Theory Equiv_Relations

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theory Equiv_Relations
imports Relation Finite_Set
begin

(*  ID:         $Id: Equiv_Relations.thy,v 1.6 2005/09/22 21:56:15 nipkow Exp $
    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge
*)

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

theory Equiv_Relations
imports Relation Finite_Set
begin

subsection {* Equivalence relations *}

locale equiv =
  fixes A and r
  assumes refl: "refl 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_comp_subset: "refl A r ==> r ⊆ r¯ O r"
  by (unfold refl_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_comp_subset)+
  done

text {* Second half. *}

lemma comp_equivI:
    "r¯ O r = r ==> Domain r = A ==> equiv A r"
  apply (unfold equiv_def refl_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_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_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_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 *}

constdefs
  quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
  "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_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 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"

syntax
  RESPECTS ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects" 80)

translations
  "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*}
syntax
  RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool"  (infixr "respects2 " 80)

translations
  "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_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_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_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 (assumption | rule congt |
         erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
  done


subsection {* Cardinality results *}

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])
   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

ML
{*
val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
val UN_constant_eq = thm "UN_constant_eq";
val UN_equiv_class = thm "UN_equiv_class";
val UN_equiv_class2 = thm "UN_equiv_class2";
val UN_equiv_class_inject = thm "UN_equiv_class_inject";
val UN_equiv_class_type = thm "UN_equiv_class_type";
val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
val Union_quotient = thm "Union_quotient";
val comp_equivI = thm "comp_equivI";
val congruent2I = thm "congruent2I";
val congruent2_commuteI = thm "congruent2_commuteI";
val congruent2_def = thm "congruent2_def";
val congruent2_implies_congruent = thm "congruent2_implies_congruent";
val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
val congruent_def = thm "congruent_def";
val eq_equiv_class = thm "eq_equiv_class";
val eq_equiv_class_iff = thm "eq_equiv_class_iff";
val equiv_class_eq = thm "equiv_class_eq";
val equiv_class_eq_iff = thm "equiv_class_eq_iff";
val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
val equiv_class_self = thm "equiv_class_self";
val equiv_comp_eq = thm "equiv_comp_eq";
val equiv_def = thm "equiv_def";
val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
val equiv_type = thm "equiv_type";
val finite_equiv_class = thm "finite_equiv_class";
val finite_quotient = thm "finite_quotient";
val quotientE = thm "quotientE";
val quotientI = thm "quotientI";
val quotient_def = thm "quotient_def";
val quotient_disj = thm "quotient_disj";
val refl_comp_subset = thm "refl_comp_subset";
val subset_equiv_class = thm "subset_equiv_class";
val sym_trans_comp_subset = thm "sym_trans_comp_subset";
*}

end

Equivalence relations

lemma sym_trans_comp_subset:

  [| sym r; trans r |] ==> r^-1 O rr

lemma refl_comp_subset:

  refl A r ==> rr^-1 O r

lemma equiv_comp_eq:

  equiv A r ==> r^-1 O r = r

lemma comp_equivI:

  [| r^-1 O r = r; Domain r = A |] ==> equiv A r

Equivalence classes

lemma equiv_class_subset:

  [| equiv A r; (a, b) ∈ r |] ==> r `` {a} ⊆ r `` {b}

theorem equiv_class_eq:

  [| equiv A r; (a, b) ∈ r |] ==> r `` {a} = r `` {b}

lemma equiv_class_self:

  [| equiv A r; aA |] ==> ar `` {a}

lemma subset_equiv_class:

  [| equiv A r; r `` {b} ⊆ r `` {a}; bA |] ==> (a, b) ∈ r

lemma eq_equiv_class:

  [| r `` {a} = r `` {b}; equiv A r; bA |] ==> (a, b) ∈ r

lemma equiv_class_nondisjoint:

  [| equiv A r; xr `` {a} ∩ r `` {b} |] ==> (a, b) ∈ r

lemma equiv_type:

  equiv A r ==> rA × A

theorem equiv_class_eq_iff:

  equiv A r ==> ((x, y) ∈ r) = (r `` {x} = r `` {y} ∧ xAyA)

theorem eq_equiv_class_iff:

  [| equiv A r; xA; yA |] ==> (r `` {x} = r `` {y}) = ((x, y) ∈ r)

Quotients

lemma quotientI:

  xA ==> r `` {x} ∈ A // r

lemma quotientE:

  [| XA // r; !!x. [| X = r `` {x}; xA |] ==> P |] ==> P

lemma Union_quotient:

  equiv A r ==> Union (A // r) = A

lemma quotient_disj:

  [| equiv A r; XA // r; YA // r |] ==> X = YXY = {}

lemma quotient_eqI:

  [| equiv A r; XA // r; YA // r; xX; yY; (x, y) ∈ r |] ==> X = Y

lemma quotient_eq_iff:

  [| equiv A r; XA // r; YA // r; xX; yY |] ==> (X = Y) = ((x, y) ∈ r)

lemma quotient_empty:

  {} // r = {}

lemma quotient_is_empty:

  (A // r = {}) = (A = {})

lemma quotient_is_empty2:

  ({} = A // r) = (A = {})

lemma singleton_quotient:

  {x} // r = {r `` {x}}

lemma quotient_diff1:

  [| inj_on (%a. {a} // r) A; aA |] ==> (A - {a}) // r = A // r - {a} // r

Defining unary operations upon equivalence classes

lemma UN_constant_eq:

  [| aA; ∀yA. f y = c |] ==> (UN y:A. f y) = c

lemma UN_equiv_class:

  [| equiv A r; f respects r; aA |] ==> (UN x:r `` {a}. f x) = f a

lemma UN_equiv_class_type:

  [| equiv A r; f respects r; XA // r; !!x. xA ==> f xB |]
  ==> (UN x:X. f x) ∈ B

lemma UN_equiv_class_inject:

  [| equiv A r; f respects r; (UN x:X. f x) = (UN y:Y. f y); XA // r;
     YA // r; !!x y. [| xA; yA; f x = f y |] ==> (x, y) ∈ r |]
  ==> X = Y

Defining binary operations upon equivalence classes

lemma congruent2_implies_congruent:

  [| equiv A r1.0; congruent2 r1.0 r2.0 f; aA |] ==> f a respects r2.0

lemma congruent2_implies_congruent_UN:

  [| equiv A1.0 r1.0; equiv A2.0 r2.0; congruent2 r1.0 r2.0 f; aA2.0 |]
  ==> (%x1. UN x2:r2.0 `` {a}. f x1 x2) respects r1.0

lemma UN_equiv_class2:

  [| equiv A1.0 r1.0; equiv A2.0 r2.0; congruent2 r1.0 r2.0 f; a1.0A1.0;
     a2.0A2.0 |]
  ==> (UN x1:r1.0 `` {a1.0}. UN x2:r2.0 `` {a2.0}. f x1 x2) = f a1.0 a2.0

lemma UN_equiv_class_type2:

  [| equiv A1.0 r1.0; equiv A2.0 r2.0; congruent2 r1.0 r2.0 f;
     X1.0A1.0 // r1.0; X2.0A2.0 // r2.0;
     !!x1 x2. [| x1A1.0; x2A2.0 |] ==> f x1 x2B |]
  ==> (UN x1:X1.0. UN x2:X2.0. f x1 x2) ∈ B

lemma UN_UN_split_split_eq:

  (UN (x1, x2):X. UN (y1, y2):Y. A x1 x2 y1 y2) =
  (UN x:X. UN y:Y. (%(x1, x2). split (A x1 x2) y) x)

lemma congruent2I:

  [| equiv A1.0 r1.0; equiv A2.0 r2.0;
     !!y z w. [| wA2.0; (y, z) ∈ r1.0 |] ==> f y w = f z w;
     !!y z w. [| wA1.0; (y, z) ∈ r2.0 |] ==> f w y = f w z |]
  ==> congruent2 r1.0 r2.0 f

lemma congruent2_commuteI:

  [| equiv A r; !!y z. [| yA; zA |] ==> f y z = f z y;
     !!y z w. [| wA; (y, z) ∈ r |] ==> f w y = f w z |]
  ==> congruent2 r r f

Cardinality results

lemma finite_quotient:

  [| finite A; rA × A |] ==> finite (A // r)

lemma finite_equiv_class:

  [| finite A; rA × A; XA // r |] ==> finite X

lemma equiv_imp_dvd_card:

  [| finite A; equiv A r; ∀XA // r. k dvd card X |] ==> k dvd card A

lemma card_quotient_disjoint:

  [| finite A; inj_on (%x. {x} // r) A |] ==> card (A // r) = card A