author  blanchet 
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parent 69605  3dda49e08b9d 
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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>\<open>S43pi\<close>, dummyT) $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6; 
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in [(\<^syntax_const>\<open>_S43pi\<close>, 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>\<open>_S43pi\<close>, dummyT) $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6; 
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in [(\<^const_syntax>\<open>S43pi\<close>, 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...<>Pi1,<>Pi+1,..<>Pk,Pi, $G * \<turnstile> $H *, []Q1...[]Qm *) 
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(* and Sj == <>P1...<>Pk, $G * \<turnstile> $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; $Rdia" and 
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S43pi1: 
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"\<lbrakk>(S43pi <>P,$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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(* 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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(* 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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end 