src/Sequents/S43.thy
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(*  Title:      Sequents/S43.thy
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    Author:     Martin Coen
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    Copyright   1991  University of Cambridge
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This implements Rajeev Gore's sequent calculus for S43.
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*)
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theory S43
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imports Modal0
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begin
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consts
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  S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
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             seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
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syntax
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  "_S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
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                         ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
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parse_translation {*
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  let
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    val tr  = seq_tr;
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    fun s43pi_tr [s1, s2, s3, s4, s5, s6] =
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      Const (@{const_syntax S43pi}, dummyT) $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6;
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  in [(@{syntax_const "_S43pi"}, s43pi_tr)] end
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*}
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print_translation {*
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let
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  val tr' = seq_tr';
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  fun s43pi_tr' [s1, s2, s3, s4, s5, s6] =
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    Const(@{syntax_const "_S43pi"}, dummyT) $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6;
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in [(@{const_syntax S43pi}, s43pi_tr')] end
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*}
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axioms
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(* Definition of the star operation using a set of Horn clauses  *)
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(* For system S43: gamma * == {[]P | []P : gamma}                *)
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(*                 delta * == {<>P | <>P : delta}                *)
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  lstar0:         "|L>"
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  lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H"
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  lstar2:         "$G |L> $H ==>   P, $G |L>      $H"
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  rstar0:         "|R>"
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  rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H"
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  rstar2:         "$G |R> $H ==>   P, $G |R>      $H"
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(* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
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(* ie                                                                        *)
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(*           S1...Sk,Sk+1...Sk+m                                             *)
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(*     ----------------------------------                                    *)
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(*     <>P1...<>Pk, $G |- $H, []Q1...[]Qm                                    *)
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(*                                                                           *)
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(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm    *)
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(*    and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
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(*    and 1<=i<=k and k<j<=k+m                                               *)
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  S43pi0:         "S43pi $L;; $R;; $Lbox; $Rdia"
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  S43pi1:
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   "[| (S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox |- $R,$Rdia |] ==>
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       S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia"
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  S43pi2:
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   "[| (S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox |- $R',P,$R,$Rdia |] ==>
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       S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia"
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(* Rules for [] and <> for S43 *)
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  boxL:           "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
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  diaR:           "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
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  pi1:
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   "[| $L1,<>P,$L2 |L> $Lbox;  $L1,<>P,$L2 |R> $Ldia;  $R |L> $Rbox;  $R |R> $Rdia;
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      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
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   $L1, <>P, $L2 |- $R"
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  pi2:
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   "[| $L |L> $Lbox;  $L |R> $Ldia;  $R1,[]P,$R2 |L> $Rbox;  $R1,[]P,$R2 |R> $Rdia;
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      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
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   $L |- $R1, []P, $R2"
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ML {*
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structure S43_Prover = Modal_ProverFun
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(
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  val rewrite_rls = @{thms rewrite_rls}
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  val safe_rls = @{thms safe_rls}
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  val unsafe_rls = @{thms unsafe_rls} @ [@{thm pi1}, @{thm pi2}]
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  val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]
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  val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},
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    @{thm rstar1}, @{thm rstar2}, @{thm S43pi0}, @{thm S43pi1}, @{thm S43pi2}]
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)
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*}
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method_setup S43_solve = {*
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  Scan.succeed (K (SIMPLE_METHOD
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    (S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3)))
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*}
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(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
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lemma "|- []P --> P" by S43_solve
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lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve   (* normality*)
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lemma "|- (P--<Q) --> []P --> []Q" by S43_solve
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lemma "|- P --> <>P" by S43_solve
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lemma "|-  [](P & Q) <-> []P & []Q" by S43_solve
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lemma "|-  <>(P | Q) <-> <>P | <>Q" by S43_solve
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lemma "|-  [](P<->Q) <-> (P>-<Q)" by S43_solve
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lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S43_solve
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lemma "|-        []P <-> ~<>(~P)" by S43_solve
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lemma "|-     [](~P) <-> ~<>P" by S43_solve
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lemma "|-       ~[]P <-> <>(~P)" by S43_solve
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lemma "|-      [][]P <-> ~<><>(~P)" by S43_solve
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lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solve
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lemma "|- []P | []Q --> [](P | Q)" by S43_solve
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lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
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lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
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lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
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lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
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lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solve
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lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solve
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lemma "|- []P --> <>Q --> <>(P & Q)" by S43_solve
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(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
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lemma "|- []A --> A" by S43_solve             (* refexivity *)
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lemma "|- []A --> [][]A" by S43_solve         (* transitivity *)
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lemma "|- []A --> <>A" by S43_solve           (* seriality *)
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lemma "|- <>[](<>A --> []<>A)" by S43_solve
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lemma "|- <>[](<>[]A --> []A)" by S43_solve
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lemma "|- []P <-> [][]P" by S43_solve
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lemma "|- <>P <-> <><>P" by S43_solve
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lemma "|- <>[]<>P --> <>P" by S43_solve
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lemma "|- []<>P <-> []<>[]<>P" by S43_solve
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lemma "|- <>[]P <-> <>[]<>[]P" by S43_solve
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(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
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lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solve
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lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve
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(* These are from Hailpern, LNCS 129 *)
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lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
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lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
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lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solve
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lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solve
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lemma "|- []P --> []<>P" by S43_solve
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lemma "|- <>[]P --> <>P" by S43_solve
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lemma "|- []P | []Q --> [](P | Q)" by S43_solve
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lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
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lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
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lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
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87ac12bed1ab converted legacy ML scripts;
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(* Theorems of system S43 *)
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lemma "|- <>[]P --> []<>P" by S43_solve
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lemma "|- <>[]P --> [][]<>P" by S43_solve
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lemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solve
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lemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solve
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lemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solve
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lemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solve
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lemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
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lemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solve
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lemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
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lemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solve
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lemma "|- <>[]<>P <-> []<>P" by S43_solve
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lemma "|- []<>[]P <-> <>[]P" by S43_solve
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(* These are from Hailpern, LNCS 129 *)
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87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
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lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
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lemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solve
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87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solve
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lemma "|- []P --> []<>P" by S43_solve
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lemma "|- <>[]P --> <>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- []<>[]P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>[]P --> <>[]<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>[]P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- []<>[]P <-> <>[]P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>[]<>P <-> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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87ac12bed1ab converted legacy ML scripts;
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lemma "|- []P | []Q --> [](P | Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
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lemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solve
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
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fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
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end