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src/Sequents/T.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/T.thy

2 ID: $Id$

3 Author: Martin Coen

4 Copyright 1991 University of Cambridge

5 *)

7 theory T

8 imports Modal0

9 begin

11 axioms

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

13 (* For system T: 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 |- $G ==> $E, []P, $F |- $G"

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

30 diaL:

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

32 $E', P, $F'|- $G'|] ==> $E, <>P, $F |- $G"

34 ML {*

35 structure T_Prover = Modal_ProverFun

36 (

37 val rewrite_rls = thms "rewrite_rls"

38 val safe_rls = thms "safe_rls"

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

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

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

42 thm "rstar1", thm "rstar2"]

43 )

44 *}

46 method_setup T_solve =

47 {* Scan.succeed (K (SIMPLE_METHOD (T_Prover.solve_tac 2))) *} "T solver"

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

52 lemma "|- []P --> P" by T_solve

53 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by T_solve (* normality*)

54 lemma "|- (P--<Q) --> []P --> []Q" by T_solve

55 lemma "|- P --> <>P" by T_solve

57 lemma "|- [](P & Q) <-> []P & []Q" by T_solve

58 lemma "|- <>(P | Q) <-> <>P | <>Q" by T_solve

59 lemma "|- [](P<->Q) <-> (P>-<Q)" by T_solve

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

61 lemma "|- []P <-> ~<>(~P)" by T_solve

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

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

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

65 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by T_solve

67 lemma "|- []P | []Q --> [](P | Q)" by T_solve

68 lemma "|- <>(P & Q) --> <>P & <>Q" by T_solve

69 lemma "|- [](P | Q) --> []P | <>Q" by T_solve

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

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

72 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by T_solve

73 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by T_solve

74 lemma "|- []P --> <>Q --> <>(P & Q)" by T_solve

76 end