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adamc@269
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1 (* Copyright (c) 2008-2010, Adam Chlipala
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2 *
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3 * This work is licensed under a
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4 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
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adamc@2
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5 * Unported License.
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6 * The license text is available at:
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7 * http://creativecommons.org/licenses/by-nc-nd/3.0/
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adamc@2
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8 *)
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9
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adamc@3
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10 (* begin hide *)
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adam@277
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11 Require Import Arith Bool List.
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12
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adamc@2
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13 Require Import Tactics.
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adamc@14
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14
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adamc@14
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15 Set Implicit Arguments.
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adamc@3
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16 (* end hide *)
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17
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18
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adamc@25
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19 (** %\chapter{Some Quick Examples}% *)
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adamc@25
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20
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adamc@25
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21
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adam@279
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22 (** 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.
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23
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24 I assume that you have installed Coq and Proof General. The code in this book is tested with Coq version 8.2pl2, though parts may work with other versions.
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25
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adam@278
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26 To set up your Proof General environment to process the source to this chapter, a few simple steps are required.
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27
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adam@278
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28 %\begin{enumerate}%#<ol>#
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29
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adam@278
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30 %\item %#<li>#Get the book source from
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31 %\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></li>#
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32
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adam@278
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33 %\item %#<li>#Unpack the tarball to some directory %\texttt{%#<tt>#DIR#</tt>#%}%.#</li>#
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34
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35 %\item %#<li>#Run %\texttt{%#<tt>#make#</tt>#%}% in %\texttt{%#<tt>#DIR#</tt>#%}%.#</li>#
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36
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adam@278
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37 %\item %#<li>#There are some minor headaches associated with getting Proof General to pass the proper command line arguments to the %\texttt{%#<tt>#coqtop#</tt>#%}% program, which provides the interactive Coq toplevel. 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:
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38 %\begin{verbatim}%#<pre>#(custom-set-variables
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adamc@25
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39 ...
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adam@278
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40 '(coq-prog-args '("-I" "DIR/src"))
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adamc@25
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41 ...
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42 )#</pre>#%\end{verbatim}%
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43 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.#</li>#
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44
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45 #</ol>#%\end{enumerate}%
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46
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47 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 [Require Import Arith Bool List Tactics.] and [Set Implicit Arguments.] 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 DIR/src#</tt>#%}%. If you have installed Proof General properly, it should start automatically when you visit a %\texttt{%#<tt>#.v#</tt>#%}% buffer in Emacs.
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48
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49 With Proof General, the portion of a buffer that Coq has processed is highlighted in some way, like being given a blue background. You step through Coq source files by positioning the point at the position you want Coq to run to and pressing C-C C-RET. This can be used both for normal step-by-step coding, by placing the point inside some command past the end of the highlighted region; and for undoing, by placing the point inside the highlighted region. *)
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adamc@9
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50
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adamc@9
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51
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adamc@20
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52 (** * Arithmetic Expressions Over Natural Numbers *)
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adamc@2
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53
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adamc@40
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54 (** We will begin with that staple of compiler textbooks, arithmetic expressions over a single type of numbers. *)
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adamc@9
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55
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adamc@20
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56 (** ** Source Language *)
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57
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adamc@9
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58 (** We begin with the syntax of the source language. *)
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adamc@2
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59
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adamc@4
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60 Inductive binop : Set := Plus | Times.
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61
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62 (** Our first line of Coq code should be unsurprising to ML and Haskell programmers. We define an algebraic datatype [binop] to stand for the binary operators of our source language. There are just two wrinkles compared to ML and Haskell. First, we use the keyword [Inductive], in place of %\texttt{%#<tt>#data#</tt>#%}%, %\texttt{%#<tt>#datatype#</tt>#%}%, or %\texttt{%#<tt>#type#</tt>#%}%. This is not just a trivial surface syntax difference; inductive types in Coq are much more expressive than garden variety algebraic datatypes, essentially enabling us to encode all of mathematics, though we begin humbly in this chapter. Second, there is the [: Set] fragment, which declares that we are defining a datatype that should be thought of as a constituent of programs. Later, we will see other options for defining datatypes in the universe of proofs or in an infinite hierarchy of universes, encompassing both programs and proofs, that is useful in higher-order constructions. *)
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63
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64 Inductive exp : Set :=
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65 | Const : nat -> exp
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adamc@4
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66 | Binop : binop -> exp -> exp -> exp.
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67
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adamc@9
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68 (** Now we define the type of arithmetic expressions. We write that a constant may be built from one argument, a natural number; and a binary operation may be built from a choice of operator and two operand expressions.
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69
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adam@277
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70 A note for readers following along in the PDF version: coqdoc supports pretty-printing of tokens in LaTeX or HTML. Where you see a right arrow character, the source contains the ASCII text %\texttt{%#<tt>#->#</tt>#%}%. Other examples of this substitution appearing in this chapter are a double right arrow for %\texttt{%#<tt>#=>#</tt>#%}%, the inverted 'A' symbol for %\texttt{%#<tt>#forall#</tt>#%}%, and the Cartesian product 'X' for %\texttt{%#<tt>#*#</tt>#%}%. When in doubt about the ASCII version of a symbol, you can consult the chapter source code.
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adamc@9
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71
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adamc@9
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72 %\medskip%
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73
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adam@292
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74 Now we are ready to say what these programs mean. We will do this by writing an interpreter that can be thought of as a trivial operational or denotational semantics. (If you are not familiar with these semantic techniques, no need to worry; we will stick to %``%#"#common sense#"#%''% constructions.) *)
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75
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adamc@4
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76 Definition binopDenote (b : binop) : nat -> nat -> nat :=
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adamc@4
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77 match b with
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adamc@4
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78 | Plus => plus
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adamc@4
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79 | Times => mult
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adamc@4
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80 end.
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adamc@2
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81
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adamc@9
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82 (** 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:
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adamc@9
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83
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adamc@9
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84 [[
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adamc@9
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85 Definition binopDenote : binop -> nat -> nat -> nat := fun (b : binop) =>
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adamc@9
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86 match b with
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adamc@9
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87 | Plus => plus
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adamc@9
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88 | Times => mult
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adamc@9
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89 end.
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adamc@9
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90
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adamc@205
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91 ]]
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adamc@205
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92
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adamc@9
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93 In this example, we could also omit all of the type annotations, arriving at:
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adamc@9
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94
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adamc@9
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95 [[
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adamc@9
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96 Definition binopDenote := fun b =>
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adamc@9
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97 match b with
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adamc@9
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98 | Plus => plus
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adamc@9
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99 | Times => mult
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adamc@9
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100 end.
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adamc@9
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101
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adamc@205
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102 ]]
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103
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adam@292
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104 Languages like Haskell and ML have a convenient %\textit{%#<i>#principal typing#</i>#%}% 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.
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105
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adam@292
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106 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 %\textit{%#<i>#Calculus of Inductive Constructions (CIC)#</i>#%}%, which is an extension of the older %\textit{%#<i>#Calculus of Constructions (CoC)#</i>#%}%. 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 %\textit{%#<i>#strong normalization#</i>#%}%, meaning that every program (and, more importantly, every proof term) terminates; and %\textit{%#<i>#relative consistency#</i>#%}% with systems like versions of 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.
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107
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108 Coq is actually based on an extension of CIC called %\textit{%#<i>#Gallina#</i>#%}%. 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.
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adamc@9
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109
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110 Commands like [Inductive] and [Definition] are part of %\textit{%#<i>#the vernacular#</i>#%}%, which includes all sorts of useful queries and requests to the Coq system.
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adamc@9
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111
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112 Finally, there is %\textit{%#<i>#Ltac#</i>#%}%, 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.
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113
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adamc@9
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114 %\medskip%
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115
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adamc@9
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116 We can give a simple definition of the meaning of an expression: *)
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117
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adamc@4
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118 Fixpoint expDenote (e : exp) : nat :=
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adamc@4
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119 match e with
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120 | Const n => n
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adamc@4
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121 | Binop b e1 e2 => (binopDenote b) (expDenote e1) (expDenote e2)
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adamc@4
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122 end.
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adamc@2
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123
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adamc@9
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124 (** We declare explicitly that this is a recursive definition, using the keyword [Fixpoint]. The rest should be old hat for functional programmers. *)
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adamc@2
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125
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adamc@16
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126 (** It is convenient to be able to test definitions before starting to prove things about them. We can verify that our semantics is sensible by evaluating some sample uses. *)
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adamc@16
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127
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adamc@16
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128 Eval simpl in expDenote (Const 42).
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adamc@205
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129 (** [= 42 : nat] *)
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adamc@205
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130
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adamc@16
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131 Eval simpl in expDenote (Binop Plus (Const 2) (Const 2)).
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adamc@205
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132 (** [= 4 : nat] *)
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adamc@205
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133
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adamc@16
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134 Eval simpl in expDenote (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
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adamc@205
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135 (** [= 28 : nat] *)
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adamc@9
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136
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adamc@20
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137 (** ** Target Language *)
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adamc@4
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138
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adamc@10
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139 (** We will compile our source programs onto a simple stack machine, whose syntax is: *)
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adamc@2
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140
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adamc@4
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141 Inductive instr : Set :=
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adamc@4
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142 | IConst : nat -> instr
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adamc@4
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143 | IBinop : binop -> instr.
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adamc@2
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144
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adamc@4
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145 Definition prog := list instr.
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adamc@4
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146 Definition stack := list nat.
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adamc@2
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147
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adamc@10
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148 (** An instruction either pushes a constant onto the stack or pops two arguments, applies a binary operator to them, and pushes the result onto the stack. A program is a list of instructions, and a stack is a list of natural numbers.
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adamc@10
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149
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adam@292
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150 We can give instructions meanings as functions from stacks to optional stacks, where running an instruction results in [None] in case of a stack underflow and results in [Some s'] when the result of execution is the new stack [s']. [::] is the %``%#"#list cons#"#%''% operator from the Coq standard library. *)
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adamc@10
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151
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adamc@4
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152 Definition instrDenote (i : instr) (s : stack) : option stack :=
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adamc@4
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153 match i with
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adamc@4
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154 | IConst n => Some (n :: s)
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adamc@4
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155 | IBinop b =>
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adamc@4
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156 match s with
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adamc@4
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157 | arg1 :: arg2 :: s' => Some ((binopDenote b) arg1 arg2 :: s')
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adamc@4
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158 | _ => None
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adamc@4
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159 end
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adamc@4
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160 end.
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adamc@2
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161
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adamc@206
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162 (** With [instrDenote] defined, it is easy to define a function [progDenote], which iterates application of [instrDenote] through a whole program.
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adamc@10
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163
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adamc@206
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164 [[
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adamc@4
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165 Fixpoint progDenote (p : prog) (s : stack) {struct p} : option stack :=
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adamc@4
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166 match p with
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adamc@4
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167 | nil => Some s
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adamc@4
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168 | i :: p' =>
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adamc@4
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169 match instrDenote i s with
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adamc@4
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170 | None => None
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adamc@4
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171 | Some s' => progDenote p' s'
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adamc@4
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172 end
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adamc@4
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173 end.
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adamc@2
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174
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adamc@206
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175 ]]
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adamc@206
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176
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adamc@206
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177 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.
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adamc@206
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178
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adamc@206
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179 Recent versions of Coq will also infer a termination argument, so that we may write simply: *)
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adamc@206
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180
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adamc@206
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181 Fixpoint progDenote (p : prog) (s : stack) : option stack :=
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adamc@206
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182 match p with
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adamc@206
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183 | nil => Some s
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adamc@206
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184 | i :: p' =>
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adamc@206
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185 match instrDenote i s with
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adamc@206
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186 | None => None
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adamc@206
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187 | Some s' => progDenote p' s'
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adamc@206
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188 end
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adamc@206
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189 end.
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adamc@2
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190
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adamc@4
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191
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adamc@9
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192 (** ** Translation *)
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adamc@4
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193
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adamc@10
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194 (** Our compiler itself is now unsurprising. [++] is the list concatenation operator from the Coq standard library. *)
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adamc@2
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195
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adamc@4
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196 Fixpoint compile (e : exp) : prog :=
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adamc@4
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197 match e with
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adamc@4
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198 | Const n => IConst n :: nil
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adamc@4
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199 | Binop b e1 e2 => compile e2 ++ compile e1 ++ IBinop b :: nil
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adamc@4
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200 end.
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adamc@2
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201
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adamc@2
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202
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adamc@16
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203 (** Before we set about proving that this compiler is correct, we can try a few test runs, using our sample programs from earlier. *)
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adamc@16
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204
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adamc@16
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205 Eval simpl in compile (Const 42).
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adamc@206
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206 (** [= IConst 42 :: nil : prog] *)
|
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adamc@206
|
207
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adamc@16
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208 Eval simpl in compile (Binop Plus (Const 2) (Const 2)).
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adamc@206
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209 (** [= IConst 2 :: IConst 2 :: IBinop Plus :: nil : prog] *)
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adamc@206
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210
|
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adamc@16
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211 Eval simpl in compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7)).
|
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adamc@206
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212 (** [= IConst 7 :: IConst 2 :: IConst 2 :: IBinop Plus :: IBinop Times :: nil : prog] *)
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adamc@16
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213
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adamc@40
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214 (** We can also run our compiled programs and check that they give the right results. *)
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adamc@16
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215
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adamc@16
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216 Eval simpl in progDenote (compile (Const 42)) nil.
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adamc@206
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217 (** [= Some (42 :: nil) : option stack] *)
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adamc@206
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218
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adamc@16
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219 Eval simpl in progDenote (compile (Binop Plus (Const 2) (Const 2))) nil.
|
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adamc@206
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220 (** [= Some (4 :: nil) : option stack] *)
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adamc@206
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221
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adamc@16
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222 Eval simpl in progDenote (compile (Binop Times (Binop Plus (Const 2) (Const 2)) (Const 7))) nil.
|
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adamc@206
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223 (** [= Some (28 :: nil) : option stack] *)
|
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adamc@16
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224
|
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adamc@16
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225
|
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adamc@20
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226 (** ** Translation Correctness *)
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adamc@4
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227
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adamc@11
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228 (** We are ready to prove that our compiler is implemented correctly. We can use a new vernacular command [Theorem] to start a correctness proof, in terms of the semantics we defined earlier: *)
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adamc@11
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229
|
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adamc@26
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230 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
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adamc@11
|
231 (* begin hide *)
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adamc@11
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232 Abort.
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adamc@11
|
233 (* end hide *)
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adamc@22
|
234 (* begin thide *)
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adamc@11
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235
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adam@292
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236 (** 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:
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adamc@11
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237 *)
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adamc@2
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238
|
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adamc@206
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239 Lemma compile_correct' : forall e p s,
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adamc@206
|
240 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s).
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|
adamc@11
|
241
|
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adamc@11
|
242 (** 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:
|
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adamc@11
|
243
|
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adamc@11
|
244 [[
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adamc@11
|
245 1 subgoal
|
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adamc@11
|
246
|
|
adamc@11
|
247 ============================
|
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adamc@15
|
248 forall (e : exp) (p : list instr) (s : stack),
|
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adamc@15
|
249 progDenote (compile e ++ p) s = progDenote p (expDenote e :: s)
|
|
adamc@206
|
250
|
|
adamc@11
|
251 ]]
|
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adamc@11
|
252
|
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adamc@11
|
253 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.
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adamc@11
|
254
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adamc@11
|
255 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 hypotheses, if we had any. Below the line is the conclusion, which, in general, is to be proved from the hypotheses.
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adamc@11
|
256
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adamc@11
|
257 We manipulate the proof state by running commands called %\textit{%#<i>#tactics#</i>#%}%. Let us start out by running one of the most important tactics:
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adamc@11
|
258 *)
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adamc@11
|
259
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adamc@4
|
260 induction e.
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adamc@2
|
261
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adamc@11
|
262 (** We declare that this proof will proceed by induction on the structure of the expression [e]. This swaps out our initial subgoal for two new subgoals, one for each case of the inductive proof:
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adamc@11
|
263
|
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adamc@11
|
264 [[
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adamc@11
|
265 2 subgoals
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adamc@11
|
266
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adamc@11
|
267 n : nat
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adamc@11
|
268 ============================
|
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adamc@11
|
269 forall (s : stack) (p : list instr),
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adamc@11
|
270 progDenote (compile (Const n) ++ p) s =
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adamc@11
|
271 progDenote p (expDenote (Const n) :: s)
|
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adamc@11
|
272 ]]
|
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adamc@11
|
273 [[
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adamc@11
|
274 subgoal 2 is:
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adamc@11
|
275 forall (s : stack) (p : list instr),
|
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adamc@11
|
276 progDenote (compile (Binop b e1 e2) ++ p) s =
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adamc@11
|
277 progDenote p (expDenote (Binop b e1 e2) :: s)
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adamc@206
|
278
|
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adamc@11
|
279 ]]
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adamc@11
|
280
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adamc@11
|
281 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.
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adamc@11
|
282
|
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adamc@11
|
283 We begin the first case with another very common tactic.
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adamc@11
|
284 *)
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adamc@11
|
285
|
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adamc@4
|
286 intros.
|
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adamc@11
|
287
|
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adamc@11
|
288 (** The current subgoal changes to:
|
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adamc@11
|
289 [[
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adamc@11
|
290
|
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adamc@11
|
291 n : nat
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adamc@11
|
292 s : stack
|
|
adamc@11
|
293 p : list instr
|
|
adamc@11
|
294 ============================
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|
adamc@11
|
295 progDenote (compile (Const n) ++ p) s =
|
|
adamc@11
|
296 progDenote p (expDenote (Const n) :: s)
|
|
adamc@206
|
297
|
|
adamc@11
|
298 ]]
|
|
adamc@11
|
299
|
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adamc@11
|
300 We see that [intros] changes [forall]-bound variables at the beginning of a goal into free variables.
|
|
adamc@11
|
301
|
|
adamc@11
|
302 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.
|
|
adamc@11
|
303 *)
|
|
adamc@11
|
304
|
|
adamc@4
|
305 unfold compile.
|
|
adamc@11
|
306 (** [[
|
|
adamc@11
|
307 n : nat
|
|
adamc@11
|
308 s : stack
|
|
adamc@11
|
309 p : list instr
|
|
adamc@11
|
310 ============================
|
|
adamc@11
|
311 progDenote ((IConst n :: nil) ++ p) s =
|
|
adamc@11
|
312 progDenote p (expDenote (Const n) :: s)
|
|
adamc@206
|
313
|
|
adamc@11
|
314 ]] *)
|
|
adamc@11
|
315
|
|
adamc@4
|
316 unfold expDenote.
|
|
adamc@11
|
317 (** [[
|
|
adamc@11
|
318 n : nat
|
|
adamc@11
|
319 s : stack
|
|
adamc@11
|
320 p : list instr
|
|
adamc@11
|
321 ============================
|
|
adamc@11
|
322 progDenote ((IConst n :: nil) ++ p) s = progDenote p (n :: s)
|
|
adamc@206
|
323
|
|
adamc@11
|
324 ]]
|
|
adamc@11
|
325
|
|
adamc@11
|
326 We only need to unfold the first occurrence of [progDenote] to prove the goal: *)
|
|
adamc@11
|
327
|
|
adamc@11
|
328 unfold progDenote at 1.
|
|
adamc@11
|
329
|
|
adamc@11
|
330 (** [[
|
|
adamc@11
|
331
|
|
adamc@11
|
332 n : nat
|
|
adamc@11
|
333 s : stack
|
|
adamc@11
|
334 p : list instr
|
|
adamc@11
|
335 ============================
|
|
adamc@11
|
336 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
|
|
adamc@11
|
337 option stack :=
|
|
adamc@11
|
338 match p0 with
|
|
adamc@11
|
339 | nil => Some s0
|
|
adamc@11
|
340 | i :: p' =>
|
|
adamc@11
|
341 match instrDenote i s0 with
|
|
adamc@11
|
342 | Some s' => progDenote p' s'
|
|
adamc@11
|
343 | None => None (A:=stack)
|
|
adamc@11
|
344 end
|
|
adamc@11
|
345 end) ((IConst n :: nil) ++ p) s =
|
|
adamc@11
|
346 progDenote p (n :: s)
|
|
adamc@206
|
347
|
|
adamc@11
|
348 ]]
|
|
adamc@11
|
349
|
|
adamc@11
|
350 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:
|
|
adamc@11
|
351 *)
|
|
adamc@11
|
352
|
|
adamc@4
|
353 simpl.
|
|
adamc@11
|
354 (** [[
|
|
adamc@11
|
355 n : nat
|
|
adamc@11
|
356 s : stack
|
|
adamc@11
|
357 p : list instr
|
|
adamc@11
|
358 ============================
|
|
adamc@11
|
359 (fix progDenote (p0 : prog) (s0 : stack) {struct p0} :
|
|
adamc@11
|
360 option stack :=
|
|
adamc@11
|
361 match p0 with
|
|
adamc@11
|
362 | nil => Some s0
|
|
adamc@11
|
363 | i :: p' =>
|
|
adamc@11
|
364 match instrDenote i s0 with
|
|
adamc@11
|
365 | Some s' => progDenote p' s'
|
|
adamc@11
|
366 | None => None (A:=stack)
|
|
adamc@11
|
367 end
|
|
adamc@11
|
368 end) p (n :: s) = progDenote p (n :: s)
|
|
adamc@206
|
369
|
|
adamc@11
|
370 ]]
|
|
adamc@11
|
371
|
|
adamc@11
|
372 Now we can unexpand the definition of [progDenote]:
|
|
adamc@11
|
373 *)
|
|
adamc@11
|
374
|
|
adamc@11
|
375 fold progDenote.
|
|
adamc@11
|
376
|
|
adamc@11
|
377 (** [[
|
|
adamc@11
|
378
|
|
adamc@11
|
379 n : nat
|
|
adamc@11
|
380 s : stack
|
|
adamc@11
|
381 p : list instr
|
|
adamc@11
|
382 ============================
|
|
adamc@11
|
383 progDenote p (n :: s) = progDenote p (n :: s)
|
|
adamc@206
|
384
|
|
adamc@11
|
385 ]]
|
|
adamc@11
|
386
|
|
adamc@11
|
387 It looks like we are at the end of this case, since we have a trivial equality. Indeed, a single tactic finishes us off:
|
|
adamc@11
|
388 *)
|
|
adamc@11
|
389
|
|
adamc@4
|
390 reflexivity.
|
|
adamc@2
|
391
|
|
adamc@11
|
392 (** On to the second inductive case:
|
|
adamc@11
|
393
|
|
adamc@11
|
394 [[
|
|
adamc@11
|
395 b : binop
|
|
adamc@11
|
396 e1 : exp
|
|
adamc@11
|
397 IHe1 : forall (s : stack) (p : list instr),
|
|
adamc@11
|
398 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
|
|
adamc@11
|
399 e2 : exp
|
|
adamc@11
|
400 IHe2 : forall (s : stack) (p : list instr),
|
|
adamc@11
|
401 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
|
|
adamc@11
|
402 ============================
|
|
adamc@11
|
403 forall (s : stack) (p : list instr),
|
|
adamc@11
|
404 progDenote (compile (Binop b e1 e2) ++ p) s =
|
|
adamc@11
|
405 progDenote p (expDenote (Binop b e1 e2) :: s)
|
|
adamc@206
|
406
|
|
adamc@11
|
407 ]]
|
|
adamc@11
|
408
|
|
adamc@11
|
409 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.
|
|
adamc@11
|
410
|
|
adamc@11
|
411 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. *)
|
|
adamc@11
|
412
|
|
adamc@4
|
413 intros.
|
|
adamc@4
|
414 unfold compile.
|
|
adamc@4
|
415 fold compile.
|
|
adamc@4
|
416 unfold expDenote.
|
|
adamc@4
|
417 fold expDenote.
|
|
adamc@11
|
418
|
|
adamc@44
|
419 (** 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.
|
|
adamc@11
|
420
|
|
adamc@11
|
421 [[
|
|
adamc@11
|
422 b : binop
|
|
adamc@11
|
423 e1 : exp
|
|
adamc@11
|
424 IHe1 : forall (s : stack) (p : list instr),
|
|
adamc@11
|
425 progDenote (compile e1 ++ p) s = progDenote p (expDenote e1 :: s)
|
|
adamc@11
|
426 e2 : exp
|
|
adamc@11
|
427 IHe2 : forall (s : stack) (p : list instr),
|
|
adamc@11
|
428 progDenote (compile e2 ++ p) s = progDenote p (expDenote e2 :: s)
|
|
adamc@11
|
429 s : stack
|
|
adamc@11
|
430 p : list instr
|
|
adamc@11
|
431 ============================
|
|
adamc@11
|
432 progDenote ((compile e2 ++ compile e1 ++ IBinop b :: nil) ++ p) s =
|
|
adamc@11
|
433 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
|
|
adamc@206
|
434
|
|
adamc@11
|
435 ]]
|
|
adamc@11
|
436
|
|
adam@277
|
437 What we need is the associative law of list concatenation, which is available as a theorem [app_ass] in the standard library. *)
|
|
adamc@11
|
438
|
|
adamc@11
|
439 Check app_ass.
|
|
adamc@11
|
440 (** [[
|
|
adamc@11
|
441 app_ass
|
|
adamc@11
|
442 : forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
|
|
adamc@206
|
443
|
|
adamc@11
|
444 ]]
|
|
adamc@11
|
445
|
|
adam@277
|
446 If we did not already know the name of the theorem, we could use the [SearchRewrite] command to find it, based on a pattern that we would like to rewrite: *)
|
|
adam@277
|
447
|
|
adam@277
|
448 SearchRewrite ((_ ++ _) ++ _).
|
|
adam@277
|
449 (** [[
|
|
adam@277
|
450 app_ass: forall (A : Type) (l m n : list A), (l ++ m) ++ n = l ++ m ++ n
|
|
adam@277
|
451 ass_app: forall (A : Type) (l m n : list A), l ++ m ++ n = (l ++ m) ++ n
|
|
adam@277
|
452
|
|
adam@277
|
453 ]]
|
|
adam@277
|
454
|
|
adamc@11
|
455 We use it to perform a rewrite: *)
|
|
adamc@11
|
456
|
|
adamc@4
|
457 rewrite app_ass.
|
|
adamc@11
|
458
|
|
adamc@206
|
459 (** changing the conclusion to:
|
|
adamc@11
|
460
|
|
adamc@206
|
461 [[
|
|
adamc@11
|
462 progDenote (compile e2 ++ (compile e1 ++ IBinop b :: nil) ++ p) s =
|
|
adamc@11
|
463 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
|
|
adamc@206
|
464
|
|
adamc@11
|
465 ]]
|
|
adamc@11
|
466
|
|
adamc@11
|
467 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: *)
|
|
adamc@11
|
468
|
|
adamc@4
|
469 rewrite IHe2.
|
|
adamc@11
|
470 (** [[
|
|
adamc@11
|
471 progDenote ((compile e1 ++ IBinop b :: nil) ++ p) (expDenote e2 :: s) =
|
|
adamc@11
|
472 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
|
|
adamc@206
|
473
|
|
adamc@11
|
474 ]]
|
|
adamc@11
|
475
|
|
adamc@11
|
476 The same process lets us apply the remaining hypothesis. *)
|
|
adamc@11
|
477
|
|
adamc@4
|
478 rewrite app_ass.
|
|
adamc@4
|
479 rewrite IHe1.
|
|
adamc@11
|
480 (** [[
|
|
adamc@11
|
481 progDenote ((IBinop b :: nil) ++ p) (expDenote e1 :: expDenote e2 :: s) =
|
|
adamc@11
|
482 progDenote p (binopDenote b (expDenote e1) (expDenote e2) :: s)
|
|
adamc@206
|
483
|
|
adamc@11
|
484 ]]
|
|
adamc@11
|
485
|
|
adamc@11
|
486 Now we can apply a similar sequence of tactics to that that ended the proof of the first case.
|
|
adamc@11
|
487 *)
|
|
adamc@11
|
488
|
|
adamc@11
|
489 unfold progDenote at 1.
|
|
adamc@4
|
490 simpl.
|
|
adamc@11
|
491 fold progDenote.
|
|
adamc@4
|
492 reflexivity.
|
|
adamc@11
|
493
|
|
adamc@11
|
494 (** And the proof is completed, as indicated by the message:
|
|
adamc@11
|
495
|
|
adamc@11
|
496 [[
|
|
adamc@11
|
497 Proof completed.
|
|
adamc@11
|
498
|
|
adamc@205
|
499 ]]
|
|
adamc@205
|
500
|
|
adamc@11
|
501 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.
|
|
adamc@11
|
502 *)
|
|
adamc@11
|
503
|
|
adamc@4
|
504 Abort.
|
|
adamc@2
|
505
|
|
adamc@26
|
506 Lemma compile_correct' : forall e s p, progDenote (compile e ++ p) s =
|
|
adamc@4
|
507 progDenote p (expDenote e :: s).
|
|
adamc@4
|
508 induction e; crush.
|
|
adamc@4
|
509 Qed.
|
|
adamc@2
|
510
|
|
adamc@11
|
511 (** 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 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.
|
|
adamc@11
|
512
|
|
adamc@210
|
513 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.
|
|
adamc@210
|
514
|
|
adamc@11
|
515 The proof of our main theorem is now easy. We prove it with four period-terminated tactics, though separating them with semicolons would work as well; the version here is easier to step through. *)
|
|
adamc@11
|
516
|
|
adamc@26
|
517 Theorem compile_correct : forall e, progDenote (compile e) nil = Some (expDenote e :: nil).
|
|
adamc@11
|
518 intros.
|
|
adamc@11
|
519 (** [[
|
|
adamc@11
|
520 e : exp
|
|
adamc@11
|
521 ============================
|
|
adamc@11
|
522 progDenote (compile e) nil = Some (expDenote e :: nil)
|
|
adamc@206
|
523
|
|
adamc@11
|
524 ]]
|
|
adamc@11
|
525
|
|
adamc@26
|
526 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: *)
|
|
adamc@11
|
527
|
|
adamc@11
|
528 Check app_nil_end.
|
|
adamc@11
|
529 (** [[
|
|
adamc@11
|
530 app_nil_end
|
|
adamc@11
|
531 : forall (A : Type) (l : list A), l = l ++ nil
|
|
adamc@11
|
532 ]] *)
|
|
adamc@11
|
533
|
|
adamc@4
|
534 rewrite (app_nil_end (compile e)).
|
|
adamc@11
|
535
|
|
adamc@11
|
536 (** 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.
|
|
adamc@11
|
537
|
|
adamc@11
|
538 [[
|
|
adamc@11
|
539 e : exp
|
|
adamc@11
|
540 ============================
|
|
adamc@11
|
541 progDenote (compile e ++ nil) nil = Some (expDenote e :: nil)
|
|
adamc@206
|
542
|
|
adamc@11
|
543 ]]
|
|
adamc@11
|
544
|
|
adamc@11
|
545 Now we can apply the lemma. *)
|
|
adamc@11
|
546
|
|
adamc@26
|
547 rewrite compile_correct'.
|
|
adamc@11
|
548 (** [[
|
|
adamc@11
|
549 e : exp
|
|
adamc@11
|
550 ============================
|
|
adamc@11
|
551 progDenote nil (expDenote e :: nil) = Some (expDenote e :: nil)
|
|
adamc@206
|
552
|
|
adamc@11
|
553 ]]
|
|
adamc@11
|
554
|
|
adamc@11
|
555 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. *)
|
|
adamc@11
|
556
|
|
adamc@4
|
557 reflexivity.
|
|
adamc@4
|
558 Qed.
|
|
adamc@22
|
559 (* end thide *)
|
|
adamc@14
|
560
|
|
adamc@14
|
561
|
|
adamc@20
|
562 (** * Typed Expressions *)
|
|
adamc@14
|
563
|
|
adamc@14
|
564 (** In this section, we will build on the initial example by adding additional expression forms that depend on static typing of terms for safety. *)
|
|
adamc@14
|
565
|
|
adamc@20
|
566 (** ** Source Language *)
|
|
adamc@14
|
567
|
|
adamc@15
|
568 (** We define a trivial language of types to classify our expressions: *)
|
|
adamc@15
|
569
|
|
adamc@14
|
570 Inductive type : Set := Nat | Bool.
|
|
adamc@14
|
571
|
|
adam@277
|
572 (** Like most programming languages, Coq uses case-sensitive variable names, so that our user-defined type [type] is distinct from the [Type] keyword that we have already seen appear in the statement of a polymorphic theorem (and that we will meet in more detail later), and our constructor names [Nat] and [Bool] are distinct from the types [nat] and [bool] in the standard library.
|
|
adam@277
|
573
|
|
adam@277
|
574 Now we define an expanded set of binary operators. *)
|
|
adamc@15
|
575
|
|
adamc@14
|
576 Inductive tbinop : type -> type -> type -> Set :=
|
|
adamc@14
|
577 | TPlus : tbinop Nat Nat Nat
|
|
adamc@14
|
578 | TTimes : tbinop Nat Nat Nat
|
|
adamc@14
|
579 | TEq : forall t, tbinop t t Bool
|
|
adamc@14
|
580 | TLt : tbinop Nat Nat Bool.
|
|
adamc@14
|
581
|
|
adamc@15
|
582 (** The definition of [tbinop] is different from [binop] in an important way. Where we declared that [binop] has type [Set], here we declare that [tbinop] has type [type -> type -> type -> Set]. We define [tbinop] as an %\textit{%#<i>#indexed type family#</i>#%}%. Indexed inductive types are at the heart of Coq's expressive power; almost everything else of interest is defined in terms of them.
|
|
adamc@15
|
583
|
|
adamc@15
|
584 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.
|
|
adamc@15
|
585
|
|
adamc@15
|
586 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]. %\textit{%#<i>#Generalized algebraic datatypes (GADTs)#</i>#%}% are a popular feature in GHC Haskell and other languages that removes this first restriction.
|
|
adamc@15
|
587
|
|
adam@292
|
588 The second restriction is not lifted by GADTs. In ML and Haskell, indices of types must be types and may not be %\textit{%#<i>#expressions#</i>#%}%. 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 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.
|
|
adamc@15
|
589 *)
|
|
adamc@15
|
590
|
|
adamc@15
|
591 (** We can define a similar type family for typed expressions. *)
|
|
adamc@15
|
592
|
|
adamc@14
|
593 Inductive texp : type -> Set :=
|
|
adamc@14
|
594 | TNConst : nat -> texp Nat
|
|
adamc@14
|
595 | TBConst : bool -> texp Bool
|
|
adamc@14
|
596 | TBinop : forall arg1 arg2 res, tbinop arg1 arg2 res -> texp arg1 -> texp arg2 -> texp res.
|
|
adamc@14
|
597
|
|
adamc@15
|
598 (** Thanks to our use of dependent types, every well-typed [texp] represents a well-typed source expression, by construction. This turns out to be very convenient for many things we might want to do with expressions. For instance, it is easy to adapt our interpreter approach to defining semantics. We start by defining a function mapping the types of our languages into Coq types: *)
|
|
adamc@15
|
599
|
|
adamc@14
|
600 Definition typeDenote (t : type) : Set :=
|
|
adamc@14
|
601 match t with
|
|
adamc@14
|
602 | Nat => nat
|
|
adamc@14
|
603 | Bool => bool
|
|
adamc@14
|
604 end.
|
|
adamc@14
|
605
|
|
adamc@15
|
606 (** It can take a few moments to come to terms with the fact that [Set], the type of types of programs, is itself a first-class type, and that we can write functions that return [Set]s. Past that wrinkle, the definition of [typeDenote] is trivial, relying on the [nat] and [bool] types from the Coq standard library.
|
|
adamc@15
|
607
|
|
adam@292
|
608 We need to define one auxiliary function, implementing a boolean binary %``%#"#less-than#"#%''% operator, which only appears in the standard library with a type fancier than what we are prepared to deal with here. The code is entirely standard and ML-like, with the one caveat being that the Coq [nat] type uses a unary representation, where [O] is zero and [S n] is the successor of [n].
|
|
adamc@15
|
609 *)
|
|
adamc@15
|
610
|
|
adam@279
|
611 Fixpoint lessThan (n1 n2 : nat) : bool :=
|
|
adamc@14
|
612 match n1, n2 with
|
|
adamc@14
|
613 | O, S _ => true
|
|
adam@279
|
614 | S n1', S n2' => lessThan n1' n2'
|
|
adamc@14
|
615 | _, _ => false
|
|
adamc@14
|
616 end.
|
|
adamc@14
|
617
|
|
adam@277
|
618 (** Now we can interpret binary operators, relying on standard-library equality test functions [eqb] and [beq_nat] for booleans and naturals, respectively: *)
|
|
adamc@15
|
619
|
|
adamc@14
|
620 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
|
|
adamc@14
|
621 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
|
|
adamc@207
|
622 match b in (tbinop arg1 arg2 res)
|
|
adamc@207
|
623 return (typeDenote arg1 -> typeDenote arg2 -> typeDenote res) with
|
|
adamc@14
|
624 | TPlus => plus
|
|
adamc@14
|
625 | TTimes => mult
|
|
adam@277
|
626 | TEq Nat => beq_nat
|
|
adam@277
|
627 | TEq Bool => eqb
|
|
adam@279
|
628 | TLt => lessThan
|
|
adamc@14
|
629 end.
|
|
adamc@14
|
630
|
|
adamc@207
|
631 (** 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 %\textit{%#<i>#dependent pattern match#</i>#%}% to come up with a definition of this function that type-checks. In each branch of the [match], we need to use branch-specific information about the indices to [tbinop]. General type inference that takes such information into account is undecidable, so it is often necessary to write annotations, like we see above on the line with [match].
|
|
adamc@15
|
632
|
|
adamc@273
|
633 The [in] annotation restates the type of the term being case-analyzed. Though we use the same names for the indices as we use in the type of the original argument binder, these are actually fresh variables, and they are %\textit{%#<i>#binding occurrences#</i>#%}%. Their scope is the [return] clause. That is, [arg1], [arg2], and [res] are new bound variables bound only within the return clause [typeDenote arg1 -> typeDenote arg2 -> typeDenote res]. By being explicit about the functional relationship between the type indices and the match result, we regain decidable type inference.
|
|
adamc@15
|
634
|
|
adamc@207
|
635 In fact, recent Coq versions use some heuristics that can save us the trouble of writing [match] annotations, and those heuristics get the job done in this case. We can get away with writing just: *)
|
|
adamc@207
|
636
|
|
adamc@207
|
637 (* begin hide *)
|
|
adamc@207
|
638 Reset tbinopDenote.
|
|
adamc@207
|
639 (* end hide *)
|
|
adamc@207
|
640 Definition tbinopDenote arg1 arg2 res (b : tbinop arg1 arg2 res)
|
|
adamc@207
|
641 : typeDenote arg1 -> typeDenote arg2 -> typeDenote res :=
|
|
adamc@207
|
642 match b with
|
|
adamc@207
|
643 | TPlus => plus
|
|
adamc@207
|
644 | TTimes => mult
|
|
adam@277
|
645 | TEq Nat => beq_nat
|
|
adam@277
|
646 | TEq Bool => eqb
|
|
adam@279
|
647 | TLt => lessThan
|
|
adamc@207
|
648 end.
|
|
adamc@207
|
649
|
|
adamc@207
|
650 (**
|
|
adamc@15
|
651 The same tricks suffice to define an expression denotation function in an unsurprising way:
|
|
adamc@15
|
652 *)
|
|
adamc@15
|
653
|
|
adamc@207
|
654 Fixpoint texpDenote t (e : texp t) : typeDenote t :=
|
|
adamc@207
|
655 match e with
|
|
adamc@14
|
656 | TNConst n => n
|
|
adamc@14
|
657 | TBConst b => b
|
|
adamc@14
|
658 | TBinop _ _ _ b e1 e2 => (tbinopDenote b) (texpDenote e1) (texpDenote e2)
|
|
adamc@14
|
659 end.
|
|
adamc@14
|
660
|
|
adamc@17
|
661 (** We can evaluate a few example programs to convince ourselves that this semantics is correct. *)
|
|
adamc@17
|
662
|
|
adamc@17
|
663 Eval simpl in texpDenote (TNConst 42).
|
|
adamc@207
|
664 (** [= 42 : typeDenote Nat] *)
|
|
adamc@207
|
665
|
|
adamc@17
|
666 Eval simpl in texpDenote (TBConst true).
|
|
adamc@207
|
667 (** [= true : typeDenote Bool] *)
|
|
adamc@207
|
668
|
|
adamc@17
|
669 Eval simpl in texpDenote (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
|
|
adamc@207
|
670 (** [= 28 : typeDenote Nat] *)
|
|
adamc@207
|
671
|
|
adamc@17
|
672 Eval simpl in texpDenote (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
|
|
adamc@207
|
673 (** [= false : typeDenote Bool] *)
|
|
adamc@207
|
674
|
|
adamc@17
|
675 Eval simpl in texpDenote (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)).
|
|
adamc@207
|
676 (** [= true : typeDenote Bool] *)
|
|
adamc@17
|
677
|
|
adamc@14
|
678
|
|
adamc@20
|
679 (** ** Target Language *)
|
|
adamc@14
|
680
|
|
adam@292
|
681 (** Now we want to define a suitable stack machine target for compilation. In the example of the untyped language, stack machine programs could encounter stack underflows and %``%#"#get stuck.#"#%''% This was unfortunate, since we had to deal with this complication even though we proved that our compiler never produced underflowing programs. We could have used dependent types to force all stack machine programs to be underflow-free.
|
|
adamc@18
|
682
|
|
adamc@18
|
683 For our new languages, besides underflow, we also have the problem of stack slots with naturals instead of bools or vice versa. This time, we will use indexed typed families to avoid the need to reason about potential failures.
|
|
adamc@18
|
684
|
|
adamc@18
|
685 We start by defining stack types, which classify sets of possible stacks. *)
|
|
adamc@18
|
686
|
|
adamc@14
|
687 Definition tstack := list type.
|
|
adamc@14
|
688
|
|
adamc@18
|
689 (** Any stack classified by a [tstack] must have exactly as many elements, and each stack element must have the type found in the same position of the stack type.
|
|
adamc@18
|
690
|
|
adamc@18
|
691 We can define instructions in terms of stack types, where every instruction's type tells us what initial stack type it expects and what final stack type it will produce. *)
|
|
adamc@18
|
692
|
|
adamc@14
|
693 Inductive tinstr : tstack -> tstack -> Set :=
|
|
adamc@14
|
694 | TINConst : forall s, nat -> tinstr s (Nat :: s)
|
|
adamc@14
|
695 | TIBConst : forall s, bool -> tinstr s (Bool :: s)
|
|
adamc@14
|
696 | TIBinop : forall arg1 arg2 res s,
|
|
adamc@14
|
697 tbinop arg1 arg2 res
|
|
adamc@14
|
698 -> tinstr (arg1 :: arg2 :: s) (res :: s).
|
|
adamc@14
|
699
|
|
adamc@18
|
700 (** Stack machine programs must be a similar inductive family, since, if we again used the [list] type family, we would not be able to guarantee that intermediate stack types match within a program. *)
|
|
adamc@18
|
701
|
|
adamc@14
|
702 Inductive tprog : tstack -> tstack -> Set :=
|
|
adamc@14
|
703 | TNil : forall s, tprog s s
|
|
adamc@14
|
704 | TCons : forall s1 s2 s3,
|
|
adamc@14
|
705 tinstr s1 s2
|
|
adamc@14
|
706 -> tprog s2 s3
|
|
adamc@14
|
707 -> tprog s1 s3.
|
|
adamc@14
|
708
|
|
adamc@18
|
709 (** Now, to define the semantics of our new target language, we need a representation for stacks at runtime. We will again take advantage of type information to define types of value stacks that, by construction, contain the right number and types of elements. *)
|
|
adamc@18
|
710
|
|
adamc@14
|
711 Fixpoint vstack (ts : tstack) : Set :=
|
|
adamc@14
|
712 match ts with
|
|
adamc@14
|
713 | nil => unit
|
|
adamc@14
|
714 | t :: ts' => typeDenote t * vstack ts'
|
|
adamc@14
|
715 end%type.
|
|
adamc@14
|
716
|
|
adamc@210
|
717 (** This is another [Set]-valued function. This time it is recursive, which is perfectly valid, since [Set] is not treated specially in determining which functions may be written. We say that the value stack of an empty stack type is any value of type [unit], which has just a single value, [tt]. A nonempty stack type leads to a value stack that is a pair, whose first element has the proper type and whose second element follows the representation for the remainder of the stack type. We write [%type] so that Coq knows to interpret [*] as Cartesian product rather than multiplication.
|
|
adamc@18
|
718
|
|
adamc@207
|
719 This idea of programming with types can take a while to internalize, but it enables a very simple definition of instruction denotation. Our definition is like what you might expect from a Lisp-like version of ML that ignored type information. Nonetheless, the fact that [tinstrDenote] passes the type-checker guarantees that our stack machine programs can never go wrong. *)
|
|
adamc@18
|
720
|
|
adamc@14
|
721 Definition tinstrDenote ts ts' (i : tinstr ts ts') : vstack ts -> vstack ts' :=
|
|
adamc@207
|
722 match i with
|
|
adamc@14
|
723 | TINConst _ n => fun s => (n, s)
|
|
adamc@14
|
724 | TIBConst _ b => fun s => (b, s)
|
|
adamc@14
|
725 | TIBinop _ _ _ _ b => fun s =>
|
|
adamc@14
|
726 match s with
|
|
adamc@14
|
727 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
|
|
adamc@14
|
728 end
|
|
adamc@14
|
729 end.
|
|
adamc@14
|
730
|
|
adamc@18
|
731 (** Why do we choose to use an anonymous function to bind the initial stack in every case of the [match]? Consider this well-intentioned but invalid alternative version:
|
|
adamc@18
|
732
|
|
adamc@18
|
733 [[
|
|
adamc@18
|
734 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
|
|
adamc@207
|
735 match i with
|
|
adamc@18
|
736 | TINConst _ n => (n, s)
|
|
adamc@18
|
737 | TIBConst _ b => (b, s)
|
|
adamc@18
|
738 | TIBinop _ _ _ _ b =>
|
|
adamc@18
|
739 match s with
|
|
adamc@18
|
740 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
|
|
adamc@18
|
741 end
|
|
adamc@18
|
742 end.
|
|
adamc@18
|
743
|
|
adamc@205
|
744 ]]
|
|
adamc@205
|
745
|
|
adamc@18
|
746 The Coq type-checker complains that:
|
|
adamc@18
|
747
|
|
adamc@18
|
748 [[
|
|
adamc@18
|
749 The term "(n, s)" has type "(nat * vstack ts)%type"
|
|
adamc@207
|
750 while it is expected to have type "vstack ?119".
|
|
adamc@207
|
751
|
|
adamc@207
|
752 ]]
|
|
adamc@207
|
753
|
|
adamc@207
|
754 The text [?119] stands for a unification variable. We can try to help Coq figure out the value of this variable with an explicit annotation on our [match] expression.
|
|
adamc@207
|
755
|
|
adamc@207
|
756 [[
|
|
adamc@207
|
757 Definition tinstrDenote ts ts' (i : tinstr ts ts') (s : vstack ts) : vstack ts' :=
|
|
adamc@207
|
758 match i in tinstr ts ts' return vstack ts' with
|
|
adamc@207
|
759 | TINConst _ n => (n, s)
|
|
adamc@207
|
760 | TIBConst _ b => (b, s)
|
|
adamc@207
|
761 | TIBinop _ _ _ _ b =>
|
|
adamc@207
|
762 match s with
|
|
adamc@207
|
763 (arg1, (arg2, s')) => ((tbinopDenote b) arg1 arg2, s')
|
|
adamc@207
|
764 end
|
|
adamc@207
|
765 end.
|
|
adamc@207
|
766
|
|
adamc@207
|
767 ]]
|
|
adamc@207
|
768
|
|
adamc@207
|
769 Now the error message changes.
|
|
adamc@207
|
770
|
|
adamc@207
|
771 [[
|
|
adamc@207
|
772 The term "(n, s)" has type "(nat * vstack ts)%type"
|
|
adamc@207
|
773 while it is expected to have type "vstack (Nat :: t)".
|
|
adamc@207
|
774
|
|
adamc@18
|
775 ]]
|
|
adamc@18
|
776
|
|
adamc@18
|
777 Recall from our earlier discussion of [match] annotations that we write the annotations to express to the type-checker the relationship between the type indices of the case object and the result type of the [match]. Coq chooses to assign to the wildcard [_] after [TINConst] the name [t], and the type error is telling us that the type checker cannot prove that [t] is the same as [ts]. By moving [s] out of the [match], we lose the ability to express, with [in] and [return] clauses, the relationship between the shared index [ts] of [s] and [i].
|
|
adamc@18
|
778
|
|
adamc@18
|
779 There %\textit{%#<i>#are#</i>#%}% reasonably general ways of getting around this problem without pushing binders inside [match]es. However, the alternatives are significantly more involved, and the technique we use here is almost certainly the best choice, whenever it applies.
|
|
adamc@18
|
780
|
|
adamc@18
|
781 *)
|
|
adamc@18
|
782
|
|
adamc@18
|
783 (** We finish the semantics with a straightforward definition of program denotation. *)
|
|
adamc@18
|
784
|
|
adamc@207
|
785 Fixpoint tprogDenote ts ts' (p : tprog ts ts') : vstack ts -> vstack ts' :=
|
|
adamc@207
|
786 match p with
|
|
adamc@14
|
787 | TNil _ => fun s => s
|
|
adamc@14
|
788 | TCons _ _ _ i p' => fun s => tprogDenote p' (tinstrDenote i s)
|
|
adamc@14
|
789 end.
|
|
adamc@14
|
790
|
|
adamc@14
|
791
|
|
adamc@14
|
792 (** ** Translation *)
|
|
adamc@14
|
793
|
|
adamc@19
|
794 (** To define our compilation, it is useful to have an auxiliary function for concatenating two stack machine programs. *)
|
|
adamc@19
|
795
|
|
adamc@207
|
796 Fixpoint tconcat ts ts' ts'' (p : tprog ts ts') : tprog ts' ts'' -> tprog ts ts'' :=
|
|
adamc@207
|
797 match p with
|
|
adamc@14
|
798 | TNil _ => fun p' => p'
|
|
adamc@14
|
799 | TCons _ _ _ i p1 => fun p' => TCons i (tconcat p1 p')
|
|
adamc@14
|
800 end.
|
|
adamc@14
|
801
|
|
adamc@19
|
802 (** With that function in place, the compilation is defined very similarly to how it was before, modulo the use of dependent typing. *)
|
|
adamc@19
|
803
|
|
adamc@207
|
804 Fixpoint tcompile t (e : texp t) (ts : tstack) : tprog ts (t :: ts) :=
|
|
adamc@207
|
805 match e with
|
|
adamc@14
|
806 | TNConst n => TCons (TINConst _ n) (TNil _)
|
|
adamc@14
|
807 | TBConst b => TCons (TIBConst _ b) (TNil _)
|
|
adamc@14
|
808 | TBinop _ _ _ b e1 e2 => tconcat (tcompile e2 _)
|
|
adamc@14
|
809 (tconcat (tcompile e1 _) (TCons (TIBinop _ b) (TNil _)))
|
|
adamc@14
|
810 end.
|
|
adamc@14
|
811
|
|
adamc@40
|
812 (** One interesting feature of the definition is the underscores appearing to the right of [=>] arrows. Haskell and ML programmers are quite familiar with compilers that infer type parameters to polymorphic values. In Coq, it is possible to go even further and ask the system to infer arbitrary terms, by writing underscores in place of specific values. You may have noticed that we have been calling functions without specifying all of their arguments. For instance, the recursive calls here to [tcompile] omit the [t] argument. Coq's %\textit{%#<i>#implicit argument#</i>#%}% mechanism automatically inserts underscores for arguments that it will probably be able to infer. Inference of such values is far from complete, though; generally, it only works in cases similar to those encountered with polymorphic type instantiation in Haskell and ML.
|
|
adamc@19
|
813
|
|
adamc@19
|
814 The underscores here are being filled in with stack types. That is, the Coq type inferencer is, in a sense, inferring something about the flow of control in the translated programs. We can take a look at exactly which values are filled in: *)
|
|
adamc@19
|
815
|
|
adamc@14
|
816 Print tcompile.
|
|
adamc@19
|
817 (** [[
|
|
adamc@19
|
818 tcompile =
|
|
adamc@19
|
819 fix tcompile (t : type) (e : texp t) (ts : tstack) {struct e} :
|
|
adamc@19
|
820 tprog ts (t :: ts) :=
|
|
adamc@19
|
821 match e in (texp t0) return (tprog ts (t0 :: ts)) with
|
|
adamc@19
|
822 | TNConst n => TCons (TINConst ts n) (TNil (Nat :: ts))
|
|
adamc@19
|
823 | TBConst b => TCons (TIBConst ts b) (TNil (Bool :: ts))
|
|
adamc@19
|
824 | TBinop arg1 arg2 res b e1 e2 =>
|
|
adamc@19
|
825 tconcat (tcompile arg2 e2 ts)
|
|
adamc@19
|
826 (tconcat (tcompile arg1 e1 (arg2 :: ts))
|
|
adamc@19
|
827 (TCons (TIBinop ts b) (TNil (res :: ts))))
|
|
adamc@19
|
828 end
|
|
adamc@19
|
829 : forall t : type, texp t -> forall ts : tstack, tprog ts (t :: ts)
|
|
adamc@19
|
830 ]] *)
|
|
adamc@19
|
831
|
|
adamc@19
|
832
|
|
adamc@19
|
833 (** We can check that the compiler generates programs that behave appropriately on our sample programs from above: *)
|
|
adamc@19
|
834
|
|
adamc@19
|
835 Eval simpl in tprogDenote (tcompile (TNConst 42) nil) tt.
|
|
adamc@207
|
836 (** [= (42, tt) : vstack (Nat :: nil)] *)
|
|
adamc@207
|
837
|
|
adamc@19
|
838 Eval simpl in tprogDenote (tcompile (TBConst true) nil) tt.
|
|
adamc@207
|
839 (** [= (true, tt) : vstack (Bool :: nil)] *)
|
|
adamc@207
|
840
|
|
adamc@19
|
841 Eval simpl in tprogDenote (tcompile (TBinop TTimes (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
|
|
adamc@207
|
842 (** [= (28, tt) : vstack (Nat :: nil)] *)
|
|
adamc@207
|
843
|
|
adamc@19
|
844 Eval simpl in tprogDenote (tcompile (TBinop (TEq Nat) (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
|
|
adamc@207
|
845 (** [= (false, tt) : vstack (Bool :: nil)] *)
|
|
adamc@207
|
846
|
|
adamc@19
|
847 Eval simpl in tprogDenote (tcompile (TBinop TLt (TBinop TPlus (TNConst 2) (TNConst 2)) (TNConst 7)) nil) tt.
|
|
adamc@207
|
848 (** [= (true, tt) : vstack (Bool :: nil)] *)
|
|
adamc@19
|
849
|
|
adamc@14
|
850
|
|
adamc@20
|
851 (** ** Translation Correctness *)
|
|
adamc@20
|
852
|
|
adamc@20
|
853 (** We can state a correctness theorem similar to the last one. *)
|
|
adamc@20
|
854
|
|
adamc@207
|
855 Theorem tcompile_correct : forall t (e : texp t),
|
|
adamc@207
|
856 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
|
|
adamc@20
|
857 (* begin hide *)
|
|
adamc@20
|
858 Abort.
|
|
adamc@20
|
859 (* end hide *)
|
|
adamc@22
|
860 (* begin thide *)
|
|
adamc@20
|
861
|
|
adamc@20
|
862 (** Again, we need to strengthen the theorem statement so that the induction will go through. This time, I will develop an alternative approach to this kind of proof, stating the key lemma as: *)
|
|
adamc@14
|
863
|
|
adamc@207
|
864 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
|
|
adamc@207
|
865 tprogDenote (tcompile e ts) s = (texpDenote e, s).
|
|
adamc@20
|
866
|
|
adam@292
|
867 (** While lemma [compile_correct'] quantified over a program that is the %``%#"#continuation#"#%''% for the expression we are considering, here we avoid drawing in any extra syntactic elements. In addition to the source expression and its type, we also quantify over an initial stack type and a stack compatible with it. Running the compilation of the program starting from that stack, we should arrive at a stack that differs only in having the program's denotation pushed onto it.
|
|
adamc@20
|
868
|
|
adamc@20
|
869 Let us try to prove this theorem in the same way that we settled on in the last section. *)
|
|
adamc@20
|
870
|
|
adamc@14
|
871 induction e; crush.
|
|
adamc@20
|
872
|
|
adamc@20
|
873 (** We are left with this unproved conclusion:
|
|
adamc@20
|
874
|
|
adamc@20
|
875 [[
|
|
adamc@20
|
876 tprogDenote
|
|
adamc@20
|
877 (tconcat (tcompile e2 ts)
|
|
adamc@20
|
878 (tconcat (tcompile e1 (arg2 :: ts))
|
|
adamc@20
|
879 (TCons (TIBinop ts t) (TNil (res :: ts))))) s =
|
|
adamc@20
|
880 (tbinopDenote t (texpDenote e1) (texpDenote e2), s)
|
|
adamc@207
|
881
|
|
adamc@20
|
882 ]]
|
|
adamc@20
|
883
|
|
adamc@20
|
884 We need an analogue to the [app_ass] theorem that we used to rewrite the goal in the last section. We can abort this proof and prove such a lemma about [tconcat].
|
|
adamc@20
|
885 *)
|
|
adamc@207
|
886
|
|
adamc@14
|
887 Abort.
|
|
adamc@14
|
888
|
|
adamc@26
|
889 Lemma tconcat_correct : forall ts ts' ts'' (p : tprog ts ts') (p' : tprog ts' ts'')
|
|
adamc@14
|
890 (s : vstack ts),
|
|
adamc@14
|
891 tprogDenote (tconcat p p') s
|
|
adamc@14
|
892 = tprogDenote p' (tprogDenote p s).
|
|
adamc@14
|
893 induction p; crush.
|
|
adamc@14
|
894 Qed.
|
|
adamc@14
|
895
|
|
adamc@20
|
896 (** This one goes through completely automatically.
|
|
adamc@20
|
897
|
|
adamc@26
|
898 Some code behind the scenes registers [app_ass] for use by [crush]. We must register [tconcat_correct] similarly to get the same effect: *)
|
|
adamc@20
|
899
|
|
adamc@26
|
900 Hint Rewrite tconcat_correct : cpdt.
|
|
adamc@14
|
901
|
|
adamc@26
|
902 (** We ask 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]. Now we are ready to return to [tcompile_correct'], proving it automatically this time. *)
|
|
adamc@20
|
903
|
|
adamc@207
|
904 Lemma tcompile_correct' : forall t (e : texp t) ts (s : vstack ts),
|
|
adamc@207
|
905 tprogDenote (tcompile e ts) s = (texpDenote e, s).
|
|
adamc@14
|
906 induction e; crush.
|
|
adamc@14
|
907 Qed.
|
|
adamc@14
|
908
|
|
adamc@20
|
909 (** We can register this main lemma as another hint, allowing us to prove the final theorem trivially. *)
|
|
adamc@20
|
910
|
|
adamc@26
|
911 Hint Rewrite tcompile_correct' : cpdt.
|
|
adamc@14
|
912
|
|
adamc@207
|
913 Theorem tcompile_correct : forall t (e : texp t),
|
|
adamc@207
|
914 tprogDenote (tcompile e nil) tt = (texpDenote e, tt).
|
|
adamc@14
|
915 crush.
|
|
adamc@14
|
916 Qed.
|
|
adamc@22
|
917 (* end thide *)
|
