1 (* Title: Sequents/S4.thy
3 Copyright 1991 University of Cambridge
11 (* Definition of the star operation using a set of Horn clauses *)
12 (* For system S4: gamma * == {[]P | []P : gamma} *)
13 (* delta * == {<>P | <>P : delta} *)
16 lstar1: "$G |L> $H ==> []P, $G |L> []P, $H"
17 lstar2: "$G |L> $H ==> P, $G |L> $H"
19 rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H"
20 rstar2: "$G |R> $H ==> P, $G |R> $H"
22 (* Rules for [] and <> *)
25 "[| $E |L> $E'; $F |R> $F'; $G |R> $G';
26 $E' |- $F', P, $G'|] ==> $E |- $F, []P, $G"
27 boxL: "$E,P,$F,[]P |- $G ==> $E, []P, $F |- $G"
29 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G"
31 "[| $E |L> $E'; $F |L> $F'; $G |R> $G';
32 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G"
35 structure S4_Prover = Modal_ProverFun
37 val rewrite_rls = @{thms rewrite_rls}
38 val safe_rls = @{thms safe_rls}
39 val unsafe_rls = @{thms unsafe_rls} @ [@{thm boxR}, @{thm diaL}]
40 val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]
41 val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},
42 @{thm rstar1}, @{thm rstar2}]
46 method_setup S4_solve = {* Scan.succeed (K (SIMPLE_METHOD (S4_Prover.solve_tac 2))) *}
49 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
51 lemma "|- []P --> P" by S4_solve
52 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S4_solve (* normality*)
53 lemma "|- (P--<Q) --> []P --> []Q" by S4_solve
54 lemma "|- P --> <>P" by S4_solve
56 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve
57 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve
58 lemma "|- [](P<->Q) <-> (P>-<Q)" by S4_solve
59 lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S4_solve
60 lemma "|- []P <-> ~<>(~P)" by S4_solve
61 lemma "|- [](~P) <-> ~<>P" by S4_solve
62 lemma "|- ~[]P <-> <>(~P)" by S4_solve
63 lemma "|- [][]P <-> ~<><>(~P)" by S4_solve
64 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S4_solve
66 lemma "|- []P | []Q --> [](P | Q)" by S4_solve
67 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve
68 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve
69 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
70 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve
71 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S4_solve
72 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S4_solve
73 lemma "|- []P --> <>Q --> <>(P & Q)" by S4_solve
76 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
78 lemma "|- []A --> A" by S4_solve (* refexivity *)
79 lemma "|- []A --> [][]A" by S4_solve (* transitivity *)
80 lemma "|- []A --> <>A" by S4_solve (* seriality *)
81 lemma "|- <>[](<>A --> []<>A)" by S4_solve
82 lemma "|- <>[](<>[]A --> []A)" by S4_solve
83 lemma "|- []P <-> [][]P" by S4_solve
84 lemma "|- <>P <-> <><>P" by S4_solve
85 lemma "|- <>[]<>P --> <>P" by S4_solve
86 lemma "|- []<>P <-> []<>[]<>P" by S4_solve
87 lemma "|- <>[]P <-> <>[]<>[]P" by S4_solve
89 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
91 lemma "|- []P | []Q <-> []([]P | []Q)" by S4_solve
92 lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S4_solve
94 (* These are from Hailpern, LNCS 129 *)
96 lemma "|- [](P & Q) <-> []P & []Q" by S4_solve
97 lemma "|- <>(P | Q) <-> <>P | <>Q" by S4_solve
98 lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S4_solve
100 lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S4_solve
101 lemma "|- []P --> []<>P" by S4_solve
102 lemma "|- <>[]P --> <>P" by S4_solve
104 lemma "|- []P | []Q --> [](P | Q)" by S4_solve
105 lemma "|- <>(P & Q) --> <>P & <>Q" by S4_solve
106 lemma "|- [](P | Q) --> []P | <>Q" by S4_solve
107 lemma "|- <>P & []Q --> <>(P & Q)" by S4_solve
108 lemma "|- [](P | Q) --> <>P | []Q" by S4_solve