(* 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