src/HOL/Equiv_Relations.thy
 author nipkow Sun Nov 21 15:44:20 2004 +0100 (2004-11-21) changeset 15303 eedbb8d22ca2 parent 15302 a643fcbc3468 child 15392 290bc97038c7 permissions -rw-r--r--
1 (*  ID:         $Id$
2     Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1996  University of Cambridge
4 *)
6 header {* Equivalence Relations in Higher-Order Set Theory *}
8 theory Equiv_Relations
9 imports Relation Finite_Set
10 begin
12 subsection {* Equivalence relations *}
14 locale equiv =
15   fixes A and r
16   assumes refl: "refl A r"
17     and sym: "sym r"
18     and trans: "trans r"
20 text {*
21   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
22   r = r"}.
24   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
25 *}
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
31 lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
32   by (unfold refl_def) blast
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
41 text {* Second half. *}
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
53 subsection {* Equivalence classes *}
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
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
66 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"
67   by (unfold equiv_def refl_def) blast
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
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)
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
82 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
83   by (unfold equiv_def refl_def) blast
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)
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)
94 subsection {* Quotients *}
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 *}
100 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"
101   by (unfold quotient_def) blast
103 lemma quotientE:
104   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"
105   by (unfold quotient_def) blast
107 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
108   by (unfold equiv_def refl_def quotient_def) blast
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
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
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
136 lemma quotient_empty [simp]: "{}//r = {}"
137 by(simp add: quotient_def)
139 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
140 by(simp add: quotient_def)
142 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
143 by(simp add: quotient_def)
146 lemma singleton_quotient: "{x}//r = {r  {x}}"
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
155 subsection {* Defining unary operations upon equivalence classes *}
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"
162 syntax
163   RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)
165 translations
166   "f respects r" == "congruent r f"
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
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
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
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 *}
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
213 subsection {* Defining binary operations upon equivalence classes *}
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"
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)
225 translations
226   "f respects2 r" => "congruent2 r r f"
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
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
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)
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
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
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
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  commute [THEN trans, symmetric])
287        apply (rule_tac  sym)
288        apply (assumption | rule congt |
289          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
290   done
293 subsection {* Cardinality results *}
295 text {*Suggested by Florian Kamm�ller*}
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  finite_Pow_iff [THEN iffD2])
301   apply (unfold quotient_def)
302   apply blast
303   done
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
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 4 apply (blast dest: quotient_disj)
320     apply (simp_all add: Union_quotient equiv_type finite_quotient)
321   done
323 lemma card_quotient_disjoint:
324  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
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_nat)
331 done
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 *}
371 end