(* 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