(* Title: Relation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
Relation = Prod +
consts
O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
converse :: "('a*'b) set => ('b*'a) set" ("(_^-1)" [1000] 999)
"^^" :: "[('a*'b) set,'a set] => 'b set" (infixl 90)
defs
comp_def "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
converse_def "r^-1 == {(y,x). (x,y):r}"
Image_def "r ^^ s == {y. ? x:s. (x,y):r}"
constdefs
Id :: "('a * 'a)set" (*the identity relation*)
"Id == {p. ? x. p = (x,x)}"
diag :: "'a set => ('a * 'a)set"
"diag(A) == UN x:A. {(x,x)}"
Domain :: "('a*'b) set => 'a set"
"Domain(r) == {x. ? y. (x,y):r}"
Range :: "('a*'b) set => 'b set"
"Range(r) == Domain(r^-1)"
refl :: "['a set, ('a*'a) set] => bool" (*reflexivity over a set*)
"refl A r == r <= A Times A & (ALL x: A. (x,x) : r)"
sym :: "('a*'a) set=>bool" (*symmetry predicate*)
"sym(r) == ALL x y. (x,y): r --> (y,x): r"
antisym:: "('a * 'a)set => bool" (*antisymmetry predicate*)
"antisym(r) == ALL x y. (x,y):r --> (y,x):r --> x=y"
trans :: "('a * 'a)set => bool" (*transitivity predicate*)
"trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
Univalent :: "('a * 'b)set => bool"
"Univalent r == !x y. (x,y):r --> (!z. (x,z):r --> y=z)"
syntax
reflexive :: "('a * 'a)set => bool" (*reflexivity over a type*)
translations
"reflexive" == "refl UNIV"
end