src/HOL/Equiv_Relations.thy
 author haftmann Tue Jul 10 17:30:43 2007 +0200 (2007-07-10) changeset 23705 315c638d5856 parent 21749 3f0e86c92ff3 child 24728 e2b3a1065676 permissions -rw-r--r--
moved finite lemmas to Finite_Set.thy
 paulson@15300 1 (* ID: $Id$ paulson@15300 2 Authors: Lawrence C Paulson, Cambridge University Computer Laboratory paulson@15300 3 Copyright 1996 University of Cambridge paulson@15300 4 *) paulson@15300 5 paulson@15300 6 header {* Equivalence Relations in Higher-Order Set Theory *} paulson@15300 7 paulson@15300 8 theory Equiv_Relations haftmann@23705 9 imports Relation paulson@15300 10 begin paulson@15300 11 paulson@15300 12 subsection {* Equivalence relations *} paulson@15300 13 paulson@15300 14 locale equiv = paulson@15300 15 fixes A and r paulson@15300 16 assumes refl: "refl A r" paulson@15300 17 and sym: "sym r" paulson@15300 18 and trans: "trans r" paulson@15300 19 paulson@15300 20 text {* paulson@15300 21 Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\ O paulson@15300 22 r = r"}. paulson@15300 23 paulson@15300 24 First half: @{text "equiv A r ==> r\ O r = r"}. paulson@15300 25 *} paulson@15300 26 paulson@15300 27 lemma sym_trans_comp_subset: paulson@15300 28 "sym r ==> trans r ==> r\ O r \ r" paulson@15300 29 by (unfold trans_def sym_def converse_def) blast paulson@15300 30 paulson@15300 31 lemma refl_comp_subset: "refl A r ==> r \ r\ O r" paulson@15300 32 by (unfold refl_def) blast paulson@15300 33 paulson@15300 34 lemma equiv_comp_eq: "equiv A r ==> r\ O r = r" paulson@15300 35 apply (unfold equiv_def) paulson@15300 36 apply clarify paulson@15300 37 apply (rule equalityI) nipkow@17589 38 apply (iprover intro: sym_trans_comp_subset refl_comp_subset)+ paulson@15300 39 done paulson@15300 40 paulson@15300 41 text {* Second half. *} paulson@15300 42 paulson@15300 43 lemma comp_equivI: paulson@15300 44 "r\ O r = r ==> Domain r = A ==> equiv A r" paulson@15300 45 apply (unfold equiv_def refl_def sym_def trans_def) paulson@15300 46 apply (erule equalityE) paulson@15300 47 apply (subgoal_tac "\x y. (x, y) \ r --> (y, x) \ r") paulson@15300 48 apply fast paulson@15300 49 apply fast paulson@15300 50 done paulson@15300 51 paulson@15300 52 paulson@15300 53 subsection {* Equivalence classes *} paulson@15300 54 paulson@15300 55 lemma equiv_class_subset: paulson@15300 56 "equiv A r ==> (a, b) \ r ==> r{a} \ r{b}" paulson@15300 57 -- {* lemma for the next result *} paulson@15300 58 by (unfold equiv_def trans_def sym_def) blast paulson@15300 59 paulson@15300 60 theorem equiv_class_eq: "equiv A r ==> (a, b) \ r ==> r{a} = r{b}" paulson@15300 61 apply (assumption | rule equalityI equiv_class_subset)+ paulson@15300 62 apply (unfold equiv_def sym_def) paulson@15300 63 apply blast paulson@15300 64 done paulson@15300 65 paulson@15300 66 lemma equiv_class_self: "equiv A r ==> a \ A ==> a \ r{a}" paulson@15300 67 by (unfold equiv_def refl_def) blast paulson@15300 68 paulson@15300 69 lemma subset_equiv_class: paulson@15300 70 "equiv A r ==> r{b} \ r{a} ==> b \ A ==> (a,b) \ r" paulson@15300 71 -- {* lemma for the next result *} paulson@15300 72 by (unfold equiv_def refl_def) blast paulson@15300 73 paulson@15300 74 lemma eq_equiv_class: paulson@15300 75 "r{a} = r{b} ==> equiv A r ==> b \ A ==> (a, b) \ r" nipkow@17589 76 by (iprover intro: equalityD2 subset_equiv_class) paulson@15300 77 paulson@15300 78 lemma equiv_class_nondisjoint: paulson@15300 79 "equiv A r ==> x \ (r{a} \ r{b}) ==> (a, b) \ r" paulson@15300 80 by (unfold equiv_def trans_def sym_def) blast paulson@15300 81 paulson@15300 82 lemma equiv_type: "equiv A r ==> r \ A \ A" paulson@15300 83 by (unfold equiv_def refl_def) blast paulson@15300 84 paulson@15300 85 theorem equiv_class_eq_iff: paulson@15300 86 "equiv A r ==> ((x, y) \ r) = (r{x} = r{y} & x \ A & y \ A)" paulson@15300 87 by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type) paulson@15300 88 paulson@15300 89 theorem eq_equiv_class_iff: paulson@15300 90 "equiv A r ==> x \ A ==> y \ A ==> (r{x} = r{y}) = ((x, y) \ r)" paulson@15300 91 by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type) paulson@15300 92 paulson@15300 93 paulson@15300 94 subsection {* Quotients *} paulson@15300 95 paulson@15300 96 constdefs paulson@15300 97 quotient :: "['a set, ('a*'a) set] => 'a set set" (infixl "'/'/" 90) paulson@15300 98 "A//r == \x \ A. {r{x}}" -- {* set of equiv classes *} paulson@15300 99 paulson@15300 100 lemma quotientI: "x \ A ==> r{x} \ A//r" paulson@15300 101 by (unfold quotient_def) blast paulson@15300 102 paulson@15300 103 lemma quotientE: paulson@15300 104 "X \ A//r ==> (!!x. X = r{x} ==> x \ A ==> P) ==> P" paulson@15300 105 by (unfold quotient_def) blast paulson@15300 106 paulson@15300 107 lemma Union_quotient: "equiv A r ==> Union (A//r) = A" paulson@15300 108 by (unfold equiv_def refl_def quotient_def) blast paulson@15300 109 paulson@15300 110 lemma quotient_disj: paulson@15300 111 "equiv A r ==> X \ A//r ==> Y \ A//r ==> X = Y | (X \ Y = {})" paulson@15300 112 apply (unfold quotient_def) paulson@15300 113 apply clarify paulson@15300 114 apply (rule equiv_class_eq) paulson@15300 115 apply assumption paulson@15300 116 apply (unfold equiv_def trans_def sym_def) paulson@15300 117 apply blast paulson@15300 118 done paulson@15300 119 paulson@15300 120 lemma quotient_eqI: paulson@15300 121 "[|equiv A r; X \ A//r; Y \ A//r; x \ X; y \ Y; (x,y) \ r|] ==> X = Y" paulson@15300 122 apply (clarify elim!: quotientE) paulson@15300 123 apply (rule equiv_class_eq, assumption) paulson@15300 124 apply (unfold equiv_def sym_def trans_def, blast) paulson@15300 125 done paulson@15300 126 paulson@15300 127 lemma quotient_eq_iff: paulson@15300 128 "[|equiv A r; X \ A//r; Y \ A//r; x \ X; y \ Y|] ==> (X = Y) = ((x,y) \ r)" paulson@15300 129 apply (rule iffI) paulson@15300 130 prefer 2 apply (blast del: equalityI intro: quotient_eqI) paulson@15300 131 apply (clarify elim!: quotientE) paulson@15300 132 apply (unfold equiv_def sym_def trans_def, blast) paulson@15300 133 done paulson@15300 134 nipkow@18493 135 lemma eq_equiv_class_iff2: nipkow@18493 136 "\ equiv A r; x \ A; y \ A \ \ ({x}//r = {y}//r) = ((x,y) : r)" nipkow@18493 137 by(simp add:quotient_def eq_equiv_class_iff) nipkow@18493 138 paulson@15300 139 paulson@15300 140 lemma quotient_empty [simp]: "{}//r = {}" paulson@15300 141 by(simp add: quotient_def) paulson@15300 142 paulson@15300 143 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})" paulson@15300 144 by(simp add: quotient_def) paulson@15300 145 paulson@15300 146 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})" paulson@15300 147 by(simp add: quotient_def) paulson@15300 148 paulson@15300 149 nipkow@15302 150 lemma singleton_quotient: "{x}//r = {r  {x}}" nipkow@15302 151 by(simp add:quotient_def) nipkow@15302 152 nipkow@15302 153 lemma quotient_diff1: nipkow@15302 154 "\ inj_on (%a. {a}//r) A; a \ A \ \ (A - {a})//r = A//r - {a}//r" nipkow@15302 155 apply(simp add:quotient_def inj_on_def) nipkow@15302 156 apply blast nipkow@15302 157 done nipkow@15302 158 paulson@15300 159 subsection {* Defining unary operations upon equivalence classes *} paulson@15300 160 paulson@15300 161 text{*A congruence-preserving function*} paulson@15300 162 locale congruent = paulson@15300 163 fixes r and f paulson@15300 164 assumes congruent: "(y,z) \ r ==> f y = f z" paulson@15300 165 wenzelm@19363 166 abbreviation wenzelm@21404 167 RESPECTS :: "('a => 'b) => ('a * 'a) set => bool" wenzelm@21404 168 (infixr "respects" 80) where wenzelm@19363 169 "f respects r == congruent r f" paulson@15300 170 paulson@15300 171 paulson@15300 172 lemma UN_constant_eq: "a \ A ==> \y \ A. f y = c ==> (\y \ A. f(y))=c" paulson@15300 173 -- {* lemma required to prove @{text UN_equiv_class} *} paulson@15300 174 by auto paulson@15300 175 paulson@15300 176 lemma UN_equiv_class: paulson@15300 177 "equiv A r ==> f respects r ==> a \ A paulson@15300 178 ==> (\x \ r{a}. f x) = f a" paulson@15300 179 -- {* Conversion rule *} paulson@15300 180 apply (rule equiv_class_self [THEN UN_constant_eq], assumption+) paulson@15300 181 apply (unfold equiv_def congruent_def sym_def) paulson@15300 182 apply (blast del: equalityI) paulson@15300 183 done paulson@15300 184 paulson@15300 185 lemma UN_equiv_class_type: paulson@15300 186 "equiv A r ==> f respects r ==> X \ A//r ==> paulson@15300 187 (!!x. x \ A ==> f x \ B) ==> (\x \ X. f x) \ B" paulson@15300 188 apply (unfold quotient_def) paulson@15300 189 apply clarify paulson@15300 190 apply (subst UN_equiv_class) paulson@15300 191 apply auto paulson@15300 192 done paulson@15300 193 paulson@15300 194 text {* paulson@15300 195 Sufficient conditions for injectiveness. Could weaken premises! paulson@15300 196 major premise could be an inclusion; bcong could be @{text "!!y. y \ paulson@15300 197 A ==> f y \ B"}. paulson@15300 198 *} paulson@15300 199 paulson@15300 200 lemma UN_equiv_class_inject: paulson@15300 201 "equiv A r ==> f respects r ==> paulson@15300 202 (\x \ X. f x) = (\y \ Y. f y) ==> X \ A//r ==> Y \ A//r paulson@15300 203 ==> (!!x y. x \ A ==> y \ A ==> f x = f y ==> (x, y) \ r) paulson@15300 204 ==> X = Y" paulson@15300 205 apply (unfold quotient_def) paulson@15300 206 apply clarify paulson@15300 207 apply (rule equiv_class_eq) paulson@15300 208 apply assumption paulson@15300 209 apply (subgoal_tac "f x = f xa") paulson@15300 210 apply blast paulson@15300 211 apply (erule box_equals) paulson@15300 212 apply (assumption | rule UN_equiv_class)+ paulson@15300 213 done paulson@15300 214 paulson@15300 215 paulson@15300 216 subsection {* Defining binary operations upon equivalence classes *} paulson@15300 217 paulson@15300 218 text{*A congruence-preserving function of two arguments*} paulson@15300 219 locale congruent2 = paulson@15300 220 fixes r1 and r2 and f paulson@15300 221 assumes congruent2: paulson@15300 222 "(y1,z1) \ r1 ==> (y2,z2) \ r2 ==> f y1 y2 = f z1 z2" paulson@15300 223 paulson@15300 224 text{*Abbreviation for the common case where the relations are identical*} nipkow@19979 225 abbreviation wenzelm@21404 226 RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool" wenzelm@21749 227 (infixr "respects2" 80) where nipkow@19979 228 "f respects2 r == congruent2 r r f" nipkow@19979 229 paulson@15300 230 paulson@15300 231 lemma congruent2_implies_congruent: paulson@15300 232 "equiv A r1 ==> congruent2 r1 r2 f ==> a \ A ==> congruent r2 (f a)" paulson@15300 233 by (unfold congruent_def congruent2_def equiv_def refl_def) blast paulson@15300 234 paulson@15300 235 lemma congruent2_implies_congruent_UN: paulson@15300 236 "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \ A2 ==> paulson@15300 237 congruent r1 (\x1. \x2 \ r2{a}. f x1 x2)" paulson@15300 238 apply (unfold congruent_def) paulson@15300 239 apply clarify paulson@15300 240 apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+) paulson@15300 241 apply (simp add: UN_equiv_class congruent2_implies_congruent) paulson@15300 242 apply (unfold congruent2_def equiv_def refl_def) paulson@15300 243 apply (blast del: equalityI) paulson@15300 244 done paulson@15300 245 paulson@15300 246 lemma UN_equiv_class2: paulson@15300 247 "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \ A1 ==> a2 \ A2 paulson@15300 248 ==> (\x1 \ r1{a1}. \x2 \ r2{a2}. f x1 x2) = f a1 a2" paulson@15300 249 by (simp add: UN_equiv_class congruent2_implies_congruent paulson@15300 250 congruent2_implies_congruent_UN) paulson@15300 251 paulson@15300 252 lemma UN_equiv_class_type2: paulson@15300 253 "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f paulson@15300 254 ==> X1 \ A1//r1 ==> X2 \ A2//r2 paulson@15300 255 ==> (!!x1 x2. x1 \ A1 ==> x2 \ A2 ==> f x1 x2 \ B) paulson@15300 256 ==> (\x1 \ X1. \x2 \ X2. f x1 x2) \ B" paulson@15300 257 apply (unfold quotient_def) paulson@15300 258 apply clarify paulson@15300 259 apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN paulson@15300 260 congruent2_implies_congruent quotientI) paulson@15300 261 done paulson@15300 262 paulson@15300 263 lemma UN_UN_split_split_eq: paulson@15300 264 "(\(x1, x2) \ X. \(y1, y2) \ Y. A x1 x2 y1 y2) = paulson@15300 265 (\x \ X. \y \ Y. (\(x1, x2). (\(y1, y2). A x1 x2 y1 y2) y) x)" paulson@15300 266 -- {* Allows a natural expression of binary operators, *} paulson@15300 267 -- {* without explicit calls to @{text split} *} paulson@15300 268 by auto paulson@15300 269 paulson@15300 270 lemma congruent2I: paulson@15300 271 "equiv A1 r1 ==> equiv A2 r2 paulson@15300 272 ==> (!!y z w. w \ A2 ==> (y,z) \ r1 ==> f y w = f z w) paulson@15300 273 ==> (!!y z w. w \ A1 ==> (y,z) \ r2 ==> f w y = f w z) paulson@15300 274 ==> congruent2 r1 r2 f" paulson@15300 275 -- {* Suggested by John Harrison -- the two subproofs may be *} paulson@15300 276 -- {* \emph{much} simpler than the direct proof. *} paulson@15300 277 apply (unfold congruent2_def equiv_def refl_def) paulson@15300 278 apply clarify paulson@15300 279 apply (blast intro: trans) paulson@15300 280 done paulson@15300 281 paulson@15300 282 lemma congruent2_commuteI: paulson@15300 283 assumes equivA: "equiv A r" paulson@15300 284 and commute: "!!y z. y \ A ==> z \ A ==> f y z = f z y" paulson@15300 285 and congt: "!!y z w. w \ A ==> (y,z) \ r ==> f w y = f w z" paulson@15300 286 shows "f respects2 r" paulson@15300 287 apply (rule congruent2I [OF equivA equivA]) paulson@15300 288 apply (rule commute [THEN trans]) paulson@15300 289 apply (rule_tac [3] commute [THEN trans, symmetric]) paulson@15300 290 apply (rule_tac [5] sym) paulson@15300 291 apply (assumption | rule congt | paulson@15300 292 erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+ paulson@15300 293 done paulson@15300 294 paulson@15300 295 end