--- a/src/Intro.v	Wed Sep 05 15:22:13 2012 -0400
+++ b/src/Intro.v	Wed Sep 26 16:35:35 2012 -0400
@@ -20,9 +20,9 @@
 
 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"!)
+Many other recent projects have attracted attention by proving important theorems using computer proof assistant software.  For instance, the L4.verified project%~\cite{seL4}% led by Gerwin Klein 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"!)
 
-The idea of %\index{certified program}% _certified program_ features prominently in this book's title.  Here the word "certified" does _not_ refer to governmental rules for how the reliability of engineered systems may be demonstrated to sufficiently high standards.  Rather, this concept of certification, a standard one in the programming languages and formal methods communities, has to do with the idea of a _certificate_, or formal mathematical artifact proving that a program meets its specification.  Government certification procedures rarely provide strong mathematical guarantees, while certified programming provides guarantees about as strong as anything we could hope for.  We trust the definition of a foundational mathematical logic, we trust an implementation of the logic, and we trust that we have encoded our informal intent properly in formal specifications, but little else is left open as an opportunity to certify incorrect software.  For programs like compilers that run in batch mode, the notion of a %\index{certifying program}% _certifying_ program is also common, where each run of the program outputs both an answer and a proof that it is correct.  Certified software can be considered to subsume certifying software, and this book focuses on the certified case, while also introducing principles and techniques of general interest for stating and proving theorems in Coq.
+The idea of %\index{certified program}% _certified program_ features prominently in this book's title.  Here the word "certified" does _not_ refer to governmental rules for how the reliability of engineered systems may be demonstrated to sufficiently high standards.  Rather, this concept of certification, a standard one in the programming languages and formal methods communities, has to do with the idea of a _certificate_, or formal mathematical artifact proving that a program meets its specification.  Government certification procedures rarely provide strong mathematical guarantees, while certified programming provides guarantees about as strong as anything we could hope for.  We trust the definition of a foundational mathematical logic, we trust an implementation of the logic, and we trust that we have encoded our informal intent properly in formal specifications, but little else is left open as an opportunity to certify incorrect software.  For programs like compilers that run in batch mode, the notion of a %\index{certifying program}% _certifying_ program is also common, where each run of the program outputs both an answer and a proof that the answer is correct.  Certified software can be considered to subsume certifying software, and this book focuses on the certified case, while also introducing principles and techniques of general interest for stating and proving theorems in Coq.
 
 %\medskip%
 
@@ -79,7 +79,7 @@
 
 %\index{ACL2}%ACL2 and %\index{HOL}%HOL lack dependent types outright.  %\index{PVS}%PVS and %\index{Twelf}%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 _computations inside types_ can be.  This restriction is important for the soundness argument behind Twelf's approach to representing and checking proofs.
 
-In contrast, %\index{PVS}%PVS's dependent types are much more general, but they are squeezed inside the single mechanism of _subset types_, 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 7 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.
+In contrast, %\index{PVS}%PVS's dependent types are much more general, but they are squeezed inside the single mechanism of _subset types_, 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 _without writing anything that looks like a proof_.  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.
 
@@ -90,7 +90,7 @@
 (**
 %\index{de Bruijn criterion}%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 _proof terms_ 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 (e.g., ZF set theory).  To believe a proof, we can ignore the possibility of bugs during _search_ and just rely on a (relatively small) proof-checking kernel that we apply to the _result_ of the search.
 
-Coq meets the de Bruijn criterion, while %\index{ACL2}%ACL2 and %\index{PVS}%PVS do not, as they employ fancy decision procedures that produce no "evidence trails" justifying their results.
+Coq meets the de Bruijn criterion, while %\index{ACL2}%ACL2 and %\index{PVS}%PVS do not, as they employ fancy decision procedures that produce no "evidence trails" justifying their results.  The HOL implementations also meet the de Bruijn criterion; for Twelf, the situation is murkier.
 *)
 
 (** ** Convenient Programmable Proof Automation *)
@@ -108,7 +108,7 @@
 (** ** Proof by Reflection *)
 
 (**
-%\index{reflection}\index{proof by reflection}%A surprising wealth of benefits follow from choosing a proof language that integrates a rich notion of computation.  Coq includes programs and proof terms in the same syntactic class.  This makes it easy to write programs that compute proofs.  With rich enough dependent types, such programs are _certified decision procedures_.  In such cases, these certified procedures can be put to good use _without ever running them_!  Their types guarantee that, if we did bother to run them, we would receive proper "ground" proofs.
+%\index{reflection}\index{proof by reflection}%A surprising wealth of benefits follows from choosing a proof language that integrates a rich notion of computation.  Coq includes programs and proof terms in the same syntactic class.  This makes it easy to write programs that compute proofs.  With rich enough dependent types, such programs are _certified decision procedures_.  In such cases, these certified procedures can be put to good use _without ever running them_!  Their types guarantee that, if we did bother to run them, we would receive proper "ground" proofs.
 
 The critical ingredient for this technique, many of whose instances are referred to as _proof by reflection_, is a way of inducing non-trivial computation inside of logical propositions during proof checking.  Further, most of these instances require dependent types to make it possible to state the appropriate theorems.  Of the proof assistants I listed, only Coq really provides this support.
 *)
@@ -165,7 +165,7 @@
 (** ** Reading This Book *)
 
 (**
-For experts in functional programming or formal methods, learning to use Coq is not hard, in a sense.  The Coq manual%~\cite{CoqManual}% and the textbook by Bertot and Cast%\'%eran%~\cite{CoqArt}% have helped many people become productive Coq users.  However, I believe that the best ways to manage significant Coq developments are far from settled.  In this book, I mean to propose my own techniques, and, rather than treating them as advanced material for a final chapter or two, I employ them from the very beginning.  After a first chapter showing off what can be done with dependent types, I retreat into simpler programming styles for the first part of the book.  The other main thrust of the book, Ltac proof automation, I adopt more or less from the very start of the technical exposition.
+For experts in functional programming or formal methods, learning to use Coq is not hard, in a sense.  The Coq manual%~\cite{CoqManual}% and the textbook by Bertot and Cast%\'%eran%~\cite{CoqArt}% have helped many people become productive Coq users.  However, I believe that the best ways to manage significant Coq developments are far from settled.  In this book, I mean to propose my own techniques, and, rather than treating them as advanced material for a final chapter or two, I employ them from the very beginning.  After a first chapter showing off what can be done with dependent types, I retreat into simpler programming styles for the first part of the book.  I adopt the other main thrust of the book, Ltac proof automation, more or less from the very start of the technical exposition.
 
 Some readers have suggested that I give multiple recommended reading orders in this introduction, targeted at people with different levels of Coq expertise.  It is certainly true that Part I of the book devotes significant space to basic concepts that most Coq users already know quite well.  However, as I am introducing these concepts, I am also developing my preferred automated proof style, so I think even the chapters on basics are worth reading for experienced Coq hackers.