(* 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 +
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 <= A Times 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