src/HOL/Equiv_Relations.thy
 author wenzelm Sat, 07 Apr 2012 16:41:59 +0200 changeset 47389 e8552cba702d parent 46752 e9e7209eb375 child 51112 da97167e03f7 permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;

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

header {* Equivalence Relations in Higher-Order Set Theory *}

theory Equiv_Relations
imports Big_Operators Relation Plain
begin

subsection {* Equivalence relations -- set version *}

definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
"equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"

lemma equivI:
"refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"

lemma equivE:
assumes "equiv A r"
obtains "refl_on A r" and "sym r" and "trans r"
using assms by (simp add: equiv_def)

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_unfold) 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
"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)"

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*}

definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
"congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"

lemma congruentI:
"(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"

lemma congruentD:
"congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> 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*}

definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
"congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"

lemma congruent2I':
assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
shows "congruent2 r1 r2 f"
using assms by (auto simp add: congruent2_def)

lemma congruent2D:
"congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
using assms by (auto simp add: congruent2_def)

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 (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"
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 KammÃ¼ller*}

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)
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
apply simp
done

subsection {* Equivalence relations -- predicate version *}

text {* Partial equivalences *}

definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
-- {* John-Harrison-style characterization *}

lemma part_equivpI:
"(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
by (auto simp add: part_equivp_def) (auto elim: sympE transpE)

lemma part_equivpE:
assumes "part_equivp R"
obtains x where "R x x" and "symp R" and "transp R"
proof -
from assms have 1: "\<exists>x. R x x"
and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
by (unfold part_equivp_def) blast+
from 1 obtain x where "R x x" ..
moreover have "symp R"
proof (rule sympI)
fix x y
assume "R x y"
with 2 [of x y] show "R y x" by auto
qed
moreover have "transp R"
proof (rule transpI)
fix x y z
assume "R x y" and "R y z"
with 2 [of x y] 2 [of y z] show "R x z" by auto
qed
ultimately show thesis by (rule that)
qed

lemma part_equivp_refl_symp_transp:
"part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
by (auto intro: part_equivpI elim: part_equivpE)

lemma part_equivp_symp:
"part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
by (erule part_equivpE, erule sympE)

lemma part_equivp_transp:
"part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
by (erule part_equivpE, erule transpE)

lemma part_equivp_typedef:
"part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
by (auto elim: part_equivpE)

text {* Total equivalences *}

definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}

lemma equivpI:
"reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
by (auto elim: reflpE sympE transpE simp add: equivp_def)

lemma equivpE:
assumes "equivp R"
obtains "reflp R" and "symp R" and "transp R"
using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)

lemma equivp_implies_part_equivp:
"equivp R \<Longrightarrow> part_equivp R"
by (auto intro: part_equivpI elim: equivpE reflpE)

lemma equivp_equiv:
"equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])

lemma equivp_reflp_symp_transp:
shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
by (auto intro: equivpI elim: equivpE)

lemma identity_equivp:
"equivp (op =)"
by (auto intro: equivpI reflpI sympI transpI)

lemma equivp_reflp:
"equivp R \<Longrightarrow> R x x"
by (erule equivpE, erule reflpE)

lemma equivp_symp:
"equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
by (erule equivpE, erule sympE)

lemma equivp_transp:
"equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
by (erule equivpE, erule transpE)

end