15300

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


147 


148 
text{*A congruencepreserving function*}


149 
locale congruent =


150 
fixes r and f


151 
assumes congruent: "(y,z) \<in> r ==> f y = f z"


152 


153 
syntax


154 
RESPECTS ::"['a => 'b, ('a * 'a) set] => bool" (infixr "respects" 80)


155 


156 
translations


157 
"f respects r" == "congruent r f"


158 


159 


160 
lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"


161 
 {* lemma required to prove @{text UN_equiv_class} *}


162 
by auto


163 


164 
lemma UN_equiv_class:


165 
"equiv A r ==> f respects r ==> a \<in> A


166 
==> (\<Union>x \<in> r``{a}. f x) = f a"


167 
 {* Conversion rule *}


168 
apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)


169 
apply (unfold equiv_def congruent_def sym_def)


170 
apply (blast del: equalityI)


171 
done


172 


173 
lemma UN_equiv_class_type:


174 
"equiv A r ==> f respects r ==> X \<in> A//r ==>


175 
(!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"


176 
apply (unfold quotient_def)


177 
apply clarify


178 
apply (subst UN_equiv_class)


179 
apply auto


180 
done


181 


182 
text {*


183 
Sufficient conditions for injectiveness. Could weaken premises!


184 
major premise could be an inclusion; bcong could be @{text "!!y. y \<in>


185 
A ==> f y \<in> B"}.


186 
*}


187 


188 
lemma UN_equiv_class_inject:


189 
"equiv A r ==> f respects r ==>


190 
(\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r


191 
==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)


192 
==> X = Y"


193 
apply (unfold quotient_def)


194 
apply clarify


195 
apply (rule equiv_class_eq)


196 
apply assumption


197 
apply (subgoal_tac "f x = f xa")


198 
apply blast


199 
apply (erule box_equals)


200 
apply (assumption  rule UN_equiv_class)+


201 
done


202 


203 


204 
subsection {* Defining binary operations upon equivalence classes *}


205 


206 
text{*A congruencepreserving function of two arguments*}


207 
locale congruent2 =


208 
fixes r1 and r2 and f


209 
assumes congruent2:


210 
"(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"


211 


212 
text{*Abbreviation for the common case where the relations are identical*}


213 
syntax


214 
RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool" (infixr "respects2 " 80)


215 


216 
translations


217 
"f respects2 r" => "congruent2 r r f"


218 


219 
lemma congruent2_implies_congruent:


220 
"equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"


221 
by (unfold congruent_def congruent2_def equiv_def refl_def) blast


222 


223 
lemma congruent2_implies_congruent_UN:


224 
"equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>


225 
congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"


226 
apply (unfold congruent_def)


227 
apply clarify


228 
apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)


229 
apply (simp add: UN_equiv_class congruent2_implies_congruent)


230 
apply (unfold congruent2_def equiv_def refl_def)


231 
apply (blast del: equalityI)


232 
done


233 


234 
lemma UN_equiv_class2:


235 
"equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2


236 
==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"


237 
by (simp add: UN_equiv_class congruent2_implies_congruent


238 
congruent2_implies_congruent_UN)


239 


240 
lemma UN_equiv_class_type2:


241 
"equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f


242 
==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2


243 
==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)


244 
==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"


245 
apply (unfold quotient_def)


246 
apply clarify


247 
apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN


248 
congruent2_implies_congruent quotientI)


249 
done


250 


251 
lemma UN_UN_split_split_eq:


252 
"(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =


253 
(\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"


254 
 {* Allows a natural expression of binary operators, *}


255 
 {* without explicit calls to @{text split} *}


256 
by auto


257 


258 
lemma congruent2I:


259 
"equiv A1 r1 ==> equiv A2 r2


260 
==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)


261 
==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)


262 
==> congruent2 r1 r2 f"


263 
 {* Suggested by John Harrison  the two subproofs may be *}


264 
 {* \emph{much} simpler than the direct proof. *}


265 
apply (unfold congruent2_def equiv_def refl_def)


266 
apply clarify


267 
apply (blast intro: trans)


268 
done


269 


270 
lemma congruent2_commuteI:


271 
assumes equivA: "equiv A r"


272 
and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"


273 
and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"


274 
shows "f respects2 r"


275 
apply (rule congruent2I [OF equivA equivA])


276 
apply (rule commute [THEN trans])


277 
apply (rule_tac [3] commute [THEN trans, symmetric])


278 
apply (rule_tac [5] sym)


279 
apply (assumption  rule congt 


280 
erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+


281 
done


282 


283 


284 
subsection {* Cardinality results *}


285 


286 
text {*Suggested by Florian Kammüller*}


287 


288 
lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"


289 
 {* recall @{thm equiv_type} *}


290 
apply (rule finite_subset)


291 
apply (erule_tac [2] finite_Pow_iff [THEN iffD2])


292 
apply (unfold quotient_def)


293 
apply blast


294 
done


295 


296 
lemma finite_equiv_class:


297 
"finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"


298 
apply (unfold quotient_def)


299 
apply (rule finite_subset)


300 
prefer 2 apply assumption


301 
apply blast


302 
done


303 


304 
lemma equiv_imp_dvd_card:


305 
"finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X


306 
==> k dvd card A"


307 
apply (rule Union_quotient [THEN subst])


308 
apply assumption


309 
apply (rule dvd_partition)


310 
prefer 4 apply (blast dest: quotient_disj)


311 
apply (simp_all add: Union_quotient equiv_type finite_quotient)


312 
done


313 


314 
ML


315 
{*


316 
val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";


317 
val UN_constant_eq = thm "UN_constant_eq";


318 
val UN_equiv_class = thm "UN_equiv_class";


319 
val UN_equiv_class2 = thm "UN_equiv_class2";


320 
val UN_equiv_class_inject = thm "UN_equiv_class_inject";


321 
val UN_equiv_class_type = thm "UN_equiv_class_type";


322 
val UN_equiv_class_type2 = thm "UN_equiv_class_type2";


323 
val Union_quotient = thm "Union_quotient";


324 
val comp_equivI = thm "comp_equivI";


325 
val congruent2I = thm "congruent2I";


326 
val congruent2_commuteI = thm "congruent2_commuteI";


327 
val congruent2_def = thm "congruent2_def";


328 
val congruent2_implies_congruent = thm "congruent2_implies_congruent";


329 
val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";


330 
val congruent_def = thm "congruent_def";


331 
val eq_equiv_class = thm "eq_equiv_class";


332 
val eq_equiv_class_iff = thm "eq_equiv_class_iff";


333 
val equiv_class_eq = thm "equiv_class_eq";


334 
val equiv_class_eq_iff = thm "equiv_class_eq_iff";


335 
val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";


336 
val equiv_class_self = thm "equiv_class_self";


337 
val equiv_comp_eq = thm "equiv_comp_eq";


338 
val equiv_def = thm "equiv_def";


339 
val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";


340 
val equiv_type = thm "equiv_type";


341 
val finite_equiv_class = thm "finite_equiv_class";


342 
val finite_quotient = thm "finite_quotient";


343 
val quotientE = thm "quotientE";


344 
val quotientI = thm "quotientI";


345 
val quotient_def = thm "quotient_def";


346 
val quotient_disj = thm "quotient_disj";


347 
val refl_comp_subset = thm "refl_comp_subset";


348 
val subset_equiv_class = thm "subset_equiv_class";


349 
val sym_trans_comp_subset = thm "sym_trans_comp_subset";


350 
*}


351 


352 
end
