--- a/src/HOL/Integ/Equiv.ML Thu Aug 08 23:46:51 2002 +0200
+++ b/src/HOL/Integ/Equiv.ML Thu Aug 08 23:47:41 2002 +0200
@@ -1,262 +1,38 @@
-(* Title: Equiv.ML
- ID: $Id$
- Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1996 University of Cambridge
-Equivalence relations in HOL Set Theory
-*)
-
-(*** Suppes, Theorem 70: r is an equiv relation iff r^-1 O r = r ***)
-
-(** first half: equiv A r ==> r^-1 O r = r **)
-
-Goalw [trans_def,sym_def,converse_def]
- "[| sym(r); trans(r) |] ==> r^-1 O r <= r";
-by (Blast_tac 1);
-qed "sym_trans_comp_subset";
-
-Goalw [refl_def] "refl A r ==> r <= r^-1 O r";
-by (Blast_tac 1);
-qed "refl_comp_subset";
-
-Goalw [equiv_def] "equiv A r ==> r^-1 O r = r";
-by (Clarify_tac 1);
-by (rtac equalityI 1);
-by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1));
-qed "equiv_comp_eq";
-
-(*second half*)
-Goalw [equiv_def,refl_def,sym_def,trans_def]
- "[| r^-1 O r = r; Domain(r) = A |] ==> equiv A r";
-by (etac equalityE 1);
-by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
-by (ALLGOALS Fast_tac);
-qed "comp_equivI";
-
-(** Equivalence classes **)
-
-(*Lemma for the next result*)
-Goalw [equiv_def,trans_def,sym_def]
- "[| equiv A r; (a,b): r |] ==> r``{a} <= r``{b}";
-by (Blast_tac 1);
-qed "equiv_class_subset";
-
-Goal "[| equiv A r; (a,b): r |] ==> r``{a} = r``{b}";
-by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
-by (rewrite_goals_tac [equiv_def,sym_def]);
-by (Blast_tac 1);
-qed "equiv_class_eq";
-
-Goalw [equiv_def,refl_def] "[| equiv A r; a: A |] ==> a: r``{a}";
-by (Blast_tac 1);
-qed "equiv_class_self";
-
-(*Lemma for the next result*)
-Goalw [equiv_def,refl_def]
- "[| equiv A r; r``{b} <= r``{a}; b: A |] ==> (a,b): r";
-by (Blast_tac 1);
-qed "subset_equiv_class";
-
-Goal "[| r``{a} = r``{b}; equiv A r; b: A |] ==> (a,b): r";
-by (REPEAT (ares_tac [equalityD2, subset_equiv_class] 1));
-qed "eq_equiv_class";
-
-(*thus r``{a} = r``{b} as well*)
-Goalw [equiv_def,trans_def,sym_def]
- "[| equiv A r; x: (r``{a} Int r``{b}) |] ==> (a,b): r";
-by (Blast_tac 1);
-qed "equiv_class_nondisjoint";
-
-Goalw [equiv_def,refl_def] "equiv A r ==> r <= A <*> A";
-by (Blast_tac 1);
-qed "equiv_type";
-
-Goal "equiv A r ==> ((x,y): r) = (r``{x} = r``{y} & x:A & y:A)";
-by (blast_tac (claset() addSIs [equiv_class_eq]
- addDs [eq_equiv_class, equiv_type]) 1);
-qed "equiv_class_eq_iff";
-
-Goal "[| equiv A r; x: A; y: A |] ==> (r``{x} = r``{y}) = ((x,y): r)";
-by (blast_tac (claset() addSIs [equiv_class_eq]
- addDs [eq_equiv_class, equiv_type]) 1);
-qed "eq_equiv_class_iff";
-
-(*** Quotients ***)
-
-(** Introduction/elimination rules -- needed? **)
-
-Goalw [quotient_def] "x:A ==> r``{x}: A//r";
-by (Blast_tac 1);
-qed "quotientI";
-
-val [major,minor] = Goalw [quotient_def]
- "[| X:(A//r); !!x. [| X = r``{x}; x:A |] ==> P |] \
-\ ==> P";
-by (resolve_tac [major RS UN_E] 1);
-by (rtac minor 1);
-by (assume_tac 2);
-by (Fast_tac 1); (*Blast_tac FAILS to prove it*)
-qed "quotientE";
-
-Goalw [equiv_def,refl_def,quotient_def]
- "equiv A r ==> Union(A//r) = A";
-by (Blast_tac 1);
-qed "Union_quotient";
-
-Goalw [quotient_def]
- "[| equiv A r; X: A//r; Y: A//r |] ==> X=Y | (X Int Y = {})";
-by (Clarify_tac 1);
-by (rtac equiv_class_eq 1);
-by (assume_tac 1);
-by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
-by (Blast_tac 1);
-qed "quotient_disj";
-
-
-(**** Defining unary operations upon equivalence classes ****)
-
-(* theorem needed to prove UN_equiv_class *)
-Goal "[| a:A; ALL y:A. b(y)=c |] ==> (UN y:A. b(y))=c";
-by Auto_tac;
-qed "UN_constant_eq";
-
-
-(** Could introduce a LOCALE with the assumptions
- equiv A r; congruent r b
-**)
+(* legacy ML bindings *)
-(*Conversion rule*)
-Goal "[| equiv A r; congruent r b; a: A |] \
-\ ==> (UN x:r``{a}. b(x)) = b(a)";
-by (rtac (equiv_class_self RS UN_constant_eq) 1 THEN REPEAT (assume_tac 1));
-by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
-by (blast_tac (claset() delrules [equalityI]) 1);
-qed "UN_equiv_class";
-
-(*type checking of UN x:r`{a}. b(x) *)
-val prems = Goalw [quotient_def]
- "[| equiv A r; congruent r b; X: A//r; \
-\ !!x. x : A ==> b(x) : B |] \
-\ ==> (UN x:X. b(x)) : B";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (stac UN_equiv_class 1);
-by (REPEAT (ares_tac prems 1));
-qed "UN_equiv_class_type";
-
-(*Sufficient conditions for injectiveness. Could weaken premises!
- major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
-*)
-val prems = Goalw [quotient_def]
- "[| equiv A r; congruent r b; \
-\ (UN x:X. b(x))=(UN y:Y. b(y)); X: A//r; Y: A//r; \
-\ !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |] \
-\ ==> X=Y";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (rtac equiv_class_eq 1);
-by (REPEAT (ares_tac prems 1));
-by (etac box_equals 1);
-by (REPEAT (ares_tac [UN_equiv_class] 1));
-qed "UN_equiv_class_inject";
-
-
-(**** Defining binary operations upon equivalence classes ****)
-
-
-Goalw [congruent_def,congruent2_def,equiv_def,refl_def]
- "[| equiv A r; congruent2 r b; a: A |] ==> congruent r (b a)";
-by (Blast_tac 1);
-qed "congruent2_implies_congruent";
-
-Goalw [congruent_def]
- "[| equiv A r; congruent2 r b; a: A |] ==> \
-\ congruent r (%x1. UN x2:r``{a}. b x1 x2)";
-by (Clarify_tac 1);
-by (rtac (equiv_type RS subsetD RS SigmaE2) 1 THEN REPEAT (assume_tac 1));
-by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
- congruent2_implies_congruent]) 1);
-by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
-by (blast_tac (claset() delrules [equalityI]) 1);
-qed "congruent2_implies_congruent_UN";
-
-Goal "[| equiv A r; congruent2 r b; a1: A; a2: A |] \
-\ ==> (UN x1:r``{a1}. UN x2:r``{a2}. b x1 x2) = b a1 a2";
-by (asm_simp_tac (simpset() addsimps [UN_equiv_class,
- congruent2_implies_congruent,
- congruent2_implies_congruent_UN]) 1);
-qed "UN_equiv_class2";
-
-(*type checking*)
-val prems = Goalw [quotient_def]
- "[| equiv A r; congruent2 r b; \
-\ X1: A//r; X2: A//r; \
-\ !!x1 x2. [| x1: A; x2: A |] ==> b x1 x2 : B |] \
-\ ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
- congruent2_implies_congruent_UN,
- congruent2_implies_congruent, quotientI]) 1));
-qed "UN_equiv_class_type2";
-
-(*Allows a natural expression of binary operators, without explicit calls
- to "split"*)
-Goal "(UN (x1,x2):X. UN (y1,y2):Y. A x1 x2 y1 y2) = \
-\ (UN x:X. UN y:Y. (%(x1, x2). (%(y1, y2). A x1 x2 y1 y2) y) x)";
-by Auto_tac;
-qed "UN_UN_split_split_eq";
-
-(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
- than the direct proof*)
-val prems = Goalw [congruent2_def,equiv_def,refl_def]
- "[| equiv A r; \
-\ !! y z w. [| w: A; (y,z) : r |] ==> b y w = b z w; \
-\ !! y z w. [| w: A; (y,z) : r |] ==> b w y = b w z \
-\ |] ==> congruent2 r b";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (blast_tac (claset() addIs (trans::prems)) 1);
-qed "congruent2I";
-
-val [equivA,commute,congt] = Goal
- "[| equiv A r; \
-\ !! y z. [| y: A; z: A |] ==> b y z = b z y; \
-\ !! y z w. [| w: A; (y,z): r |] ==> b w y = b w z \
-\ |] ==> congruent2 r b";
-by (resolve_tac [equivA RS congruent2I] 1);
-by (rtac (commute RS trans) 1);
-by (rtac (commute RS trans RS sym) 3);
-by (rtac sym 5);
-by (REPEAT (ares_tac [congt] 1
- ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
-qed "congruent2_commuteI";
-
-
-(*** Cardinality results suggested by Florian Kammueller ***)
-
-(*Recall that equiv A r implies r <= A <*> A (equiv_type) *)
-Goal "[| finite A; r <= A <*> A |] ==> finite (A//r)";
-by (rtac finite_subset 1);
-by (etac (finite_Pow_iff RS iffD2) 2);
-by (rewtac quotient_def);
-by (Blast_tac 1);
-qed "finite_quotient";
-
-Goalw [quotient_def]
- "[| finite A; r <= A <*> A; X : A//r |] ==> finite X";
-by (rtac finite_subset 1);
-by (assume_tac 2);
-by (Blast_tac 1);
-qed "finite_equiv_class";
-
-Goal "[| finite A; equiv A r; ALL X : A//r. k dvd card(X) |] \
-\ ==> k dvd card(A)";
-by (rtac (Union_quotient RS subst) 1);
-by (assume_tac 1);
-by (rtac dvd_partition 1);
-by (blast_tac (claset() addDs [quotient_disj]) 4);
-by (ALLGOALS
- (asm_simp_tac (simpset() addsimps [Union_quotient, equiv_type,
- finite_quotient])));
-qed "equiv_imp_dvd_card";
+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_class_subset = thm "equiv_class_subset";
+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";
--- a/src/HOL/Integ/Equiv.thy Thu Aug 08 23:46:51 2002 +0200
+++ b/src/HOL/Integ/Equiv.thy Thu Aug 08 23:47:41 2002 +0200
@@ -1,23 +1,273 @@
-(* Title: Equiv.thy
+(* Title: HOL/Integ/Equiv.thy
ID: $Id$
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
-
-Equivalence relations in Higher-Order Set Theory
*)
-Equiv = Relation + Finite_Set +
+header {* Equivalence relations in Higher-Order Set Theory *}
+
+theory Equiv = Relation + Finite_Set:
+
+subsection {* Equiv 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
+
+lemma 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
+
+lemma 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)
+
+lemma 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
- equiv :: "['a set, ('a*'a) set] => bool"
- "equiv A r == refl A r & sym(r) & trans(r)"
+ 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
- quotient :: "['a set, ('a*'a) set] => 'a set set" (infixl "'/'/" 90)
- "A//r == UN x:A. {r``{x}}" (*set of equiv classes*)
+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
+
+
+subsection {* Defining unary operations upon equivalence classes *}
+
+locale congruent =
+ fixes r and b
+ assumes congruent: "(y, z) \<in> r ==> b y = b z"
+
+lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. b y = c ==> (\<Union>y \<in> A. b(y))=c"
+ -- {* lemma required to prove @{text UN_equiv_class} *}
+ by auto
+
+lemma UN_equiv_class:
+ "equiv A r ==> congruent r b ==> a \<in> A
+ ==> (\<Union>x \<in> r``{a}. b x) = b 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
- congruent :: "[('a*'a) set, 'a=>'b] => bool"
- "congruent r b == ALL y z. (y,z):r --> b(y)=b(z)"
+lemma UN_equiv_class_type:
+ "equiv A r ==> congruent r b ==> X \<in> A//r ==>
+ (!!x. x \<in> A ==> b x : B) ==> (\<Union>x \<in> X. b x) : 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 ==> b y \<in> B"}.
+*}
+
+lemma UN_equiv_class_inject:
+ "equiv A r ==> congruent r b ==>
+ (\<Union>x \<in> X. b x) = (\<Union>y \<in> Y. b y) ==> X \<in> A//r ==> Y \<in> A//r
+ ==> (!!x y. x \<in> A ==> y \<in> A ==> b x = b y ==> (x, y) \<in> r)
+ ==> X = Y"
+ apply (unfold quotient_def)
+ apply clarify
+ apply (rule equiv_class_eq)
+ apply assumption
+ apply (subgoal_tac "b x = b xa")
+ apply blast
+ apply (erule box_equals)
+ apply (assumption | rule UN_equiv_class)+
+ done
+
+
+subsection {* Defining binary operations upon equivalence classes *}
+
+locale congruent2 =
+ fixes r and b
+ assumes congruent2:
+ "(y1, z1) \<in> r ==> (y2, z2) \<in> r ==> b y1 y2 = b z1 z2"
+
+lemma congruent2_implies_congruent:
+ "equiv A r ==> congruent2 r b ==> a \<in> A ==> congruent r (b a)"
+ by (unfold congruent_def congruent2_def equiv_def refl_def) blast
+
+lemma congruent2_implies_congruent_UN:
+ "equiv A r ==> congruent2 r b ==> a \<in> A ==>
+ congruent r (\<lambda>x1. \<Union>x2 \<in> r``{a}. b 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 A r ==> congruent2 r b ==> a1 \<in> A ==> a2 \<in> A
+ ==> (\<Union>x1 \<in> r``{a1}. \<Union>x2 \<in> r``{a2}. b x1 x2) = b a1 a2"
+ by (simp add: UN_equiv_class congruent2_implies_congruent
+ congruent2_implies_congruent_UN)
- congruent2 :: "[('a*'a) set, ['a,'a]=>'b] => bool"
- "congruent2 r b == ALL y1 z1 y2 z2.
- (y1,z1):r --> (y2,z2):r --> b y1 y2 = b z1 z2"
+lemma UN_equiv_class_type2:
+ "equiv A r ==> congruent2 r b ==> X1 \<in> A//r ==> X2 \<in> A//r
+ ==> (!!x1 x2. x1 \<in> A ==> x2 \<in> A ==> b x1 x2 \<in> B)
+ ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. b 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 A r
+ ==> (!!y z w. w \<in> A ==> (y, z) \<in> r ==> b y w = b z w)
+ ==> (!!y z w. w \<in> A ==> (y, z) \<in> r ==> b w y = b w z)
+ ==> congruent2 r b"
+ -- {* 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 ==> b y z = b z y"
+ and congt: "!!y z w. w \<in> A ==> (y, z) \<in> r ==> b w y = b w z"
+ shows "congruent2 r b"
+ apply (rule equivA [THEN congruent2I])
+ 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
+
end