Gentle Introduction to Dependent Types with Idris
Last updated on 20180820
About the Book
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Dependent types are a powerful concept that allows us to write proofcarrying 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 selfcontained.
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 whitepapers, 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 are welcome on GitHub.
Table of Contents

 Preface and 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. Firstorder logic
 2.1.3. Higherorder logic
 2.2. Set theory abstractions

2.3. Substitution and mathematical proofs
 2.3.1. Proofs by truth tables
 2.3.2. Threecolumn proofs
 2.3.3. Formal proofs
 2.3.4. Proof techniques
 2.3.5. 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

1. Formal systems

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. Higherorder 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. Interfaces and implementations
 4.2. CurryHoward isomorphism

4.1. Basic syntax and definitions

5. Proving in Idris

5.1. Weekdays
 5.1.1. First proof (autoinference)
 5.1.2. Second proof (rewrite)
 5.1.3. Third proof (impossible)

5.2. Natural numbers
 5.2.1. First proof (autoinference 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

5.4. Trees
 5.4.1. Depth
 5.4.2. Map and size
 5.4.3. Length of mapped trees

5.1. Weekdays
 Conclusion
 Further reading
 Appendix A: Metamath

Appendix B: IO, Codegen targets, compilation, and FFI
 IO
 Codegen
 Compilation
 Foreign Function Interface
 About the author

4. Programming in Idris
 Notes
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