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ˉ
. Because the pLTS is
ˉ
-respecting, in fact
ʘ
⃒
ʘ
and so again we have $
s
=
that
ʘ
is also a distribution. Hence, we have $
ʘ
=
ˉ
∈
V
(
ʘ
).
e
FS
)
†
Now for the general case we suppose
ʔ
(
ʘ
. It is not hard to show that we
can decompose
ʘ
into
s
∈
ʔ
e
ʔ
(
s
)
·
ʘ
s
such that
s
FS
ʘ
s
fo
r
each
s
∈
ʔ
, and
recall that each such state
s
is s
ta
ble. From above we have that $
s
∈
V
(
ʘ
s
) for those
=
∈
ʔ
∈
s
∈
ʔ
s
, and so $
ʔ
ʔ
(
s
)
·
$
s
ʔ
(
s
)
·
V
(
ʘ
s
)
=
V
(
ʘ
).
Lemma 6.48
Let ʔ and ʘ be distributions in an ˉ-respecting convergent pLTS
S
,
ʩ
˄
,
ₒ
.Ifʘ
FS
ʔ, then it holds that
V
(
ʘ
)
ↇ
V
(
ʔ
)
.
Proof
Let
ʔ
and
ʘ
be distributions in an
ˉ
-respecting convergent pLTS given by
S
,
ʩ
˄
,
ₒ
. We note that
ʔ
then
(
ʔ
)
(i) If
ʔ
⃒
V
ↆ
V
(
ʔ
).
e
FS
)
†
ʘ
, then we have
(ii) If
ʔ
(
V
(
ʔ
)
ↆ
V
(
ʘ
).
e
FS
)
†
Here (i) follows from Lemma
6.32
. For (ii), let us assume
ʔ
(
ʘ
. For any
e
FS
)
†
ʘ
.It
ʔ
we have the matching transition
ʘ
ʘ
such that
ʔ
(
ʔ
⃒
⃒
follows from Lemmas
6.47
and (i) that $
ʔ
∈
V
(
ʘ
)
ↆ
V
(
ʘ
). Consequently, we
V
ↆ
V
obtain
(
ʘ
).
Now suppose
ʘ
(
ʔ
)
FS
ʔ
. By definition there exists some
ʘ
such that
ʘ
⃒
ʘ
e
FS
)†
ʘ
. By (i) and (ii) above we obtain
and
ʔ
(
V
(
ʔ
)
ↆ
V
(
ʘ
)
ↆ
V
(
ʘ
).
Theorem 6.22
For any finitary convergent processes P and Q,ifP
FS
Q then
rr
must
Q.
Proof
P
We reason as follows.
P
FS
Q
implies
[
P
|
Act
T
]
FS
[
Q
|
Act
T
]
Lemma
6.30
, for any
ʩ
-test
T
implies
V
([
P
|
Act
T
])
ↇ
V
([
Q
|
Act
T
])
[
·
]is
ˉ
-respecting; Lemma
6.48
d
(
T
,
P
)
d
(
T
,
Q
)
iff
A
ↇ
A
Definitions
6.24
and
6.10
iff
A
(
T
,
P
)
ↇ
A
(
T
,
Q
)
Corollary
6.3
·
A
ↇ
·
A
∈
−
1, 1]
ʩ
implies
h
(
T
,
P
)
h
(
T
,
Q
)
for any
h
[
1, 1]
ʩ
implies
h
·
A
(
T
,
P
)
≤
h
·
A
(
T
,
Q
)
for any
h
∈
[
−
rr
must
Q.
Note that in the second line above, both [
P
iff
P
|
Act
T
] are convergent, since
for any convergent process
R
and very finite process
T
, by induction on the structure
of
T
, it can be shown that the composition
R
|
Act
T
] and [
Q
We are now ready to prove the main result of the section which states that
nonnegative-reward must testing is as discriminating as real-reward must testing.
Theorem 6.23
|
Act
T
is also convergent.
For any finitary convergent processes P , Q, it holds that P
rr
must
Q
if and only if P
nr
must
Q.
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