(* Title: 91/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