Transformation of Applicative Specifications into Imperative ...

CHAPTER 7. CORRECTNESS OF TRANSFORMATION RULES
for all (ve1, x, ve2) ∈ Value A ∗ (te1×tv×te2):
mOP (f)(vearg)([ ]) ≡
case m ′ OP (f)(vearg)([ v ↦→ dv ]) of
⊥ → ⊥,
((ve1, ve2), [ v ↦→ dv ′ ]) → ((ve1, dv ′ , ve2), [ ])
end
Note that dom [v ↦→ dv] = dom [v ↦→ dv ′ ] = {v}.
Explanation: mOP (f) returns chaos if m ′ OP (f) returns chaos. Otherwise
mOP (f) returns the same result as m ′ OP (f), but with the actual
value dv ′ of v included. When evaluating m ′ OP (f) a store s = [v ↦→ dv]
is kept reflecting the actual value dv of v. This value is altered during
the evaluation of f, since f is a generator.
3. Combined observer and generator
f : te1 × tv × te2 ∼ → te3 × tv × te4
σ
f : te1 × te2 ∼ → read v write v te3 × te4
for all (ve1, x, ve2) ∈ Value A ∗ (te1×tv×te2), (ve3, x, ve4) ∈ Value A ∗ (te3×tv×te4):
mOP (f)(ve1, x, ve2)([ ]) ≡
case m ′ OP (f)(ve1, ve2)([ v ↦→ x ]) of
⊥ → ⊥,
((ve3, ve4), [ v ↦→ x ′ ]) → ((ve3, x ′ , ve4), [ ])
end
Explanation: mOP (f) returns chaos if m ′ OP (f) returns chaos. Otherwise
mOP (f) returns the same result as m ′ OP (f), but with the actual
value of x ′ of v included. When evaluating m ′ OP (f) a store s = [v ↦→ x]
is kept reflecting the actual value x of v. This value is altered during
the evaluation of f, since f is a generator.
Model Category
Composition is defined the obvious way. Identities are defined such that a
model is mapped to itself.
Model Functor
The functor Mod : Sign → Cat is a functor that maps Sign to the category
Mod(Σ) of Σ-models and maps each signature morphism σ : Σ → Σ ′ in Sign
to the model morphism Mod(σ) : Mod(Σ ′ ) → Mod(Σ) which is described
above.
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