src/ZF/Rel.thy
 author clasohm Tue, 06 Feb 1996 12:27:17 +0100 changeset 1478 2b8c2a7547ab parent 1401 0c439768f45c child 2469 b50b8c0eec01 permissions -rw-r--r--
expanded tabs
```
(*  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
```