# Theory S4

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theory S4
imports Modal0
`(*  Title:      Sequents/S4.thy    Author:     Martin Coen    Copyright   1991  University of Cambridge*)theory S4imports Modal0beginaxioms(* 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))) *}(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)lemma "|- []P --> P" by S4_solvelemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve   (* normality*)lemma "|- (P--<Q) --> []P --> []Q" by S4_solvelemma "|- P --> <>P" by S4_solvelemma "|-  [](P & Q) <-> []P & []Q" by S4_solvelemma "|-  <>(P | Q) <-> <>P | <>Q" by S4_solvelemma "|-  [](P<->Q) <-> (P>-<Q)" by S4_solvelemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S4_solvelemma "|-        []P <-> ~<>(~P)" by S4_solvelemma "|-     [](~P) <-> ~<>P" by S4_solvelemma "|-       ~[]P <-> <>(~P)" by S4_solvelemma "|-      [][]P <-> ~<><>(~P)" by S4_solvelemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solvelemma "|- []P | []Q --> [](P | Q)" by S4_solvelemma "|- <>(P & Q) --> <>P & <>Q" by S4_solvelemma "|- [](P | Q) --> []P | <>Q" by S4_solvelemma "|- <>P & []Q --> <>(P & Q)" by S4_solvelemma "|- [](P | Q) --> <>P | []Q" by S4_solvelemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solvelemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solvelemma "|- []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_solvelemma "|- <>[](<>[]A --> []A)" by S4_solvelemma "|- []P <-> [][]P" by S4_solvelemma "|- <>P <-> <><>P" by S4_solvelemma "|- <>[]<>P --> <>P" by S4_solvelemma "|- []<>P <-> []<>[]<>P" by S4_solvelemma "|- <>[]P <-> <>[]<>[]P" by S4_solve(* Theorems for system S4 from Hughes and Cresswell, p.60 *)lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solvelemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve(* These are from Hailpern, LNCS 129 *)lemma "|- [](P & Q) <-> []P & []Q" by S4_solvelemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solvelemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solvelemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solvelemma "|- []P --> []<>P" by S4_solvelemma "|- <>[]P --> <>P" by S4_solvelemma "|- []P | []Q --> [](P | Q)" by S4_solvelemma "|- <>(P & Q) --> <>P & <>Q" by S4_solvelemma "|- [](P | Q) --> []P | <>Q" by S4_solvelemma "|- <>P & []Q --> <>(P & Q)" by S4_solvelemma "|- [](P | Q) --> <>P | []Q" by S4_solveend`