Theory S4

(*  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 (fn ctxt => SIMPLE_METHOD (S4_Prover.solve_tac ctxt 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