src/Sequents/S4.thy
 author nipkow Wed, 04 Aug 2004 11:25:08 +0200 changeset 15106 e8cef6993701 parent 2073 fb0655539d05 child 17481 75166ebb619b permissions -rw-r--r--
aded comment
```
(*  Title:      Modal/S4
ID:         \$Id\$
Author:     Martin Coen
Copyright   1991  University of Cambridge
*)

S4 = Modal0 +
rules
(* Definition of the star operation using a set of Horn clauses *)
(* For system S4:  gamma * == {[]P | []P : gamma}               *)
(*                 delta * == {<>P | <>P : delta}               *)

lstar0         "|L>"
lstar1         "\$G |L> \$H ==> []P, \$G |L> []P, \$H"
lstar2         "\$G |L> \$H ==>   P, \$G |L>      \$H"
rstar0         "|R>"
rstar1         "\$G |R> \$H ==> <>P, \$G |R> <>P, \$H"
rstar2         "\$G |R> \$H ==>   P, \$G |R>      \$H"

(* Rules for [] and <> *)

boxR
"[| \$E |L> \$E';  \$F |R> \$F';  \$G |R> \$G';
\$E'         |- \$F', P, \$G'|] ==> \$E          |- \$F, []P, \$G"
boxL     "\$E,P,\$F,[]P |-         \$G    ==> \$E, []P, \$F |-          \$G"

diaR     "\$E          |- \$F,P,\$G,<>P   ==> \$E          |- \$F, <>P, \$G"
diaL
"[| \$E |L> \$E';  \$F |L> \$F';  \$G |R> \$G';
\$E', P, \$F' |-         \$G'|] ==> \$E, <>P, \$F |- \$G"
end
```