src/Sequents/S4.thy
 changeset 21426 87ac12bed1ab parent 17481 75166ebb619b child 21590 ef7278f553eb
```--- a/src/Sequents/S4.thy	Mon Nov 20 21:23:12 2006 +0100
+++ b/src/Sequents/S4.thy	Mon Nov 20 23:47:10 2006 +0100
@@ -32,6 +32,81 @@
"[| \$E |L> \$E';  \$F |L> \$F';  \$G |R> \$G';
\$E', P, \$F' |-         \$G'|] ==> \$E, <>P, \$F |- \$G"

-ML {* use_legacy_bindings (the_context ()) *}
+ML {*
+structure S4_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 S4_solve =
+{* Method.no_args (Method.SIMPLE_METHOD (S4_Prover.solve_tac 2)) *} "S4 solver"
+
+
+(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
+
+lemma "|- []P --> P" by S4_solve
+lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve   (* normality*)
+lemma "|- (P--<Q) --> []P --> []Q" by S4_solve
+lemma "|- P --> <>P" by S4_solve
+
+lemma "|-  [](P & Q) <-> []P & []Q" by S4_solve
+lemma "|-  <>(P | Q) <-> <>P | <>Q" by S4_solve
+lemma "|-  [](P<->Q) <-> (P>-<Q)" by S4_solve
+lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S4_solve
+lemma "|-        []P <-> ~<>(~P)" by S4_solve
+lemma "|-     [](~P) <-> ~<>P" by S4_solve
+lemma "|-       ~[]P <-> <>(~P)" by S4_solve
+lemma "|-      [][]P <-> ~<><>(~P)" by S4_solve
+lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve
+
+lemma "|- []P | []Q --> [](P | Q)" by S4_solve
+lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve
+lemma "|- [](P | Q) --> []P | <>Q" by S4_solve
+lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
+lemma "|- [](P | Q) --> <>P | []Q" by S4_solve
+lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve
+lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve
+lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve
+
+
+(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
+
+lemma "|- []A --> A" by S4_solve             (* refexivity *)
+lemma "|- []A --> [][]A" by S4_solve         (* transitivity *)
+lemma "|- []A --> <>A" by S4_solve           (* seriality *)
+lemma "|- <>[](<>A --> []<>A)" by S4_solve
+lemma "|- <>[](<>[]A --> []A)" by S4_solve
+lemma "|- []P <-> [][]P" by S4_solve
+lemma "|- <>P <-> <><>P" by S4_solve
+lemma "|- <>[]<>P --> <>P" by S4_solve
+lemma "|- []<>P <-> []<>[]<>P" by S4_solve
+lemma "|- <>[]P <-> <>[]<>[]P" by S4_solve
+
+(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
+
+lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve
+lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve
+
+(* These are from Hailpern, LNCS 129 *)
+
+lemma "|- [](P & Q) <-> []P & []Q" by S4_solve
+lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve
+lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve
+
+lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve
+lemma "|- []P --> []<>P" by S4_solve
+lemma "|- <>[]P --> <>P" by S4_solve
+
+lemma "|- []P | []Q --> [](P | Q)" by S4_solve
+lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve
+lemma "|- [](P | Q) --> []P | <>Q" by S4_solve
+lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
+lemma "|- [](P | Q) --> <>P | []Q" by S4_solve

end```