This is the entry point to a series of electronic textbooks on various aspects of Software Foundations, the mathematical underpinnings of reliable software. Topics in the series include basic concepts of logic, computer-assisted theorem proving, the Coq proof assistant, functional programming, operational semantics, logics and techniques for reasoning about programs, static type systems, property-based random testing, and verification of practical C code. The exposition is intended for a broad range of readers, from advanced undergraduates to PhD students and researchers. No specific background in logic or programming languages is assumed, though a degree of mathematical maturity will be helpful.
The principal novelty of the series is that it is one hundred percent formalized and machine-checked: each text is literally a script for Coq. The books are intended to be read alongside (or inside) an interactive session with Coq. All the details in the text are fully formalized in Coq, and most of the exercises are designed to be worked using Coq.
The files in each book are organized into a sequence of core chapters, covering about one semester's worth of material and organized into a coherent linear narrative, plus a number of "offshoot" chapters covering additional topics. All the core chapters are suitable for both upper-level undergraduate and graduate students.
This book, Logical Foundations, lays groundwork for the others, introducing the reader to the basic ideas of functional programming, constructive logic, and the Coq proof assistant.


Building reliable software is really hard -- really hard. The scale and complexity of modern systems, the number of people involved, and the range of demands placed on them make it challenging to build software that is even more-or-less correct, much less 100% correct. At the same time, the increasing degree to which information processing is woven into every aspect of society greatly amplifies the cost of bugs and insecurities.
Computer scientists and software engineers have responded to these challenges by developing a host of techniques for improving software reliability, ranging from recommendations about managing software projects teams (e.g., extreme programming) to design philosophies for libraries (e.g., model-view-controller, publish-subscribe, etc.) and programming languages (e.g., object-oriented programming, aspect-oriented programming, functional programming, ...) to mathematical techniques for specifying and reasoning about properties of software and tools for helping validate these properties. The Software Foundations series is focused on this last set of tools.
This volume weaves together three conceptual threads:
(1) basic tools from logic for making and justifying precise claims about programs;
(2) the use of proof assistants to construct rigorous logical arguments;
(3) functional programming, both as a method of programming that simplifies reasoning about programs and as a bridge between programming and logic.


Logic is the field of study whose subject matter is proofs -- unassailable arguments for the truth of particular propositions. Volumes have been written about the central role of logic in computer science. Manna and Waldinger called it "the calculus of computer science," while Halpern et al.'s paper On the Unusual Effectiveness of Logic in Computer Science catalogs scores of ways in which logic offers critical tools and insights. Indeed, they observe that, "As a matter of fact, logic has turned out to be significantly more effective in computer science than it has been in mathematics. This is quite remarkable, especially since much of the impetus for the development of logic during the past one hundred years came from mathematics."
In particular, the fundamental tools of inductive proof are ubiquitous in all of computer science. You have surely seen them before, perhaps in a course on discrete math or analysis of algorithms, but in this course we will examine them more deeply than you have probably done so far.

Proof Assistants

The flow of ideas between logic and computer science has not been unidirectional: CS has also made important contributions to logic. One of these has been the development of software tools for helping construct proofs of logical propositions. These tools fall into two broad categories:
  • Automated theorem provers provide "push-button" operation: you give them a proposition and they return either true or false (or, sometimes, don't know: ran out of time). Although their capabilities are still limited to specific domains, they have matured tremendously in recent years and are used now in a multitude of settings. Examples of such tools include SAT solvers, SMT solvers, and model checkers.
  • Proof assistants are hybrid tools that automate the more routine aspects of building proofs while depending on human guidance for more difficult aspects. Widely used proof assistants include Isabelle, Agda, Twelf, ACL2, PVS, and Coq, among many others.
This course is based around Coq, a proof assistant that has been under development since 1983 and that in recent years has attracted a large community of users in both research and industry. Coq provides a rich environment for interactive development of machine-checked formal reasoning. The kernel of the Coq system is a simple proof-checker, which guarantees that only correct deduction steps are ever performed. On top of this kernel, the Coq environment provides high-level facilities for proof development, including a large library of common definitions and lemmas, powerful tactics for constructing complex proofs semi-automatically, and a special-purpose programming language for defining new proof-automation tactics for specific situations.
Coq has been a critical enabler for a huge variety of work across computer science and mathematics:
  • As a platform for modeling programming languages, it has become a standard tool for researchers who need to describe and reason about complex language definitions. It has been used, for example, to check the security of the JavaCard platform, obtaining the highest level of common criteria certification, and for formal specifications of the x86 and LLVM instruction sets and programming languages such as C.
  • As an environment for developing formally certified software and hardware, Coq has been used, for example, to build CompCert, a fully-verified optimizing compiler for C, and CertiKOS, a fully verified hypervisor, for proving the correctness of subtle algorithms involving floating point numbers, and as the basis for CertiCrypt, an environment for reasoning about the security of cryptographic algorithms. It is also being used to build verified implementations of the open-source RISC-V processor architecture.
  • As a realistic environment for functional programming with dependent types, it has inspired numerous innovations. For example, the Ynot system embeds "relational Hoare reasoning" (an extension of the Hoare Logic we will see later in this course) in Coq.
  • As a proof assistant for higher-order logic, it has been used to validate a number of important results in mathematics. For example, its ability to include complex computations inside proofs made it possible to develop the first formally verified proof of the 4-color theorem. This proof had previously been controversial among mathematicians because it required checking a large number of configurations using a program. In the Coq formalization, everything is checked, including the correctness of the computational part. More recently, an even more massive effort led to a Coq formalization of the Feit-Thompson Theorem, the first major step in the classification of finite simple groups.
By the way, in case you're wondering about the name, here's what the official Coq web site at INRIA (the French national research lab where Coq has mostly been developed) says about it: "Some French computer scientists have a tradition of naming their software as animal species: Caml, Elan, Foc or Phox are examples of this tacit convention. In French, 'coq' means rooster, and it sounds like the initials of the Calculus of Constructions (CoC) on which it is based." The rooster is also the national symbol of France, and C-o-q are the first three letters of the name of Thierry Coquand, one of Coq's early developers.

Functional Programming

The term functional programming refers both to a collection of programming idioms that can be used in almost any programming language and to a family of programming languages designed to emphasize these idioms, including Haskell, OCaml, Standard ML, F#, Scala, Scheme, Racket, Common Lisp, Clojure, Erlang, and Coq.
Functional programming has been developed over many decades -- indeed, its roots go back to Church's lambda-calculus, which was invented in the 1930s, well before the first electronic computers! But since the early '90s it has enjoyed a surge of interest among industrial engineers and language designers, playing a key role in high-value systems at companies like Jane Street Capital, Microsoft, Facebook, Twitter, and Ericsson.
The most basic tenet of functional programming is that, as much as possible, computation should be pure, in the sense that the only effect of execution should be to produce a result: it should be free from side effects such as I/O, assignments to mutable variables, redirecting pointers, etc. For example, whereas an imperative sorting function might take a list of numbers and rearrange its pointers to put the list in order, a pure sorting function would take the original list and return a new list containing the same numbers in sorted order.
A significant benefit of this style of programming is that it makes programs easier to understand and reason about. If every operation on a data structure yields a new data structure, leaving the old one intact, then there is no need to worry about how that structure is being shared and whether a change by one part of the program might break an invariant relied on by another part of the program. These considerations are particularly critical in concurrent systems, where every piece of mutable state that is shared between threads is a potential source of pernicious bugs. Indeed, a large part of the recent interest in functional programming in industry is due to its simpler behavior in the presence of concurrency.
Another reason for the current excitement about functional programming is related to the first: functional programs are often much easier to parallelize and physically distribute than their imperative counterparts. If running a computation has no effect other than producing a result, then it does not matter where it is run. Similarly, if a data structure is never modified destructively, then it can be copied freely, across cores or across the network. Indeed, the "Map-Reduce" idiom, which lies at the heart of massively distributed query processors like Hadoop and is used by Google to index the entire web is a classic example of functional programming.
For purposes of this course, functional programming has yet another significant attraction: it serves as a bridge between logic and computer science. Indeed, Coq itself can be viewed as a combination of a small but extremely expressive functional programming language plus a set of tools for stating and proving logical assertions. Moreover, when we come to look more closely, we find that these two sides of Coq are actually aspects of the very same underlying machinery -- i.e., proofs are programs.

Further Reading

This text is intended to be self contained, but readers looking for a deeper treatment of particular topics will find some suggestions for further reading in the Postscript chapter. Bibliographic information for all cited works can be found in the file Bib.


System Requirements

Coq runs on Windows, Linux, and macOS. The files in this book have been tested with Coq 8.17.1.
You will need:
  • A current installation of Coq, available from the Coq home page. The "Coq Platform" usually offers the smoothest installation experience.
    If you use the VSCode + Docker option described below, you don't need to install Coq separately.
  • An IDE for interacting with Coq. There are several choices:
    • The VSCoq extension for Visual Studio Code offers a simple interface via a familiar IDE. This option is the recommended default.
      VSCoq can be used as an ordinary IDE or it can be combined with Docker (see below) for a lightweight installation experience.
    • Proof General is an Emacs-based IDE. It tends to be preferred by users who are already comfortable with Emacs. It requires a separate installation (google "Proof General").
      Adventurous users of Coq within Emacs may want to check out extensions such as company-coq and control-lock.
    • CoqIDE is a simpler stand-alone IDE. It is distributed with Coq, so it should be available once you have Coq installed. It can also be compiled from scratch, but on some platforms this may involve installing additional packages for GUI libraries and such.
      Users who like CoqIDE should consider running it with the "asynchronous" and "error resilience" modes disabled:
                coqide -async-proofs off \
                -async-proofs-command-error-resilience off Foo.v & ]] *)

Using Coq with VSCode and Docker

The Visual Studio Code IDE can cooperate with the Docker virtualization platform to compile Coq scripts without the need for any separate Coq installation. To get things set up, follow these steps:
  • Install Docker from or make sure your existing installation is up to date.
  • Make sure Docker is running.
  • Install VSCode from and start it running.
  • Install VSCode's Remote Containers Extention from
  • Set up a directory for this SF volume by downloading the provided .tgz file. Besides the .v file for each chapter, this directory will contain a .devcontainer subdirectory with instructions for VSCode about where to find an appropriate Docker image and a _CoqProject file, whose presence triggers the VSCoq extension.
  • In VSCode, use File > Open Folder to open the new directory. VSCode should ask you whether you want to run the project in the associated Docker container. (If it does not ask you, you can open the command palette by pressing F1 and run the command “Dev Containers: Reopen in Container”.)
  • Check that VSCoq is working by double-clicking the file Basics.v from the list on the left (you should see a blinking cursor in the window that opens; if not you can click in that window to select it), and pressing alt+downarrow (on MacOS, control+option+downarrow) a few times. You should see the cursor move through the file and the region above the cursor get highlighted.
  • To see what other key bindings are available, press F1 and then type Coq:, or visit the VSCoq web pages:


Each chapter includes numerous exercises. Each is marked with a "star rating," which can be interpreted as follows:
  • One star: easy exercises that underscore points in the text and that, for most readers, should take only a minute or two. Get in the habit of working these as you reach them.
  • Two stars: straightforward exercises (five or ten minutes).
  • Three stars: exercises requiring a bit of thought (ten minutes to half an hour).
  • Four and five stars: more difficult exercises (half an hour and up).
Those using SF in a classroom setting should note that the autograder assigns extra points to harder exercises:
      1 star = 1 point
      2 stars = 2 points
      3 stars = 3 points
      4 stars = 6 points
      5 stars = 10 points
Some exercises are marked "advanced," and some are marked "optional." Doing just the non-optional, non-advanced exercises should provide good coverage of the core material. Optional exercises provide a bit of extra practice with key concepts and introduce secondary themes that may be of interest to some readers. Advanced exercises are for readers who want an extra challenge and a deeper cut at the material.
Please do not post solutions to the exercises in a public place. Software Foundations is widely used both for self-study and for university courses. Having solutions easily available makes it much less useful for courses, which typically have graded homework assignments. We especially request that readers not post solutions to the exercises anyplace where they can be found by search engines.

Downloading the Coq Files

A tar file containing the full sources for the "release version" of this book (as a collection of Coq scripts and HTML files) is available at
If you are using the book as part of a class, your professor may give you access to a locally modified version of the files; you should use that one instead of the public release version, so that you get any local updates during the semester.

Chapter Dependencies

A diagram of the dependencies between chapters and some suggested paths through the material can be found in the file deps.html.

Recommended Citation Format

If you want to refer to this volume in your own writing, please do so as follows:
    @book {Pierce:SF1,
    author = {Benjamin C. Pierce and
                      Arthur Azevedo de Amorim and
                      Chris Casinghino and
                      Marco Gaboardi and
                      Michael Greenberg and
                      Cătălin Hriţcu and
                      Vilhelm Sjöberg and
                      Brent Yorgey},
    editor = {Benjamin C. Pierce},
    title = "Logical Foundations",
    series = "Software Foundations",
    volume = "1",
    year = "2024",
    publisher = "Electronic textbook",
    note = {Version 6.6, \URL{}}


Sample Exams

A large compendium of exams from many offerings of CIS5000 ("Software Foundations") at the University of Pennsylvania can be found at There has been some drift of notations over the years, but most of the problems are still relevant to the current text.

Lecture Videos

Lectures for two intensive summer courses based on Logical Foundations (part of the DeepSpec summer school series) can be found at The video quality in the 2017 lectures is poor at the beginning but gets better in the later lectures.

Note for Instructors and Contributors

If you plan to use these materials in your own teaching, or if you are using software foundations for self study and are finding things you'd like to help add or improve, your contributions are welcome! You are warmly invited to join the private SF git repo.
In order to keep the legalities simple and to have a single point of responsibility in case the need should ever arise to adjust the license terms, sublicense, etc., we ask all contributors (i.e., everyone with access to the developers' repository) to assign copyright in their contributions to the appropriate "author of record," as follows:
  • I hereby assign copyright in my past and future contributions to the Software Foundations project to the Author of Record of each volume or component, to be licensed under the same terms as the rest of Software Foundations. I understand that, at present, the Authors of Record are as follows: For Volumes 1 and 2, known until 2016 as "Software Foundations" and from 2016 as (respectively) "Logical Foundations" and "Programming Foundations," and for Volume 4, "QuickChick: Property-Based Testing in Coq," the Author of Record is Benjamin C. Pierce. For Volume 3, "Verified Functional Algorithms," and volume 5, "Verifiable C," the Author of Record is Andrew W. Appel. For Volume 6, "Separation Logic Foundations," the author of record is Arthur Chargueraud. For components outside of designated volumes (e.g., typesetting and grading tools and other software infrastructure), the Author of Record is Benjamin Pierce.
To get started, please send an email to Benjamin Pierce, describing yourself and how you plan to use the materials and including (1) the above copyright transfer text and (2) your github username.
We'll set you up with access to the git repository and developers' mailing lists. In the repository you'll find the files INSTRUCTORS and CONTRIBUTING with further instructions.


Thanks to the efforts of a team of volunteer translators, Software Foundations can be enjoyed in Japanese at A Chinese translation is also underway; you can preview it at


Development of the Software Foundations series has been supported, in part, by the National Science Foundation under the NSF Expeditions grant 1521523, The Science of Deep Specification.

(* 2024-04-27 10:26 *)