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. S``X) ` (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. S``X"])
   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. S``X"])
   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 Kammüller\<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