--- a/src/Intro.v	Wed Aug 03 10:34:53 2011 -0400
+++ b/src/Intro.v	Thu Aug 25 11:46:56 2011 -0400
@@ -9,7 +9,12 @@
 
 (** %\fi
 
-\usepackage{hyperref}
+\usepackage{makeidx,hyperref}
+
+\title{Certified Programming with Dependent Types}
+\author{Adam Chlipala}
+
+\makeindex
 
 \begin{document}
 
@@ -47,9 +52,15 @@
 
 (**
 
-We would all like to have programs check that our programs are correct.  Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits.  The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science.  Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant.
+We would all like to have programs check that our programs are correct.  Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits.  The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science.  Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant.  Almost every subject covered is also relevant to interactive computer theorem-proving in general, such as for traditional mathematical theorems.  In fact, I hope to demonstrate how verified programs are useful as building blocks in all sorts of formalizations.
 
-There are a good number of (though definitely not %``%#"#many#"#%''%) tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs.  This is my attempt at an exhaustive list of interactive %``%#"#proof assistants#"#%''% satisfying a few criteria.  First, the authors of each tool must intend for it to be put to use for software-related applications.  Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it.  A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team.
+Research into mechanized theorem-proving began around the 1970's, and some of the earliest practical work involved Nqthm%~\cite{Nqthm}\index{Nqthm}%, the %``%#"#Boyer-Moore Theorem Prover,#"#%''% which was used to prove such theorems as correctness of a complete hardware and software stack%~\cite{Piton}%.  ACL2%~\cite{CAR}\index{ACL2}%, Nqthm's successor, has seen significant industry adoption, for instance, by AMD to verify correctness of floating-point division units%~\cite{AMD}%.
+
+Around the beginning of the 21st century, the pace of progress in practical applications of interactive theorem-proving accelerated significantly.  Several well-known formal developments have been carried out in Coq, the system that this book deals with.  In the realm of pure mathematics, Georges Gonthier built a machine-checked proof of the four color theorem%~\cite{4C}%, a mathematical problem first posed more than a hundred years before, where the only previous proofs had required trusting ad-hoc software to do brute-force checking of key facts.  In the realm of program verification, Xavier Leroy led the CompCert project to produce a verified C compiler back-end%~\cite{CompCert}% robust enough to use with real embedded software.
+
+Many other recent projects have attracted attention by proving important theorems using computer proof assistant software.  For instance, the L4.verified project%~\cite{seL4}% has given a mechanized proof of correctness for a realistic microkernel, using the Isabelle/HOL proof assistant%~\cite{Isabelle/HOL}\index{Isabelle/HOL}%.  The amount of ongoing work in the area is so large that I cannot hope to list all the recent successes, so from this point I will assume that the reader is convinced both that we ought to want machine-checked proofs and that they seem to be feasible to produce.  (To readers not yet convinced, I suggest a Web search for %``%#"#machine-checked proof#"#%''%!)
+
+There are a good number of (though definitely not %``%#"#many#"#%''%) tools that are in wide use today for building machine-checked mathematical proofs and machine-certified programs.  The following is my attempt at an exhaustive list of interactive %``%#"#proof assistants#"#%''% satisfying a few criteria.  First, the authors of each tool must intend for it to be put to use for software-related applications.  Second, there must have been enough engineering effort put into the tool that someone not doing research on the tool itself would feel his time was well spent using it.  A third criterion is more of an empirical validation of the second: the tool must have a significant user community outside of its own development team.
 
 %
 \medskip
@@ -74,7 +85,7 @@
 </table>
 #
 
-Isabelle/HOL, implemented with the %``%#"#proof assistant development framework#"#%''% Isabelle, is the most popular proof assistant for the HOL logic.  The other implementations of HOL can be considered equivalent for purposes of the discussion here.
+Isabelle/HOL, implemented with the %``%#"#proof assistant development framework#"#%''% Isabelle%~\cite{Isabelle}%, is the most popular proof assistant for the HOL logic.  The other implementations of HOL can be considered equivalent for purposes of the discussion here.
 
 *)
 
@@ -102,7 +113,7 @@
 
 ACL2 and HOL lack dependent types outright.  PVS and Twelf each supports a different strict subset of Coq's dependent type language.  Twelf's type language is restricted to a bare-bones, monomorphic lambda calculus, which places serious restrictions on how complicated %\textit{%#<i>#computations inside types#</i>#%}% can be.  This restriction is important for the soundness argument behind Twelf's approach to representing and checking proofs.
 
-In contrast, PVS's dependent types are much more general, but they are squeezed inside the single mechanism of %\textit{%#<i>#subset types#</i>#%}%, where a normal type is refined by attaching a predicate over its elements.  Each member of the subset type is an element of the base type that satisfies the predicate.
+In contrast, PVS's dependent types are much more general, but they are squeezed inside the single mechanism of %\textit{%#<i>#subset types#</i>#%}%, where a normal type is refined by attaching a predicate over its elements.  Each member of the subset type is an element of the base type that satisfies the predicate.  Chapter 6 of this book introduces that style of programming in Coq, while the remaining chapters of Part II deal with features of dependent typing in Coq that go beyond what PVS supports.
 
 Dependent types are not just useful because they help you express correctness properties in types.  Dependent types also often let you write certified programs %\textit{%#<i>#without writing anything that looks like a proof#</i>#%}%.  Even with subset types, which for many contexts can be used to express any relevant property with enough acrobatics, the human driving the proof assistant usually has to build some proofs explicitly.  Writing formal proofs is hard, so we want to avoid it as far as possible, so dependent types are invaluable.
 
@@ -111,9 +122,9 @@
 (** ** An Easy-to-Check Kernel Proof Language *)
 
 (**
-Scores of automated decision procedures are useful in practical theorem proving, but it is unfortunate to have to trust in the correct implementation of each procedure.  Proof assistants satisfying the %``%#"#de Bruijn criterion#"#%''% may use complicated and extensible procedures to seek out proofs, but in the end they produce %\textit{%#<i>#proof terms#</i>#%}% in kernel languages.  These core languages have feature complexity on par with what you find in proposals for formal foundations for mathematics.  To believe a proof, we can ignore the possibility of bugs during %\textit{%#<i>#search#</i>#%}% and just rely on a (relatively small) proof-checking kernel that we apply to the %\textit{%#<i>#result#</i>#%}% of the search.
+Scores of automated decision procedures are useful in practical theorem proving, but it is unfortunate to have to trust in the correct implementation of each procedure.  Proof assistants satisfy the %``%#"#de Bruijn criterion#"#%''% when they produce %\textit{%#<i>#proof terms#</i>#%}% in small kernel languages, even when they use complicated and extensible procedures to seek out proofs in the first place.  These core languages have feature complexity on par with what you find in proposals for formal foundations for mathematics.  To believe a proof, we can ignore the possibility of bugs during %\textit{%#<i>#search#</i>#%}% and just rely on a (relatively small) proof-checking kernel that we apply to the %\textit{%#<i>#result#</i>#%}% of the search.
 
-ACL2 and PVS do not meet the de Bruijn criterion, employing fancy decision procedures that produce no %``%#"#evidence trails#"#%''% justifying their results.
+Coq meets the de Bruijn criterion, while ACL2 and PVS do not, as they employ fancy decision procedures that produce no %``%#"#evidence trails#"#%''% justifying their results.
 *)
 
 (** ** Convenient Programmable Proof Automation *)
@@ -137,14 +148,14 @@
 *)
 
 
-(** * Why Not a Different Dependently-Typed Language? *)
+(** * Why Not a Different Dependently Typed Language? *)
 
 (**
 The logic and programming language behind Coq belongs to a type-theory ecosystem with a good number of other thriving members.  %Agda\footnote{\url{http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php}}%#<a href="http://appserv.cs.chalmers.se/users/ulfn/wiki/agda.php">Agda</a># and %Epigram\footnote{\url{http://www.e-pig.org/}}%#<a href="http://www.e-pig.org/">Epigram</a># are the most developed tools among the alternatives to Coq, and there are others that are earlier in their lifecycles.  All of the languages in this family feel sort of like different historical offshoots of Latin.  The hardest conceptual epiphanies are, for the most part, portable among all the languages.  Given this, why choose Coq for certified programming?
 
-I think the answer is simple.  None of the competition has well-developed systems for tactic-based theorem proving.  Agda and Epigram are designed and marketed more as programming languages than proof assistants.  Dependent types are great, because they often help you prove deep theorems without doing anything that feels like proving.  Nonetheless, almost any interesting certified programming project will benefit from some activity that deserves to be called proving, and many interesting projects absolutely require semi-automated proving, if the sanity of the programmer is to be safeguarded.  Informally, proving is unavoidable when any correctness proof for a program has a structure that does not mirror the structure of the program itself.  An example is a compiler correctness proof, which probably proceeds by induction on program execution traces, which have no simple relationship with the structure of the compiler or the structure of the programs it compiles.  In building such proofs, a mature system for scripted proof automation is invaluable.
+I think the answer is simple.  None of the competition has well-developed systems for tactic-based theorem proving.  Agda and Epigram are designed and marketed more as programming languages than proof assistants.  Dependent types are great, because they often help you prove deep theorems without doing anything that feels like proving.  Nonetheless, almost any interesting certified programming project will benefit from some activity that deserves to be called proving, and many interesting projects absolutely require semi-automated proving, to protect the sanity of the programmer.  Informally, proving is unavoidable when any correctness proof for a program has a structure that does not mirror the structure of the program itself.  An example is a compiler correctness proof, which probably proceeds by induction on program execution traces, which have no simple relationship with the structure of the compiler or the structure of the programs it compiles.  In building such proofs, a mature system for scripted proof automation is invaluable.
 
-On the other hand, Agda, Epigram, and similar tools have less implementation baggage associated with them, and so they tend to be the default first homes of innovations in practical type theory.  Some significant kinds of dependently-typed programs are much easier to write in Agda and Epigram than in Coq.  The former tools may very well be superior choices for projects that do not involve any %``%#"#proving.#"#%''%  Anecdotally, I have gotten the impression that manual proving is orders of magnitudes more costly than manual coping with Coq's lack of programming bells and whistles.  In this book, I will devote significant time to patterns for programming with dependent types in Coq as it is today.  We can hope that the type theory community is tending towards convergence on the right set of features for practical programming with dependent types, and that we will eventually have a single tool embodying those features.
+On the other hand, Agda, Epigram, and similar tools have less implementation baggage associated with them, and so they tend to be the default first homes of innovations in practical type theory.  Some significant kinds of dependently typed programs are much easier to write in Agda and Epigram than in Coq.  The former tools may very well be superior choices for projects that do not involve any %``%#"#proving.#"#%''%  Anecdotally, I have gotten the impression that manual proving is orders of magnitudes more costly than manual coping with Coq's lack of programming bells and whistles.  In this book, I will devote significant time to patterns for programming with dependent types in Coq as it is today.  We can hope that the type theory community is tending towards convergence on the right set of features for practical programming with dependent types, and that we will eventually have a single tool embodying those features.
 *)
 
 
@@ -155,32 +166,34 @@
 
 I will go out on a limb and guess that the reader is a dedicated fan of some functional programming language or another, and that he may even have been involved in teaching that language to undergraduates.  I want to propose an analogy between two attitudes: coming to a negative conclusion about Coq after reading common Coq developments in the wild, and coming to a negative conclusion about Your Favorite Language after looking at the programs undergraduates write in it in the first week of class.  The pragmatics of mechanized proving and program verification have been under serious study for much less time than the pragmatics of programming have been.  The computer theorem proving community is still developing the key insights that correspond to those that functional programming texts and instructors impart to their students, to help those students get over that critical hump where using the language stops being more trouble than it is worth.  Most of the insights for Coq are barely even disseminated among the experts, let alone set down in a tutorial form.  I hope to use this book to go a long way towards remedying that.
 
-If I do that job well, then this book should be of interest even to people who have participated in classes or tutorials specifically about Coq.  The book should even be useful to people who have been using Coq for years but who are mystified when their Coq developments prove impenetrable by colleagues.  The crucial angle in this book is that there are %``%#"#design patterns#"#%''% for reliably avoiding the really grungy parts of theorem proving, and consistent use of these patterns can get you over the hump to the point where it is worth your while to use Coq to prove your theorems and certify your programs, even if formal verification is not your main concern in a project.  We will follow this theme by pursuing two main methods for replacing manual proofs with more understandable artifacts: dependently-typed functions and custom Ltac decision procedures.
+If I do that job well, then this book should be of interest even to people who have participated in classes or tutorials specifically about Coq.  The book should even be useful to people who have been using Coq for years but who are mystified when their Coq developments prove impenetrable by colleagues.  The crucial angle in this book is that there are %``%#"#design patterns#"#%''% for reliably avoiding the really grungy parts of theorem proving, and consistent use of these patterns can get you over the hump to the point where it is worth your while to use Coq to prove your theorems and certify your programs, even if formal verification is not your main concern in a project.  We will follow this theme by pursuing two main methods for replacing manual proofs with more understandable artifacts: dependently typed functions and custom Ltac decision procedures.
 *)
 
 
 (** * Prerequisites *)
 
 (**
-I try to keep the required background knowledge to a minimum in this book.  I will assume familiarity with the material from usual discrete math and logic courses taken by all undergraduate computer science majors, and I will assume that readers have significant experience programming in one of the ML dialects, in Haskell, or in some other, closely-related language.  Experience with only dynamically-typed functional languages might lead to befuddlement in some places, but a reader who has come to grok Scheme deeply will probably be fine.
+I try to keep the required background knowledge to a minimum in this book.  I will assume familiarity with the material from usual discrete math and logic courses taken by all undergraduate computer science majors, and I will assume that readers have significant experience programming in one of the ML dialects, in Haskell, or in some other, closely related language.  Experience with only dynamically typed functional languages might lead to befuddlement in some places, but a reader who has come to understand Scheme deeply will probably be fine.
 
-A good portion of the book is about how to formalize programming languages, compilers, and proofs about them.  To understand those parts, it will be helpful to have a basic knowledge of formal type systems, operational semantics, and the theorems usually proved about such systems.  As a reference on these topics, I recommend %\emph{%#<i>#Types and Programming Languages#</i>#%}%, by Benjamin C. Pierce.
+Part IV of this manuscript is about formalizing programming languages and compilers.  That part certainly depends on basic knowledge of formal type systems, operational semantics, and the theorems usually proved about such systems.  As a reference on these topics, I recommend %\emph{%#<i>#Types and Programming Languages#</i>#%}~\cite{TAPL}%, by Benjamin C. Pierce.  However, my current plan is to break Part IV into a separate, online-only document, since I expect the formalization interests of many readers of the book to lie outside of programming languages.  I do often use examples from programming languages in the earlier parts of the book, but I have tried to design them to be comprehensible on the basis of ML or Haskell experience alone.
 *)
 
 
 (** * Using This Book *)
 
 (**
-This book is generated automatically from Coq source files using the wonderful coqdoc program.  The latest PDF version is available at:
+This book is generated automatically from Coq source files using the wonderful coqdoc program.  The latest PDF version, with hyperlinks from identifier uses to the corresponding definitions, is available at:
 %\begin{center}\url{http://adam.chlipala.net/cpdt/cpdt.pdf}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/cpdt.pdf">http://adam.chlipala.net/cpdt/cpdt.pdf</a></tt></blockquote>#
-There is also an online HTML version available, with a hyperlink from each use of an identifier to that identifier's definition:
+There is also an online HTML version available, which of course also provides hyperlinks:
 %\begin{center}\url{http://adam.chlipala.net/cpdt/html/toc.html}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/html/toc.html">http://adam.chlipala.net/cpdt/html/toc.html</a></tt></blockquote>#
 The source code to the book is also freely available at:
 %\begin{center}\url{http://adam.chlipala.net/cpdt/cpdt.tgz}\end{center}%#<blockquote><tt><a href="http://adam.chlipala.net/cpdt/cpdt.tgz">http://adam.chlipala.net/cpdt/cpdt.tgz</a></tt></blockquote>#
 
 There, you can find all of the code appearing in this book, with prose interspersed in comments, in exactly the order that you find in this document.  You can step through the code interactively with your chosen graphical Coq interface.  The code also has special comments indicating which parts of the chapters make suitable starting points for interactive class sessions, where the class works together to construct the programs and proofs.  The included Makefile has a target %\texttt{%#<tt>#templates#</tt>#%}% for building a fresh set of class template files automatically from the book source.
 
-I believe that a good graphical interface to Coq is crucial for using it productively.  I use the %Proof General\footnote{\url{http://proofgeneral.inf.ed.ac.uk/}}%#<a href="http://proofgeneral.inf.ed.ac.uk/">Proof General</a># mode for Emacs, which supports a number of other proof assistants besides Coq.  There is also the standalone CoqIDE program developed by the Coq team.  I like being able to combine certified programming and proving with other kinds of work inside the same full-featured editor, and CoqIDE has had a good number of crashes and other annoying bugs in recent history, though I hear that it is improving.  In the initial part of this book, I will reference Proof General procedures explicitly, in introducing how to use Coq, but most of the book will be interface-agnostic, so feel free to use CoqIDE if you prefer it.  The one issue with CoqIDE, regarding running through the book source, is that I will sometimes begin a proof attempt but cancel it with the Coq [Abort] or [Restart] commands, which CoqIDE does not support.  It would be bad form to leave such commands lying around in a real, finished development, but I find these commands helpful in writing single source files that trace a user's thought process in designing a proof.
+A traditional printed version of the book is slated to appear from MIT Press in the future.  The online versions will remain available at no cost even after the printed book is released, and I intend to keep the source code up-to-date with bug fixes and compatibility changes to track new Coq releases.
+
+I believe that a good graphical interface to Coq is crucial for using it productively.  I use the %Proof General\footnote{\url{http://proofgeneral.inf.ed.ac.uk/}}%#<a href="http://proofgeneral.inf.ed.ac.uk/">Proof General</a># mode for Emacs, which supports a number of other proof assistants besides Coq.  There is also the standalone CoqIDE program developed by the Coq team.  I like being able to combine certified programming and proving with other kinds of work inside the same full-featured editor, and CoqIDE has had a good number of crashes and other annoying bugs in recent history, though I hear that it is improving.  In the initial part of this book, I will reference Proof General procedures explicitly, in introducing how to use Coq, but most of the book will be interface-agnostic, so feel free to use CoqIDE if you prefer it.  The one issue with CoqIDE, regarding running through the book source, is that I will sometimes begin a proof attempt but cancel it with the Coq [Abort] or #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Restart%}%#</span></span># commands, which CoqIDE does not support.  It would be bad form to leave such commands lying around in a real, finished development, but I find these commands helpful in writing single source files that trace a user's thought process in designing a proof.
 
 The next chapter will introduce Coq with some simple examples, and the chapter starts with step-by-step instructions on getting set up to run the book chapters interactively, assuming that you have Coq and Proof General installed.
 *)