# Theory T

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theory T
imports Modal0
`(*  Title:      Sequents/T.thy    Author:     Martin Coen    Copyright   1991  University of Cambridge*)theory Timports Modal0beginaxioms(* Definition of the star operation using a set of Horn clauses *)(* For system T:  gamma * == {P | []P : gamma}                  *)(*                delta * == {P | <>P : delta}                  *)  lstar0:         "|L>"  lstar1:         "\$G |L> \$H ==> []P, \$G |L> P, \$H"  lstar2:         "\$G |L> \$H ==>   P, \$G |L>    \$H"  rstar0:         "|R>"  rstar1:         "\$G |R> \$H ==> <>P, \$G |R> P, \$H"  rstar2:         "\$G |R> \$H ==>   P, \$G |R>    \$H"(* Rules for [] and <> *)  boxR:   "[| \$E |L> \$E';  \$F |R> \$F';  \$G |R> \$G';               \$E'        |- \$F', P, \$G'|] ==> \$E          |- \$F, []P, \$G"  boxL:     "\$E, P, \$F  |-         \$G    ==> \$E, []P, \$F |-          \$G"  diaR:     "\$E         |- \$F, P,  \$G    ==> \$E          |- \$F, <>P, \$G"  diaL:   "[| \$E |L> \$E';  \$F |L> \$F';  \$G |R> \$G';               \$E', P, \$F'|-         \$G'|] ==> \$E, <>P, \$F |-          \$G"ML {*structure T_Prover = Modal_ProverFun(  val rewrite_rls = @{thms rewrite_rls}  val safe_rls = @{thms safe_rls}  val unsafe_rls = @{thms unsafe_rls} @ [@{thm boxR}, @{thm diaL}]  val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]  val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},    @{thm rstar1}, @{thm rstar2}])*}method_setup T_solve = {* Scan.succeed (K (SIMPLE_METHOD (T_Prover.solve_tac 2))) *}(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)lemma "|- []P --> P" by T_solvelemma "|- [](P-->Q) --> ([]P-->[]Q)" by T_solve   (* normality*)lemma "|- (P--<Q) --> []P --> []Q" by T_solvelemma "|- P --> <>P" by T_solvelemma "|-  [](P & Q) <-> []P & []Q" by T_solvelemma "|-  <>(P | Q) <-> <>P | <>Q" by T_solvelemma "|-  [](P<->Q) <-> (P>-<Q)" by T_solvelemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by T_solvelemma "|-        []P <-> ~<>(~P)" by T_solvelemma "|-     [](~P) <-> ~<>P" by T_solvelemma "|-       ~[]P <-> <>(~P)" by T_solvelemma "|-      [][]P <-> ~<><>(~P)" by T_solvelemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by T_solvelemma "|- []P | []Q --> [](P | Q)" by T_solvelemma "|- <>(P & Q) --> <>P & <>Q" by T_solvelemma "|- [](P | Q) --> []P | <>Q" by T_solvelemma "|- <>P & []Q --> <>(P & Q)" by T_solvelemma "|- [](P | Q) --> <>P | []Q" by T_solvelemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by T_solvelemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by T_solvelemma "|- []P --> <>Q --> <>(P & Q)" by T_solveend`