(* Title: Modal/S4.thy ID: $Id: S4.thy,v 1.4 2006/11/29 14:44:57 wenzelm Exp $ 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 = {* 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
lemma
|- []P --> P
lemma
|- [](P --> Q) --> []P --> []Q
lemma
|- (P --< Q) --> []P --> []Q
lemma
|- P --> <>P
lemma
|- [](P ∧ Q) <-> []P ∧ []Q
lemma
|- <>(P ∨ Q) <-> <>P ∨ <>Q
lemma
|- [](P <-> Q) <-> P >-< Q
lemma
|- <>(P --> Q) <-> []P --> <>Q
lemma
|- []P <-> ¬ <>(¬ P)
lemma
|- [](¬ P) <-> ¬ <>P
lemma
|- ¬ []P <-> <>(¬ P)
lemma
|- [][]P <-> ¬ <><>(¬ P)
lemma
|- ¬ <>(P ∨ Q) <-> ¬ <>P ∧ ¬ <>Q
lemma
|- []P ∨ []Q --> [](P ∨ Q)
lemma
|- <>(P ∧ Q) --> <>P ∧ <>Q
lemma
|- [](P ∨ Q) --> []P ∨ <>Q
lemma
|- <>P ∧ []Q --> <>(P ∧ Q)
lemma
|- [](P ∨ Q) --> <>P ∨ []Q
lemma
|- <>(P --> Q ∧ R) --> ([]P --> <>Q) ∧ ([]P --> <>R)
lemma
|- (P --< Q) ∧ (Q --< R) --> P --< R
lemma
|- []P --> <>Q --> <>(P ∧ Q)
lemma
|- []A --> A
lemma
|- []A --> [][]A
lemma
|- []A --> <>A
lemma
|- <>[](<>A --> []<>A)
lemma
|- <>[](<>[]A --> []A)
lemma
|- []P <-> [][]P
lemma
|- <>P <-> <><>P
lemma
|- <>[]<>P --> <>P
lemma
|- []<>P <-> []<>[]<>P
lemma
|- <>[]P <-> <>[]<>[]P
lemma
|- []P ∨ []Q <-> []([]P ∨ []Q)
lemma
|- ((P >-< Q) --< R) --> (P >-< Q) --< []R
lemma
|- [](P ∧ Q) <-> []P ∧ []Q
lemma
|- <>(P ∨ Q) <-> <>P ∨ <>Q
lemma
|- <>(P --> Q) <-> []P --> <>Q
lemma
|- [](P --> Q) --> <>P --> <>Q
lemma
|- []P --> []<>P
lemma
|- <>[]P --> <>P
lemma
|- []P ∨ []Q --> [](P ∨ Q)
lemma
|- <>(P ∧ Q) --> <>P ∧ <>Q
lemma
|- [](P ∨ Q) --> []P ∨ <>Q
lemma
|- <>P ∧ []Q --> <>(P ∧ Q)
lemma
|- [](P ∨ Q) --> <>P ∨ []Q