--- a/src/Intro.v	Mon Nov 09 11:48:27 2009 -0500
+++ b/src/Intro.v	Mon Nov 09 13:18:46 2009 -0500
@@ -91,7 +91,7 @@
 (** ** Dependent Types *)
 
 (**
-A language of %\textit{%#<i>#dependent types#</i>#%}% may include references to programs inside of types.  For instance, the type of an array might include a program expression giving the size of the array, making it possible to verify lack of out-of-bounds accesses statically.  Dependent types can go even further than this, effectively capturing any correctness property in a type.  For instance, later in this book, we will see how to give a Mini-ML compiler a type that guarantees that it maps well-typed source programs to well-typed target programs.
+A language of %\textit{%#<i>#dependent types#</i>#%}% may include references to programs inside of types.  For instance, the type of an array might include a program expression giving the size of the array, making it possible to verify absence of out-of-bounds accesses statically.  Dependent types can go even further than this, effectively capturing any correctness property in a type.  For instance, later in this book, we will see how to give a Mini-ML compiler a type that guarantees that it maps well-typed source programs to well-typed target programs.
 
 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.
 
@@ -137,7 +137,7 @@
 
 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.
 
-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 then 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.
 *)