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¯ O
r = r"}.
First half: @{text "equiv A r ==> r¯ O r = r"}.
*}
lemma sym_trans_comp_subset:
"sym r ==> trans r ==> r¯ O r ⊆ r"
by (unfold trans_def sym_def converse_def) blast
lemma refl_on_comp_subset: "refl_on A r ==> r ⊆ r¯ O r"
by (unfold refl_on_def) blast
lemma equiv_comp_eq: "equiv A r ==> r¯ 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¯ 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 "∀x y. (x, y) ∈ r --> (y, x) ∈ r")
apply fast
apply fast
done
subsection {* Equivalence classes *}
lemma equiv_class_subset:
"equiv A r ==> (a, b) ∈ r ==> r``{a} ⊆ 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) ∈ 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 ∈ A ==> a ∈ r``{a}"
by (unfold equiv_def refl_on_def) blast
lemma subset_equiv_class:
"equiv A r ==> r``{b} ⊆ r``{a} ==> b ∈ A ==> (a,b) ∈ 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 ∈ A ==> (a, b) ∈ r"
by (iprover intro: equalityD2 subset_equiv_class)
lemma equiv_class_nondisjoint:
"equiv A r ==> x ∈ (r``{a} ∩ r``{b}) ==> (a, b) ∈ r"
by (unfold equiv_def trans_def sym_def) blast
lemma equiv_type: "equiv A r ==> r ⊆ A × A"
by (unfold equiv_def refl_on_def) blast
theorem equiv_class_eq_iff:
"equiv A r ==> ((x, y) ∈ r) = (r``{x} = r``{y} & x ∈ A & y ∈ A)"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
theorem eq_equiv_class_iff:
"equiv A r ==> x ∈ A ==> y ∈ A ==> (r``{x} = r``{y}) = ((x, y) ∈ r)"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
subsection {* Quotients *}
definition quotient :: "'a set => ('a × 'a) set => 'a set set" (infixl "'/'/" 90) where
[code del]: "A//r = (\<Union>x ∈ A. {r``{x}})" -- {* set of equiv classes *}
lemma quotientI: "x ∈ A ==> r``{x} ∈ A//r"
by (unfold quotient_def) blast
lemma quotientE:
"X ∈ A//r ==> (!!x. X = r``{x} ==> x ∈ 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 ∈ A//r ==> Y ∈ A//r ==> X = Y | (X ∩ 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 ∈ A//r; Y ∈ A//r; x ∈ X; y ∈ Y; (x,y) ∈ 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 ∈ A//r; Y ∈ A//r; x ∈ X; y ∈ Y|] ==> (X = Y) = ((x,y) ∈ 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:
"[| equiv A r; x ∈ A; y ∈ A |] ==> ({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:
"[| inj_on (%a. {a}//r) A; a ∈ A |] ==> (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) ∈ 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 ∈ A ==> ∀y ∈ A. f y = c ==> (\<Union>y ∈ 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 ∈ A
==> (\<Union>x ∈ 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 ∈ A//r ==>
(!!x. x ∈ A ==> f x ∈ B) ==> (\<Union>x ∈ X. f 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 ∈
A ==> f y ∈ B"}.
*}
lemma UN_equiv_class_inject:
"equiv A r ==> f respects r ==>
(\<Union>x ∈ X. f x) = (\<Union>y ∈ Y. f y) ==> X ∈ A//r ==> Y ∈ A//r
==> (!!x y. x ∈ A ==> y ∈ A ==> f x = f y ==> (x, y) ∈ 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) ∈ r1 ==> (y2,z2) ∈ 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 ∈ 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 ∈ A2 ==>
congruent r1 (λx1. \<Union>x2 ∈ 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 ∈ A1 ==> a2 ∈ A2
==> (\<Union>x1 ∈ r1``{a1}. \<Union>x2 ∈ 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 ∈ A1//r1 ==> X2 ∈ A2//r2
==> (!!x1 x2. x1 ∈ A1 ==> x2 ∈ A2 ==> f x1 x2 ∈ B)
==> (\<Union>x1 ∈ X1. \<Union>x2 ∈ X2. f x1 x2) ∈ 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) ∈ X. \<Union>(y1, y2) ∈ Y. A x1 x2 y1 y2) =
(\<Union>x ∈ X. \<Union>y ∈ Y. (λ(x1, x2). (λ(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 ∈ A2 ==> (y,z) ∈ r1 ==> f y w = f z w)
==> (!!y z w. w ∈ A1 ==> (y,z) ∈ 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 ∈ A ==> z ∈ A ==> f y z = f z y"
and congt: "!!y z w. w ∈ A ==> (y,z) ∈ 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 ⊆ A × 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 ⊆ A × A ==> X ∈ 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 ==> ∀X ∈ A//r. k dvd card X
==> k dvd card A"
apply (rule Union_quotient [THEN subst [where P="λA. 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:
"[| finite A; inj_on (λx. {x} // r) A |] ==> 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