Lean is an open-source proof assistant language that supports formal verification of mathematical proofs and computer programs. It was developed by [[Leonardo de Moura]] and his team at Microsoft Research. Lean is based on a dependent type theory called the Calculus of Inductive Constructions. It has a rich type system that allows users to express complex mathematical concepts and encode precise specifications. This makes Lean suitable for formalizing mathematics, verifying software correctness, and conducting formal reasoning. One of the notable features of Lean is its support for dependent types. Dependent types allow types to depend on values, enabling more expressive and precise specifications. This feature makes it possible to encode intricate mathematical structures in Lean, such as dependent pairs, dependent functions, and inductive families. Lean provides an interactive environment where users can write proofs incrementally. It offers a tactic-based system where users can apply pre-defined tactics to guide the proof process automatically. Additionally, Lean supports high-level proof automation through the use of proof search procedures. Lean also includes powerful [[metaprogramming]] capabilities that enable users to extend the language itself. It allows for the definition of new tactics and automated proof procedures, making it possible to customize the proof assistant according to specific needs. Furthermore, Lean has a growing library of formalized mathematics called "mathlib." Mathlib covers various areas of mathematics like algebra, analysis, topology, number theory, and more. This library serves as a valuable resource for researchers and users interested in formalizing mathematics using Lean. # Tools and Resources Can you tell me some GPT related resource for learning LEAN the proof assistant and programming language? Sure! Here are some GPT-related resources for learning Lean, the proof assistant and programming language: 1. The official Lean website: leanprover.github.io provides comprehensive documentation, tutorials, and examples to get started with Lean. 2. The Lean Community website: leanprover.github.io/community offers a collection of community-contributed resources such as blogs, videos, and libraries that can help you learn Lean. 3. The Lean tutorial repository on GitHub: github.com/leanprover/tutorial is a great resource for beginners to learn the basics of Lean through interactive exercises and examples. 4. The "Theorem Proving in Lean" book: leanprover.github.io/theorem_proving_in_lean is an excellent resource for learning how to use Lean as a proof assistant. It covers both foundational concepts and advanced topics. 5. The "Programming Language Foundations in Agda" book: plfa.github.io is not specifically about Lean, but it provides a similar approach to teaching programming language theory using the Agda proof assistant. Many concepts covered in this book are applicable to Lean as well. 6. Online course on "Logic and Proof" by Imperial College London: imperial.ac.uk/computing/people/academic/p.dawson/logic-and-proof covers formal logic and proof techniques using the Lean proof assistant as a teaching tool. 7. The Lean Zulip chat platform: [leanprover.zulipchat.com](http://leanprover.zulipchat.com) is an active community space where you can ask questions, engage in discussions, and seek guidance from experienced users of the Lean proof assistant. ## Answer from Bing - [**Lean’s official website**: You can find the official documentation and tutorials for Lean on their website](https://en.wikipedia.org/wiki/Lean_%28proof_assistant%29) [1](https://en.wikipedia.org/wiki/Lean_%28proof_assistant%29). They also have a community forum where you can ask questions and get help from other users. - [**Lean on GitHub**: Lean is an open-source project hosted on GitHub](https://en.wikipedia.org/wiki/Lean_%28proof_assistant%29) [1](https://en.wikipedia.org/wiki/Lean_%28proof_assistant%29). You can find the source code, documentation, and other resources related to Lean on their GitHub page. - [**LeanDojo**: LeanDojo is a research project that aims to improve theorem proving with retrieval-augmented language models](https://arxiv.org/abs/2306.15626) [2](https://arxiv.org/abs/2306.15626). It contains fine-grained annotations of premises in proofs, providing valuable data for premise selection: a key bottleneck in theorem proving. - [**Generative language modeling for automated theorem proving**: This paper presents an automated prover and proof assistant, GPT-f, for the Metamath formalization language](https://openai.com/research/generative-language-modeling-for-automated-theorem-proving) [3](https://openai.com/research/generative-language-modeling-for-automated-theorem-proving). It found new short proofs that were accepted into the main Metamath library. - [**Proof Artifact Co-training for Theorem Proving with Language Models**: This paper applies a methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date](https://arxiv.org/abs/2102.06203) [4](https://arxiv.org/abs/2102.06203). It shows that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%. - These resources should provide you with a solid foundation for learning both GPT-related topics and how to use the Lean proof assistant effectively. # Conclusion Overall, Lean provides a modern and powerful platform for formal reasoning and verification. Its expressive type system, interactive environment, customizable features, and growing library make it an attractive choice for researchers in mathematics and computer science seeking rigorous formalization and verification capabilities. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Lean") or contains(title, "Lean") ```