src/HOL/Equiv_Relations.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 46752 e9e7209eb375 child 51112 da97167e03f7 permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
1 (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
2     Copyright   1996  University of Cambridge
3 *)
5 header {* Equivalence Relations in Higher-Order Set Theory *}
7 theory Equiv_Relations
8 imports Big_Operators Relation Plain
9 begin
11 subsection {* Equivalence relations -- set version *}
13 definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
14   "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
16 lemma equivI:
17   "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
18   by (simp add: equiv_def)
20 lemma equivE:
21   assumes "equiv A r"
22   obtains "refl_on A r" and "sym r" and "trans r"
23   using assms by (simp add: equiv_def)
25 text {*
26   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
27   r = r"}.
29   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
30 *}
32 lemma sym_trans_comp_subset:
33     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
34   by (unfold trans_def sym_def converse_unfold) blast
36 lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
37   by (unfold refl_on_def) blast
39 lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
40   apply (unfold equiv_def)
41   apply clarify
42   apply (rule equalityI)
43    apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+
44   done
46 text {* Second half. *}
48 lemma comp_equivI:
49     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
50   apply (unfold equiv_def refl_on_def sym_def trans_def)
51   apply (erule equalityE)
52   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
53    apply fast
54   apply fast
55   done
58 subsection {* Equivalence classes *}
60 lemma equiv_class_subset:
61   "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"
62   -- {* lemma for the next result *}
63   by (unfold equiv_def trans_def sym_def) blast
65 theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r{a} = r{b}"
66   apply (assumption | rule equalityI equiv_class_subset)+
67   apply (unfold equiv_def sym_def)
68   apply blast
69   done
71 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"
72   by (unfold equiv_def refl_on_def) blast
74 lemma subset_equiv_class:
75     "equiv A r ==> r{b} \<subseteq> r{a} ==> b \<in> A ==> (a,b) \<in> r"
76   -- {* lemma for the next result *}
77   by (unfold equiv_def refl_on_def) blast
79 lemma eq_equiv_class:
80     "r{a} = r{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
81   by (iprover intro: equalityD2 subset_equiv_class)
83 lemma equiv_class_nondisjoint:
84     "equiv A r ==> x \<in> (r{a} \<inter> r{b}) ==> (a, b) \<in> r"
85   by (unfold equiv_def trans_def sym_def) blast
87 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
88   by (unfold equiv_def refl_on_def) blast
90 theorem equiv_class_eq_iff:
91   "equiv A r ==> ((x, y) \<in> r) = (r{x} = r{y} & x \<in> A & y \<in> A)"
92   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
94 theorem eq_equiv_class_iff:
95   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r{x} = r{y}) = ((x, y) \<in> r)"
96   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
99 subsection {* Quotients *}
101 definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90) where
102   "A//r = (\<Union>x \<in> A. {r{x}})"  -- {* set of equiv classes *}
104 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"
105   by (unfold quotient_def) blast
107 lemma quotientE:
108   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"
109   by (unfold quotient_def) blast
111 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
112   by (unfold equiv_def refl_on_def quotient_def) blast
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
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
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
139 lemma eq_equiv_class_iff2:
140   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
141 by(simp add:quotient_def eq_equiv_class_iff)
144 lemma quotient_empty [simp]: "{}//r = {}"
145 by(simp add: quotient_def)
147 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
148 by(simp add: quotient_def)
150 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
151 by(simp add: quotient_def)
154 lemma singleton_quotient: "{x}//r = {r  {x}}"
155 by(simp add:quotient_def)
157 lemma quotient_diff1:
158   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
159 apply(simp add:quotient_def inj_on_def)
160 apply blast
161 done
163 subsection {* Defining unary operations upon equivalence classes *}
165 text{*A congruence-preserving function*}
167 definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
168   "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
170 lemma congruentI:
171   "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
172   by (auto simp add: congruent_def)
174 lemma congruentD:
175   "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
176   by (auto simp add: congruent_def)
178 abbreviation
179   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
180     (infixr "respects" 80) where
181   "f respects r == congruent r f"
184 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
185   -- {* lemma required to prove @{text UN_equiv_class} *}
186   by auto
188 lemma UN_equiv_class:
189   "equiv A r ==> f respects r ==> a \<in> A
190     ==> (\<Union>x \<in> r{a}. f x) = f a"
191   -- {* Conversion rule *}
192   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
193   apply (unfold equiv_def congruent_def sym_def)
194   apply (blast del: equalityI)
195   done
197 lemma UN_equiv_class_type:
198   "equiv A r ==> f respects r ==> X \<in> A//r ==>
199     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
200   apply (unfold quotient_def)
201   apply clarify
202   apply (subst UN_equiv_class)
203      apply auto
204   done
206 text {*
207   Sufficient conditions for injectiveness.  Could weaken premises!
208   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
209   A ==> f y \<in> B"}.
210 *}
212 lemma UN_equiv_class_inject:
213   "equiv A r ==> f respects r ==>
214     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
215     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
216     ==> X = Y"
217   apply (unfold quotient_def)
218   apply clarify
219   apply (rule equiv_class_eq)
220    apply assumption
221   apply (subgoal_tac "f x = f xa")
222    apply blast
223   apply (erule box_equals)
224    apply (assumption | rule UN_equiv_class)+
225   done
228 subsection {* Defining binary operations upon equivalence classes *}
230 text{*A congruence-preserving function of two arguments*}
232 definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
233   "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
235 lemma congruent2I':
236   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
237   shows "congruent2 r1 r2 f"
238   using assms by (auto simp add: congruent2_def)
240 lemma congruent2D:
241   "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
242   using assms by (auto simp add: congruent2_def)
244 text{*Abbreviation for the common case where the relations are identical*}
245 abbreviation
246   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
247     (infixr "respects2" 80) where
248   "f respects2 r == congruent2 r r f"
251 lemma congruent2_implies_congruent:
252     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
253   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
255 lemma congruent2_implies_congruent_UN:
256   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
257     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2{a}. f x1 x2)"
258   apply (unfold congruent_def)
259   apply clarify
260   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
261   apply (simp add: UN_equiv_class congruent2_implies_congruent)
262   apply (unfold congruent2_def equiv_def refl_on_def)
263   apply (blast del: equalityI)
264   done
266 lemma UN_equiv_class2:
267   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
268     ==> (\<Union>x1 \<in> r1{a1}. \<Union>x2 \<in> r2{a2}. f x1 x2) = f a1 a2"
269   by (simp add: UN_equiv_class congruent2_implies_congruent
270     congruent2_implies_congruent_UN)
272 lemma UN_equiv_class_type2:
273   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
274     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
275     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
276     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
277   apply (unfold quotient_def)
278   apply clarify
279   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
280     congruent2_implies_congruent quotientI)
281   done
283 lemma UN_UN_split_split_eq:
284   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
285     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
286   -- {* Allows a natural expression of binary operators, *}
287   -- {* without explicit calls to @{text split} *}
288   by auto
290 lemma congruent2I:
291   "equiv A1 r1 ==> equiv A2 r2
292     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
293     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
294     ==> congruent2 r1 r2 f"
295   -- {* Suggested by John Harrison -- the two subproofs may be *}
296   -- {* \emph{much} simpler than the direct proof. *}
297   apply (unfold congruent2_def equiv_def refl_on_def)
298   apply clarify
299   apply (blast intro: trans)
300   done
302 lemma congruent2_commuteI:
303   assumes equivA: "equiv A r"
304     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
305     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
306   shows "f respects2 r"
307   apply (rule congruent2I [OF equivA equivA])
308    apply (rule commute [THEN trans])
309      apply (rule_tac [3] commute [THEN trans, symmetric])
310        apply (rule_tac [5] sym)
311        apply (rule congt | assumption |
312          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
313   done
316 subsection {* Quotients and finiteness *}
318 text {*Suggested by Florian KammÃ¼ller*}
320 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
321   -- {* recall @{thm equiv_type} *}
322   apply (rule finite_subset)
323    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
324   apply (unfold quotient_def)
325   apply blast
326   done
328 lemma finite_equiv_class:
329   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
330   apply (unfold quotient_def)
331   apply (rule finite_subset)
332    prefer 2 apply assumption
333   apply blast
334   done
336 lemma equiv_imp_dvd_card:
337   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
338     ==> k dvd card A"
339   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
340    apply assumption
341   apply (rule dvd_partition)
342      prefer 3 apply (blast dest: quotient_disj)
343     apply (simp_all add: Union_quotient equiv_type)
344   done
346 lemma card_quotient_disjoint:
347  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
348 apply(simp add:quotient_def)
349 apply(subst card_UN_disjoint)
350    apply assumption
351   apply simp
352  apply(fastforce simp add:inj_on_def)
353 apply simp
354 done
357 subsection {* Equivalence relations -- predicate version *}
359 text {* Partial equivalences *}
361 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
362   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
363     -- {* John-Harrison-style characterization *}
365 lemma part_equivpI:
366   "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
367   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
369 lemma part_equivpE:
370   assumes "part_equivp R"
371   obtains x where "R x x" and "symp R" and "transp R"
372 proof -
373   from assms have 1: "\<exists>x. R x x"
374     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
375     by (unfold part_equivp_def) blast+
376   from 1 obtain x where "R x x" ..
377   moreover have "symp R"
378   proof (rule sympI)
379     fix x y
380     assume "R x y"
381     with 2 [of x y] show "R y x" by auto
382   qed
383   moreover have "transp R"
384   proof (rule transpI)
385     fix x y z
386     assume "R x y" and "R y z"
387     with 2 [of x y] 2 [of y z] show "R x z" by auto
388   qed
389   ultimately show thesis by (rule that)
390 qed
392 lemma part_equivp_refl_symp_transp:
393   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
394   by (auto intro: part_equivpI elim: part_equivpE)
396 lemma part_equivp_symp:
397   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
398   by (erule part_equivpE, erule sympE)
400 lemma part_equivp_transp:
401   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
402   by (erule part_equivpE, erule transpE)
404 lemma part_equivp_typedef:
405   "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
406   by (auto elim: part_equivpE)
409 text {* Total equivalences *}
411 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
412   "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}
414 lemma equivpI:
415   "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
416   by (auto elim: reflpE sympE transpE simp add: equivp_def)
418 lemma equivpE:
419   assumes "equivp R"
420   obtains "reflp R" and "symp R" and "transp R"
421   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
423 lemma equivp_implies_part_equivp:
424   "equivp R \<Longrightarrow> part_equivp R"
425   by (auto intro: part_equivpI elim: equivpE reflpE)
427 lemma equivp_equiv:
428   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
429   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
431 lemma equivp_reflp_symp_transp:
432   shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
433   by (auto intro: equivpI elim: equivpE)
435 lemma identity_equivp:
436   "equivp (op =)"
437   by (auto intro: equivpI reflpI sympI transpI)
439 lemma equivp_reflp:
440   "equivp R \<Longrightarrow> R x x"
441   by (erule equivpE, erule reflpE)
443 lemma equivp_symp:
444   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
445   by (erule equivpE, erule sympE)
447 lemma equivp_transp:
448   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
449   by (erule equivpE, erule transpE)
451 end