(* 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] = Const (@{const_syntax S43pi}, dummyT) $ 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] = Const(@{syntax_const "_S43pi"}, dummyT) $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6; in [(@{const_syntax 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 |- $H, []Q1...[]Qm *) (* *) (* where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm *) (* and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *) (* and 1<=i<=k and k(S43pi <>P,$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>-(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