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 D: Implementing a formal system
-
Appendix A: Writing a simple type checker in Haskell
- About the author
- Notes
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