Next Steps

This book introduces the very basics of functional programming in Lean, including a tiny amount of interactive theorem proving. Using dependently-typed functional languages like Lean is a deep topic, and much can be said. Depending on your interests, the following resources might be useful for learning Lean 4.

Learning Lean

Lean 4 itself is described in the following resources:

  • Theorem Proving in Lean 4 is a tutorial on writing proofs using Lean.
  • The Lean 4 Manual provides a reference for the language and its features. At the time of writing, it is still incomplete, but it describes many aspects of Lean in greater detail than this book.
  • How To Prove It With Lean is a Lean-based accompaniment to the well-regarded textbook How To Prove It that provides an introduction to writing paper-and-pencil mathematical proofs.
  • Metaprogramming in Lean 4 provides an overview of Lean's extension mechanisms, from infix operators and notations to macros, custom tactics, and full-on custom embedded languages.
  • Functional Programming in Lean may be interesting to readers who enjoy jokes about recursion.

However, the best way to continue learning Lean is to start reading and writing code, consulting the documentation when you get stuck. Additionally, the Lean Zulip is an excellent place to meet other Lean users, ask for help, and help others.

The Standard Library

Out of the box, Lean itself includes a fairly minimal library. Lean is self-hosted, and the included code is just enough to implement Lean itself. For many applications, a larger standard library is needed.

std4 is an in-progress standard library that includes many data structures, tactics, type class instances, and functions that are out of scope for the Lean compiler itself. To use std4, the first step is to find a commit in its history that's compatible with the version of Lean 4 that you're using (that is, one in which the lean-toolchain file matches the one in your project). Then, add the following to the top level of your lakefile.lean, where COMMIT_HASH is the appropriate version:

require std from git
  "" @ "COMMIT_HASH"

Mathematics in Lean

Most resources for mathematicians are written for Lean 3. A wide selection are available at the community site. To get started doing mathematics in Lean 4, it is probably easiest to participate in the process of porting the mathematics library mathlib from Lean 3 to Lean 4. Please see the mathlib4 README for further information.

Using Dependent Types in Computer Science

Coq is a language that has a lot in common with Lean. For computer scientists, the Software Foundations series of interactive textbooks provides an excellent introduction to applications of Coq in computer science. The fundamental ideas of Lean and Coq are very similar, and skills are readily transferable between the systems.

Programming with Dependent Types

For programmers who are interested in learning to use indexed families and dependent types to structure programs, Edwin Brady's Type Driven Development with Idris provides an excellent introduction. Like Coq, Idris is a close cousin of Lean, though it lacks tactics.

Understanding Dependent Types

The Little Typer is a book for programmers who haven't formally studied logic or the theory of programming languages, but who want to build an understanding of the core ideas of dependent type theory. While all of the above resources aim to be as practical as possible, The Little Typer presents an approach to dependent type theory where the very basics are built up from scratch, using only concepts from programming. Disclaimer: the author of Functional Programming in Lean is also an author of The Little Typer.