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S
`
E
(Φ
;
p
)
S
p
`
E
(Φ)
S
`
E
(Φ
;:
p
)
S
p
`
E
(Φ)
R
A
+
:
R
A
−
:
S
`
E
(Φ
;
f
U
g
)
S
`
E
(Φ
;
g
)
S
`
E
(Φ
;
f
^
g
)
S
`
E
(Φ
;
f
;
g
)
R
U
:
R
^
:
S
`
E
(Φ
;
f
;
X
f
U
g
)
S
`
E
(Φ
;
f
R
g
)
S
`
E
(Φ
;
f
;
g
)
S
`
E
(Φ
;
f
_
g
)
S
`
E
(Φ
;
f
)
R
R
:
R
_
:
S
`
E
(Φ
;
g
;
X
f
R
g
)
S
`
E
(Φ
;
g
)
S
`
E
(Φ
;
F
f
)
S
`
E
(Φ
;
f
)
S
`
E
(Φ
;
G
f
)
S
`
E
(Φ
;
f
;
XG
f
)
R
F
:
R
G
:
S
`
E
(Φ
;
XF
f
)
S
`
E
(
Φ
)
S
=
S
1
[ [
S
k
;
R
split
:
S
i
6
=
fg;
for
i
= 1
:::
k
S
1
`
E
(Φ)
S
k
`
E
(Φ)
R
X
:
S
`
E
(
X
Φ
1
;:::;
X
Φ
n
)
T
`
E
(Φ
1
;:::;
Φ
n
)
T
=Img(
S
)
Fig. 2.
M-Rules: Multiple state tableau rules ('
S
p
'and'
S
p
' are defined in the text).
M-Sequent
holds
in a Kripke structure
K
iff
S
j
=
Φ
1
^^
Φ
n
,i.e.thereexistsapath
π
in
K
with
Φ
1
^^
Φ
n
. An M-Tableau is a rooted finite directed graph
of nodes labeled with M-Sequents that are connected via the rules shown in figure 2,
where we define the following short hand for
p
π
(0)
2
S
and
π
j
=
A
:
2
S
p
:=
S
p
:=
f
2
j
2 `
g;
f
2
j
62 `
g
s
S
p
(
s
)
s
S
p
(
s
)
In the split rule
R
split
the set of states
S
on the LHS is partitioned into a nonempty
pairwise disjunctive list of sets
S
1
;:::;
S
k
that cover
S
. For M-Tableaux we require every
node to be labeled with a unique M-Sequent. M-Paths, successful M-Path, and SCC are
defined exactly as in the single state case of the last section. The only exception is a
finite M-Path ending with an M-Sequent
), with an empty set of states on
the left side. By definition, such an M-Path is always unsuccessful even if the list of
formulae
fg `
E
(
Φ
is empty.
To lift Theorem 1, Theorem 2, and in particular Lemma 3 to M-Tableaux we first
note that the definitions of successful paths, SCCs, and successful tableau do only de-
pend on the RHS of the sequents, in both cases, for S-Tableaux and M-Tableaux. As an
immediate consequence we have:
Φ
Theorem 4.
A partially unsuccessful M-Tableau is successful iff it contains a success-
ful SCC.
Lemma 5.
An M-Tableau contains an infinite successful M-Path iff it contains a suc-
cessful SCC.
The second step is to map an M-Tableau to a
set
of S-Tableau, where the M-Tableau is
successful iff one S-Tableau is successful. The mapping
Ψ
0
is defined along the graph
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