(* Title: CCL/Trancl.thy
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
header {* Transitive closure of a relation *}
theory Trancl
imports CCL
begin
consts
trans :: "i set => o" (*transitivity predicate*)
id :: "i set"
rtrancl :: "i set => i set" ("(_^*)" [100] 100)
trancl :: "i set => i set" ("(_^+)" [100] 100)
O :: "[i set,i set] => i set" (infixr 60)
axioms
trans_def: "trans(r) == (ALL x y z. <x,y>:r --> <y,z>:r --> <x,z>:r)"
comp_def: (*composition of relations*)
"r O s == {xz. EX x y z. xz = <x,z> & <x,y>:s & <y,z>:r}"
id_def: (*the identity relation*)
"id == {p. EX x. p = <x,x>}"
rtrancl_def: "r^* == lfp(%s. id Un (r O s))"
trancl_def: "r^+ == r O rtrancl(r)"
ML {* use_legacy_bindings (the_context ()) *}
end