Verified Functional Programming in Agda

Abstract

First edition. Agda is an advanced programming language based on Type Theory. Agda's type system is expressive enough to support full functional verification of programs, in two styles. In external verification, we write pure functional programs and then write proofs of properties about them. The proofs are separate external artifacts, typically using structural induction. In internal verification, we specify properties of programs through rich types for the programs themselves. This often necessitates including proofs inside code, to show the type checker that the specified properties hold. The power to prove properties of programs in these two styles is a profound addition to the practice of programming, giving programmers the power to guarantee the absence of bugs, and thus improve the quality of software more than previously possible. The book begins with an introduction to functional programming through familiar examples like booleans, natural numbers, and lists, and techniques for external verification. Internal verification is considered through the examples of vectors, binary search trees, and Braun trees. More advanced material on type-level computation, explicit reasoning about termination, and normalization by evaluation is also included. The book also includes a medium-sized case study on Huffman encoding and decoding. 1. Functional programming with the Booleans -- 1.1 Declaring the datatype of Booleans -- 1.2 First steps interacting with Agda -- 1.3 Syntax declarations -- 1.4 Defining Boolean operations by pattern matching: negation -- 1.5 Defining Boolean operations by pattern matching: and, or -- 1.6 The if-then-else operation -- 1.7 Conclusion -- Exercises -- 2. Introduction to constructive proof -- 2.1 A first theorem about the Booleans -- 2.2 Universal theorems -- 2.3 Another example, and more on implicit arguments -- 2.4 Theorems with hypotheses -- 2.5 Going deeper: Curry-Howard and constructivity -- 2.6 Further examples -- 2.7 Conclusion -- Exercises -- 3. Natural numbers -- 3.1 Peano natural numbers -- 3.2 Addition -- 3.3 Multiplication -- 3.4 Arithmetic comparison -- 3.5 Even/odd and mutually recursive definitions -- 3.6 Conclusion -- Exercises -- 4. Lists -- 4.1 The list datatype and type parameters -- 4.2 Basic operations on lists -- 4.3 Reasoning about list operations -- 4.4 Conclusion -- Exercises -- 5. Internal verification -- 5.1 Vectors -- 5.2 Binary search trees -- 5.3 Sigma types -- 5.4 Braun trees -- 5.5 Discussion: internal vs. external verification -- 5.6 Conclusion -- Exercises -- 6. Type-level computation -- 6.1 Integers -- 6.2 Formatted printing -- 6.3 Proof by reflection -- 6.4 Conclusion -- Exercises -- 7. Generating Agda parsers with gratr -- 7.1 A primer on grammars -- 7.2 Generating parsers with gratr -- 7.3 Conclusion -- Exercises -- 8. A case study: Huffman encoding and decoding -- 8.1 The files -- 8.2 The input formats -- 8.3 Encoding textual input -- 8.4 Decoding encoded text -- 8.5 Conclusion -- Exercises -- 9. Reasoning about termination -- 9.1 Termination proofs -- 9.2 Operational semantics for SK combinators -- 9.3 Conclusion -- Exercises -- 10. Intuitionistic logic and Kripke semantics -- 10.1 Positive propositional intuitionistic logic (PPIL) -- 10.2 Kripke structures -- 10.3 Kripke semantics for PPIL -- 10.4 Soundness of PPIL -- 10.5 Completeness -- 10.6 Conclusion -- Exercises -- Appendix A. Quick guide to symbols -- Appendix B. Commonly used Emacs control commands -- Appendix C. Some extra Emacs definitions -- References -- Index -- Author's biography.