src/Sequents/S4.thy
 author wenzelm Wed Nov 29 15:44:57 2006 +0100 (2006-11-29) changeset 21590 ef7278f553eb parent 21426 87ac12bed1ab child 30510 4120fc59dd85 permissions -rw-r--r--
```     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
```