(* Title: Modal/S43.thy
ID: $Id$
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)
ML {*
val S43pi = "S43pi";
val SS43pi = "@S43pi";
val tr = seq_tr;
val tr' = seq_tr';
fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6;
fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =
Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6;
*}
parse_translation {* [(SS43pi,s43pi_tr)] *}
print_translation {* [(S43pi,s43pi_tr')] *}
axioms
(* Definition of the star operation using a set of Horn clauses *)
(* For system S43: gamma * == {[]P | []P : gamma} *)
(* delta * == {<>P | <>P : delta} *)
lstar0: "|L>"
lstar1: "$G |L> $H ==> []P, $G |L> []P, $H"
lstar2: "$G |L> $H ==> P, $G |L> $H"
rstar0: "|R>"
rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H"
rstar2: "$G |R> $H ==> P, $G |R> $H"
(* 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 * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *)
(* and 1<=i<=k and kP,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox |- $R,$Rdia |] ==>
S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia"
S43pi2:
"[| (S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox |- $R',P,$R,$Rdia |] ==>
S43pi $L';; $R'; []P,$R; $Lbox;$Rdia"
(* Rules for [] and <> for S43 *)
boxL: "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
diaR: "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
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"
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 {* use_legacy_bindings (the_context ()) *}
end