author  wenzelm 
Thu, 23 Jul 2015 14:25:05 +0200  
changeset 60770  240563fbf41d 
parent 54742  7a86358a3c0b 
child 61385  538100cc4399 
permissions  rwrr 
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(* Title: Sequents/T.thy 
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Author: Martin Coen 
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Copyright 1991 University of Cambridge 
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*) 
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theory T 
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imports Modal0 

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begin 

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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 T: 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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(* Rules for [] and <> *) 
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boxR: 
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"[ $E L> $E'; $F R> $F'; $G R> $G'; 

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$E'  $F', P, $G'] ==> $E  $F, []P, $G" and 
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boxL: "$E, P, $F  $G ==> $E, []P, $F  $G" and 

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diaR: "$E  $F, P, $G ==> $E  $F, <>P, $G" and 

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diaL: 
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"[ $E L> $E'; $F L> $F'; $G R> $G'; 

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$E', P, $F' $G'] ==> $E, <>P, $F  $G" 
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ML \<open> 
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structure T_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 boxR}, @{thm diaL}] 

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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}] 

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) 
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\<close> 
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method_setup T_solve = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2))\<close> 
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(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) 

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lemma " []P > P" by T_solve 

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lemma " [](P>Q) > ([]P>[]Q)" by T_solve (* normality*) 

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lemma " (P<Q) > []P > []Q" by T_solve 

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lemma " P > <>P" by T_solve 

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lemma " [](P & Q) <> []P & []Q" by T_solve 

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lemma " <>(P  Q) <> <>P  <>Q" by T_solve 

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lemma " [](P<>Q) <> (P><Q)" by T_solve 

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lemma " <>(P>Q) <> ([]P><>Q)" by T_solve 

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lemma " []P <> ~<>(~P)" by T_solve 

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lemma " [](~P) <> ~<>P" by T_solve 

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lemma " ~[]P <> <>(~P)" by T_solve 

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lemma " [][]P <> ~<><>(~P)" by T_solve 

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lemma " ~<>(P  Q) <> ~<>P & ~<>Q" by T_solve 

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lemma " []P  []Q > [](P  Q)" by T_solve 

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lemma " <>(P & Q) > <>P & <>Q" by T_solve 

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lemma " [](P  Q) > []P  <>Q" by T_solve 

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lemma " <>P & []Q > <>(P & Q)" by T_solve 

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lemma " [](P  Q) > <>P  []Q" by T_solve 

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lemma " <>(P>(Q & R)) > ([]P > <>Q) & ([]P><>R)" by T_solve 

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lemma " (P<Q) & (Q<R) > (P<R)" by T_solve 

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lemma " []P > <>Q > <>(P & Q)" by T_solve 

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end 