(* 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 Finite_Set Relation Plain
begin
subsection {* Equivalence relations *}
locale equiv =
fixes A and r
assumes refl_on: "refl_on 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_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
by (unfold refl_on_def) blast
lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
apply (unfold equiv_def)
apply clarify
apply (rule equalityI)
apply (iprover intro: sym_trans_comp_subset refl_on_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_on_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_on_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_on_def) blast
lemma eq_equiv_class:
"r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
by (iprover 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_on_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 *}
definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set" (infixl "'/'/" 90) where
[code del]: "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_on_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 eq_equiv_class_iff2:
"\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
by(simp add:quotient_def eq_equiv_class_iff)
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)
lemma singleton_quotient: "{x}//r = {r `` {x}}"
by(simp add:quotient_def)
lemma quotient_diff1:
"\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
apply(simp add:quotient_def inj_on_def)
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"
abbreviation
RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
(infixr "respects" 80) where
"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*}
abbreviation
RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
(infixr "respects2" 80) where
"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_on_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_on_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_on_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 (rule congt | assumption |
erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
done
subsection {* Quotients and finiteness *}
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 [where P="\<lambda>A. k dvd card A"]])
apply assumption
apply (rule dvd_partition)
prefer 3 apply (blast dest: quotient_disj)
apply (simp_all add: Union_quotient equiv_type)
done
lemma card_quotient_disjoint:
"\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
apply(simp add:quotient_def)
apply(subst card_UN_disjoint)
apply assumption
apply simp
apply(fastsimp simp add:inj_on_def)
apply (simp add:setsum_constant)
done
end