Validating Mathematical Structures

Sakaguchi, Kazuhiko

doi:10.48550/arxiv.2002.00620

Cited by 2 publications

(3 citation statements)

References 23 publications

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“…Another approach to building the algebraic hierarchy in Coq is using packed classes [Garillot et al, 2009] which mainly solves the problem of multiple inheritance. This approach has been extended in [Cohen et al, 2020] and [Sakaguchi, 2020] to overcome the complexity of using it to build and maintain the hierarchy. [Cohen et al, 2020] creates an ELPI [Dunchev et al, 2015;Tassi, 2018] plugin to Coq introducing a language for building the algebraic hierarchy whose expressions are elaborated into packed classes.…”

Section: Discussionmentioning

confidence: 99%

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Leveraging the Information Contained in Theory Presentations

Carette

Farmer

Sharoda

2020

Lecture Notes in Computer Science

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Building a large library of mathematical knowledge is a complex and labour-intensive task. By examining current libraries of mathematics, we see that the human effort put in building them is not entirely directed towards tasks that need human creativity.Instead, a non-trivial amount of work is spent on providing definitions that could have been mechanically derived.In this work, we propose a generative approach to library building, so definitions that can be automatically derived are computed by meta-programs. We focus our attention on libraries of algebraic structures, like monoids, groups, and rings. These structures are highly inter-related and their commonalities have been well-studied in universal algebra. We use theory presentation combinators to build a library of algebraic structures. Definitions from universal algebra and programming languages meta-theory are used to derive library definitions of constructions, like homomorphisms and term languages, from algebraic theory presentations. The result is an interpreter that, given 227 theory expressions, builds a library of over 5000 definitions. This library is, then, exported to Agda and Lean.iii To my family, You are my greatest blessing iv I would like to express my sincere thanks to my supervisors Dr. Jacques Carette and Dr. William Farmer for their continuous support to my learning journey. Your expertise and feedback were invaluable in shaping this research direction and throughout my studies. I learned from you a lot about how to do research and communicate it. I

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Section: Discussionmentioning

confidence: 99%

“…it makes it possible to add new structures and connections between them, while keeping the older ones. [Sakaguchi, 2020] provides invariants and algorithms to validate the structure of the library. Type classes has been used to build the algebraic hierarchy in Coq and Lean.…”

Section: Discussionmentioning

confidence: 99%

Leveraging the Information Contained in Theory Presentations

Carette

Farmer

Sharoda

2020

Lecture Notes in Computer Science

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“…Much has been written about different methods for defining and maintaining hierarchies of algebraic structures in proof assistants [12,19,24,27]. In some sense, our project is orthogonal to this literature: we work at a single fixed point within mathlib's type class hierarchy.…”

Section: Concluding Thoughtsmentioning

confidence: 99%

Formalizing the Ring of Witt Vectors

Commelin,

Lewis

2020

Preprint

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The ring of Witt vectors W over a base ring is an important tool in algebraic number theory and lies at the foundations of modern -adic Hodge theory. W has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic > 1, and it can be used more specifically to construct from a finite field containing Z/ Z the corresponding unramified field extension of the -adic numbers Q (which is unique up to isomorphism).We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime , the ring of Witt vectors over Z/ Z is isomorphic to the ring of -adic integers Z . We develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature.

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Validating Mathematical Structures

Cited by 2 publications

References 23 publications

Leveraging the Information Contained in Theory Presentations

Leveraging the Information Contained in Theory Presentations

Formalizing the Ring of Witt Vectors

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