5 |
5 |
6 theory S4 |
6 theory S4 |
7 imports Modal0 |
7 imports Modal0 |
8 begin |
8 begin |
9 |
9 |
10 axioms |
10 axiomatization where |
11 (* Definition of the star operation using a set of Horn clauses *) |
11 (* Definition of the star operation using a set of Horn clauses *) |
12 (* For system S4: gamma * == {[]P | []P : gamma} *) |
12 (* For system S4: gamma * == {[]P | []P : gamma} *) |
13 (* delta * == {<>P | <>P : delta} *) |
13 (* delta * == {<>P | <>P : delta} *) |
14 |
14 |
15 lstar0: "|L>" |
15 lstar0: "|L>" and |
16 lstar1: "$G |L> $H ==> []P, $G |L> []P, $H" |
16 lstar1: "$G |L> $H ==> []P, $G |L> []P, $H" and |
17 lstar2: "$G |L> $H ==> P, $G |L> $H" |
17 lstar2: "$G |L> $H ==> P, $G |L> $H" and |
18 rstar0: "|R>" |
18 rstar0: "|R>" and |
19 rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H" |
19 rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H" and |
20 rstar2: "$G |R> $H ==> P, $G |R> $H" |
20 rstar2: "$G |R> $H ==> P, $G |R> $H" and |
21 |
21 |
22 (* Rules for [] and <> *) |
22 (* Rules for [] and <> *) |
23 |
23 |
24 boxR: |
24 boxR: |
25 "[| $E |L> $E'; $F |R> $F'; $G |R> $G'; |
25 "[| $E |L> $E'; $F |R> $F'; $G |R> $G'; |
26 $E' |- $F', P, $G'|] ==> $E |- $F, []P, $G" |
26 $E' |- $F', P, $G'|] ==> $E |- $F, []P, $G" and |
27 boxL: "$E,P,$F,[]P |- $G ==> $E, []P, $F |- $G" |
27 boxL: "$E,P,$F,[]P |- $G ==> $E, []P, $F |- $G" and |
28 |
28 |
29 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G" |
29 diaR: "$E |- $F,P,$G,<>P ==> $E |- $F, <>P, $G" and |
30 diaL: |
30 diaL: |
31 "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; |
31 "[| $E |L> $E'; $F |L> $F'; $G |R> $G'; |
32 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" |
32 $E', P, $F' |- $G'|] ==> $E, <>P, $F |- $G" |
33 |
33 |
34 ML {* |
34 ML {* |