src/Sequents/S4.thy

+(*  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