src/HOL/Equiv_Relations.thy
 author blanchet Thu Jan 16 21:22:01 2014 +0100 (2014-01-16) changeset 55024 05cc0dbf3a50 parent 55022 eeba3ba73486 child 58889 5b7a9633cfa8 permissions -rw-r--r--
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     1 (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory

     2     Copyright   1996  University of Cambridge

     3 *)

     4

     5 header {* Equivalence Relations in Higher-Order Set Theory *}

     6

     7 theory Equiv_Relations

     8 imports Groups_Big Relation

     9 begin

    10

    11 subsection {* Equivalence relations -- set version *}

    12

    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"

    15

    16 lemma equivI:

    17   "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"

    18   by (simp add: equiv_def)

    19

    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)

    24

    25 text {*

    26   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O

    27   r = r"}.

    28

    29   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.

    30 *}

    31

    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

    35

    36 lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"

    37   by (unfold refl_on_def) blast

    38

    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

    45

    46 text {* Second half. *}

    47

    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

    56

    57

    58 subsection {* Equivalence classes *}

    59

    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

    64

    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

    70

    71 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"

    72   by (unfold equiv_def refl_on_def) blast

    73

    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

    78

    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)

    82

    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

    86

    87 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"

    88   by (unfold equiv_def refl_on_def) blast

    89

    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)

    93

    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)

    97

    98

    99 subsection {* Quotients *}

   100

   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 *}

   103

   104 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"

   105   by (unfold quotient_def) blast

   106

   107 lemma quotientE:

   108   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"

   109   by (unfold quotient_def) blast

   110

   111 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"

   112   by (unfold equiv_def refl_on_def quotient_def) blast

   113

   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

   123

   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

   130

   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

   138

   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)

   142

   143

   144 lemma quotient_empty [simp]: "{}//r = {}"

   145 by(simp add: quotient_def)

   146

   147 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"

   148 by(simp add: quotient_def)

   149

   150 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"

   151 by(simp add: quotient_def)

   152

   153

   154 lemma singleton_quotient: "{x}//r = {r  {x}}"

   155 by(simp add:quotient_def)

   156

   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

   162

   163

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

   165

   166 text{*A congruence-preserving function*}

   167

   168 definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where

   169   "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"

   170

   171 lemma congruentI:

   172   "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"

   173   by (auto simp add: congruent_def)

   174

   175 lemma congruentD:

   176   "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"

   177   by (auto simp add: congruent_def)

   178

   179 abbreviation

   180   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"

   181     (infixr "respects" 80) where

   182   "f respects r == congruent r f"

   183

   184

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

   186   -- {* lemma required to prove @{text UN_equiv_class} *}

   187   by auto

   188

   189 lemma UN_equiv_class:

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

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

   192   -- {* Conversion rule *}

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

   194   apply (unfold equiv_def congruent_def sym_def)

   195   apply (blast del: equalityI)

   196   done

   197

   198 lemma UN_equiv_class_type:

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

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

   201   apply (unfold quotient_def)

   202   apply clarify

   203   apply (subst UN_equiv_class)

   204      apply auto

   205   done

   206

   207 text {*

   208   Sufficient conditions for injectiveness.  Could weaken premises!

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

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

   211 *}

   212

   213 lemma UN_equiv_class_inject:

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

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

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

   217     ==> X = Y"

   218   apply (unfold quotient_def)

   219   apply clarify

   220   apply (rule equiv_class_eq)

   221    apply assumption

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

   223    apply blast

   224   apply (erule box_equals)

   225    apply (assumption | rule UN_equiv_class)+

   226   done

   227

   228

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

   230

   231 text{*A congruence-preserving function of two arguments*}

   232

   233 definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where

   234   "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"

   235

   236 lemma congruent2I':

   237   assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"

   238   shows "congruent2 r1 r2 f"

   239   using assms by (auto simp add: congruent2_def)

   240

   241 lemma congruent2D:

   242   "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"

   243   using assms by (auto simp add: congruent2_def)

   244

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

   246 abbreviation

   247   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"

   248     (infixr "respects2" 80) where

   249   "f respects2 r == congruent2 r r f"

   250

   251

   252 lemma congruent2_implies_congruent:

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

   254   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast

   255

   256 lemma congruent2_implies_congruent_UN:

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

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

   259   apply (unfold congruent_def)

   260   apply clarify

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

   262   apply (simp add: UN_equiv_class congruent2_implies_congruent)

   263   apply (unfold congruent2_def equiv_def refl_on_def)

   264   apply (blast del: equalityI)

   265   done

   266

   267 lemma UN_equiv_class2:

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

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

   270   by (simp add: UN_equiv_class congruent2_implies_congruent

   271     congruent2_implies_congruent_UN)

   272

   273 lemma UN_equiv_class_type2:

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

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

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

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

   278   apply (unfold quotient_def)

   279   apply clarify

   280   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN

   281     congruent2_implies_congruent quotientI)

   282   done

   283

   284 lemma UN_UN_split_split_eq:

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

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

   287   -- {* Allows a natural expression of binary operators, *}

   288   -- {* without explicit calls to @{text split} *}

   289   by auto

   290

   291 lemma congruent2I:

   292   "equiv A1 r1 ==> equiv A2 r2

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

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

   295     ==> congruent2 r1 r2 f"

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

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

   298   apply (unfold congruent2_def equiv_def refl_on_def)

   299   apply clarify

   300   apply (blast intro: trans)

   301   done

   302

   303 lemma congruent2_commuteI:

   304   assumes equivA: "equiv A r"

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

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

   307   shows "f respects2 r"

   308   apply (rule congruent2I [OF equivA equivA])

   309    apply (rule commute [THEN trans])

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

   311        apply (rule_tac [5] sym)

   312        apply (rule congt | assumption |

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

   314   done

   315

   316

   317 subsection {* Quotients and finiteness *}

   318

   319 text {*Suggested by Florian KammÃ¼ller*}

   320

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

   322   -- {* recall @{thm equiv_type} *}

   323   apply (rule finite_subset)

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

   325   apply (unfold quotient_def)

   326   apply blast

   327   done

   328

   329 lemma finite_equiv_class:

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

   331   apply (unfold quotient_def)

   332   apply (rule finite_subset)

   333    prefer 2 apply assumption

   334   apply blast

   335   done

   336

   337 lemma equiv_imp_dvd_card:

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

   339     ==> k dvd card A"

   340   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])

   341    apply assumption

   342   apply (rule dvd_partition)

   343      prefer 3 apply (blast dest: quotient_disj)

   344     apply (simp_all add: Union_quotient equiv_type)

   345   done

   346

   347 lemma card_quotient_disjoint:

   348  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"

   349 apply(simp add:quotient_def)

   350 apply(subst card_UN_disjoint)

   351    apply assumption

   352   apply simp

   353  apply(fastforce simp add:inj_on_def)

   354 apply simp

   355 done

   356

   357

   358 subsection {* Projection *}

   359

   360 definition proj where "proj r x = r  {x}"

   361

   362 lemma proj_preserves:

   363 "x \<in> A \<Longrightarrow> proj r x \<in> A//r"

   364 unfolding proj_def by (rule quotientI)

   365

   366 lemma proj_in_iff:

   367 assumes "equiv A r"

   368 shows "(proj r x \<in> A//r) = (x \<in> A)"

   369 apply(rule iffI, auto simp add: proj_preserves)

   370 unfolding proj_def quotient_def proof clarsimp

   371   fix y assume y: "y \<in> A" and "r  {x} = r  {y}"

   372   moreover have "y \<in> r  {y}" using assms y unfolding equiv_def refl_on_def by blast

   373   ultimately have "(x,y) \<in> r" by blast

   374   thus "x \<in> A" using assms unfolding equiv_def refl_on_def by blast

   375 qed

   376

   377 lemma proj_iff:

   378 "\<lbrakk>equiv A r; {x,y} \<subseteq> A\<rbrakk> \<Longrightarrow> (proj r x = proj r y) = ((x,y) \<in> r)"

   379 by (simp add: proj_def eq_equiv_class_iff)

   380

   381 (*

   382 lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"

   383 unfolding proj_def equiv_def refl_on_def by blast

   384 *)

   385

   386 lemma proj_image: "(proj r)  A = A//r"

   387 unfolding proj_def[abs_def] quotient_def by blast

   388

   389 lemma in_quotient_imp_non_empty:

   390 "\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<noteq> {}"

   391 unfolding quotient_def using equiv_class_self by fast

   392

   393 lemma in_quotient_imp_in_rel:

   394 "\<lbrakk>equiv A r; X \<in> A//r; {x,y} \<subseteq> X\<rbrakk> \<Longrightarrow> (x,y) \<in> r"

   395 using quotient_eq_iff[THEN iffD1] by fastforce

   396

   397 lemma in_quotient_imp_closed:

   398 "\<lbrakk>equiv A r; X \<in> A//r; x \<in> X; (x,y) \<in> r\<rbrakk> \<Longrightarrow> y \<in> X"

   399 unfolding quotient_def equiv_def trans_def by blast

   400

   401 lemma in_quotient_imp_subset:

   402 "\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<subseteq> A"

   403 using assms in_quotient_imp_in_rel equiv_type by fastforce

   404

   405

   406 subsection {* Equivalence relations -- predicate version *}

   407

   408 text {* Partial equivalences *}

   409

   410 definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   411   "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)"

   412     -- {* John-Harrison-style characterization *}

   413

   414 lemma part_equivpI:

   415   "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"

   416   by (auto simp add: part_equivp_def) (auto elim: sympE transpE)

   417

   418 lemma part_equivpE:

   419   assumes "part_equivp R"

   420   obtains x where "R x x" and "symp R" and "transp R"

   421 proof -

   422   from assms have 1: "\<exists>x. R x x"

   423     and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"

   424     by (unfold part_equivp_def) blast+

   425   from 1 obtain x where "R x x" ..

   426   moreover have "symp R"

   427   proof (rule sympI)

   428     fix x y

   429     assume "R x y"

   430     with 2 [of x y] show "R y x" by auto

   431   qed

   432   moreover have "transp R"

   433   proof (rule transpI)

   434     fix x y z

   435     assume "R x y" and "R y z"

   436     with 2 [of x y] 2 [of y z] show "R x z" by auto

   437   qed

   438   ultimately show thesis by (rule that)

   439 qed

   440

   441 lemma part_equivp_refl_symp_transp:

   442   "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"

   443   by (auto intro: part_equivpI elim: part_equivpE)

   444

   445 lemma part_equivp_symp:

   446   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"

   447   by (erule part_equivpE, erule sympE)

   448

   449 lemma part_equivp_transp:

   450   "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"

   451   by (erule part_equivpE, erule transpE)

   452

   453 lemma part_equivp_typedef:

   454   "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"

   455   by (auto elim: part_equivpE)

   456

   457

   458 text {* Total equivalences *}

   459

   460 definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where

   461   "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}

   462

   463 lemma equivpI:

   464   "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"

   465   by (auto elim: reflpE sympE transpE simp add: equivp_def)

   466

   467 lemma equivpE:

   468   assumes "equivp R"

   469   obtains "reflp R" and "symp R" and "transp R"

   470   using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)

   471

   472 lemma equivp_implies_part_equivp:

   473   "equivp R \<Longrightarrow> part_equivp R"

   474   by (auto intro: part_equivpI elim: equivpE reflpE)

   475

   476 lemma equivp_equiv:

   477   "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"

   478   by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])

   479

   480 lemma equivp_reflp_symp_transp:

   481   shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"

   482   by (auto intro: equivpI elim: equivpE)

   483

   484 lemma identity_equivp:

   485   "equivp (op =)"

   486   by (auto intro: equivpI reflpI sympI transpI)

   487

   488 lemma equivp_reflp:

   489   "equivp R \<Longrightarrow> R x x"

   490   by (erule equivpE, erule reflpE)

   491

   492 lemma equivp_symp:

   493   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"

   494   by (erule equivpE, erule sympE)

   495

   496 lemma equivp_transp:

   497   "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"

   498   by (erule equivpE, erule transpE)

   499

   500 hide_const (open) proj

   501

   502 end
`