Decidability in Intuitionistic Type Theory is Functionally Decidable

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x : A], i.e. to require that the predicate (∀x ∈ A)B(x)∨¬B(x) is provable, is equivalent, when working within the framework of Martin-Löf’s Intuitionistic Type Theory, to require that there exists a decision function φ : A → Boole such that (∀x ∈ A) (φ(x) =Boole true) ↔ B(x). Since we will also show that the proposition x =Boole true [x : Boole] is decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate B(x) prop [x : A] can be effectively reduced by a function φ ∈ A → Boole to the decidability of the predicate φ(x) =Boole true [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function φ, together with a proof of its correctness, is effectively constructed as a function of the proof of (∀x ∈ A)B(x)∨¬B(x).