Gentle Introduction to Dependent Types with Idris
Gentle Introduction to Dependent Types with Idris
About the Book
You can purchase a printed edition of this book on Amazon
Dependent types are a powerful concept that allows us to write proof-carrying code. Idris is a programming language that supports dependent types. We will learn about the mathematical foundations, and then write correct software and mathematically prove properties about it.
This book aims to be accessible to novices that have no prior experience beyond high school mathematics. Thus, this book is designed to be self-contained.
The reason for writing this book is that I could not find a book that explained how things work, so I had to do a lot of research on the internet through white-papers, forums, and example code in order to come up with a complete picture of what dependent types are and what they are good for.
The first part of this book serves as an introduction to the theory behind Idris, while the second part is a practical introduction to Idris with examples.
Contributions to the book are welcome on GitHub.
About the Author
Table of Contents
- Preface and acknowledgments
- Introduction
-
1. Formal systems
- 1.1. MU puzzle example
-
2. Classical mathematical logic
-
2.1. Hierarchy of mathematical logic and definitions
- 2.1.1. Propositional logic
- 2.1.2. First-order logic
- 2.1.3. Higher-order logic
- 2.2. Set theory abstractions
-
2.3. Substitution and mathematical proofs
- 2.3.1. Proofs by truth tables
- 2.3.2. Three-column proofs
- 2.3.3. Formal proofs
- 2.3.4. Mathematical induction
-
2.1. Hierarchy of mathematical logic and definitions
-
3. Type theory
-
3.1. Lambda calculus
- 3.1.1. Term reduction
- 3.2. Lambda calculus with types
- 3.3. Dependent types
-
3.4. Intuitionistic theory of types
- 3.4.1. Intuitionistic logic
-
3.1. Lambda calculus
-
4. Programming in Idris
-
4.1. Basic syntax and definitions
- 4.1.1. Defining functions
- 4.1.2. Defining and inferring types
- 4.1.3. Anonymous lambda functions
- 4.1.4. Recursive functions
- 4.1.5. Recursive data types
- 4.1.6. Total and partial functions
- 4.1.7. Higher-order functions
- 4.1.8. Dependent types
- 4.1.9. Implicit parameters
- 4.1.10. Pattern matching expressions
- 4.1.11. Interfaces and implementations
- 4.2. Curry-Howard isomorphism
- 4.3. Quantitative Type Theory
-
4.1. Basic syntax and definitions
-
5. Proving in Idris
-
5.1. Weekdays
- 5.1.1. First proof (auto-inference)
- 5.1.2. Second proof (rewrite)
- 5.1.3. Third proof (impossible)
-
5.2. Natural numbers
- 5.2.1. First proof (auto-inference and existence)
- 5.2.2. Second proof (introduction of a new given)
- 5.2.3. Third proof (induction)
- 5.2.4. Ordering
- 5.2.5. Safe division
- 5.2.6. Maximum of two numbers
- 5.2.7. List of even naturals
- 5.2.8. Partial orders
-
5.3. Computations as types
- 5.3.1. Same elements in a list (vector)
- 5.3.2. Evenness of numbers
-
5.4. Trees
- 5.4.1. Depth
- 5.4.2. Map and size
- 5.4.3. Length of mapped trees
-
5.1. Weekdays
- Conclusion
- Further reading
-
Appendices
-
Appendix A: Writing a simple type checker in Haskell
- Evaluator
- Type checker
- Environments
-
Appendix B: Theorem provers
- Metamath
- Simple Theorem Prover
-
Appendix C: IO, Codegen targets, compilation, and FFI
- IO
- Codegen
- Compilation
- Foreign Function Interface
-
Appendix A: Writing a simple type checker in Haskell
- About the author
- Notes
The Leanpub 45-day 100% Happiness Guarantee
Within 45 days of purchase you can get a 100% refund on any Leanpub purchase, in two clicks.
See full terms
Do Well. Do Good.
Authors have earned$10,788,735writing, publishing and selling on Leanpub, earning 80% royalties while saving up to 25 million pounds of CO2 and up to 46,000 trees.
Learn more about writing on Leanpub
Free Updates. DRM Free.
If you buy a Leanpub book, you get free updates for as long as the author updates the book! Many authors use Leanpub to publish their books in-progress, while they are writing them. All readers get free updates, regardless of when they bought the book or how much they paid (including free).
Most Leanpub books are available in PDF (for computers), EPUB (for phones and tablets) and MOBI (for Kindle). The formats that a book includes are shown at the top right corner of this page.
Finally, Leanpub books don't have any DRM copy-protection nonsense, so you can easily read them on any supported device.
Learn more about Leanpub's ebook formats and where to read them
Top Books
- #1
Aprendiendo Git
Miguel Angel Durán GarcíaGit no es complicado... ¡Si lo entiendes! 😜
¿Sientes que sabes usarlo porque has memorizado todos los comandos que necesitas? ¡Pero no entiendes qué hace cada cosa y por qué! Así es normal que, cuando exista un problema, te cueste resolverlo.
¡Con este libro vas a entender de una vez por todas todo lo que es Git y cómo sacarle provecho!
- #2
C++20
Rainer GrimmC++20 is the next big C++ standard after C++11. As C++11 did it, C++20 changes the way we program modern C++. This change is, in particular, due to the big four of C++20: ranges, coroutines, concepts, and modules.
- #3
Introducing EventStorming
Alberto BrandoliniThe deepest tutorial and explanation about EventStorming, straight from the inventor.
- #4
Rector - The Power of Automated Refactoring
Matthias Noback and Tomas VotrubaLearn how to automatically and continuously upgrade and improve your PHP code base
- #5
Atomic Kotlin
Bruce Eckel and Svetlana IsakovaFor both beginning and experienced programmers! From the author of the multi-award-winning Thinking in C++ and Thinking in Java and a Kotlin team member comes a book that breaks concepts into small, easy-to-digest "atoms," along with exercises supported by hints and solutions directly inside IntelliJ IDEA! Full support at www.AtomicKotlin.com.
- #6
Functional Programming Made Easier
Charles ScalfaniA Functional Programming book from beginner to advanced without skipping a single step along the way.
In my 40 years of programming, I've felt that programming books always let me down, especially Functional Programming books. So, I wrote the book I wish I had 5 years ago.
Functional Programming will never be easy, but it can be easier.
- #7
Ansible for DevOps
Jeff GeerlingAnsible is a simple, but powerful, server and configuration management tool. Learn to use Ansible effectively, whether you manage one server—or thousands.
- #8
On Java 8
Bruce EckelBy the author of "Thinking in Java," covering version 8 of the Java programming language. This book is intended for “dedicated beginners.” It doesn’t assume you know anything about programming and takes you through the material step-by-step. However, I assume you are able to figure things out. It’s not a “dummies” or “idiot’s” guide.
- #9
OpenIntro Statistics
David Diez, Christopher Barr, Mine Cetinkaya-Rundel, and OpenIntroA complete foundation for Statistics, also serving as a foundation for Data Science.
Leanpub revenue supports OpenIntro (US-based nonprofit) so we can provide free desk copies to teachers interested in using OpenIntro Statistics in the classroom and expand the project to support free textbooks in other subjects.
More resources: openintro.org.
- #10
Effective Kotlin
Marcin MoskałaEffective Kotlin summarizes the best practices and experiences of the Kotlin community, together with a deep explanation of some lesser-known Kotlin functionalities. All of the best practices are presented as simple rules with detailed explanations.
Top Bundles
- #1
CCIE Service Provider Ultimate Study Bundle
2 Books
Piotr Jablonski, Lukasz Bromirski, and Nick Russo have joined forces to deliver the only CCIE Service Provider training resource you'll ever need. This bundle contains a detailed and challenging collection of workbook labs, plus an extensively detailed technical reference guide. All of us have earned the CCIE Service Provider certification... - #2
Software Architecture for Developers: Volumes 1 & 2 - Technical leadership and communication
2 Books
"Software Architecture for Developers" is a practical and pragmatic guide to modern, lightweight software architecture, specifically aimed at developers. You'll learn:The essence of software architecture.Why the software architecture role should include coding, coaching and collaboration.The things that you really need to think about before... - #3
Modern C++ Collection
3 Books
Get All about Modern C++C++ Standard Library, including C++20Concurrency with Modern C++, including C++20C++20Each book has about 200 complete code examples. Updates are included. When I update one of the books, you immediately get the updated bundle. You can expect significant updates to each new C++ standard (C++23, C++26, .. ) and also... - #4
The Python Craftsman
3 Books
The Python Craftsman series comprises The Python Apprentice, The Python Journeyman, and The Python Master. The first book is primarily suitable for for programmers with some experience of programming in another language. If you don't have any experience with programming this book may be a bit daunting. You'll be learning not just a programming... - #6
"The C++ Standard Library" and "Concurrency with Modern C++"
2 Books
Get my books "The C++ Standard Library" and "Concurrency with Modern C++" in a bundle. The first book gives you the details you should know about the C++ standard library; the second one dives deeper into concurrency with modern C++. In sum, you get more than 600 pages full of modern C++ and about 250 source files presenting the standard library... - #7
10 Books Bundle - A New Career in Tech 🚀
10 Books
Do you want to create yourself a new and profitable skillset - in one year or less? Do you want to position yourself on the right side of change finally? Do you want to work from home and set your own work schedule while choosing fun and interesting coding projects? Software developers and freelance developersearn six figuresin the US, according... - #9
Pattern-Oriented Memory Forensics and Malware Detection
2 Books
This training bundle for security engineers and researchers, malware and memory forensics analysts includes two accelerated training courses for Windows memory dump analysis using WinDbg. It is also useful for technical support and escalation engineers who analyze memory dumps from complex software environments and need to check for possible...
