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

Proofs of Propositions and Transformations 427
For transformation equivalence we additionally have to show that Reset( BSpece 1 , envs1
, Sysp Sys , ) ¢ Reset( BSpece 1 ¡ ¥ envs1¡ Sys ¥ p Sys ) implies Reset( ¥ BSpece 1 L , envs1 , Sysp ,
) Sys ¢ Reset( BSpece 1 ¡ L envs1¡ , , Sysp Sys , ). But this immediately follows from the
definition of Reset and the substitutivity of ¢ under channel hiding.
This concludes the ¨
proof of Proposition 4.
£
Proof of Transformation 1
We will first prove the observation equivalence part by showing that =
( BSpece ¥ § Id envs , Sysp
envs¥ ¥ ¥ Sys )¦ ¥ Sys , e p BSpec Sys is a weak bisimulation. It is
clear that (i) of the definition of weak bisimulations is satisfied. For (ii) let ( BSpece Id §
, envs , Sysp
Sys ¥ ¡ ¥ Sys e p , BSpec , envs , Sys ) and assume that BSpece § Id , envs ,
a ¢
Sysp conf Sys £ ¥ p . Then BSpece , envs , Sysp
¡ Sys
envs¡ £ ¥
e BSpec , , Sysp ¥ Sys , conf p
= BSpece envs¡ , , ¡ § Id Sysp ¡ ¡
a¡ , and ¥ a¡ ¡
Sys
¡
if a¡ a and Id(a) if a . So a
and clearly ( BSpece envs¡ ¥ ¡ § Id ¥ Sysp ¥ Sys , BSpece , envs , Sysp Sys ¥ ¡ ) . Item (iii) of the
definition of weak bisimulations is proved in a similar way. For the transformation
equivalence part observe that (Reset( BSpece § Id , envs , Sysp ¥ Sys ), Reset( BSpece , envs ,
Sysp ¥ Sys )) = ( Reset(BSpece
)§ Id ¥ , Sysp ¥ Sys , Reset(BSpece
)¥ , Sysp Sys ¥ ¡ ) . And
thus Reset( BSpece § Id , envs , Sysp ¥ Sys ) ¢ Reset( BSpece ¥ envs, Sysp ¥ Sys ).
This concludes the proof of Transformation ¨
1.
Proofs of Transformations 2 and 3
Both proofs consist of exhibiting appropriate weak bisimulations and by applying the
definition of function Reset. The proofs are of a complexity similar to that of Transformation
¨
1.
Proof of Transformation 4
The observation equivalence part of the proof consists of showing that = £
( BSpece §
envs¥ , Sys
f p ¥ Sys ,
BSpec e ¥ Sys p )¦ is a weak bisimulation. To prove this we need properties (iv) and (vi)
of Proposition 2 stating that the channel of any non-silent action performed by some
configuration is part of the ChSort of that configuration, and that the channel sort of a
configuration equals the channel sort of any of its derivatives. To prove the transformation
equivalence part we need the fact that the channel sort of any extended behaviour
specification equals the channel sort of the reset of that specification. This fact is subject
of Lemma 4.
a
BSpec e § f ¡ ¥ envs¥ Sys p ¥ Sys ) ¡ f ¤ ChSort(BSpec e ¥ Sys p ) = f ¡ ¤ ChSort(
This proves Transformation ¨
4.
Lemma 4
Let BSpece BSpecifications ¡ e . Then ChSort(BSpece ) = ChSort(Reset(BSpece ¨
)).
Proof of Lemma 4
The lemma follows directly from the definitions of functions Reset and ChSort and from
Lemma 1. ¨