Theory T

(*  Title:      Sequents/T.thy
    Author:     Martin Coen
    Copyright   1991  University of Cambridge
*)

theory T
imports Modal0
begin

axiomatization where
(* Definition of the star operation using a set of Horn clauses *)
(* For system T:  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           $G     $E, []P, $F           $G" and
  diaR:     "$E          $F, P,  $G     $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 T_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 T_solve = Scan.succeed (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2))


(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)

lemma " []P  P" by T_solve
lemma " [](P  Q)  ([]P  []Q)" by T_solve   (* normality*)
lemma " (P --< Q)  []P  []Q" by T_solve
lemma " P  <>P" by T_solve

lemma "  [](P  Q)  []P  []Q" by T_solve
lemma "  <>(P  Q)  <>P  <>Q" by T_solve
lemma "  [](P  Q)  (P >-< Q)" by T_solve
lemma "  <>(P  Q)  ([]P  <>Q)" by T_solve
lemma "        []P  ¬ <>(¬ P)" by T_solve
lemma "     [](¬ P)  ¬ <>P" by T_solve
lemma "       ¬ []P  <>(¬ P)" by T_solve
lemma "      [][]P  ¬ <><>(¬ P)" by T_solve
lemma " ¬ <>(P  Q)  ¬ <>P  ¬ <>Q" by T_solve

lemma " []P  []Q  [](P  Q)" by T_solve
lemma " <>(P  Q)  <>P  <>Q" by T_solve
lemma " [](P  Q)  []P  <>Q" by T_solve
lemma " <>P  []Q  <>(P  Q)" by T_solve
lemma " [](P  Q)  <>P  []Q" by T_solve
lemma " <>(P  (Q  R))  ([]P  <>Q)  ([]P  <>R)" by T_solve
lemma " (P --< Q)  (Q --< R )  (P --< R)" by T_solve
lemma " []P  <>Q  <>(P  Q)" by T_solve

end