(* 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