# Theory Equiv_Relations

Up to index of Isabelle/HOL

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 r ⊆ r

lemma refl_comp_subset:

      refl A r ==> r ⊆ r^-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; a ∈ A |] ==> a ∈ r `` {a}

lemma subset_equiv_class:

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

lemma eq_equiv_class:

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

lemma equiv_class_nondisjoint:

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

lemma equiv_type:

      equiv A r ==> r ⊆ A × A

theorem equiv_class_eq_iff:

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

theorem eq_equiv_class_iff:

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

### Quotients

lemma quotientI:

      x ∈ A ==> r `` {x} ∈ A // r

lemma quotientE:

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

lemma Union_quotient:

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

lemma quotient_disj:

      [| equiv A r; X ∈ A // r; Y ∈ A // r |] ==> X = Y ∨ X ∩ Y = {}

lemma quotient_eqI:

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

lemma quotient_eq_iff:

      [| equiv A r; X ∈ A // r; Y ∈ A // r; x ∈ X; y ∈ Y |] ==> (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; a ∈ A |] ==> (A - {a}) // r = A // r - {a} // r

### Defining unary operations upon equivalence classes

lemma UN_constant_eq:

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

lemma UN_equiv_class:

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

lemma UN_equiv_class_type:

      [| equiv A r; f respects r; X ∈ A // r; !!x. x ∈ A ==> f x ∈ B |]
      ==> (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); X ∈ A // r;
         Y ∈ A // r; !!x y. [| x ∈ A; y ∈ A; 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; a ∈ A |] ==> 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; a ∈ A2.0 |]
      ==> (%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; a1.0 ∈ A1.0;
         a2.0 ∈ A2.0 |]
      ==> (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;
         X1.0 ∈ A1.0 // r1.0; X2.0 ∈ A2.0 // r2.0;
         !!x1 x2. [| x1 ∈ A1.0; x2 ∈ A2.0 |] ==> f x1 x2 ∈ B |]
      ==> (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. [| w ∈ A2.0; (y, z) ∈ r1.0 |] ==> f y w = f z w;
         !!y z w. [| w ∈ A1.0; (y, z) ∈ r2.0 |] ==> f w y = f w z |]
      ==> congruent2 r1.0 r2.0 f

lemma congruent2_commuteI:

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

### Cardinality results

lemma finite_quotient:

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

lemma finite_equiv_class:

      [| finite A; r ⊆ A × A; X ∈ A // r |] ==> finite X

lemma equiv_imp_dvd_card:

      [| finite A; equiv A r; ∀X∈A // 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