj:
   111   "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
   112   apply (unfold quotient_def)
   113   apply clarify
   114   apply (rule equiv_class_eq)
   115    apply assumption
   116   apply (unfold equiv_def trans_def sym_def)
   117   apply blast
   118   done
   119 
   120 lemma quotient_eqI:
   121   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y" 
   122   apply (clarify elim!: quotientE)
   123   apply (rule equiv_class_eq, assumption)
   124   apply (unfold equiv_def sym_def trans_def, blast)
   125   done
   126 
   127 lemma quotient_eq_iff:
   128   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)" 
   129   apply (rule iffI)  
   130    prefer 2 apply (blast del: equalityI intro: quotient_eqI) 
   131   apply (clarify elim!: quotientE)
   132   apply (unfold equiv_def sym_def trans_def, blast)
   133   done
   134 
   135 
   136 lemma quotient_empty [simp]: "{}//r = {}"
   137 by(simp add: quotient_def)
   138 
   139 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
   140 by(simp add: quotient_def)
   141 
   142 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
   143 by(simp add: quotient_def)
   144 
   145 
   146 lemma singleton_quotient: "{x}//r = {r `` {x}}"
   147 by(simp add:quotient_def)
   148 
   149 lemma quotient_diff1:
   150   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
   151 apply(simp add:quotient_def inj_on_def)
   152 apply blast
   153 done
   154 
   155 subsection {* Defining unary operations upon equivalence classes *}
   156 
   157 text{*A congruence-preserving function*}
   158 locale congruent =
   159   fixes r and f
   160   assumes congruent: "(y,z) \<in> r ==> f y = f z"
   161 
   162 syntax
   163   RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)
   164 
   165 translations
   166   "f respects r" == "congruent r f"
   167 
   168 
   169 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
   170   -- {* lemma required to prove @{text UN_equiv_class} *}
   171   by auto
   172 
   173 lemma UN_equiv_class:
   174   "equiv A r ==> f respects r ==> a \<in> A
   175     ==> (\<Union>x \<in> r``{a}. f x) = f a"
   176   -- {* Conversion rule *}
   177   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
   178   apply (unfold equiv_def congruent_def sym_def)
   179   apply (blast del: equalityI)
   180   done
   181 
   182 lemma UN_equiv_class_type:
   183   "equiv A r ==> f respects r ==> X \<in> A//r ==>
   184     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
   185   apply (unfold quotient_def)
   186   apply clarify
   187   apply (subst UN_equiv_class)
   188      apply auto
   189   done
   190 
   191 text {*
   192   Sufficient conditions for injectiveness.  Could weaken premises!
   193   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
   194   A ==> f y \<in> B"}.
   195 *}
   196 
   197 lemma UN_equiv_class_inject:
   198   "equiv A r ==> f respects r ==>
   199     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
   200     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
   201     ==> X = Y"
   202   apply (unfold quotient_def)
   203   apply clarify
   204   apply (rule equiv_class_eq)
   205    apply assumption
   206   apply (subgoal_tac "f x = f xa")
   207    apply blast
   208   apply (erule box_equals)
   209    apply (assumption | rule UN_equiv_class)+
   210   done
   211 
   212 
   213 subsection {* Defining binary operations upon equivalence classes *}
   214 
   215 text{*A congruence-preserving function of two arguments*}
   216 locale congruent2 =
   217   fixes r1 and r2 and f
   218   assumes congruent2:
   219     "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
   220 
   221 text{*Abbreviation for the common case where the relations are identical*}
   222 syntax
   223   RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects2 " 80)
   224 
   225 translations
   226   "f respects2 r" => "congruent2 r r f"
   227 
   228 lemma congruent2_implies_congruent:
   229     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
   230   by (unfold congruent_def congruent2_def equiv_def refl_def) blast
   231 
   232 lemma congruent2_implies_congruent_UN:
   233   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
   234     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
   235   apply (unfold congruent_def)
   236   apply clarify
   237   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
   238   apply (simp add: UN_equiv_class congruent2_implies_congruent)
   239   apply (unfold congruent2_def equiv_def refl_def)
   240   apply (blast del: equalityI)
   241   done
   242 
   243 lemma UN_equiv_class2:
   244   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
   245     ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
   246   by (simp add: UN_equiv_class congruent2_implies_congruent
   247     congruent2_implies_congruent_UN)
   248 
   249 lemma UN_equiv_class_type2:
   250   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
   251     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
   252     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
   253     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
   254   apply (unfold quotient_def)
   255   apply clarify
   256   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
   257     congruent2_implies_congruent quotientI)
   258   done
   259 
   260 lemma UN_UN_split_split_eq:
   261   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
   262     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
   263   -- {* Allows a natural expression of binary operators, *}
   264   -- {* without explicit calls to @{text split} *}
   265   by auto
   266 
   267 lemma congruent2I:
   268   "equiv A1 r1 ==> equiv A2 r2
   269     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
   270     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
   271     ==> congruent2 r1 r2 f"
   272   -- {* Suggested by John Harrison -- the two subproofs may be *}
   273   -- {* \emph{much} simpler than the direct proof. *}
   274   apply (unfold congruent2_def equiv_def refl_def)
   275   apply clarify
   276   apply (blast intro: trans)
   277   done
   278 
   279 lemma congruent2_commuteI:
   280   assumes equivA: "equiv A r"
   281     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
   282     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
   283   shows "f respects2 r"
   284   apply (rule congruent2I [OF equivA equivA])
   285    apply (rule commute [THEN trans])
   286      apply (rule_tac [3] commute [THEN trans, symmetric])
   287        apply (rule_tac [5] sym)
   288        apply (assumption | rule congt |
   289          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
   290   done
   291 
   292 
   293 subsection {* Cardinality results *}
   294 
   295 text {*Suggested by Florian Kammüller*}
   296 
   297 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
   298   -- {* recall @{thm equiv_type} *}
   299   apply (rule finite_subset)
   300    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
   301   apply (unfold quotient_def)
   302   apply blast
   303   done
   304 
   305 lemma finite_equiv_class:
   306   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
   307   apply (unfold quotient_def)
   308   apply (rule finite_subset)
   309    prefer 2 apply assumption
   310   apply blast
   311   done
   312 
   313 lemma equiv_imp_dvd_card:
   314   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
   315     ==> k dvd card A"
   316   apply (rule Union_quotient [THEN subst])
   317    apply assumption
   318   apply (rule dvd_partition)
   319      prefer 3 apply (blast dest: quotient_disj)
   320     apply (simp_all add: Union_quotient equiv_type)
   321   done
   322 
   323 lemma card_quotient_disjoint:
   324  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
   325 apply(simp add:quotient_def)
   326 apply(subst card_UN_disjoint)
   327    apply assumption
   328   apply simp
   329  apply(fastsimp simp add:inj_on_def)
   330 apply (simp add:setsum_constant)
   331 done
   332 
   333 ML
   334 {*
   335 val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
   336 val UN_constant_eq = thm "UN_constant_eq";
   337 val UN_equiv_class = thm "UN_equiv_class";
   338 val UN_equiv_class2 = thm "UN_equiv_class2";
   339 val UN_equiv_class_inject = thm "UN_equiv_class_inject";
   340 val UN_equiv_class_type = thm "UN_equiv_class_type";
   341 val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
   342 val Union_quotient = thm "Union_quotient";
   343 val comp_equivI = thm "comp_equivI";
   344 val congruent2I = thm "congruent2I";
   345 val congruent2_commuteI = thm "congruent2_commuteI";
   346 val congruent2_def = thm "congruent2_def";
   347 val congruent2_implies_congruent = thm "congruent2_implies_congruent";
   348 val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
   349 val congruent_def = thm "congruent_def";
   350 val eq_equiv_class = thm "eq_equiv_class";
   351 val eq_equiv_class_iff = thm "eq_equiv_class_iff";
   352 val equiv_class_eq = thm "equiv_class_eq";
   353 val equiv_class_eq_iff = thm "equiv_class_eq_iff";
   354 val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
   355 val equiv_class_self = thm "equiv_class_self";
   356 val equiv_comp_eq = thm "equiv_comp_eq";
   357 val equiv_def = thm "equiv_def";
   358 val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
   359 val equiv_type = thm "equiv_type";
   360 val finite_equiv_class = thm "finite_equiv_class";
   361 val finite_quotient = thm "finite_quotient";
   362 val quotientE = thm "quotientE";
   363 val quotientI = thm "quotientI";
   364 val quotient_def = thm "quotient_def";
   365 val quotient_disj = thm "quotient_disj";
   366 val refl_comp_subset = thm "refl_comp_subset";
   367 val subset_equiv_class = thm "subset_equiv_class";
   368 val sym_trans_comp_subset = thm "sym_trans_comp_subset";
   369 *}
   370 
   371 end