# Theory S43

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theory S43
imports Modal0
uses [S43.ML]
begin

    (*  Title:      Modal/S43.thy
        ID:         $Id: S43.thy,v 1.3 2005/09/18 13:20:09 wenzelm Exp $
        Author:     Martin Coen
        Copyright   1991  University of Cambridge

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

    theory S43
    imports Modal0
    begin

    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)

    ML {*
      val S43pi  = "S43pi";
      val SS43pi = "@S43pi";

      val tr  = seq_tr;
      val tr' = 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;

    *}

    parse_translation {* [(SS43pi,s43pi_tr)] *}
    print_translation {* [(S43pi,s43pi_tr')] *}

    axioms
    (* 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"

    ML {* use_legacy_bindings (the_context ()) *}

    end