src/Sequents/S4.thy
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(*  Title:      Modal/S4
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    ID:         $Id$
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    Author:     Martin Coen
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    Copyright   1991  University of Cambridge
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*)
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S4 = Modal0 +
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rules
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(* Definition of the star operation using a set of Horn clauses *)
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(* For system S4:  gamma * == {[]P | []P : gamma}               *)
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(*                 delta * == {<>P | <>P : delta}               *)
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  lstar0         "|L>"
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  lstar1         "$G |L> $H ==> []P, $G |L> []P, $H"
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  lstar2         "$G |L> $H ==>   P, $G |L>      $H"
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  rstar0         "|R>"
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  rstar1         "$G |R> $H ==> <>P, $G |R> <>P, $H"
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  rstar2         "$G |R> $H ==>   P, $G |R>      $H"
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(* Rules for [] and <> *)
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  boxR
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   "[| $E |L> $E';  $F |R> $F';  $G |R> $G';  
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           $E'         |- $F', P, $G'|] ==> $E          |- $F, []P, $G"
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  boxL     "$E,P,$F,[]P |-         $G    ==> $E, []P, $F |-          $G"
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  diaR     "$E          |- $F,P,$G,<>P   ==> $E          |- $F, <>P, $G"
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  diaL
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   "[| $E |L> $E';  $F |L> $F';  $G |R> $G';  
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           $E', P, $F' |-         $G'|] ==> $E, <>P, $F |- $G"
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end