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2.
0
∈
A
d
(
˄.ˉ
,
ʔ
)
iff ʔ
⃒
ʵ.
3.
0
∈
A
A
d
(
.
4. Suppose the action ˉ does not occur in the test T . Then o
a
∈
X
a.ˉ
,
ʔ
)
iff ʔ
⃒
−ₒ
d
(
˄.ˉ
∈
A
a.T
,
ʔ
)
a
⃒
0
iff there is a ʔ
∈
D
sub
(
sCSP)
with ʔ
ʔ
and o
d
(
T
,
ʔ
)
.
with o
(
ˉ
)
=
∈
A
d
(
T
1
p
↕
d
(
T
i
,
ʔ
)
.
5. o
∈
A
T
2
,
ʔ
)
iff o
=
p
·
o
1
+
(1
−
p
)
·
o
2
for certain o
i
∈
A
d
(
T
1
6. o
∈
A
T
2
,
ʔ
)
if there are a q
∈
[0, 1]
and ʔ
1
,
ʔ
2
∈
D
sub
(
sCSP)
such that
d
(
T
i
,
ʔ
i
)
.
ʔ
⃒
q
·
ʔ
1
+
(1
−
q
)
·
ʔ
2
and o
=
q
·
o
1
+
(1
−
q
)
·
o
2
for certain o
i
∈
A
ˉ
−ₒ
Proof
, the states in the support of [
ˉ
|
Act
ʔ
] have no other
outgoing transitions than
ˉ
. Therefore, [
ˉ
|
Act
ʔ
] is the unique extreme derivative of
itself, and as $[
ˉ
1. Since
ˉ
|
Act
ʔ
ˉ
we have
d
(
ˉ
,
ʔ
)
|
Act
ʔ
]
=|
ʔ
|·
A
={|
ʔ
|·
ˉ
}
.
2. (
|
Act
ʵ
.
All states involved in this derivation (that is, all states
u
in the support of the
intermediate distributions
ʔ
i
⃐
) Assume
ʔ
⃒
ʵ
. By Lemma
6.27
(1), we have
˄.ˉ
|
Act
ʔ
⃒
˄.ˉ
and
ʔ
i
of Definition
6.4
) must have the form
ˉ
˄.ˉ
|
Act
s
, and thus satisfy
u
−ₒ
for all
ˉ
∈
ʩ
. Therefore, it follows that
[
˄.ˉ
|
Act
ʔ
]
⃒
[
˄.ˉ
|
Act
ʵ
]. Trivially, [
˄.ˉ
|
Act
ʵ
]
=
ʵ
is stable, and hence an
0
,so
0
∈
A
d
(
˄.ˉ
,
ʔ
).
extreme derivative of [
˄.ˉ
|
Act
ʔ
]. Moreover, $
ʵ
=
) Suppose
0
d
(
˄.ˉ
,
ʔ
), that is, there is some extreme derivative
ʓ
of
(
⃒
∈
A
0 . Given the operational semantics of
rpCSP
, all
[
˄.ˉ
|
Act
ʔ
] such that $
ʓ
=
states
u
∈
ʓ
must have one of the forms
u
=
[
˄.ˉ
|
Act
t
]or
u
=
[
ˉ
|
Act
t
].
0
, the latter possibility cannot occur. It follows that all transitions
contributing to the derivation [
˄.ˉ
|
Act
ʔ
]
As $
ʓ
=
⃒
ʓ
do not require any action
|
Act
ʔ
] for some distribution
ʔ
with
from
˄.ˉ
, and, in fact,
ʓ
has the form [
˄.ˉ
ʔ
.As
ʓ
must be stable, yet none of the states in its support are, it follows
ʔ
⃒
, that is,
ʔ
=
that
ʓ
=∅
ʵ
.
3. Let
T
:
=
a
∈
X
a.ˉ
.
X
|
Act
ʔ
by
Lemma
6.27
(1), and by the same argument as in the previous case, we have
[
T
ʔ
for some
ʔ
. Then,
T
(
⃐
) Assume
ʔ
⃒
−ₒ
|
Act
ʔ
⃒
T
|
Act
ʔ
]. All states in the support of
T
|
Act
ʔ
are deadlocked. So
|
Act
ʔ
]
⃒
[
T
0
. Thus, we have
0
∈
A
d
(
T
,
ʔ
).
[
T
|
Act
ʔ
]
⃒
[
T
|
Act
ʔ
] and $(
T
|
Act
ʔ
)
=
) Suppose
0
d
(
T
,
ʔ
). By the very same reasoning as in Case 2 we find
(
⃒
∈
A
X
ʔ
for some
ʔ
such that
T
|
Act
ʔ
is stable. This implies
ʔ
.
4. Let
T
be a test in which the success action
ˉ
does not occur, and let
U
be an
abbreviation for
˄.ˉ
that
ʔ
⃒
−ₒ
a.T
.
a
⃒
) Assume there is a
ʔ
∈
D
sub
(
sCSP
) with
ʔ
ʔ
and
o
d
(
T
,
ʔ
).
(
⃐
∈
A
a
−ₒ
ʔ
.
Using Lemma
6.27
(1) and (3), and the same reasoning as in the previous cases,
[
U
ʔ
pre
ʔ
post
Without loss of generality we may assume that
ʔ
⃒
⃒
˄
−ₒ
|
Act
ʔ
]
⃒
|
Act
ʔ
pre
]
|
Act
ʔ
post
]
⃒
|
Act
ʔ
]
⃒
[
U
[
T
[
T
ʓ
for a stable
=
∈
A
d
(
U
,
ʔ
).
subdistribution
ʓ
with $
ʓ
o
. It follows that
o
(
⃒
) Suppose
o
∈
A
d
(
U
,
ʔ
) with
o
(
ˉ
)
=
0. Then, there is a stable subdistribution
ʓ
such that [
U
|
Act
ʔ
]
⃒
ʓ
and $
ʓ
=
o
. Since
o
(
ˉ
)
=
0, there is no state in
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