Pruebas de la Solución

The Logic In this chapter we introduce the system NAω, which builds on G¨odel’s System T and is a negative fragment of Heyting’s Arithmetic with finite types. We carry out the proofs in minimal logic, but when not clear from the context or when we want to emphasize in which system the formulas are valid, we use the following notation:

`_m for derivability in minimal logic

`_i for derivability in intuitionistic logic, i.e., minimal logic +efq⊥

`c for derivability in classical logic, i.e., intuitionistic logic +Stab⊥

A discussion at length onefq⊥_A and the reason why we are excluding it in addition toStab⊥, in connection with the observations made in Subsection 2.3.2, is carried out in Section 8.1.

In our (minimal) setting, the distinction between classical and construc- tive proofs relies (only) in the semantics that we fix a-priori for the existen- tial quantifier. As seen in this chapter, the classical existential quantifier is viewed as an abbreviation and thus proofs of ˜∃-formulas consist in deriving ⊥and therefore they can be regarded as proofs by contradiction. To recover the computational (or constructive) content in the classical proofs, we rely on the fact that the Π0₂-formulas are equiderivable in both systems and use the translation mechanism introduced in the following chapter.

Summary of this chapter. Among the key notions used throughout this thesis, we define in Section 2.1 the types, terms, formulas and proof terms. Since the terms are built using also the recursion operators which will be associated to the induction axioms when extracting programs from proofs, we have presented in this chapter the corresponding reduction rules. We have given the main notations and abbreviations used in this thesis.

Special care is payed to the weak operators ˜∨, ˜∃, ˜∧ and ˜¬, which we see only as abbreviation and treat in Section 2.2. We distinguish among the logical falsity⊥and the arithmetical F, since we want to allow substitution of the former when A-translating the proofs. Consequently, the axioms efq

and efq⊥ require special treatment, as presented in Section 2.3.2.

Proofs are presented both as derivation trees in natural deduction style and as λ-terms. In Section 2.4 we have presented the connection between the two.

A-Translation and its

Refined Version

By working in a fragment of classical logic in which ∃ is not explicit, but rather an abbreviation, the proofs of Π0₂-formulas do not provide a witness in an explicit manner. For this reason, recovering the computational content of existential goals cannot be done in a direct way by the Curry-Howard correspondence. Therefore, we need to apply first a transformation method on the proofs, in order to lift them to a form in which the computational content becomes explicit. In order to do this, we have to consider a special treatment of the negation, which plays in this process the role of a place holder for “strong” instances of the existential goal formula.

This chapter is organized as follows: Section 3.1 gives a brief exposition of the G¨odel-Gentzen translation which, combined with Friedman’s trick, results in the so-called “A-Translation” presented in Section 3.2. In this dissertation we work with a refinement due to (Berger et al., 2002), which we overview in detail in Section 3.3. The key idea evolves around the concept of definite/goal formulas, which are restricted classes introduced in order to improve the original A-Translation. We will present them in Section 3.3.1 in a manner close to (Ishihara, 2000)’s approach, since we believe that this illustrates better their structure. Once the proofs are lifted to their constructive counterparts, we are able to unravel their computational content by the extraction rules summarized in Section 3.4. We conclude this chapter by comparing in Section 3.5 the classes defined for the purpose of refining the A-Translation and the ones introduced by (Ishihara, 2000) in order to determine which classical formulas are provable in constructive logic.

3.1

The Double Negation Translation

A first step towards identifying the computational content in classical proofs consists in translating them into constructively valid ones. In order to achieve this, variants of the Double-Negation Translation have been pro- posed in the literature. The first such translation, also called the Nega- tive Translation, has been introduced (independently) by (G¨odel, 1933) and (Gentzen, 1936). The method consists in double negating all atomic formu- las, a process by which also the translation of classical principles become constructively valid.

Definition 3.1. G¨odel-Gentzen’s Negative Translation ·g _{is given by}

⊥g :=⊥, ψg :=¬¬ψ, ψ atomic (ψ1∧ψ2)g :=ψ1g∧ψ g 2, (ψ1 →ψ2)g :=ψ₁g→ψ₂g, (∀xψ)g :=∀xψg.

Lemma 3.1. For any formula A of NAω,

`m(Stab⊥A)g and `m(efq⊥A)g.

Proof. By induction on A. Details in (Schwichtenberg and Wainer, 2011).

This translation preserves the classical validity of the formulas, as shown by the following theorem. More importantly, as a consequence of Lemma 3.1 the double-negated formulas becomeNAω-valid.

Theorem 3.1. For any formula A of NAω we have

(a) `cA↔Ag,

(b) Γ`cA iff Γg `m Ag, where Γg:={Bg |for all B∈Γ}.

Proof. By structural induction on A for (a) and induction on the derivation for (b). The latter follows for each rule from the induction hypothesis using again the same rule, because the Negative Translation acts as a homomor- phism for the connectives ofNAω (Troelstra and Schwichtenberg, 2000).

However, in the case whenA is a Π0₂-formula, the translation does not help us recover the constructive meaning of the existential quantifier. As pointed out in the previous chapter (see Section 2.2 for an explanation on the weak operators), a classical proof of ˜∃xAconsists in deriving a contradiction

from the assumption ∀x.A → ⊥. It is in general not easy to read of the

means for constructing x, so the aim is to translate this proof in such a way that the information on how to obtain the witness for x is recovered. The following section presents a solution to this problem.