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

Proofs of Propositions and Transformations 429
¢
BSpec e
¢
¡
¡
£
£
application of 7
because of (1) ChSort(BSpec e ¥ Sys p ) §
application of 4
because of (2) f ¤ ChSort(BSpec e
L§ f ¥ envs¥ Sys p ¥ Sys
e
£ BSpec
application of 1 ¦
¢
¡
e BSpec
L§ Id ¥ envs¥ Sys p ¥ Sys
£
¡ ¡ ¡ ¡ ¤ (L)¦ ¦
ch L f (ch) f
L¥ Sys p ) = Id ¤ ChSort(BSpec e
L¥ Sys p ) ¦
L¥ envs¥ Sys p ¥ Sys
This ends the proof of Transformation 10. Note that the proof implicitly uses the
congruence property of ¢
¡ as stated in Proposition 4. ¨
Proof of Transformation 11
The proof of the observation equivalence part consists of showing that = £
( (BSpece 1 ¢
BSpece 2) envs1 ¥ envs2 Sys L¥ p
¥
e (BSpec1 L) ¢ (BSpec ¥ Sys e 2 envs1 ¥ envs2 Sys L)¥ p ) ¡
£
¥ Sys
Chan(la ) l ¡ a ¡ AASort(BSpece 1 ¥ Sysp ) la ¡ AASort(BSpece 2 ¥ Sysp L ¤ ¦ is a weak
)¦ §
bisimulation. The proof requires (iii) and (v) of Proposition 2. The transformation
equivalence part follows from the definition of Reset and from ¨
Lemma 2.
Proof of Transformation 12
The result for transformation equivalence directly follows from the definition of Reset
and from the result for observation equivalence. For observation equivalence we will
show that = £
( BSpece ¥ envs¥ Sys p
1 ¥ Sys , BSpece ¥ envs¥ Sys p
2 ¡ ) Sysp1
and Sysp2
are non-
¥ Sys
conflicting is a weak bisimulation. Let ¦ (conf p p
p a
1 conf ¡ 2 ) and suppose £ conf1 ¥ conf p
1 ¡ .
p a
The proof is by rule induction on the shape of the derivation £ tree of conf conf p
1 ¡ . We
argue by cases on the applied axioms and rules.
Case axiom (9’)
Then conf p
1 = Cc
¡ ¥ Er)¥ ¥ (E1 Sys ¥¡ ¡ p
1 ¥ Sys , a = ¡ , conf p
1 ¡ =
BSpec § p P1 ¥¡ ¡ ¡ Er© Pr
§ § E1©
Cc
¡£ Er¢ ¥ ¥ Sys E1£ ¡ ¡ p
p
1 , Sysp1
CD1 ¥ Sys ¡ ¡ CDp ¡ ¡ CD p r , CDp cluster class Cc ¡ ¥ Pr P1 ¥¡ ¡
behaviour specification ¡ ¡ BSpecp , BSpecp P1 ¥¡ ¡ ¡ ¥ Er© Pr ¡ BSpecifications, and conf § E1© § p
= Cc
¥ Er)¥ ¥ ¥¡ ¡ ¡ (E1 Sys p
Sys ¥ 2 . Since Sysp1
and Sys p
2 are non-conflicting, Sys p
tains CDp and thus by axiom (9’) conf p
£ 2 conf p
2 ¡ =
¥ Sys p
¡
¤ Sys ¥
p
2 . But then also conf2 conf p
2 ¡ and clearly (conf p
1 ¡ ¥ conf p
.
2 ¡ ) ¡
2
2 also con-
§ BSpec p § § E1© P1 ¥¡ ¡ ¡ Er© Pr C c E1£ ¡ ¡ ¡ £ Er¢ ,
Case rule (s’)
Then conf p
1 = BSpece 1 ¢ BSpece 2 ¥ envs1 envs2 ¥ Sys p
p
1 , conf1 ¥ Sys ¡ = BSpece 1 ¡ ¢ BSpece2 ,
envs2 envs1¡ Sys ¥ p
1 ¥ Sys , and BSpece 1 ¥ envs1 Sys ¥ p a
£
e
1 BSpec ¡ ¥ envs1¡ ¥ 1 Sys ¥ Sys
p
1 . Since
¥ Sys
( BSpece 1 ¥ envs1 Sys ¥ p
e
1 , BSpec ¥ 1 ¥ envs1 Sys ¥ Sys p
2 ¡ ) , we have by induction that
¥ Sys
e BSpec1 , envs1 , Sys p
¡
a
¤ 2 , conf Sys
p , for some conf p with ( BSpece 1 ¡ , envs1¡ , Sys p
1 ,
Sys ¥ conf p ) . But then necessarily ¡
conf p = BSpece 1 ¡ ¥ envs1¡ Sys ¥ p
2 . By applying rule
¥ Sys
(s’) we then have BSpece 1 ¢ BSpece2 ¥ envs1 envs2 ¥ Sys p
¡
a
¤ 2 conf ¥ Sys
p
2 ¡ = BSpece 1 ¡ ¢ BSpece2 ,
envs2 envs1¡ Sys ¥ p
p
2 , and evidently (conf1 ¥ Sys ¡ conf ¥ p
2 ¡ ) ¡ .
The proofs of the remaining (and symmetric) cases are of a similar complexity. This ends