src/Sequents/S43.thy
 author wenzelm Fri, 21 May 2004 21:14:18 +0200 changeset 14765 bafb24c150c1 parent 2073 fb0655539d05 child 17481 75166ebb619b permissions -rw-r--r--
proper use of 'syntax';
```
(*  Title:      Modal/S43
ID:         \$Id\$
Author:     Martin Coen

This implements Rajeev Gore's sequent calculus for S43.
*)

S43 = Modal0 +

consts
S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
syntax
"@S43pi" :: "[seq, seq, seq, seq, seq, seq] => 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  = Sequents.seq_tr;
val tr' = Sequents.seq_tr';

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'[s1,s2,s3,s4,s5,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
```