(* Title: 91/Modal/S43
ID: $Id$
Author: Martin Coen
Copyright 1991 University of Cambridge
This implements Rajeev Gore's sequent calculus for S43.
*)
S43 = Modal0 +
consts
S43pi :: "[sobj=>sobj, sobj=>sobj, sobj=>sobj,\
\ sobj=>sobj, sobj=>sobj, sobj=>sobj] => prop"
"@S43pi" :: "[sequence, sequence, sequence, sequence, sequence,\
\ sequence] => prop" ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
rules
(* Definition of the star operation using a set of Horn clauses *)
(* For system S43: 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"
(* Set of Horn clauses to generate the antecedents for the S43 pi rule *)
(* ie *)
(* S1...Sk,Sk+1...Sk+m *)
(* ---------------------------------- *)
(* <>P1...<>Pk, $G |- $H, []Q1...[]Qm *)
(* *)
(* where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm *)
(* and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *)
(* and 1<=i<=k and k<j<=k+m *)
S43pi0 "S43pi $L;; $R;; $Lbox; $Rdia"
S43pi1
"[| (S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox |- $R,$Rdia |] ==> \
\ S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia"
S43pi2
"[| (S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox |- $R',P,$R,$Rdia |] ==> \
\ S43pi $L';; $R'; []P,$R; $Lbox;$Rdia"
(* Rules for [] and <> for S43 *)
boxL "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
diaR "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
pi1
"[| $L1,<>P,$L2 |L> $Lbox; $L1,<>P,$L2 |R> $Ldia; $R |L> $Rbox; $R |R> $Rdia; \
\ S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==> \
\ $L1, <>P, $L2 |- $R"
pi2
"[| $L |L> $Lbox; $L |R> $Ldia; $R1,[]P,$R2 |L> $Rbox; $R1,[]P,$R2 |R> $Rdia; \
\ S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==> \
\ $L |- $R1, []P, $R2"
end
ML
local
val S43pi = "S43pi";
val SS43pi = "@S43pi";
val tr = LK.seq_tr1;
val tr' = LK.seq_tr1';
fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6;
fun s43pi_tr'[Abs(_,_,s1),Abs(_,_,s2),Abs(_,_,s3),
Abs(_,_,s4),Abs(_,_,s5),Abs(_,_,s6)] =
Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6;
in
val parse_translation = [(SS43pi,s43pi_tr)];
val print_translation = [(S43pi,s43pi_tr')]
end