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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=iso-8859-1"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>MoreProp: More about 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">MoreProp<span class="subtitle">More about 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> "Prop".<br/>
<br/>
</div>
<div class="doc">
<a name="lab181"></a><h1 class="section">Relations</h1>
<div class="paragraph"> </div>
A proposition parameterized by a number (such as <span class="inlinecode"><span class="id" type="var">ev</span></span> or
<span class="inlinecode"><span class="id" type="var">beautiful</span></span>) can be thought of as a <i>property</i> — i.e., it defines
a subset of <span class="inlinecode"><span class="id" type="var">nat</span></span>, namely those numbers for which the proposition
is provable. In the same way, a two-argument proposition can be
thought of as a <i>relation</i> — i.e., it defines a set of pairs for
which the proposition is provable.
</div>
<div class="code code-tight">
<br/>
<br/>
</div>
<div class="doc">
One useful example is the "less than or equal to"
relation on numbers.
<div class="paragraph"> </div>
The following definition should be fairly intuitive. It
says that there are two ways to give evidence that one number is
less than or equal to another: either observe that they are the
same number, or give evidence that the first is less than or equal
to the predecessor of the second.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">le</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">le_n</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">le</span> <span class="id" type="var">n</span> <span class="id" type="var">n</span><br/>
| <span class="id" type="var">le_S</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">le</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">le</span> <span class="id" type="var">n</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>)).<br/>
<br/>
<span class="id" type="keyword">Notation</span> "m <= n" := (<span class="id" type="var">le</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span>).<br/>
<br/>
</div>
<div class="doc">
Proofs of facts about <span class="inlinecode"><=</span> using the constructors <span class="inlinecode"><span class="id" type="var">le_n</span></span> and
<span class="inlinecode"><span class="id" type="var">le_S</span></span> follow the same patterns as proofs about properties, like
<span class="inlinecode"><span class="id" type="var">ev</span></span> in chapter <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>. We can <span class="inlinecode"><span class="id" type="tactic">apply</span></span> the constructors to prove <span class="inlinecode"><=</span>
goals (e.g., to show that <span class="inlinecode">3<=3</span> or <span class="inlinecode">3<=6</span>), and we can use
tactics like <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> to extract information from <span class="inlinecode"><=</span>
hypotheses in the context (e.g., to prove that <span class="inlinecode">(2</span> <span class="inlinecode"><=</span> <span class="inlinecode">1)</span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode">2+2=5</span>.)
<div class="paragraph"> </div>
Here are some sanity checks on the definition. (Notice that,
although these are the same kind of simple "unit tests" as we gave
for the testing functions we wrote in the first few lectures, we
must construct their proofs explicitly — <span class="inlinecode"><span class="id" type="tactic">simpl</span></span> and
<span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> don't do the job, because the proofs aren't just a
matter of simplifying computations.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">test_le1</span> :<br/>
3 <= 3.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">test_le2</span> :<br/>
3 <= 6.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">test_le3</span> :<br/>
(2 <= 1) <span style="font-family: arial;">→</span> 2 + 2 = 5.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* WORKED IN CLASS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The "strictly less than" relation <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><</span> <span class="inlinecode"><span class="id" type="var">m</span></span> can now be defined
in terms of <span class="inlinecode"><span class="id" type="var">le</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">lt</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span>:<span class="id" type="var">nat</span>) := <span class="id" type="var">le</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) <span class="id" type="var">m</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "m < n" := (<span class="id" type="var">lt</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span>).<br/>
<br/>
</div>
<div class="doc">
Here are a few more simple relations on numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">square_of</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">sq</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">square_of</span> <span class="id" type="var">n</span> (<span class="id" type="var">n</span> * <span class="id" type="var">n</span>).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">next_nat</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</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">nn</span> : <span class="id" type="var">next_nat</span> <span class="id" type="var">n</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>).<br/>
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">next_even</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</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">ne_1</span> : <span class="id" type="var">ev</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">next_even</span> <span class="id" type="var">n</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)<br/>
| <span class="id" type="var">ne_2</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>)) <span style="font-family: arial;">→</span> <span class="id" type="var">next_even</span> <span class="id" type="var">n</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">
<a name="lab182"></a><h4 class="section">Exercise: 2 stars (total_relation)</h4>
Define an inductive binary relation <span class="inlinecode"><span class="id" type="var">total_relation</span></span> that holds
between every pair of natural numbers.
</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="lab183"></a><h4 class="section">Exercise: 2 stars (empty_relation)</h4>
Define an inductive binary relation <span class="inlinecode"><span class="id" type="var">empty_relation</span></span> (on numbers)
that never holds.
</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="lab184"></a><h4 class="section">Exercise: 2 stars, optional (le_exercises)</h4>
Here are a number of facts about the <span class="inlinecode"><=</span> and <span class="inlinecode"><</span> relations that
we are going to need later in the course. The proofs make good
practice exercises.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">le_trans</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>, <span class="id" type="var">m</span> <= <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> <= <span class="id" type="var">o</span> <span style="font-family: arial;">→</span> <span class="id" type="var">m</span> <= <span class="id" type="var">o</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">O_le_n</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
0 <= <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/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">n_le_m__Sn_le_Sm</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">n</span> <= <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <span class="id" type="var">S</span> <span class="id" type="var">n</span> <= <span class="id" type="var">S</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/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">Sn_le_Sm__n_le_m</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">S</span> <span class="id" type="var">n</span> <= <span class="id" type="var">S</span> <span class="id" type="var">m</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="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">le_plus_l</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">a</span> <span class="id" type="var">b</span>,<br/>
<span class="id" type="var">a</span> <= <span class="id" type="var">a</span> + <span class="id" type="var">b</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">plus_lt</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n1</span> <span class="id" type="var">n2</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">n1</span> + <span class="id" type="var">n2</span> < <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n1</span> < <span class="id" type="var">m</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">n2</span> < <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">lt</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">lt_S</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">n</span> < <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">n</span> < <span class="id" type="var">S</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/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ble_nat_true</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">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">true</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="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">le_ble_nat</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">n</span> <= <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* Hint: This may be easiest to prove by induction on <span class="inlinecode"><span class="id" type="var">m</span></span>. *)</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">ble_nat_true_trans</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">o</span>,<br/>
<span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ble_nat</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span> = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* Hint: This theorem can be easily proved without using <span class="inlinecode"><span class="id" type="tactic">induction</span></span>. *)</span><br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab185"></a><h4 class="section">Exercise: 3 stars (R_provability)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Module</span> <span class="id" type="var">R</span>.<br/>
</div>
<div class="doc">
We can define three-place relations, four-place relations,
etc., in just the same way as binary relations. For example,
consider the following three-place relation on numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">R</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">c1</span> : <span class="id" type="var">R</span> 0 0 0 <br/>
| <span class="id" type="var">c2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>, <span class="id" type="var">R</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>) <span class="id" type="var">n</span> (<span class="id" type="var">S</span> <span class="id" type="var">o</span>)<br/>
| <span class="id" type="var">c3</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>, <span class="id" type="var">R</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span> <span class="id" type="var">m</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) (<span class="id" type="var">S</span> <span class="id" type="var">o</span>)<br/>
| <span class="id" type="var">c4</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>, <span class="id" type="var">R</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>) (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">o</span>)) <span style="font-family: arial;">→</span> <span class="id" type="var">R</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span><br/>
| <span class="id" type="var">c5</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>, <span class="id" type="var">R</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span> <span style="font-family: arial;">→</span> <span class="id" type="var">R</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span>.<br/>
<br/>
</div>
<div class="doc">
<div class="paragraph"> </div>
<ul class="doclist">
<li> Which of the following propositions are provable?
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode">1</span> <span class="inlinecode">1</span> <span class="inlinecode">2</span>
</li>
<li> <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode">2</span> <span class="inlinecode">2</span> <span class="inlinecode">6</span>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If we dropped constructor <span class="inlinecode"><span class="id" type="var">c5</span></span> from the definition of <span class="inlinecode"><span class="id" type="var">R</span></span>,
would the set of provable propositions change? Briefly (1
sentence) explain your answer.
<div class="paragraph"> </div>
</li>
<li> If we dropped constructor <span class="inlinecode"><span class="id" type="var">c4</span></span> from the definition of <span class="inlinecode"><span class="id" type="var">R</span></span>,
would the set of provable propositions change? Briefly (1
sentence) explain your answer.
</li>
</ul>
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab186"></a><h4 class="section">Exercise: 3 stars, optional (R_fact)</h4>
Relation <span class="inlinecode"><span class="id" type="var">R</span></span> actually encodes a familiar function. State and prove two
theorems that formally connects the relation and the function.
That is, if <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">o</span></span> is true, what can we say about <span class="inlinecode"><span class="id" type="var">m</span></span>,
<span class="inlinecode"><span class="id" type="var">n</span></span>, and <span class="inlinecode"><span class="id" type="var">o</span></span>, and vice versa?
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">R</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab187"></a><h1 class="section">Programming with Propositions</h1>
<div class="paragraph"> </div>
A <i>proposition</i> is a statement expressing a factual claim,
like "two plus two equals four." In Coq, propositions are written
as expressions of type <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>. Although we haven't discussed this
explicitly, we have already seen numerous examples.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (2 + 2 = 4).<br/>
<span class="comment">(* ===> 2 + 2 = 4 : Prop *)</span><br/>
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">ble_nat</span> 3 2 = <span class="id" type="var">false</span>).<br/>
<span class="comment">(* ===> ble_nat 3 2 = false : Prop *)</span><br/>
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">beautiful</span> 8).<br/>
<span class="comment">(* ===> beautiful 8 : Prop *)</span><br/>
<br/>
</div>
<div class="doc">
Both provable and unprovable claims are perfectly good
propositions. Simply <i>being</i> a proposition is one thing; being
<i>provable</i> is something else!
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (2 + 2 = 5).<br/>
<span class="comment">(* ===> 2 + 2 = 5 : Prop *)</span><br/>
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">beautiful</span> 4).<br/>
<span class="comment">(* ===> beautiful 4 : Prop *)</span><br/>
<br/>
</div>
<div class="doc">
Both <span class="inlinecode">2</span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode">=</span> <span class="inlinecode">4</span> and <span class="inlinecode">2</span> <span class="inlinecode">+</span> <span class="inlinecode">2</span> <span class="inlinecode">=</span> <span class="inlinecode">5</span> are legal expressions
of type <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>.
<div class="paragraph"> </div>
We've mainly seen one place that propositions can appear in Coq: in
<span class="inlinecode"><span class="id" type="keyword">Theorem</span></span> (and <span class="inlinecode"><span class="id" type="keyword">Lemma</span></span> and <span class="inlinecode"><span class="id" type="keyword">Example</span></span>) declarations.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_2_2_is_4</span> : <br/>
2 + 2 = 4.<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">
But they can be used in many other ways. For example, we have also seen that
we can give 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.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">plus_fact</span> : <span class="id" type="keyword">Prop</span> := 2 + 2 = 4.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">plus_fact</span>.<br/>
<span class="comment">(* ===> plus_fact : Prop *)</span><br/>
<br/>
</div>
<div class="doc">
We can later use this name in any situation where a proposition is
expected — for example, as the claim in a <span class="inlinecode"><span class="id" type="keyword">Theorem</span></span> declaration.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_fact_is_true</span> : <br/>
<span class="id" type="var">plus_fact</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
We've seen several ways of constructing propositions.
<div class="paragraph"> </div>
<ul class="doclist">
<li> We can define a new proposition primitively using <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>.
<div class="paragraph"> </div>
</li>
<li> Given two expressions <span class="inlinecode"><span class="id" type="var">e1</span></span> and <span class="inlinecode"><span class="id" type="var">e2</span></span> of the same type, we can
form the proposition <span class="inlinecode"><span class="id" type="var">e1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">e2</span></span>, which states that their
values are equal.
<div class="paragraph"> </div>
</li>
<li> We can combine propositions using implication and
quantification.
</li>
</ul>
<div class="paragraph"> </div>
We have also seen <i>parameterized propositions</i>, such as <span class="inlinecode"><span class="id" type="var">even</span></span> and
<span class="inlinecode"><span class="id" type="var">beautiful</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">even</span> 4).<br/>
<span class="comment">(* ===> even 4 : Prop *)</span><br/>
<span class="id" type="keyword">Check</span> (<span class="id" type="var">even</span> 3).<br/>
<span class="comment">(* ===> even 3 : Prop *)</span><br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">even</span>.<br/>
<span class="comment">(* ===> even : nat -> Prop *)</span><br/>
<br/>
</div>
<div class="doc">
The type of <span class="inlinecode"><span class="id" type="var">even</span></span>, i.e., <span class="inlinecode"><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span></span>, can be pronounced in
three equivalent ways: (1) "<span class="inlinecode"><span class="id" type="var">even</span></span> is a <i>function</i> from numbers to
propositions," (2) "<span class="inlinecode"><span class="id" type="var">even</span></span> is a <i>family</i> of propositions, indexed
by a number <span class="inlinecode"><span class="id" type="var">n</span></span>," or (3) "<span class="inlinecode"><span class="id" type="var">even</span></span> is a <i>property</i> of numbers."
<div class="paragraph"> </div>
Propositions — including parameterized propositions — are
first-class citizens in Coq. For example, we can define functions
from numbers to propositions...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">between</span> (<span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span>: <span class="id" type="var">nat</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">andb</span> (<span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">o</span>) (<span class="id" type="var">ble_nat</span> <span class="id" type="var">o</span> <span class="id" type="var">m</span>) = <span class="id" type="var">true</span>.<br/>
<br/>
</div>
<div class="doc">
... and then partially apply them:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">teen</span> : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span> := <span class="id" type="var">between</span> 13 19.<br/>
<br/>
</div>
<div class="doc">
We can even pass propositions — including parameterized
propositions — as arguments to functions:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">true_for_zero</span> (<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="var">P</span> 0.<br/>
<br/>
</div>
<div class="doc">
Here are two more examples of passing parameterized propositions
as arguments to a function.
<div class="paragraph"> </div>
The first function, <span class="inlinecode"><span class="id" type="var">true_for_all_numbers</span></span>, takes a proposition
<span class="inlinecode"><span class="id" type="var">P</span></span> as argument and builds the proposition that <span class="inlinecode"><span class="id" type="var">P</span></span> is true for
all natural numbers.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">true_for_all_numbers</span> (<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">P</span> <span class="id" type="var">n</span>.<br/>
<br/>
</div>
<div class="doc">
The second, <span class="inlinecode"><span class="id" type="var">preserved_by_S</span></span>, takes <span class="inlinecode"><span class="id" type="var">P</span></span> and builds the proposition
that, if <span class="inlinecode"><span class="id" type="var">P</span></span> is true for some natural number <span class="inlinecode"><span class="id" type="var">n'</span></span>, then it is also
true by the successor of <span class="inlinecode"><span class="id" type="var">n'</span></span> — i.e. that <span class="inlinecode"><span class="id" type="var">P</span></span> is <i>preserved by
successor</i>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">preserved_by_S</span> (<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n'</span>, <span class="id" type="var">P</span> <span class="id" type="var">n'</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">S</span> <span class="id" type="var">n'</span>).<br/>
<br/>
</div>
<div class="doc">
Finally, we can put these ingredients together to define
a proposition stating that induction is valid for natural numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">natural_number_induction_valid</span> : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">P</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">true_for_zero</span> <span class="id" type="var">P</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">preserved_by_S</span> <span class="id" type="var">P</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">true_for_all_numbers</span> <span class="id" type="var">P</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab188"></a><h4 class="section">Exercise: 3 stars (combine_odd_even)</h4>
Complete the definition of the <span class="inlinecode"><span class="id" type="var">combine_odd_even</span></span> function
below. It takes as arguments two properties of numbers <span class="inlinecode"><span class="id" type="var">Podd</span></span> and
<span class="inlinecode"><span class="id" type="var">Peven</span></span>. As its result, it should return a new property <span class="inlinecode"><span class="id" type="var">P</span></span> such
that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is equivalent to <span class="inlinecode"><span class="id" type="var">Podd</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> when <span class="inlinecode"><span class="id" type="var">n</span></span> is odd, and
equivalent to <span class="inlinecode"><span class="id" type="var">Peven</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> otherwise.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">combine_odd_even</span> (<span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">admit</span>.<br/>
<br/>
</div>
<div class="doc">
To test your definition, see whether you can prove the following
facts:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">combine_odd_even_intro</span> : <br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
(<span class="id" type="var">oddb</span> <span class="id" type="var">n</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Podd</span> <span class="id" type="var">n</span>) <span style="font-family: arial;">→</span><br/>
(<span class="id" type="var">oddb</span> <span class="id" type="var">n</span> = <span class="id" type="var">false</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Peven</span> <span class="id" type="var">n</span>) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">combine_odd_even</span> <span class="id" type="var">Podd</span> <span class="id" type="var">Peven</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/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">combine_odd_even_elim_odd</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">combine_odd_even</span> <span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">oddb</span> <span class="id" type="var">n</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Podd</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/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">combine_odd_even_elim_even</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">combine_odd_even</span> <span class="id" type="var">Podd</span> <span class="id" type="var">Peven</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">oddb</span> <span class="id" type="var">n</span> = <span class="id" type="var">false</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Peven</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/>
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<font size=-2>☐</font>
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One more quick digression, for adventurous souls: if we can define
parameterized propositions using <span class="inlinecode"><span class="id" type="keyword">Definition</span></span>, then can we also
define them using <span class="inlinecode"><span class="id" type="keyword">Fixpoint</span></span>? Of course we can! However, this
kind of "recursive parameterization" doesn't correspond to
anything very familiar from everyday mathematics. The following
exercise gives a slightly contrived example.
<div class="paragraph"> </div>
<a name="lab189"></a><h4 class="section">Exercise: 4 stars, optional (true_upto_n__true_everywhere)</h4>
Define a recursive function
<span class="inlinecode"><span class="id" type="var">true_upto_n__true_everywhere</span></span> that makes
<span class="inlinecode"><span class="id" type="var">true_upto_n_example</span></span> work.
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<br/>
<span class="comment">(* <br/>
Fixpoint true_upto_n__true_everywhere<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
<br/>
Example true_upto_n_example :<br/>
(true_upto_n__true_everywhere 3 (fun n => even n))<br/>
= (even 3 -> even 2 -> even 1 -> forall m : nat, even m).<br/>
Proof. reflexivity. Qed.<br/>
*)</span><br/>
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<font size=-2>☐</font>
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<span class="comment">(* $Date: 2013-07-17 16:19:11 -0400 (Wed, 17 Jul 2013) $ *)</span><br/>
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