(* Title: HOL/trancl.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Transitive closure of a relation
rtrancl is refl/transitive closure; trancl is transitive closure
*)
Trancl = Lfp +
consts
trans :: "('a * 'a)set => bool" (*transitivity predicate*)
id :: "('a * 'a)set"
rtrancl :: "('a * 'a)set => ('a * 'a)set" ("(_^*)" [100] 100)
trancl :: "('a * 'a)set => ('a * 'a)set" ("(_^+)" [100] 100)
O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
rules
trans_def "trans(r) == (!x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
comp_def (*composition of relations*)
"r O s == {xz. ? x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
id_def (*the identity relation*)
"id == {p. ? x. p = <x,x>}"
rtrancl_def "r^* == lfp(%s. id Un (r O s))"
trancl_def "r^+ == r O rtrancl(r)"
end