Transformation of Applicative Specifications into Imperative ...

7.5. EXAMPLE
• s ′ init = [t ↦→ tinit] where tinit ∈ B that is t can be mapped to either
true or false. This means that there are two models of I_SPEC only
distinguishable by the initial value of the store.
The semantics of the satisfaction relation is defined such that m ′ |=Σ
I_SP EC.
Then m ′′ = 〈m ′′ A , m′′ OP , s′′ init 〉 = Mod(σ)(m′ ) is calculated:
• m ′′ A = Mod(σ)(m′ A ) = m′ A = [T ↦→ ValueBool] = [T ↦→ B]
• for m ′′ OP
the following holds:
– dom m ′′ OP = dom m′ OP = {f}
– m ′′ OP (f)(x)([ ]) ≡
case m ′ OP (f)(Unit)([ t ↦→ x ]) of
(Unit, [ t ↦→ ∼ x ]) → (∼ x, [ ])
end
where the case ⊥ is left out because f is total. This means that
m ′′ OP = [f ↦→ λx : B · ∼ x]
• s ′′
init = Mod(σ)(s′ init ) = s′ init \(dom V ′ \dom V ) = [t ↦→ tinit]\({t}\{}) =
[t ↦→ tinit]\{t} = [ ]
As can be seen
This implies that
m = m ′′ = Mod(σ)(m ′ )
Mod(σ)(m ′ ) |=Σ e and m ′ |=Σ e ′
where e ′ = Sen(σ)(e). This means that the satisfaction condition holds
for the instance of a sentence and a model considered.
In order to prove that it holds for the entire RSLI subset, similar considerations
should be made for the entire RSLI subset, e.g. all sentences and
all models. This can be obtained by basing the proofs on the structure of
the sentences as done in [Lin04].
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