(* Title: ZF/Rel.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Relations in Zermelo-Fraenkel Set Theory
*)
Rel = ZF +
consts
refl,irrefl,equiv :: "[i,i]=>o"
sym,asym,antisym,trans :: "i=>o"
trans_on :: "[i,i]=>o" ("trans[_]'(_')")
rules
refl_def "refl(A,r) == (ALL x: A. <x,x> : r)"
irrefl_def "irrefl(A,r) == ALL x: A. <x,x> ~: r"
sym_def "sym(r) == ALL x y. <x,y>: r --> <y,x>: r"
asym_def "asym(r) == ALL x y. <x,y>:r --> ~ <y,x>:r"
antisym_def "antisym(r) == ALL x y.<x,y>:r --> <y,x>:r --> x=y"
trans_def "trans(r) == ALL x y z. <x,y>: r --> <y,z>: r --> <x,z>: r"
trans_on_def "trans[A](r) == ALL x:A. ALL y:A. ALL z:A. \
\ <x,y>: r --> <y,z>: r --> <x,z>: r"
equiv_def "equiv(A,r) == r <= A*A & refl(A,r) & sym(r) & trans(r)"
end