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

424 Proofs of Propositions and Transformations
definition of Conf p and from conf p
1 ¡ Conf p of Reset. (iii) holds vacuously since
. (ii) is an immediate result of the definition
¡ ¡ ¡ . Further AASort(conf p
2 ) = AASort(BSpece 2 ¥ Sysp )
= AASort(Cp Sys ¥ p ) = AASort(BSpece 1 ¥ Sysp ) = AASort(conf p
1 ), and thus (v) is also satisfied.
Case axiom (2’)
Then BSpec e 1 § ch!m(E1 ¥¡ ¡ ¡ ¥ En) C p E¢1 £ ¡ ¡ ¡£ E¢r¢ , BSpec e 2 § C p E¢1 £ ¡ ¡ ¡£ E¢r¢ , and a = ch!m§ data with
£
ch!m§ n ¡
Abs(a) =
rule (3) of the definition of Conf p , we ¡ derive Abs(a) AASort(Cp ¥ Sysp ), and ¡
thus Abs(a)
AASort(BSpece 1 ¥ Sysp ) = AASort(conf p
1 ), so the result follows.
ch!m§ n ¦ = AASort(ch!m(E1 ¥¡ ¡ ¡ ¥ En)). Then, using the condition of
¡ data ¡ = n. (i), (ii), and (v) are proved as in case axiom (1’). For (iii) we notice that
Case axiom (3’)
Analogous to the case of axiom (2’).
Case axiom (4’)
Then BSpec e 1 § m(E1 ¥¡ ¡ ¡ ¥ Em)(p1¥¡ ¡ ¡ ¥ pn) C p E ¢1 £ ¡ ¡ ¡£ E ¢r¢ , BSpec e 2 § m p1£ ¡ ¡ ¡ £ pn¢ S p C p E ¢1 £ ¡ ¡ ¡£ E ¢r¢ with S p
being the body of method m, and a = ¡ . (ii), (iii), and (v) are proved as in case axiom (1’).
For (i) we have to show that the condition of rule (3) of the definition of Conf p is satisfied.
The first part of the definition directly follows from conf p
1 ¡ Conf p . For the second part we
use context condition (10’) together with the definition of AASort, and deduce AASort(S p )
¥ AASort(C p ¥ Sys p ). From this the result follows since AASort( m p1£ ¡ ¡ ¡ £ pn¢ S p ) = AASort(S p )
¥ AASort(C p ¥ Sys p ).
Cases axioms (5’),(6’) and (7’)
Are proved analogous to the case of axiom (4’).
Case axiom (8’)
Then BSpec e 1 C p (E1 ¥¡ ¡ ¡ ¥ Er), BSpec e 2 § m(E¡1 ¥¡ ¡ ¡ ¥ E¡q)() C p E1£ ¡ ¡ ¡£ Er¢ , and a = ¡ . (i), (ii), and (iii)
are proved as in case axiom (1’). Item (v) holds since AASort(conf p
2 ) = AASort(BSpece 2 ¥ Sysp )
= AASort(C p ¥ Sys p ) = AASort(BSpec e 1 ¥ Sysp ) = AASort(conf p
1 ).
Case axiom (9’)
Then BSpec e 1 C c (E1 ¥¡ ¡ ¡ ¥ Er), BSpec e 2 § BSpec p § § E1© P1 ¥¡ ¡ ¡ ¥ Er © Pr C c E1£ ¡ ¡ ¡£ Er¢ where BSpec p
is the behaviour specification and where P1 ¥¡ ¡ ¡ ¥ Pr are the expression parameters of the
cluster class with name C c , a = ¡ , and envs = envs¡ =
of the definition of Reset. (iii) holds vacuously. (v) is satisfied since AASort(Conf p
2 ) =
AASort(BSpece 2 ¥ Sysp ) = AASort(Cc Sys ¥ p ) = AASort(BSpece 1 ¥ Sysp ) = AASort(conf p
. (ii) is an immediate consequence
1 ). For (i)
we first have to show that BSpecp
E1© § ¥¡ ¡ ¡ ¥ § © Pr ¥ ¥ P1 Er Sysp ¥ Sys¡ Conf p . Using context
condition (13’) it easily follows that BSpecp E1© § ¥¡ ¡ ¡ ¥ Er© Pr ¡ § P1 BSpecifications, and thus
that BSpecp
E1© § ¥¡ ¡ ¡ ¥ § © Pr ¥ ¥ P1 Er Sysp ¥ Sys¡
¥
SSpecifications. From this the result follows
because SSpecification Conf p . The next thing we have to show for (i) is that the condition
of rule (4) of the definition of Conf p holds. The first part of the condition follows from
conf p
¡ 1 Conf p . For the second part we have to show that Reset(BSpecp E1© § ¡ § ¡ Er© P1 , , Pr
) BSpec p § E1© P1 ¥¡ ¡ ¡ ¥ Er© Pr . But this follows from the fact that BSpec § p § E1© P1 ¥¡ ¡ ¡ ¥ Er© Pr
§
BSpecifications by applying Lemma 1.
¡