Gentle Introduction to Dependent Types with Idris

✍ Scribed by Boro Sitnikovski

• Publisher: CreateSpace Independent Publishing Platform
• Year: 2018
• Tongue: English
• Leaves: 130
• Category: Library

✦ Synopsis

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 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.

✦ Table of Contents

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
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
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
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
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
About the author