1 (* Title: 91/Modal/ex/S4thms |
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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 |
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7 (* Theorems of system S4 from Hughes and Cresswell, p.46 *) |
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8 |
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9 try "|- []A --> A"; (* refexivity *) |
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10 try "|- []A --> [][]A"; (* transitivity *) |
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11 try "|- []A --> <>A"; (* seriality *) |
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12 try "|- <>[](<>A --> []<>A)"; |
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13 try "|- <>[](<>[]A --> []A)"; |
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14 try "|- []P <-> [][]P"; |
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15 try "|- <>P <-> <><>P"; |
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16 try "|- <>[]<>P --> <>P"; |
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17 try "|- []<>P <-> []<>[]<>P"; |
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18 try "|- <>[]P <-> <>[]<>[]P"; |
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19 |
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20 (* Theorems for system S4 from Hughes and Cresswell, p.60 *) |
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21 |
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22 try "|- []P | []Q <-> []([]P | []Q)"; |
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23 try "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)"; |
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24 |
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25 (* These are from Hailpern, LNCS 129 *) |
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26 |
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27 try "|- [](P & Q) <-> []P & []Q"; |
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28 try "|- <>(P | Q) <-> <>P | <>Q"; |
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29 try "|- <>(P --> Q) <-> ([]P --> <>Q)"; |
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30 |
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31 try "|- [](P --> Q) --> (<>P --> <>Q)"; |
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32 try "|- []P --> []<>P"; |
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33 try "|- <>[]P --> <>P"; |
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34 |
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35 try "|- []P | []Q --> [](P | Q)"; |
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36 try "|- <>(P & Q) --> <>P & <>Q"; |
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37 try "|- [](P | Q) --> []P | <>Q"; |
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38 try "|- <>P & []Q --> <>(P & Q)"; |
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39 try "|- [](P | Q) --> <>P | []Q"; |
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40 |
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