(* 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 begin
definition
quotient :: "[i,i]=>i" (infixl "'/'/" 90) (*set of equiv classes*) where
"A//r == {r``{x} . x \<in> A}"
definition
congruent :: "[i,i=>i]=>o" where
"congruent(r,b) == \<forall>y z. <y,z>:r \<longrightarrow> b(y)=b(z)"
definition
congruent2 :: "[i,i,[i,i]=>i]=>o" where
"congruent2(r1,r2,b) == \<forall>y1 z1 y2 z2.
<y1,z1>:r1 \<longrightarrow> <y2,z2>:r2 \<longrightarrow> 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 \<subseteq> r"
by (unfold trans_def sym_def, blast)
lemma refl_comp_subset:
"[| refl(A,r); r \<subseteq> A*A |] ==> r \<subseteq> 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 "\<forall>x y. <x,y> \<in> r \<longrightarrow> <y,x> \<in> r", blast+)
done
(** Equivalence classes **)
(*Lemma for the next result*)
lemma equiv_class_subset:
"[| sym(r); trans(r); <a,b>: r |] ==> r``{a} \<subseteq> 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 \<in> A |] ==> a \<in> r``{a}"
by (unfold equiv_def refl_def, blast)
(*Lemma for the next result*)
lemma subset_equiv_class:
"[| equiv(A,r); r``{b} \<subseteq> r``{a}; b \<in> A |] ==> <a,b>: r"
by (unfold equiv_def refl_def, blast)
lemma eq_equiv_class: "[| r``{a} = r``{b}; equiv(A,r); b \<in> 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} \<inter> r``{b}) |] ==> <a,b>: r"
by (unfold equiv_def trans_def sym_def, blast)
lemma equiv_type: "equiv(A,r) ==> r \<subseteq> A*A"
by (unfold equiv_def, blast)
lemma equiv_class_eq_iff:
"equiv(A,r) ==> <x,y>: r \<longleftrightarrow> r``{x} = r``{y} & x \<in> A & y \<in> A"
by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
lemma eq_equiv_class_iff:
"[| equiv(A,r); x \<in> A; y \<in> A |] ==> r``{x} = r``{y} \<longleftrightarrow> <x,y>: r"
by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type)
(*** Quotients ***)
(** Introduction/elimination rules -- needed? **)
lemma quotientI [TC]: "x \<in> A ==> r``{x}: A//r"
apply (unfold quotient_def)
apply (erule RepFunI)
done
lemma quotientE:
"[| X \<in> A//r; !!x. [| X = r``{x}; x \<in> 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 \<in> A//r; Y \<in> A//r |] ==> X=Y | (X \<inter> Y \<subseteq> 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 \<in> A |] ==> (\<Union>x\<in>r``{a}. b(x)) = b(a)"
apply (subgoal_tac "\<forall>x \<in> 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\<in>r``{a}. b(x)"} *)
lemma UN_equiv_class_type:
"[| equiv(A,r); b respects r; X \<in> A//r; !!x. x \<in> A ==> b(x) \<in> B |]
==> (\<Union>x\<in>X. b(x)) \<in> 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 \<in> A ==> b(y):B
*)
lemma UN_equiv_class_inject:
"[| equiv(A,r); b respects r;
(\<Union>x\<in>X. b(x))=(\<Union>y\<in>Y. b(y)); X \<in> A//r; Y \<in> A//r;
!!x y. [| x \<in> A; y \<in> 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 \<in> 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 \<in> A2 |] ==>
congruent(r1, %x1. \<Union>x2 \<in> 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 \<in> r1``{a1}. \<Union>x2 \<in> 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) \<in> B
|] ==> (\<Union>x1\<in>X1. \<Union>x2\<in>X2. b(x1,x2)) \<in> 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 \<in> A2; <y,z> \<in> r1 |] ==> b(y,w) = b(z,w);
!! y z w. [| w \<in> A1; <y,z> \<in> 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 \<in> A; z \<in> A |] ==> b(y,z) = b(z,y)"
and congt: "!! y z w. [| w \<in> 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 \<in> A//r;
!!w. [| w \<in> A |] ==> congruent(r, %z. b(w,z));
!!x y. [| x \<in> A; y \<in> A |] ==> b(y,x) = b(x,y)
|] ==> congruent(r, %w. \<Union>z\<in>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