Theory S43

(*  Title:      Sequents/S43.thy
    Author:     Martin Coen
    Copyright   1991  University of Cambridge

This implements Rajeev Gore's sequent calculus for S43.
*)

theory S43
imports Modal0
begin

consts
  S43pi :: "[seq'seq', seq'seq', seq'seq',
             seq'seq', seq'seq', seq'seq']  prop"
syntax
  "_S43pi" :: "[seq, seq, seq, seq, seq, seq]  prop"
                         ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)

parse_translation 
  let
    val tr  = seq_tr;
    fun s43pi_tr [s1, s2, s3, s4, s5, s6] =
      Syntax.const const_syntaxS43pi $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6;
  in [(syntax_const‹_S43pi›, K s43pi_tr)] end


print_translation 
let
  val tr' = seq_tr';
  fun s43pi_tr' [s1, s2, s3, s4, s5, s6] =
    Syntax.const syntax_const‹_S43pi› $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6;
in [(const_syntaxS43pi, K s43pi_tr')] end


axiomatization where
(* Definition of the star operation using a set of Horn clauses  *)
(* For system S43: 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

(* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
(* ie                                                                        *)
(*           S1...Sk,Sk+1...Sk+m                                             *)
(*     ----------------------------------                                    *)
(*     <>P1...<>Pk, $G ⊢ $H, []Q1...[]Qm                                    *)
(*                                                                           *)
(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * ⊢ $H *, []Q1...[]Qm    *)
(*    and Sj == <>P1...<>Pk, $G * ⊢ $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
(*    and 1<=i<=k and k<j<=k+m                                               *)

  S43pi0:         "S43pi $L;; $R;; $Lbox; $Rdia" and
  S43pi1:
   "(S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox  $R,$Rdia 
       S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia" and
  S43pi2:
   "(S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox  $R',P,$R,$Rdia 
       S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia" and

(* Rules for [] and <> for S43 *)

  boxL:           "$E, P, $F, []P  $G  $E, []P, $F  $G" and
  diaR:           "$E  $F, P, $G, <>P  $E  $F, <>P, $G" and
  pi1:
   "$L1,<>P,$L2 |L> $Lbox;  $L1,<>P,$L2 |R> $Ldia;  $R |L> $Rbox;  $R |R> $Rdia;
      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia 
   $L1, <>P, $L2  $R" and
  pi2:
   "$L |L> $Lbox;  $L |R> $Ldia;  $R1,[]P,$R2 |L> $Rbox;  $R1,[]P,$R2 |R> $Rdia;
      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia 
   $L  $R1, []P, $R2"


ML 
structure S43_Prover = Modal_ProverFun
(
  val rewrite_rls = @{thms rewrite_rls}
  val safe_rls = @{thms safe_rls}
  val unsafe_rls = @{thms unsafe_rls} @ [@{thm pi1}, @{thm pi2}]
  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}, @{thm S43pi0}, @{thm S43pi1}, @{thm S43pi2}]
)



method_setup S43_solve = 
  Scan.succeed (fn ctxt => SIMPLE_METHOD
    (S43_Prover.solve_tac ctxt 2 ORELSE S43_Prover.solve_tac ctxt 3))



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

lemma " []P  P" by S43_solve
lemma " [](P  Q)  ([]P  []Q)" by S43_solve   (* normality*)
lemma " (P--<Q)  []P  []Q" by S43_solve
lemma " P  <>P" by S43_solve

lemma "  [](P  Q)  []P  []Q" by S43_solve
lemma "  <>(P  Q)  <>P  <>Q" by S43_solve
lemma "  [](P  Q)  (P>-<Q)" by S43_solve
lemma "  <>(P  Q)  ([]P  <>Q)" by S43_solve
lemma "        []P  ¬ <>(¬ P)" by S43_solve
lemma "     [](¬P)  ¬ <>P" by S43_solve
lemma "       ¬ []P  <>(¬ P)" by S43_solve
lemma "      [][]P  ¬ <><>(¬ P)" by S43_solve
lemma " ¬ <>(P  Q)  ¬ <>P  ¬ <>Q" by S43_solve

lemma " []P  []Q  [](P  Q)" by S43_solve
lemma " <>(P  Q)  <>P  <>Q" by S43_solve
lemma " [](P  Q)  []P  <>Q" by S43_solve
lemma " <>P  []Q  <>(P  Q)" by S43_solve
lemma " [](P  Q)  <>P  []Q" by S43_solve
lemma " <>(P  (Q  R))  ([]P  <>Q)  ([]P  <>R)" by S43_solve
lemma " (P --< Q)  (Q --<R )  (P --< R)" by S43_solve
lemma " []P  <>Q  <>(P  Q)" by S43_solve


(* Theorems of system S4 from Hughes and Cresswell, p.46 *)

lemma " []A  A" by S43_solve             (* refexivity *)
lemma " []A  [][]A" by S43_solve         (* transitivity *)
lemma " []A  <>A" by S43_solve           (* seriality *)
lemma " <>[](<>A  []<>A)" by S43_solve
lemma " <>[](<>[]A  []A)" by S43_solve
lemma " []P  [][]P" by S43_solve
lemma " <>P  <><>P" by S43_solve
lemma " <>[]<>P  <>P" by S43_solve
lemma " []<>P  []<>[]<>P" by S43_solve
lemma " <>[]P  <>[]<>[]P" by S43_solve

(* Theorems for system S4 from Hughes and Cresswell, p.60 *)

lemma " []P  []Q  []([]P  []Q)" by S43_solve
lemma " ((P >-< Q) --< R)  ((P >-< Q) --< []R)" by S43_solve

(* These are from Hailpern, LNCS 129 *)

lemma " [](P  Q)  []P  []Q" by S43_solve
lemma " <>(P  Q)  <>P  <>Q" by S43_solve
lemma " <>(P  Q)  ([]P  <>Q)" by S43_solve

lemma " [](P  Q)  (<>P  <>Q)" by S43_solve
lemma " []P  []<>P" by S43_solve
lemma " <>[]P  <>P" by S43_solve

lemma " []P  []Q  [](P  Q)" by S43_solve
lemma " <>(P  Q)  <>P  <>Q" by S43_solve
lemma " [](P  Q)  []P  <>Q" by S43_solve
lemma " <>P  []Q  <>(P  Q)" by S43_solve
lemma " [](P  Q)  <>P  []Q" by S43_solve


(* Theorems of system S43 *)

lemma " <>[]P  []<>P" by S43_solve
lemma " <>[]P  [][]<>P" by S43_solve
lemma " [](<>P  <>Q)  []<>P  []<>Q" by S43_solve
lemma " <>[]P  <>[]Q  <>([]P  []Q)" by S43_solve
lemma " []([]P  []Q)  []([]Q  []P)" by S43_solve
lemma " [](<>P  <>Q)  [](<>Q  <>P)" by S43_solve
lemma " []([]P  Q)  []([]Q  P)" by S43_solve
lemma " [](P  <>Q)  [](Q  <>P)" by S43_solve
lemma " [](P  []Q  R)  [](P  ([]R  Q))" by S43_solve
lemma " [](P  (Q  <>C))  [](P  C  <>Q)" by S43_solve
lemma " []([]P  Q)  [](P  []Q)  []P  []Q" by S43_solve
lemma " <>P  <>Q  <>(<>P  Q)  <>(P  <>Q)" by S43_solve
lemma " [](P  Q)  []([]P  Q)  [](P  []Q)  []P  []Q" by S43_solve
lemma " <>P  <>Q  <>(P  Q)  <>(<>P  Q)  <>(P  <>Q)" by S43_solve
lemma " <>[]<>P  []<>P" by S43_solve
lemma " []<>[]P  <>[]P" by S43_solve

(* These are from Hailpern, LNCS 129 *)

lemma " [](P  Q)  []P  []Q" by S43_solve
lemma " <>(P  Q)  <>P  <>Q" by S43_solve
lemma " <>(P  Q)  []P  <>Q" by S43_solve

lemma " [](P  Q)  <>P  <>Q" by S43_solve
lemma " []P  []<>P" by S43_solve
lemma " <>[]P  <>P" by S43_solve
lemma " []<>[]P  []<>P" by S43_solve
lemma " <>[]P  <>[]<>P" by S43_solve
lemma " <>[]P  []<>P" by S43_solve
lemma " []<>[]P  <>[]P" by S43_solve
lemma " <>[]<>P  []<>P" by S43_solve

lemma " []P  []Q  [](P  Q)" by S43_solve
lemma " <>(P  Q)  <>P  <>Q" by S43_solve
lemma " [](P  Q)  []P  <>Q" by S43_solve
lemma " <>P  []Q  <>(P  Q)" by S43_solve
lemma " [](P  Q)  <>P  []Q" by S43_solve
lemma " [](P  Q)  []<>P  []<>Q" by S43_solve
lemma " <>[]P  <>[]Q  <>(P  Q)" by S43_solve
lemma " <>[](P  Q)  <>[]P  <>[]Q" by S43_solve
lemma " []<>(P  Q)  []<>P  []<>Q" by S43_solve

end