9 Relation = Prod + |
9 Relation = Prod + |
10 consts |
10 consts |
11 id :: "('a * 'a)set" (*the identity relation*) |
11 id :: "('a * 'a)set" (*the identity relation*) |
12 O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60) |
12 O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60) |
13 trans :: "('a * 'a)set => bool" (*transitivity predicate*) |
13 trans :: "('a * 'a)set => bool" (*transitivity predicate*) |
14 converse :: "('a * 'b)set => ('b * 'a)set" |
14 converse :: "('a*'b) set => ('b*'a) set" |
15 "^^" :: "[('a * 'b) set, 'a set] => 'b set" (infixl 90) |
15 "^^" :: "[('a*'b) set,'a set] => 'b set" (infixl 90) |
16 Domain :: "('a * 'b) set => 'a set" |
16 Domain :: "('a*'b) set => 'a set" |
17 Range :: "('a * 'b) set => 'b set" |
17 Range :: "('a*'b) set => 'b set" |
18 defs |
18 defs |
19 id_def "id == {p. ? x. p = (x,x)}" |
19 id_def "id == {p. ? x. p = (x,x)}" |
20 comp_def "r O s == {(x,z). ? y. (x,y):s & (y,z):r}" |
20 comp_def "r O s == {(x,z). ? y. (x,y):s & (y,z):r}" |
21 trans_def "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)" |
21 trans_def "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)" |
22 converse_def "converse(r) == {(y,x). (x,y):r}" |
22 converse_def "converse(r) == {(y,x). (x,y):r}" |