Quelle  S43.thy   Sprache: Isabelle

(*  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 🍋‹S43pi› $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6;
  in [(🍋‹_S43pi›, K s43pi_tr)] end
›

print_translation ‹
let
  val tr' = seq_tr';
  fun s43pi_tr' [s1, s2, s3, s4, s5, s6] =
    Syntax.const 🍋‹_S43pi› $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6;
in [(🍋‹S43pi›, 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 \<turnstile> $H, []Q1...[]Qm                                    *)
(*                                                                           *)
(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * \<turnstile> $H *, []Q1...[]Qm    *)
(*    and Sj == <>P1...<>Pk, $G * \<turnstile> $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-- []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>- 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 -- (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