1 (* Title: 91/Modal/S43 |
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2 ID: $Id$ |
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3 Author: Martin Coen |
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4 Copyright 1991 University of Cambridge |
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5 |
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6 This implements Rajeev Gore's sequent calculus for S43. |
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7 *) |
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8 |
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9 S43 = Modal0 + |
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10 |
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11 consts |
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12 S43pi :: "[sobj=>sobj, sobj=>sobj, sobj=>sobj, |
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13 sobj=>sobj, sobj=>sobj, sobj=>sobj] => prop" |
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14 "@S43pi" :: "[sequence, sequence, sequence, sequence, sequence, |
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15 sequence] => prop" ("S43pi((_);(_);(_);(_);(_);(_))" [] 5) |
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16 |
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17 rules |
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18 (* Definition of the star operation using a set of Horn clauses *) |
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19 (* For system S43: gamma * == {[]P | []P : gamma} *) |
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20 (* delta * == {<>P | <>P : delta} *) |
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21 |
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22 lstar0 "|L>" |
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23 lstar1 "$G |L> $H ==> []P, $G |L> []P, $H" |
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24 lstar2 "$G |L> $H ==> P, $G |L> $H" |
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25 rstar0 "|R>" |
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26 rstar1 "$G |R> $H ==> <>P, $G |R> <>P, $H" |
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27 rstar2 "$G |R> $H ==> P, $G |R> $H" |
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28 |
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29 (* Set of Horn clauses to generate the antecedents for the S43 pi rule *) |
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30 (* ie *) |
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31 (* S1...Sk,Sk+1...Sk+m *) |
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32 (* ---------------------------------- *) |
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33 (* <>P1...<>Pk, $G |- $H, []Q1...[]Qm *) |
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34 (* *) |
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35 (* where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm *) |
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36 (* and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *) |
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37 (* and 1<=i<=k and k<j<=k+m *) |
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38 |
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39 S43pi0 "S43pi $L;; $R;; $Lbox; $Rdia" |
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40 S43pi1 |
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41 "[| (S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox |- $R,$Rdia |] ==> |
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42 S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia" |
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43 S43pi2 |
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44 "[| (S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox |- $R',P,$R,$Rdia |] ==> |
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45 S43pi $L';; $R'; []P,$R; $Lbox;$Rdia" |
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46 |
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47 (* Rules for [] and <> for S43 *) |
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48 |
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49 boxL "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G" |
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50 diaR "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G" |
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51 pi1 |
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52 "[| $L1,<>P,$L2 |L> $Lbox; $L1,<>P,$L2 |R> $Ldia; $R |L> $Rbox; $R |R> $Rdia; |
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53 S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==> |
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54 $L1, <>P, $L2 |- $R" |
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55 pi2 |
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56 "[| $L |L> $Lbox; $L |R> $Ldia; $R1,[]P,$R2 |L> $Rbox; $R1,[]P,$R2 |R> $Rdia; |
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57 S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==> |
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58 $L |- $R1, []P, $R2" |
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59 end |
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60 |
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61 ML |
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62 |
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63 local |
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64 |
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65 val S43pi = "S43pi"; |
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66 val SS43pi = "@S43pi"; |
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67 |
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68 val tr = LK.seq_tr1; |
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69 val tr' = LK.seq_tr1'; |
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70 |
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71 fun s43pi_tr[s1,s2,s3,s4,s5,s6]= |
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72 Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6; |
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73 fun s43pi_tr'[Abs(_,_,s1),Abs(_,_,s2),Abs(_,_,s3), |
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74 Abs(_,_,s4),Abs(_,_,s5),Abs(_,_,s6)] = |
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75 Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6; |
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76 in |
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77 val parse_translation = [(SS43pi,s43pi_tr)]; |
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78 val print_translation = [(S43pi,s43pi_tr')] |
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79 end |
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