🤖 AI Summary
This work presents the first assumption-free formalization of Cauchy real numbers in Cubical Agda, circumventing reliance on the axiom of choice, setoid bookkeeping, or explicit universe-level management—issues that commonly hinder constructive real number constructions in intuitionistic mathematics. Building upon the higher inductive-inductive types introduced in Homotopy Type Theory, the construction leverages Cubical Agda’s native support for higher inductive types to yield a fully type-checked, non-vacuous, and postulate-free development of the reals. This approach not only resolves longstanding challenges related to redundancy and universe complexity but also establishes a robust foundation for machine-verified constructive analysis.
📝 Abstract
Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping overhead on every definition and theorem, and Dedekind cuts force cumbersome universe-level tracking in predicative type theory. The Homotopy Type Theory (HoTT) book presents an alternative construction of the Cauchy real numbers as a higher inductive-inductive type family, avoiding all three compromises. We formalize the HoTT book reals in Cubical Agda, a proof assistant whose native support for higher inductive types allows the construction to be expressed directly. The code type-checks without postulates or holes, providing a foundation for further machine-assisted work in constructive analysis.