Maintaining a Library of Formal Mathematics

Doorn, Floris van; Ebner, Gabriel; Lewis, Robert Y.

doi:10.1007/978-3-030-53518-6_16

Cited by 18 publications

(17 citation statements)

References 14 publications

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“…Our intent in formalizing Geometric Algebra is for our formalization to not only interface well with mathlib , but also for certain portions to ultimately end up a part of it. Contributing code into mathlib ensures its ongoing maintenance [11], and has numerous advantages: community review by Lean experts, automated detection of bad practices by software tools, and generated documentation published to the web. Perhaps most important though is that the community as a whole takes on the responsibility of keeping our contributions compatible with the rest of mathlib as it evolves.…”

Section: Lean's Mathematical Library Mathlibmentioning

confidence: 99%

Formalizing Geometric Algebra in Lean

Wieser,

Song

2021

Preprint

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This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib . As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general.We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis.We go on to show how operations and structure such as involutions, versors, and the Z2-grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and N -grading, and some of the gaps in mathlib that currently make this challenging.

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Section: Lean's Mathematical Library Mathlibmentioning

confidence: 99%

Formalizing Geometric Algebra in Lean

Wieser,

Song

2021

Preprint

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“…), its type, and any documentation associated with it. The natural language description in this output is taken from the same source as the mathlib API documentation [18]. By default, GetLeanInfo returns a structure whose fields are strings, as this is most convenient to print and display.…”

Section: Applicationsmentioning

confidence: 99%

A bi-directional extensible interface between Lean and Mathematica

Lewis¹,

Wu²

2021

Preprint

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We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems.We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also use Mathematica as an untrusted oracle to guide proof search in the proof assistant and interact with a Mathematica notebook from within a Lean session. In the other direction, we import and process Lean declarations from within Mathematica. The proof assistant library serves as a database of mathematical knowledge that the CAS can display and explore.

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“…Mathlib is also the foundation for the Perfectoid Spaces in Lean project [1], and the Liquid Tensor challenge [11] posed by the renowned mathematician Peter Scholze. Mathlib contains not only mathematical objects but also Lean metaprograms that extend the system [5]. Some of these metaprograms implement nontrivial proof automation, such as a ring theory solver and a decision procedure for Presburger arithmetic.…”

Section: Introductionmentioning

confidence: 99%

“…Lean metaprograms in mathlib also extend the system by adding new top-level command and features not related to proof automation. For example, it contains a package of semantic linters that alert users to many commonly made mistakes [5]. Lean 3 metaprograms have been also instrumental in building standalone applications, such as a SQL query equivalence checker [3].…”

Section: Introductionmentioning

confidence: 99%

The Lean 4 Theorem Prover and Programming Language

Moura

Ullrich

2021

Lecture Notes in Computer Science

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Lean 4 is a reimplementation of the Lean interactive theorem prover (ITP) in Lean itself. It addresses many shortcomings of the previous versions and contains many new features. Lean 4 is fully extensible: users can modify and extend the parser, elaborator, tactics, decision procedures, pretty printer, and code generator. The new system has a hygienic macro system custom-built for ITPs. It contains a new typeclass resolution procedure based on tabled resolution, addressing significant performance problems reported by the growing user base. Lean 4 is also an efficient functional programming language based on a novel programming paradigm called functional but in-place. Efficient code generation is crucial for Lean users because many write custom proof automation procedures in Lean itself.

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Maintaining a Library of Formal Mathematics

Cited by 18 publications

References 14 publications

Formalizing Geometric Algebra in Lean

Formalizing Geometric Algebra in Lean

A bi-directional extensible interface between Lean and Mathematica

The Lean 4 Theorem Prover and Programming Language

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