(*  Title:      Sequents/S4.thy

    Author:     Martin Coen

    Copyright   1991  University of Cambridge

*)

theory S4

imports Modal0

begin

axiomatization where

(* Definition of the star operation using a set of Horn clauses *)

(* For system S4:  gamma * == {[]P | []P : gamma}               *)

(*                 delta * == {<>P | <>P : delta}               *)

  lstar0:         "|L>" and

  lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H" and

  lstar2:         "$G |L> $H ==>   P, $G |L>      $H" and

  rstar0:         "|R>" and

  rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H" and

  rstar2:         "$G |R> $H ==>   P, $G |R>      $H" and

(* Rules for [] and <> *)

  boxR:

   "[| $E |L> $E';  $F |R> $F';  $G |R> $G';

           $E'         |- $F', P, $G'|] ==> $E          |- $F, []P, $G" and

  boxL:     "$E,P,$F,[]P |-         $G    ==> $E, []P, $F |-          $G" and

  diaR:     "$E          |- $F,P,$G,<>P   ==> $E          |- $F, <>P, $G" and

  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_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