Theory EquivClass

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theory EquivClass
imports Trancl
begin

(*  Title:      ZF/EquivClass.thy
    ID:         $Id: EquivClass.thy,v 1.9 2007/10/07 19:19:32 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

*)

header{*Equivalence Relations*}

theory EquivClass imports Trancl Perm begin

definition
  quotient   :: "[i,i]=>i"    (infixl "'/'/" 90)  (*set of equiv classes*)  where
      "A//r == {r``{x} . x:A}"

definition
  congruent  :: "[i,i=>i]=>o"  where
      "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"

definition
  congruent2 :: "[i,i,[i,i]=>i]=>o"  where
      "congruent2(r1,r2,b) == ALL y1 z1 y2 z2.
           <y1,z1>:r1 --> <y2,z2>:r2 --> b(y1,y2) = b(z1,z2)"

abbreviation
  RESPECTS ::"[i=>i, i] => o"  (infixr "respects" 80) where
  "f respects r == congruent(r,f)"

abbreviation
  RESPECTS2 ::"[i=>i=>i, i] => o"  (infixr "respects2 " 80) where
  "f respects2 r == congruent2(r,r,f)"
    --{*Abbreviation for the common case where the relations are identical*}


subsection{*Suppes, Theorem 70:
    @{term r} is an equiv relation iff @{term "converse(r) O r = r"}*}

(** first half: equiv(A,r) ==> converse(r) O r = r **)

lemma sym_trans_comp_subset:
    "[| sym(r); trans(r) |] ==> converse(r) O r <= r"
by (unfold trans_def sym_def, blast)

lemma refl_comp_subset:
    "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r"
by (unfold refl_def, blast)

lemma equiv_comp_eq:
    "equiv(A,r) ==> converse(r) O r = r"
apply (unfold equiv_def)
apply (blast del: subsetI intro!: sym_trans_comp_subset refl_comp_subset)
done

(*second half*)
lemma comp_equivI:
    "[| converse(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 "ALL x y. <x,y> : r --> <y,x> : r", blast+)
done

(** Equivalence classes **)

(*Lemma for the next result*)
lemma equiv_class_subset:
    "[| sym(r);  trans(r);  <a,b>: r |] ==> r``{a} <= r``{b}"
by (unfold trans_def sym_def, blast)

lemma equiv_class_eq:
    "[| equiv(A,r);  <a,b>: r |] ==> r``{a} = r``{b}"
apply (unfold equiv_def)
apply (safe del: subsetI intro!: equalityI equiv_class_subset)
apply (unfold sym_def, blast)
done

lemma equiv_class_self:
    "[| equiv(A,r);  a: A |] ==> a: r``{a}"
by (unfold equiv_def refl_def, blast)

(*Lemma for the next result*)
lemma subset_equiv_class:
    "[| equiv(A,r);  r``{b} <= r``{a};  b: A |] ==> <a,b>: r"
by (unfold equiv_def refl_def, blast)

lemma eq_equiv_class: "[| r``{a} = r``{b};  equiv(A,r);  b: A |] ==> <a,b>: r"
by (assumption | rule equalityD2 subset_equiv_class)+

(*thus r``{a} = r``{b} as well*)
lemma equiv_class_nondisjoint:
    "[| equiv(A,r);  x: (r``{a} Int 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, blast)

lemma equiv_class_eq_iff:
     "equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A"
by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)

lemma eq_equiv_class_iff:
     "[| equiv(A,r);  x: A;  y: A |] ==> r``{x} = r``{y} <-> <x,y>: r"
by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)

(*** Quotients ***)

(** Introduction/elimination rules -- needed? **)

lemma quotientI [TC]: "x:A ==> r``{x}: A//r"
apply (unfold quotient_def)
apply (erule RepFunI)
done

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 Int Y <= 0)"
apply (unfold quotient_def)
apply (safe intro!: equiv_class_eq, assumption)
apply (unfold equiv_def trans_def sym_def, blast)
done

subsection{*Defining Unary Operations upon Equivalence Classes*}

(** Could have a locale with the premises equiv(A,r)  and  congruent(r,b)
**)

(*Conversion rule*)
lemma UN_equiv_class:
    "[| equiv(A,r);  b respects r;  a: A |] ==> (UN x:r``{a}. b(x)) = b(a)"
apply (subgoal_tac "∀x ∈ r``{a}. b(x) = b(a)") 
 apply simp
 apply (blast intro: equiv_class_self)  
apply (unfold equiv_def sym_def congruent_def, blast)
done

(*type checking of  UN x:r``{a}. b(x) *)
lemma UN_equiv_class_type:
    "[| equiv(A,r);  b respects r;  X: A//r;  !!x.  x : A ==> b(x) : B |]
     ==> (UN x:X. b(x)) : B"
apply (unfold quotient_def, safe)
apply (simp (no_asm_simp) add: UN_equiv_class)
done

(*Sufficient conditions for injectiveness.  Could weaken premises!
  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
*)
lemma UN_equiv_class_inject:
    "[| equiv(A,r);   b respects r;
        (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;
        !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |]
     ==> X=Y"
apply (unfold quotient_def, safe)
apply (rule equiv_class_eq, assumption)
apply (simp add: UN_equiv_class [of A r b])  
done


subsection{*Defining Binary Operations upon Equivalence Classes*}

lemma congruent2_implies_congruent:
    "[| equiv(A,r1);  congruent2(r1,r2,b);  a: A |] ==> congruent(r2,b(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,b);  a: A2 |] ==>
     congruent(r1, %x1. \<Union>x2 ∈ r2``{a}. b(x1,x2))"
apply (unfold congruent_def, safe)
apply (frule equiv_type [THEN subsetD], assumption)
apply clarify 
apply (simp add: UN_equiv_class congruent2_implies_congruent)
apply (unfold congruent2_def equiv_def refl_def, blast)
done

lemma UN_equiv_class2:
    "[| equiv(A1,r1);  equiv(A2,r2);  congruent2(r1,r2,b);  a1: A1;  a2: A2 |]
     ==> (\<Union>x1 ∈ r1``{a1}. \<Union>x2 ∈ r2``{a2}. b(x1,x2)) = b(a1,a2)"
by (simp add: UN_equiv_class congruent2_implies_congruent
              congruent2_implies_congruent_UN)

(*type checking*)
lemma UN_equiv_class_type2:
    "[| equiv(A,r);  b respects2 r;
        X1: A//r;  X2: A//r;
        !!x1 x2.  [| x1: A; x2: A |] ==> b(x1,x2) : B
     |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B"
apply (unfold quotient_def, safe)
apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN 
                    congruent2_implies_congruent quotientI)
done


(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
  than the direct proof*)
lemma congruent2I:
    "[|  equiv(A1,r1);  equiv(A2,r2);  
        !! y z w. [| w ∈ A2;  <y,z> ∈ r1 |] ==> b(y,w) = b(z,w);
        !! y z w. [| w ∈ A1;  <y,z> ∈ r2 |] ==> b(w,y) = b(w,z)
     |] ==> congruent2(r1,r2,b)"
apply (unfold congruent2_def equiv_def refl_def, safe)
apply (blast intro: trans) 
done

lemma congruent2_commuteI:
 assumes equivA: "equiv(A,r)"
     and commute: "!! y z. [| y: A;  z: A |] ==> b(y,z) = b(z,y)"
     and congt:   "!! y z w. [| w: A;  <y,z>: r |] ==> b(w,y) = b(w,z)"
 shows "b respects2 r"
apply (insert equivA [THEN equiv_type, THEN subsetD]) 
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 (blast intro: congt)+
done

(*Obsolete?*)
lemma congruent_commuteI:
    "[| equiv(A,r);  Z: A//r;
        !!w. [| w: A |] ==> congruent(r, %z. b(w,z));
        !!x y. [| x: A;  y: A |] ==> b(y,x) = b(x,y)
     |] ==> congruent(r, %w. UN z: Z. b(w,z))"
apply (simp (no_asm) add: congruent_def)
apply (safe elim!: quotientE)
apply (frule equiv_type [THEN subsetD], assumption)
apply (simp add: UN_equiv_class [of A r]) 
apply (simp add: congruent_def) 
done

end

Suppes, Theorem 70: @{term r} is an equiv relation iff @{term "converse(r) O r = r"}

lemma sym_trans_comp_subset:

  [| sym(r); trans(r) |] ==> converse(r) O rr

lemma refl_comp_subset:

  [| refl(A, r); rA × A |] ==> r ⊆ converse(r) O r

lemma equiv_comp_eq:

  equiv(A, r) ==> converse(r) O r = r

lemma comp_equivI:

  [| converse(r) O r = r; domain(r) = A |] ==> equiv(A, r)

lemma equiv_class_subset:

  [| sym(r); trans(r); ⟨a, b⟩ ∈ r |] ==> r `` {a} ⊆ r `` {b}

lemma 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

lemma equiv_class_eq_iff:

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

lemma eq_equiv_class_iff:

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

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 ⊆ 0

Defining Unary Operations upon Equivalence Classes

lemma UN_equiv_class:

  [| equiv(A, r); b respects r; aA |] ==> (\<Union>xr `` {a}. b(x)) = b(a)

lemma UN_equiv_class_type:

  [| equiv(A, r); b respects r; XA // r; !!x. xA ==> b(x) ∈ B |]
  ==> (\<Union>xX. b(x)) ∈ B

lemma UN_equiv_class_inject:

  [| equiv(A, r); b respects r; (\<Union>xX. b(x)) = (\<Union>yY. b(y));
     XA // r; YA // r;
     !!x y. [| xA; yA; b(x) = b(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, b); aA |] ==> b(a) respects r2.0

lemma congruent2_implies_congruent_UN:

  [| equiv(A1.0, r1.0); equiv(A2.0, r2.0); congruent2(r1.0, r2.0, b); aA2.0 |]
  ==> (λx1. \<Union>x2r2.0 `` {a}. b(x1, x2)) respects r1.0

lemma UN_equiv_class2:

  [| equiv(A1.0, r1.0); equiv(A2.0, r2.0); congruent2(r1.0, r2.0, b); a1.0A1.0;
     a2.0A2.0 |]
  ==> (\<Union>x1r1.0 `` {a1.0}. \<Union>x2r2.0 `` {a2.0}. b(x1, x2)) =
      b(a1.0, a2.0)

lemma UN_equiv_class_type2:

  [| equiv(A, r); b respects2  r; X1.0A // r; X2.0A // r;
     !!x1 x2. [| x1A; x2A |] ==> b(x1, x2) ∈ B |]
  ==> (\<Union>x1X1.0. \<Union>x2X2.0. b(x1, x2)) ∈ B

lemma congruent2I:

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

lemma congruent2_commuteI:

  [| equiv(A, r); !!y z. [| yA; zA |] ==> b(y, z) = b(z, y);
     !!y z w. [| wA; ⟨y, z⟩ ∈ r |] ==> b(w, y) = b(w, z) |]
  ==> b respects2  r

lemma congruent_commuteI:

  [| equiv(A, r); ZA // r; !!w. wA ==> (λz. b(w, z)) respects r;
     !!x y. [| xA; yA |] ==> b(y, x) = b(x, y) |]
  ==> (λw. \<Union>zZ. b(w, z)) respects r