src/ZF/Rel.thy
 author lcp Tue, 29 Nov 1994 00:31:31 +0100 changeset 753 ec86863e87c8 parent 435 ca5356bd315a child 1155 928a16e02f9f permissions -rw-r--r--
replaced "rules" by "defs"
```
(*  Title: 	ZF/Rel.thy
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

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[_]'(_')")

defs
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
```