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'\<Rightarrow>seq', seq'\<Rightarrow>seq', seq'\<Rightarrow>seq',
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             seq'\<Rightarrow>seq', seq'\<Rightarrow>seq', seq'\<Rightarrow>seq'] \<Rightarrow> prop"
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syntax
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  "_S43pi" :: "[seq, seq, seq, seq, seq, seq] \<Rightarrow> prop"
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                         ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
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parse_translation \<open>
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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"}, K s43pi_tr)] end
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\<close>
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print_translation \<open>
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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}, K s43pi_tr')] end
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\<close>
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axiomatization where
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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>" and
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  lstar1:         "$G |L> $H \<Longrightarrow> []P, $G |L> []P, $H" and
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  lstar2:         "$G |L> $H \<Longrightarrow>   P, $G |L>      $H" and
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  rstar0:         "|R>" and
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  rstar1:         "$G |R> $H \<Longrightarrow> <>P, $G |R> <>P, $H" and
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  rstar2:         "$G |R> $H \<Longrightarrow>   P, $G |R>      $H" and
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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 \<turnstile> $H, []Q1...[]Qm                                    *)
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(*                                                                           *)
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(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * \<turnstile> $H *, []Q1...[]Qm    *)
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(*    and Sj == <>P1...<>Pk, $G * \<turnstile> $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; $Rdia" and
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  S43pi1:
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   "\<lbrakk>(S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox \<turnstile> $R,$Rdia\<rbrakk> \<Longrightarrow>
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       S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia" and
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  S43pi2:
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   "\<lbrakk>(S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox \<turnstile> $R',P,$R,$Rdia\<rbrakk> \<Longrightarrow>
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       S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia" and
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(* Rules for [] and <> for S43 *)
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  boxL:           "$E, P, $F, []P \<turnstile> $G \<Longrightarrow> $E, []P, $F \<turnstile> $G" and
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  diaR:           "$E \<turnstile> $F, P, $G, <>P \<Longrightarrow> $E \<turnstile> $F, <>P, $G" and
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  pi1:
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   "\<lbrakk>$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\<rbrakk> \<Longrightarrow>
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   $L1, <>P, $L2 \<turnstile> $R" and
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  pi2:
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   "\<lbrakk>$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\<rbrakk> \<Longrightarrow>
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   $L \<turnstile> $R1, []P, $R2"
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ML \<open>
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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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\<close>
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method_setup S43_solve = \<open>
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  Scan.succeed (fn ctxt => SIMPLE_METHOD
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    (S43_Prover.solve_tac ctxt 2 ORELSE S43_Prover.solve_tac ctxt 3))
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\<close>
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(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
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lemma "\<turnstile> []P \<longrightarrow> P" by S43_solve
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lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by S43_solve   (* normality*)
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lemma "\<turnstile> (P--<Q) \<longrightarrow> []P \<longrightarrow> []Q" by S43_solve
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lemma "\<turnstile> P \<longrightarrow> <>P" by S43_solve
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lemma "\<turnstile>  [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
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lemma "\<turnstile>  <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
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lemma "\<turnstile>  [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P>-<Q)" by S43_solve
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lemma "\<turnstile>  <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S43_solve
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lemma "\<turnstile>        []P \<longleftrightarrow> \<not> <>(\<not> P)" by S43_solve
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lemma "\<turnstile>     [](\<not>P) \<longleftrightarrow> \<not> <>P" by S43_solve
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lemma "\<turnstile>       \<not> []P \<longleftrightarrow> <>(\<not> P)" by S43_solve
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lemma "\<turnstile>      [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by S43_solve
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lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by S43_solve
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lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
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lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
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lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
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lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by S43_solve
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lemma "\<turnstile> (P --< Q) \<and> (Q --<R ) \<longrightarrow> (P --< R)" by S43_solve
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lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
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(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
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lemma "\<turnstile> []A \<longrightarrow> A" by S43_solve             (* refexivity *)
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lemma "\<turnstile> []A \<longrightarrow> [][]A" by S43_solve         (* transitivity *)
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lemma "\<turnstile> []A \<longrightarrow> <>A" by S43_solve           (* seriality *)
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lemma "\<turnstile> <>[](<>A \<longrightarrow> []<>A)" by S43_solve
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lemma "\<turnstile> <>[](<>[]A \<longrightarrow> []A)" by S43_solve
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lemma "\<turnstile> []P \<longleftrightarrow> [][]P" by S43_solve
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lemma "\<turnstile> <>P \<longleftrightarrow> <><>P" by S43_solve
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lemma "\<turnstile> <>[]<>P \<longrightarrow> <>P" by S43_solve
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lemma "\<turnstile> []<>P \<longleftrightarrow> []<>[]<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longleftrightarrow> <>[]<>[]P" by S43_solve
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(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
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lemma "\<turnstile> []P \<or> []Q \<longleftrightarrow> []([]P \<or> []Q)" by S43_solve
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lemma "\<turnstile> ((P >-< Q) --< R) \<longrightarrow> ((P >-< Q) --< []R)" by S43_solve
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(* These are from Hailpern, LNCS 129 *)
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lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
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lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
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lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S43_solve
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lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> (<>P \<longrightarrow> <>Q)" by S43_solve
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lemma "\<turnstile> []P \<longrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S43_solve
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lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
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lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
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lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
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87ac12bed1ab converted legacy ML scripts;
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87ac12bed1ab converted legacy ML scripts;
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(* Theorems of system S43 *)
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lemma "\<turnstile> <>[]P \<longrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longrightarrow> [][]<>P" by S43_solve
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lemma "\<turnstile> [](<>P \<or> <>Q) \<longrightarrow> []<>P \<or> []<>Q" by S43_solve
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lemma "\<turnstile> <>[]P \<and> <>[]Q \<longrightarrow> <>([]P \<and> []Q)" by S43_solve
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lemma "\<turnstile> []([]P \<longrightarrow> []Q) \<or> []([]Q \<longrightarrow> []P)" by S43_solve
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lemma "\<turnstile> [](<>P \<longrightarrow> <>Q) \<or> [](<>Q \<longrightarrow> <>P)" by S43_solve
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lemma "\<turnstile> []([]P \<longrightarrow> Q) \<or> []([]Q \<longrightarrow> P)" by S43_solve
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lemma "\<turnstile> [](P \<longrightarrow> <>Q) \<or> [](Q \<longrightarrow> <>P)" by S43_solve
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lemma "\<turnstile> [](P \<longrightarrow> []Q \<longrightarrow> R) \<or> [](P \<or> ([]R \<longrightarrow> Q))" by S43_solve
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lemma "\<turnstile> [](P \<or> (Q \<longrightarrow> <>C)) \<or> [](P \<longrightarrow> C \<longrightarrow> <>Q)" by S43_solve
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lemma "\<turnstile> []([]P \<or> Q) \<and> [](P \<or> []Q) \<longrightarrow> []P \<or> []Q" by S43_solve
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lemma "\<turnstile> <>P \<and> <>Q \<longrightarrow> <>(<>P \<and> Q) \<or> <>(P \<and> <>Q)" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<and> []([]P \<or> Q) \<and> [](P \<or> []Q) \<longrightarrow> []P \<or> []Q" by S43_solve
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lemma "\<turnstile> <>P \<and> <>Q \<longrightarrow> <>(P \<and> Q) \<or> <>(<>P \<and> Q) \<or> <>(P \<and> <>Q)" by S43_solve
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lemma "\<turnstile> <>[]<>P \<longleftrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> []<>[]P \<longleftrightarrow> <>[]P" by S43_solve
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87ac12bed1ab converted legacy ML scripts;
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(* These are from Hailpern, LNCS 129 *)
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lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
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lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
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lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> []P \<longrightarrow> <>Q" by S43_solve
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lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> <>P \<longrightarrow> <>Q" by S43_solve
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lemma "\<turnstile> []P \<longrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S43_solve
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lemma "\<turnstile> []<>[]P \<longrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longrightarrow> <>[]<>P" by S43_solve
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lemma "\<turnstile> <>[]P \<longrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> []<>[]P \<longleftrightarrow> <>[]P" by S43_solve
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lemma "\<turnstile> <>[]<>P \<longleftrightarrow> []<>P" by S43_solve
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lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
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lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
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lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
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lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []<>P \<or> []<>Q" by S43_solve
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lemma "\<turnstile> <>[]P \<and> <>[]Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
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lemma "\<turnstile> <>[](P \<and> Q) \<longleftrightarrow> <>[]P \<and> <>[]Q" by S43_solve
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lemma "\<turnstile> []<>(P \<or> Q) \<longleftrightarrow> []<>P \<or> []<>Q" by S43_solve
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fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
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end