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Lean 4: Programming, Proof, and Formal Verification

A Deep Guide to Functional Programming and Interactive Theorem Proving

Lean 4: Programming, Proof, and Formal Verification
This book is 100% completeLast updated on 2026-10-02
Steve Publications

Go beyond the basics of Lean 4. This practical deep dive explores functional programming, dependent types, theorem proving and formal verification. Learn how Lean works under the hood, build reliable proofs and automation, and tackle larger formal projects with confidence.

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About the Book

This book provides a thorough, technically precise exploration of Lean 4 as a functional programming language and interactive theorem prover. It is intended for software engineers, mathematicians and researchers who want to understand Lean at a deep level: its type-theoretic foundations, its architecture, its facilities for programming and verification and its ecosystem for building large formal projects. By the end of this book you will be able to write executable Lean programs, develop machine-checked proofs, construct custom automation and language extensions and organize substantial formal verification projects. The material assumes familiarity with programming and basic mathematics but no prior background in type theory, dependent types, or theorem provers.

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Steve Publications

Steve is a technology professional with more than 20 years of experience spanning software development, server infrastructure, cybersecurity, vulnerability research and reverse engineering. His work focuses on understanding complex systems, uncovering security weaknesses and developing practical solutions that make technology more secure, reliable and effective.

Through Steve Publications, he collaborates with experienced researchers, engineers and industry experts to create practical, technically rigorous books on various topics not limited to self-improvement, software development, cybersecurity, cloud computing, DevOps, artificial intelligence and enterprise technologies. His mission is to make complex technical subjects accessible through clear explanations, real-world examples and hands-on exercises that help IT professionals, students and technology enthusiasts build valuable, practical skills.

Steve Publications embraces AI as a supporting tool for translation, proofreading and editorial refinement, not as a substitute for original research, technical expertise, or hands-on experience. Some contributors choose to publish under pseudonyms to protect their privacy or maintain professional boundaries.

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Contents

Table of Contents

A Deep Guide to Functional Programming and Interactive Theorem Proving

Chapter 1: Why Lean

  1. The Problem of Correctness in Software and Mathematics
  2. Unifying Programs and Proofs
  3. A Brief History of Lean and Its Design Philosophy
  4. What Lean Can Do: Use Cases and Examples
  5. How to Read This Book

Chapter 2: Getting Started with Lean 4

  1. Installing Lean and Lake
  2. Your First Lean Project
  3. Editors, LSP and the Lean Environment
  4. Running and Interacting with Lean
  5. The Lake Build System and Package Manager
  6. Project Structure and Configuration

Chapter 3: The Lean Syntax and Basic Types

  1. Modules, Namespaces and Imports
  2. Identifiers and Declarations
  3. Basic Types and Literals
  4. Variables and Local Definitions
  5. Functions and Applications
  6. Pattern Matching

Chapter 4: Dependent Types and the Foundations

  1. From Simple Types to Dependent Types
  2. The Calculus of Constructions
  3. Universes and Polymorphism
  4. Pi Types and Dependent Functions
  5. Dependent Pairs and Sigma Types
  6. Definitional Equality and Conversion

Chapter 5: Inductive Types and Data Structures

  1. Defining Inductive Types
  2. Pattern Matching and Recursion
  3. Lists, Vectors and Indexed Types
  4. Inductive Families
  5. Algebraic Data Types in Practice
  6. Recursive and Well-founded Functions

Chapter 6: Functional Programming in Lean

  1. Higher-order Functions and Composition
  2. Type Classes and Generic Programming
  3. Structures and Records
  4. Monads and Do Notation
  5. Standard Library Collections
  6. Performance and Mutation

Chapter 7: Propositions as Types

  1. Prop and Type: Two Universes of Discourse
  2. Logical Connectives as Types
  3. Quantifiers as Dependent Types
  4. Proof Terms and Proof Checking
  5. Classical and Constructive Logic
  6. Decidability and Computation

Chapter 8: Writing Proofs in Lean

  1. Term-Style Proofs
  2. Tactic Mode Basics
  3. Core Tactics: intro, apply, exact, refine
  4. Breaking Down Goals: constructor, cases, induction
  5. Structuring Proofs: have, show, calc
  6. When to Use Tactics Versus Terms

Chapter 9: Equality and Rewriting

  1. Definitional and Propositional Equality
  2. The Equality Type and Its Eliminator
  3. The rw and subst Tactics
  4. Calc Blocks for Chain Reasoning
  5. Function Extensionality and Congruence
  6. Quotients and Equivalence Relations

Chapter 10: Automation and Simplification

  1. The Simplifier: simp and simp_lemmas
  2. Rewrite Tactics and Strategies
  3. Decision Procedures: decide and native_decide
  4. Arithmetic Automation
  5. Proof Search: aesop and Other Tools
  6. When Automation Helps and Hurts

Chapter 11: The Lean Type System Deep Dive

  1. Lean’s Type Theory: A Formal View
  2. Inductive Type Recursors and Eliminators
  3. Computation Rules and Reduction
  4. Termination and Well-founded Recursion
  5. Universe Levels and Size Constraints
  6. Type Checking and Soundness

Chapter 12: Implicit Arguments and Type Class Resolution

  1. Implicit Arguments and Inference
  2. Type Classes: Concept and Mechanism
  3. Instance Resolution and Search
  4. Coercions and Canonical Constructions
  5. Type Class Hierarchies and Algebra
  6. Debugging Type Class Failures

Chapter 13: The Lean Architecture

  1. The Trusted Computing Base
  2. From Source to Proof Term: The Elaboration Pipeline
  3. The Parser and Syntax Trees
  4. The Kernel: Proof Checking
  5. The Compiler and Runtime
  6. Untrusted but Useful: Automation and Tooling

Chapter 14: Metaprogramming and Extensibility

  1. Why Extend Lean?
  2. Syntax Extensions and Custom Notation
  3. Hygienic Macros
  4. Elaborators and Commands
  5. Tactic Implementation
  6. Building Domain-Specific Languages

Chapter 15: Formal Verification and Verified Programming

  1. Specification: Pre-, Post- and Invariants
  2. Verifying Algorithms and Data Structures
  3. Functional Correctness and Properties
  4. Verified Compilers and Translations
  5. Cryptographic and Security Properties
  6. Case Study: A Verified Data Structure

Chapter 16: Mathematical Formalization with Mathlib

  1. Lean Core, Stdlib and Mathlib
  2. Working with Existing Formalizations
  3. Algebra and Type Class Hierarchies
  4. Analysis and Topology
  5. Discrete Mathematics and Combinatorics
  6. Engineering Large Formalizations

Chapter 17: Practical Engineering and Best Practices

  1. Project Organization at Scale
  2. Namespace and Module Design
  3. Performance Engineering
  4. Documentation and Readability
  5. Testing and Debugging Proofs
  6. Common Pitfalls and Solutions

Chapter 18: The Lean Ecosystem and Beyond

  1. Lean 4 Version History and Releases
  2. Tooling and Integrations
  3. Community and Contribution
  4. Interoperability and FFIs
  5. Real-World Deployments
  6. Where Lean Is Headed

Conclusion

References

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