Theory Equiv_Relations

theory Equiv_Relations
imports Groups_Big
```(*  Title:      HOL/Equiv_Relations.thy
Author:     Lawrence C Paulson, 1996 Cambridge University Computer Laboratory
*)

section ‹Equivalence Relations in Higher-Order Set Theory›

theory Equiv_Relations
imports Groups_Big
begin

subsection ‹Equivalence relations -- set version›

definition equiv :: "'a set ⇒ ('a × 'a) set ⇒ bool"
where "equiv A r ⟷ refl_on A r ∧ sym r ∧ trans r"

lemma equivI: "refl_on A r ⟹ sym r ⟹ trans r ⟹ 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 ‹
Suppes, Theorem 70: ‹r› is an equiv relation iff ‹r¯ O r = r›.

First half: ‹equiv A r ⟹ r¯ O r = r›.
›

lemma sym_trans_comp_subset: "sym r ⟹ trans r ⟹ r¯ O r ⊆ r"
unfolding trans_def sym_def converse_unfold by blast

lemma refl_on_comp_subset: "refl_on A r ⟹ r ⊆ r¯ O r"
unfolding refl_on_def by 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›
unfolding equiv_def trans_def sym_def by 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}"
unfolding equiv_def refl_on_def by blast

lemma subset_equiv_class: "equiv A r ⟹ r``{b} ⊆ r``{a} ⟹ b ∈ A ⟹ (a, b) ∈ r"
― ‹lemma for the next result›
unfolding equiv_def refl_on_def by 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"
unfolding equiv_def trans_def sym_def by blast

lemma equiv_type: "equiv A r ⟹ r ⊆ A × A"
unfolding equiv_def refl_on_def by blast

lemma 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)

lemma 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 "A//r = (⋃x ∈ A. {r``{x}})"  ― ‹set of equiv classes›

lemma quotientI: "x ∈ A ==> r``{x} ∈ A//r"
unfolding quotient_def by blast

lemma quotientE: "X ∈ A//r ⟹ (⋀x. X = r``{x} ⟹ x ∈ A ⟹ P) ⟹ P"
unfolding quotient_def by blast

lemma Union_quotient: "equiv A r ⟹ ⋃(A//r) = A"
unfolding equiv_def refl_on_def quotient_def by 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)
apply assumption
apply (unfold equiv_def sym_def trans_def)
apply 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)
apply 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 (λa. {a}//r) A ⟹ a ∈ A ⟹ (A - {a})//r = A//r - {a}//r"
unfolding quotient_def inj_on_def by blast

subsection ‹Refinement of one equivalence relation WRT another›

lemma refines_equiv_class_eq: "R ⊆ S ⟹ equiv A R ⟹ equiv A S ⟹ R``(S``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)

lemma refines_equiv_class_eq2: "R ⊆ S ⟹ equiv A R ⟹ equiv A S ⟹ S``(R``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)

lemma refines_equiv_image_eq: "R ⊆ S ⟹ equiv A R ⟹ equiv A S ⟹ (λX. S``X) ` (A//R) = A//S"
by (auto simp: quotient_def image_UN refines_equiv_class_eq2)

lemma finite_refines_finite:
"finite (A//R) ⟹ R ⊆ S ⟹ equiv A R ⟹ equiv A S ⟹ finite (A//S)"
by (erule finite_surj [where f = "λX. S``X"]) (simp add: refines_equiv_image_eq)

lemma finite_refines_card_le:
"finite (A//R) ⟹ R ⊆ S ⟹ equiv A R ⟹ equiv A S ⟹ card (A//S) ≤ card (A//R)"
by (subst refines_equiv_image_eq [of R S A, symmetric])
(auto simp: card_image_le [where f = "λX. S``X"])

subsection ‹Defining unary operations upon equivalence classes›

text ‹A congruence-preserving function.›

definition congruent :: "('a × 'a) set ⇒ ('a ⇒ 'b) ⇒ bool"
where "congruent r f ⟷ (∀(y, z) ∈ r. f y = f z)"

lemma congruentI: "(⋀y z. (y, z) ∈ r ⟹ f y = f z) ⟹ congruent r f"
by (auto simp add: congruent_def)

lemma congruentD: "congruent r f ⟹ (y, z) ∈ r ⟹ f y = f z"
by (auto simp add: congruent_def)

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 ⟹ (⋃y ∈ A. f y) = c"
― ‹lemma required to prove ‹UN_equiv_class››
by auto

lemma UN_equiv_class: "equiv A r ⟹ f respects r ⟹ a ∈ A ⟹ (⋃x ∈ r``{a}. f x) = f a"
― ‹Conversion rule›
apply (rule equiv_class_self [THEN UN_constant_eq])
apply assumption
apply 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) ⟹ (⋃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
‹⋀y. y ∈ A ⟹ f y ∈ B›.
›

lemma UN_equiv_class_inject:
"equiv A r ⟹ f respects r ⟹
(⋃x ∈ X. f x) = (⋃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.›

definition congruent2 :: "('a × 'a) set ⇒ ('b × 'b) set ⇒ ('a ⇒ 'b ⇒ 'c) ⇒ bool"
where "congruent2 r1 r2 f ⟷ (∀(y1, z1) ∈ r1. ∀(y2, z2) ∈ r2. f y1 y2 = f z1 z2)"

lemma congruent2I':
assumes "⋀y1 z1 y2 z2. (y1, z1) ∈ r1 ⟹ (y2, z2) ∈ r2 ⟹ 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 ⟹ (y1, z1) ∈ r1 ⟹ (y2, z2) ∈ r2 ⟹ f y1 y2 = f z1 z2"
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 ∈ A ⟹ congruent r2 (f a)"
unfolding congruent_def congruent2_def equiv_def refl_on_def by blast

lemma congruent2_implies_congruent_UN:
"equiv A1 r1 ⟹ equiv A2 r2 ⟹ congruent2 r1 r2 f ⟹ a ∈ A2 ⟹
congruent r1 (λx1. ⋃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 ⟹
(⋃x1 ∈ r1``{a1}. ⋃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)
⟹ (⋃x1 ∈ X1. ⋃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:
"(⋃(x1, x2) ∈ X. ⋃(y1, y2) ∈ Y. A x1 x2 y1 y2) =
(⋃x ∈ X. ⋃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 ‹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›
― ‹∗‹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 KammÃ¼ller›

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 (fastforce simp add:inj_on_def)
apply simp
done

subsection ‹Projection›

definition proj :: "('b × 'a) set ⇒ 'b ⇒ 'a set"
where "proj r x = r `` {x}"

lemma proj_preserves: "x ∈ A ⟹ proj r x ∈ A//r"
unfolding proj_def by (rule quotientI)

lemma proj_in_iff:
assumes "equiv A r"
shows "proj r x ∈ A//r ⟷ x ∈ A"
(is "?lhs ⟷ ?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 ∈ A" and "r `` {x} = r `` {y}"
moreover have "y ∈ r `` {y}"
using assms y unfolding equiv_def refl_on_def by blast
ultimately have "(x, y) ∈ r" by blast
then show "x ∈ A"
using assms unfolding equiv_def refl_on_def by blast
qed
qed

lemma proj_iff: "equiv A r ⟹ {x, y} ⊆ A ⟹ proj r x = proj r y ⟷ (x, y) ∈ r"
by (simp add: proj_def eq_equiv_class_iff)

(*
lemma in_proj: "⟦equiv A r; x ∈ A⟧ ⟹ x ∈ 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 ⟹ X ∈ A//r ⟹ X ≠ {}"
unfolding quotient_def using equiv_class_self by fast

lemma in_quotient_imp_in_rel: "equiv A r ⟹ X ∈ A//r ⟹ {x, y} ⊆ X ⟹ (x, y) ∈ r"
using quotient_eq_iff[THEN iffD1] by fastforce

lemma in_quotient_imp_closed: "equiv A r ⟹ X ∈ A//r ⟹ x ∈ X ⟹ (x, y) ∈ r ⟹ y ∈ X"
unfolding quotient_def equiv_def trans_def by blast

lemma in_quotient_imp_subset: "equiv A r ⟹ X ∈ A//r ⟹ X ⊆ A"
using in_quotient_imp_in_rel equiv_type by fastforce

subsection ‹Equivalence relations -- predicate version›

text ‹Partial equivalences.›

definition part_equivp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
where "part_equivp R ⟷ (∃x. R x x) ∧ (∀x y. R x y ⟷ R x x ∧ R y y ∧ R x = R y)"
― ‹John-Harrison-style characterization›

lemma part_equivpI: "∃x. R x x ⟹ symp R ⟹ transp R ⟹ 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: "∃x. R x x"
and 2: "⋀x y. R x y ⟷ R x x ∧ R y y ∧ 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 ⟷ (∃x. R x x) ∧ symp R ∧ transp R"
by (auto intro: part_equivpI elim: part_equivpE)

lemma part_equivp_symp: "part_equivp R ⟹ R x y ⟹ R y x"
by (erule part_equivpE, erule sympE)

lemma part_equivp_transp: "part_equivp R ⟹ R x y ⟹ R y z ⟹ R x z"
by (erule part_equivpE, erule transpE)

lemma part_equivp_typedef: "part_equivp R ⟹ ∃d. d ∈ {c. ∃x. R x x ∧ c = Collect (R x)}"
by (auto elim: part_equivpE)

text ‹Total equivalences.›

definition equivp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
where "equivp R ⟷ (∀x y. R x y = (R x = R y))" ― ‹John-Harrison-style characterization›

lemma equivpI: "reflp R ⟹ symp R ⟹ transp R ⟹ 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 ⟹ part_equivp R"
by (auto intro: part_equivpI elim: equivpE reflpE)

lemma equivp_equiv: "equiv UNIV A ⟷ equivp (λx y. (x, y) ∈ A)"
by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])

lemma equivp_reflp_symp_transp: "equivp R ⟷ reflp R ∧ symp R ∧ 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 ⟹ R x x"
by (erule equivpE, erule reflpE)

lemma equivp_symp: "equivp R ⟹ R x y ⟹ R y x"
by (erule equivpE, erule sympE)

lemma equivp_transp: "equivp R ⟹ R x y ⟹ R y z ⟹ R x z"
by (erule equivpE, erule transpE)

hide_const (open) proj

end
```