src/Sequents/S4.thy
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(*  Title:      Sequents/S4.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 S4
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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 S4:  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,[]P |-         $G    ==> $E, []P, $F |-          $G"
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  diaR:     "$E          |- $F,P,$G,<>P   ==> $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 S4_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 S4_solve =
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  {* Scan.succeed (K (SIMPLE_METHOD (S4_Prover.solve_tac 2))) *} "S4 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 S4_solve
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lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve   (* normality*)
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lemma "|- (P--<Q) --> []P --> []Q" by S4_solve
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lemma "|- P --> <>P" by S4_solve
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lemma "|-  [](P & Q) <-> []P & []Q" by S4_solve
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lemma "|-  <>(P | Q) <-> <>P | <>Q" by S4_solve
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lemma "|-  [](P<->Q) <-> (P>-<Q)" by S4_solve
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lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S4_solve
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lemma "|-        []P <-> ~<>(~P)" by S4_solve
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lemma "|-     [](~P) <-> ~<>P" by S4_solve
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lemma "|-       ~[]P <-> <>(~P)" by S4_solve
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lemma "|-      [][]P <-> ~<><>(~P)" by S4_solve
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lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve
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lemma "|- []P | []Q --> [](P | Q)" by S4_solve
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lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve
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lemma "|- [](P | Q) --> []P | <>Q" by S4_solve
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lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
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lemma "|- [](P | Q) --> <>P | []Q" by S4_solve
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lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve
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lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve
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lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve
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(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
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lemma "|- []A --> A" by S4_solve             (* refexivity *)
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lemma "|- []A --> [][]A" by S4_solve         (* transitivity *)
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lemma "|- []A --> <>A" by S4_solve           (* seriality *)
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lemma "|- <>[](<>A --> []<>A)" by S4_solve
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lemma "|- <>[](<>[]A --> []A)" by S4_solve
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lemma "|- []P <-> [][]P" by S4_solve
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lemma "|- <>P <-> <><>P" by S4_solve
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lemma "|- <>[]<>P --> <>P" by S4_solve
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lemma "|- []<>P <-> []<>[]<>P" by S4_solve
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lemma "|- <>[]P <-> <>[]<>[]P" by S4_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 S4_solve
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lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve
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(* These are from Hailpern, LNCS 129 *)
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lemma "|- [](P & Q) <-> []P & []Q" by S4_solve
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lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve
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lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve
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lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve
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lemma "|- []P --> []<>P" by S4_solve
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lemma "|- <>[]P --> <>P" by S4_solve
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lemma "|- []P | []Q --> [](P | Q)" by S4_solve
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lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
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lemma "|- [](P | Q) --> <>P | []Q" by S4_solve
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end