--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Integ/Equiv.thy Mon Feb 27 16:36:17 1995 +0100
@@ -0,0 +1,28 @@
+(* Title: Equiv.thy
+ ID: $Id$
+ Authors: Riccardo Mattolini, Dip. Sistemi e Informatica
+ Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 Universita' di Firenze
+ Copyright 1993 University of Cambridge
+
+Equivalence relations in Higher-Order Set Theory
+*)
+
+Equiv = Relation +
+consts
+ refl,equiv :: "['a set,('a*'a) set]=>bool"
+ sym :: "('a*'a) set=>bool"
+ "'/" :: "['a set,('a*'a) set]=>'a set set" (infixl 90)
+ (*set of equiv classes*)
+ congruent :: "[('a*'a) set,'a=>'b]=>bool"
+ congruent2 :: "[('a*'a) set,['a,'a]=>'b]=>bool"
+
+defs
+ refl_def "refl(A,r) == r <= Sigma(A,%x.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 == UN x:A. {r^^{x}}"
+ 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