src/HOL/Equiv_Relations.thy
 author nipkow Thu, 09 Dec 2004 18:30:59 +0100 changeset 15392 290bc97038c7 parent 15303 eedbb8d22ca2 child 15539 333a88244569 permissions -rw-r--r--
First step in reorganizing Finite_Set

(*  ID:         $Id$
Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
*)

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 = {}"

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

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

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

lemma quotient_diff1:
"\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
apply blast
done

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 (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"
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  commute [THEN trans, symmetric])
apply (rule_tac  sym)
apply (assumption | rule congt |
erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
done

subsection {* Cardinality results *}

text {*Suggested by Florian Kamm�ller*}

lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
-- {* recall @{thm equiv_type} *}
apply (rule finite_subset)
apply (erule_tac  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 3 apply (blast dest: quotient_disj)
done

lemma card_quotient_disjoint:
"\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
apply(subst card_UN_disjoint)
apply assumption
apply simp
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