# HG changeset patch
# User paulson
# Date 1100881909 -3600
# Node ID 7dd5853a4812392e42d17d4e60152254a86d1cc6
# Parent  576fd0b65ed859befa04552146724cccaea4d929
moved and renamed Integ/Equiv.thy

diff -r 576fd0b65ed8 -r 7dd5853a4812 src/HOL/Equiv_Relations.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Equiv_Relations.thy	Fri Nov 19 17:31:49 2004 +0100
@@ -0,0 +1,352 @@
+(*  ID:         $Id$
+    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1996  University of Cambridge
+*)
+
+header {* Equivalence Relations in Higher-Order Set Theory *}
+
+theory Equiv_Relations
+imports Relation Finite_Set
+begin
+
+subsection {* Equivalence relations *}
+
+locale equiv =
+  fixes A and r
+  assumes refl: "refl A r"
+    and sym: "sym r"
+    and trans: "trans r"
+
+text {*
+  Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
+  r = r"}.
+
+  First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
+*}
+
+lemma sym_trans_comp_subset:
+    "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
+  by (unfold trans_def sym_def converse_def) blast
+
+lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
+  by (unfold refl_def) blast
+
+lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
+  apply (unfold equiv_def)
+  apply clarify
+  apply (rule equalityI)
+   apply (rules intro: sym_trans_comp_subset refl_comp_subset)+
+  done
+
+text {* Second half. *}
+
+lemma comp_equivI:
+    "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
+  apply (unfold equiv_def refl_def sym_def trans_def)
+  apply (erule equalityE)
+  apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
+   apply fast
+  apply fast
+  done
+
+
+subsection {* Equivalence classes *}
+
+lemma equiv_class_subset:
+  "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
+  -- {* lemma for the next result *}
+  by (unfold equiv_def trans_def sym_def) blast
+
+theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
+  apply (assumption | rule equalityI equiv_class_subset)+
+  apply (unfold equiv_def sym_def)
+  apply blast
+  done
+
+lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
+  by (unfold equiv_def refl_def) blast
+
+lemma subset_equiv_class:
+    "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
+  -- {* lemma for the next result *}
+  by (unfold equiv_def refl_def) blast
+
+lemma eq_equiv_class:
+    "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
+  by (rules intro: equalityD2 subset_equiv_class)
+
+lemma equiv_class_nondisjoint:
+    "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
+  by (unfold equiv_def trans_def sym_def) blast
+
+lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
+  by (unfold equiv_def refl_def) blast
+
+theorem equiv_class_eq_iff:
+  "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
+  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
+
+theorem eq_equiv_class_iff:
+  "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
+  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
+
+
+subsection {* Quotients *}
+
+constdefs
+  quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
+  "A//r == \<Union>x \<in> A. {r``{x}}"  -- {* set of equiv classes *}
+
+lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
+  by (unfold quotient_def) blast
+
+lemma quotientE:
+  "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
+  by (unfold quotient_def) blast
+
+lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
+  by (unfold equiv_def refl_def quotient_def) blast
+
+lemma quotient_disj:
+  "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (rule equiv_class_eq)
+   apply assumption
+  apply (unfold equiv_def trans_def sym_def)
+  apply blast
+  done
+
+lemma quotient_eqI:
+  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y" 
+  apply (clarify elim!: quotientE)
+  apply (rule equiv_class_eq, assumption)
+  apply (unfold equiv_def sym_def trans_def, blast)
+  done
+
+lemma quotient_eq_iff:
+  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)" 
+  apply (rule iffI)  
+   prefer 2 apply (blast del: equalityI intro: quotient_eqI) 
+  apply (clarify elim!: quotientE)
+  apply (unfold equiv_def sym_def trans_def, blast)
+  done
+
+
+lemma quotient_empty [simp]: "{}//r = {}"
+by(simp add: quotient_def)
+
+lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
+by(simp add: quotient_def)
+
+lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
+by(simp add: quotient_def)
+
+
+subsection {* Defining unary operations upon equivalence classes *}
+
+text{*A congruence-preserving function*}
+locale congruent =
+  fixes r and f
+  assumes congruent: "(y,z) \<in> r ==> f y = f z"
+
+syntax
+  RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)
+
+translations
+  "f respects r" == "congruent r f"
+
+
+lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
+  -- {* lemma required to prove @{text UN_equiv_class} *}
+  by auto
+
+lemma UN_equiv_class:
+  "equiv A r ==> f respects r ==> a \<in> A
+    ==> (\<Union>x \<in> r``{a}. f x) = f a"
+  -- {* Conversion rule *}
+  apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
+  apply (unfold equiv_def congruent_def sym_def)
+  apply (blast del: equalityI)
+  done
+
+lemma UN_equiv_class_type:
+  "equiv A r ==> f respects r ==> X \<in> A//r ==>
+    (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (subst UN_equiv_class)
+     apply auto
+  done
+
+text {*
+  Sufficient conditions for injectiveness.  Could weaken premises!
+  major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
+  A ==> f y \<in> B"}.
+*}
+
+lemma UN_equiv_class_inject:
+  "equiv A r ==> f respects r ==>
+    (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
+    ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
+    ==> X = Y"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (rule equiv_class_eq)
+   apply assumption
+  apply (subgoal_tac "f x = f xa")
+   apply blast
+  apply (erule box_equals)
+   apply (assumption | rule UN_equiv_class)+
+  done
+
+
+subsection {* Defining binary operations upon equivalence classes *}
+
+text{*A congruence-preserving function of two arguments*}
+locale congruent2 =
+  fixes r1 and r2 and f
+  assumes congruent2:
+    "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
+
+text{*Abbreviation for the common case where the relations are identical*}
+syntax
+  RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects2 " 80)
+
+translations
+  "f respects2 r" => "congruent2 r r f"
+
+lemma congruent2_implies_congruent:
+    "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
+  by (unfold congruent_def congruent2_def equiv_def refl_def) blast
+
+lemma congruent2_implies_congruent_UN:
+  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
+    congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
+  apply (unfold congruent_def)
+  apply clarify
+  apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
+  apply (simp add: UN_equiv_class congruent2_implies_congruent)
+  apply (unfold congruent2_def equiv_def refl_def)
+  apply (blast del: equalityI)
+  done
+
+lemma UN_equiv_class2:
+  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
+    ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
+  by (simp add: UN_equiv_class congruent2_implies_congruent
+    congruent2_implies_congruent_UN)
+
+lemma UN_equiv_class_type2:
+  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
+    ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
+    ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
+    ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
+    congruent2_implies_congruent quotientI)
+  done
+
+lemma UN_UN_split_split_eq:
+  "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
+    (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
+  -- {* Allows a natural expression of binary operators, *}
+  -- {* without explicit calls to @{text split} *}
+  by auto
+
+lemma congruent2I:
+  "equiv A1 r1 ==> equiv A2 r2
+    ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
+    ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
+    ==> congruent2 r1 r2 f"
+  -- {* Suggested by John Harrison -- the two subproofs may be *}
+  -- {* \emph{much} simpler than the direct proof. *}
+  apply (unfold congruent2_def equiv_def refl_def)
+  apply clarify
+  apply (blast intro: trans)
+  done
+
+lemma congruent2_commuteI:
+  assumes equivA: "equiv A r"
+    and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
+    and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
+  shows "f respects2 r"
+  apply (rule congruent2I [OF equivA equivA])
+   apply (rule commute [THEN trans])
+     apply (rule_tac [3] commute [THEN trans, symmetric])
+       apply (rule_tac [5] sym)
+       apply (assumption | rule congt |
+         erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
+  done
+
+
+subsection {* Cardinality results *}
+
+text {*Suggested by Florian Kammüller*}
+
+lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
+  -- {* recall @{thm equiv_type} *}
+  apply (rule finite_subset)
+   apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
+  apply (unfold quotient_def)
+  apply blast
+  done
+
+lemma finite_equiv_class:
+  "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
+  apply (unfold quotient_def)
+  apply (rule finite_subset)
+   prefer 2 apply assumption
+  apply blast
+  done
+
+lemma equiv_imp_dvd_card:
+  "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
+    ==> k dvd card A"
+  apply (rule Union_quotient [THEN subst])
+   apply assumption
+  apply (rule dvd_partition)
+     prefer 4 apply (blast dest: quotient_disj)
+    apply (simp_all add: Union_quotient equiv_type finite_quotient)
+  done
+
+ML
+{*
+val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
+val UN_constant_eq = thm "UN_constant_eq";
+val UN_equiv_class = thm "UN_equiv_class";
+val UN_equiv_class2 = thm "UN_equiv_class2";
+val UN_equiv_class_inject = thm "UN_equiv_class_inject";
+val UN_equiv_class_type = thm "UN_equiv_class_type";
+val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
+val Union_quotient = thm "Union_quotient";
+val comp_equivI = thm "comp_equivI";
+val congruent2I = thm "congruent2I";
+val congruent2_commuteI = thm "congruent2_commuteI";
+val congruent2_def = thm "congruent2_def";
+val congruent2_implies_congruent = thm "congruent2_implies_congruent";
+val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
+val congruent_def = thm "congruent_def";
+val eq_equiv_class = thm "eq_equiv_class";
+val eq_equiv_class_iff = thm "eq_equiv_class_iff";
+val equiv_class_eq = thm "equiv_class_eq";
+val equiv_class_eq_iff = thm "equiv_class_eq_iff";
+val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
+val equiv_class_self = thm "equiv_class_self";
+val equiv_comp_eq = thm "equiv_comp_eq";
+val equiv_def = thm "equiv_def";
+val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
+val equiv_type = thm "equiv_type";
+val finite_equiv_class = thm "finite_equiv_class";
+val finite_quotient = thm "finite_quotient";
+val quotientE = thm "quotientE";
+val quotientI = thm "quotientI";
+val quotient_def = thm "quotient_def";
+val quotient_disj = thm "quotient_disj";
+val refl_comp_subset = thm "refl_comp_subset";
+val subset_equiv_class = thm "subset_equiv_class";
+val sym_trans_comp_subset = thm "sym_trans_comp_subset";
+*}
+
+end
diff -r 576fd0b65ed8 -r 7dd5853a4812 src/HOL/Integ/Equiv.thy
--- a/src/HOL/Integ/Equiv.thy	Fri Nov 19 15:05:10 2004 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,353 +0,0 @@
-(*  Title:      HOL/Integ/Equiv.thy
-    ID:         $Id$
-    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1996  University of Cambridge
-*)
-
-header {* Equivalence relations in Higher-Order Set Theory *}
-
-theory Equiv
-imports Relation Finite_Set
-begin
-
-subsection {* Equivalence relations *}
-
-locale equiv =
-  fixes A and r
-  assumes refl: "refl A r"
-    and sym: "sym r"
-    and trans: "trans r"
-
-text {*
-  Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
-  r = r"}.
-
-  First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
-*}
-
-lemma sym_trans_comp_subset:
-    "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
-  by (unfold trans_def sym_def converse_def) blast
-
-lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
-  by (unfold refl_def) blast
-
-lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
-  apply (unfold equiv_def)
-  apply clarify
-  apply (rule equalityI)
-   apply (rules intro: sym_trans_comp_subset refl_comp_subset)+
-  done
-
-text {* Second half. *}
-
-lemma comp_equivI:
-    "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
-  apply (unfold equiv_def refl_def sym_def trans_def)
-  apply (erule equalityE)
-  apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
-   apply fast
-  apply fast
-  done
-
-
-subsection {* Equivalence classes *}
-
-lemma equiv_class_subset:
-  "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
-  -- {* lemma for the next result *}
-  by (unfold equiv_def trans_def sym_def) blast
-
-theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
-  apply (assumption | rule equalityI equiv_class_subset)+
-  apply (unfold equiv_def sym_def)
-  apply blast
-  done
-
-lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
-  by (unfold equiv_def refl_def) blast
-
-lemma subset_equiv_class:
-    "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
-  -- {* lemma for the next result *}
-  by (unfold equiv_def refl_def) blast
-
-lemma eq_equiv_class:
-    "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
-  by (rules intro: equalityD2 subset_equiv_class)
-
-lemma equiv_class_nondisjoint:
-    "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
-  by (unfold equiv_def trans_def sym_def) blast
-
-lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
-  by (unfold equiv_def refl_def) blast
-
-theorem equiv_class_eq_iff:
-  "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
-  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
-
-theorem eq_equiv_class_iff:
-  "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
-  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
-
-
-subsection {* Quotients *}
-
-constdefs
-  quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
-  "A//r == \<Union>x \<in> A. {r``{x}}"  -- {* set of equiv classes *}
-
-lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
-  by (unfold quotient_def) blast
-
-lemma quotientE:
-  "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
-  by (unfold quotient_def) blast
-
-lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
-  by (unfold equiv_def refl_def quotient_def) blast
-
-lemma quotient_disj:
-  "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
-  apply (unfold quotient_def)
-  apply clarify
-  apply (rule equiv_class_eq)
-   apply assumption
-  apply (unfold equiv_def trans_def sym_def)
-  apply blast
-  done
-
-lemma quotient_eqI:
-  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y" 
-  apply (clarify elim!: quotientE)
-  apply (rule equiv_class_eq, assumption)
-  apply (unfold equiv_def sym_def trans_def, blast)
-  done
-
-lemma quotient_eq_iff:
-  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)" 
-  apply (rule iffI)  
-   prefer 2 apply (blast del: equalityI intro: quotient_eqI) 
-  apply (clarify elim!: quotientE)
-  apply (unfold equiv_def sym_def trans_def, blast)
-  done
-
-
-lemma quotient_empty [simp]: "{}//r = {}"
-by(simp add: quotient_def)
-
-lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
-by(simp add: quotient_def)
-
-lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
-by(simp add: quotient_def)
-
-
-subsection {* Defining unary operations upon equivalence classes *}
-
-text{*A congruence-preserving function*}
-locale congruent =
-  fixes r and f
-  assumes congruent: "(y,z) \<in> r ==> f y = f z"
-
-syntax
-  RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)
-
-translations
-  "f respects r" == "congruent r f"
-
-
-lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
-  -- {* lemma required to prove @{text UN_equiv_class} *}
-  by auto
-
-lemma UN_equiv_class:
-  "equiv A r ==> f respects r ==> a \<in> A
-    ==> (\<Union>x \<in> r``{a}. f x) = f a"
-  -- {* Conversion rule *}
-  apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
-  apply (unfold equiv_def congruent_def sym_def)
-  apply (blast del: equalityI)
-  done
-
-lemma UN_equiv_class_type:
-  "equiv A r ==> f respects r ==> X \<in> A//r ==>
-    (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
-  apply (unfold quotient_def)
-  apply clarify
-  apply (subst UN_equiv_class)
-     apply auto
-  done
-
-text {*
-  Sufficient conditions for injectiveness.  Could weaken premises!
-  major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
-  A ==> f y \<in> B"}.
-*}
-
-lemma UN_equiv_class_inject:
-  "equiv A r ==> f respects r ==>
-    (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
-    ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
-    ==> X = Y"
-  apply (unfold quotient_def)
-  apply clarify
-  apply (rule equiv_class_eq)
-   apply assumption
-  apply (subgoal_tac "f x = f xa")
-   apply blast
-  apply (erule box_equals)
-   apply (assumption | rule UN_equiv_class)+
-  done
-
-
-subsection {* Defining binary operations upon equivalence classes *}
-
-text{*A congruence-preserving function of two arguments*}
-locale congruent2 =
-  fixes r1 and r2 and f
-  assumes congruent2:
-    "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
-
-text{*Abbreviation for the common case where the relations are identical*}
-syntax
-  RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects2 " 80)
-
-translations
-  "f respects2 r" => "congruent2 r r f"
-
-lemma congruent2_implies_congruent:
-    "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
-  by (unfold congruent_def congruent2_def equiv_def refl_def) blast
-
-lemma congruent2_implies_congruent_UN:
-  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
-    congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
-  apply (unfold congruent_def)
-  apply clarify
-  apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
-  apply (simp add: UN_equiv_class congruent2_implies_congruent)
-  apply (unfold congruent2_def equiv_def refl_def)
-  apply (blast del: equalityI)
-  done
-
-lemma UN_equiv_class2:
-  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
-    ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
-  by (simp add: UN_equiv_class congruent2_implies_congruent
-    congruent2_implies_congruent_UN)
-
-lemma UN_equiv_class_type2:
-  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
-    ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
-    ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
-    ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
-  apply (unfold quotient_def)
-  apply clarify
-  apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
-    congruent2_implies_congruent quotientI)
-  done
-
-lemma UN_UN_split_split_eq:
-  "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
-    (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
-  -- {* Allows a natural expression of binary operators, *}
-  -- {* without explicit calls to @{text split} *}
-  by auto
-
-lemma congruent2I:
-  "equiv A1 r1 ==> equiv A2 r2
-    ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
-    ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
-    ==> congruent2 r1 r2 f"
-  -- {* Suggested by John Harrison -- the two subproofs may be *}
-  -- {* \emph{much} simpler than the direct proof. *}
-  apply (unfold congruent2_def equiv_def refl_def)
-  apply clarify
-  apply (blast intro: trans)
-  done
-
-lemma congruent2_commuteI:
-  assumes equivA: "equiv A r"
-    and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
-    and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
-  shows "f respects2 r"
-  apply (rule congruent2I [OF equivA equivA])
-   apply (rule commute [THEN trans])
-     apply (rule_tac [3] commute [THEN trans, symmetric])
-       apply (rule_tac [5] sym)
-       apply (assumption | rule congt |
-         erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
-  done
-
-
-subsection {* Cardinality results *}
-
-text {*Suggested by Florian Kammüller*}
-
-lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
-  -- {* recall @{thm equiv_type} *}
-  apply (rule finite_subset)
-   apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
-  apply (unfold quotient_def)
-  apply blast
-  done
-
-lemma finite_equiv_class:
-  "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
-  apply (unfold quotient_def)
-  apply (rule finite_subset)
-   prefer 2 apply assumption
-  apply blast
-  done
-
-lemma equiv_imp_dvd_card:
-  "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
-    ==> k dvd card A"
-  apply (rule Union_quotient [THEN subst])
-   apply assumption
-  apply (rule dvd_partition)
-     prefer 4 apply (blast dest: quotient_disj)
-    apply (simp_all add: Union_quotient equiv_type finite_quotient)
-  done
-
-ML
-{*
-val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
-val UN_constant_eq = thm "UN_constant_eq";
-val UN_equiv_class = thm "UN_equiv_class";
-val UN_equiv_class2 = thm "UN_equiv_class2";
-val UN_equiv_class_inject = thm "UN_equiv_class_inject";
-val UN_equiv_class_type = thm "UN_equiv_class_type";
-val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
-val Union_quotient = thm "Union_quotient";
-val comp_equivI = thm "comp_equivI";
-val congruent2I = thm "congruent2I";
-val congruent2_commuteI = thm "congruent2_commuteI";
-val congruent2_def = thm "congruent2_def";
-val congruent2_implies_congruent = thm "congruent2_implies_congruent";
-val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
-val congruent_def = thm "congruent_def";
-val eq_equiv_class = thm "eq_equiv_class";
-val eq_equiv_class_iff = thm "eq_equiv_class_iff";
-val equiv_class_eq = thm "equiv_class_eq";
-val equiv_class_eq_iff = thm "equiv_class_eq_iff";
-val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
-val equiv_class_self = thm "equiv_class_self";
-val equiv_comp_eq = thm "equiv_comp_eq";
-val equiv_def = thm "equiv_def";
-val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
-val equiv_type = thm "equiv_type";
-val finite_equiv_class = thm "finite_equiv_class";
-val finite_quotient = thm "finite_quotient";
-val quotientE = thm "quotientE";
-val quotientI = thm "quotientI";
-val quotient_def = thm "quotient_def";
-val quotient_disj = thm "quotient_disj";
-val refl_comp_subset = thm "refl_comp_subset";
-val subset_equiv_class = thm "subset_equiv_class";
-val sym_trans_comp_subset = thm "sym_trans_comp_subset";
-*}
-
-end
diff -r 576fd0b65ed8 -r 7dd5853a4812 src/HOL/Integ/IntDef.thy
--- a/src/HOL/Integ/IntDef.thy	Fri Nov 19 15:05:10 2004 +0100
+++ b/src/HOL/Integ/IntDef.thy	Fri Nov 19 17:31:49 2004 +0100
@@ -8,7 +8,7 @@
 header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
 
 theory IntDef
-imports Equiv NatArith
+imports Equiv_Relations NatArith
 begin
 
 constdefs
diff -r 576fd0b65ed8 -r 7dd5853a4812 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Nov 19 15:05:10 2004 +0100
+++ b/src/HOL/IsaMakefile	Fri Nov 19 17:31:49 2004 +0100
@@ -82,11 +82,11 @@
   $(SRC)/TFL/tfl.ML $(SRC)/TFL/thms.ML $(SRC)/TFL/thry.ML \
   $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML \
   Datatype.thy Datatype_Universe.ML Datatype_Universe.thy \
-  Divides.thy Extraction.thy Finite_Set.ML Finite_Set.thy \
+  Divides.thy Equiv_Relations.thy Extraction.thy Finite_Set.ML Finite_Set.thy \
   Fun.thy Gfp.ML Gfp.thy Hilbert_Choice.thy HOL.ML \
   HOL.thy HOL_lemmas.ML Inductive.thy Infinite_Set.thy Integ/Numeral.thy \
   Integ/cooper_dec.ML Integ/cooper_proof.ML \
-  Integ/Equiv.thy Integ/IntArith.thy Integ/IntDef.thy \
+  Integ/IntArith.thy Integ/IntDef.thy \
   Integ/IntDiv.thy Integ/NatBin.thy Integ/NatSimprocs.thy Integ/Parity.thy \
   Integ/int_arith1.ML Integ/int_factor_simprocs.ML Integ/nat_simprocs.ML \
   Integ/Presburger.thy Integ/presburger.ML Integ/qelim.ML \