(* Title: Sequents/T.thy
Author: Martin Coen
Copyright 1991 University of Cambridge
*)
theory T
imports Modal0
begin
axiomatization where
(* Definition of the star operation using a set of Horn clauses *)
(* For system T: gamma * == {P | []P : gamma} *)
(* delta * == {P | <>P : delta} *)
lstar0: "|L>" and
lstar1: "$G |L> $H ==> []P, $G |L> P, $H" and
lstar2: "$G |L> $H ==> P, $G |L> $H" and
rstar0: "|R>" and
rstar1: "$G |R> $H ==> <>P, $G |R> P, $H" and
rstar2: "$G |R> $H ==> P, $G |R> $H" and
(* Rules for [] and <> *)
boxR:
"[| $E |L> $E'; $F |R> $F'; $G |R> $G';
$E' |- $F', P, $G'|] ==> $E |- $F, []P, $G" and
boxL: "$E, P, $F |- $G ==> $E, []P, $F |- $G" and
diaR: "$E |- $F, P, $G ==> $E |- $F, <>P, $G" and
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 (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2)) *}
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
lemma "|- []P --> P" by T_solve
lemma "|- [](P-->Q) --> ([]P-->[]Q)" by T_solve (* normality*)
lemma "|- (P--<Q) --> []P --> []Q" by T_solve
lemma "|- P --> <>P" by T_solve
lemma "|- [](P & Q) <-> []P & []Q" by T_solve
lemma "|- <>(P | Q) <-> <>P | <>Q" by T_solve
lemma "|- [](P<->Q) <-> (P>-<Q)" by T_solve
lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by T_solve
lemma "|- []P <-> ~<>(~P)" by T_solve
lemma "|- [](~P) <-> ~<>P" by T_solve
lemma "|- ~[]P <-> <>(~P)" by T_solve
lemma "|- [][]P <-> ~<><>(~P)" by T_solve
lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by T_solve
lemma "|- []P | []Q --> [](P | Q)" by T_solve
lemma "|- <>(P & Q) --> <>P & <>Q" by T_solve
lemma "|- [](P | Q) --> []P | <>Q" by T_solve
lemma "|- <>P & []Q --> <>(P & Q)" by T_solve
lemma "|- [](P | Q) --> <>P | []Q" by T_solve
lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by T_solve
lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by T_solve
lemma "|- []P --> <>Q --> <>(P & Q)" by T_solve
end