###### Gentle Introduction to Dependent Types with Idris
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Last updated on 2018-09-06

Dependent types are a powerful concept that allow 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, and no prior experience beyond high school mathematics is needed. Thus, this book is written in a way to be self-contained.

The reason for writing this book is that I could not find a book that starts explaining things from scratch, so I had to do a lot of research on the internet through whitepapers, forums, example codes in order to come up with a complete picture on 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.

•
• Preface
• Acknowledgements
• Introduction
• Part I: Theoretical introduction
• 1. Formal systems
• 1.1. MU puzzle example
• 2. Classic 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. Proof techniques
• 2.3.5. Mathematical induction
• 3. Type theory
• 3.1. Lambda calculus
• 3.1.1. Terms reduction
• 3.2. Lambda calculus with types
• 3.3. Dependent types
• 3.4. Intuitionistic theory of types
• 3.4.1. Intuitionistic logic
• Part II: Practical introduction
• 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. Strict evaluation
• 4.1.11. Pattern matching expressions
• 4.1.12. Documentation and searching
• 4.2. Curry-Howard isomorphism
• 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
• 5.3. Trees
• 5.3.1. Depth
• 5.3.2. Map and size
• 5.3.3. Length of mapped trees
• Conclusion
• Appendix A: Metamath
• Appendix B: IO, Codegen targets, compilation, and FFI
• IO
• Codegen
• Compilation
• Foreign Function Interface
• Notes

Boro Sitnikovski

Boro Sitnikovski has 10+ years of experience working professionally as a Software Engineer. He started programming with the Assembly programming language on an Intel x86 at the age of 10. While in high school, he has won several prizes on competitive programming, varying from 4th, 3rd, and 1st place.

He has a Bachelor of Engineering in Informatics degree, and his thesis was titled “Programming in Haskell using algebraic data structures”. His research interests include: programming languages, mathematics, logic, algorithms, and writing correct software.

He is a strong believer in the open source philosophy, and contributes to various open source projects.

In his spare time he enjoys some time-off with his family.

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