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Lean from First Proofs to Formalized Mathematics

A visual, complete learning path from programming fundamentals to expert theorem proving in Lean

Read each section in order. Every title can be opened as a TheoryTrace document.

Cover1
Copyright2
How to read this book3
Introduction4
Chapter 1: What Lean Is and How Formalization Changes Mathematics5
Chapter 2: Setting Up a Lean Working Environment6
Chapter 3: The First Lean Terms, Types, and Commands7
Chapter 4: Propositions as Types and Proofs as Terms8
Chapter 5: The Calculus of Dependent Types9
Chapter 6: Quantifiers, Equality, and Rewriting10
Chapter 7: Defining Data with Inductive Types11
Chapter 8: Recursive Definitions and Termination12
Chapter 9: Pattern Matching, Functions, and Computation13
Chapter 10: Structures, Records, and Mathematical Objects14
Chapter 11: Type Classes and Instance Synthesis15
Chapter 12: Notation, Syntax, Macros, and Elaboration16
Chapter 13: Namespaces, Modules, Imports, and Large Projects17
Chapter 14: The Tactic Mode Mental Model18
Chapter 15: Core Interactive Tactics for Logic and Equality19
Chapter 16: Simplification, Normalization, and Automation20
Chapter 17: Attributes, Declarations, and the Lean Environment21
Chapter 18: Coercions, Subtypes, Quotients, and Extensionality22
Chapter 19: Universes, Sorts, and Foundational Boundaries23
Chapter 20: Classical Reasoning and Constructive Reasoning24
Chapter 21: Searching, Reading, and Reusing Mathlib25
Chapter 22: Sets, Relations, Functions, and Basic Discrete Mathematics26
Chapter 23: Algebraic Hierarchies in Lean27
Chapter 24: Order, Lattice, and Topological Interfaces28
Chapter 25: Analysis and Calculational Proof in Mathlib29
Chapter 26: Induction, Recursion, and Well-Founded Proofs at Scale30
Chapter 27: Metaprogramming Foundations in Lean31
Chapter 28: Writing Custom Tactics and Proof Automation32
Chapter 29: Software Engineering for Formalization Projects33
Chapter 30: Debugging Lean Proofs and Understanding Error Messages34
Chapter 31: Designing Visual Demonstrations for Every Lean Feature35
Chapter 32: Case Study I — Formalizing a Complete Discrete Mathematics Development36
Chapter 33: Case Study II — Building an Algebraic Theory37
Chapter 34: Case Study III — A First Formal Analysis Project38
Chapter 35: From Competence to Mastery39
Conclusion40

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