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Appendix
This appendix contains the proof of Theorem 1.
Proof:
To demonstrate Theorem 1, we assume that two overtaking messages cannot
have the same name and we use the following lemma which can be found in the
book “
Introduction to Lattices and Order
” [9](p.110):
Lemma 1.
Let
(
L, ∨, ∧
)
be a triple where
L
is a non-empty set equipped with
two binary operations
∨
and
∧
which satisfy for all
a, b, c ∈ L
:
(1)
a ∨ a
=
a
and
a ∧ a
=
a
(idempotency laws);
(2)
a ∨ b
=
b ∨ a
and
a ∧ b
=
b ∧ a
(commutative laws );
(3) (
a ∨ b
)
∨ c
=
a ∨
(
b ∨ c
)
et
(
a ∧ b
)
∧ c
=
a ∧
(
b ∧ c
)
(associative laws);
(4)
a ∨
(
a ∧ b
)=
a
et
a ∧
(
a ∨ b
)=
a
(absorption laws).
then:
;
(
ii
)
If we define ≤ by a ≤ b if a ∨ b
=
b,then≤ is an order relation;
(
(
i
)
∀a, b ∈ L, a ∨ b
=
b ⇔ a ∧ b
=
a
iii
)
With
≤
as in
(
ii
)
,
(
L, ≤
)
is a lattice such that
∀a, b ∈ L, a ∨ b
sup{a, b}
=
and a ∧ b
=
inf{a, b}.
We will show that
J
P,M
canbeequippedwithtwobinaryoperations
∨
and
∧
which verify the properties 1-4 of the lemma.
Let
J
P,M
be the set of join points corresponding to a pointcut
P
=(
I
P
,E
P
,
≤
P
,A
P
,α
P
,φ
P
, ≺
P
) in a bSD
M
=(
I,E,≤,A,α,φ,≺
). Let
∨
and
∧
be the
operators defined for each
J
i
,
J
j
of
J
P,M
by:
{e, f ∈ J
i
|e ≺ f, ∃e
∈ E
P
,e
e
)
e
)
J
i
∨ J
j
=
=
μ
i
1
(
,μ
j
1
(
≤ e}∪
{e, f ∈ J
j
|e ≺ f, ∃e
∈ E
P
,e
e
)
e
)
=
μ
j
1
(
,μ
i
1
(
≤ e}
{e, f ∈ J
i
|e ≺ f, ∃e
∈ E
P
,e
e
)
e
)
J
i
∧ J
j
=
=
μ
i
1
(
,e ≤ μ
j
1
(
}∪
{e, f ∈ J
j
|e ≺ f, ∃e
∈ E
P
,e
e
)
e
)
=
μ
j
1
(
,e ≤ μ
i
1
(
}
μ
i
=
<μ
i
0
,μ
i
1
,μ
i
2
>
and
μ
j
=
<μ
j
0
,μ
j
1
,μ
j
2
>
being the isomorphisms
associating
P
to the respective join points
J
i
and
J
j
.
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