src/Sequents/S43.thy
 changeset 2073 fb0655539d05 child 14765 bafb24c150c1
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/Sequents/S43.thy	Wed Oct 09 13:32:33 1996 +0200
1.3 @@ -0,0 +1,78 @@
1.4 +(*  Title:      Modal/S43
1.5 +    ID:         \$Id\$
1.6 +    Author:     Martin Coen
1.7 +    Copyright   1991  University of Cambridge
1.8 +
1.9 +This implements Rajeev Gore's sequent calculus for S43.
1.10 +*)
1.11 +
1.12 +S43 = Modal0 +
1.13 +
1.14 +consts
1.15 +  S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
1.16 +             seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
1.17 +  "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
1.18 +                         ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
1.19 +
1.20 +rules
1.21 +(* Definition of the star operation using a set of Horn clauses  *)
1.22 +(* For system S43: gamma * == {[]P | []P : gamma}                *)
1.23 +(*                 delta * == {<>P | <>P : delta}                *)
1.24 +
1.25 +  lstar0         "|L>"
1.26 +  lstar1         "\$G |L> \$H ==> []P, \$G |L> []P, \$H"
1.27 +  lstar2         "\$G |L> \$H ==>   P, \$G |L>      \$H"
1.28 +  rstar0         "|R>"
1.29 +  rstar1         "\$G |R> \$H ==> <>P, \$G |R> <>P, \$H"
1.30 +  rstar2         "\$G |R> \$H ==>   P, \$G |R>      \$H"
1.31 +
1.32 +(* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
1.33 +(* ie                                                                        *)
1.34 +(*           S1...Sk,Sk+1...Sk+m                                             *)
1.35 +(*     ----------------------------------                                    *)
1.36 +(*     <>P1...<>Pk, \$G |- \$H, []Q1...[]Qm                                    *)
1.37 +(*                                                                           *)
1.38 +(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, \$G * |- \$H *, []Q1...[]Qm    *)
1.39 +(*    and Sj == <>P1...<>Pk, \$G * |- \$H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
1.40 +(*    and 1<=i<=k and k<j<=k+m                                               *)
1.41 +
1.42 +  S43pi0         "S43pi \$L;; \$R;; \$Lbox; \$Rdia"
1.43 +  S43pi1
1.44 +   "[| (S43pi <>P,\$L';     \$L;; \$R; \$Lbox;\$Rdia);   \$L',P,\$L,\$Lbox |- \$R,\$Rdia |] ==>
1.45 +       S43pi     \$L'; <>P,\$L;; \$R; \$Lbox;\$Rdia"
1.46 +  S43pi2
1.47 +   "[| (S43pi \$L';; []P,\$R';     \$R; \$Lbox;\$Rdia);  \$L',\$Lbox |- \$R',P,\$R,\$Rdia |] ==>
1.48 +       S43pi \$L';;     \$R'; []P,\$R; \$Lbox;\$Rdia"
1.49 +
1.50 +(* Rules for [] and <> for S43 *)
1.51 +
1.52 +  boxL           "\$E, P, \$F, []P |- \$G ==> \$E, []P, \$F |- \$G"
1.53 +  diaR           "\$E |- \$F, P, \$G, <>P ==> \$E |- \$F, <>P, \$G"
1.54 +  pi1
1.55 +   "[| \$L1,<>P,\$L2 |L> \$Lbox;  \$L1,<>P,\$L2 |R> \$Ldia;  \$R |L> \$Rbox;  \$R |R> \$Rdia;
1.56 +      S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==>
1.57 +   \$L1, <>P, \$L2 |- \$R"
1.58 +  pi2
1.59 +   "[| \$L |L> \$Lbox;  \$L |R> \$Ldia;  \$R1,[]P,\$R2 |L> \$Rbox;  \$R1,[]P,\$R2 |R> \$Rdia;
1.60 +      S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==>
1.61 +   \$L |- \$R1, []P, \$R2"
1.62 +end
1.63 +
1.64 +ML
1.65 +
1.66 +local
1.67 +
1.68 +  val S43pi  = "S43pi";
1.69 +  val SS43pi = "@S43pi";
1.70 +
1.71 +  val tr  = Sequents.seq_tr;
1.72 +  val tr' = Sequents.seq_tr';
1.73 +
1.74 +  fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
1.75 +        Const(S43pi,dummyT)\$tr s1\$tr s2\$tr s3\$tr s4\$tr s5\$tr s6;
1.76 +  fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =
1.77 +        Const(SS43pi,dummyT)\$tr' s1\$tr' s2\$tr' s3\$tr' s4\$tr' s5\$tr' s6;
1.78 +in
1.79 +val parse_translation = [(SS43pi,s43pi_tr)];
1.80 +val print_translation = [(S43pi,s43pi_tr')]
1.81 +end
```