The Lean mathematical library

This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here

LLMSTEP: LLM proofstep suggestions in Lean

We present LLMSTEP, a tool for integrating a language model into the Lean proof assistant. LLMSTEP is a Lean 4 tactic that sends a user's proof state to a server hosting a language model. The language model generates suggestions, which are checked in Lean and displayed to a user in their development environment. We provide a baseline language model, along with code for fine-tuning and evaluation to support further development. We provide server implementations that run on CPU, a CUDA GPU, or a Google Colab notebook, as a step towards fast, effective language model suggestions for any user

Scalar actions in Lean's mathlib

Scalar actions are ubiquitous in mathematics, and therefore it is valuable to be able to write them succinctly when formalizing. In this paper we explore how Lean 3's typeclasses are used by mathlib for scalar actions with examples, illustrate some of the problems which come up when using them such as compatibility of actions and non-definitionally-equal diamonds, and note how these problems can be solved. We outline where more work is needed in mathlib in this area.Comment: 6 pages, 2 figures. For associated conference presentation slides, see https://eric-wieser.github.io/fmm-202

The Lean 4 Theorem Prover and Programming Language

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

Multiple inheritance hazards in algebraic typeclass hierarchies

Abstract algebra provides a large hierarchy of properties that a collection of objects can satisfy, such as forming an abelian group or a semiring. These classifications can arranged into a broad and typically acyclic directed graph. This graph perspective encodes naturally in the typeclass system of theorem provers such as Lean, where nodes can be represented as structures (or records) containing the requisite axioms. This design inevitably needs some form of multiple inheritance; a ring is both a semiring and an abelian group. In the presence of dependently-typed typeclasses that themselves consume typeclasses as type-parameters, such as a vector space typeclass which assumes the presence of an existing additive structure, the implementation details of structure multiple inheritance matter. The type of the outer typeclass is influenced by the path taken to resolve the typeclasses it consumes. Unless all these paths are considered judgmentally equal, this is a recipe for disaster. This paper provide a concrete explanation of how these situations arise (reduced from real examples in mathlib), compares implementation approaches for multiple inheritance by whether judgmental equality is preserved, and outlines solutions to the problems discovered.Comment: 11 pages, 3 figure