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

Proofs of Propositions and Transformations 425
Case rule (a’)
Then BSpece 1 S § p£ e
1 ; S p
2 Cp ¡ ¡£ Er¢ , BSpec E1£ ¡ e 2 S § p£ e
1 ¡ ; S p
2 Cp ¡ ¡£ Er¢ where S E1£ ¡ p£
¡ e
1 , and
§ S p£ e
1 Cp ¡ ¡£ Er¢ , env , Sys E1£ ¡ p a
£ § , S Sys
p£ e
1 ¡ Cp ¡ ¡£ Er¢ , env¡ , Sys E1£ ¡ p , . Items (ii)
Sys
and (v) are proved as in case axiom (1’). By induction we ¡ ¤ have ¡
a Abs(a)
AASort(
§ S p£ e
1 Cp ¡ ¡£ Er¢ , env , Sys E1£ ¡ p ), but then clearly a ¡ ¤ Abs(a) ¡ AASort(conf ¥ Sys p
1 ),
so (iii) holds. For (i) we have to show that the condition of rule (3) of the definition of
Conf p is satisfied. Again, the first part of the condition follows from conf p
1 ¡ Conf p . For
the second part first notice that since conf p
1 ¡ Conf p , AASort(S p£ e
1 ; S p
2) AASort(C ¥ p ¥ Sysp ).
But then AASort(S p£ e
¥ 1 ) AASort(Cp ¥ Sysp ) and thus
§ S p£ e
1 Cp ¡ ¡£ Er¢ ¥ envs¥ Sys E1£ ¡ p ¡ Conf ¥ Sys p .
By induction we then have
§ S p£ e
1 ¡ Cp ¡ ¡£ Er¢ ¥ env¡ , Sys E1£ ¡ p ¡ Conf ¥ Sys p and thus AASort(S p£ e
1 ¡ )
¥ AASort(Cp ¥ Sysp ). But then AASort(S p£ e
1 ¡ ; S p
2 ) ¥ AASort(Cp ¥ Sysp ), and consequently
conf p
2 ¡ Conf p .
Case rules (b’), ¡ ¡ ,(r’)
Are all proved analogous to the case of rule (a’).
Case rule (s’)
Then conf p
1 = BSpece 1 ¢ BSpece2 ¥ envs1 envs2 Sys ¥ p , conf ¥ Sys p
2 = BSpece 1 ¡ ¢ BSpece envs1¡ 2 ¥
envs2 Sys ¥ p , and ¥ Sys BSpece 1 ¥ envs1 ¥ Sysp a
£
e BSpec1 ¡ ¥ envs1¡ ¥ Sys ¥ Sys
p . By induction
¥ Sys
we ¡ ¤ have ¡ a Abs(a) AASort( BSpece 1 ¥ envs1 ¥ Sysp ), AASort ( ¥ Sys BSpece 1 , envs1 , Sysp , ) = AASort( Sys BSpece 1 ¡ ¥ envs1¡ ¥ Sysp ), and Reset ( ¥ Sys BSpece 1 , envs1 , Sysp , ) = Reset
Sys
( BSpece 1 ¡ ¥ envs1¡ ¥ Sysp ). From this, (ii), (iii), and (v) follow easily. For (i) observe that
¥ Sys
for conf p
1 to be member of Conf p , BSpece 1 ¥ envs1 ¥ Sysp and ¥ Sys BSpece 2 ¥ envs2¥ Sysp ¥ Sys
both must be member of Conf p (see rule (5) of the definition of Conf p ). But then by
induction BSpece 1 ¡ ¥ envs1¡ ¥ Sysp ¡ Conf ¥ Sys p , and thus (using rule (5) again) conf p
2 ¡
Conf p .
Case rules (n’), ¡ ¡ ,(w’)
Are proved in a similar way as in the case of rule (s’).
Case rule (x’)
Then conf p
1 =
§ BSpece Cc ¡£ Er¢ ¥ envs¥ E1£ ¡ ¡ Sysp Sys ¥ , conf p
2 =
§ BSpece ¡ Cc ¡£ Er¢ E1£ envs¡ ¡ ¡ , , Sysp Sys , , and BSpece envs¥ ¥ Sysp
£
¡ Sys
¥ ¥ envs¡ ¥
a
e BSpec Sysp Sys ¥ . Since conf p
¡ 1 Conf p
we have that Reset(BSpec e ) BSpec p § § E1© P1 ¥¡ ¥ Er© Pr where BSpec p denotes the behaviour
specification and where P1 ¥¡ ¡ ¡ ¥ Pn denote the expression parameters of the
cluster class with name C c . Item (ii) directly follows from the definition of Reset.
By induction, ¡ ¤ a ¡ Abs(a) AASort( BSpece envs¥ Sys ¥ p Sys ). Further we have
¥
AASort( BSpece envs¥ Sys ¥ p Sys ) = AASort(BSpec ¥ e Sys ¥ p ) = £
according to Lemma ¦ 1
AASort(Reset(BSpece )¥ Sysp ) = AASort(BSpecp E1© § ¥¡ ¡ ¡ ¥ Er© Pr ¥ § P1 Sysp ) = £
¦
according to
Lemma 3 AASort (BSpecp , Sysp ) = £
¦ context condition (10’) AASort(Cc ¥ Sysp ) =
AASort(conf p
1 ), and thus (iii) follows. (v) is true because AASort(conf p
1 ) = AASort(Cc , Sysp )
= AASort(conf p
p
2 ). Since conf1 ¡ Conf p , we have using rule (4) of the definition of Conf p
that BSpec e , envs , Sys p , Sys ¡ Conf p . By induction we then have BSpec e ¡ ¥ envs¡ ¥ Sys p ,
Sys ¡ Conf p and Reset ( BSpec e ¥ envs¥ Sys p ¥ Sys ) = Reset ( BSpec e ¡ ¥ envs¡ ¥ Sys p ¥ Sys ). Now,