src/Sequents/S4.thy
author bulwahn
Tue Aug 31 08:00:53 2010 +0200 (2010-08-31)
changeset 38950 62578950e748
parent 35762 af3ff2ba4c54
child 39159 0dec18004e75
permissions -rw-r--r--
storing options for prolog code generation in the theory
     1 (*  Title:      Sequents/S4.thy
     2     Author:     Martin Coen
     3     Copyright   1991  University of Cambridge
     4 *)
     5 
     6 theory S4
     7 imports Modal0
     8 begin
     9 
    10 axioms
    11 (* Definition of the star operation using a set of Horn clauses *)
    12 (* For system S4:  gamma * == {[]P | []P : gamma}               *)
    13 (*                 delta * == {<>P | <>P : delta}               *)
    14 
    15   lstar0:         "|L>"
    16   lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H"
    17   lstar2:         "$G |L> $H ==>   P, $G |L>      $H"
    18   rstar0:         "|R>"
    19   rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H"
    20   rstar2:         "$G |R> $H ==>   P, $G |R>      $H"
    21 
    22 (* Rules for [] and <> *)
    23 
    24   boxR:
    25    "[| $E |L> $E';  $F |R> $F';  $G |R> $G';
    26            $E'         |- $F', P, $G'|] ==> $E          |- $F, []P, $G"
    27   boxL:     "$E,P,$F,[]P |-         $G    ==> $E, []P, $F |-          $G"
    28 
    29   diaR:     "$E          |- $F,P,$G,<>P   ==> $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"
    33 
    34 ML {*
    35 structure S4_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 *}
    45 
    46 method_setup S4_solve =
    47   {* Scan.succeed (K (SIMPLE_METHOD (S4_Prover.solve_tac 2))) *} "S4 solver"
    48 
    49 
    50 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
    51 
    52 lemma "|- []P --> P" by S4_solve
    53 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve   (* normality*)
    54 lemma "|- (P--<Q) --> []P --> []Q" by S4_solve
    55 lemma "|- P --> <>P" by S4_solve
    56 
    57 lemma "|-  [](P & Q) <-> []P & []Q" 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 <-> ~<>(~P)" 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 | Q) <-> ~<>P & ~<>Q" by S4_solve
    66 
    67 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 & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve
    73 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve
    74 lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve
    75 
    76 
    77 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
    78 
    79 lemma "|- []A --> A" by S4_solve             (* refexivity *)
    80 lemma "|- []A --> [][]A" by S4_solve         (* transitivity *)
    81 lemma "|- []A --> <>A" by S4_solve           (* seriality *)
    82 lemma "|- <>[](<>A --> []<>A)" by S4_solve
    83 lemma "|- <>[](<>[]A --> []A)" by S4_solve
    84 lemma "|- []P <-> [][]P" 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 
    90 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
    91 
    92 lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve
    93 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve
    94 
    95 (* These are from Hailpern, LNCS 129 *)
    96 
    97 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve
    98 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve
    99 lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve
   100 
   101 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve
   102 lemma "|- []P --> []<>P" by S4_solve
   103 lemma "|- <>[]P --> <>P" by S4_solve
   104 
   105 lemma "|- []P | []Q --> [](P | Q)" 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 
   111 end