Specification of Reactive Hardware/Software Systems - Electronic ...

426 Proofs of Propositions and Transformations
(i) follows from rule (4) of the definition of Conf p .
This concludes the proof of Proposition ¨
2.
The proof of Proposition 2 is based on the following three lemmas:
Lemma 1
Let BSpec be a behaviour specification. Then Reset(BSpec) BSpec. ¨
Proof of Lemma 1
The proof is an easy induction on the structure of BSpec. ¨
Lemma 2
Let BSpec e be an extended behaviour specification. Then AASort(BSpec e ¥ Sys p ) =
AASort(Reset(BSpec e )¥ Sys p ). ¨
Proof of Lemma 2
The proof proceeds by structural induction. ¨
Lemma 3
Let BSpec p be a parameterised behaviour specification. Then AASort(BSpec p ¥ Sys p ) =
AASort(BSpec p § § g ¥ Sys p ). ¨
Proof of Lemma 3
Again, the proof is an easy induction on the structure of BSpec. ¨
Proof of Proposition 3
The proofs of the properties of this proposition are variants of the proofs of similar
properties given in [Mil89] for CCS. ¨
Proof of Proposition 4
We will only prove (2). (1) and (3) are proved in an analogous way. For observation
equivalence we will show that = £
( BSpece 1 L,envs1,Sysp ,Sys , BSpece 1 ¡ L,envs¡1,Sysp ¡
¡
,Sys
e ) BSpec1,envs1,Sysp ¢
¡
e ,Sys BSpec1 ,envs¡1,Sysp ¦ ¡ ,Sys is a weak bisimulation. Clearly
the Sys parts of the configurations are syntactically identical, so (i) of the definition
of weak bisimulations is satisfied. For (2) let ( BSpece 1 L,envs1,Sysp
¡ e ,Sys , BSpec1 L,envs¡1,Sysp
¡ ¡
e ,Sys ) and assume that BSpec1 L,envs1,Sysp £
a
,Sys conf p , for some
conf p . Then, using rule (v’), conf p = BSpece L,envs,Sysp ,Sys for some BSpece
and envs,
e BSpec1,envs1,Sysp ¡
¡
£
¡ ,Sys
a
e p BSpec ,envs,Sys ,Sys , and a = or Chan(a) L. But then,
since BSpece 1 , envs1 , Sysp
¢
¡ , Sys e BSpec1 ,envs¡1 ,Sysp ¡ ,Sys , we have by Proposition
3(ii) that BSpece 1 ¡ ,envs¡1 ,Sysp
¤
e ,Sys ¡ ,envs¡ BSpec ¡ ,Sysp ,Sys for some ¡ BSpece envs¡
¡ and
such that BSpece ,envs,Sysp ¢
e ,Sys ¡ ,envs¡ BSpec ,Sys
p ,Sys . From a ¡ = ¡ ¡ ¡ or Chan(a) L it
¡ a
is then easy to see that BSpece 1 ¡ L,envs¡1 ,Sysp
¤
e ,Sys ¡ L,envs¡ BSpec ¡ ,Sysp ,Sys ¡ and
clearly ( BSpece L,envs,Sysp ,Sys , BSpece ,Sys ¡ L,envs¡ p ,Sys ¡ ) . This concludes ¡ the
proof of condition (2) of the definition of weak bisimulations. Condition (3) is proved
in a symmetric way.
¡ a