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src/Sequents/S43.thy

changeset 21426 | 87ac12bed1ab |

parent 17481 | 75166ebb619b |

child 30510 | 4120fc59dd85 |

--- a/src/Sequents/S43.thy Mon Nov 20 21:23:12 2006 +0100 +++ b/src/Sequents/S43.thy Mon Nov 20 23:47:10 2006 +0100 @@ -77,6 +77,130 @@ S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==> $L |- $R1, []P, $R2" -ML {* use_legacy_bindings (the_context ()) *} + +ML {* +structure S43_Prover = Modal_ProverFun +( + val rewrite_rls = thms "rewrite_rls" + val safe_rls = thms "safe_rls" + val unsafe_rls = thms "unsafe_rls" @ [thm "pi1", thm "pi2"] + 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", thm "S43pi0", thm "S43pi1", thm "S43pi2"] +) +*} + + +method_setup S43_solve = {* + Method.no_args (Method.SIMPLE_METHOD + (S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3)) +*} "S4 solver" + + +(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) + +lemma "|- []P --> P" by S43_solve +lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve (* normality*) +lemma "|- (P--<Q) --> []P --> []Q" by S43_solve +lemma "|- P --> <>P" by S43_solve + +lemma "|- [](P & Q) <-> []P & []Q" by S43_solve +lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve +lemma "|- [](P<->Q) <-> (P>-<Q)" by S43_solve +lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S43_solve +lemma "|- []P <-> ~<>(~P)" by S43_solve +lemma "|- [](~P) <-> ~<>P" by S43_solve +lemma "|- ~[]P <-> <>(~P)" by S43_solve +lemma "|- [][]P <-> ~<><>(~P)" by S43_solve +lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solve + +lemma "|- []P | []Q --> [](P | Q)" by S43_solve +lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve +lemma "|- [](P | Q) --> []P | <>Q" by S43_solve +lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve +lemma "|- [](P | Q) --> <>P | []Q" by S43_solve +lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solve +lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solve +lemma "|- []P --> <>Q --> <>(P & Q)" by S43_solve + + +(* Theorems of system S4 from Hughes and Cresswell, p.46 *) + +lemma "|- []A --> A" by S43_solve (* refexivity *) +lemma "|- []A --> [][]A" by S43_solve (* transitivity *) +lemma "|- []A --> <>A" by S43_solve (* seriality *) +lemma "|- <>[](<>A --> []<>A)" by S43_solve +lemma "|- <>[](<>[]A --> []A)" by S43_solve +lemma "|- []P <-> [][]P" by S43_solve +lemma "|- <>P <-> <><>P" by S43_solve +lemma "|- <>[]<>P --> <>P" by S43_solve +lemma "|- []<>P <-> []<>[]<>P" by S43_solve +lemma "|- <>[]P <-> <>[]<>[]P" by S43_solve + +(* Theorems for system S4 from Hughes and Cresswell, p.60 *) + +lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solve +lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve + +(* These are from Hailpern, LNCS 129 *) + +lemma "|- [](P & Q) <-> []P & []Q" by S43_solve +lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve +lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solve + +lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solve +lemma "|- []P --> []<>P" by S43_solve +lemma "|- <>[]P --> <>P" by S43_solve + +lemma "|- []P | []Q --> [](P | Q)" by S43_solve +lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve +lemma "|- [](P | Q) --> []P | <>Q" by S43_solve +lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve +lemma "|- [](P | Q) --> <>P | []Q" by S43_solve + + +(* Theorems of system S43 *) + +lemma "|- <>[]P --> []<>P" by S43_solve +lemma "|- <>[]P --> [][]<>P" by S43_solve +lemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solve +lemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solve +lemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solve +lemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solve +lemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solve +lemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solve +lemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solve +lemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solve +lemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve +lemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solve +lemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve +lemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solve +lemma "|- <>[]<>P <-> []<>P" by S43_solve +lemma "|- []<>[]P <-> <>[]P" by S43_solve + +(* These are from Hailpern, LNCS 129 *) + +lemma "|- [](P & Q) <-> []P & []Q" by S43_solve +lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve +lemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solve + +lemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solve +lemma "|- []P --> []<>P" by S43_solve +lemma "|- <>[]P --> <>P" by S43_solve +lemma "|- []<>[]P --> []<>P" by S43_solve +lemma "|- <>[]P --> <>[]<>P" by S43_solve +lemma "|- <>[]P --> []<>P" by S43_solve +lemma "|- []<>[]P <-> <>[]P" by S43_solve +lemma "|- <>[]<>P <-> []<>P" by S43_solve + +lemma "|- []P | []Q --> [](P | Q)" by S43_solve +lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve +lemma "|- [](P | Q) --> []P | <>Q" by S43_solve +lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve +lemma "|- [](P | Q) --> <>P | []Q" by S43_solve +lemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solve +lemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solve +lemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solve +lemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solve end