(* Title: HOL/Equiv_Relations.thy
Author: Lawrence C Paulson, 1996 Cambridge University Computer Laboratory
*)
section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
theory Equiv_Relations
imports Groups_Big
begin
subsection \<open>Equivalence relations -- set version\<close>
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"
by (simp add: equiv_def)
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 \<open>
Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O r = r\<close>.
First half: \<open>equiv A r \<Longrightarrow> r\<inverse> O r = r\<close>.
\<close>
lemma sym_trans_comp_subset: "sym r \<Longrightarrow> trans r \<Longrightarrow> r\<inverse> O r \<subseteq> r"
unfolding trans_def sym_def converse_unfold by blast
lemma refl_on_comp_subset: "refl_on A r \<Longrightarrow> r \<subseteq> r\<inverse> O r"
unfolding refl_on_def by blast
lemma equiv_comp_eq: "equiv A r \<Longrightarrow> r\<inverse> O r = r"
unfolding equiv_def
by (iprover intro: sym_trans_comp_subset refl_on_comp_subset equalityI)
text \<open>Second half.\<close>
lemma comp_equivI:
assumes "r\<inverse> O r = r" "Domain r = A"
shows "equiv A r"
proof -
have *: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r"
using assms by blast
show ?thesis
unfolding equiv_def refl_on_def sym_def trans_def
using assms by (auto intro: *)
qed
subsection \<open>Equivalence classes\<close>
lemma equiv_class_subset: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} \<subseteq> r``{b}"
\<comment> \<open>lemma for the next result\<close>
unfolding equiv_def trans_def sym_def by blast
theorem equiv_class_eq: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} = r``{b}"
by (intro equalityI equiv_class_subset; force simp add: equiv_def sym_def)
lemma equiv_class_self: "equiv A r \<Longrightarrow> a \<in> A \<Longrightarrow> a \<in> r``{a}"
unfolding equiv_def refl_on_def by blast
lemma subset_equiv_class: "equiv A r \<Longrightarrow> r``{b} \<subseteq> r``{a} \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
\<comment> \<open>lemma for the next result\<close>
unfolding equiv_def refl_on_def by blast
lemma eq_equiv_class: "r``{a} = r``{b} \<Longrightarrow> equiv A r \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
by (iprover intro: equalityD2 subset_equiv_class)
lemma equiv_class_nondisjoint: "equiv A r \<Longrightarrow> x \<in> (r``{a} \<inter> r``{b}) \<Longrightarrow> (a, b) \<in> r"
unfolding equiv_def trans_def sym_def by blast
lemma equiv_type: "equiv A r \<Longrightarrow> r \<subseteq> A \<times> A"
unfolding equiv_def refl_on_def by blast
lemma equiv_class_eq_iff: "equiv A r \<Longrightarrow> (x, y) \<in> r \<longleftrightarrow> r``{x} = r``{y} \<and> x \<in> A \<and> y \<in> A"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
lemma eq_equiv_class_iff: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> r``{x} = r``{y} \<longleftrightarrow> (x, y) \<in> r"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
subsection \<open>Quotients\<close>
definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set" (infixl "'/'/" 90)
where "A//r = (\<Union>x \<in> A. {r``{x}})" \<comment> \<open>set of equiv classes\<close>
lemma quotientI: "x \<in> A \<Longrightarrow> r``{x} \<in> A//r"
unfolding quotient_def by blast
lemma quotientE: "X \<in> A//r \<Longrightarrow> (\<And>x. X = r``{x} \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
unfolding quotient_def by blast
lemma Union_quotient: "equiv A r \<Longrightarrow> \<Union>(A//r) = A"
unfolding equiv_def refl_on_def quotient_def by blast
lemma quotient_disj: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> X = Y \<or> X \<inter> Y = {}"
unfolding quotient_def equiv_def trans_def sym_def by blast
lemma quotient_eqI:
assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" "(x, y) \<in> r"
shows "X = Y"
proof -
obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
using assms by (auto elim!: quotientE)
then have "(a,b) \<in> r"
using xy \<open>equiv A r\<close> unfolding equiv_def sym_def trans_def by blast
then show ?thesis
unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
qed
lemma quotient_eq_iff:
assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y"
shows "X = Y \<longleftrightarrow> (x, y) \<in> r"
proof
assume L: "X = Y"
with assms show "(x, y) \<in> r"
unfolding equiv_def sym_def trans_def by (blast elim!: quotientE)
next
assume \<section>: "(x, y) \<in> r" show "X = Y"
by (rule quotient_eqI) (use \<section> assms in \<open>blast+\<close>)
qed
lemma eq_equiv_class_iff2: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> {x}//r = {y}//r \<longleftrightarrow> (x, y) \<in> 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 = {} \<longleftrightarrow> A = {}"
by (simp add: quotient_def)
lemma quotient_is_empty2 [iff]: "{} = A//r \<longleftrightarrow> A = {}"
by (simp add: quotient_def)
lemma singleton_quotient: "{x}//r = {r `` {x}}"
by (simp add: quotient_def)
lemma quotient_diff1: "inj_on (\<lambda>a. {a}//r) A \<Longrightarrow> a \<in> A \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
unfolding quotient_def inj_on_def by blast
subsection \<open>Refinement of one equivalence relation WRT another\<close>
lemma refines_equiv_class_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> R``(S``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)
lemma refines_equiv_class_eq2: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> S``(R``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)
lemma refines_equiv_image_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> (\<lambda>X. S``X) ` (A//R) = A//S"
by (auto simp: quotient_def image_UN refines_equiv_class_eq2)
lemma finite_refines_finite:
"finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> finite (A//S)"
by (erule finite_surj [where f = "\<lambda>X. S``X"]) (simp add: refines_equiv_image_eq)
lemma finite_refines_card_le:
"finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> card (A//S) \<le> card (A//R)"
by (subst refines_equiv_image_eq [of R S A, symmetric])
(auto simp: card_image_le [where f = "\<lambda>X. S``X"])
subsection \<open>Defining unary operations upon equivalence classes\<close>
text \<open>A congruence-preserving function.\<close>
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"
by (auto simp add: congruent_def)
lemma congruentD: "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
by (auto simp add: congruent_def)
abbreviation RESPECTS :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infixr "respects" 80)
where "f respects r \<equiv> congruent r f"
lemma UN_constant_eq: "a \<in> A \<Longrightarrow> \<forall>y \<in> A. f y = c \<Longrightarrow> (\<Union>y \<in> A. f y) = c"
\<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
by auto
lemma UN_equiv_class:
assumes "equiv A r" "f respects r" "a \<in> A"
shows "(\<Union>x \<in> r``{a}. f x) = f a"
\<comment> \<open>Conversion rule\<close>
proof -
have \<section>: "\<forall>x\<in>r `` {a}. f x = f a"
using assms unfolding equiv_def congruent_def sym_def by blast
show ?thesis
by (iprover intro: assms UN_constant_eq [OF equiv_class_self \<section>])
qed
lemma UN_equiv_class_type:
assumes r: "equiv A r" "f respects r" and X: "X \<in> A//r" and AB: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
shows "(\<Union>x \<in> X. f x) \<in> B"
using assms unfolding quotient_def
by (auto simp: UN_equiv_class [OF r])
text \<open>
Sufficient conditions for injectiveness. Could weaken premises!
major premise could be an inclusion; \<open>bcong\<close> could be
\<open>\<And>y. y \<in> A \<Longrightarrow> f y \<in> B\<close>.
\<close>
lemma UN_equiv_class_inject:
assumes "equiv A r" "f respects r"
and eq: "(\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y)"
and X: "X \<in> A//r" and Y: "Y \<in> A//r"
and fr: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> (x, y) \<in> r"
shows "X = Y"
proof -
obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
using assms by (auto elim!: quotientE)
then have "\<Union> (f ` r `` {a}) = f a" "\<Union> (f ` r `` {b}) = f b"
by (iprover intro: UN_equiv_class [OF \<open>equiv A r\<close>] assms)+
then have "f a = f b"
using eq unfolding a b by (iprover intro: trans sym)
then have "(a,b) \<in> r"
using fr \<open>a \<in> A\<close> \<open>b \<in> A\<close> by blast
then show ?thesis
unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
qed
subsection \<open>Defining binary operations upon equivalence classes\<close>
text \<open>A congruence-preserving function of two arguments.\<close>
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"
by (auto simp add: congruent2_def)
text \<open>Abbreviation for the common case where the relations are identical.\<close>
abbreviation RESPECTS2:: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infixr "respects2" 80)
where "f respects2 r \<equiv> congruent2 r r f"
lemma congruent2_implies_congruent:
"equiv A r1 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a \<in> A \<Longrightarrow> congruent r2 (f a)"
unfolding congruent_def congruent2_def equiv_def refl_on_def by blast
lemma congruent2_implies_congruent_UN:
assumes "equiv A1 r1" "equiv A2 r2" "congruent2 r1 r2 f" "a \<in> A2"
shows "congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
unfolding congruent_def
proof clarify
fix c d
assume cd: "(c,d) \<in> r1"
then have "c \<in> A1" "d \<in> A1"
using \<open>equiv A1 r1\<close> by (auto elim!: equiv_type [THEN subsetD, THEN SigmaE2])
with assms show "\<Union> (f c ` r2 `` {a}) = \<Union> (f d ` r2 `` {a})"
proof (simp add: UN_equiv_class congruent2_implies_congruent)
show "f c a = f d a"
using assms cd unfolding congruent2_def equiv_def refl_on_def by blast
qed
qed
lemma UN_equiv_class2:
"equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a1 \<in> A1 \<Longrightarrow> a2 \<in> A2 \<Longrightarrow>
(\<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 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f
\<Longrightarrow> X1 \<in> A1//r1 \<Longrightarrow> X2 \<in> A2//r2
\<Longrightarrow> (\<And>x1 x2. x1 \<in> A1 \<Longrightarrow> x2 \<in> A2 \<Longrightarrow> f x1 x2 \<in> B)
\<Longrightarrow> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
unfolding quotient_def
by (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
congruent2_implies_congruent quotientI)
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)"
\<comment> \<open>Allows a natural expression of binary operators,\<close>
\<comment> \<open>without explicit calls to \<open>split\<close>\<close>
by auto
lemma congruent2I:
"equiv A1 r1 \<Longrightarrow> equiv A2 r2
\<Longrightarrow> (\<And>y z w. w \<in> A2 \<Longrightarrow> (y,z) \<in> r1 \<Longrightarrow> f y w = f z w)
\<Longrightarrow> (\<And>y z w. w \<in> A1 \<Longrightarrow> (y,z) \<in> r2 \<Longrightarrow> f w y = f w z)
\<Longrightarrow> congruent2 r1 r2 f"
\<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<close>
\<comment> \<open>\<^emph>\<open>much\<close> simpler than the direct proof.\<close>
unfolding congruent2_def equiv_def refl_on_def
by (blast intro: trans)
lemma congruent2_commuteI:
assumes equivA: "equiv A r"
and commute: "\<And>y z. y \<in> A \<Longrightarrow> z \<in> A \<Longrightarrow> f y z = f z y"
and congt: "\<And>y z w. w \<in> A \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> f w y = f w z"
shows "f respects2 r"
proof (rule congruent2I [OF equivA equivA])
note eqv = equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2]
show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f y w = f z w"
by (iprover intro: commute [THEN trans] sym congt elim: eqv)
show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f w y = f w z"
by (iprover intro: congt elim: eqv)
qed
subsection \<open>Quotients and finiteness\<close>
text \<open>Suggested by Florian Kammüller\<close>
lemma finite_quotient:
assumes "finite A" "r \<subseteq> A \<times> A"
shows "finite (A//r)"
\<comment> \<open>recall @{thm equiv_type}\<close>
proof -
have "A//r \<subseteq> Pow A"
using assms unfolding quotient_def by blast
moreover have "finite (Pow A)"
using assms by simp
ultimately show ?thesis
by (iprover intro: finite_subset)
qed
lemma finite_equiv_class: "finite A \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> X \<in> A//r \<Longrightarrow> finite X"
unfolding quotient_def
by (erule rev_finite_subset) blast
lemma equiv_imp_dvd_card:
assumes "finite A" "equiv A r" "\<And>X. X \<in> A//r \<Longrightarrow> k dvd card X"
shows "k dvd card A"
proof (rule Union_quotient [THEN subst])
show "k dvd card (\<Union> (A // r))"
apply (rule dvd_partition)
using assms
by (auto simp: Union_quotient dest: quotient_disj)
qed (use assms in blast)
lemma card_quotient_disjoint:
assumes "finite A" "inj_on (\<lambda>x. {x} // r) A"
shows "card (A//r) = card A"
proof -
have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}"
using assms by (fastforce simp add: quotient_def inj_on_def)
with assms show ?thesis
by (simp add: quotient_def card_UN_disjoint)
qed
subsection \<open>Projection\<close>
definition proj :: "('b \<times> 'a) set \<Rightarrow> 'b \<Rightarrow> 'a set"
where "proj r x = r `` {x}"
lemma proj_preserves: "x \<in> A \<Longrightarrow> proj r x \<in> A//r"
unfolding proj_def by (rule quotientI)
lemma proj_in_iff:
assumes "equiv A r"
shows "proj r x \<in> A//r \<longleftrightarrow> x \<in> A"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?rhs
then show ?lhs by (simp add: proj_preserves)
next
assume ?lhs
then show ?rhs
unfolding proj_def quotient_def
proof clarsimp
fix y
assume y: "y \<in> A" and "r `` {x} = r `` {y}"
moreover have "y \<in> r `` {y}"
using assms y unfolding equiv_def refl_on_def by blast
ultimately have "(x, y) \<in> r" by blast
then show "x \<in> A"
using assms unfolding equiv_def refl_on_def by blast
qed
qed
lemma proj_iff: "equiv A r \<Longrightarrow> {x, y} \<subseteq> A \<Longrightarrow> proj r x = proj r y \<longleftrightarrow> (x, y) \<in> r"
by (simp add: proj_def eq_equiv_class_iff)
(*
lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
unfolding proj_def equiv_def refl_on_def by blast
*)
lemma proj_image: "proj r ` A = A//r"
unfolding proj_def[abs_def] quotient_def by blast
lemma in_quotient_imp_non_empty: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<noteq> {}"
unfolding quotient_def using equiv_class_self by fast
lemma in_quotient_imp_in_rel: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> {x, y} \<subseteq> X \<Longrightarrow> (x, y) \<in> r"
using quotient_eq_iff[THEN iffD1] by fastforce
lemma in_quotient_imp_closed: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> X"
unfolding quotient_def equiv_def trans_def by blast
lemma in_quotient_imp_subset: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<subseteq> A"
using in_quotient_imp_in_rel equiv_type by fastforce
subsection \<open>Equivalence relations -- predicate version\<close>
text \<open>Partial equivalences.\<close>
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)"
\<comment> \<open>John-Harrison-style characterization\<close>
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"
unfolding part_equivp_def by 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 \<open>Total equivalences.\<close>
definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
where "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
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: "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
by (auto intro: equivpI elim: equivpE)
lemma identity_equivp: "equivp (=)"
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)
lemma equivp_rtranclp: "symp r \<Longrightarrow> equivp r\<^sup>*\<^sup>*"
by(intro equivpI reflpI sympI transpI)(auto dest: sympD[OF symp_rtranclp])
lemmas equivp_rtranclp_symclp [simp] = equivp_rtranclp[OF symp_symclp]
lemma equivp_vimage2p: "equivp R \<Longrightarrow> equivp (vimage2p f f R)"
by(auto simp add: equivp_def vimage2p_def dest: fun_cong)
lemma equivp_imp_transp: "equivp R \<Longrightarrow> transp R"
by(simp add: equivp_reflp_symp_transp)
subsection \<open>Equivalence closure\<close>
definition equivclp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
"equivclp r = (symclp r)\<^sup>*\<^sup>*"
lemma transp_equivclp [simp]: "transp (equivclp r)"
by(simp add: equivclp_def)
lemma reflp_equivclp [simp]: "reflp (equivclp r)"
by(simp add: equivclp_def)
lemma symp_equivclp [simp]: "symp (equivclp r)"
by(simp add: equivclp_def)
lemma equivp_evquivclp [simp]: "equivp (equivclp r)"
by(simp add: equivpI)
lemma tranclp_equivclp [simp]: "(equivclp r)\<^sup>+\<^sup>+ = equivclp r"
by(simp add: equivclp_def)
lemma rtranclp_equivclp [simp]: "(equivclp r)\<^sup>*\<^sup>* = equivclp r"
by(simp add: equivclp_def)
lemma symclp_equivclp [simp]: "symclp (equivclp r) = equivclp r"
by(simp add: equivclp_def symp_symclp_eq)
lemma equivclp_symclp [simp]: "equivclp (symclp r) = equivclp r"
by(simp add: equivclp_def)
lemma equivclp_conversep [simp]: "equivclp (conversep r) = equivclp r"
by(simp add: equivclp_def)
lemma equivclp_sym [sym]: "equivclp r x y \<Longrightarrow> equivclp r y x"
by(rule sympD[OF symp_equivclp])
lemma equivclp_OO_equivclp_le_equivclp: "equivclp r OO equivclp r \<le> equivclp r"
by(rule transp_relcompp_less_eq transp_equivclp)+
lemma rtranlcp_le_equivclp: "r\<^sup>*\<^sup>* \<le> equivclp r"
unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
lemma rtranclp_conversep_le_equivclp: "r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)
lemma symclp_rtranclp_le_equivclp: "symclp r\<^sup>*\<^sup>* \<le> equivclp r"
unfolding symclp_pointfree
by(rule le_supI)(simp_all add: rtranclp_conversep[symmetric] rtranlcp_le_equivclp rtranclp_conversep_le_equivclp)
lemma r_OO_conversep_into_equivclp:
"r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>* \<le> equivclp r"
by(blast intro: order_trans[OF _ equivclp_OO_equivclp_le_equivclp] relcompp_mono rtranlcp_le_equivclp rtranclp_conversep_le_equivclp del: predicate2I)
lemma equivclp_induct [consumes 1, case_names base step, induct pred: equivclp]:
assumes a: "equivclp r a b"
and cases: "P a" "\<And>y z. equivclp r a y \<Longrightarrow> r y z \<or> r z y \<Longrightarrow> P y \<Longrightarrow> P z"
shows "P b"
using a unfolding equivclp_def
by(induction rule: rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
lemma converse_equivclp_induct [consumes 1, case_names base step]:
assumes major: "equivclp r a b"
and cases: "P b" "\<And>y z. r y z \<or> r z y \<Longrightarrow> equivclp r z b \<Longrightarrow> P z \<Longrightarrow> P y"
shows "P a"
using major unfolding equivclp_def
by(induction rule: converse_rtranclp_induct; fold equivclp_def; blast intro: cases elim: symclpE)
lemma equivclp_refl [simp]: "equivclp r x x"
by(rule reflpD[OF reflp_equivclp])
lemma r_into_equivclp [intro]: "r x y \<Longrightarrow> equivclp r x y"
unfolding equivclp_def by(blast intro: symclpI)
lemma converse_r_into_equivclp [intro]: "r y x \<Longrightarrow> equivclp r x y"
unfolding equivclp_def by(blast intro: symclpI)
lemma rtranclp_into_equivclp: "r\<^sup>*\<^sup>* x y \<Longrightarrow> equivclp r x y"
using rtranlcp_le_equivclp[of r] by blast
lemma converse_rtranclp_into_equivclp: "r\<^sup>*\<^sup>* y x \<Longrightarrow> equivclp r x y"
by(blast intro: equivclp_sym rtranclp_into_equivclp)
lemma equivclp_into_equivclp: "\<lbrakk> equivclp r a b; r b c \<or> r c b \<rbrakk> \<Longrightarrow> equivclp r a c"
unfolding equivclp_def by(erule rtranclp.rtrancl_into_rtrancl)(auto intro: symclpI)
lemma equivclp_trans [trans]: "\<lbrakk> equivclp r a b; equivclp r b c \<rbrakk> \<Longrightarrow> equivclp r a c"
using equivclp_OO_equivclp_le_equivclp[of r] by blast
hide_const (open) proj
end