[Paper Review] A preliminary univalent formalization of the p-adic numbers

Álvaro Pelayo, Vladimir Voevodsky+1 more|arXiv (Cornell University)|Feb 5, 2013

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TL;DR

This paper presents a constructive, univalent formalization of the p-adic numbers in Coq using homotopy type theory, defining p-adic integers as a quotient of formal power series over ℤ and p-adic numbers as the field of fractions of this ring. The construction employs apartness relations and Heyting fields to handle undecidable equality, providing a foundational framework for future algorithmic development in p-adic integrable systems.

ABSTRACT

In this paper we give a preliminary formalization of the p-adic numbers, in the context of the second author's univalent foundations program. We also provide the corresponding code verifying the construction in the proof assistant Coq. Because work in the univalent setting is ongoing, the structure and organization of the construction of the p-adic numbers we give in this paper is expected to change as Coq libraries are more suitably rearranged, and optimized, by the authors and other researchers in the future. So our construction here should be deemed as a first approximation which is subject to improvements.

Motivation & Objective

• To develop a constructive, univalent formalization of the p-adic numbers within homotopy type theory using Coq.
• To provide a foundation for algorithmic construction and numerical analysis in p-adic integrable systems.
• To formalize p-adic integers as a quotient of ℤ[[X]] and p-adic numbers as their field of fractions.
• To handle undecidable equality in ℤ[[X]] via apartness relations and constructively defined integral domains and Heyting fields.
• To lay the groundwork for future formalization of p-adic integrable systems and inverse spectral problems in quantum systems.

Proposed method

• Construct p-adic integers ℤₚ as a quotient of the ring of formal power series ℤ[[X]] over the integers.
• Use the univalent foundations and Coq’s type theory to formalize algebraic structures with homotopy-theoretic coherence.
• Define apartness relations to replace equality in constructive settings where equality is undecidable.
• Formalize Heyting fields as the constructive analog of fields, ensuring compatibility with constructive logic.
• Construct the field of fractions of ℤₚ as the p-adic numbers ℚₚ using a constructive version of the field of fractions construction.
• Implement all constructions in Coq, with verified code available alongside the paper, using the univalent foundations library.

Experimental results

Research questions

• RQ1 How can the p-adic numbers be constructively formalized in a univalent, homotopy type-theoretic framework?
• RQ2 What algebraic structures are needed to handle undecidable equality in formal power series rings constructively?
• RQ3 Can the field of fractions construction be adapted to Heyting fields to define ℚₚ without classical choice?
• RQ4 How can this formalization support the development of algorithms for p-adic integrable systems?
• RQ5 What is the role of apartness relations in ensuring constructive validity of algebraic operations in ℤₚ and ℚₚ?

Key findings

• The p-adic integers ℤₚ are successfully formalized as a quotient of ℤ[[X]], with a constructive proof of integrality and apartness structure.
• The field of fractions construction is adapted to Heyting fields, enabling ℚₚ to be defined as a Heyting field without classical logic.
• The formalization is implemented in Coq using the univalent foundations library, with verified code available as supplementary material.
• The construction avoids classical assumptions by relying on apartness and decidable properties in the context of constructive algebra.
• The approach provides a first approximation to a univalent formalization of p-adic numbers, expected to evolve with future library refinements.
• The formalization supports the long-term goal of extracting algorithms for inverse spectral problems in p-adic quantum integrable systems.