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 *)

     4

     5 section \<open>Equivalence Relations in Higher-Order Set Theory\<close>

     6

     7 theory Equiv_Relations

     8 imports Groups_Big Relation

     9 begin

    10

    11 subsection \<open>Equivalence relations -- set version\<close>

    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 \<open>

    26   Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O

    27   r = r\<close>.

    28

    29   First half: \<open>equiv A r ==> r\<inverse> O r = r\<close>.

    30 \<close>

    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 \<open>Second half.\<close>

    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 \<open>Equivalence classes\<close>

    59

    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

    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   \<comment> \<open>lemma for the next result\<close>

    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 \<open>Quotients\<close>

   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}})"  \<comment> \<open>set of equiv classes\<close>

   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 subsection \<open>Refinement of one equivalence relation WRT another\<close>

   164

   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)

   168

   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)

   172

   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)

   176

   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

   182

   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

   188

   189

   190 subsection \<open>Defining unary operations upon equivalence classes\<close>

   191

   192 text\<open>A congruence-preserving function\<close>

   193

   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)"

   196

   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)

   200

   201 lemma congruentD:

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

   203   by (auto simp add: congruent_def)

   204

   205 abbreviation

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

   207     (infixr "respects" 80) where

   208   "f respects r == congruent r f"

   209

   210

   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

   214

   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

   223

   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

   232

   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>

   238

   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

   253

   254

   255 subsection \<open>Defining binary operations upon equivalence classes\<close>

   256

   257 text\<open>A congruence-preserving function of two arguments\<close>

   258

   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)"

   261

   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)

   266

   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)

   270

   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"

   276

   277

   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

   281

   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

   292

   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)

   298

   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

   309

   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

   316

   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

   328

   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 [3] commute [THEN trans, symmetric])

   337        apply (rule_tac [5] sym)

   338        apply (rule congt | assumption |

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

   340   done

   341

   342

   343 subsection \<open>Quotients and finiteness\<close>

   344

   345 text \<open>Suggested by Florian KammÃ¼ller\<close>

   346

   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 [2] finite_Pow_iff [THEN iffD2])

   351   apply (unfold quotient_def)

   352   apply blast

   353   done

   354

   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

   362

   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

   372

   373 lemma card_quotient_disjoint:

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

   375 apply(simp add:quotient_def)

   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

   382

   383

   384 subsection \<open>Projection\<close>

   385

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

   387

   388 lemma proj_preserves:

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

   390 unfolding proj_def by (rule quotientI)

   391

   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

   402

   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)

   406

   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 *)

   411

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

   413 unfolding proj_def[abs_def] quotient_def by blast

   414

   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

   418

   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

   422

   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

   426

   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

   430

   431

   432 subsection \<open>Equivalence relations -- predicate version\<close>

   433

   434 text \<open>Partial equivalences\<close>

   435

   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>

   439

   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)

   443

   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

   466

   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)

   470

   471 lemma part_equivp_symp:

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

   473   by (erule part_equivpE, erule sympE)

   474

   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)

   478

   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)

   482

   483

   484 text \<open>Total equivalences\<close>

   485

   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>

   488

   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)

   492

   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)

   497

   498 lemma equivp_implies_part_equivp:

   499   "equivp R \<Longrightarrow> part_equivp R"

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

   501

   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])

   505

   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)

   509

   510 lemma identity_equivp:

   511   "equivp (op =)"

   512   by (auto intro: equivpI reflpI sympI transpI)

   513

   514 lemma equivp_reflp:

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

   516   by (erule equivpE, erule reflpE)

   517

   518 lemma equivp_symp:

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

   520   by (erule equivpE, erule sympE)

   521

   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)

   525

   526 hide_const (open) proj

   527

   528 end