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src/Sequents/S4.thy

changeset 21426 | 87ac12bed1ab |

parent 17481 | 75166ebb619b |

child 21590 | ef7278f553eb |

--- a/src/Sequents/S4.thy Mon Nov 20 21:23:12 2006 +0100 +++ b/src/Sequents/S4.thy Mon Nov 20 23:47:10 2006 +0100 @@ -32,6 +32,81 @@ "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" -ML {* use_legacy_bindings (the_context ()) *} +ML {* +structure S4_Prover = Modal_ProverFun +( + val rewrite_rls = thms "rewrite_rls" + val safe_rls = thms "safe_rls" + val unsafe_rls = thms "unsafe_rls" @ [thm "boxR", thm "diaL"] + val bound_rls = thms "bound_rls" @ [thm "boxL", thm "diaR"] + val aside_rls = [thm "lstar0", thm "lstar1", thm "lstar2", thm "rstar0", + thm "rstar1", thm "rstar2"] +) +*} + +method_setup S4_solve = +{* Method.no_args (Method.SIMPLE_METHOD (S4_Prover.solve_tac 2)) *} "S4 solver" + + +(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) + +lemma "|- []P --> P" by S4_solve +lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve (* normality*) +lemma "|- (P--<Q) --> []P --> []Q" by S4_solve +lemma "|- P --> <>P" by S4_solve + +lemma "|- [](P & Q) <-> []P & []Q" by S4_solve +lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve +lemma "|- [](P<->Q) <-> (P>-<Q)" by S4_solve +lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S4_solve +lemma "|- []P <-> ~<>(~P)" by S4_solve +lemma "|- [](~P) <-> ~<>P" by S4_solve +lemma "|- ~[]P <-> <>(~P)" by S4_solve +lemma "|- [][]P <-> ~<><>(~P)" by S4_solve +lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve + +lemma "|- []P | []Q --> [](P | Q)" by S4_solve +lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve +lemma "|- [](P | Q) --> []P | <>Q" by S4_solve +lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve +lemma "|- [](P | Q) --> <>P | []Q" by S4_solve +lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve +lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve +lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve + + +(* Theorems of system S4 from Hughes and Cresswell, p.46 *) + +lemma "|- []A --> A" by S4_solve (* refexivity *) +lemma "|- []A --> [][]A" by S4_solve (* transitivity *) +lemma "|- []A --> <>A" by S4_solve (* seriality *) +lemma "|- <>[](<>A --> []<>A)" by S4_solve +lemma "|- <>[](<>[]A --> []A)" by S4_solve +lemma "|- []P <-> [][]P" by S4_solve +lemma "|- <>P <-> <><>P" by S4_solve +lemma "|- <>[]<>P --> <>P" by S4_solve +lemma "|- []<>P <-> []<>[]<>P" by S4_solve +lemma "|- <>[]P <-> <>[]<>[]P" by S4_solve + +(* Theorems for system S4 from Hughes and Cresswell, p.60 *) + +lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve +lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve + +(* These are from Hailpern, LNCS 129 *) + +lemma "|- [](P & Q) <-> []P & []Q" by S4_solve +lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve +lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve + +lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve +lemma "|- []P --> []<>P" by S4_solve +lemma "|- <>[]P --> <>P" by S4_solve + +lemma "|- []P | []Q --> [](P | Q)" by S4_solve +lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve +lemma "|- [](P | Q) --> []P | <>Q" by S4_solve +lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve +lemma "|- [](P | Q) --> <>P | []Q" by S4_solve end