(* Title: Equiv.thy
ID: $Id$
Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Equivalence relations in Higher-Order Set Theory
*)
Equiv = Relation + Finite +
constdefs
equiv :: "['a set, ('a*'a) set] => bool"
"equiv A r == refl A r & sym(r) & trans(r)"
quotient :: "['a set, ('a*'a) set] => 'a set set" (infixl "'/'/" 90)
"A//r == UN x:A. {r^^{x}}" (*set of equiv classes*)
congruent :: "[('a*'a) set, 'a=>'b] => bool"
"congruent r b == ALL y z. (y,z):r --> b(y)=b(z)"
congruent2 :: "[('a*'a) set, ['a,'a]=>'b] => bool"
"congruent2 r b == ALL y1 z1 y2 z2.
(y1,z1):r --> (y2,z2):r --> b y1 y2 = b z1 z2"
end