A handbook to the Coq software for writing and checking mathematical proofs, with a practical engineering focus. The technology of mechanized program verification can play a supporting role in many kinds of research projects in computer science, and related tools for formal proof-checking are seeing increasing adoption in mathematics and engineering. This book provides an introduction to the Coq software for writing and checking mathematical proofs. It takes a practical engineering focus throughout, emphasizing techniques that will help users to build, understand, and maintain large Coq developments and minimize the cost of code change over time. Two topics, rarely discussed elsewhere, are covered in detail: effective dependently typed programming (making productive use of a feature at the heart of the Coq system) and construction of domain-specific proof tactics. Almost every subject covered is also relevant to interactive computer theorem proving in general, not just program verification, demonstrated through examples of verified programs applied in many different sorts of formalizations. The book develops a unique automated proof style and applies it throughout; even experienced Coq users may benefit from reading about basic Coq concepts from this novel perspective. The book also offers a library of tactics, or programs that find proofs, designed for use with examples in the book. Readers will acquire the necessary skills to reimplement these tactics in other settings by the end of the book. All of the code appearing in the book is freely available online.

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.

Dependent types are a concept that allows developers to write proof-carrying code. Idris is a programming language that supports dependent types. This book will teach you the mathematical foundations of Idris as well as how to use it to write software and mathematically prove properties. The first part of the book serves as an introduction to the language's underlying theories. It starts by reviewing formal systems and mathematical logical systems as foundational building blocks, then gradually builds up to dependent types. Next, you'll learn type theory for dependent types. Following this, you'll explore the Idris programming language and conclude by exploring the depths of formal systems and type checkers by implementing them. Introduction to Dependent Types with Idris will walk you through simple examples through more advanced techniques, stepping up the difficulty as you gain more knowledge. Every chapter includes a set of exercises based on what it covered to further cement your learning. No specialized knowledge of mathematics is expected beyond the basics, so it is perfect for novices. You will: Understand Lambda calculus and dependent types Gain insight into functional programming Write mathematical proofs with Idris.

This volume constitutes the post-proceedings of the 18th International Workshop on Implementation and Applications of Functional Languages. Fifteen full papers are presented. Each one was submitted to two rounds of reviews to ensure accuracy, thoroughness, and readability. The papers address all current theoretical and methodological issues in functional and function-based languages.

This volume contains the proceedings of the 4th International Conference on Verified Software: Theories, Tools, and Experiments, VSTTE 2012, held in Philadelphia, PA, USA, in January 2012. The 20 revised full papers presented together with 2 invited talks and 2 tutorials were carefully revised and selected from 54 initial submissions for inclusion in the book. The goal of the VSTTE conference is to advance the state of the art through the interaction of theory development, tool evolution, and experimental validation. The papers address topics such as: specification and verification techniques, tool support for specification languages, tool for various design methodologies, tool integration and plug-ins, automation in formal verification, tool comparisons and benchmark repositories, combination of tools and techniques, customizing tools for particular applications, challenge problems, refinement methodologies, requirements modeling, specification languages, specification/verification case-studies, software design methods, and program logic.