(* Title: Relation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
Relation = Prod +
consts
id :: "('a * 'a)set" (*the identity relation*)
O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
trans :: "('a * 'a)set => bool" (*transitivity predicate*)
converse :: "('a*'b) set => ('b*'a) set"
"^^" :: "[('a*'b) set,'a set] => 'b set" (infixl 90)
Domain :: "('a*'b) set => 'a set"
Range :: "('a*'b) set => 'b set"
defs
id_def "id == {p. ? x. p = (x,x)}"
comp_def "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
trans_def "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
converse_def "converse(r) == {(y,x). (x,y):r}"
Domain_def "Domain(r) == {x. ? y. (x,y):r}"
Range_def "Range(r) == Domain(converse(r))"
Image_def "r ^^ s == {y. ? x:s. (x,y):r}"
end