30 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G" |
30 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G" |
31 diaL: |
31 diaL: |
32 "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; |
32 "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; |
33 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" |
33 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" |
34 |
34 |
35 ML {* use_legacy_bindings (the_context ()) *} |
35 ML {* |
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36 structure S4_Prover = Modal_ProverFun |
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37 ( |
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38 val rewrite_rls = thms "rewrite_rls" |
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39 val safe_rls = thms "safe_rls" |
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40 val unsafe_rls = thms "unsafe_rls" @ [thm "boxR", thm "diaL"] |
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41 val bound_rls = thms "bound_rls" @ [thm "boxL", thm "diaR"] |
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42 val aside_rls = [thm "lstar0", thm "lstar1", thm "lstar2", thm "rstar0", |
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43 thm "rstar1", thm "rstar2"] |
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44 ) |
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45 *} |
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46 |
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47 method_setup S4_solve = |
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48 {* Method.no_args (Method.SIMPLE_METHOD (S4_Prover.solve_tac 2)) *} "S4 solver" |
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49 |
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50 |
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51 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) |
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52 |
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53 lemma "|- []P --> P" by S4_solve |
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54 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve (* normality*) |
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55 lemma "|- (P--<Q) --> []P --> []Q" by S4_solve |
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56 lemma "|- P --> <>P" by S4_solve |
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57 |
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58 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve |
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59 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve |
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60 lemma "|- [](P<->Q) <-> (P>-<Q)" by S4_solve |
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61 lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S4_solve |
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62 lemma "|- []P <-> ~<>(~P)" by S4_solve |
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63 lemma "|- [](~P) <-> ~<>P" by S4_solve |
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64 lemma "|- ~[]P <-> <>(~P)" by S4_solve |
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65 lemma "|- [][]P <-> ~<><>(~P)" by S4_solve |
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66 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve |
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67 |
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68 lemma "|- []P | []Q --> [](P | Q)" by S4_solve |
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69 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve |
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70 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve |
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71 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve |
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72 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve |
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73 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve |
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74 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve |
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75 lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve |
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76 |
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77 |
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78 (* Theorems of system S4 from Hughes and Cresswell, p.46 *) |
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79 |
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80 lemma "|- []A --> A" by S4_solve (* refexivity *) |
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81 lemma "|- []A --> [][]A" by S4_solve (* transitivity *) |
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82 lemma "|- []A --> <>A" by S4_solve (* seriality *) |
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83 lemma "|- <>[](<>A --> []<>A)" by S4_solve |
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84 lemma "|- <>[](<>[]A --> []A)" by S4_solve |
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85 lemma "|- []P <-> [][]P" by S4_solve |
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86 lemma "|- <>P <-> <><>P" by S4_solve |
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87 lemma "|- <>[]<>P --> <>P" by S4_solve |
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88 lemma "|- []<>P <-> []<>[]<>P" by S4_solve |
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89 lemma "|- <>[]P <-> <>[]<>[]P" by S4_solve |
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90 |
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91 (* Theorems for system S4 from Hughes and Cresswell, p.60 *) |
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92 |
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93 lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve |
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94 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve |
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95 |
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96 (* These are from Hailpern, LNCS 129 *) |
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97 |
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98 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve |
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99 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve |
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100 lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve |
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101 |
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102 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve |
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103 lemma "|- []P --> []<>P" by S4_solve |
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104 lemma "|- <>[]P --> <>P" by S4_solve |
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105 |
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106 lemma "|- []P | []Q --> [](P | Q)" by S4_solve |
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107 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve |
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108 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve |
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109 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve |
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110 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve |
36 |
111 |
37 end |
112 end |