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Prop.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>Prop: Propositions and Evidence</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">Prop<span class="subtitle">Propositions and Evidence</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="var">Logic</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab232"></a><h2 class="section">From Boolean Functions to Propositions</h2>
<div class="paragraph"> </div>
In chapter <span class="inlinecode"><span class="id" type="var">Basics</span></span> we defined a <i>function</i> <span class="inlinecode"><span class="id" type="var">evenb</span></span> that tests a
number for evenness, yielding <span class="inlinecode"><span class="id" type="var">true</span></span> if so. We can use this
function to define the <i>proposition</i> that some number <span class="inlinecode"><span class="id" type="var">n</span></span> is
even:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">even</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="keyword">Prop</span> := <br/>
<span class="id" type="var">evenb</span> <span class="id" type="var">n</span> = <span class="id" type="var">true</span>.<br/>
<br/>
</div>
<div class="doc">
That is, we can define "<span class="inlinecode"><span class="id" type="var">n</span></span> is even" to mean "the function <span class="inlinecode"><span class="id" type="var">evenb</span></span>
returns <span class="inlinecode"><span class="id" type="var">true</span></span> when applied to <span class="inlinecode"><span class="id" type="var">n</span></span>."
<div class="paragraph"> </div>
Note that here we have given a name
to a proposition using a <span class="inlinecode"><span class="id" type="keyword">Definition</span></span>, just as we have
given names to expressions of other sorts. This isn't a fundamentally
new kind of proposition; it is still just an equality.
<div class="paragraph"> </div>
Another alternative is to define the concept of evenness
directly. Instead of going via the <span class="inlinecode"><span class="id" type="var">evenb</span></span> function ("a number is
even if a certain computation yields <span class="inlinecode"><span class="id" type="var">true</span></span>"), we can say what the
concept of evenness means by giving two different ways of
presenting <i>evidence</i> that a number is even.
<div class="paragraph"> </div>
<a name="lab233"></a><h2 class="section">Inductively Defined Propositions</h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ev</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ev_0</span> : <span class="id" type="var">ev</span> <span class="id" type="var">O</span><br/>
| <span class="id" type="var">ev_SS</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, <span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)).<br/>
<br/>
</div>
<div class="doc">
This definition says that there are two ways to give
evidence that a number <span class="inlinecode"><span class="id" type="var">m</span></span> is even. First, <span class="inlinecode">0</span> is even, and
<span class="inlinecode"><span class="id" type="var">ev_0</span></span> is evidence for this. Second, if <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> for some
<span class="inlinecode"><span class="id" type="var">n</span></span> and we can give evidence <span class="inlinecode"><span class="id" type="var">e</span></span> that <span class="inlinecode"><span class="id" type="var">n</span></span> is even, then <span class="inlinecode"><span class="id" type="var">m</span></span> is
also even, and <span class="inlinecode"><span class="id" type="var">ev_SS</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">e</span></span> is the evidence.
<div class="paragraph"> </div>
<a name="lab234"></a><h4 class="section">Exercise: 1 star (double_even)</h4>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">double_even</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> (<span class="id" type="var">double</span> <span class="id" type="var">n</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="lab235"></a><h3 class="section">Discussion: Computational vs. Inductive Definitions</h3>
<div class="paragraph"> </div>
We have seen that the proposition "<span class="inlinecode"><span class="id" type="var">n</span></span> is even" can be
phrased in two different ways — indirectly, via a boolean testing
function <span class="inlinecode"><span class="id" type="var">evenb</span></span>, or directly, by inductively describing what
constitutes evidence for evenness. These two ways of defining
evenness are about equally easy to state and work with. Which we
choose is basically a question of taste.
<div class="paragraph"> </div>
However, for many other properties of interest, the direct
inductive definition is preferable, since writing a testing
function may be awkward or even impossible.
<div class="paragraph"> </div>
One such property is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>. This is a perfectly sensible
definition of a set of numbers, but we cannot translate its
definition directly into a Coq Fixpoint (or into a recursive
function in any other common programming language). We might be
able to find a clever way of testing this property using a
<span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> (indeed, it is not too hard to find one in this case),
but in general this could require arbitrarily deep thinking. In
fact, if the property we are interested in is uncomputable, then
we cannot define it as a <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span> no matter how hard we try,
because Coq requires that all <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>s correspond to
terminating computations.
<div class="paragraph"> </div>
On the other hand, writing an inductive definition of what it
means to give evidence for the property <span class="inlinecode"><span class="id" type="var">beautiful</span></span> is
straightforward.
<div class="paragraph"> </div>
<a name="lab236"></a><h4 class="section">Exercise: 1 star (ev__even)</h4>
Here is a proof that the inductive definition of evenness implies
the computational one.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev__even</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">even</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">E</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span> <span class="id" type="var">E'</span>].<br/>
<span class="id" type="var">Case</span> "E = ev_0".<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">even</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "E = ev_SS n' E'".<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">even</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHE'</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Could this proof also be carried out by induction on <span class="inlinecode"><span class="id" type="var">n</span></span> instead
of <span class="inlinecode"><span class="id" type="var">E</span></span>? If not, why not?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
The induction principle for inductively defined propositions does
not follow quite the same form as that of inductively defined
sets. For now, you can take the intuitive view that induction on
evidence <span class="inlinecode"><span class="id" type="var">ev</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is similar to induction on <span class="inlinecode"><span class="id" type="var">n</span></span>, but restricts our
attention to only those numbers for which evidence <span class="inlinecode"><span class="id" type="var">ev</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> could be
generated. We'll look at the induction principle of <span class="inlinecode"><span class="id" type="var">ev</span></span> in more
depth below, to explain what's really going on.
<div class="paragraph"> </div>
<a name="lab237"></a><h4 class="section">Exercise: 1 star (l_fails)</h4>
The following proof attempt will not succeed.
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">l</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="var">Case</span> "O". <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">ev_0</span>.<br/>
<span class="id" type="var">Case</span> "S".<br/>
...
<div class="paragraph"> </div>
</div>
Intuitively, we expect the proof to fail because not every
number is even. However, what exactly causes the proof to fail?
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab238"></a><h4 class="section">Exercise: 2 stars (ev_sum)</h4>
Here's another exercise requiring induction.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev_sum</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</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="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="lab239"></a><h1 class="section">Inductively Defined Propositions</h1>
<div class="paragraph"> </div>
As a running example, let's
define a simple property of natural numbers — we'll call it
"<span class="inlinecode"><span class="id" type="var">beautiful</span></span>."
<div class="paragraph"> </div>
Informally, a number is <span class="inlinecode"><span class="id" type="var">beautiful</span></span> if it is <span class="inlinecode">0</span>, <span class="inlinecode">3</span>, <span class="inlinecode">5</span>, or the
sum of two <span class="inlinecode"><span class="id" type="var">beautiful</span></span> numbers.
<div class="paragraph"> </div>
More pedantically, we can define <span class="inlinecode"><span class="id" type="var">beautiful</span></span> numbers by giving four
rules:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Rule <span class="inlinecode"><span class="id" type="var">b_0</span></span>: The number <span class="inlinecode">0</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</li>
<li> Rule <span class="inlinecode"><span class="id" type="var">b_3</span></span>: The number <span class="inlinecode">3</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</li>
<li> Rule <span class="inlinecode"><span class="id" type="var">b_5</span></span>: The number <span class="inlinecode">5</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</li>
<li> Rule <span class="inlinecode"><span class="id" type="var">b_sum</span></span>: If <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span> are both <span class="inlinecode"><span class="id" type="var">beautiful</span></span>, then so is
their sum.
</li>
</ul>
<a name="lab240"></a><h2 class="section">Inference Rules</h2>
We will see many definitions like this one during the rest
of the course, and for purposes of informal discussions, it is
helpful to have a lightweight notation that makes them easy to
read and write. <i>Inference rules</i> are one such notation:
<div class="paragraph"> </div>
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(b_0)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">beautiful 0</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(b_3)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">beautiful 3</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(b_5)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">beautiful 5</td>
<td></td>
</td>
</table></center><center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule">beautiful n beautiful m</td>
<td class="infrulenamecol" rowspan="3">
(b_sum)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule">beautiful (n+m)</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
<a name="lab241"></a><h3 class="section"> </h3>
Each of the textual rules above is reformatted here as an
inference rule; the intended reading is that, if the <i>premises</i>
above the line all hold, then the <i>conclusion</i> below the line
follows. For example, the rule <span class="inlinecode"><span class="id" type="var">b_sum</span></span> says that, if <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span>
are both <span class="inlinecode"><span class="id" type="var">beautiful</span></span> numbers, then it follows that <span class="inlinecode"><span class="id" type="var">n</span>+<span class="id" type="var">m</span></span> is
<span class="inlinecode"><span class="id" type="var">beautiful</span></span> too. If a rule has no premises above the line, then
its conclusion holds unconditionally.
<div class="paragraph"> </div>
These rules <i>define</i> the property <span class="inlinecode"><span class="id" type="var">beautiful</span></span>. That is, if we
want to convince someone that some particular number is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>,
our argument must be based on these rules. For a simple example,
suppose we claim that the number <span class="inlinecode">5</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>. To support
this claim, we just need to point out that rule <span class="inlinecode"><span class="id" type="var">b_5</span></span> says so.
Or, if we want to claim that <span class="inlinecode">8</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>, we can support our
claim by first observing that <span class="inlinecode">3</span> and <span class="inlinecode">5</span> are both <span class="inlinecode"><span class="id" type="var">beautiful</span></span> (by
rules <span class="inlinecode"><span class="id" type="var">b_3</span></span> and <span class="inlinecode"><span class="id" type="var">b_5</span></span>) and then pointing out that their sum, <span class="inlinecode">8</span>,
is therefore <span class="inlinecode"><span class="id" type="var">beautiful</span></span> by rule <span class="inlinecode"><span class="id" type="var">b_sum</span></span>. This argument can be
expressed graphically with the following <i>proof tree</i>:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="code code-tight">
----------- (<span class="id" type="var">b_3</span>) ----------- (<span class="id" type="var">b_5</span>)<br/>
<span class="id" type="var">beautiful</span> 3 <span class="id" type="var">beautiful</span> 5<br/>
------------------------------- (<span class="id" type="var">b_sum</span>)<br/>
<span class="id" type="var">beautiful</span> 8
<div class="paragraph"> </div>
</div>
<a name="lab242"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
Of course, there are other ways of using these rules to argue that
<span class="inlinecode">8</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>, for instance:
<div class="paragraph"> </div>
<div class="code code-tight">
----------- (<span class="id" type="var">b_5</span>) ----------- (<span class="id" type="var">b_3</span>)<br/>
<span class="id" type="var">beautiful</span> 5 <span class="id" type="var">beautiful</span> 3<br/>
------------------------------- (<span class="id" type="var">b_sum</span>)<br/>
<span class="id" type="var">beautiful</span> 8
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<a name="lab243"></a><h4 class="section">Exercise: 1 star (varieties_of_beauty)</h4>
How many different ways are there to show that <span class="inlinecode">8</span> is <span class="inlinecode"><span class="id" type="var">beautiful</span></span>?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab244"></a><h3 class="section"> </h3>
In Coq, we can express the definition of <span class="inlinecode"><span class="id" type="var">beautiful</span></span> as
follows:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">beautiful</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">b_0</span> : <span class="id" type="var">beautiful</span> 0<br/>
| <span class="id" type="var">b_3</span> : <span class="id" type="var">beautiful</span> 3<br/>
| <span class="id" type="var">b_5</span> : <span class="id" type="var">beautiful</span> 5<br/>
| <span class="id" type="var">b_sum</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">beautiful</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> (<span class="id" type="var">n</span>+<span class="id" type="var">m</span>).<br/>
<br/>
</div>
<div class="doc">
The first line declares that <span class="inlinecode"><span class="id" type="var">beautiful</span></span> is a proposition — or,
more formally, a family of propositions "indexed by" natural
numbers. (That is, for each number <span class="inlinecode"><span class="id" type="var">n</span></span>, the claim that "<span class="inlinecode"><span class="id" type="var">n</span></span> is
<span class="inlinecode"><span class="id" type="var">beautiful</span></span>" is a proposition.) Such a family of propositions is
often called a <i>property</i> of numbers. Each of the remaining lines
embodies one of the rules for <span class="inlinecode"><span class="id" type="var">beautiful</span></span> numbers.
<a name="lab245"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
The rules introduced this way have the same status as proven
theorems; that is, they are true axiomatically.
So we can use Coq's <span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic with the rule names to prove
that particular numbers are <span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">three_is_beautiful</span>: <span class="id" type="var">beautiful</span> 3.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* This simply follows from the rule <span class="inlinecode"><span class="id" type="var">b_3</span></span>. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_3</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">eight_is_beautiful</span>: <span class="id" type="var">beautiful</span> 8.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* First we use the rule <span class="inlinecode"><span class="id" type="var">b_sum</span></span>, telling Coq how to<br/>
instantiate <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">m</span></span>. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_sum</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">n</span>:=3) (<span class="id" type="var">m</span>:=5).<br/>
<span class="comment">(* To solve the subgoals generated by <span class="inlinecode"><span class="id" type="var">b_sum</span></span>, we must provide<br/>
evidence of <span class="inlinecode"><span class="id" type="var">beautiful</span></span> <span class="inlinecode">3</span> and <span class="inlinecode"><span class="id" type="var">beautiful</span></span> <span class="inlinecode">5</span>. Fortunately we<br/>
have rules for both. *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_3</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_5</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab246"></a><h3 class="section"> </h3>
As you would expect, we can also prove theorems that have
hypotheses about <span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beautiful_plus_eight</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> (8+<span class="id" type="var">n</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">B</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_sum</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">n</span>:=8) (<span class="id" type="var">m</span>:=<span class="id" type="var">n</span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">eight_is_beautiful</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">B</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab247"></a><h4 class="section">Exercise: 2 stars (b_times2)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">b_times2</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> (2×<span class="id" type="var">n</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="lab248"></a><h4 class="section">Exercise: 3 stars (b_timesm)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">b_timesm</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">beautiful</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> (<span class="id" type="var">m</span>×<span class="id" type="var">n</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="lab249"></a><h2 class="section">Induction Over Evidence</h2>
<div class="paragraph"> </div>
Besides <i>constructing</i> evidence that numbers are beautiful, we can
also <i>reason about</i> such evidence.
<div class="paragraph"> </div>
The fact that we introduced <span class="inlinecode"><span class="id" type="var">beautiful</span></span> with an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>
declaration tells Coq not only that the constructors <span class="inlinecode"><span class="id" type="var">b_0</span></span>, <span class="inlinecode"><span class="id" type="var">b_3</span></span>,
<span class="inlinecode"><span class="id" type="var">b_5</span></span> and <span class="inlinecode"><span class="id" type="var">b_sum</span></span> are ways to build evidence, but also that these
four constructors are the <i>only</i> ways to build evidence that
numbers are beautiful.
<div class="paragraph"> </div>
In other words, if someone gives us evidence <span class="inlinecode"><span class="id" type="var">E</span></span> for the assertion
<span class="inlinecode"><span class="id" type="var">beautiful</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>, then we know that <span class="inlinecode"><span class="id" type="var">E</span></span> must have one of four shapes:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">E</span></span> is <span class="inlinecode"><span class="id" type="var">b_0</span></span> (and <span class="inlinecode"><span class="id" type="var">n</span></span> is <span class="inlinecode"><span class="id" type="var">O</span></span>),
</li>
<li> <span class="inlinecode"><span class="id" type="var">E</span></span> is <span class="inlinecode"><span class="id" type="var">b_3</span></span> (and <span class="inlinecode"><span class="id" type="var">n</span></span> is <span class="inlinecode">3</span>),
</li>
<li> <span class="inlinecode"><span class="id" type="var">E</span></span> is <span class="inlinecode"><span class="id" type="var">b_5</span></span> (and <span class="inlinecode"><span class="id" type="var">n</span></span> is <span class="inlinecode">5</span>), or
</li>
<li> <span class="inlinecode"><span class="id" type="var">E</span></span> is <span class="inlinecode"><span class="id" type="var">b_sum</span></span> <span class="inlinecode"><span class="id" type="var">n1</span></span> <span class="inlinecode"><span class="id" type="var">n2</span></span> <span class="inlinecode"><span class="id" type="var">E1</span></span> <span class="inlinecode"><span class="id" type="var">E2</span></span> (and <span class="inlinecode"><span class="id" type="var">n</span></span> is <span class="inlinecode"><span class="id" type="var">n1</span>+<span class="id" type="var">n2</span></span>, where <span class="inlinecode"><span class="id" type="var">E1</span></span> is
evidence that <span class="inlinecode"><span class="id" type="var">n1</span></span> is beautiful and <span class="inlinecode"><span class="id" type="var">E2</span></span> is evidence that <span class="inlinecode"><span class="id" type="var">n2</span></span>
is beautiful).
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab250"></a><h3 class="section"> </h3>
This permits us to <i>analyze</i> any hypothesis of the form <span class="inlinecode"><span class="id" type="var">beautiful</span></span>
<span class="inlinecode"><span class="id" type="var">n</span></span> to see how it was constructed, using the tactics we already
know. In particular, we can use the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic that we
have already seen for reasoning about inductively defined <i>data</i>
to reason about inductively defined <i>evidence</i>.
<div class="paragraph"> </div>
To illustrate this, let's define another property of numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">gorgeous</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">g_0</span> : <span class="id" type="var">gorgeous</span> 0<br/>
| <span class="id" type="var">g_plus3</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> (3+<span class="id" type="var">n</span>)<br/>
| <span class="id" type="var">g_plus5</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> (5+<span class="id" type="var">n</span>).<br/>
<br/>
</div>
<div class="doc">
<a name="lab251"></a><h4 class="section">Exercise: 1 star (gorgeous_tree)</h4>
Write out the definition of <span class="inlinecode"><span class="id" type="var">gorgeous</span></span> numbers using inference rule
notation.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab252"></a><h4 class="section">Exercise: 1 star (gorgeous_plus13)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">gorgeous_plus13</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <br/>
<span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> (13+<span class="id" type="var">n</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="lab253"></a><h3 class="section"> </h3>
It seems intuitively obvious that, although <span class="inlinecode"><span class="id" type="var">gorgeous</span></span> and
<span class="inlinecode"><span class="id" type="var">beautiful</span></span> are presented using slightly different rules, they are
actually the same property in the sense that they are true of the
same numbers. Indeed, we can prove this.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">gorgeous__beautiful</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <br/>
<span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">n'</span>|<span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "g_0".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_0</span>.<br/>
<span class="id" type="var">Case</span> "g_plus3".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_sum</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">b_3</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHgorgeous</span>.<br/>
<span class="id" type="var">Case</span> "g_plus5".<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">b_sum</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">b_5</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHgorgeous</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Notice that the argument proceeds by induction on the <i>evidence</i> <span class="inlinecode"><span class="id" type="var">H</span></span>!
<div class="paragraph"> </div>
Let's see what happens if we try to prove this by induction on <span class="inlinecode"><span class="id" type="var">n</span></span>
instead of induction on the evidence <span class="inlinecode"><span class="id" type="var">H</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">gorgeous__beautiful_FAILED</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <br/>
<span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "n = 0". <span class="id" type="tactic">apply</span> <span class="id" type="var">b_0</span>.<br/>
<span class="id" type="var">Case</span> "n = S n'". <span class="comment">(* We are stuck! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
The problem here is that doing induction on <span class="inlinecode"><span class="id" type="var">n</span></span> doesn't yield a
useful induction hypothesis. Knowing how the property we are
interested in behaves on the predecessor of <span class="inlinecode"><span class="id" type="var">n</span></span> doesn't help us
prove that it holds for <span class="inlinecode"><span class="id" type="var">n</span></span>. Instead, we would like to be able to
have induction hypotheses that mention other numbers, such as <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">-</span>
<span class="inlinecode">3</span> and <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">-</span> <span class="inlinecode">5</span>. This is given precisely by the shape of the
constructors for <span class="inlinecode"><span class="id" type="var">gorgeous</span></span>.
<div class="paragraph"> </div>
<a name="lab254"></a><h4 class="section">Exercise: 2 stars (gorgeous_sum)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">gorgeous_sum</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</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="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="lab255"></a><h4 class="section">Exercise: 3 stars, advanced (beautiful__gorgeous)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beautiful__gorgeous</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> <span class="id" type="var">n</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="lab256"></a><h4 class="section">Exercise: 3 stars, optional (g_times2)</h4>
Prove the <span class="inlinecode"><span class="id" type="var">g_times2</span></span> theorem below without using <span class="inlinecode"><span class="id" type="var">gorgeous__beautiful</span></span>.
You might find the following helper lemma useful.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">helper_g_times2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>, <span class="id" type="var">x</span> + (<span class="id" type="var">z</span> + <span class="id" type="var">y</span>)= <span class="id" type="var">z</span> + <span class="id" type="var">x</span> + <span class="id" type="var">y</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">g_times2</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">gorgeous</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">gorgeous</span> (2×<span class="id" type="var">n</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</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="lab257"></a><h2 class="section"><span class="inlinecode"><span class="id" type="var">Inversion</span></span> on Evidence</h2>
<div class="paragraph"> </div>
Another situation where we want to analyze evidence for evenness
is when proving that, if <span class="inlinecode"><span class="id" type="var">n</span></span> is even, then <span class="inlinecode"><span class="id" type="var">pred</span></span> <span class="inlinecode">(<span class="id" type="var">pred</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> is
too. In this case, we don't need to do an inductive proof. The
right tactic turns out to be <span class="inlinecode"><span class="id" type="tactic">inversion</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev_minus2</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> (<span class="id" type="var">pred</span> (<span class="id" type="var">pred</span> <span class="id" type="var">n</span>)).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span> <span class="id" type="var">E'</span>].<br/>
<span class="id" type="var">Case</span> "E = ev_0". <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">ev_0</span>.<br/>
<span class="id" type="var">Case</span> "E = ev_SS n' E'". <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E'</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab258"></a><h4 class="section">Exercise: 1 star, optional (ev_minus2_n)</h4>
What happens if we try to use <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> on <span class="inlinecode"><span class="id" type="var">n</span></span> instead of <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> on <span class="inlinecode"><span class="id" type="var">E</span></span>?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab259"></a><h3 class="section"> </h3>
Another example, in which <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> helps narrow down to
the relevant cases.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">SSev__even</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span> <span class="id" type="var">E'</span>].<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">E'</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab260"></a><h2 class="section"><span class="inlinecode"><span class="id" type="tactic">inversion</span></span> revisited</h2>
<div class="paragraph"> </div>
These uses of <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> may seem a bit mysterious at first.
Until now, we've only used <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> on equality
propositions, to utilize injectivity of constructors or to
discriminate between different constructors. But we see here
that <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> can also be applied to analyzing evidence
for inductively defined propositions.
<div class="paragraph"> </div>
(You might also expect that <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> would be a more suitable
tactic to use here. Indeed, it is possible to use <span class="inlinecode"><span class="id" type="tactic">destruct</span></span>, but
it often throws away useful information, and the <span class="inlinecode"><span class="id" type="var">eqn</span>:</span> qualifier
doesn't help much in this case.)
<div class="paragraph"> </div>
Here's how <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> works in general. Suppose the name
<span class="inlinecode"><span class="id" type="var">I</span></span> refers to an assumption <span class="inlinecode"><span class="id" type="var">P</span></span> in the current context, where
<span class="inlinecode"><span class="id" type="var">P</span></span> has been defined by an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> declaration. Then,
for each of the constructors of <span class="inlinecode"><span class="id" type="var">P</span></span>, <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">I</span></span> generates
a subgoal in which <span class="inlinecode"><span class="id" type="var">I</span></span> has been replaced by the exact,
specific conditions under which this constructor could have
been used to prove <span class="inlinecode"><span class="id" type="var">P</span></span>. Some of these subgoals will be
self-contradictory; <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> throws these away. The ones
that are left represent the cases that must be proved to
establish the original goal.
<div class="paragraph"> </div>
In this particular case, the <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> analyzed the construction
<span class="inlinecode"><span class="id" type="var">ev</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>))</span>, determined that this could only have been
constructed using <span class="inlinecode"><span class="id" type="var">ev_SS</span></span>, and generated a new subgoal with the
arguments of that constructor as new hypotheses. (It also
produced an auxiliary equality, which happens to be useless here.)
We'll begin exploring this more general behavior of inversion in
what follows.
<div class="paragraph"> </div>
<a name="lab261"></a><h4 class="section">Exercise: 1 star (inversion_practice)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">SSSSev__even</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">ev</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)))) <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic can also be used to derive goals by showing
the absurdity of a hypothesis.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">even5_nonsense</span> : <br/>
<span class="id" type="var">ev</span> 5 <span style="font-family: arial;">→</span> 2 + 2 = 9.<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="lab262"></a><h4 class="section">Exercise: 3 stars, advanced (ev_ev__ev)</h4>
Finding the appropriate thing to do induction on is a
bit tricky here:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev_ev__ev</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">ev</span> (<span class="id" type="var">n</span>+<span class="id" type="var">m</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> <span class="id" type="var">m</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="lab263"></a><h4 class="section">Exercise: 3 stars, optional (ev_plus_plus)</h4>
Here's an exercise that just requires applying existing lemmas. No
induction or even case analysis is needed, but some of the rewriting
may be tedious.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev_plus_plus</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">p</span>,<br/>
<span class="id" type="var">ev</span> (<span class="id" type="var">n</span>+<span class="id" type="var">m</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> (<span class="id" type="var">n</span>+<span class="id" type="var">p</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> (<span class="id" type="var">m</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="lab264"></a><h1 class="section">Additional Exercises</h1>
<div class="paragraph"> </div>
<a name="lab265"></a><h4 class="section">Exercise: 4 stars (palindromes)</h4>
A palindrome is a sequence that reads the same backwards as
forwards.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Define an inductive proposition <span class="inlinecode"><span class="id" type="var">pal</span></span> on <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> that
captures what it means to be a palindrome. (Hint: You'll need
three cases. Your definition should be based on the structure
of the list; just having a single constructor
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">c</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">l</span>, <span class="id" type="var">l</span> = <span class="id" type="var">rev</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">pal</span> <span class="id" type="var">l</span>
<div class="paragraph"> </div>
</div>
may seem obvious, but will not work very well.)
<div class="paragraph"> </div>
</li>
<li> Prove that
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="font-family: arial;">∀</span><span class="id" type="var">l</span>, <span class="id" type="var">pal</span> (<span class="id" type="var">l</span> ++ <span class="id" type="var">rev</span> <span class="id" type="var">l</span>).
<div class="paragraph"> </div>
</div>
</li>
<li> Prove that
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="font-family: arial;">∀</span><span class="id" type="var">l</span>, <span class="id" type="var">pal</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">l</span> = <span class="id" type="var">rev</span> <span class="id" type="var">l</span>.
<div class="paragraph"> </div>
</div>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab266"></a><h4 class="section">Exercise: 5 stars, optional (palindrome_converse)</h4>
Using your definition of <span class="inlinecode"><span class="id" type="var">pal</span></span> from the previous exercise, prove
that
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="font-family: arial;">∀</span><span class="id" type="var">l</span>, <span class="id" type="var">l</span> = <span class="id" type="var">rev</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">pal</span> <span class="id" type="var">l</span>.
<div class="paragraph"> </div>
</div>
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab267"></a><h4 class="section">Exercise: 4 stars, advanced (subsequence)</h4>
A list is a <i>subsequence</i> of another list if all of the elements
in the first list occur in the same order in the second list,
possibly with some extra elements in between. For example,
<div class="paragraph"> </div>
<div class="code code-tight">
[1,2,3]
<div class="paragraph"> </div>
</div>
is a subsequence of each of the lists
<div class="paragraph"> </div>
<div class="code code-tight">
[1,2,3]<br/>
[1,1,1,2,2,3]<br/>
[1,2,7,3]<br/>
[5,6,1,9,9,2,7,3,8]
<div class="paragraph"> </div>
</div>
but it is <i>not</i> a subsequence of any of the lists
<div class="paragraph"> </div>
<div class="code code-tight">
[1,2]<br/>
[1,3]<br/>
[5,6,2,1,7,3,8]
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<ul class="doclist">
<li> Define an inductive proposition <span class="inlinecode"><span class="id" type="var">subseq</span></span> on <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">nat</span></span> that
captures what it means to be a subsequence. (Hint: You'll need