# Theory EquivClass

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theory EquivClass
imports Trancl
`(*  Title:      ZF/EquivClass.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1994  University of Cambridge*)header{*Equivalence Relations*}theory EquivClass imports Trancl Perm begindefinition  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) == ∀y z. <y,z>:r --> b(y)=b(z)"definition  congruent2 :: "[i,i,[i,i]=>i]=>o"  where      "congruent2(r1,r2,b) == ∀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 "∀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)donelemma 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} ∩ 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)donelemma 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 ⊆ 0)"apply (unfold quotient_def)apply (safe intro!: equiv_class_eq, assumption)apply (unfold equiv_def trans_def sym_def, blast)donesubsection{*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 |] ==> (\<Union>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  @{term"\<Union>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 |]     ==> (\<Union>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;        (\<Union>x∈X. b(x))=(\<Union>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])donesubsection{*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 clarifyapply (simp add: UN_equiv_class congruent2_implies_congruent)apply (unfold congruent2_def equiv_def refl_def, blast)donelemma 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     |] ==> (\<Union>x1∈X1. \<Union>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)donelemma 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. \<Union>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)doneend`