(* Version of 4/21/2010 *)

These are notes for a one-semester course on Software Foundations -- the mathematical theory of programming and programming languages -- suitable for graduate or upper-level undergraduate students. They develop basic concepts of functional programming, logic, operational semantics, lambda-calculus, and static type systems, using the Coq proof assistant.
The main novelty of the course is that the development is formalized and machine-checked: the text is literally a script for the Coq proof assistant. It is intended to be read hand-in-hand with the accompanying Coq source file in an interactive session with Coq. All the details of the lectures are fully developed in Coq, and the exercises are designed to be worked using Coq.
The files are organized into a sequence of "core chapters," covering about one semester's worth of material and organized into a coherent linear narrative, plus some "appendices" covering additional topics.
The notes come in two variants: a "terse version" for use in lectures and a "full version" that can be read outside of class and used as a template for homework assignments.


Some fundamental themes of the course:
  • Logic: The mathematical basis for software engineering...
                    logic                        calculus
             --------------------   =   ----------------------------
             software engineering       mechanical/civil engineering
    • In particular, inductively defined sets and relations and inductive proofs about them are ubiquitous in all of computer science.
  • Functional programming: An increasingly important part of the software developer's bag of tricks
    • Advanced programming idioms in mainstream software development methodologies are increasingly incorporating ideas from functional programming.
    • In particular, using persistent data structures and avoiding mutable state enormously simplifies many concurrent programming tasks.
  • Foundations of programming languages (the second part of the course):
    • Notations and techniques for rigorously describing and stress-testing new programming languages and language features. (This is a surprisingly common activity! Most large software systems include subsystems that are basically programming languages -- think of regular expressions, command-line formats, preference and configuration files, SQL, Flash, PDF, etc., etc.)
    • A more sophisticated understanding of the everyday tools used to build software... what's going on under the hood of <your favorite programming language>.
  • Coq: An industrial-strength proof assistant
    • Proof assistants are becoming more and more popular in both software and (especially) hardware industries. Coq is not the only one in widespread use, but learning one thoroughly will give you a big advantage in coming to grips with another.


Required Background

These notes are intended to be accessible to a broad range of readers, from advanced undergraduates to PhD students. They assume little specific background in programming languages or logic. However, a degree of mathematical maturity will be helpful.


Our laboratory for this course is the Coq proof assistant. Coq can be seen as a combination of two things:
  • a simple and slightly idiosyncratic (but, in its way, extremely expressive) programming language, together with
  • a set of tools for stating logical assertions (including assertions about the behavior of programs) and marshalling evidence of their truth.
We will be investigating both aspects in tandem.

System Requirements

Coq runs on Windows and pretty much all flavors of Unix (including Linux and OS X). You will need:
  • A current (8.2) installation of Coq (available from the Coq home page).
  • An IDE for interacting with Coq. Currently, there are two choices:
    • 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").
    • CoqIDE is a simpler stand-alone IDE. It is distributed with Coq, but on some platforms compiling it involves installing additional packages for GUI libraries and such.

Access to the Coq files

A tar file containing the full sources for the "release version" of these notes (as a collection of Coq scripts and HTML files) is available here:
If you are using the notes as part of a class, you may be given access to a locally extended version of the files, which you should then use instead of the release version.


Each chapter of the notes includes numerous exercises. Some are marked "optional"; those that are not should be considered as recommended. Doing just the recommended exercises should provide good coverage of the material in approximately six or eight hours of study time.
The "star ratings" for the exercises can be interpreted as follows:
  • One star: easy exercises that most readers should take only a minute or two. None of these are explicitly marked "Recommended"; rather, all of them should be considered as recommended: readers should be in the habit of working them as they reach them.
  • Two stars: straightforward exercises (five or ten minutes).
  • Three stars: exercises requiring a bit of thought (fifteen minutes to half an hour).
  • Four stars: more difficult exercises (an hour or two).

Recommended reading

Students who want additional texts to supplement and deepen what they find here can have a look at these (among many others):
  • Types and Programming Languages, by Benjamin C. Pierce. MIT Press, 2002.
  • Interactive Theorem Proving and Program Development: Coq'Art: The Calculus of Inductive Constructions, by Yves Bertot and Pierre Castéran. Springer-Verlag, 2004.
  • Certified Programming with Dependent Types, by Adam Chlipala. A draft textbook on practical proof engineering with Coq, available from his web page.
  • Practical Foundations for Programming Languages, by Robert Harper. Manuscript, available from his web page.
  • The Formal Semantics of Programming Languages: An Introduction, by Glynn Winskel. MIT Press, 1993.
  • Foundations for Programming Languages, by John C. Mitchell. MIT Press, 1996.

For Instructors

If you are an instructor and intend to use these materials in your own course, you will undoubtedly find things you'd like to change, improve, or add. Your contributions are welcome!
Please send an email to Benjamin Pierce, and we can set you up with read/write access to our subversion repository and developers' mailing list; in the repository you'll find a README with further instructions.