Mercurial > cpdt > repo
changeset 399:5986e9fd40b5
Start figuring out which coqdoc changes will be needed to produce a pretty final version
author | Adam Chlipala <adam@chlipala.net> |
---|---|
date | Fri, 08 Jun 2012 11:25:11 -0400 |
parents | 05efde66559d |
children | d27c0be2c3d4 |
files | src/Conclusion.v src/Exercises.v src/Intro.v src/Predicates.v src/StackMachine.v |
diffstat | 5 files changed, 37 insertions(+), 71 deletions(-) [+] |
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--- a/src/Conclusion.v Wed Jun 06 11:25:13 2012 -0400 +++ b/src/Conclusion.v Fri Jun 08 11:25:11 2012 -0400 @@ -11,7 +11,7 @@ (** I have designed this book to present the key ideas needed to get started with productive use of Coq. Many people have learned to use Coq through a variety of resources, yet there is a distinct lack of agreement on structuring principles and techniques for easing the evolution of Coq developments over time. Here I have emphasized two unusual techniques: programming with dependent types, and proving with scripted proof automation. I have also tried to present other material following my own take on how to keep Coq code beautiful and scalable. - Part of the attraction of Coq and similar tools is that their logical foundations are small. A few pages of %\LaTeX{}%#LaTeX# code suffice to define CIC, Coq's logic, yet there do not seem to be any practical limits on which mathematical concepts may be encoded on top of this modest base. At the same time, the %\emph{%#<i>#pragmatic#</i>#%}% foundation of Coq is vast, encompassing tactics, libraries, and design patterns for programs, theorem statements, and proof scripts. I hope the preceding chapters have given a sense of just how much there is to learn before it is possible to drive Coq with the same ease with which many readers write informal proofs! The pay-off of this learning process is that many proofs, especially those with many details to check, become much easier to write than they are on paper. Further, the truth of such theorems may be established with much greater confidence, even without reading proof details. + Part of the attraction of Coq and similar tools is that their logical foundations are small. A few pages of %\LaTeX{}%#LaTeX# code suffice to define CIC, Coq's logic, yet there do not seem to be any practical limits on which mathematical concepts may be encoded on top of this modest base. At the same time, the _pragmatic_ foundation of Coq is vast, encompassing tactics, libraries, and design patterns for programs, theorem statements, and proof scripts. I hope the preceding chapters have given a sense of just how much there is to learn before it is possible to drive Coq with the same ease with which many readers write informal proofs! The pay-off of this learning process is that many proofs, especially those with many details to check, become much easier to write than they are on paper. Further, the truth of such theorems may be established with much greater confidence, even without reading proof details. As Coq has so many moving parts to catalogue mentally, I have not attempted to describe most of them here; nor have I attempted to give exhaustive descriptions of the few I devote space to. To those readers who have made it this far through the book, my advice is: read through the Coq manual, front to back, at some level of detail. Get a sense for which bits of functionality are available. Dig more into those categories that sound relevant to the developments you want to build, and keep the rest in mind in case they come in handy later.
--- a/src/Exercises.v Wed Jun 06 11:25:13 2012 -0400 +++ b/src/Exercises.v Fri Jun 08 11:25:11 2012 -0400 @@ -197,7 +197,7 @@ %\item%#<li># Define a recursive function [liftVar : forall ts1 ts2 t t', member t (ts1 ++ ts2) -> member t (ts1 ++ t' :: ts2)]. This function should %``%#"#lift#"#%''% a de Bruijn variable so that its type refers to a new variable inserted somewhere in the index list.#</li># %\item%#<li># Define a recursive function [lift' : forall ts t (e : exp ts t) ts1 ts2 t', ts = ts1 ++ ts2 -> exp (ts1 ++ t' :: ts2) t] which performs a similar lifting on an [exp]. The convoluted type is to get around restrictions on [match] annotations. We delay %``%#"#realizing#"#%''% that the first index of [e] is built with list concatenation until after a dependent [match], and the new explicit proof argument must be used to cast some terms that come up in the [match] body.#</li># %\item%#<li># Define a function [lift : forall ts t t', exp ts t -> exp (t' :: ts) t], which handles simpler top-level lifts. This should be an easy one-liner based on [lift'].#</li># - %\item%#<li># Define a recursive function [substVar : forall ts1 ts2 t t', member t (ts1 ++ t' :: ts2) -> (t' = t) + member t (ts1 ++ ts2)]. This function is the workhorse behind substitution applied to a variable. It returns [inl] to indicate that the variable we pass to it is the variable that we are substituting for, and it returns [inr] to indicate that the variable we are examining is %\textit{%#<i>#not#</i>#%}% the one we are substituting for. In the first case, we get a proof that the necessary typing relationship holds, and, in the second case, we get the original variable modified to reflect the removal of the substitutee from the typing context.#</li># + %\item%#<li># Define a recursive function [substVar : forall ts1 ts2 t t', member t (ts1 ++ t' :: ts2) -> (t' = t) + member t (ts1 ++ ts2)]. This function is the workhorse behind substitution applied to a variable. It returns [inl] to indicate that the variable we pass to it is the variable that we are substituting for, and it returns [inr] to indicate that the variable we are examining is _not_ the one we are substituting for. In the first case, we get a proof that the necessary typing relationship holds, and, in the second case, we get the original variable modified to reflect the removal of the substitutee from the typing context.#</li># %\item%#<li># Define a recursive function [subst' : forall ts t (e : exp ts t) ts1 t' ts2, ts = ts1 ++ t' :: ts2 -> exp (ts1 ++ ts2) t' -> exp (ts1 ++ ts2) t]. This is the workhorse of substitution in expressions, employing the same proof-passing trick as for [lift']. You will probably want to use [lift] somewhere in the definition of [subst'].#</li># %\item%#<li># Now [subst] should be a one-liner, defined in terms of [subst'].#</li># %\item%#<li># Prove a correctness theorem for each auxiliary function, leading up to the proof of [subst] correctness.#</li># @@ -269,7 +269,7 @@ | _ => assert True by constructor; eapply mult_both end. -(** This hint has the effect of applying [mult_both] %\emph{%#<i>#at most once#</i>#%}% during a proof. After the next chapter, it should be clear why the hint has that effect, but for now treat it as a useful black box. Simply using [Hint Resolve mult_both] would increase proof search time unacceptably, because there are just too many ways to use [mult_both] repeatedly within a proof. +(** This hint has the effect of applying [mult_both] _at most once_ during a proof. After the next chapter, it should be clear why the hint has that effect, but for now treat it as a useful black box. Simply using [Hint Resolve mult_both] would increase proof search time unacceptably, because there are just too many ways to use [mult_both] repeatedly within a proof. The order of the theorems above is itself a meta-level hint, since I found that order to work well for allowing the use of earlier theorems as hints in the proofs of later theorems.
--- a/src/Intro.v Wed Jun 06 11:25:13 2012 -0400 +++ b/src/Intro.v Fri Jun 08 11:25:11 2012 -0400 @@ -22,7 +22,7 @@ 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#"#%''%!) -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 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. %\medskip%
--- a/src/Predicates.v Wed Jun 06 11:25:13 2012 -0400 +++ b/src/Predicates.v Fri Jun 08 11:25:11 2012 -0400 @@ -58,7 +58,7 @@ The essence of the argument is roughly this: to an engineer, not all functions of type [A -> B] are created equal, but all proofs of a proposition [P -> Q] are. This idea is known as %\index{proof irrelevance}%_proof irrelevance_, and its formalizations in logics prevent us from distinguishing between alternate proofs of the same proposition. Proof irrelevance is compatible with, but not derivable in, Gallina. Apart from this theoretical concern, I will argue that it is most effective to do engineering with Coq by employing different techniques for programs versus proofs. Most of this book is organized around that distinction, describing how to program, by applying standard functional programming techniques in the presence of dependent types; and how to prove, by writing custom Ltac decision procedures. -With that perspective in mind, this chapter is sort of a mirror image of the last chapter, introducing how to define predicates with inductive definitions. We will point out similarities in places, but much of the effective Coq user's bag of tricks is disjoint for predicates versus %``%#"#datatypes.#"#%''% This chapter is also a covert introduction to dependent types, which are the foundation on which interesting inductive predicates are built, though we will rely on tactics to build dependently-typed proof terms for us for now. A future chapter introduces more manual application of dependent types. *) +With that perspective in mind, this chapter is sort of a mirror image of the last chapter, introducing how to define predicates with inductive definitions. We will point out similarities in places, but much of the effective Coq user's bag of tricks is disjoint for predicates versus %``%#"#datatypes.#"#%''% This chapter is also a covert introduction to dependent types, which are the foundation on which interesting inductive predicates are built, though we will rely on tactics to build dependently typed proof terms for us for now. A future chapter introduces more manual application of dependent types. *) (** * Propositional Logic *) @@ -385,7 +385,7 @@ Hence the distinction between [bool] and [Prop]. Programs of type [bool] are computational by construction; we can always run them to determine their results. Many [Prop]s are undecidable, and so we can write more expressive formulas with [Prop]s than with [bool]s, but the inevitable consequence is that we cannot simply %``%#"#run a [Prop] to determine its truth.#"#%''% -Constructive logic lets us define all of the logical connectives in an aesthetically-appealing way, with orthogonal inductive definitions. That is, each connective is defined independently using a simple, shared mechanism. Constructivity also enables a trick called %\index{program extraction}%_program extraction_, where we write programs by phrasing them as theorems to be proved. Since our proofs are just functional programs, we can extract executable programs from our final proofs, which we could not do as naturally with classical proofs. +Constructive logic lets us define all of the logical connectives in an aesthetically appealing way, with orthogonal inductive definitions. That is, each connective is defined independently using a simple, shared mechanism. Constructivity also enables a trick called %\index{program extraction}%_program extraction_, where we write programs by phrasing them as theorems to be proved. Since our proofs are just functional programs, we can extract executable programs from our final proofs, which we could not do as naturally with classical proofs. We will see more about Coq's program extraction facility in a later chapter. However, I think it is worth interjecting another warning at this point, following up on the prior warning about taking the Curry-Howard correspondence too literally. It is possible to write programs by theorem-proving methods in Coq, but hardly anyone does it. It is almost always most useful to maintain the distinction between programs and proofs. If you write a program by proving a theorem, you are likely to run into algorithmic inefficiencies that you introduced in your proof to make it easier to prove. It is a shame to have to worry about such situations while proving tricky theorems, and it is a happy state of affairs that you almost certainly will not need to, with the ideal of extracting programs from proofs being confined mostly to theoretical studies. *)
--- a/src/StackMachine.v Wed Jun 06 11:25:13 2012 -0400 +++ b/src/StackMachine.v Fri Jun 08 11:25:11 2012 -0400 @@ -19,17 +19,17 @@ (** %\chapter{Some Quick Examples}% *) -(** I will start off by jumping right in to a fully-worked set of examples, building certified compilers from increasingly complicated source languages to stack machines. We will meet a few useful tactics and see how they can be used in manual proofs, and we will also see how easily these proofs can be automated instead. This chapter is not meant to give full explanations of the features that are employed. Rather, it is meant more as an advertisement of what is possible. Later chapters will introduce all of the concepts in bottom-up fashion. +(** I will start off by jumping right in to a fully worked set of examples, building certified compilers from increasingly complicated source languages to stack machines. We will meet a few useful tactics and see how they can be used in manual proofs, and we will also see how easily these proofs can be automated instead. This chapter is not meant to give full explanations of the features that are employed. Rather, it is meant more as an advertisement of what is possible. Later chapters will introduce all of the concepts in bottom-up fashion. As always, you can step through the source file %\texttt{%#<tt>#StackMachine.v#</tt>#%}% for this chapter interactively in Proof General. Alternatively, to get a feel for the whole lifecycle of creating a Coq development, you can enter the pieces of source code in this chapter in a new %\texttt{%#<tt>#.v#</tt>#%}% file in an Emacs buffer. If you do the latter, include two lines -%\index{Vernacular commands!Require}%[Require Import Bool] #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%Arith%}%#</span></span># #<span class="inlinecode"><span class="id" type="var">#%\coqdocconstructor{%List%}%#</span></span># [CpdtTactics.] +%\index{Vernacular commands!Require}%[Require Import Bool Arith List CpdtTactics.] %\noindent{}%and -%\index{Vernacular commands!Set Implicit Arguments}%[Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.] +%\index{Vernacular commands!Set Implicit Arguments}%[Set Implicit Arguments.] -%\noindent{}%at the start of the file, to match some code hidden in this rendering of the chapter source. In general, similar commands will be hidden in the book rendering of each chapter's source code, so you will need to insert them in from-scratch replayings of the code that is presented. To be more specific, every chapter begins with some imports of other modules, followed by [Set Implicit] #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Arguments%}%#</span></span>#[.], where the latter affects the default behavior of definitions regarding type inference. +%\noindent{}%at the start of the file, to match some code hidden in this rendering of the chapter source. In general, similar commands will be hidden in the book rendering of each chapter's source code, so you will need to insert them in from-scratch replayings of the code that is presented. To be more specific, every chapter begins with some imports of other modules, followed by [Set Implicit Arguments.], where the latter affects the default behavior of definitions regarding type inference. *) @@ -64,36 +64,32 @@ end. (** The meaning of a binary operator is a binary function over naturals, defined with pattern-matching notation analogous to the %\texttt{%#<tt>#case#</tt>#%}% and %\texttt{%#<tt>#match#</tt>#%}% of ML and Haskell, and referring to the functions [plus] and [mult] from the Coq standard library. The keyword [Definition] is Coq's all-purpose notation for binding a term of the programming language to a name, with some associated syntactic sugar, like the notation we see here for defining a function. That sugar could be expanded to yield this definition: - [[ Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) => match b with | Plus => plus | Times => mult end. - ]] In this example, we could also omit all of the type annotations, arriving at: - [[ Definition binopDenote := fun b => match b with | Plus => plus | Times => mult end. - ]] Languages like Haskell and ML have a convenient %\index{principal types}\index{type inference}%_principal types_ property, which gives us strong guarantees about how effective type inference will be. Unfortunately, Coq's type system is so expressive that any kind of %``%#"#complete#"#%''% type inference is impossible, and the task even seems to be hard heuristically in practice. Nonetheless, Coq includes some very helpful heuristics, many of them copying the workings of Haskell and ML type-checkers for programs that fall in simple fragments of Coq's language. -This is as good a time as any to mention the preponderance of different languages associated with Coq. The theoretical foundation of Coq is a formal system called the %\index{Calculus of Inductive Constructions}\index{CIC|see{Calculus of Inductive Constructions}}%_Calculus of Inductive Constructions (CIC)_ %\cite{CIC}%, which is an extension of the older %\index{Calculus of Constructions}\index{CoC|see{Calculus of Constructions}}%_Calculus of Constructions (CoC)_ %\cite{CoC}%. CIC is quite a spartan foundation, which is helpful for proving metatheory but not so helpful for real development. Still, it is nice to know that it has been proved that CIC enjoys properties like %\index{strong normalization}%_strong normalization_ %\cite{CIC}%, meaning that every program (and, more importantly, every proof term) terminates; and %\index{relative consistency}%_relative consistency_ %\cite{SetsInTypes}% with systems like versions of %\index{Zermelo-Fraenkel set theory}%Zermelo-Fraenkel set theory, which roughly means that you can believe that Coq proofs mean that the corresponding propositions are %``%#"#really true,#"#%''% if you believe in set theory. +This is as good a time as any to mention the preponderance of different languages associated with Coq. The theoretical foundation of Coq is a formal system called the%\index{Calculus of Inductive Constructions}\index{CIC|see{Calculus of Inductive Constructions}}% _Calculus of Inductive Constructions_ (CIC)%~\cite{CIC}%, which is an extension of the older%\index{Calculus of Constructions}\index{CoC|see{Calculus of Constructions}}% _Calculus of Constructions_ (CoC)%~\cite{CoC}%. CIC is quite a spartan foundation, which is helpful for proving metatheory but not so helpful for real development. Still, it is nice to know that it has been proved that CIC enjoys properties like%\index{strong normalization}% _strong normalization_ %\cite{CIC}%, meaning that every program (and, more importantly, every proof term) terminates; and%\index{relative consistency}% _relative consistency_ %\cite{SetsInTypes}% with systems like versions of %\index{Zermelo-Fraenkel set theory}%Zermelo-Fraenkel set theory, which roughly means that you can believe that Coq proofs mean that the corresponding propositions are %``%#"#really true,#"#%''% if you believe in set theory. -Coq is actually based on an extension of CIC called %\index{Gallina}%_Gallina_. The text after the [:=] and before the period in the last code example is a term of Gallina. Gallina adds many useful features that are not compiled internally to more primitive CIC features. The important metatheorems about CIC have not been extended to the full breadth of these features, but most Coq users do not seem to lose much sleep over this omission. +Coq is actually based on an extension of CIC called%\index{Gallina}% _Gallina_. The text after the [:=] and before the period in the last code example is a term of Gallina. Gallina adds many useful features that are not compiled internally to more primitive CIC features. The important metatheorems about CIC have not been extended to the full breadth of these features, but most Coq users do not seem to lose much sleep over this omission. -Next, there is %\index{Ltac}%_Ltac_, Coq's domain-specific language for writing proofs and decision procedures. We will see some basic examples of Ltac later in this chapter, and much of this book is devoted to more involved Ltac examples. +Next, there is%\index{Ltac}% _Ltac_, Coq's domain-specific language for writing proofs and decision procedures. We will see some basic examples of Ltac later in this chapter, and much of this book is devoted to more involved Ltac examples. -Finally, commands like [Inductive] and [Definition] are part of %\index{Vernacular commands}%_the Vernacular_, which includes all sorts of useful queries and requests to the Coq system. Every Coq source file is a series of vernacular commands, where many command forms take arguments that are Gallina or Ltac programs. (Actually, Coq source files are more like _trees_ of vernacular commands, thanks to various nested scoping constructs.) +Finally, commands like [Inductive] and [Definition] are part of%\index{Vernacular commands}% _the Vernacular_, which includes all sorts of useful queries and requests to the Coq system. Every Coq source file is a series of vernacular commands, where many command forms take arguments that are Gallina or Ltac programs. (Actually, Coq source files are more like _trees_ of vernacular commands, thanks to various nested scoping constructs.) %\medskip% @@ -203,12 +199,12 @@ (* end hide *) (* begin thide *) -(** Though a pencil-and-paper proof might clock out at this point, writing %``%#"#by a routine induction on [e],#"#%''% it turns out not to make sense to attack this proof directly. We need to use the standard trick of %\index{strengthening the induction hypothesis}%_strengthening the induction hypothesis_. We do that by proving an auxiliary lemma, using the command [Lemma] that is a synonym for [Theorem], conventionally used for less important theorems that appear in the proofs of primary theorems.%\index{Vernacular commands!Lemma}% *) +(** Though a pencil-and-paper proof might clock out at this point, writing %``%#"#by a routine induction on [e],#"#%''% it turns out not to make sense to attack this proof directly. We need to use the standard trick of%\index{strengthening the induction hypothesis}% _strengthening the induction hypothesis_. We do that by proving an auxiliary lemma, using the command [Lemma] that is a synonym for [Theorem], conventionally used for less important theorems that appear in the proofs of primary theorems.%\index{Vernacular commands!Lemma}% *) Lemma compile_correct' : forall e p s, progDenote (compile e ++ p) s = progDenote p (expDenote e :: s). -(** After the period in the [Lemma] command, we are in %\index{interactive proof-editing mode}%_the interactive proof-editing mode_. We find ourselves staring at this ominous screen of text: +(** After the period in the [Lemma] command, we are in%\index{interactive proof-editing mode}% _the interactive proof-editing mode_. We find ourselves staring at this ominous screen of text: [[ 1 subgoal @@ -223,7 +219,7 @@ Next in the output, we see our single subgoal described in full detail. There is a double-dashed line, above which would be our free variables and %\index{hypotheses}%hypotheses, if we had any. Below the line is the %\index{conclusion}%conclusion, which, in general, is to be proved from the hypotheses. -We manipulate the proof state by running commands called %\index{tactics}%_tactics_. Let us start out by running one of the most important tactics:%\index{tactics!induction}% +We manipulate the proof state by running commands called%\index{tactics}% _tactics_. Let us start out by running one of the most important tactics:%\index{tactics!induction}% *) induction e. @@ -271,10 +267,7 @@ To progress further, we need to use the definitions of some of the functions appearing in the goal. The [unfold] tactic replaces an identifier with its definition.%\index{tactics!unfold}% *) -(* begin hide *) unfold compile. -(* end hide *) -(** %\coqdockw{unfold} \coqdocdefinition{compile}.%#<tt>unfold compile.</tt># *) (** [[ n : nat s : stack @@ -286,10 +279,7 @@ ]] *) -(* begin hide *) unfold expDenote. -(* end hide *) -(** %\coqdockw{unfold} \coqdocdefinition{expDenote}.%#<tt>unfold expDenote.</tt># *) (** [[ n : nat s : stack @@ -301,10 +291,7 @@ We only need to unfold the first occurrence of [progDenote] to prove the goal. An [at] clause used with [unfold] specifies a particular occurrence of an identifier to unfold, where we count occurrences from left to right.%\index{tactics!unfold}% *) -(* begin hide *) unfold progDenote at 1. -(* end hide *) -(** %\coqdockw{unfold} \coqdocdefinition{progDenote} \coqdoctac{at} 1.%#<tt>unfold progDenote at 1.</tt># *) (** [[ n : nat s : stack @@ -324,7 +311,7 @@ ]] -This last [unfold] has left us with an anonymous fixpoint version of [progDenote], which will generally happen when unfolding recursive definitions. Note that Coq has automatically renamed the [fix] arguments [p] and [s] to [p0] and [s0], to avoid clashes with our local free variables. There is also a subterm %\coqdocconstructor{None} (\coqdocvar{A}:=\coqdocdefinition{stack})%#<tt>None (A:=stack)</tt>#, which has an annotation specifying that the type of the term ought to be [option A]. This is phrased as an explicit instantiation of a named type parameter [A] from the definition of [option]. +This last [unfold] has left us with an anonymous fixpoint version of [progDenote], which will generally happen when unfolding recursive definitions. Note that Coq has automatically renamed the [fix] arguments [p] and [s] to [p0] and [s0], to avoid clashes with our local free variables. There is also a subterm [None (A:=stack)], which has an annotation specifying that the type of the term ought to be [option A]. This is phrased as an explicit instantiation of a named type parameter [A] from the definition of [option]. Fortunately, in this case, we can eliminate the complications of anonymous recursion right away, since the structure of the argument ([iConst n :: nil) ++ p] is known, allowing us to simplify the internal pattern match with the [simpl] tactic, which applies the same reduction strategy that we used earlier with [Eval] (and whose details we still postpone).%\index{tactics!simpl}% *) @@ -351,10 +338,7 @@ Now we can unexpand the definition of [progDenote]:%\index{tactics!fold}% *) -(* begin hide *) fold progDenote. -(* end hide *) -(** %\coqdockw{fold} \coqdocdefinition{progDenote}.%#<tt>fold progDenote.</tt># *) (** [[ n : nat s : stack @@ -388,20 +372,13 @@ We see our first example of %\index{hypotheses}%hypotheses above the double-dashed line. They are the inductive hypotheses [IHe1] and [IHe2] corresponding to the subterms [e1] and [e2], respectively. -We start out the same way as before, introducing new free variables and unfolding and folding the appropriate definitions. The seemingly frivolous [unfold]/%\coqdockw{fold}%#<tt>fold</tt># pairs are actually accomplishing useful work, because [unfold] will sometimes perform easy simplifications. %\index{tactics!intros}\index{tactics!unfold}\index{tactics!fold}% *) +We start out the same way as before, introducing new free variables and unfolding and folding the appropriate definitions. The seemingly frivolous [unfold]/[fold] pairs are actually accomplishing useful work, because [unfold] will sometimes perform easy simplifications. %\index{tactics!intros}\index{tactics!unfold}\index{tactics!fold}% *) intros. -(* begin hide *) unfold compile. fold compile. unfold expDenote. fold expDenote. -(* end hide *) -(** %\coqdockw{unfold} \coqdocdefinition{compile}. -\coqdockw{fold} \coqdocdefinition{compile}. -\coqdockw{unfold} \coqdocdefinition{expDenote}. -\coqdockw{fold} \coqdocdefinition{expDenote}.% - #<tt>unfold compile. fold compile. unfold expDenote. fold expDenote.</tt># *) (** Now we arrive at a point where the tactics we have seen so far are insufficient. No further definition unfoldings get us anywhere, so we will need to try something different. @@ -431,12 +408,9 @@ ]] -If we did not already know the name of the theorem, we could use the %\index{Vernacular commands!SearchRewrite}\coqdockw{%#<tt>#SearchRewrite#</tt>#%}% command to find it, based on a pattern that we would like to rewrite: *) +If we did not already know the name of the theorem, we could use the %\index{Vernacular commands!SearchRewrite}%[SearchRewrite] command to find it, based on a pattern that we would like to rewrite: *) -(* begin hide *) SearchRewrite ((_ ++ _) ++ _). -(* end hide *) -(** %\coqdockw{%#<tt>#SearchRewrite#</tt>#%}% [((_ ++ _) ++ _).] *) (** [[ app_assoc_reverse: forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n @@ -481,17 +455,18 @@ Now we can apply a similar sequence of tactics to the one that ended the proof of the first case.%\index{tactics!unfold}\index{tactics!simpl}\index{tactics!fold}\index{tactics!reflexivity}% *) -(* begin hide *) unfold progDenote at 1. simpl. fold progDenote. reflexivity. -(* end hide *) -(** %\coqdockw{unfold} \coqdocdefinition{progDenote} \coqdoctac{at} 1. \coqdockw{simpl}. \coqdockw{fold} \coqdocdefinition{progDenote}. \coqdockw{reflexivity}.%#<tt>unfold progDenote at 1. simpl. fold progDenote. reflexivity.</tt># *) (** And the proof is completed, as indicated by the message: *) -(** %\coqdockw{Proof} \coqdockw{completed}.%#<tt>Proof completed.</tt># *) +(** +<< + Proof completed. +>> +*) (** And there lies our first proof. Already, even for simple theorems like this, the final proof script is unstructured and not very enlightening to readers. If we extend this approach to more serious theorems, we arrive at the unreadable proof scripts that are the favorite complaints of opponents of tactic-based proving. Fortunately, Coq has rich support for scripted automation, and we can take advantage of such a scripted tactic (defined elsewhere) to make short work of this lemma. We abort the old proof attempt and start again.%\index{Vernacular commands!Abort}% *) @@ -507,7 +482,7 @@ (** We need only to state the basic inductive proof scheme and call a tactic that automates the tedious reasoning in between. In contrast to the period tactic terminator from our last proof, the %\index{tactics!semicolon}%semicolon tactic separator supports structured, compositional proofs. The tactic [t1; t2] has the effect of running [t1] and then running [t2] on each remaining subgoal. The semicolon is one of the most fundamental building blocks of effective proof automation. The period terminator is very useful for exploratory proving, where you need to see intermediate proof states, but final proofs of any serious complexity should have just one period, terminating a single compound tactic that probably uses semicolons. -The [crush] tactic comes from the library associated with this book and is not part of the Coq standard library. The book's library contains a number of other tactics that are especially helpful in highly-automated proofs. +The [crush] tactic comes from the library associated with this book and is not part of the Coq standard library. The book's library contains a number of other tactics that are especially helpful in highly automated proofs. The %\index{Vernacular commands!Qed}%[Qed] command checks that the proof is finished and, if so, saves it. The tactic commands we have written above are an example of a _proof script_, or a series of Ltac programs; while [Qed] uses the result of the script to generate a _proof term_, a well-typed term of Gallina. To believe that a theorem is true, we only need to trust that the (relatively simple) checker for proof terms is correct; the use of proof scripts is immaterial. Part I of this book will introduce the principles behind encoding all proofs as terms of Gallina. @@ -587,12 +562,12 @@ ML and Haskell have indexed algebraic datatypes. For instance, their list types are indexed by the type of data that the list carries. However, compared to Coq, ML and Haskell 98 place two important restrictions on datatype definitions. -First, the indices of the range of each data constructor must be type variables bound at the top level of the datatype definition. There is no way to do what we did here, where we, for instance, say that [TPlus] is a constructor building a [tbinop] whose indices are all fixed at [Nat]. %\index{generalized algebraic datatypes}\index{GADTs|see{generalized algebraic datatypes}}%_Generalized algebraic datatypes (GADTs)_ %\cite{GADT}% are a popular feature in %\index{GHC Haskell}%GHC Haskell and other languages that removes this first restriction. +First, the indices of the range of each data constructor must be type variables bound at the top level of the datatype definition. There is no way to do what we did here, where we, for instance, say that [TPlus] is a constructor building a [tbinop] whose indices are all fixed at [Nat]. %\index{generalized algebraic datatypes}\index{GADTs|see{generalized algebraic datatypes}}% _Generalized algebraic datatypes_ (GADT's)%~\cite{GADT}% are a popular feature in %\index{GHC Haskell}%GHC Haskell and other languages that removes this first restriction. -The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be _expressions_. In Coq, types may be indexed by arbitrary Gallina terms. Type indices can live in the same universe as programs, and we can compute with them just like regular programs. Haskell supports a hobbled form of computation in type indices based on %\index{Haskell}%multi-parameter type classes, and recent extensions like type functions bring Haskell programming even closer to %``%#"#real#"#%''% functional programming with types, but, without dependent typing, there must always be a gap between how one programs with types and how one programs normally. +The second restriction is not lifted by GADT's. In ML and Haskell, indices of types must be types and may not be _expressions_. In Coq, types may be indexed by arbitrary Gallina terms. Type indices can live in the same universe as programs, and we can compute with them just like regular programs. Haskell supports a hobbled form of computation in type indices based on %\index{Haskell}%multi-parameter type classes, and recent extensions like type functions bring Haskell programming even closer to %``%#"#real#"#%''% functional programming with types, but, without dependent typing, there must always be a gap between how one programs with types and how one programs normally. *) -(** We can define a similar type family for typed expressions, where a term of type [texp t] can be assigned object language type [t]. (It is conventional in the world of interactive theorem proving to call the language of the proof assistant the %\index{meta language}%_meta language_ and a language being formalized the %\index{object language}%_object language_.) *) +(** We can define a similar type family for typed expressions, where a term of type [texp t] can be assigned object language type [t]. (It is conventional in the world of interactive theorem proving to call the language of the proof assistant the%\index{meta language}% _meta language_ and a language being formalized the%\index{object language}% _object language_.) *) Inductive texp : type -> Set := | TNConst : nat -> texp Nat @@ -619,7 +594,7 @@ | TLt => leb end. -(** This function has just a few differences from the denotation functions we saw earlier. First, [tbinop] is an indexed type, so its indices become additional arguments to [tbinopDenote]. Second, we need to perform a genuine %\index{dependent pattern matching}%_dependent pattern match_, where the necessary _type_ of each case body depends on the _value_ that has been matched. At this early stage, we will not go into detail on the many subtle aspects of Gallina that support dependent pattern-matching, but the subject is central to Part II of the book. +(** This function has just a few differences from the denotation functions we saw earlier. First, [tbinop] is an indexed type, so its indices become additional arguments to [tbinopDenote]. Second, we need to perform a genuine%\index{dependent pattern matching}% _dependent pattern match_, where the necessary _type_ of each case body depends on the _value_ that has been matched. At this early stage, we will not go into detail on the many subtle aspects of Gallina that support dependent pattern-matching, but the subject is central to Part II of the book. The same tricks suffice to define an expression denotation function in an unsurprising way: *) @@ -645,7 +620,7 @@ Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)). -(** [= ] %\coqdocconstructor{%#<tt>#false#</tt>#%}% [ : typeDenote Bool] *) +(** [= false : typeDenote Bool] *) Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)). @@ -780,22 +755,22 @@ (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *) Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt. -(** [= (42, tt) : vstack (][Nat :: nil)] *) +(** [= (42, tt) : vstack (Nat :: nil)] *) Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt. -(** [= (][true][, tt) : vstack (][Bool :: nil)] *) +(** [= (true, tt) : vstack (Bool :: nil)] *) Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt. -(** [= (28, tt) : vstack (][Nat :: nil)] *) +(** [= (28, tt) : vstack (Nat :: nil)] *) Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt. -(** [= (]%\coqdocconstructor{%#<tt>#false#</tt>#%}%[, tt) : vstack (][Bool :: nil)] *) +(** [= (false, tt) : vstack (Bool :: nil)] *) Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt. -(** [= (][true][, tt) : vstack (][Bool :: nil)] *) +(** [= (true, tt) : vstack (Bool :: nil)] *) (** ** Translation Correctness *) @@ -847,10 +822,7 @@ Some code behind the scenes registers [app_assoc_reverse] for use by [crush]. We must register [tconcat_correct] similarly to get the same effect:%\index{Vernacular commands!Hint Rewrite}% *) -(* begin hide *) Hint Rewrite tconcat_correct. -(* end hide *) -(** %\noindent%[Hint] %\coqdockw{%#<tt>#Rewrite#</tt>#%}% [tconcat_correct.] *) (** Here we meet the pervasive concept of a _hint_. Many proofs can be found through exhaustive enumerations of combinations of possible proof steps; hints provide the set of steps to consider. The tactic [crush] is applying such brute force search for us silently, and it will consider more possibilities as we add more hints. This particular hint asks that the lemma be used for left-to-right rewriting, and we ask for the hint to be added to the hint database called [cpdt], which is the database used by [crush]. In general, segragating hints into different databases is helpful to control the performance of proof search, in cases where domain knowledge allows us to narrow the set of proof steps to be considered in brute force search. Part III of this book considers such pragmatic aspects of proof search in much more detail. @@ -863,10 +835,7 @@ (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *) -(* begin hide *) Hint Rewrite tcompile_correct'. -(* end hide *) -(** %\noindent%[Hint ]%\coqdockw{%#<tt>#Rewrite#</tt>#%}%[ tcompile_correct'.] *) Theorem tcompile_correct : forall t (e : texp t), tprogDenote (tcompile e nil) tt = (texpDenote e, tt). @@ -874,12 +843,9 @@ Qed. (* end thide *) -(** It is probably worth emphasizing that we are doing more than building mathematical models. Our compilers are functional programs that can be executed efficiently. One strategy for doing so is based on %\index{program extraction}%_program extraction_, which generates OCaml code from Coq developments. For instance, we run a command to output the OCaml version of [tcompile]:%\index{Vernacular commands!Extraction}% *) +(** It is probably worth emphasizing that we are doing more than building mathematical models. Our compilers are functional programs that can be executed efficiently. One strategy for doing so is based on%\index{program extraction}% _program extraction_, which generates OCaml code from Coq developments. For instance, we run a command to output the OCaml version of [tcompile]:%\index{Vernacular commands!Extraction}% *) -(* begin hide *) Extraction tcompile. -(* end hide *) -(** %\noindent\coqdockw{%#<tt>#Extraction#</tt>#%}%[ tcompile.] *) (** << let rec tcompile t e ts =