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Poly.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>Poly: Polymorphism and Higher-Order Functions</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Poly<span class="subtitle">Polymorphism and Higher-Order Functions</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
<div class="paragraph"> </div>
In this chapter we continue our development of basic
concepts of functional programming. The critical new ideas are
<i>polymorphism</i> (abstracting functions over the types of the data
they manipulate) and <i>higher-order functions</i> (treating functions
as data).
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Lists</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab102"></a><h1 class="section">Polymorphism</h1>
</div>
<div class="code code-space">
</div>
<div class="doc">
<a name="lab103"></a><h2 class="section">Polymorphic Lists</h2>
<div class="paragraph"> </div>
For the last couple of chapters, we've been working just
with lists of numbers. Obviously, interesting programs also need
to be able to manipulate lists with elements from other types —
lists of strings, lists of booleans, lists of lists, etc. We
<i>could</i> just define a new inductive datatype for each of these,
for example...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">boollist</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">bool_nil</span> : <span class="id" type="var">boollist</span><br/>
| <span class="id" type="var">bool_cons</span> : <span class="id" type="var">bool</span> <span style="font-family: arial;">→</span> <span class="id" type="var">boollist</span> <span style="font-family: arial;">→</span> <span class="id" type="var">boollist</span>.<br/>
<br/>
</div>
<div class="doc">
... but this would quickly become tedious, partly because we
have to make up different constructor names for each datatype, but
mostly because we would also need to define new versions of all
our list manipulating functions (<span class="inlinecode"><span class="id" type="var">length</span></span>, <span class="inlinecode"><span class="id" type="var">rev</span></span>, etc.) for each
new datatype definition.
<div class="paragraph"> </div>
<a name="lab104"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
To avoid all this repetition, Coq supports <i>polymorphic</i>
inductive type definitions. For example, here is a <i>polymorphic
list</i> datatype.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">list</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nil</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">cons</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span>.<br/>
<br/>
</div>
<div class="doc">
This is exactly like the definition of <span class="inlinecode"><span class="id" type="var">natlist</span></span> from the
previous chapter, except that the <span class="inlinecode"><span class="id" type="var">nat</span></span> argument to the <span class="inlinecode"><span class="id" type="var">cons</span></span>
constructor has been replaced by an arbitrary type <span class="inlinecode"><span class="id" type="var">X</span></span>, a binding
for <span class="inlinecode"><span class="id" type="var">X</span></span> has been added to the header, and the occurrences of
<span class="inlinecode"><span class="id" type="var">natlist</span></span> in the types of the constructors have been replaced by
<span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>. (We can re-use the constructor names <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span>
because the earlier definition of <span class="inlinecode"><span class="id" type="var">natlist</span></span> was inside of a
<span class="inlinecode"><span class="id" type="keyword">Module</span></span> definition that is now out of scope.)
<div class="paragraph"> </div>
What sort of thing is <span class="inlinecode"><span class="id" type="var">list</span></span> itself? One good way to think
about it is that <span class="inlinecode"><span class="id" type="var">list</span></span> is a <i>function</i> from <span class="inlinecode"><span class="id" type="keyword">Type</span></span>s to
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definitions; or, to put it another way, <span class="inlinecode"><span class="id" type="var">list</span></span> is a
function from <span class="inlinecode"><span class="id" type="keyword">Type</span></span>s to <span class="inlinecode"><span class="id" type="keyword">Type</span></span>s. For any particular type <span class="inlinecode"><span class="id" type="var">X</span></span>,
the type <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> is an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>ly defined set of lists whose
elements are things of type <span class="inlinecode"><span class="id" type="var">X</span></span>.
<div class="paragraph"> </div>
With this definition, when we use the constructors <span class="inlinecode"><span class="id" type="var">nil</span></span> and
<span class="inlinecode"><span class="id" type="var">cons</span></span> to build lists, we need to tell Coq the type of the
elements in the lists we are building — that is, <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span>
are now <i>polymorphic constructors</i>. Observe the types of these
constructors:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">nil</span>.<br/>
<span class="comment">(* ===> nil : forall X : Type, list X *)</span><br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">cons</span>.<br/>
<span class="comment">(* ===> cons : forall X : Type, X -> list X -> list X *)</span><br/>
<br/>
</div>
<div class="doc">
The "<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>" in these types can be read as an additional
argument to the constructors that determines the expected types of
the arguments that follow. When <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span> are used, these
arguments are supplied in the same way as the others. For
example, the list containing <span class="inlinecode">2</span> and <span class="inlinecode">1</span> is written like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 2 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 1 (<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>))).<br/>
<br/>
</div>
<div class="doc">
(We've gone back to writing <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span> explicitly here
because we haven't yet defined the <span class="inlinecode"></span> <span class="inlinecode">[]</span> <span class="inlinecode"></span> and <span class="inlinecode">::</span> notations for
the new version of lists. We'll do that in a bit.)
<div class="paragraph"> </div>
We can now go back and make polymorphic (or "generic")
versions of all the list-processing functions that we wrote
before. Here is <span class="inlinecode"><span class="id" type="var">length</span></span>, for example:
<div class="paragraph"> </div>
<a name="lab105"></a><h3 class="section"> </h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">length</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ 0<br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">S</span> (<span class="id" type="var">length</span> <span class="id" type="var">X</span> <span class="id" type="var">t</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Note that the uses of <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span> in <span class="inlinecode"><span class="id" type="keyword">match</span></span> patterns
do not require any type annotations: Coq already knows that the list
<span class="inlinecode"><span class="id" type="var">l</span></span> contains elements of type <span class="inlinecode"><span class="id" type="var">X</span></span>, so there's no reason to include
<span class="inlinecode"><span class="id" type="var">X</span></span> in the pattern. (More precisely, the type <span class="inlinecode"><span class="id" type="var">X</span></span> is a parameter
of the whole definition of <span class="inlinecode"><span class="id" type="var">list</span></span>, not of the individual
constructors. We'll come back to this point later.)
<div class="paragraph"> </div>
As with <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span>, we can use <span class="inlinecode"><span class="id" type="var">length</span></span> by applying it first
to a type and then to its list argument:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_length1</span> :<br/>
<span class="id" type="var">length</span> <span class="id" type="var">nat</span> (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 1 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 2 (<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>))) = 2.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
To use our length with other kinds of lists, we simply
instantiate it with an appropriate type parameter:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_length2</span> :<br/>
<span class="id" type="var">length</span> <span class="id" type="var">bool</span> (<span class="id" type="var">cons</span> <span class="id" type="var">bool</span> <span class="id" type="var">true</span> (<span class="id" type="var">nil</span> <span class="id" type="var">bool</span>)) = 1.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab106"></a><h3 class="section"> </h3>
Let's close this subsection by re-implementing a few other
standard list functions on our new polymorphic lists:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">app</span> (<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>)<br/>
: (<span class="id" type="var">list</span> <span class="id" type="var">X</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l1</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ <span class="id" type="var">l2</span><br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">cons</span> <span class="id" type="var">X</span> <span class="id" type="var">h</span> (<span class="id" type="var">app</span> <span class="id" type="var">X</span> <span class="id" type="var">t</span> <span class="id" type="var">l2</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">snoc</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">v</span>:<span class="id" type="var">X</span>) : (<span class="id" type="var">list</span> <span class="id" type="var">X</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ <span class="id" type="var">cons</span> <span class="id" type="var">X</span> <span class="id" type="var">v</span> (<span class="id" type="var">nil</span> <span class="id" type="var">X</span>)<br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">cons</span> <span class="id" type="var">X</span> <span class="id" type="var">h</span> (<span class="id" type="var">snoc</span> <span class="id" type="var">X</span> <span class="id" type="var">t</span> <span class="id" type="var">v</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">rev</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">list</span> <span class="id" type="var">X</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ <span class="id" type="var">nil</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">snoc</span> <span class="id" type="var">X</span> (<span class="id" type="var">rev</span> <span class="id" type="var">X</span> <span class="id" type="var">t</span>) <span class="id" type="var">h</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_rev1</span> :<br/>
<span class="id" type="var">rev</span> <span class="id" type="var">nat</span> (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 1 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 2 (<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>)))<br/>
= (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 2 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 1 (<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>))).<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_rev2</span>:<br/>
<span class="id" type="var">rev</span> <span class="id" type="var">bool</span> (<span class="id" type="var">nil</span> <span class="id" type="var">bool</span>) = <span class="id" type="var">nil</span> <span class="id" type="var">bool</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">MumbleBaz</span>.<br/>
</div>
<div class="doc">
<a name="lab107"></a><h4 class="section">Exercise: 2 stars (mumble_grumble)</h4>
Consider the following two inductively defined types.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">mumble</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">a</span> : <span class="id" type="var">mumble</span><br/>
| <span class="id" type="var">b</span> : <span class="id" type="var">mumble</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">mumble</span><br/>
| <span class="id" type="var">c</span> : <span class="id" type="var">mumble</span>.<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">grumble</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">d</span> : <span class="id" type="var">mumble</span> <span style="font-family: arial;">→</span> <span class="id" type="var">grumble</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">e</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">grumble</span> <span class="id" type="var">X</span>.<br/>
<br/>
</div>
<div class="doc">
Which of the following are well-typed elements of <span class="inlinecode"><span class="id" type="var">grumble</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> for
some type <span class="inlinecode"><span class="id" type="var">X</span></span>?
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">d</span></span> <span class="inlinecode">(<span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode">5)</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">d</span></span> <span class="inlinecode"><span class="id" type="var">mumble</span></span> <span class="inlinecode">(<span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode">5)</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">d</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode">(<span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode">5)</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">e</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode"><span class="id" type="var">true</span></span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">e</span></span> <span class="inlinecode"><span class="id" type="var">mumble</span></span> <span class="inlinecode">(<span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode">0)</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">e</span></span> <span class="inlinecode"><span class="id" type="var">bool</span></span> <span class="inlinecode">(<span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode">0)</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">c</span></span>
</li>
</ul>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab108"></a><h4 class="section">Exercise: 2 stars (baz_num_elts)</h4>
Consider the following inductive definition:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">baz</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">x</span> : <span class="id" type="var">baz</span> <span style="font-family: arial;">→</span> <span class="id" type="var">baz</span><br/>
| <span class="id" type="var">y</span> : <span class="id" type="var">baz</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bool</span> <span style="font-family: arial;">→</span> <span class="id" type="var">baz</span>.<br/>
<br/>
</div>
<div class="doc">
How <i>many</i> elements does the type <span class="inlinecode"><span class="id" type="var">baz</span></span> have?
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">MumbleBaz</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab109"></a><h3 class="section">Type Annotation Inference</h3>
<div class="paragraph"> </div>
Let's write the definition of <span class="inlinecode"><span class="id" type="var">app</span></span> again, but this time we won't
specify the types of any of the arguments. Will Coq still accept
it?
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">app'</span> <span class="id" type="var">X</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l1</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ <span class="id" type="var">l2</span><br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">cons</span> <span class="id" type="var">X</span> <span class="id" type="var">h</span> (<span class="id" type="var">app'</span> <span class="id" type="var">X</span> <span class="id" type="var">t</span> <span class="id" type="var">l2</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Indeed it will. Let's see what type Coq has assigned to <span class="inlinecode"><span class="id" type="var">app'</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">app'</span>.<br/>
<span class="comment">(* ===> forall X : Type, list X -> list X -> list X *)</span><br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">app</span>.<br/>
<span class="comment">(* ===> forall X : Type, list X -> list X -> list X *)</span><br/>
<br/>
</div>
<div class="doc">
It has exactly the same type type as <span class="inlinecode"><span class="id" type="var">app</span></span>. Coq was able to
use a process called <i>type inference</i> to deduce what the types of
<span class="inlinecode"><span class="id" type="var">X</span></span>, <span class="inlinecode"><span class="id" type="var">l1</span></span>, and <span class="inlinecode"><span class="id" type="var">l2</span></span> must be, based on how they are used. For
example, since <span class="inlinecode"><span class="id" type="var">X</span></span> is used as an argument to <span class="inlinecode"><span class="id" type="var">cons</span></span>, it must be a
<span class="inlinecode"><span class="id" type="keyword">Type</span></span>, since <span class="inlinecode"><span class="id" type="var">cons</span></span> expects a <span class="inlinecode"><span class="id" type="keyword">Type</span></span> as its first argument;
matching <span class="inlinecode"><span class="id" type="var">l1</span></span> with <span class="inlinecode"><span class="id" type="var">nil</span></span> and <span class="inlinecode"><span class="id" type="var">cons</span></span> means it must be a <span class="inlinecode"><span class="id" type="var">list</span></span>; and
so on.
<div class="paragraph"> </div>
This powerful facility means we don't always have to write
explicit type annotations everywhere, although explicit type
annotations are still quite useful as documentation and sanity
checks. You should try to find a balance in your own code between
too many type annotations (so many that they clutter and distract)
and too few (which forces readers to perform type inference in
their heads in order to understand your code).
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab110"></a><h3 class="section">Type Argument Synthesis</h3>
<div class="paragraph"> </div>
Whenever we use a polymorphic function, we need to pass it
one or more types in addition to its other arguments. For
example, the recursive call in the body of the <span class="inlinecode"><span class="id" type="var">length</span></span> function
above must pass along the type <span class="inlinecode"><span class="id" type="var">X</span></span>. But just like providing
explicit type annotations everywhere, this is heavy and verbose.
Since the second argument to <span class="inlinecode"><span class="id" type="var">length</span></span> is a list of <span class="inlinecode"><span class="id" type="var">X</span></span>s, it seems
entirely obvious that the first argument can only be <span class="inlinecode"><span class="id" type="var">X</span></span> — why
should we have to write it explicitly?
<div class="paragraph"> </div>
Fortunately, Coq permits us to avoid this kind of redundancy. In
place of any type argument we can write the "implicit argument"
<span class="inlinecode"><span class="id" type="var">_</span></span>, which can be read as "Please figure out for yourself what
type belongs here." More precisely, when Coq encounters a <span class="inlinecode"><span class="id" type="var">_</span></span>, it
will attempt to <i>unify</i> all locally available information — the
type of the function being applied, the types of the other
arguments, and the type expected by the context in which the
application appears — to determine what concrete type should
replace the <span class="inlinecode"><span class="id" type="var">_</span></span>.
<div class="paragraph"> </div>
This may sound similar to type annotation inference — and,
indeed, the two procedures rely on the same underlying mechanisms.
Instead of simply omitting the types of some arguments to a
function, like
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">app'</span> <span class="id" type="var">X</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span> :=
<div class="paragraph"> </div>
</div>
we can also replace the types with <span class="inlinecode"><span class="id" type="var">_</span></span>, like
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">app'</span> (<span class="id" type="var">X</span> : <span class="id" type="var">_</span>) (<span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">_</span>) : <span class="id" type="var">list</span> <span class="id" type="var">X</span> :=
<div class="paragraph"> </div>
</div>
which tells Coq to attempt to infer the missing information, just
as with argument synthesis.
<div class="paragraph"> </div>
Using implicit arguments, the <span class="inlinecode"><span class="id" type="var">length</span></span> function can be written
like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">length'</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ 0<br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">S</span> (<span class="id" type="var">length'</span> <span class="id" type="var">_</span> <span class="id" type="var">t</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
In this instance, we don't save much by writing <span class="inlinecode"><span class="id" type="var">_</span></span> instead of
<span class="inlinecode"><span class="id" type="var">X</span></span>. But in many cases the difference can be significant. For
example, suppose we want to write down a list containing the
numbers <span class="inlinecode">1</span>, <span class="inlinecode">2</span>, and <span class="inlinecode">3</span>. Instead of writing this...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">list123</span> :=<br/>
<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 1 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 2 (<span class="id" type="var">cons</span> <span class="id" type="var">nat</span> 3 (<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>))).<br/>
<br/>
</div>
<div class="doc">
...we can use argument synthesis to write this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">list123'</span> := <span class="id" type="var">cons</span> <span class="id" type="var">_</span> 1 (<span class="id" type="var">cons</span> <span class="id" type="var">_</span> 2 (<span class="id" type="var">cons</span> <span class="id" type="var">_</span> 3 (<span class="id" type="var">nil</span> <span class="id" type="var">_</span>))).<br/>
<br/>
</div>
<div class="doc">
<a name="lab111"></a><h3 class="section">Implicit Arguments</h3>
<div class="paragraph"> </div>
In fact, we can go further. To avoid having to sprinkle <span class="inlinecode"><span class="id" type="var">_</span></span>'s
throughout our programs, we can tell Coq <i>always</i> to infer the
type argument(s) of a given function. The <span class="inlinecode"><span class="id" type="var">Arguments</span></span> directive
specifies the name of the function or constructor, and then lists
its argument names, with curly braces around any arguments to be
treated as implicit.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">nil</span> {<span class="id" type="var">X</span>}.<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">cons</span> {<span class="id" type="var">X</span>} <span class="id" type="var">_</span> <span class="id" type="var">_</span>. <span class="comment">(* use underscore for argument position that has no name *)</span><br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">length</span> {<span class="id" type="var">X</span>} <span class="id" type="var">l</span>.<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">app</span> {<span class="id" type="var">X</span>} <span class="id" type="var">l1</span> <span class="id" type="var">l2</span>.<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">rev</span> {<span class="id" type="var">X</span>} <span class="id" type="var">l</span>.<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">snoc</span> {<span class="id" type="var">X</span>} <span class="id" type="var">l</span> <span class="id" type="var">v</span>.<br/>
<br/>
<span class="comment">(* note: no _ arguments required... *)</span><br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">list123''</span> := <span class="id" type="var">cons</span> 1 (<span class="id" type="var">cons</span> 2 (<span class="id" type="var">cons</span> 3 <span class="id" type="var">nil</span>)).<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">length</span> <span class="id" type="var">list123''</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab112"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Alternatively, we can declare an argument to be implicit while
defining the function itself, by surrounding the argument in curly
braces. For example:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">length''</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">nat</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">nil</span> ⇒ 0<br/>
| <span class="id" type="var">cons</span> <span class="id" type="var">h</span> <span class="id" type="var">t</span> ⇒ <span class="id" type="var">S</span> (<span class="id" type="var">length''</span> <span class="id" type="var">t</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
(Note that we didn't even have to provide a type argument to
the recursive call to <span class="inlinecode"><span class="id" type="var">length''</span></span>; indeed, it is invalid to provide
one.) We will use this style whenever possible, although we will
continue to use use explicit <span class="inlinecode"><span class="id" type="var">Argument</span></span> declarations for
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> constructors.
<div class="paragraph"> </div>
<a name="lab113"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
One small problem with declaring arguments <span class="inlinecode"><span class="id" type="keyword">Implicit</span></span> is
that, occasionally, Coq does not have enough local information to
determine a type argument; in such cases, we need to tell Coq that
we want to give the argument explicitly this time, even though
we've globally declared it to be <span class="inlinecode"><span class="id" type="keyword">Implicit</span></span>. For example, suppose we
write this:
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Definition mynil := nil. *)</span><br/>
<br/>
</div>
<div class="doc">
If we uncomment this definition, Coq will give us an error,
because it doesn't know what type argument to supply to <span class="inlinecode"><span class="id" type="var">nil</span></span>. We
can help it by providing an explicit type declaration (so that Coq
has more information available when it gets to the "application"
of <span class="inlinecode"><span class="id" type="var">nil</span></span>):
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mynil</span> : <span class="id" type="var">list</span> <span class="id" type="var">nat</span> := <span class="id" type="var">nil</span>.<br/>
<br/>
</div>
<div class="doc">
Alternatively, we can force the implicit arguments to be explicit by
prefixing the function name with <span class="inlinecode">@</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> @<span class="id" type="var">nil</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mynil'</span> := @<span class="id" type="var">nil</span> <span class="id" type="var">nat</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab114"></a><h3 class="section"> </h3>
Using argument synthesis and implicit arguments, we can
define convenient notation for lists, as before. Since we have
made the constructor type arguments implicit, Coq will know to
automatically infer these when we use the notations.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "x :: y" := (<span class="id" type="var">cons</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>)<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 60, <span class="id" type="var">right</span> <span class="id" type="var">associativity</span>).<br/>
<span class="id" type="keyword">Notation</span> "[ ]" := <span class="id" type="var">nil</span>.<br/>
<span class="id" type="keyword">Notation</span> "[ x ; .. ; y ]" := (<span class="id" type="var">cons</span> <span class="id" type="var">x</span> .. (<span class="id" type="var">cons</span> <span class="id" type="var">y</span> []) ..).<br/>
<span class="id" type="keyword">Notation</span> "x ++ y" := (<span class="id" type="var">app</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>)<br/>
(<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 60, <span class="id" type="var">right</span> <span class="id" type="var">associativity</span>).<br/>
<br/>
</div>
<div class="doc">
Now lists can be written just the way we'd hope:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">list123'''</span> := [1; 2; 3].<br/>
<br/>
</div>
<div class="doc">
<a name="lab115"></a><h3 class="section">Exercises: Polymorphic Lists</h3>
<div class="paragraph"> </div>
<a name="lab116"></a><h4 class="section">Exercise: 2 stars, optional (poly_exercises)</h4>
Here are a few simple exercises, just like ones in the <span class="inlinecode"><span class="id" type="var">Lists</span></span>
chapter, for practice with polymorphism. Fill in the definitions
and complete the proofs below.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="tactic">repeat</span> {<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">n</span> : <span class="id" type="var">X</span>) (<span class="id" type="var">count</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">list</span> <span class="id" type="var">X</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_repeat1</span>:<br/>
<span class="id" type="tactic">repeat</span> <span class="id" type="var">true</span> 2 = <span class="id" type="var">cons</span> <span class="id" type="var">true</span> (<span class="id" type="var">cons</span> <span class="id" type="var">true</span> <span class="id" type="var">nil</span>).<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">nil_app</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>, <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>:<span class="id" type="var">list</span> <span class="id" type="var">X</span>,<br/>
<span class="id" type="var">app</span> [] <span class="id" type="var">l</span> = <span class="id" type="var">l</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">rev_snoc</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>,<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">v</span> : <span class="id" type="var">X</span>,<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">s</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>,<br/>
<span class="id" type="var">rev</span> (<span class="id" type="var">snoc</span> <span class="id" type="var">s</span> <span class="id" type="var">v</span>) = <span class="id" type="var">v</span> :: (<span class="id" type="var">rev</span> <span class="id" type="var">s</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">rev_involutive</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>, <span style="font-family: arial;">∀</span><span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>,<br/>
<span class="id" type="var">rev</span> (<span class="id" type="var">rev</span> <span class="id" type="var">l</span>) = <span class="id" type="var">l</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">snoc_with_append</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>,<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>,<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">v</span> : <span class="id" type="var">X</span>,<br/>
<span class="id" type="var">snoc</span> (<span class="id" type="var">l1</span> ++ <span class="id" type="var">l2</span>) <span class="id" type="var">v</span> = <span class="id" type="var">l1</span> ++ (<span class="id" type="var">snoc</span> <span class="id" type="var">l2</span> <span class="id" type="var">v</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab117"></a><h2 class="section">Polymorphic Pairs</h2>
<div class="paragraph"> </div>
Following the same pattern, the type definition we gave in
the last chapter for pairs of numbers can be generalized to
<i>polymorphic pairs</i> (or <i>products</i>):
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">prod</span> (<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
<span class="id" type="var">pair</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Y</span> <span style="font-family: arial;">→</span> <span class="id" type="var">prod</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>.<br/>
<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">pair</span> {<span class="id" type="var">X</span>} {<span class="id" type="var">Y</span>} <span class="id" type="var">_</span> <span class="id" type="var">_</span>.<br/>
<br/>
</div>
<div class="doc">
As with lists, we make the type arguments implicit and define the
familiar concrete notation.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "( x , y )" := (<span class="id" type="var">pair</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>).<br/>
<br/>
</div>
<div class="doc">
We can also use the <span class="inlinecode"><span class="id" type="keyword">Notation</span></span> mechanism to define the standard
notation for pair <i>types</i>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "X × Y" := (<span class="id" type="var">prod</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>) : <span class="id" type="var">type_scope</span>.<br/>
<br/>
</div>
<div class="doc">
(The annotation <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">type_scope</span></span> tells Coq that this abbreviation
should be used when parsing types. This avoids a clash with the
multiplication symbol.)
<div class="paragraph"> </div>
<a name="lab118"></a><h3 class="section"> </h3>
A note of caution: it is easy at first to get <span class="inlinecode">(<span class="id" type="var">x</span>,<span class="id" type="var">y</span>)</span> and
<span class="inlinecode"><span class="id" type="var">X</span>×<span class="id" type="var">Y</span></span> confused. Remember that <span class="inlinecode">(<span class="id" type="var">x</span>,<span class="id" type="var">y</span>)</span> is a <i>value</i> built from two
other values; <span class="inlinecode"><span class="id" type="var">X</span>×<span class="id" type="var">Y</span></span> is a <i>type</i> built from two other types. If
<span class="inlinecode"><span class="id" type="var">x</span></span> has type <span class="inlinecode"><span class="id" type="var">X</span></span> and <span class="inlinecode"><span class="id" type="var">y</span></span> has type <span class="inlinecode"><span class="id" type="var">Y</span></span>, then <span class="inlinecode">(<span class="id" type="var">x</span>,<span class="id" type="var">y</span>)</span> has type
<span class="inlinecode"><span class="id" type="var">X</span>×<span class="id" type="var">Y</span></span>.
<div class="paragraph"> </div>
The first and second projection functions now look pretty
much as they would in any functional programming language.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">fst</span> {<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">p</span> : <span class="id" type="var">X</span> × <span class="id" type="var">Y</span>) : <span class="id" type="var">X</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">x</span>,<span class="id" type="var">y</span>) ⇒ <span class="id" type="var">x</span> <span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">snd</span> {<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">p</span> : <span class="id" type="var">X</span> × <span class="id" type="var">Y</span>) : <span class="id" type="var">Y</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">x</span>,<span class="id" type="var">y</span>) ⇒ <span class="id" type="var">y</span> <span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
The following function takes two lists and combines them
into a list of pairs. In many functional programming languages,
it is called <span class="inlinecode"><span class="id" type="var">zip</span></span>. We call it <span class="inlinecode"><span class="id" type="var">combine</span></span> for consistency with
Coq's standard library. Note that the pair notation can be used both in expressions and in
patterns...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">combine</span> {<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">lx</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ly</span> : <span class="id" type="var">list</span> <span class="id" type="var">Y</span>)<br/>
: <span class="id" type="var">list</span> (<span class="id" type="var">X</span>×<span class="id" type="var">Y</span>) :=<br/>
<span class="id" type="keyword">match</span> (<span class="id" type="var">lx</span>,<span class="id" type="var">ly</span>) <span class="id" type="keyword">with</span><br/>
| ([],_) ⇒ []<br/>
| (<span class="id" type="var">_</span>,[]) ⇒ []<br/>
| (<span class="id" type="var">x</span>::<span class="id" type="var">tx</span>, <span class="id" type="var">y</span>::<span class="id" type="var">ty</span>) ⇒ (<span class="id" type="var">x</span>,<span class="id" type="var">y</span>) :: (<span class="id" type="var">combine</span> <span class="id" type="var">tx</span> <span class="id" type="var">ty</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab119"></a><h4 class="section">Exercise: 1 star, optional (combine_checks)</h4>
Try answering the following questions on paper and
checking your answers in coq:
<div class="paragraph"> </div>
<ul class="doclist">
<li> What is the type of <span class="inlinecode"><span class="id" type="var">combine</span></span> (i.e., what does <span class="inlinecode"><span class="id" type="keyword">Check</span></span>
<span class="inlinecode">@<span class="id" type="var">combine</span></span> print?)
</li>
<li> What does
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Eval</span> <span class="id" type="tactic">compute</span> <span class="id" type="keyword">in</span> (<span class="id" type="var">combine</span> [1;2] [<span class="id" type="var">false</span>;<span class="id" type="var">false</span>;<span class="id" type="var">true</span>;<span class="id" type="var">true</span>]).
<div class="paragraph"> </div>
</div>
print? <font size=-2>☐</font>
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab120"></a><h4 class="section">Exercise: 2 stars (split)</h4>
The function <span class="inlinecode"><span class="id" type="tactic">split</span></span> is the right inverse of combine: it takes a
list of pairs and returns a pair of lists. In many functional
programing languages, this function is called <span class="inlinecode"><span class="id" type="var">unzip</span></span>.
<div class="paragraph"> </div>
Uncomment the material below and fill in the definition of
<span class="inlinecode"><span class="id" type="tactic">split</span></span>. Make sure it passes the given unit tests.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="tactic">split</span><br/>
{<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">l</span> : <span class="id" type="var">list</span> (<span class="id" type="var">X</span>×<span class="id" type="var">Y</span>))<br/>
: (<span class="id" type="var">list</span> <span class="id" type="var">X</span>) × (<span class="id" type="var">list</span> <span class="id" type="var">Y</span>) :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_split</span>:<br/>
<span class="id" type="tactic">split</span> [(1,<span class="id" type="var">false</span>);(2,<span class="id" type="var">false</span>)] = ([1;2],[<span class="id" type="var">false</span>;<span class="id" type="var">false</span>]).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab121"></a><h2 class="section">Polymorphic Options</h2>
<div class="paragraph"> </div>
One last polymorphic type for now: <i>polymorphic options</i>.
The type declaration generalizes the one for <span class="inlinecode"><span class="id" type="var">natoption</span></span> in the
previous chapter:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">option</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">Some</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">option</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">None</span> : <span class="id" type="var">option</span> <span class="id" type="var">X</span>.<br/>
<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">Some</span> {<span class="id" type="var">X</span>} <span class="id" type="var">_</span>.<br/>
<span class="id" type="var">Arguments</span> <span class="id" type="var">None</span> {<span class="id" type="var">X</span>}.<br/>
<br/>
</div>
<div class="doc">
<a name="lab122"></a><h3 class="section"> </h3>
We can now rewrite the <span class="inlinecode"><span class="id" type="var">index</span></span> function so that it works
with any type of lists.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">index</span> {<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>)<br/>
(<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">option</span> <span class="id" type="var">X</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">l</span> <span class="id" type="keyword">with</span><br/>
| [] ⇒ <span class="id" type="var">None</span><br/>
| <span class="id" type="var">a</span> :: <span class="id" type="var">l'</span> ⇒ <span class="id" type="keyword">if</span> <span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">O</span> <span class="id" type="keyword">then</span> <span class="id" type="var">Some</span> <span class="id" type="var">a</span> <span class="id" type="keyword">else</span> <span class="id" type="var">index</span> (<span class="id" type="var">pred</span> <span class="id" type="var">n</span>) <span class="id" type="var">l'</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_index1</span> : <span class="id" type="var">index</span> 0 [4;5;6;7] = <span class="id" type="var">Some</span> 4.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_index2</span> : <span class="id" type="var">index</span> 1 [[1];[2]] = <span class="id" type="var">Some</span> [2].<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_index3</span> : <span class="id" type="var">index</span> 2 [<span class="id" type="var">true</span>] = <span class="id" type="var">None</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab123"></a><h4 class="section">Exercise: 1 star, optional (hd_opt_poly)</h4>
Complete the definition of a polymorphic version of the
<span class="inlinecode"><span class="id" type="var">hd_opt</span></span> function from the last chapter. Be sure that it
passes the unit tests below.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">hd_opt</span> {<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>} (<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) : <span class="id" type="var">option</span> <span class="id" type="var">X</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
</div>
<div class="doc">
Once again, to force the implicit arguments to be explicit,
we can use <span class="inlinecode">@</span> before the name of the function.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> @<span class="id" type="var">hd_opt</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_hd_opt1</span> : <span class="id" type="var">hd_opt</span> [1;2] = <span class="id" type="var">Some</span> 1.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_hd_opt2</span> : <span class="id" type="var">hd_opt</span> [[1];[2]] = <span class="id" type="var">Some</span> [1].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab124"></a><h1 class="section">Functions as Data</h1>
</div>
<div class="code code-space">
</div>
<div class="doc">
<a name="lab125"></a><h2 class="section">Higher-Order Functions</h2>
<div class="paragraph"> </div>
Like many other modern programming languages — including
all <i>functional languages</i> (ML, Haskell, Scheme, etc.) — Coq
treats functions as first-class citizens, allowing functions to be
passed as arguments to other functions, returned as results,
stored in data structures, etc.
<div class="paragraph"> </div>
Functions that manipulate other functions are often called
<i>higher-order</i> functions. Here's a simple one:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">doit3times</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">f</span>:<span class="id" type="var">X</span><span style="font-family: arial;">→</span><span class="id" type="var">X</span>) (<span class="id" type="var">n</span>:<span class="id" type="var">X</span>) : <span class="id" type="var">X</span> :=<br/>
<span class="id" type="var">f</span> (<span class="id" type="var">f</span> (<span class="id" type="var">f</span> <span class="id" type="var">n</span>)).<br/>
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<div class="doc">
The argument <span class="inlinecode"><span class="id" type="var">f</span></span> here is itself a function (from <span class="inlinecode"><span class="id" type="var">X</span></span> to
<span class="inlinecode"><span class="id" type="var">X</span></span>); the body of <span class="inlinecode"><span class="id" type="var">doit3times</span></span> applies <span class="inlinecode"><span class="id" type="var">f</span></span> three times to some
value <span class="inlinecode"><span class="id" type="var">n</span></span>.
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<span class="id" type="keyword">Check</span> @<span class="id" type="var">doit3times</span>.<br/>
<span class="comment">(* ===> doit3times : forall X : Type, (X -> X) -> X -> X *)</span><br/>
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<span class="id" type="keyword">Example</span> <span class="id" type="var">test_doit3times</span>: <span class="id" type="var">doit3times</span> <span class="id" type="var">minustwo</span> 9 = 3.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
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<span class="id" type="keyword">Example</span> <span class="id" type="var">test_doit3times'</span>: <span class="id" type="var">doit3times</span> <span class="id" type="var">negb</span> <span class="id" type="var">true</span> = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
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<div class="doc">
<a name="lab126"></a><h2 class="section">Partial Application</h2>
<div class="paragraph"> </div>
In fact, the multiple-argument functions we have already
seen are also examples of passing functions as data. To see why,
recall the type of <span class="inlinecode"><span class="id" type="var">plus</span></span>.
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<span class="id" type="keyword">Check</span> <span class="id" type="var">plus</span>.<br/>
<span class="comment">(* ==> nat -> nat -> nat *)</span><br/>