Transformation of Applicative Specifications into Imperative ...

CHAPTER 7. CORRECTNESS OF TRANSFORMATION RULES
• mA = [T ↦→ ValueBool] = [T ↦→ B]
• mOP = [f ↦→ λx : B • ∼ x]
• sinit = [ ], as there are no variable definitions in A_SPEC
The semantics of the satisfaction relation is defined such that m |=Σ
A_SP EC.
Then the imperative specification I_SPEC is obtained by transforming
A_SPEC using the transformation rules defined in Chapter 6 such that
A_SPEC ✄ I_SPEC. The type of interest is T and the corresponding variable
is named t.
Specification 7.2 – Imperative specification I_SPEC
scheme I_SPEC =
class
type T = Bool
variable t : T
value
f : Unit → read t write t Unit
f() ≡ t := ∼ t
end
I_SPEC can be regarded as a signature Σ ′ = σ(Σ) and a sentence e ′ =
Sen(σ)(e).
The signature Σ ′ = 〈A ′ , OP ′ , V ′ 〉 of I_SPEC is defined as follows:
• A ′ = [T ↦→ Bool], reflecting the type of T
• OP ′ = [f ↦→ Unit → read t write t Unit], reflecting the type of the
function f
• V ′ = [t ↦→ T ], reflecting the type of the variable t
The sentence is defined as e ′ =✷ f() ≡ t := ∼ t
Then the model m ′ = 〈m ′ A , m′ OP , s′ init 〉 of I_SPEC can be defined:
• m ′ A = [T ↦→ ValueBool] = [T ↦→ B]
• m ′ OP = [f ↦→ λu : Unit • s ′ : Store • ([t ↦→∼ s ′ (t)], d_unit)]
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