src/HOL/Equiv_Relations.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 61952 546958347e05 child 63092 a949b2a5f51d permissions -rw-r--r--
generalize more theorems to support enat and ennreal
1 (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
2     Copyright   1996  University of Cambridge
3 *)
5 section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
7 theory Equiv_Relations
8 imports Groups_Big Relation
9 begin
11 subsection \<open>Equivalence relations -- set version\<close>
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 \<open>
26   Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O
27   r = r\<close>.
29   First half: \<open>equiv A r ==> r\<inverse> O r = r\<close>.
30 \<close>
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 \<open>Second half.\<close>
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 \<open>Equivalence classes\<close>
60 lemma equiv_class_subset:
61   "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"
62   \<comment> \<open>lemma for the next result\<close>
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   \<comment> \<open>lemma for the next result\<close>
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 \<open>Quotients\<close>
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}})"  \<comment> \<open>set of equiv classes\<close>
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}}"
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 \<open>Refinement of one equivalence relation WRT another\<close>
165 lemma refines_equiv_class_eq:
166    "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> R(S{a}) = S{a}"
167   by (auto simp: equiv_class_eq_iff)
169 lemma refines_equiv_class_eq2:
170    "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> S(R{a}) = S{a}"
171   by (auto simp: equiv_class_eq_iff)
173 lemma refines_equiv_image_eq:
174    "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> (\<lambda>X. SX)  (A//R) = A//S"
175    by (auto simp: quotient_def image_UN refines_equiv_class_eq2)
177 lemma finite_refines_finite:
178    "\<lbrakk>finite (A//R); R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> finite (A//S)"
179     apply (erule finite_surj [where f = "\<lambda>X. SX"])
180     apply (simp add: refines_equiv_image_eq)
181     done
183 lemma finite_refines_card_le:
184    "\<lbrakk>finite (A//R); R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> card (A//S) \<le> card (A//R)"
185   apply (subst refines_equiv_image_eq [of R S A, symmetric])
186   apply (auto simp: card_image_le [where f = "\<lambda>X. SX"])
187   done
190 subsection \<open>Defining unary operations upon equivalence classes\<close>
192 text\<open>A congruence-preserving function\<close>
194 definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
195   "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
197 lemma congruentI:
198   "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
199   by (auto simp add: congruent_def)
201 lemma congruentD:
202   "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
203   by (auto simp add: congruent_def)
205 abbreviation
206   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
207     (infixr "respects" 80) where
208   "f respects r == congruent r f"
211 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
212   \<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
213   by auto
215 lemma UN_equiv_class:
216   "equiv A r ==> f respects r ==> a \<in> A
217     ==> (\<Union>x \<in> r{a}. f x) = f a"
218   \<comment> \<open>Conversion rule\<close>
219   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
220   apply (unfold equiv_def congruent_def sym_def)
221   apply (blast del: equalityI)
222   done
224 lemma UN_equiv_class_type:
225   "equiv A r ==> f respects r ==> X \<in> A//r ==>
226     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
227   apply (unfold quotient_def)
228   apply clarify
229   apply (subst UN_equiv_class)
230      apply auto
231   done
233 text \<open>
234   Sufficient conditions for injectiveness.  Could weaken premises!
235   major premise could be an inclusion; bcong could be \<open>!!y. y \<in>
236   A ==> f y \<in> B\<close>.
237 \<close>
239 lemma UN_equiv_class_inject:
240   "equiv A r ==> f respects r ==>
241     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
242     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
243     ==> X = Y"
244   apply (unfold quotient_def)
245   apply clarify
246   apply (rule equiv_class_eq)
247    apply assumption
248   apply (subgoal_tac "f x = f xa")
249    apply blast
250   apply (erule box_equals)
251    apply (assumption | rule UN_equiv_class)+
252   done
255 subsection \<open>Defining binary operations upon equivalence classes\<close>
257 text\<open>A congruence-preserving function of two arguments\<close>
259 definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
260   "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
262 lemma congruent2I':
263   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
264   shows "congruent2 r1 r2 f"
265   using assms by (auto simp add: congruent2_def)
267 lemma congruent2D:
268   "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
269   using assms by (auto simp add: congruent2_def)
271 text\<open>Abbreviation for the common case where the relations are identical\<close>
272 abbreviation
273   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
274     (infixr "respects2" 80) where
275   "f respects2 r == congruent2 r r f"
278 lemma congruent2_implies_congruent:
279     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
280   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
282 lemma congruent2_implies_congruent_UN:
283   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
284     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2{a}. f x1 x2)"
285   apply (unfold congruent_def)
286   apply clarify
287   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
288   apply (simp add: UN_equiv_class congruent2_implies_congruent)
289   apply (unfold congruent2_def equiv_def refl_on_def)
290   apply (blast del: equalityI)
291   done
293 lemma UN_equiv_class2:
294   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
295     ==> (\<Union>x1 \<in> r1{a1}. \<Union>x2 \<in> r2{a2}. f x1 x2) = f a1 a2"
296   by (simp add: UN_equiv_class congruent2_implies_congruent
297     congruent2_implies_congruent_UN)
299 lemma UN_equiv_class_type2:
300   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
301     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
302     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
303     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
304   apply (unfold quotient_def)
305   apply clarify
306   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
307     congruent2_implies_congruent quotientI)
308   done
310 lemma UN_UN_split_split_eq:
311   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
312     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
313   \<comment> \<open>Allows a natural expression of binary operators,\<close>
314   \<comment> \<open>without explicit calls to \<open>split\<close>\<close>
315   by auto
317 lemma congruent2I:
318   "equiv A1 r1 ==> equiv A2 r2
319     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
320     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
321     ==> congruent2 r1 r2 f"
322   \<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<close>
323   \<comment> \<open>\emph{much} simpler than the direct proof.\<close>
324   apply (unfold congruent2_def equiv_def refl_on_def)
325   apply clarify
326   apply (blast intro: trans)
327   done
329 lemma congruent2_commuteI:
330   assumes equivA: "equiv A r"
331     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
332     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
333   shows "f respects2 r"
334   apply (rule congruent2I [OF equivA equivA])
335    apply (rule commute [THEN trans])
336      apply (rule_tac  commute [THEN trans, symmetric])
337        apply (rule_tac  sym)
338        apply (rule congt | assumption |
339          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
340   done
343 subsection \<open>Quotients and finiteness\<close>
345 text \<open>Suggested by Florian Kammüller\<close>
347 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
348   \<comment> \<open>recall @{thm equiv_type}\<close>
349   apply (rule finite_subset)
350    apply (erule_tac  finite_Pow_iff [THEN iffD2])
351   apply (unfold quotient_def)
352   apply blast
353   done
355 lemma finite_equiv_class:
356   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
357   apply (unfold quotient_def)
358   apply (rule finite_subset)
359    prefer 2 apply assumption
360   apply blast
361   done
363 lemma equiv_imp_dvd_card:
364   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
365     ==> k dvd card A"
366   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
367    apply assumption
368   apply (rule dvd_partition)
369      prefer 3 apply (blast dest: quotient_disj)
370     apply (simp_all add: Union_quotient equiv_type)
371   done
373 lemma card_quotient_disjoint:
374  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
376 apply(subst card_UN_disjoint)
377    apply assumption
378   apply simp
379  apply(fastforce simp add:inj_on_def)
380 apply simp
381 done
384 subsection \<open>Projection\<close>
386 definition proj where "proj r x = r  {x}"
388 lemma proj_preserves:
389 "x \<in> A \<Longrightarrow> proj r x \<in> A//r"
390 unfolding proj_def by (rule quotientI)
392 lemma proj_in_iff:
393 assumes "equiv A r"
394 shows "(proj r x \<in> A//r) = (x \<in> A)"
395 apply(rule iffI, auto simp add: proj_preserves)
396 unfolding proj_def quotient_def proof clarsimp
397   fix y assume y: "y \<in> A" and "r  {x} = r  {y}"
398   moreover have "y \<in> r  {y}" using assms y unfolding equiv_def refl_on_def by blast
399   ultimately have "(x,y) \<in> r" by blast
400   thus "x \<in> A" using assms unfolding equiv_def refl_on_def by blast
401 qed
403 lemma proj_iff:
404 "\<lbrakk>equiv A r; {x,y} \<subseteq> A\<rbrakk> \<Longrightarrow> (proj r x = proj r y) = ((x,y) \<in> r)"
405 by (simp add: proj_def eq_equiv_class_iff)
407 (*
408 lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
409 unfolding proj_def equiv_def refl_on_def by blast
410 *)
412 lemma proj_image: "(proj r)  A = A//r"
413 unfolding proj_def[abs_def] quotient_def by blast
415 lemma in_quotient_imp_non_empty:
416 "\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<noteq> {}"
417 unfolding quotient_def using equiv_class_self by fast
419 lemma in_quotient_imp_in_rel:
420 "\<lbrakk>equiv A r; X \<in> A//r; {x,y} \<subseteq> X\<rbrakk> \<Longrightarrow> (x,y) \<in> r"
421 using quotient_eq_iff[THEN iffD1] by fastforce
423 lemma in_quotient_imp_closed:
424 "\<lbrakk>equiv A r; X \<in> A//r; x \<in> X; (x,y) \<in> r\<rbrakk> \<Longrightarrow> y \<in> X"
425 unfolding quotient_def equiv_def trans_def by blast
427 lemma in_quotient_imp_subset:
428 "\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<subseteq> A"
429 using assms in_quotient_imp_in_rel equiv_type by fastforce
432 subsection \<open>Equivalence relations -- predicate version\<close>
434 text \<open>Partial equivalences\<close>
436 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
437   "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)"
438     \<comment> \<open>John-Harrison-style characterization\<close>
440 lemma part_equivpI:
441   "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
442   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
444 lemma part_equivpE:
445   assumes "part_equivp R"
446   obtains x where "R x x" and "symp R" and "transp R"
447 proof -
448   from assms have 1: "\<exists>x. R x x"
449     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
450     by (unfold part_equivp_def) blast+
451   from 1 obtain x where "R x x" ..
452   moreover have "symp R"
453   proof (rule sympI)
454     fix x y
455     assume "R x y"
456     with 2 [of x y] show "R y x" by auto
457   qed
458   moreover have "transp R"
459   proof (rule transpI)
460     fix x y z
461     assume "R x y" and "R y z"
462     with 2 [of x y] 2 [of y z] show "R x z" by auto
463   qed
464   ultimately show thesis by (rule that)
465 qed
467 lemma part_equivp_refl_symp_transp:
468   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
469   by (auto intro: part_equivpI elim: part_equivpE)
471 lemma part_equivp_symp:
472   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
473   by (erule part_equivpE, erule sympE)
475 lemma part_equivp_transp:
476   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
477   by (erule part_equivpE, erule transpE)
479 lemma part_equivp_typedef:
480   "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
481   by (auto elim: part_equivpE)
484 text \<open>Total equivalences\<close>
486 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
487   "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
489 lemma equivpI:
490   "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
491   by (auto elim: reflpE sympE transpE simp add: equivp_def)
493 lemma equivpE:
494   assumes "equivp R"
495   obtains "reflp R" and "symp R" and "transp R"
496   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
498 lemma equivp_implies_part_equivp:
499   "equivp R \<Longrightarrow> part_equivp R"
500   by (auto intro: part_equivpI elim: equivpE reflpE)
502 lemma equivp_equiv:
503   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
504   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
506 lemma equivp_reflp_symp_transp:
507   shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
508   by (auto intro: equivpI elim: equivpE)
510 lemma identity_equivp:
511   "equivp (op =)"
512   by (auto intro: equivpI reflpI sympI transpI)
514 lemma equivp_reflp:
515   "equivp R \<Longrightarrow> R x x"
516   by (erule equivpE, erule reflpE)
518 lemma equivp_symp:
519   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
520   by (erule equivpE, erule sympE)
522 lemma equivp_transp:
523   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
524   by (erule equivpE, erule transpE)
526 hide_const (open) proj
528 end