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(* Title: 91/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
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