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src/Sequents/S4.thy

author | haftmann |

Thu, 08 Jul 2010 16:19:24 +0200 | |

changeset 37744 | 3daaf23b9ab4 |

parent 35762 | af3ff2ba4c54 |

child 39159 | 0dec18004e75 |

permissions | -rw-r--r-- |

tuned titles

(* Title: Sequents/S4.thy Author: Martin Coen Copyright 1991 University of Cambridge *) theory S4 imports Modal0 begin axioms (* Definition of the star operation using a set of Horn clauses *) (* For system S4: gamma * == {[]P | []P : gamma} *) (* delta * == {<>P | <>P : delta} *) lstar0: "|L>" lstar1: "$G |L> $H ==> []P, $G |L> []P, $H" lstar2: "$G |L> $H ==> P, $G |L> $H" rstar0: "|R>" rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H" rstar2: "$G |R> $H ==> P, $G |R> $H" (* Rules for [] and <> *) boxR: "[| $E |L> $E'; $F |R> $F'; $G |R> $G'; $E' |- $F', P, $G'|] ==> $E |- $F, []P, $G" boxL: "$E,P,$F,[]P |- $G ==> $E, []P, $F |- $G" diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G" diaL: "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" 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 = {* Scan.succeed (K (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