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Lists.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>Lists: Working with Structured Data</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">Lists<span class="subtitle">Working with Structured Data</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</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="keyword">Induction</span>.<br/>
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">NatList</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab61"></a><h1 class="section">Pairs of Numbers</h1>
<div class="paragraph"> </div>
In an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> type definition, each constructor can take
any number of arguments — none (as with <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">O</span></span>), one (as
with <span class="inlinecode"><span class="id" type="var">S</span></span>), or more than one, as in this definition:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">natprod</span> : <span class="id" type="keyword">Type</span> :=<br/>
<span class="id" type="var">pair</span> : <span class="id" type="var">nat</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">natprod</span>.<br/>
<br/>
</div>
<div class="doc">
This declaration can be read: "There is just one way to
construct a pair of numbers: by applying the constructor <span class="inlinecode"><span class="id" type="var">pair</span></span> to
two arguments of type <span class="inlinecode"><span class="id" type="var">nat</span></span>."
<div class="paragraph"> </div>
We can construct an element of <span class="inlinecode"><span class="id" type="var">natprod</span></span> like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">pair</span> 3 5).<br/>
<br/>
</div>
<div class="doc">
<a name="lab62"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Here are two simple function definitions for extracting the
first and second components of a pair. (The definitions also
illustrate how to do pattern matching on two-argument
constructors.)
</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">p</span> : <span class="id" type="var">natprod</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span><br/>
| <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">x</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">snd</span> (<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span><br/>
| <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">y</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<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">fst</span> (<span class="id" type="var">pair</span> 3 5)).<br/>
<span class="comment">(* ===> 3 *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab63"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Since pairs are used quite a bit, it is nice to be able to
write them with the standard mathematical notation <span class="inlinecode">(<span class="id" type="var">x</span>,<span class="id" type="var">y</span>)</span> instead
of <span class="inlinecode"><span class="id" type="var">pair</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span>. We can tell Coq to allow this with a <span class="inlinecode"><span class="id" type="keyword">Notation</span></span>
declaration.
</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">
The new notation can be used both in expressions and in
pattern matches (indeed, we've seen it already in the previous
chapter — this notation is provided as part of the standard
library):
</div>
<div class="code code-tight">
<br/>
<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">fst</span> (3,5)).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">fst'</span> (<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span><br/>
| (<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">end</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">snd'</span> (<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span><br/>
| (<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">end</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">swap_pair</span> (<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>) : <span class="id" type="var">natprod</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">p</span> <span class="id" type="keyword">with</span><br/>
| (<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="var">x</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab64"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Let's try and prove a few simple facts about pairs. If we
state the lemmas in a particular (and slightly peculiar) way, we
can prove them with just reflexivity (and its built-in
simplification):
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">surjective_pairing'</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>),<br/>
(<span class="id" type="var">n</span>,<span class="id" type="var">m</span>) = (<span class="id" type="var">fst</span> (<span class="id" type="var">n</span>,<span class="id" type="var">m</span>), <span class="id" type="var">snd</span> (<span class="id" type="var">n</span>,<span class="id" type="var">m</span>)).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Note that <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> is not enough if we state the lemma in a
more natural way:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">surjective_pairing_stuck</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>),<br/>
<span class="id" type="var">p</span> = (<span class="id" type="var">fst</span> <span class="id" type="var">p</span>, <span class="id" type="var">snd</span> <span class="id" type="var">p</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* Doesn't reduce anything! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab65"></a><h3 class="section"> </h3>
We have to expose the structure of <span class="inlinecode"><span class="id" type="var">p</span></span> so that <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> can
perform the pattern match in <span class="inlinecode"><span class="id" type="var">fst</span></span> and <span class="inlinecode"><span class="id" type="var">snd</span></span>. We can do this with
<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>.
<div class="paragraph"> </div>
Notice that, unlike for <span class="inlinecode"><span class="id" type="var">nat</span></span>s, <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> doesn't generate an
extra subgoal here. That's because <span class="inlinecode"><span class="id" type="var">natprod</span></span>s can only be
constructed in one way.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">surjective_pairing</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>),<br/>
<span class="id" type="var">p</span> = (<span class="id" type="var">fst</span> <span class="id" type="var">p</span>, <span class="id" type="var">snd</span> <span class="id" type="var">p</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">p</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">n</span> <span class="id" type="var">m</span>]. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab66"></a><h4 class="section">Exercise: 1 star (snd_fst_is_swap)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">snd_fst_is_swap</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>),<br/>
(<span class="id" type="var">snd</span> <span class="id" type="var">p</span>, <span class="id" type="var">fst</span> <span class="id" type="var">p</span>) = <span class="id" type="var">swap_pair</span> <span class="id" type="var">p</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 class="paragraph"> </div>
<a name="lab67"></a><h4 class="section">Exercise: 1 star, optional (fst_swap_is_snd)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">fst_swap_is_snd</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <span class="id" type="var">natprod</span>),<br/>
<span class="id" type="var">fst</span> (<span class="id" type="var">swap_pair</span> <span class="id" type="var">p</span>) = <span class="id" type="var">snd</span> <span class="id" type="var">p</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="lab68"></a><h1 class="section">Lists of Numbers</h1>
<div class="paragraph"> </div>
Generalizing the definition of pairs a little, we can
describe the type of <i>lists</i> of numbers like this: "A list is
either the empty list or else a pair of a number and another
list."
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">natlist</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nil</span> : <span class="id" type="var">natlist</span><br/>
| <span class="id" type="var">cons</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">natlist</span> <span style="font-family: arial;">→</span> <span class="id" type="var">natlist</span>.<br/>
<br/>
</div>
<div class="doc">
For example, here is a three-element list:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mylist</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/>
<br/>
</div>
<div class="doc">
<a name="lab69"></a><h3 class="section"> </h3>
As with pairs, it is more convenient to write lists in
familiar programming notation. The following two declarations
allow us to use <span class="inlinecode">::</span> as an infix <span class="inlinecode"><span class="id" type="var">cons</span></span> operator and square
brackets as an "outfix" notation for constructing lists.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "x :: l" := (<span class="id" type="var">cons</span> <span class="id" type="var">x</span> <span class="id" type="var">l</span>) (<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> <span class="id" type="var">nil</span>) ..).<br/>
<br/>
</div>
<div class="doc">
It is not necessary to fully understand these declarations,
but in case you are interested, here is roughly what's going on.
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="var">right</span></span> <span class="inlinecode"><span class="id" type="var">associativity</span></span> annotation tells Coq how to parenthesize
expressions involving several uses of <span class="inlinecode">::</span> so that, for example,
the next three declarations mean exactly the same thing:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mylist1</span> := 1 :: (2 :: (3 :: <span class="id" type="var">nil</span>)).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mylist2</span> := 1 :: 2 :: 3 :: <span class="id" type="var">nil</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">mylist3</span> := [1;2;3].<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">at</span></span> <span class="inlinecode"><span class="id" type="var">level</span></span> <span class="inlinecode">60</span> part tells Coq how to parenthesize
expressions that involve both <span class="inlinecode">::</span> and some other infix operator.
For example, since we defined <span class="inlinecode">+</span> as infix notation for the <span class="inlinecode"><span class="id" type="var">plus</span></span>
function at level 50,
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Notation</span> "x + y" := (<span class="id" type="var">plus</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> 50, <span class="id" type="var">left</span> <span class="id" type="var">associativity</span>).
<div class="paragraph"> </div>
</div>
The <span class="inlinecode">+</span> operator will bind tighter than <span class="inlinecode">::</span>, so <span class="inlinecode">1</span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode">::</span> <span class="inlinecode">[3]</span>
will be parsed, as we'd expect, as <span class="inlinecode">(1</span> <span class="inlinecode">+</span> <span class="inlinecode">2)</span> <span class="inlinecode">::</span> <span class="inlinecode">[3]</span> rather than <span class="inlinecode">1</span>
<span class="inlinecode">+</span> <span class="inlinecode">(2</span> <span class="inlinecode">::</span> <span class="inlinecode">[3])</span>.
<div class="paragraph"> </div>
(By the way, it's worth noting in passing that expressions like "<span class="inlinecode">1</span>
<span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode">::</span> <span class="inlinecode">[3]</span>" can be a little confusing when you read them in a .v
file. The inner brackets, around 3, indicate a list, but the outer
brackets, which are invisible in the HTML rendering, are there to
instruct the "coqdoc" tool that the bracketed part should be
displayed as Coq code rather than running text.)
<div class="paragraph"> </div>
The second and third <span class="inlinecode"><span class="id" type="keyword">Notation</span></span> declarations above introduce the
standard square-bracket notation for lists; the right-hand side of
the third one illustrates Coq's syntax for declaring n-ary
notations and translating them to nested sequences of binary
constructors.
<div class="paragraph"> </div>
<a name="lab70"></a><h3 class="section">Repeat</h3>
A number of functions are useful for manipulating lists.
For example, the <span class="inlinecode"><span class="id" type="tactic">repeat</span></span> function takes a number <span class="inlinecode"><span class="id" type="var">n</span></span> and a
<span class="inlinecode"><span class="id" type="var">count</span></span> and returns a list of length <span class="inlinecode"><span class="id" type="var">count</span></span> where every element
is <span class="inlinecode"><span class="id" type="var">n</span></span>.
</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">n</span> <span class="id" type="var">count</span> : <span class="id" type="var">nat</span>) : <span class="id" type="var">natlist</span> := <br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">count</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> ⇒ <span class="id" type="var">nil</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">count'</span> ⇒ <span class="id" type="var">n</span> :: (<span class="id" type="tactic">repeat</span> <span class="id" type="var">n</span> <span class="id" type="var">count'</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab71"></a><h3 class="section">Length</h3>
The <span class="inlinecode"><span class="id" type="var">length</span></span> function calculates the length of a list.
</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">l</span>:<span class="id" type="var">natlist</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> ⇒ <span class="id" type="var">O</span><br/>
| <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">
<a name="lab72"></a><h3 class="section">Append</h3>
The <span class="inlinecode"><span class="id" type="var">app</span></span> ("append") function concatenates two 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">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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">h</span> :: <span class="id" type="var">t</span> ⇒ <span class="id" type="var">h</span> :: (<span class="id" type="var">app</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">
Actually, <span class="inlinecode"><span class="id" type="var">app</span></span> will be used a lot in some parts of what
follows, so it is convenient to have an infix operator for it.
</div>
<div class="code code-tight">
<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="var">right</span> <span class="id" type="var">associativity</span>, <span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 60).<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_app1</span>: [1;2;3] ++ [4;5] = [1;2;3;4;5].<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_app2</span>: <span class="id" type="var">nil</span> ++ [4;5] = [4;5].<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_app3</span>: [1;2;3] ++ <span class="id" type="var">nil</span> = [1;2;3].<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">
Here are two smaller examples of programming with lists.
The <span class="inlinecode"><span class="id" type="var">hd</span></span> function returns the first element (the "head") of the
list, while <span class="inlinecode"><span class="id" type="var">tl</span></span> returns everything but the first
element (the "tail").
Of course, the empty list has no first element, so we
must pass a default value to be returned in that case.
<div class="paragraph"> </div>
<a name="lab73"></a><h3 class="section">Head (with default) and Tail</h3>
</div>
<div class="code code-space">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">hd</span> (<span class="id" type="var">default</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</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> ⇒ <span class="id" type="var">default</span><br/>
| <span class="id" type="var">h</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">Definition</span> <span class="id" type="var">tl</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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> <br/>
| <span class="id" type="var">h</span> :: <span class="id" type="var">t</span> ⇒ <span class="id" type="var">t</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_hd1</span>: <span class="id" type="var">hd</span> 0 [1;2;3] = 1.<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_hd2</span>: <span class="id" type="var">hd</span> 0 [] = 0.<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_tl</span>: <span class="id" type="var">tl</span> [1;2;3] = [2;3].<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="lab74"></a><h4 class="section">Exercise: 2 stars (list_funs)</h4>
Complete the definitions of <span class="inlinecode"><span class="id" type="var">nonzeros</span></span>, <span class="inlinecode"><span class="id" type="var">oddmembers</span></span> and
<span class="inlinecode"><span class="id" type="var">countoddmembers</span></span> below. Have a look at the tests to understand
what these functions should do.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">nonzeros</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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_nonzeros</span>: <span class="id" type="var">nonzeros</span> [0;1;0;2;3;0;0] = [1;2;3].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">oddmembers</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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_oddmembers</span>: <span class="id" type="var">oddmembers</span> [0;1;0;2;3;0;0] = [1;3].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">countoddmembers</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) : <span class="id" type="var">nat</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_countoddmembers1</span>: <span class="id" type="var">countoddmembers</span> [1;0;3;1;4;5] = 4.<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_countoddmembers2</span>: <span class="id" type="var">countoddmembers</span> [0;2;4] = 0.<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_countoddmembers3</span>: <span class="id" type="var">countoddmembers</span> <span class="id" type="var">nil</span> = 0.<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 class="paragraph"> </div>
<a name="lab75"></a><h4 class="section">Exercise: 3 stars, advanced (alternate)</h4>
Complete the definition of <span class="inlinecode"><span class="id" type="var">alternate</span></span>, which "zips up" two lists
into one, alternating between elements taken from the first list
and elements from the second. See the tests below for more
specific examples.
<div class="paragraph"> </div>
Note: one natural and elegant way of writing <span class="inlinecode"><span class="id" type="var">alternate</span></span> will fail
to satisfy Coq's requirement that all <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> definitions be
"obviously terminating." If you find yourself in this rut, look
for a slightly more verbose solution that considers elements of
both lists at the same time. (One possible solution requires
defining a new kind of pairs, but this is not the only way.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">alternate</span> (<span class="id" type="var">l1</span> <span class="id" type="var">l2</span> : <span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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_alternate1</span>: <span class="id" type="var">alternate</span> [1;2;3] [4;5;6] = [1;4;2;5;3;6].<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_alternate2</span>: <span class="id" type="var">alternate</span> [1] [4;5;6] = [1;4;5;6].<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_alternate3</span>: <span class="id" type="var">alternate</span> [1;2;3] [4] = [1;4;2;3].<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_alternate4</span>: <span class="id" type="var">alternate</span> [] [20;30] = [20;30].<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="lab76"></a><h2 class="section">Bags via Lists</h2>
<div class="paragraph"> </div>
A <span class="inlinecode"><span class="id" type="var">bag</span></span> (or <span class="inlinecode"><span class="id" type="var">multiset</span></span>) is like a set, but each element can appear
multiple times instead of just once. One reasonable
implementation of bags is to represent a bag of numbers as a
list.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">bag</span> := <span class="id" type="var">natlist</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab77"></a><h4 class="section">Exercise: 3 stars (bag_functions)</h4>
Complete the following definitions for the functions
<span class="inlinecode"><span class="id" type="var">count</span></span>, <span class="inlinecode"><span class="id" type="var">sum</span></span>, <span class="inlinecode"><span class="id" type="var">add</span></span>, and <span class="inlinecode"><span class="id" type="var">member</span></span> for bags.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">count</span> (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">nat</span> := <br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
</div>
<div class="doc">
All these proofs can be done just by <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">test_count1</span>: <span class="id" type="var">count</span> 1 [1;2;3;1;4;1] = 3.<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_count2</span>: <span class="id" type="var">count</span> 6 [1;2;3;1;4;1] = 0.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
Multiset <span class="inlinecode"><span class="id" type="var">sum</span></span> is similar to set <span class="inlinecode"><span class="id" type="var">union</span></span>: <span class="inlinecode"><span class="id" type="var">sum</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> contains
all the elements of <span class="inlinecode"><span class="id" type="var">a</span></span> and of <span class="inlinecode"><span class="id" type="var">b</span></span>. (Mathematicians usually
define <span class="inlinecode"><span class="id" type="var">union</span></span> on multisets a little bit differently, which
is why we don't use that name for this operation.)
For <span class="inlinecode"><span class="id" type="var">sum</span></span> we're giving you a header that does not give explicit
names to the arguments. Moreover, it uses the keyword
<span class="inlinecode"><span class="id" type="keyword">Definition</span></span> instead of <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>, so even if you had names for
the arguments, you wouldn't be able to process them recursively.
The point of stating the question this way is to encourage you to
think about whether <span class="inlinecode"><span class="id" type="var">sum</span></span> can be implemented in another way —
perhaps by using functions that have already been defined.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">sum</span> : <span class="id" type="var">bag</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bag</span> <span style="font-family: arial;">→</span> <span class="id" type="var">bag</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_sum1</span>: <span class="id" type="var">count</span> 1 (<span class="id" type="var">sum</span> [1;2;3] [1;4;1]) = 3.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">add</span> (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">bag</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_add1</span>: <span class="id" type="var">count</span> 1 (<span class="id" type="var">add</span> 1 [1;4;1]) = 3.<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_add2</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">add</span> 1 [1;4;1]) = 0.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">member</span> (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">bool</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_member1</span>: <span class="id" type="var">member</span> 1 [1;4;1] = <span class="id" type="var">true</span>.<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_member2</span>: <span class="id" type="var">member</span> 2 [1;4;1] = <span class="id" type="var">false</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 class="paragraph"> </div>
<a name="lab78"></a><h4 class="section">Exercise: 3 stars, optional (bag_more_functions)</h4>
Here are some more bag functions for you to practice with.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">remove_one</span> (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">bag</span> :=<br/>
<span class="comment">(* When remove_one is applied to a bag without the number to remove,<br/>
it should return the same bag unchanged. *)</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_remove_one1</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_one</span> 5 [2;1;5;4;1]) = 0.<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_remove_one2</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_one</span> 5 [2;1;4;1]) = 0.<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_remove_one3</span>: <span class="id" type="var">count</span> 4 (<span class="id" type="var">remove_one</span> 5 [2;1;4;5;1;4]) = 2.<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_remove_one4</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_one</span> 5 [2;1;5;4;5;1;4]) = 1.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">remove_all</span> (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) (<span class="id" type="var">s</span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">bag</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_remove_all1</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_all</span> 5 [2;1;5;4;1]) = 0.<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_remove_all2</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_all</span> 5 [2;1;4;1]) = 0.<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_remove_all3</span>: <span class="id" type="var">count</span> 4 (<span class="id" type="var">remove_all</span> 5 [2;1;4;5;1;4]) = 2.<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_remove_all4</span>: <span class="id" type="var">count</span> 5 (<span class="id" type="var">remove_all</span> 5 [2;1;5;4;5;1;4;5;1;4]) = 0.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">subset</span> (<span class="id" type="var">s<sub>1</sub></span>:<span class="id" type="var">bag</span>) (<span class="id" type="var">s<sub>2</sub></span>:<span class="id" type="var">bag</span>) : <span class="id" type="var">bool</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_subset1</span>: <span class="id" type="var">subset</span> [1;2] [2;1;4;1] = <span class="id" type="var">true</span>.<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_subset2</span>: <span class="id" type="var">subset</span> [1;2;2] [2;1;4;1] = <span class="id" type="var">false</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 class="paragraph"> </div>
<a name="lab79"></a><h4 class="section">Exercise: 3 stars (bag_theorem)</h4>
Write down an interesting theorem about bags involving the
functions <span class="inlinecode"><span class="id" type="var">count</span></span> and <span class="inlinecode"><span class="id" type="var">add</span></span>, and prove it. Note that, since this
problem is somewhat open-ended, it's possible that you may come up
with a theorem which is true, but whose proof requires techniques
you haven't learned yet. Feel free to ask for help if you get
stuck!
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab80"></a><h1 class="section">Reasoning About Lists</h1>
<div class="paragraph"> </div>
Just as with numbers, simple facts about list-processing
functions can sometimes be proved entirely by simplification. For
example, the simplification performed by <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> is enough
for this theorem...
</div>
<div class="code code-tight">
<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">l</span>:<span class="id" type="var">natlist</span>,<br/>
[] ++ <span class="id" type="var">l</span> = <span class="id" type="var">l</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">
... because the <span class="inlinecode">[]</span> is substituted into the match position
in the definition of <span class="inlinecode"><span class="id" type="var">app</span></span>, allowing the match itself to be
simplified.
<div class="paragraph"> </div>
Also, as with numbers, it is sometimes helpful to perform case
analysis on the possible shapes (empty or non-empty) of an unknown
list.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">tl_length_pred</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>,<br/>
<span class="id" type="var">pred</span> (<span class="id" type="var">length</span> <span class="id" type="var">l</span>) = <span class="id" type="var">length</span> (<span class="id" type="var">tl</span> <span class="id" type="var">l</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">l</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">l</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n</span> <span class="id" type="var">l'</span>].<br/>
<span class="id" type="var">Case</span> "l = nil".<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "l = cons n l'".<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Here, the <span class="inlinecode"><span class="id" type="var">nil</span></span> case works because we've chosen to define
<span class="inlinecode"><span class="id" type="var">tl</span></span> <span class="inlinecode"><span class="id" type="var">nil</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">nil</span></span>. Notice that the <span class="inlinecode"><span class="id" type="keyword">as</span></span> annotation on the <span class="inlinecode"><span class="id" type="tactic">destruct</span></span>
tactic here introduces two names, <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">l'</span></span>, corresponding to
the fact that the <span class="inlinecode"><span class="id" type="var">cons</span></span> constructor for lists takes two
arguments (the head and tail of the list it is constructing).
<div class="paragraph"> </div>
Usually, though, interesting theorems about lists require
induction for their proofs.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab81"></a><h2 class="section">Micro-Sermon</h2>
<div class="paragraph"> </div>
Simply reading example proofs will not get you very far! It is
very important to work through the details of each one, using Coq
and thinking about what each step of the proof achieves.
Otherwise it is more or less guaranteed that the exercises will
make no sense.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab82"></a><h2 class="section">Induction on Lists</h2>
<div class="paragraph"> </div>
Proofs by induction over datatypes like <span class="inlinecode"><span class="id" type="var">natlist</span></span> are
perhaps a little less familiar than standard natural number
induction, but the basic idea is equally simple. Each <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>
declaration defines a set of data values that can be built up from
the declared constructors: a boolean can be either <span class="inlinecode"><span class="id" type="var">true</span></span> or
<span class="inlinecode"><span class="id" type="var">false</span></span>; a number can be either <span class="inlinecode"><span class="id" type="var">O</span></span> or <span class="inlinecode"><span class="id" type="var">S</span></span> applied to a number; a
list can be either <span class="inlinecode"><span class="id" type="var">nil</span></span> or <span class="inlinecode"><span class="id" type="var">cons</span></span> applied to a number and a list.
<div class="paragraph"> </div>
Moreover, applications of the declared constructors to one another
are the <i>only</i> possible shapes that elements of an inductively
defined set can have, and this fact directly gives rise to a way
of reasoning about inductively defined sets: a number is either
<span class="inlinecode"><span class="id" type="var">O</span></span> or else it is <span class="inlinecode"><span class="id" type="var">S</span></span> applied to some <i>smaller</i> number; a list is
either <span class="inlinecode"><span class="id" type="var">nil</span></span> or else it is <span class="inlinecode"><span class="id" type="var">cons</span></span> applied to some number and some
<i>smaller</i> list; etc. So, if we have in mind some proposition <span class="inlinecode"><span class="id" type="var">P</span></span>
that mentions a list <span class="inlinecode"><span class="id" type="var">l</span></span> and we want to argue that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for
<i>all</i> lists, we can reason as follows:
<div class="paragraph"> </div>
<ul class="doclist">
<li> First, show that <span class="inlinecode"><span class="id" type="var">P</span></span> is true of <span class="inlinecode"><span class="id" type="var">l</span></span> when <span class="inlinecode"><span class="id" type="var">l</span></span> is <span class="inlinecode"><span class="id" type="var">nil</span></span>.
<div class="paragraph"> </div>
</li>
<li> Then show that <span class="inlinecode"><span class="id" type="var">P</span></span> is true of <span class="inlinecode"><span class="id" type="var">l</span></span> when <span class="inlinecode"><span class="id" type="var">l</span></span> is <span class="inlinecode"><span class="id" type="var">cons</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">l'</span></span> for
some number <span class="inlinecode"><span class="id" type="var">n</span></span> and some smaller list <span class="inlinecode"><span class="id" type="var">l'</span></span>, assuming that <span class="inlinecode"><span class="id" type="var">P</span></span>
is true for <span class="inlinecode"><span class="id" type="var">l'</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
Since larger lists can only be built up from smaller ones,
eventually reaching <span class="inlinecode"><span class="id" type="var">nil</span></span>, these two things together establish the
truth of <span class="inlinecode"><span class="id" type="var">P</span></span> for all lists <span class="inlinecode"><span class="id" type="var">l</span></span>. Here's a concrete example:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">app_assoc</span> : <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">l3</span> : <span class="id" type="var">natlist</span>, <br/>
(<span class="id" type="var">l1</span> ++ <span class="id" type="var">l2</span>) ++ <span class="id" type="var">l3</span> = <span class="id" type="var">l1</span> ++ (<span class="id" type="var">l2</span> ++ <span class="id" type="var">l3</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span> <span class="id" type="var">l3</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">l1</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n</span> <span class="id" type="var">l1'</span>].<br/>
<span class="id" type="var">Case</span> "l1 = nil".<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "l1 = cons n l1'".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHl1'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Again, this Coq proof is not especially illuminating as a
static written document — it is easy to see what's going on if
you are reading the proof in an interactive Coq session and you
can see the current goal and context at each point, but this state
is not visible in the written-down parts of the Coq proof. So a
natural-language proof — one written for human readers — will
need to include more explicit signposts; in particular, it will
help the reader stay oriented if we remind them exactly what the
induction hypothesis is in the second case.
<div class="paragraph"> </div>
<a name="lab83"></a><h3 class="section">Informal version</h3>
<div class="paragraph"> </div>
<i>Theorem</i>: For all lists <span class="inlinecode"><span class="id" type="var">l1</span></span>, <span class="inlinecode"><span class="id" type="var">l2</span></span>, and <span class="inlinecode"><span class="id" type="var">l3</span></span>,
<span class="inlinecode">(<span class="id" type="var">l1</span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l2</span>)</span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l3</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">l1</span></span> <span class="inlinecode">++</span> <span class="inlinecode">(<span class="id" type="var">l2</span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l3</span>)</span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">l1</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> First, suppose <span class="inlinecode"><span class="id" type="var">l1</span></span> <span class="inlinecode">=</span> <span class="inlinecode">[]</span>. We must show
<div class="paragraph"> </div>
<div class="code code-tight">
([] ++ <span class="id" type="var">l2</span>) ++ <span class="id" type="var">l3</span> = [] ++ (<span class="id" type="var">l2</span> ++ <span class="id" type="var">l3</span>),
<div class="paragraph"> </div>
</div>
which follows directly from the definition of <span class="inlinecode">++</span>.
<div class="paragraph"> </div>
</li>
<li> Next, suppose <span class="inlinecode"><span class="id" type="var">l1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span>::<span class="id" type="var">l1'</span></span>, with
<div class="paragraph"> </div>
<div class="code code-tight">
(<span class="id" type="var">l1'</span> ++ <span class="id" type="var">l2</span>) ++ <span class="id" type="var">l3</span> = <span class="id" type="var">l1'</span> ++ (<span class="id" type="var">l2</span> ++ <span class="id" type="var">l3</span>)
<div class="paragraph"> </div>
</div>
(the induction hypothesis). We must show
<div class="paragraph"> </div>
<div class="code code-tight">
((<span class="id" type="var">n</span> :: <span class="id" type="var">l1'</span>) ++ <span class="id" type="var">l2</span>) ++ <span class="id" type="var">l3</span> = (<span class="id" type="var">n</span> :: <span class="id" type="var">l1'</span>) ++ (<span class="id" type="var">l2</span> ++ <span class="id" type="var">l3</span>).
<div class="paragraph"> </div>
</div>
By the definition of <span class="inlinecode">++</span>, this follows from
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">n</span> :: ((<span class="id" type="var">l1'</span> ++ <span class="id" type="var">l2</span>) ++ <span class="id" type="var">l3</span>) = <span class="id" type="var">n</span> :: (<span class="id" type="var">l1'</span> ++ (<span class="id" type="var">l2</span> ++ <span class="id" type="var">l3</span>)),
<div class="paragraph"> </div>
</div>
which is immediate from the induction hypothesis. <font size=-2>☐</font>
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab84"></a><h3 class="section">Another example</h3>
<div class="paragraph"> </div>
Here is a similar example to be worked together in class:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">app_length</span> : <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">natlist</span>, <br/>
<span class="id" type="var">length</span> (<span class="id" type="var">l1</span> ++ <span class="id" type="var">l2</span>) = (<span class="id" type="var">length</span> <span class="id" type="var">l1</span>) + (<span class="id" type="var">length</span> <span class="id" type="var">l2</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">l1</span> <span class="id" type="var">l2</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">l1</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n</span> <span class="id" type="var">l1'</span>].<br/>
<span class="id" type="var">Case</span> "l1 = nil".<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "l1 = cons".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHl1'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab85"></a><h3 class="section">Reversing a list</h3>
For a slightly more involved example of an inductive proof
over lists, suppose we define a "cons on the right" function
<span class="inlinecode"><span class="id" type="var">snoc</span></span> like this...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">snoc</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) (<span class="id" type="var">v</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">natlist</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">v</span>]<br/>
| <span class="id" type="var">h</span> :: <span class="id" type="var">t</span> ⇒ <span class="id" type="var">h</span> :: (<span class="id" type="var">snoc</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/>
</div>
<div class="doc">
... and use it to define a list-reversing function <span class="inlinecode"><span class="id" type="var">rev</span></span>
like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">rev</span> (<span class="id" type="var">l</span>:<span class="id" type="var">natlist</span>) : <span class="id" type="var">natlist</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><br/>
| <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">rev</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>: <span class="id" type="var">rev</span> [1;2;3] = [3;2;1].<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_rev2</span>: <span class="id" type="var">rev</span> <span class="id" type="var">nil</span> = <span class="id" type="var">nil</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="lab86"></a><h3 class="section">Proofs about reverse</h3>
Now let's prove some more list theorems using our newly
defined <span class="inlinecode"><span class="id" type="var">snoc</span></span> and <span class="inlinecode"><span class="id" type="var">rev</span></span>. For something a little more challenging
than the inductive proofs we've seen so far, let's prove that
reversing a list does not change its length. Our first attempt at
this proof gets stuck in the successor case...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">rev_length_firsttry</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span> : <span class="id" type="var">natlist</span>,<br/>
<span class="id" type="var">length</span> (<span class="id" type="var">rev</span> <span class="id" type="var">l</span>) = <span class="id" type="var">length</span> <span class="id" type="var">l</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">l</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">l</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n</span> <span class="id" type="var">l'</span>].<br/>
<span class="id" type="var">Case</span> "l = []".<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "l = n :: l'".<br/>
<span class="comment">(* This is the tricky case. Let's begin as usual <br/>
by simplifying. *)</span><br/>
<span class="id" type="tactic">simpl</span>.<br/>
<span class="comment">(* Now we seem to be stuck: the goal is an equality <br/>