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src/Sequents/S4.thy

author | haftmann |

Fri Jun 19 21:08:07 2009 +0200 (2009-06-19) | |

changeset 31726 | ffd2dc631d88 |

parent 30549 | d2d7874648bd |

child 35762 | af3ff2ba4c54 |

permissions | -rw-r--r-- |

merged

1 (* Title: Modal/S4.thy

2 ID: $Id$

3 Author: Martin Coen

4 Copyright 1991 University of Cambridge

5 *)

7 theory S4

8 imports Modal0

9 begin

11 axioms

12 (* Definition of the star operation using a set of Horn clauses *)

13 (* For system S4: gamma * == {[]P | []P : gamma} *)

14 (* delta * == {<>P | <>P : delta} *)

16 lstar0: "|L>"

17 lstar1: "$G |L> $H ==> []P, $G |L> []P, $H"

18 lstar2: "$G |L> $H ==> P, $G |L> $H"

19 rstar0: "|R>"

20 rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H"

21 rstar2: "$G |R> $H ==> P, $G |R> $H"

23 (* Rules for [] and <> *)

25 boxR:

26 "[| $E |L> $E'; $F |R> $F'; $G |R> $G';

27 $E' |- $F', P, $G'|] ==> $E |- $F, []P, $G"

28 boxL: "$E,P,$F,[]P |- $G ==> $E, []P, $F |- $G"

30 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G"

31 diaL:

32 "[| $E |L> $E'; $F |L> $F'; $G |R> $G';

33 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G"

35 ML {*

36 structure S4_Prover = Modal_ProverFun

37 (

38 val rewrite_rls = thms "rewrite_rls"

39 val safe_rls = thms "safe_rls"

40 val unsafe_rls = thms "unsafe_rls" @ [thm "boxR", thm "diaL"]

41 val bound_rls = thms "bound_rls" @ [thm "boxL", thm "diaR"]

42 val aside_rls = [thm "lstar0", thm "lstar1", thm "lstar2", thm "rstar0",

43 thm "rstar1", thm "rstar2"]

44 )

45 *}

47 method_setup S4_solve =

48 {* Scan.succeed (K (SIMPLE_METHOD (S4_Prover.solve_tac 2))) *} "S4 solver"

51 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)

53 lemma "|- []P --> P" by S4_solve

54 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve (* normality*)

55 lemma "|- (P--<Q) --> []P --> []Q" by S4_solve

56 lemma "|- P --> <>P" by S4_solve

58 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve

59 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve

60 lemma "|- [](P<->Q) <-> (P>-<Q)" by S4_solve

61 lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S4_solve

62 lemma "|- []P <-> ~<>(~P)" by S4_solve

63 lemma "|- [](~P) <-> ~<>P" by S4_solve

64 lemma "|- ~[]P <-> <>(~P)" by S4_solve

65 lemma "|- [][]P <-> ~<><>(~P)" by S4_solve

66 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve

68 lemma "|- []P | []Q --> [](P | Q)" by S4_solve

69 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve

70 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve

71 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve

72 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve

73 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve

74 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve

75 lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve

78 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)

80 lemma "|- []A --> A" by S4_solve (* refexivity *)

81 lemma "|- []A --> [][]A" by S4_solve (* transitivity *)

82 lemma "|- []A --> <>A" by S4_solve (* seriality *)

83 lemma "|- <>[](<>A --> []<>A)" by S4_solve

84 lemma "|- <>[](<>[]A --> []A)" by S4_solve

85 lemma "|- []P <-> [][]P" by S4_solve

86 lemma "|- <>P <-> <><>P" by S4_solve

87 lemma "|- <>[]<>P --> <>P" by S4_solve

88 lemma "|- []<>P <-> []<>[]<>P" by S4_solve

89 lemma "|- <>[]P <-> <>[]<>[]P" by S4_solve

91 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)

93 lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve

94 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve

96 (* These are from Hailpern, LNCS 129 *)

98 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve

99 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve

100 lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve

102 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve

103 lemma "|- []P --> []<>P" by S4_solve

104 lemma "|- <>[]P --> <>P" by S4_solve

106 lemma "|- []P | []Q --> [](P | Q)" by S4_solve

107 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve

108 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve

109 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve

110 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve

112 end