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 *)
     6 
     7 theory T
     8 imports Modal0
     9 begin
    10 
    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}                  *)
    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  |-         $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"
    33 
    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 *}
    45 
    46 method_setup T_solve =
    47   {* Scan.succeed (K (SIMPLE_METHOD (T_Prover.solve_tac 2))) *} "T solver"
    48 
    49 
    50 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
    51 
    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
    56 
    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
    66 
    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
    75 
    76 end