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src/Sequents/S43.thy

author | wenzelm |

Thu, 23 Jul 2015 14:25:05 +0200 | |

changeset 60770 | 240563fbf41d |

parent 54742 | 7a86358a3c0b |

child 61385 | 538100cc4399 |

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

isabelle update_cartouches;

(* 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 \<open> 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 \<close> print_translation \<open> 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 \<close> 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<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 \<open> 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}] ) \<close> method_setup S43_solve = \<open> Scan.succeed (fn ctxt => SIMPLE_METHOD (S43_Prover.solve_tac ctxt 2 ORELSE S43_Prover.solve_tac ctxt 3)) \<close> (* 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--<Q) --> []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>-<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 | 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--<R) --> (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