# Theory S43

Up to index of Isabelle/Sequents

theory S43
imports Modal0
`(*  Title:      Sequents/S43.thy    Author:     Martin Coen    Copyright   1991  University of CambridgeThis implements Rajeev Gore's sequent calculus for S43.*)theory S43imports Modal0beginconsts  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"}, 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}, s43pi_tr')] end*}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 {*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 (K (SIMPLE_METHOD    (S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3)))*}(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)lemma "|- []P --> P" by S43_solvelemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve   (* normality*)lemma "|- (P--<Q) --> []P --> []Q" by S43_solvelemma "|- P --> <>P" by S43_solvelemma "|-  [](P & Q) <-> []P & []Q" by S43_solvelemma "|-  <>(P | Q) <-> <>P | <>Q" by S43_solvelemma "|-  [](P<->Q) <-> (P>-<Q)" by S43_solvelemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S43_solvelemma "|-        []P <-> ~<>(~P)" by S43_solvelemma "|-     [](~P) <-> ~<>P" by S43_solvelemma "|-       ~[]P <-> <>(~P)" by S43_solvelemma "|-      [][]P <-> ~<><>(~P)" by S43_solvelemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solvelemma "|- []P | []Q --> [](P | Q)" by S43_solvelemma "|- <>(P & Q) --> <>P & <>Q" by S43_solvelemma "|- [](P | Q) --> []P | <>Q" by S43_solvelemma "|- <>P & []Q --> <>(P & Q)" by S43_solvelemma "|- [](P | Q) --> <>P | []Q" by S43_solvelemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solvelemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solvelemma "|- []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_solvelemma "|- <>[](<>[]A --> []A)" by S43_solvelemma "|- []P <-> [][]P" by S43_solvelemma "|- <>P <-> <><>P" by S43_solvelemma "|- <>[]<>P --> <>P" by S43_solvelemma "|- []<>P <-> []<>[]<>P" by S43_solvelemma "|- <>[]P <-> <>[]<>[]P" by S43_solve(* Theorems for system S4 from Hughes and Cresswell, p.60 *)lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solvelemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve(* These are from Hailpern, LNCS 129 *)lemma "|- [](P & Q) <-> []P & []Q" by S43_solvelemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solvelemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solvelemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solvelemma "|- []P --> []<>P" by S43_solvelemma "|- <>[]P --> <>P" by S43_solvelemma "|- []P | []Q --> [](P | Q)" by S43_solvelemma "|- <>(P & Q) --> <>P & <>Q" by S43_solvelemma "|- [](P | Q) --> []P | <>Q" by S43_solvelemma "|- <>P & []Q --> <>(P & Q)" by S43_solvelemma "|- [](P | Q) --> <>P | []Q" by S43_solve(* Theorems of system S43 *)lemma "|- <>[]P --> []<>P" by S43_solvelemma "|- <>[]P --> [][]<>P" by S43_solvelemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solvelemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solvelemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solvelemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solvelemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solvelemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solvelemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solvelemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solvelemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solvelemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solvelemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solvelemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solvelemma "|- <>[]<>P <-> []<>P" by S43_solvelemma "|- []<>[]P <-> <>[]P" by S43_solve(* These are from Hailpern, LNCS 129 *)lemma "|- [](P & Q) <-> []P & []Q" by S43_solvelemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solvelemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solvelemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solvelemma "|- []P --> []<>P" by S43_solvelemma "|- <>[]P --> <>P" by S43_solvelemma "|- []<>[]P --> []<>P" by S43_solvelemma "|- <>[]P --> <>[]<>P" by S43_solvelemma "|- <>[]P --> []<>P" by S43_solvelemma "|- []<>[]P <-> <>[]P" by S43_solvelemma "|- <>[]<>P <-> []<>P" by S43_solvelemma "|- []P | []Q --> [](P | Q)" by S43_solvelemma "|- <>(P & Q) --> <>P & <>Q" by S43_solvelemma "|- [](P | Q) --> []P | <>Q" by S43_solvelemma "|- <>P & []Q --> <>(P & Q)" by S43_solvelemma "|- [](P | Q) --> <>P | []Q" by S43_solvelemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solvelemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solvelemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solvelemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solveend`