--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/ex/Equiv.thy Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,28 @@
+(* Title: ZF/ex/equiv.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+Equivalence relations in Zermelo-Fraenkel Set Theory
+*)
+
+Equiv = Trancl +
+consts
+ refl,equiv :: "[i,i]=>o"
+ sym :: "i=>o"
+ "'/" :: "[i,i]=>i" (infixl 90) (*set of equiv classes*)
+ congruent :: "[i,i=>i]=>o"
+ congruent2 :: "[i,[i,i]=>i]=>o"
+
+rules
+ refl_def "refl(A,r) == r <= (A*A) & (ALL x: A. <x,x> : r)"
+ sym_def "sym(r) == ALL x y. <x,y>: r --> <y,x>: r"
+ equiv_def "equiv(A,r) == refl(A,r) & sym(r) & trans(r)"
+ quotient_def "A/r == {r``{x} . x:A}"
+ congruent_def "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
+
+ congruent2_def
+ "congruent2(r,b) == ALL y1 z1 y2 z2. \
+\ <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)"
+
+end