14 id :: "('a * 'a)set" |
14 id :: "('a * 'a)set" |
15 rtrancl :: "('a * 'a)set => ('a * 'a)set" ("(_^*)" [100] 100) |
15 rtrancl :: "('a * 'a)set => ('a * 'a)set" ("(_^*)" [100] 100) |
16 trancl :: "('a * 'a)set => ('a * 'a)set" ("(_^+)" [100] 100) |
16 trancl :: "('a * 'a)set => ('a * 'a)set" ("(_^+)" [100] 100) |
17 O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60) |
17 O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60) |
18 defs |
18 defs |
19 trans_def "trans(r) == (!x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)" |
19 trans_def "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)" |
20 comp_def (*composition of relations*) |
20 comp_def (*composition of relations*) |
21 "r O s == {xz. ? x y z. xz = <x,z> & <x,y>:s & <y,z>:r}" |
21 "r O s == {xz. ? x y z. xz = (x,z) & (x,y):s & (y,z):r}" |
22 id_def (*the identity relation*) |
22 id_def (*the identity relation*) |
23 "id == {p. ? x. p = <x,x>}" |
23 "id == {p. ? x. p = (x,x)}" |
24 rtrancl_def "r^* == lfp(%s. id Un (r O s))" |
24 rtrancl_def "r^* == lfp(%s. id Un (r O s))" |
25 trancl_def "r^+ == r O rtrancl(r)" |
25 trancl_def "r^+ == r O rtrancl(r)" |
26 end |
26 end |