(* Title: Modal/S43.thy ID: $Id: S43.thy,v 1.3 2005/09/18 13:20:09 wenzelm Exp $ 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 k<j<=k+m *) S43pi0: "S43pi $L;; $R;; $Lbox; $Rdia" S43pi1: "[| (S43pi <>P,$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