Transformation of Applicative Specifications into Imperative ...

Satisfaction Relation
7.5. EXAMPLE
Then the satisfaction relation |=Σ must be defined, such that the satisfaction
condition can be formulated. The dynamic semantics of mRSL value
expressions was defined in [Lin04]. This should be extended to cover the
whole subset of RSLI.
Proof of Satisfaction Condition
The last step is to prove that the satisfaction condition holds. The satisfaction
condition is proved in [Lin04] for the institution of mRSL. Most likely
this can be carried over to the institution of RSLI.
If this is the case the transformation rules are correct. This means that
the transformation of an applicative RSL specification into an imperative
RSL specification using the transformation rules will be a correct development
step.
7.5 Example
The following simple example illustrates the ideas of this chapter.
First an applicative specification A_SPEC is defined:
Specification 7.1 – Applicative specification A_SPEC
scheme A_SPEC =
class
type T = Bool
value
f : T → T
f(x) ≡ ∼ x
end
A_SPEC can be regarded as a signature Σ and a sentence e.
The signature Σ = 〈A, OP, V 〉 of A_SPEC is defined as follows:
• A = [T ↦→ Bool], reflecting the type of T
• OP = [f ↦→ T → T ], reflecting the type of the function f
• V = [ ], as there are no variable definitions in A_SPEC
The sentence is defined as e =∀ x : T • f(x) ≡ ∼ x
Then the model m = 〈mA, mOP , sinit〉 of A_SPEC can be defined:
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