Information Technology Reference
In-Depth Information
FS
)
†
ʘ
implies that
ʔ
=
i
∈
I
p
i
·
s
i
,
s
i
e
e
Proof
By Lemma
6.1
ʔ
(
FS
ʘ
i
, as well
=
i
∈
I
p
i
·
ʱ
⃒
ʔ
as
ʘ
ʘ
i
. By Corollary
6.7
and Proposition
6.2
we know from
ʔ
for
ʔ
i
∈
D
sub
(
S
) such that
ʔ
=
i
∈
I
p
i
·
ʱ
⃒
ʔ
i
ʔ
i
. For each
i
that
s
i
∈
I
we
ʱ
⃒
ʱ
⃒
e
ʔ
i
that there is a
ʘ
i
∈
D
sub
(
S
) with
ʘ
i
ʘ
i
infer from
s
i
FS
ʘ
i
and
s
i
and
=
i
∈
I
p
i
·
FS
)
†
ʘ
. Let
ʘ
:
ʔ
i
e
ʘ
i
. Then Definition
6.2
(2) and Theorem
6.5
(i)
(
ʱ
⃒
FS
)
†
ʘ
and
ʘ
yield
ʔ
(
e
ʘ
.
FS
)
†
ʘ and ʔ
A
A
e
Lemma 6.22
Suppose ʔ
(
⃒
−ₒ
. Then ʘ
⃒
−ₒ
.
A
e
FS
)
†
ʔ
Proof
Suppose
ʔ
(
ʘ
and
ʔ
⃒
−ₒ
. By Lemma
6.21
there exists
some subdistribution
ʘ
such t
ha
t
ʘ
ʘ
and
ʔ
(
e
FS
)
†
ʘ
. From Lemma
6.1
⃒
we know that
ʔ
=
i
∈
I
p
i
·
FS
ʘ
i
,
ʘ
=
i
∈
I
p
i
·
e
ʔ
s
i
,
s
i
ʘ
i
, with
s
i
∈
A
A
for all
i
∈
I
. Since
ʔ
−ₒ
, we have that
s
i
−ₒ
for all
i
∈
I
. It follows from
. By Theorem
6.5
(i), we obtain that
i
∈
I
p
i
·
ʘ
i
⃒
A
e
FS
ʘ
i
that
ʘ
i
⃒
ʘ
i
s
i
−ₒ
i
∈
I
p
i
·
A
A
ʘ
i
The next result shows how the failure simulation preorder can alternatively be
defined in terms of failure similarity. This is consistent with Definition 5.5 for finite
processes.
Proposition 6.7
Fo r ʔ
,
ʘ
∈
D
sub
(
S
)
we have ʘ
FS
ʔ just when there is a ʘ
match
with ʘ
⃒
ʘ
match
−ₒ
. By the transitivity of
⃒
we have that
ʘ
⃒
−ₒ
.
e
FS
)
†
ʘ
match
.
and ʔ
(
FS
ↆ
S
×
D
sub
(
S
) be the relation given by
s
FS
ʘ
iff
ʘ
FS
s
. Then
FS
Proof
Let
=
i
p
i
·
FS
ↆ
e
is a failure simulation; hence
FS
. Now supp
os
e
ʘ
FS
ʔ
. Let
ʔ
:
s
i
.
⃒
i
p
i
·
ʘ
i
and
ʘ
i
FS
s
i
for each
i
, whence
s
i
FS
ʘ
i
,
Then there are
ʘ
i
with
ʘ
=
i
p
i
·
e
FS
ʘ
i
. Take
ʘ
match
e
FS
)
†
ʘ
match
.
and thus
s
i
:
ʘ
i
. Definition
6.2
yields
ʔ
(
e
FS
)
†
For the other direction, it suffices to show that the relation (
·⃐
satisfies
e
FS
)
†
the two clauses of Definition
6.19
, yielding (
·⃐ↆ
FS
. Here we write
⃐
∈
D
sub
(
S
), there is a
ʘ
match
with
for the inverse of
⃒
. So suppose, for given
ʔ
,
ʘ
FS
)
†
ʘ
match
.
ʘ
match
e
ʘ
⃒
and
ʔ
(
⃒
i
∈
I
p
i
·
ʱ
ʔ
i
Suppose
ʔ
for some
ʱ
∈
Act
˄
. By Lemma
6.21
, there is some
ʘ
and (
i
∈
I
p
i
·
ʱ
⃒
e
FS
)
†
ʘ
. From Proposition
6.2
,
ʘ
such that
ʘ
match
ʔ
i
)(
we know that
ʘ
=
i
∈
I
p
i
·
e
FS
)
†
ʘ
i
for subdistributions
ʘ
i
such that
ʔ
i
(
ʘ
i
⃒
i
p
i
·
ʱ
ʘ
i
∈
⃒
for
i
I
. Thus
ʘ
by the transitivity of
(Theorem
6.6
) and
ʔ
i
((
e
FS
)
†
)
ʘ
i
·⃐
for each
i
∈
I
by the reflexivity of
⃐
.
A
A
. By Lemma
6.22
we have
ʘ
match
Suppose
ʔ
⃒
−ₒ
⃒
−ₒ
. It follows that
A
ʘ
⃒
−ₒ
by the transitivity of
⃒
.
ʘ
match
in Proposition
6.7
immediately above. For the same reason explained in Example 5.14, defining
Note the appearance of the “anterior step”
ʘ
⃒
FS
FS
)
†
e
simply to be (
(i.e. without anterior step) would not have been suitable.
e
Remark 6.3
FS
s
; here no
anterior step is needed. One direction of this statement has been obtained in the
beginning of the proof of Proposition
6.7
; for the other note that
s
For
s
∈
S
and
ʘ
∈
D
sub
(
S
), we have
s
FS
ʘ
iff
ʘ
e
FS
ʘ
implies
Search WWH ::
Custom Search