src/Sequents/T.thy
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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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axioms
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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>"
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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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(* 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, $G"
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  boxL:     "$E, P, $F  |-         $G    ==> $E, []P, $F |-          $G"
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  diaR:     "$E         |- $F, P,  $G    ==> $E          |- $F, <>P, $G"
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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 {*
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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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*}
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method_setup T_solve =
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  {* Scan.succeed (K (SIMPLE_METHOD (T_Prover.solve_tac 2))) *} "T solver"
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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