changeset 206:3f4576f15130

Revising for 8.2 through first big example
author Adam Chlipala Fri, 06 Nov 2009 16:44:06 -0500 f05514cc6c0d a7be5d9a2fd4 Makefile src/Intro.v src/StackMachine.v 3 files changed, 57 insertions(+), 48 deletions(-) [+]
line wrap: on
line diff
--- a/Makefile	Fri Nov 06 12:15:05 2009 -0500
+++ b/Makefile	Fri Nov 06 16:44:06 2009 -0500
@@ -29,7 +29,7 @@
doc: latex/cpdt.dvi latex/cpdt.pdf html

latex/cpdt.tex: Makefile $(VS) - cd src ; coqdoc --latex -s$(VS_DOC) \
+	cd src ; coqdoc --interpolate --latex -s $(VS_DOC) \ -p "\usepackage{url}" \ -p "\title{Certified Programming with Dependent Types}" \ -p "\author{Adam Chlipala}" \ @@ -44,7 +44,7 @@ html: Makefile$(VS) src/toc.html
mkdir -p html
-	cd src ; coqdoc $(VS_DOC) \ + cd src ; coqdoc --interpolate$(VS_DOC) \
--glob-from ../\$(GLOBALS) \
-d ../html
cp src/toc.html html/
--- a/src/Intro.v	Fri Nov 06 12:15:05 2009 -0500
+++ b/src/Intro.v	Fri Nov 06 16:44:06 2009 -0500
@@ -1,4 +1,4 @@
*
* Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
@@ -19,7 +19,7 @@

(**

Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
@@ -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, and I will also try to mention related innovations in Agda and Epigram.  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 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.
*)

@@ -157,7 +157,7 @@
(**
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.

-A good portion of the book is about how to formalize programming languages, compilers, and proofs about them.  I depart significantly from today's most popular methodology for pencil-and-paper formalism among programming languages researchers.  There is no need to be familiar with operational semantics, preservation and progress theorems, or any of the material found in courses on programming language semantics but not in basic discrete math and logic courses.  I will use operational semantics very sparingly, and there will be no preservation or progress proofs.  Instead, I will use a style that seems to work much better in Coq, which can be given the fancy-sounding name %\textit{%#<i>#foundational type-theoretic semantics#</i>#%}% or the more populist name %\textit{%#<i>#semantics by definitional compilers#</i>#%}%.
+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.
*)


--- a/src/StackMachine.v	Fri Nov 06 12:15:05 2009 -0500
+++ b/src/StackMachine.v	Fri Nov 06 16:44:06 2009 -0500
@@ -1,4 +1,4 @@
*
* Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
@@ -19,14 +19,14 @@
(** %\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.  I assume that you have installed Coq and Proof General.  The code in this book is tested with Coq version 8.1pl3, though parts may work with other versions.
+(** 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.  I assume that you have installed Coq and Proof General.  The code in this book is tested with Coq version 8.2pl1, though parts may work with other versions.

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 a line [Require Import List Tactics.] at the start of the file, to match some code hidden in this rendering of the chapter source, and be sure to run the Coq binary %\texttt{%#<tt>#coqtop#</tt>#%}% with the command-line argument %\texttt{%#<tt>#-I SRC#</tt>#%}%, where %\texttt{%#<tt>#SRC#</tt>#%}% is the path to a directory containing the source for this book.  In either case, you will need to run %\texttt{%#<tt>#make#</tt>#%}% in the root directory of the source distribution for the book before getting started.  If you have installed Proof General properly, it should start automatically when you visit a %\texttt{%#<tt>#.v#</tt>#%}% buffer in Emacs.

There are some minor headaches associated with getting Proof General to pass the proper command line arguments to the %\texttt{%#<tt>#coqtop#</tt>#%}% program.  The best way to add settings that will be shared by many source files is to add a custom variable setting to your %\texttt{%#<tt>#.emacs#</tt>#%}% file, like this:
%\begin{verbatim}%#<pre>#(custom-set-variables
...
-  '(coq-prog-args '("-impredicative-set" "-I" "/path/to/cpdt/src"))
+  '(coq-prog-args '("-I" "/path/to/cpdt/src"))
...
)#</pre>#%\end{verbatim}%
The extra arguments demonstrated here are the proper choices for working with the code for this book.  The ellipses stand for other Emacs customization settings you may already have.  It can be helpful to save several alternate sets of flags in your %\texttt{%#<tt>#.emacs#</tt>#%}% file, with all but one commented out within the %\texttt{%#<tt>#custom-set-variables#</tt>#%}% block at any given time.
@@ -144,8 +144,9 @@
end
end.

-(** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program. *)
+(** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program.

+   [[
Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
match p with
| nil => Some s
@@ -156,7 +157,21 @@
end
end.

-(** There is one interesting difference compared to our previous example of a [Fixpoint].  This recursive function takes two arguments, [p] and [s].  It is critical for the soundness of Coq that every program terminate, so a shallow syntactic termination check is imposed on every recursive function definition.  One of the function parameters must be designated to decrease monotonically across recursive calls.  That is, every recursive call must use a version of that argument that has been pulled out of the current argument by some number of [match] expressions.  [expDenote] has only one argument, so we did not need to specify which of its arguments decreases.  For [progDenote], we resolve the ambiguity by writing [{struct p}] to indicate that argument [p] decreases structurally. *)
+  ]]
+
+  There is one interesting difference compared to our previous example of a [Fixpoint].  This recursive function takes two arguments, [p] and [s].  It is critical for the soundness of Coq that every program terminate, so a shallow syntactic termination check is imposed on every recursive function definition.  One of the function parameters must be designated to decrease monotonically across recursive calls.  That is, every recursive call must use a version of that argument that has been pulled out of the current argument by some number of [match] expressions.  [expDenote] has only one argument, so we did not need to specify which of its arguments decreases.  For [progDenote], we resolve the ambiguity by writing [{struct p}] to indicate that argument [p] decreases structurally.
+
+   Recent versions of Coq will also infer a termination argument, so that we may write simply: *)
+
+Fixpoint progDenote (p : prog) (s : stack) : option stack :=
+  match p with
+    | nil => Some s
+    | i :: p' =>
+      match instrDenote i s with
+        | None => None
+        | Some s' => progDenote p' s'
+      end
+  end.

(** ** Translation *)
@@ -173,32 +188,24 @@
(** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)

Eval simpl in compile (Const 42).
-(** [[
-= IConst 42 :: nil : prog
-]] *)
+(** [= IConst 42 :: nil : prog] *)
+
Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
-(** [[
-= IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog
-]] *)
+(** [= IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog] *)
+
Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
-(** [[
-= IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog
-]] *)
+(** [= IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog] *)

(** We can also run our compiled programs and check that they give the right results. *)

Eval simpl in progDenote (compile (Const 42)) nil.
-(** [[
-= Some (42 :: nil) : option stack
-]] *)
+(** [= Some (42 :: nil) : option stack] *)
+
Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
-(** [[
-= Some (4 :: nil) : option stack
-]] *)
+(** [= Some (4 :: nil) : option stack] *)
+
Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7))) nil.
-(** [[
-= Some (28 :: nil) : option stack
-]] *)
+(** [= Some (28 :: nil) : option stack] *)

(** ** Translation Correctness *)
@@ -214,7 +221,8 @@
(** 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 %\textit{%#<i>#strengthening the induction hypothesis#</i>#%}%.  We do that by proving an auxiliary lemma:
*)

-Lemma compile_correct' : forall e p s, progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
+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 %\textit{%#<i>#the interactive proof-editing mode#</i>#%}%.  We find ourselves staring at this ominous screen of text:

@@ -224,6 +232,7 @@
============================
forall (e : exp) (p : list instr) (s : stack),
progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
+
]]

Coq seems to be restating the lemma for us.  What we are seeing is a limited case of a more general protocol for describing where we are in a proof.  We are told that we have a single subgoal.  In general, during a proof, we can have many pending subgoals, each of which is a logical proposition to prove.  Subgoals can be proved in any order, but it usually works best to prove them in the order that Coq chooses.
@@ -251,6 +260,7 @@
forall (s : stack) (p : list instr),
progDenote (compile (Binop b e1 e2) ++ p) s =
progDenote p (expDenote (Binop b e1 e2) :: s)
+
]]

The first and current subgoal is displayed with the double-dashed line below free variables and hypotheses, while later subgoals are only summarized with their conclusions.  We see an example of a free variable in the first subgoal; [n] is a free variable of type [nat].  The conclusion is the original theorem statement where [e] has been replaced by [Const n].  In a similar manner, the second case has [e] replaced by a generalized invocation of the [Binop] expression constructor.  We can see that proving both cases corresponds to a standard proof by structural induction.
@@ -269,6 +279,7 @@
============================
progDenote (compile (Const n) ++ p) s =
progDenote p (expDenote (Const n) :: s)
+
]]

We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
@@ -284,6 +295,7 @@
============================
progDenote ((IConst n :: nil) ++ p) s =
progDenote p (expDenote (Const n) :: s)
+
]] *)

unfold expDenote.
@@ -293,6 +305,7 @@
p : list instr
============================
progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
+
]]

We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
@@ -316,15 +329,14 @@
end
end) ((IConst n :: nil) ++ p) s =
progDenote p (n :: s)
+
]]

This last [unfold] has left us with an anonymous fixpoint version of [progDenote], which will generally happen when unfolding recursive definitions.  Fortunately, in this case, we can eliminate such complications 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:
*)

simpl.
-
(** [[
-
n : nat
s : stack
p : list instr
@@ -339,6 +351,7 @@
| None => None (A:=stack)
end
end) p (n :: s) = progDenote p (n :: s)
+
]]

Now we can unexpand the definition of [progDenote]:
@@ -353,6 +366,7 @@
p : list instr
============================
progDenote p (n :: s) = progDenote p (n :: s)
+
]]

It looks like we are at the end of this case, since we have a trivial equality.  Indeed, a single tactic finishes us off:
@@ -363,7 +377,6 @@
(** On to the second inductive case:

[[
-
b : binop
e1 : exp
IHe1 : forall (s : stack) (p : list instr),
@@ -375,6 +388,7 @@
forall (s : stack) (p : list instr),
progDenote (compile (Binop b e1 e2) ++ p) s =
progDenote p (expDenote (Binop b e1 e2) :: s)
+
]]

We see our first example of hypotheses above the double-dashed line.  They are the inductive hypotheses [IHe1] and [IHe2] corresponding to the subterms [e1] and [e2], respectively.
@@ -390,7 +404,6 @@
(** 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.

[[
-
b : binop
e1 : exp
IHe1 : forall (s : stack) (p : list instr),
@@ -403,47 +416,47 @@
============================
progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
+
]]

What we need is the associative law of list concatenation, available as a theorem [app_ass] in the standard library. *)

Check app_ass.
-
(** [[
-
app_ass
: forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
+
]]

We use it to perform a rewrite: *)

rewrite app_ass.

-(** changing the conclusion to: [[
+(** changing the conclusion to:

+[[
progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
+
]]

Now we can notice that the lefthand side of the equality matches the lefthand side of the second inductive hypothesis, so we can rewrite with that hypothesis, too: *)

rewrite IHe2.
-
(** [[
-
progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
+
]]

The same process lets us apply the remaining hypothesis. *)

rewrite app_ass.
rewrite IHe1.
-
(** [[
-
progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
+
]]

Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
@@ -477,20 +490,17 @@

Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
intros.
-
(** [[
-
e : exp
============================
progDenote (compile e) nil = Some (expDenote e :: nil)
+
]]

At this point, we want to massage the lefthand side to match the statement of [compile_correct'].  A theorem from the standard library is useful: *)

Check app_nil_end.
-
(** [[
-
app_nil_end
: forall (A : Type) (l : list A), l = l ++ nil
]] *)
@@ -500,21 +510,20 @@
(** This time, we explicitly specify the value of the variable [l] from the theorem statement, since multiple expressions of list type appear in the conclusion.  [rewrite] might choose the wrong place to rewrite if we did not specify which we want.

[[
-
e : exp
============================
progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
+
]]

Now we can apply the lemma. *)

rewrite compile_correct'.
-
(** [[
-
e : exp
============================
progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
+
]]

We are almost done.  The lefthand and righthand sides can be seen to match by simple symbolic evaluation.  That means we are in luck, because Coq identifies any pair of terms as equal whenever they normalize to the same result by symbolic evaluation.  By the definition of [progDenote], that is the case here, but we do not need to worry about such details.  A simple invocation of [reflexivity] does the normalization and checks that the two results are syntactically equal. *)