The document provides an introduction to using the Lean theorem prover for interactive theorem proving. It covers fundamental concepts, tactics, and techniques for verifying logical statements and systems. The guide explains how to use Lean for proving properties about functional and imperative programming languages, mathematics, and other domains. It emphasizes the importance of hands-on practice and understanding the underlying logic of proof assistants. The document also introduces key Lean tactics such as `apply`, `exact`, `assumption`, and `cc`, and discusses the use of Hoare logic for specifying and verifying program correctness. The guide is designed to help readers develop practical skills in formal verification and prepare them for applying these techniques in academic or industrial settings.

The document provides an introduction to Lean, a dependent type theory implementation. It covers Lean's use as a specification, programming, assertion, proof, automation, and metaprogramming language. The text explains key concepts like Lean's kernel, tactics, and automation, and provides examples of defining types, writing proofs, and using automation for reasoning. It also discusses Lean's role in theorem proving and programming, including recursive definitions and inductive types.

文档《Programming in Lean Release 3.4.2》系统介绍了Lean编程语言的基础知识和核心概念。Lean基于归纳构造演算（CIC）的逻辑框架，支持函数式编程和类型理论。文档详细讲解了基本类型（如自然数、整数、列表和布尔值）、函数定义、单子的使用（如option、list、state和IO单子）以及输入输出操作。内容涵盖编程基础、高级主题（如策略和自动化工具的编写），并展示了如何在Lean环境中进行交互式编程和数学证明。

文档详细介绍了Lean的发展历程及其核心理论基础。从2013年的Lean 1到202x年的Lean 4，Lean经历了多次重大更新，逐步成为一个功能强大的编程语言和定理证明器。文档重点阐述了依赖类型理论的优势，包括内置的计算解释、统一的证明和术语表示，以及减少代码重复的能力。此外，还介绍了Lean 4的实现特点，如通过Lean本身实现的编译器、白色盒自动化、类型推断和决策过程等技术细节。

The document provides a comprehensive reference manual for Lean 3.3.0, detailing its usage across various editors like VSCode and Emacs, package management, lexical structure, expressions, and declarations. It covers essential commands, tactics, and configurations, offering guidance on setting up projects and integrating Lean with different development environments.

《Theorem Proving in Lean Release 3.23.0》是一本关于使用Lean系统进行定理证明的教程。文档介绍了Lean项目，由Leonardo de Moura于2013年发起，采用Apache 2.0许可证，支持网页和本地安装两种使用方式。教程基于依赖类型理论，解释了如何定义数学对象、表达断言以及使用该语言进行证明。书中详细讨论了命题与证明、量词与等式、策略等核心内容，并通过示例展示了如何在Lean中证明命题逻辑的有效命题。

文档主要介绍了《Lean 4》的实现及其核心特性。Lean 4 是在 Lean 中实现的，用户可以自定义所有模块，并且保持逻辑一致性。编译器生成 C 代码，支持混合编译和解释代码。文档还提到了 Lean 4 的可扩展性，用户可以通过 Lean 访问内部功能，如类型推理、统一器和简化器等。此外，文档还展示了 Lean 4 在数学证明和应用领域的潜力，例如在完美空间和 IMO 大挑战中的应用。

文档《Lean 2 Quick Reference》提供了Lean 2编程语言的快速参考指南，涵盖了各种命令和战术的使用方法。内容包括检查和评估表达式、打印信息、设置选项、目标管理、组合战术、等价推理、归纳与情况处理以及特殊用途战术等。文档还介绍了与Lean集成的Emacs模式命令和Unicode符号的使用方法。