src/Sequents/S43.thy
author haftmann
Fri Jun 19 21:08:07 2009 +0200 (2009-06-19)
changeset 31726 ffd2dc631d88
parent 30549 d2d7874648bd
child 35113 1a0c129bb2e0
permissions -rw-r--r--
merged
     1 (*  Title:      Modal/S43.thy
     2     ID:         $Id$
     3     Author:     Martin Coen
     4     Copyright   1991  University of Cambridge
     5 
     6 This implements Rajeev Gore's sequent calculus for S43.
     7 *)
     8 
     9 theory S43
    10 imports Modal0
    11 begin
    12 
    13 consts
    14   S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
    15              seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
    16 syntax
    17   "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
    18                          ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
    19 
    20 ML {*
    21   val S43pi  = "S43pi";
    22   val SS43pi = "@S43pi";
    23 
    24   val tr  = seq_tr;
    25   val tr' = seq_tr';
    26 
    27   fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
    28         Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6;
    29   fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =
    30         Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6;
    31 
    32 *}
    33 
    34 parse_translation {* [(SS43pi,s43pi_tr)] *}
    35 print_translation {* [(S43pi,s43pi_tr')] *}
    36 
    37 axioms
    38 (* Definition of the star operation using a set of Horn clauses  *)
    39 (* For system S43: gamma * == {[]P | []P : gamma}                *)
    40 (*                 delta * == {<>P | <>P : delta}                *)
    41 
    42   lstar0:         "|L>"
    43   lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H"
    44   lstar2:         "$G |L> $H ==>   P, $G |L>      $H"
    45   rstar0:         "|R>"
    46   rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H"
    47   rstar2:         "$G |R> $H ==>   P, $G |R>      $H"
    48 
    49 (* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
    50 (* ie                                                                        *)
    51 (*           S1...Sk,Sk+1...Sk+m                                             *)
    52 (*     ----------------------------------                                    *)
    53 (*     <>P1...<>Pk, $G |- $H, []Q1...[]Qm                                    *)
    54 (*                                                                           *)
    55 (*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm    *)
    56 (*    and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
    57 (*    and 1<=i<=k and k<j<=k+m                                               *)
    58 
    59   S43pi0:         "S43pi $L;; $R;; $Lbox; $Rdia"
    60   S43pi1:
    61    "[| (S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox |- $R,$Rdia |] ==>
    62        S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia"
    63   S43pi2:
    64    "[| (S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox |- $R',P,$R,$Rdia |] ==>
    65        S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia"
    66 
    67 (* Rules for [] and <> for S43 *)
    68 
    69   boxL:           "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
    70   diaR:           "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
    71   pi1:
    72    "[| $L1,<>P,$L2 |L> $Lbox;  $L1,<>P,$L2 |R> $Ldia;  $R |L> $Rbox;  $R |R> $Rdia;
    73       S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
    74    $L1, <>P, $L2 |- $R"
    75   pi2:
    76    "[| $L |L> $Lbox;  $L |R> $Ldia;  $R1,[]P,$R2 |L> $Rbox;  $R1,[]P,$R2 |R> $Rdia;
    77       S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
    78    $L |- $R1, []P, $R2"
    79 
    80 
    81 ML {*
    82 structure S43_Prover = Modal_ProverFun
    83 (
    84   val rewrite_rls = thms "rewrite_rls"
    85   val safe_rls = thms "safe_rls"
    86   val unsafe_rls = thms "unsafe_rls" @ [thm "pi1", thm "pi2"]
    87   val bound_rls = thms "bound_rls" @ [thm "boxL", thm "diaR"]
    88   val aside_rls = [thm "lstar0", thm "lstar1", thm "lstar2", thm "rstar0",
    89     thm "rstar1", thm "rstar2", thm "S43pi0", thm "S43pi1", thm "S43pi2"]
    90 )
    91 *}
    92 
    93 
    94 method_setup S43_solve = {*
    95   Scan.succeed (K (SIMPLE_METHOD
    96     (S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3)))
    97 *} "S4 solver"
    98 
    99 
   100 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
   101 
   102 lemma "|- []P --> P" by S43_solve
   103 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve   (* normality*)
   104 lemma "|- (P--<Q) --> []P --> []Q" by S43_solve
   105 lemma "|- P --> <>P" by S43_solve
   106 
   107 lemma "|-  [](P & Q) <-> []P & []Q" by S43_solve
   108 lemma "|-  <>(P | Q) <-> <>P | <>Q" by S43_solve
   109 lemma "|-  [](P<->Q) <-> (P>-<Q)" by S43_solve
   110 lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S43_solve
   111 lemma "|-        []P <-> ~<>(~P)" by S43_solve
   112 lemma "|-     [](~P) <-> ~<>P" by S43_solve
   113 lemma "|-       ~[]P <-> <>(~P)" by S43_solve
   114 lemma "|-      [][]P <-> ~<><>(~P)" by S43_solve
   115 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solve
   116 
   117 lemma "|- []P | []Q --> [](P | Q)" by S43_solve
   118 lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
   119 lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
   120 lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
   121 lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
   122 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solve
   123 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solve
   124 lemma "|- []P --> <>Q --> <>(P & Q)" by S43_solve
   125 
   126 
   127 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
   128 
   129 lemma "|- []A --> A" by S43_solve             (* refexivity *)
   130 lemma "|- []A --> [][]A" by S43_solve         (* transitivity *)
   131 lemma "|- []A --> <>A" by S43_solve           (* seriality *)
   132 lemma "|- <>[](<>A --> []<>A)" by S43_solve
   133 lemma "|- <>[](<>[]A --> []A)" by S43_solve
   134 lemma "|- []P <-> [][]P" by S43_solve
   135 lemma "|- <>P <-> <><>P" by S43_solve
   136 lemma "|- <>[]<>P --> <>P" by S43_solve
   137 lemma "|- []<>P <-> []<>[]<>P" by S43_solve
   138 lemma "|- <>[]P <-> <>[]<>[]P" by S43_solve
   139 
   140 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
   141 
   142 lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solve
   143 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve
   144 
   145 (* These are from Hailpern, LNCS 129 *)
   146 
   147 lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
   148 lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
   149 lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solve
   150 
   151 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solve
   152 lemma "|- []P --> []<>P" by S43_solve
   153 lemma "|- <>[]P --> <>P" by S43_solve
   154 
   155 lemma "|- []P | []Q --> [](P | Q)" by S43_solve
   156 lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
   157 lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
   158 lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
   159 lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
   160 
   161 
   162 (* Theorems of system S43 *)
   163 
   164 lemma "|- <>[]P --> []<>P" by S43_solve
   165 lemma "|- <>[]P --> [][]<>P" by S43_solve
   166 lemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solve
   167 lemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solve
   168 lemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solve
   169 lemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solve
   170 lemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solve
   171 lemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solve
   172 lemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solve
   173 lemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solve
   174 lemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
   175 lemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solve
   176 lemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
   177 lemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solve
   178 lemma "|- <>[]<>P <-> []<>P" by S43_solve
   179 lemma "|- []<>[]P <-> <>[]P" by S43_solve
   180 
   181 (* These are from Hailpern, LNCS 129 *)
   182 
   183 lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
   184 lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
   185 lemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solve
   186 
   187 lemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solve
   188 lemma "|- []P --> []<>P" by S43_solve
   189 lemma "|- <>[]P --> <>P" by S43_solve
   190 lemma "|- []<>[]P --> []<>P" by S43_solve
   191 lemma "|- <>[]P --> <>[]<>P" by S43_solve
   192 lemma "|- <>[]P --> []<>P" by S43_solve
   193 lemma "|- []<>[]P <-> <>[]P" by S43_solve
   194 lemma "|- <>[]<>P <-> []<>P" by S43_solve
   195 
   196 lemma "|- []P | []Q --> [](P | Q)" by S43_solve
   197 lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
   198 lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
   199 lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
   200 lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
   201 lemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solve
   202 lemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solve
   203 lemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solve
   204 lemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solve
   205 
   206 end