Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

148 CHAPTER 6. HML WITH RECURSION
Proof: Assume that q ∈ [p]∼, where p is one of p1, . . . , pn. To prove our claim, it
is sufficient to show that
q ∈ 〈·a·〉[p ′
]∼ ∩ [·a·]
[p ′
]∼
.
a,p ′ .p a → p ′
a
p ′ .p a → p ′
(Can you see why?) The proof can be divided into two parts, namely:
a) q ∈
a,p ′ .p a → p ′
b) q ∈
[·a·]
We proceed by proving these claims in turn.
a
〈·a·〉[p ′ ]∼ and
p ′ .p a → p ′
[p ′
]∼ .
a) We recall that q ∼ p. Assume that p a → p ′ for some action a and process p ′ .
Then there is a q ′ , where q a → q ′ and q ′ ∼ p ′ . We have therefore shown that,
for all a and p ′ , there is a q ′ such that
q a → q ′ and q ′ ∈ [p ′ ]∼ .
This means that, for each a and p ′ such that p a → p ′ , we have that
We may therefore conclude that
which was to be shown.
q ∈
q ∈ 〈·a·〉[p ′ ]∼ .
a,p ′ .p a → p ′
〈·a·〉[p ′ ]∼ ,
b) Let a ∈ Act and q a → q ′ .We have to show that q ′ ∈
[p ′ ]∼. To this end,
p ′ .p a → p ′
observe that, as q a → q ′ and p ∼ q, there exists a p ′ such that p a → p ′ and
p ′ ∼ q ′ . For this q ′ we have that q ′ ∈ [p ′ ]∼. We have therefore proven that,
for all a and q ′ ,
which is equivalent to
q a → q ′ ⇒ ∃p ′ . p a → p ′ and q ∈ [p ′ ]∼ ,
q ∈
[·a·]
a
p ′ .p a → p ′
[p ′
]∼ .