(* Title: Relation.thy
ID: $Id$
Author: Riccardo Mattolini, Dip. Sistemi e Informatica
and Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 Universita' di Firenze
Copyright 1993 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*'a) set => ('a*'a) set"
"^^" :: "[('a*'a) set,'a set] => 'a set" (infixl 90)
Domain :: "('a*'a) set => 'a set"
Range :: "('a*'a) set => 'a set"
defs
id_def "id == {p. ? x. p = (x,x)}"
comp_def (*composition of relations*)
"r O s == {xz. ? x y z. xz = (x,z) & (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) == {z. (? w:r. ? x y. w=(x,y) & z=(y,x))}"
Domain_def "Domain(r) == {z. ! x. (z=x --> (? y. (x,y):r))}"
Range_def "Range(r) == Domain(converse(r))"
Image_def "r ^^ s == {y. y:Range(r) & (? x:s. (x,y):r)}"
end