src/Sequents/S4.thy
author wenzelm
Thu Dec 07 00:42:04 2006 +0100 (2006-12-07)
changeset 21687 f689f729afab
parent 21590 ef7278f553eb
child 30510 4120fc59dd85
permissions -rw-r--r--
reorganized structure Goal vs. Tactic;
     1 (*  Title:      Modal/S4.thy
     2     ID:         $Id$
     3     Author:     Martin Coen
     4     Copyright   1991  University of Cambridge
     5 *)
     6 
     7 theory S4
     8 imports Modal0
     9 begin
    10 
    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}               *)
    15 
    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"
    22 
    23 (* Rules for [] and <> *)
    24 
    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 |-          $G"
    29 
    30   diaR:     "$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"
    34 
    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 *}
    46 
    47 method_setup S4_solve =
    48   {* Method.no_args (Method.SIMPLE_METHOD (S4_Prover.solve_tac 2)) *} "S4 solver"
    49 
    50 
    51 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
    52 
    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
    57 
    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
    67 
    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
    76 
    77 
    78 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
    79 
    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
    90 
    91 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
    92 
    93 lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve
    94 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve
    95 
    96 (* These are from Hailpern, LNCS 129 *)
    97 
    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
   101 
   102 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve
   103 lemma "|- []P --> []<>P" by S4_solve
   104 lemma "|- <>[]P --> <>P" by S4_solve
   105 
   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
   111 
   112 end