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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/>
<span class="id" type="keyword">Module</span> <a name="NatList"><span class="id" type="module">NatList</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab57"></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 here:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="NatList.natprod"><span class="id" type="inductive">natprod</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="NatList.pair"><span class="id" type="constructor">pair</span></a> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Lists.html#natprod"><span class="id" type="inductive">natprod</span></a>.<br/>
<br/>
</div>
<div class="doc">
This declaration can be read: "There is 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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (<a class="idref" href="Lists.html#NatList.pair"><span class="id" type="constructor">pair</span></a> 3 5).<br/>
<br/>
</div>
<div class="doc">
Here are two simple functions 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> <a name="NatList.fst"><span class="id" type="definition">fst</span></a> (<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.pair"><span class="id" type="constructor">pair</span></a> <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/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.snd"><span class="id" type="definition">snd</span></a> (<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.pair"><span class="id" type="constructor">pair</span></a> <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="var">Compute</span> (<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> (<a class="idref" href="Lists.html#NatList.pair"><span class="id" type="constructor">pair</span></a> 3 5)).<br/>
<span class="comment">(* ===> 3 *)</span><br/>
<br/>
</div>
<div class="doc">
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 )" := (<a class="idref" href="Lists.html#NatList.pair"><span class="id" type="constructor">pair</span></a> <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 works because the pair notation is actually
provided as part of the standard library):
</div>
<div class="code code-tight">
<br/>
<span class="id" type="var">Compute</span> (<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> (3,5)).<br/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.fst'"><span class="id" type="definition">fst'</span></a> (<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> <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/>
<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.snd'"><span class="id" type="definition">snd'</span></a> (<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> <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> <a name="NatList.swap_pair"><span class="id" type="definition">swap_pair</span></a> (<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>) : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> <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">
Let's try to prove a few simple facts about pairs.
<div class="paragraph"> </div>
If we state things in a particular (and slightly peculiar) way, we
can complete proofs with just reflexivity (and its built-in
simplification):
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="NatList.surjective_pairing'"><span class="id" type="lemma">surjective_pairing'</span></a> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>),<br/>
(<a class="idref" href="Lists.html#n"><span class="id" type="variable">n</span></a>,<a class="idref" href="Lists.html#m"><span class="id" type="variable">m</span></a>) = (<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> (<a class="idref" href="Lists.html#n"><span class="id" type="variable">n</span></a>,<a class="idref" href="Lists.html#m"><span class="id" type="variable">m</span></a>), <a class="idref" href="Lists.html#NatList.snd"><span class="id" type="definition">snd</span></a> (<a class="idref" href="Lists.html#n"><span class="id" type="variable">n</span></a>,<a class="idref" href="Lists.html#m"><span class="id" type="variable">m</span></a>)).<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">
But <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> <a name="NatList.surjective_pairing_stuck"><span class="id" type="lemma">surjective_pairing_stuck</span></a> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>),<br/>
<a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> = (<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>, <a class="idref" href="Lists.html#NatList.snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>).<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">
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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="NatList.surjective_pairing"><span class="id" type="lemma">surjective_pairing</span></a> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>),<br/>
<a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a> = (<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>, <a class="idref" href="Lists.html#NatList.snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>).<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">
Notice that, unlike its behavior with <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 class="paragraph"> </div>
<a name="lab58"></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> <a name="NatList.snd_fst_is_swap"><span class="id" type="lemma">snd_fst_is_swap</span></a> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>),<br/>
(<a class="idref" href="Lists.html#NatList.snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>, <a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>) = <a class="idref" href="Lists.html#NatList.swap_pair"><span class="id" type="definition">swap_pair</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>.<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 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab59"></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> <a name="NatList.fst_swap_is_snd"><span class="id" type="lemma">fst_swap_is_snd</span></a> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">p</span> : <a class="idref" href="Lists.html#NatList.natprod"><span class="id" type="inductive">natprod</span></a>),<br/>
<a class="idref" href="Lists.html#NatList.fst"><span class="id" type="definition">fst</span></a> (<a class="idref" href="Lists.html#NatList.swap_pair"><span class="id" type="definition">swap_pair</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>) = <a class="idref" href="Lists.html#NatList.snd"><span class="id" type="definition">snd</span></a> <a class="idref" href="Lists.html#p"><span class="id" type="variable">p</span></a>.<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 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab60"></a><h1 class="section">Lists of Numbers</h1>
<div class="paragraph"> </div>
Generalizing the definition of pairs, 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> <a name="NatList.natlist"><span class="id" type="inductive">natlist</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="NatList.nil"><span class="id" type="constructor">nil</span></a> : <a class="idref" href="Lists.html#natlist"><span class="id" type="inductive">natlist</span></a><br/>
| <a name="NatList.cons"><span class="id" type="constructor">cons</span></a> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Lists.html#natlist"><span class="id" type="inductive">natlist</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Lists.html#natlist"><span class="id" type="inductive">natlist</span></a>.<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> <a name="NatList.mylist"><span class="id" type="definition">mylist</span></a> := <a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> 1 (<a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> 2 (<a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> 3 <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>)).<br/>
<br/>
</div>
<div class="doc">
As with pairs, it is more convenient to write lists in
familiar programming notation. The following 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" := (<a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> <span class="id" type="var">x</span> <span class="id" type="var">l</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> "[ ]" := <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>.<br/>
<span class="id" type="keyword">Notation</span> "[ x ; .. ; y ]" := (<a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> <span class="id" type="var">x</span> .. (<a class="idref" href="Lists.html#NatList.cons"><span class="id" type="constructor">cons</span></a> <span class="id" type="var">y</span> <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>) ..).<br/>
<br/>
</div>
<div class="doc">
It is not necessary to understand the details of these
declarations, but in case you are interested, here is roughly
what's going on. 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> <a name="NatList.mylist1"><span class="id" type="definition">mylist1</span></a> := 1 :: (2 :: (3 :: <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>)).<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.mylist2"><span class="id" type="definition">mylist2</span></a> := 1 :: 2 :: 3 :: <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>.<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.mylist3"><span class="id" type="definition">mylist3</span></a> := [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>
(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="lab61"></a><h3 class="section">Repeat</h3>
<div class="paragraph"> </div>
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> <a name="NatList.repeat"><span class="id" type="definition">repeat</span></a> (<span class="id" type="var">n</span> <span class="id" type="var">count</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#count"><span class="id" type="variable">count</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> ⇒ <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">count'</span> ⇒ <a class="idref" href="Lists.html#n"><span class="id" type="variable">n</span></a> :: (<a class="idref" href="Lists.html#repeat"><span class="id" type="definition">repeat</span></a> <a class="idref" href="Lists.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="var">count'</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab62"></a><h3 class="section">Length</h3>
<div class="paragraph"> </div>
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> <a name="NatList.length"><span class="id" type="definition">length</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a><br/>
| <span class="id" type="var">h</span> :: <span class="id" type="var">t</span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> (<a class="idref" href="Lists.html#length"><span class="id" type="definition">length</span></a> <span class="id" type="var">t</span>)<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab63"></a><h3 class="section">Append</h3>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="var">app</span></span> function concatenates (appends) two lists.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="NatList.app"><span class="id" type="definition">app</span></a> (<span class="id" type="var">l<sub>1</sub></span> <span class="id" type="var">l<sub>2</sub></span> : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#l<sub>1</sub>"><span class="id" type="variable">l<sub>1</sub></span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ⇒ <a class="idref" href="Lists.html#l<sub>2</sub>"><span class="id" type="variable">l<sub>2</sub></span></a><br/>
| <span class="id" type="var">h</span> :: <span class="id" type="var">t</span> ⇒ <span class="id" type="var">h</span> :: (<a class="idref" href="Lists.html#app"><span class="id" type="definition">app</span></a> <span class="id" type="var">t</span> <a class="idref" href="Lists.html#l<sub>2</sub>"><span class="id" type="variable">l<sub>2</sub></span></a>)<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" := (<a class="idref" href="Lists.html#NatList.app"><span class="id" type="definition">app</span></a> <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> <a name="NatList.test_app1"><span class="id" type="definition">test_app1</span></a>: [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> <a name="NatList.test_app2"><span class="id" type="definition">test_app2</span></a>: <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ++ [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> <a name="NatList.test_app3"><span class="id" type="definition">test_app3</span></a>: [1;2;3] ++ <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> = [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">
<a name="lab64"></a><h3 class="section">Head (with default) and Tail</h3>
<div class="paragraph"> </div>
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>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.hd"><span class="id" type="definition">hd</span></a> (<span class="id" type="var">default</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ⇒ <a class="idref" href="Lists.html#default"><span class="id" type="variable">default</span></a><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> <a name="NatList.tl"><span class="id" type="definition">tl</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ⇒ <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a><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> <a name="NatList.test_hd<sub>1</sub>"><span class="id" type="definition">test_hd<sub>1</sub></span></a>: <a class="idref" href="Lists.html#NatList.hd"><span class="id" type="definition">hd</span></a> 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> <a name="NatList.test_hd<sub>2</sub>"><span class="id" type="definition">test_hd<sub>2</sub></span></a>: <a class="idref" href="Lists.html#NatList.hd"><span class="id" type="definition">hd</span></a> 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> <a name="NatList.test_tl"><span class="id" type="definition">test_tl</span></a>: <a class="idref" href="Lists.html#NatList.tl"><span class="id" type="definition">tl</span></a> [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="lab65"></a><h3 class="section">Exercises</h3>
<div class="paragraph"> </div>
<a name="lab66"></a><h4 class="section">Exercise: 2 stars, recommended (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> <a name="NatList.nonzeros"><span class="id" type="definition">nonzeros</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_nonzeros"><span class="id" type="definition">test_nonzeros</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.nonzeros"><span class="id" type="axiom">nonzeros</span></a> [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> <a name="NatList.oddmembers"><span class="id" type="definition">oddmembers</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_oddmembers"><span class="id" type="definition">test_oddmembers</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.oddmembers"><span class="id" type="axiom">oddmembers</span></a> [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> <a name="NatList.countoddmembers"><span class="id" type="definition">countoddmembers</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_countoddmembers1"><span class="id" type="definition">test_countoddmembers1</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.countoddmembers"><span class="id" type="axiom">countoddmembers</span></a> [1;0;3;1;4;5] = 4.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_countoddmembers2"><span class="id" type="definition">test_countoddmembers2</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.countoddmembers"><span class="id" type="axiom">countoddmembers</span></a> [0;2;4] = 0.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_countoddmembers3"><span class="id" type="definition">test_countoddmembers3</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.countoddmembers"><span class="id" type="axiom">countoddmembers</span></a> <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> = 0.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab67"></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> <a name="NatList.alternate"><span class="id" type="definition">alternate</span></a> (<span class="id" type="var">l<sub>1</sub></span> <span class="id" type="var">l<sub>2</sub></span> : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_alternate1"><span class="id" type="definition">test_alternate1</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.alternate"><span class="id" type="axiom">alternate</span></a> [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/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_alternate2"><span class="id" type="definition">test_alternate2</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.alternate"><span class="id" type="axiom">alternate</span></a> [1] [4;5;6] = [1;4;5;6].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_alternate3"><span class="id" type="definition">test_alternate3</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.alternate"><span class="id" type="axiom">alternate</span></a> [1;2;3] [4] = [1;4;2;3].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_alternate4"><span class="id" type="definition">test_alternate4</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.alternate"><span class="id" type="axiom">alternate</span></a> [] [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 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab68"></a><h3 class="section">Bags via Lists</h3>
<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, except that each element
can appear multiple times rather than just once. One possible
implementation is to represent a bag of numbers as a list.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="NatList.bag"><span class="id" type="definition">bag</span></a> := <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab69"></a><h4 class="section">Exercise: 3 stars, recommended (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> <a name="NatList.count"><span class="id" type="definition">count</span></a> (<span class="id" type="var">v</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</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> <a name="NatList.test_count1"><span class="id" type="definition">test_count1</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 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> <a name="NatList.test_count2"><span class="id" type="definition">test_count2</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 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> <a name="NatList.sum"><span class="id" type="definition">sum</span></a> : <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_sum1"><span class="id" type="definition">test_sum1</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 1 (<a class="idref" href="Lists.html#NatList.sum"><span class="id" type="axiom">sum</span></a> [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> <a name="NatList.add"><span class="id" type="definition">add</span></a> (<span class="id" type="var">v</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_add1"><span class="id" type="definition">test_add1</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 1 (<a class="idref" href="Lists.html#NatList.add"><span class="id" type="axiom">add</span></a> 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> <a name="NatList.test_add2"><span class="id" type="definition">test_add2</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.add"><span class="id" type="axiom">add</span></a> 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> <a name="NatList.member"><span class="id" type="definition">member</span></a> (<span class="id" type="var">v</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="Basics.html#bool"><span class="id" type="inductive">bool</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_member1"><span class="id" type="definition">test_member1</span></a>: <a class="idref" href="Lists.html#NatList.member"><span class="id" type="axiom">member</span></a> 1 [1;4;1] = <a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_member2"><span class="id" type="definition">test_member2</span></a>: <a class="idref" href="Lists.html#NatList.member"><span class="id" type="axiom">member</span></a> 2 [1;4;1] = <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab70"></a><h4 class="section">Exercise: 3 stars, optional (bag_more_functions)</h4>
Here are some more bag functions for you to practice with.
<div class="paragraph"> </div>
When remove_one is applied to a bag without the number to remove,
it should return the same bag unchanged.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="NatList.remove_one"><span class="id" type="definition">remove_one</span></a> (<span class="id" type="var">v</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_remove_one1"><span class="id" type="definition">test_remove_one1</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_one"><span class="id" type="axiom">remove_one</span></a> 5 [2;1;5;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">Example</span> <a name="NatList.test_remove_one2"><span class="id" type="definition">test_remove_one2</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_one"><span class="id" type="axiom">remove_one</span></a> 5 [2;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">Example</span> <a name="NatList.test_remove_one3"><span class="id" type="definition">test_remove_one3</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 4 (<a class="idref" href="Lists.html#NatList.remove_one"><span class="id" type="axiom">remove_one</span></a> 5 [2;1;4;5;1;4]) = 2.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_remove_one4"><span class="id" type="definition">test_remove_one4</span></a>:<br/>
<a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_one"><span class="id" type="axiom">remove_one</span></a> 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> <a name="NatList.remove_all"><span class="id" type="definition">remove_all</span></a> (<span class="id" type="var">v</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) (<span class="id" type="var">s</span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_remove_all1"><span class="id" type="definition">test_remove_all1</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_all"><span class="id" type="axiom">remove_all</span></a> 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> <a name="NatList.test_remove_all2"><span class="id" type="definition">test_remove_all2</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_all"><span class="id" type="axiom">remove_all</span></a> 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> <a name="NatList.test_remove_all3"><span class="id" type="definition">test_remove_all3</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 4 (<a class="idref" href="Lists.html#NatList.remove_all"><span class="id" type="axiom">remove_all</span></a> 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> <a name="NatList.test_remove_all4"><span class="id" type="definition">test_remove_all4</span></a>: <a class="idref" href="Lists.html#NatList.count"><span class="id" type="axiom">count</span></a> 5 (<a class="idref" href="Lists.html#NatList.remove_all"><span class="id" type="axiom">remove_all</span></a> 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> <a name="NatList.subset"><span class="id" type="definition">subset</span></a> (<span class="id" type="var">s<sub>1</sub></span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) (<span class="id" type="var">s<sub>2</sub></span>:<a class="idref" href="Lists.html#NatList.bag"><span class="id" type="definition">bag</span></a>) : <a class="idref" href="Basics.html#bool"><span class="id" type="inductive">bool</span></a> <br/>
<span class="comment">(* REPLACE THIS LINE WITH := _your_definition_ . *)</span> . <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_subset1"><span class="id" type="definition">test_subset1</span></a>: <a class="idref" href="Lists.html#NatList.subset"><span class="id" type="axiom">subset</span></a> [1;2] [2;1;4;1] = <a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<span class="id" type="keyword">Example</span> <a name="NatList.test_subset2"><span class="id" type="definition">test_subset2</span></a>: <a class="idref" href="Lists.html#NatList.subset"><span class="id" type="axiom">subset</span></a> [1;2;2] [2;1;4;1] = <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab71"></a><h4 class="section">Exercise: 3 stars, recommended (bag_theorem)</h4>
Write down an interesting theorem <span class="inlinecode"><span class="id" type="var">bag_theorem</span></span> 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">(*<br/>
Theorem bag_theorem : ...<br/>
Proof.<br/>
...<br/>
Qed.<br/>
*)</span><br/>
<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab72"></a><h1 class="section">Reasoning About Lists</h1>
<div class="paragraph"> </div>
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> <a name="NatList.nil_app"><span class="id" type="lemma">nil_app</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>,<br/>
[] ++ <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a> = <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a>.<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
"scrutinee" 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> <a name="NatList.tl_length_pred"><span class="id" type="lemma">tl_length_pred</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>,<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Peano.html#pred"><span class="id" type="abbreviation">pred</span></a> (<a class="idref" href="Lists.html#NatList.length"><span class="id" type="definition">length</span></a> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a>) = <a class="idref" href="Lists.html#NatList.length"><span class="id" type="definition">length</span></a> (<a class="idref" href="Lists.html#NatList.tl"><span class="id" type="definition">tl</span></a> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a>).<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="comment">(* l = nil *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l = cons n l' *)</span><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="lab73"></a><h3 class="section">Micro-Sermon</h3>
<div class="paragraph"> </div>
Simply reading example proof scripts will not get you very far!
It is important to work through the details of each one, using Coq
and thinking about what each step achieves. Otherwise it is more
or less guaranteed that the exercises will make no sense when you
get to them. 'Nuff said.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab74"></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 a
little less familiar than standard natural number induction, but
the 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 using 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 another 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 arguments 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> <a name="NatList.app_assoc"><span class="id" type="lemma">app_assoc</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">l<sub>1</sub></span> <span class="id" type="var">l<sub>2</sub></span> <span class="id" type="var">l<sub>3</sub></span> : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>,<br/>
(<a class="idref" href="Lists.html#l<sub>1</sub>"><span class="id" type="variable">l<sub>1</sub></span></a> ++ <a class="idref" href="Lists.html#l<sub>2</sub>"><span class="id" type="variable">l<sub>2</sub></span></a>) ++ <a class="idref" href="Lists.html#l<sub>3</sub>"><span class="id" type="variable">l<sub>3</sub></span></a> = <a class="idref" href="Lists.html#l<sub>1</sub>"><span class="id" type="variable">l<sub>1</sub></span></a> ++ (<a class="idref" href="Lists.html#l<sub>2</sub>"><span class="id" type="variable">l<sub>2</sub></span></a> ++ <a class="idref" href="Lists.html#l<sub>3</sub>"><span class="id" type="variable">l<sub>3</sub></span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">l<sub>1</sub></span> <span class="id" type="var">l<sub>2</sub></span> <span class="id" type="var">l<sub>3</sub></span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">l<sub>1</sub></span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n</span> <span class="id" type="var">l<sub>1</sub>'</span> <span class="id" type="var">IHl1'</span>].<br/>
- <span class="comment">(* l<sub>1</sub> = nil *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* l<sub>1</sub> = cons n l<sub>1</sub>' *)</span><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">
Notice that, as when doing induction on natural numbers, the
<span class="inlinecode"><span class="id" type="keyword">as</span>...</span> clause provided to the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic gives a name to
the induction hypothesis corresponding to the smaller list <span class="inlinecode"><span class="id" type="var">l<sub>1</sub>'</span></span>
in the <span class="inlinecode"><span class="id" type="var">cons</span></span> case. Once 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>
For comparison, here is an informal proof of the same theorem.
<div class="paragraph"> </div>
<i>Theorem</i>: For all lists <span class="inlinecode"><span class="id" type="var">l<sub>1</sub></span></span>, <span class="inlinecode"><span class="id" type="var">l<sub>2</sub></span></span>, and <span class="inlinecode"><span class="id" type="var">l<sub>3</sub></span></span>,
<span class="inlinecode">(<span class="id" type="var">l<sub>1</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l<sub>2</sub></span>)</span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l<sub>3</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">l<sub>1</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode">(<span class="id" type="var">l<sub>2</sub></span></span> <span class="inlinecode">++</span> <span class="inlinecode"><span class="id" type="var">l<sub>3</sub></span>)</span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">l<sub>1</sub></span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> First, suppose <span class="inlinecode"><span class="id" type="var">l<sub>1</sub></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">l<sub>2</sub></span>) ++ <span class="id" type="var">l<sub>3</sub></span> = [] ++ (<span class="id" type="var">l<sub>2</sub></span> ++ <span class="id" type="var">l<sub>3</sub></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">l<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span>::<span class="id" type="var">l<sub>1</sub>'</span></span>, with
<div class="paragraph"> </div>
<div class="code code-tight">
(<span class="id" type="var">l<sub>1</sub>'</span> ++ <span class="id" type="var">l<sub>2</sub></span>) ++ <span class="id" type="var">l<sub>3</sub></span> = <span class="id" type="var">l<sub>1</sub>'</span> ++ (<span class="id" type="var">l<sub>2</sub></span> ++ <span class="id" type="var">l<sub>3</sub></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">l<sub>1</sub>'</span>) ++ <span class="id" type="var">l<sub>2</sub></span>) ++ <span class="id" type="var">l<sub>3</sub></span> = (<span class="id" type="var">n</span> :: <span class="id" type="var">l<sub>1</sub>'</span>) ++ (<span class="id" type="var">l<sub>2</sub></span> ++ <span class="id" type="var">l<sub>3</sub></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">l<sub>1</sub>'</span> ++ <span class="id" type="var">l<sub>2</sub></span>) ++ <span class="id" type="var">l<sub>3</sub></span>) = <span class="id" type="var">n</span> :: (<span class="id" type="var">l<sub>1</sub>'</span> ++ (<span class="id" type="var">l<sub>2</sub></span> ++ <span class="id" type="var">l<sub>3</sub></span>)),
<div class="paragraph"> </div>
</div>
which is immediate from the induction hypothesis. <font size=-2 color="rgb(80%,80%,100%)">☐</font>
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab75"></a><h3 class="section">Reversing a list</h3>
<div class="paragraph"> </div>
For a slightly more involved example of inductive proof over
lists, suppose we use <span class="inlinecode"><span class="id" type="var">app</span></span> to define a list-reversing function
<span class="inlinecode"><span class="id" type="var">rev</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="NatList.rev"><span class="id" type="definition">rev</span></a> (<span class="id" type="var">l</span>:<a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>) : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a> :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> ⇒ <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a><br/>
| <span class="id" type="var">h</span> :: <span class="id" type="var">t</span> ⇒ <a class="idref" href="Lists.html#rev"><span class="id" type="definition">rev</span></a> <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> <a name="NatList.test_rev1"><span class="id" type="definition">test_rev1</span></a>: <a class="idref" href="Lists.html#NatList.rev"><span class="id" type="definition">rev</span></a> [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> <a name="NatList.test_rev2"><span class="id" type="definition">test_rev2</span></a>: <a class="idref" href="Lists.html#NatList.rev"><span class="id" type="definition">rev</span></a> <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a> = <a class="idref" href="Lists.html#NatList.nil"><span class="id" type="constructor">nil</span></a>.<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="lab76"></a><h3 class="section">Proofs about reverse</h3>
<div class="paragraph"> </div>
Now let's prove some theorems about our newly defined <span class="inlinecode"><span class="id" type="var">rev</span></span>.
For something a bit more challenging than what we've seen, let's
prove that reversing a list does not change its length. Our first
attempt gets stuck in the successor case...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="NatList.rev_length_firsttry"><span class="id" type="lemma">rev_length_firsttry</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span> : <a class="idref" href="Lists.html#NatList.natlist"><span class="id" type="inductive">natlist</span></a>,<br/>
<a class="idref" href="Lists.html#NatList.length"><span class="id" type="definition">length</span></a> (<a class="idref" href="Lists.html#NatList.rev"><span class="id" type="definition">rev</span></a> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a>) = <a class="idref" href="Lists.html#NatList.length"><span class="id" type="definition">length</span></a> <a class="idref" href="Lists.html#l"><span class="id" type="variable">l</span></a>.<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> <span class="id" type="var">IHl'</span>].<br/>
- <span class="comment">(* l = <span class="inlinecode"></span> *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>