Theory Equiv_Relations

(*  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 BNF_Least_Fixpoint
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"
  unfolding equiv_def
  by (iprover intro: sym_trans_comp_subset refl_on_comp_subset equalityI)

text Second half.

lemma comp_equivI:
  assumes "r¯ O r = r" "Domain r = A"
  shows "equiv A r"
proof -
  have *: "x y. (x, y)  r  (y, x)  r"
    using assms by blast
  show ?thesis
    unfolding equiv_def refl_on_def sym_def trans_def
    using assms by (auto intro: *)
qed


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}"
  by (intro equalityI equiv_class_subset; force simp add: equiv_def sym_def)

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)

lemma disjnt_equiv_class: "equiv A r  disjnt (r``{a}) (r``{b})  (a, b)  r"
  by (auto dest: equiv_class_self simp: equiv_class_eq_iff disjnt_def)


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 = {}"
  unfolding quotient_def equiv_def trans_def sym_def by blast

lemma quotient_eqI:
  assumes "equiv A r" "X  A//r" "Y  A//r" and xy: "x  X" "y  Y" "(x, y)  r"
  shows "X = Y"
proof -
  obtain a b where "a  A" and a: "X = r `` {a}" and "b  A" and b: "Y = r `` {b}"
    using assms by (auto elim!: quotientE)
  then have "(a,b)  r"
      using xy equiv A r unfolding equiv_def sym_def trans_def by blast
  then show ?thesis
    unfolding a b by (rule equiv_class_eq [OF equiv A r])
qed

lemma quotient_eq_iff:
  assumes "equiv A r" "X  A//r" "Y  A//r" and xy: "x  X" "y  Y" 
  shows "X = Y  (x, y)  r"
proof
  assume L: "X = Y" 
  with assms show "(x, y)  r" 
    unfolding equiv_def sym_def trans_def by (blast elim!: quotientE)
next
  assume §: "(x, y)  r" show "X = Y"
    by (rule quotient_eqI) (use § assms in blast+)
qed

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:
  assumes "equiv A r" "f respects r" "a  A"
  shows "(x  r``{a}. f x) = f a"
  ― ‹Conversion rule
proof -
  have §: "xr `` {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 §])
qed

lemma UN_equiv_class_type:
  assumes r: "equiv A r" "f respects r" and X: "X  A//r" and AB: "x. x  A  f x  B"
  shows "(x  X. f x)  B"
  using assms unfolding quotient_def
  by (auto simp: UN_equiv_class [OF r])

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:
  assumes "equiv A r" "f respects r"
    and eq: "(x  X. f x) = (y  Y. f y)" 
    and X: "X  A//r" and Y: "Y  A//r" 
    and fr: "x y. x  A  y  A  f x = f y  (x, y)  r"
  shows "X = Y"
proof -
  obtain a b where "a  A" and a: "X = r `` {a}" and "b  A" and b: "Y = r `` {b}"
    using assms by (auto elim!: quotientE)
  then have " (f ` r `` {a}) = f a" " (f ` r `` {b}) = f b"
    by (iprover intro: UN_equiv_class [OF equiv A r] assms)+
  then have "f a = f b"
    using eq unfolding a b by (iprover intro: trans sym)
  then have "(a,b)  r"
    using fr a  A b  A by blast
  then show ?thesis
    unfolding a b by (rule equiv_class_eq [OF equiv A r])
qed


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:
  assumes "equiv A1 r1" "equiv A2 r2" "congruent2 r1 r2 f" "a  A2" 
  shows "congruent r1 (λx1. x2  r2``{a}. f x1 x2)"
  unfolding congruent_def
proof clarify
  fix c d
  assume cd: "(c,d)  r1"
  then have "c  A1" "d  A1"
    using equiv A1 r1 by (auto elim!: equiv_type [THEN subsetD, THEN SigmaE2])
  moreover have "f c a = f d a"
    using assms cd unfolding congruent2_def equiv_def refl_on_def by blast
  ultimately show " (f c ` r2 `` {a}) =  (f d ` r2 `` {a})"
    using assms by (simp add: UN_equiv_class congruent2_implies_congruent)
qed

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"
  unfolding quotient_def
  by (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
                   congruent2_implies_congruent quotientI)


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.
  unfolding congruent2_def equiv_def refl_on_def
  by (blast intro: trans)

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"
proof (rule congruent2I [OF equivA equivA])
  note eqv = equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2]
  show "y z w. w  A; (y, z)  r  f y w = f z w"
    by (iprover intro: commute [THEN trans] sym congt elim: eqv)
  show "y z w. w  A; (y, z)  r  f w y = f w z"
    by (iprover intro: congt elim: eqv)
qed


subsection Quotients and finiteness

text Suggested by Florian Kammüller

lemma finite_quotient:
  assumes "finite A" "r  A × A"
  shows "finite (A//r)"
    ― ‹recall @{thm equiv_type}
proof -
  have "A//r  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  r  A × A  X  A//r  finite X"
  unfolding quotient_def
  by (erule rev_finite_subset) blast

lemma equiv_imp_dvd_card:
  assumes "finite A" "equiv A r" "X. X  A//r  k dvd card X"
  shows "k dvd card A"
proof (rule Union_quotient [THEN subst])
  show "k dvd card ( (A // r))"
    apply (rule dvd_partition)
    using assms
    by (auto simp: Union_quotient dest: quotient_disj)
qed (use assms in blast)


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 safe
    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 (=)"
  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)

lemma equivp_rtranclp: "symp r  equivp r**"
  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  equivp (vimage2p f f R)"
  by(auto simp add: equivp_def vimage2p_def dest: fun_cong)

lemma equivp_imp_transp: "equivp R  transp R"
  by(simp add: equivp_reflp_symp_transp)


subsection Equivalence closure

definition equivclp :: "('a  'a  bool)  'a  'a  bool" where
  "equivclp r = (symclp r)**"

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)++ = equivclp r"
  by(simp add: equivclp_def)

lemma rtranclp_equivclp [simp]: "(equivclp r)** = 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  equivclp r y x"
  by(rule sympD[OF symp_equivclp])

lemma equivclp_OO_equivclp_le_equivclp: "equivclp r OO equivclp r  equivclp r"
  by(rule transp_relcompp_less_eq transp_equivclp)+

lemma rtranlcp_le_equivclp: "r**  equivclp r"
  unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)

lemma rtranclp_conversep_le_equivclp: "r¯¯**  equivclp r"
  unfolding equivclp_def by(rule rtranclp_mono)(simp add: symclp_pointfree)

lemma symclp_rtranclp_le_equivclp: "symclp r**  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** OO r¯¯**  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" "y z. equivclp r a y  r y z  r z y  P y  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" "y z. r y z  r z y  equivclp r z b  P z  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  equivclp r x y"
  unfolding equivclp_def by(blast intro: symclpI)

lemma converse_r_into_equivclp [intro]: "r y x  equivclp r x y"
  unfolding equivclp_def by(blast intro: symclpI)

lemma rtranclp_into_equivclp: "r** x y  equivclp r x y"
  using rtranlcp_le_equivclp[of r] by blast

lemma converse_rtranclp_into_equivclp: "r** y x  equivclp r x y"
  by(blast intro: equivclp_sym rtranclp_into_equivclp)

lemma equivclp_into_equivclp: " equivclp r a b; r b c  r c b   equivclp r a c"
  unfolding equivclp_def by(erule rtranclp.rtrancl_into_rtrancl)(auto intro: symclpI)

lemma equivclp_trans [trans]: " equivclp r a b; equivclp r b c   equivclp r a c"
  using equivclp_OO_equivclp_le_equivclp[of r] by blast

hide_const (open) proj

end