author  wenzelm 
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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"}, K 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}, K s43pi_tr')] end 
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*} 
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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 ==> []P, $G L> []P, $H" and 

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lstar2: "$G L> $H ==> P, $G L> $H" and 

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rstar0: "R>" and 

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rstar1: "$G R> $H ==> <>P, $G R> <>P, $H" and 

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rstar2: "$G R> $H ==> 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  $H, []Q1...[]Qm *) 
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(* *) 
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(* where Si == <>P1...<>Pi1,<>Pi+1,..<>Pk,Pi, $G *  $H *, []Q1...[]Qm *) 
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(* and Sj == <>P1...<>Pk, $G *  $H *, []Q1...[]Qj1,[]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; $Rdia" and 
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S43pi1: 
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"[ (S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox  $R,$Rdia ] ==> 

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S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia" and 
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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" and 
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(* Rules for [] and <> for S43 *) 
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boxL: "$E, P, $F, []P  $G ==> $E, []P, $F  $G" and 
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diaR: "$E  $F, P, $G, <>P ==> $E  $F, <>P, $G" and 

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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" and 
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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 

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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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(* 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 

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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)  [](Q > <>P)" by S43_solve 

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lemma " [](P > []Q>R)  [](P  ([]R > Q))" by S43_solve 

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lemma " [](P  (Q > <>C))  [](P > C > <>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)" 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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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 > []<>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 > <>(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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end 