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src/Sequents/T.thy

author | wenzelm |

Sun, 11 Jan 2009 21:49:59 +0100 | |

changeset 29450 | ac7f67be7f1f |

parent 21590 | ef7278f553eb |

child 30510 | 4120fc59dd85 |

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

tuned categories;

(* Title: Modal/T.thy ID: $Id$ Author: Martin Coen Copyright 1991 University of Cambridge *) theory T imports Modal0 begin axioms (* Definition of the star operation using a set of Horn clauses *) (* For system T: 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 |- $G ==> $E, []P, $F |- $G" diaR: "$E |- $F, P, $G ==> $E |- $F, <>P, $G" diaL: "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; $E', P, $F'|- $G'|] ==> $E, <>P, $F |- $G" ML {* structure T_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 T_solve = {* Method.no_args (Method.SIMPLE_METHOD (T_Prover.solve_tac 2)) *} "T solver" (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) lemma "|- []P --> P" by T_solve lemma "|- [](P-->Q) --> ([]P-->[]Q)" by T_solve (* normality*) lemma "|- (P--<Q) --> []P --> []Q" by T_solve lemma "|- P --> <>P" by T_solve lemma "|- [](P & Q) <-> []P & []Q" by T_solve lemma "|- <>(P | Q) <-> <>P | <>Q" by T_solve lemma "|- [](P<->Q) <-> (P>-<Q)" by T_solve lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by T_solve lemma "|- []P <-> ~<>(~P)" by T_solve lemma "|- [](~P) <-> ~<>P" by T_solve lemma "|- ~[]P <-> <>(~P)" by T_solve lemma "|- [][]P <-> ~<><>(~P)" by T_solve lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by T_solve lemma "|- []P | []Q --> [](P | Q)" by T_solve lemma "|- <>(P & Q) --> <>P & <>Q" by T_solve lemma "|- [](P | Q) --> []P | <>Q" by T_solve lemma "|- <>P & []Q --> <>(P & Q)" by T_solve lemma "|- [](P | Q) --> <>P | []Q" by T_solve lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by T_solve lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by T_solve lemma "|- []P --> <>Q --> <>(P & Q)" by T_solve end