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A sequent calculus with dependent types for classical arithmetic
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In: LICS 2018 - 33th Annual ACM/IEEE Symposium on Logic in Computer Science ; https://hal.inria.fr/hal-01703526 ; LICS 2018 - 33th Annual ACM/IEEE Symposium on Logic in Computer Science, Jul 2018, Oxford, United Kingdom. pp.720-729, ⟨10.1145/3209108.3209199⟩ (2018)
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Abstract:
International audience ; In a recent paper, Herbelin developed a calculus dPA$^\omega$ in which constructive proofs for the axioms of countable and dependent choices could be derived via the encoding of a proof of countable universal quantification as a stream of it components. 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 $\mathbb{N} \to A$ into streams $(a_0,a_1,\ldots)$) and of lazy evaluation with sharing (for these coinductive objects).Building on previous works, we introduce in this paper a variant of dPA$^\omega$ 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, 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.
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Keyword:
[INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO]; ACM: F.: Theory of Computation/F.3: LOGICS AND MEANINGS OF PROGRAMS/F.3.2: Semantics of Programming Languages/F.3.2.2: Operational semantics; ACM: F.: Theory of Computation/F.3: LOGICS AND MEANINGS OF PROGRAMS/F.3.3: Studies of Program Constructs/F.3.3.0: Control primitives; ACM: F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.1: Mathematical Logic/F.4.1.2: Lambda calculus and related systems; ACM: F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.1: Mathematical Logic/F.4.1.7: Proof theory; classical arithmetic; classical realizability; Curry-Howard; dependent choice; dependent types; sequent calculus; side effects
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URL: https://hal.inria.fr/hal-01703526v2/file/dlpaw.pdf
https://hal.inria.fr/hal-01703526v2/document
https://doi.org/10.1145/3209108.3209199
https://hal.inria.fr/hal-01703526
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