(* Title: Modal/S4.thy
ID: $Id$
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