src/Sequents/S43.thy
author wenzelm
Sun Sep 18 15:20:08 2005 +0200 (2005-09-18)
changeset 17481 75166ebb619b
parent 14765 bafb24c150c1
child 21426 87ac12bed1ab
permissions -rw-r--r--
converted to Isar theory format;
     1 (*  Title:      Modal/S43.thy
     2     ID:         $Id$
     3     Author:     Martin Coen
     4     Copyright   1991  University of Cambridge
     5 
     6 This implements Rajeev Gore's sequent calculus for S43.
     7 *)
     8 
     9 theory S43
    10 imports Modal0
    11 begin
    12 
    13 consts
    14   S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
    15              seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
    16 syntax
    17   "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
    18                          ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
    19 
    20 ML {*
    21   val S43pi  = "S43pi";
    22   val SS43pi = "@S43pi";
    23 
    24   val tr  = seq_tr;
    25   val tr' = seq_tr';
    26 
    27   fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
    28         Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6;
    29   fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =
    30         Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6;
    31 
    32 *}
    33 
    34 parse_translation {* [(SS43pi,s43pi_tr)] *}
    35 print_translation {* [(S43pi,s43pi_tr')] *}
    36 
    37 axioms
    38 (* Definition of the star operation using a set of Horn clauses  *)
    39 (* For system S43: gamma * == {[]P | []P : gamma}                *)
    40 (*                 delta * == {<>P | <>P : delta}                *)
    41 
    42   lstar0:         "|L>"
    43   lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H"
    44   lstar2:         "$G |L> $H ==>   P, $G |L>      $H"
    45   rstar0:         "|R>"
    46   rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H"
    47   rstar2:         "$G |R> $H ==>   P, $G |R>      $H"
    48 
    49 (* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
    50 (* ie                                                                        *)
    51 (*           S1...Sk,Sk+1...Sk+m                                             *)
    52 (*     ----------------------------------                                    *)
    53 (*     <>P1...<>Pk, $G |- $H, []Q1...[]Qm                                    *)
    54 (*                                                                           *)
    55 (*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm    *)
    56 (*    and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
    57 (*    and 1<=i<=k and k<j<=k+m                                               *)
    58 
    59   S43pi0:         "S43pi $L;; $R;; $Lbox; $Rdia"
    60   S43pi1:
    61    "[| (S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox |- $R,$Rdia |] ==>
    62        S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia"
    63   S43pi2:
    64    "[| (S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox |- $R',P,$R,$Rdia |] ==>
    65        S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia"
    66 
    67 (* Rules for [] and <> for S43 *)
    68 
    69   boxL:           "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
    70   diaR:           "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
    71   pi1:
    72    "[| $L1,<>P,$L2 |L> $Lbox;  $L1,<>P,$L2 |R> $Ldia;  $R |L> $Rbox;  $R |R> $Rdia;
    73       S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
    74    $L1, <>P, $L2 |- $R"
    75   pi2:
    76    "[| $L |L> $Lbox;  $L |R> $Ldia;  $R1,[]P,$R2 |L> $Rbox;  $R1,[]P,$R2 |R> $Rdia;
    77       S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
    78    $L |- $R1, []P, $R2"
    79 
    80 ML {* use_legacy_bindings (the_context ()) *}
    81 
    82 end