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src/Sequents/S43.thy

author | paulson |

Wed, 09 Oct 1996 13:32:33 +0200 | |

changeset 2073 | fb0655539d05 |

child 14765 | bafb24c150c1 |

permissions | -rw-r--r-- |

New unified treatment of sequent calculi by Sara Kalvala
combines the old LK and Modal with the new ILL (Int. Linear Logic)

(* Title: Modal/S43 ID: $Id$ Author: Martin Coen Copyright 1991 University of Cambridge This implements Rajeev Gore's sequent calculus for S43. *) S43 = Modal0 + consts S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq', seq'=>seq', seq'=>seq', seq'=>seq'] => prop" "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop" ("S43pi((_);(_);(_);(_);(_);(_))" [] 5) rules (* 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, $R2" end ML local val S43pi = "S43pi"; val SS43pi = "@S43pi"; val tr = Sequents.seq_tr; val tr' = Sequents.seq_tr'; fun s43pi_tr[s1,s2,s3,s4,s5,s6]= Const(S43pi,dummyT)$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; in val parse_translation = [(SS43pi,s43pi_tr)]; val print_translation = [(S43pi,s43pi_tr')] end