Étienne Miquey (Gallinette, INRIA), A constructive proof of dependent choice in classical arithmetic via memoization



In 2012, Herbelin developed a calculus (dPA^ω) in which constructive proofs for the axioms of countable and dependent choices can be derived via the memoization of choice functions However, the property of normalization (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalization is due to the simultaneous presence of dependent dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type ℕ→A into streams (a₀,a₁,…)) and of lazy evaluation with sharing (for these coinductive objects). Building on previous works, we introduce a variant of dPA^ω presented as a sequent calculus. On the one hand, we take advantage of a variant of Krivine classical realizability we developed to prove the normalization of classical call-by-need. On the other hand, we benefit of dL[tp], a classical sequent calculus with dependent types in which type safety is ensured using delimited continuations together with a syntactic restriction. By combining the techniques developed in these papers, we manage to define a realizability interpretation à la Krivine of our calculus that allows us to prove normalization and soundness. This talk will go over the whole process, starting from Herbelin’s calculus dPA^ω until our introduction of its sequent calculus counterpart dLPA^ω.