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Induction.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>Induction: Proof by Induction</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">Induction<span class="subtitle">Proof by Induction</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
<div class="paragraph"> </div>
First, we import all of our definitions from the previous
chapter.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="Basics.html#"><span class="id" type="library">Basics</span></a>.<br/>
<br/>
</div>
<div class="doc">
For the <span class="inlinecode"><span class="id" type="keyword">Require</span></span> <span class="inlinecode"><span class="id" type="keyword">Export</span></span> to work, you first need to use
<span class="inlinecode"><span class="id" type="var">coqc</span></span> to compile <span class="inlinecode"><span class="id" type="var">Basics.v</span></span> into <span class="inlinecode"><span class="id" type="var">Basics.vo</span></span>. This is like
making a .class file from a .java file, or a .o file from a .c
file. There are two ways to do it:
<div class="paragraph"> </div>
<ul class="doclist">
<li> In CoqIDE:
<div class="paragraph"> </div>
Open <span class="inlinecode"><span class="id" type="var">Basics.v</span></span>. In the "Compile" menu, click on "Compile
Buffer".
<div class="paragraph"> </div>
</li>
<li> From the command line:
<div class="paragraph"> </div>
Run <span class="inlinecode"><span class="id" type="var">coqc</span></span> <span class="inlinecode"><span class="id" type="var">Basics.v</span></span>
</li>
</ul>
<div class="paragraph"> </div>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab41"></a><h1 class="section">Proof by Induction</h1>
<div class="paragraph"> </div>
We proved in the last chapter that <span class="inlinecode">0</span> is a neutral element
for <span class="inlinecode">+</span> on the left using an easy argument based on
simplification. The fact that it is also a neutral element on the
<i>right</i>...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_n_O_firsttry"><span class="id" type="lemma">plus_n_O_firsttry</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + 0.<br/>
<br/>
</div>
<div class="doc">
... cannot be proved in the same simple way. Just applying
<span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span> doesn't work, since the <span class="inlinecode"><span class="id" type="var">n</span></span> in <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> is an arbitrary
unknown number, so the <span class="inlinecode"><span class="id" type="keyword">match</span></span> in the definition of <span class="inlinecode">+</span> can't be
simplified.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* Does nothing! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
And reasoning by cases using <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> doesn't get us much
further: the branch of the case analysis where we assume <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>
goes through fine, but in the branch where <span class="inlinecode"><span class="id" type="var">n</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">n'</span></span> for some <span class="inlinecode"><span class="id" type="var">n'</span></span> we
get stuck in exactly the same way. We could use <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> to
get one step further, but, since <span class="inlinecode"><span class="id" type="var">n</span></span> can be arbitrarily large, if we
try to keep on like this we'll never be done.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_n_O_secondtry"><span class="id" type="lemma">plus_n_O_secondtry</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + 0.<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">destruct</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="comment">(* n = 0 *)</span><br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* so far so good... *)</span><br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* ...but here we are stuck again *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
To prove interesting facts about numbers, lists, and other
inductively defined sets, we usually need a more powerful
reasoning principle: <i>induction</i>.
<div class="paragraph"> </div>
Recall (from high school, a discrete math course, etc.) the
principle of induction over natural numbers: If <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> is some
proposition involving a natural number <span class="inlinecode"><span class="id" type="var">n</span></span> and we want to show
that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <i>all</i> numbers <span class="inlinecode"><span class="id" type="var">n</span></span>, we can reason like this:
<div class="paragraph"> </div>
<ul class="doclist">
<li> show that <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">O</span>)</span> holds;
</li>
<li> show that, for any <span class="inlinecode"><span class="id" type="var">n'</span></span>, if <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n'</span>)</span> holds, then so does
<span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span>;
</li>
<li> conclude that <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> holds for all <span class="inlinecode"><span class="id" type="var">n</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
In Coq, the steps are the same but the order is backwards: we
begin with the goal of proving <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> for all <span class="inlinecode"><span class="id" type="var">n</span></span> and break it
down (by applying the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic) into two separate
subgoals: first showing <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">O</span>)</span> and then showing <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n'</span>)</span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">S</span></span>
<span class="inlinecode"><span class="id" type="var">n'</span>)</span>. Here's how this works for the theorem at hand:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_n_O"><span class="id" type="lemma">plus_n_O</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>, <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + 0.<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> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span> <span class="id" type="var">IHn'</span>].<br/>
- <span class="comment">(* n = 0 *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Like <span class="inlinecode"><span class="id" type="tactic">destruct</span></span>, the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic takes an <span class="inlinecode"><span class="id" type="keyword">as</span>...</span>
clause that specifies the names of the variables to be introduced
in the subgoals. In the first branch, <span class="inlinecode"><span class="id" type="var">n</span></span> is replaced by <span class="inlinecode">0</span> and
the goal becomes <span class="inlinecode">0</span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>, which follows by simplification. In
the second, <span class="inlinecode"><span class="id" type="var">n</span></span> is replaced by <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> and the assumption <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" type="var">n'</span></span> is added to the context (with the name <span class="inlinecode"><span class="id" type="var">IHn'</span></span>, i.e., the
Induction Hypothesis for <span class="inlinecode"><span class="id" type="var">n'</span></span> — notice that this name is
explicitly chosen in the <span class="inlinecode"><span class="id" type="keyword">as</span>...</span> clause of the call to <span class="inlinecode"><span class="id" type="tactic">induction</span></span>
rather than letting Coq choose one arbitrarily). The goal in this
case becomes <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span>, which simplifies to <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">n'</span></span> <span class="inlinecode">+</span> <span class="inlinecode">0)</span>
<span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span>, which in turn follows from <span class="inlinecode"><span class="id" type="var">IHn'</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="minus_diag"><span class="id" type="lemma">minus_diag</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Peano.html#minus"><span class="id" type="abbreviation">minus</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = 0.<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">n</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> <span class="id" type="var">IHn'</span>].<br/>
- <span class="comment">(* n = 0 *)</span><br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
(The use of the <span class="inlinecode"><span class="id" type="tactic">intros</span></span> tactic in these proofs is actually
redundant. When applied to a goal that contains quantified
variables, the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic will automatically move them
into the context as needed.)
<div class="paragraph"> </div>
<a name="lab42"></a><h4 class="section">Exercise: 2 stars, recommended (basic_induction)</h4>
Prove the following using induction. You might need previously
proven results.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_0_r"><span class="id" type="lemma">mult_0_r</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * 0 = 0.<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> <a name="plus_n_Sm"><span class="id" type="lemma">plus_n_Sm</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>, <br/>
<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_comm"><span class="id" type="lemma">plus_comm</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> = <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_assoc"><span class="id" type="lemma">plus_assoc</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab43"></a><h4 class="section">Exercise: 2 stars (double_plus)</h4>
Consider the following function, which doubles its argument:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <a name="double"><span class="id" type="definition">double</span></a> (<span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) :=<br/>
<span class="id" type="keyword">match</span> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> <span class="id" type="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#O"><span class="id" type="constructor">O</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <span class="id" type="var">n'</span> ⇒ <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> (<a class="idref" href="Induction.html#double"><span class="id" type="definition">double</span></a> <span class="id" type="var">n'</span>))<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Use induction to prove this simple fact about <span class="inlinecode"><span class="id" type="var">double</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <a name="double_plus"><span class="id" type="lemma">double_plus</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <a class="idref" href="Induction.html#double"><span class="id" type="definition">double</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> .<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab44"></a><h4 class="section">Exercise: 2 stars, optional (evenb_S)</h4>
One inconveninent aspect of our definition of <span class="inlinecode"><span class="id" type="var">evenb</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is that it
may need to perform a recursive call on <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">-</span> <span class="inlinecode">2</span>. This makes proofs
about <span class="inlinecode"><span class="id" type="var">evenb</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> harder when done by induction on <span class="inlinecode"><span class="id" type="var">n</span></span>, since we may
need an induction hypothesis about <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">-</span> <span class="inlinecode">2</span>. The following lemma
gives a better characterization of <span class="inlinecode"><span class="id" type="var">evenb</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="evenb_S"><span class="id" type="lemma">evenb_S</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#evenb"><span class="id" type="definition">evenb</span></a> (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) = <a class="idref" href="Basics.html#negb"><span class="id" type="definition">negb</span></a> (<a class="idref" href="Basics.html#evenb"><span class="id" type="definition">evenb</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab45"></a><h4 class="section">Exercise: 1 star (destruct_induction)</h4>
Briefly explain the difference between the tactics <span class="inlinecode"><span class="id" type="tactic">destruct</span></span>
and <span class="inlinecode"><span class="id" type="tactic">induction</span></span>.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab46"></a><h1 class="section">Proofs Within Proofs</h1>
<div class="paragraph"> </div>
In Coq, as in informal mathematics, large proofs are often
broken into a sequence of theorems, with later proofs referring to
earlier theorems. But sometimes a proof will require some
miscellaneous fact that is too trivial and of too little general
interest to bother giving it its own top-level name. In such
cases, it is convenient to be able to simply state and prove the
needed "sub-theorem" right at the point where it is used. The
<span class="inlinecode"><span class="id" type="tactic">assert</span></span> tactic allows us to do this. For example, our earlier
proof of the <span class="inlinecode"><span class="id" type="var">mult_0_plus</span></span> theorem referred to a previous theorem
named <span class="inlinecode"><span class="id" type="var">plus_O_n</span></span>. We could instead use <span class="inlinecode"><span class="id" type="tactic">assert</span></span> to state and
prove <span class="inlinecode"><span class="id" type="var">plus_O_n</span></span> in-line:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_0_plus'"><span class="id" type="lemma">mult_0_plus'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
(0 + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) * <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>.<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">m</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H</span>: 0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>). { <span class="id" type="tactic">reflexivity</span>. }<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">assert</span></span> tactic introduces two sub-goals. The first is
the assertion itself; by prefixing it with <span class="inlinecode"><span class="id" type="var">H</span>:</span> we name the
assertion <span class="inlinecode"><span class="id" type="var">H</span></span>. (We can also name the assertion with <span class="inlinecode"><span class="id" type="keyword">as</span></span> just as
we did above with <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> and <span class="inlinecode"><span class="id" type="tactic">induction</span></span>, i.e., <span class="inlinecode"><span class="id" type="tactic">assert</span></span> <span class="inlinecode">(0</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span>
<span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode"><span class="id" type="keyword">as</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span>.) Note that we surround the proof of this assertion
with curly braces <span class="inlinecode">{</span> <span class="inlinecode">...</span> <span class="inlinecode">}</span>, both for readability and so that,
when using Coq interactively, we can see more easily when we have
finished this sub-proof. The second goal is the same as the one
at the point where we invoke <span class="inlinecode"><span class="id" type="tactic">assert</span></span> except that, in the context,
we now have the assumption <span class="inlinecode"><span class="id" type="var">H</span></span> that <span class="inlinecode">0</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span>. That is,
<span class="inlinecode"><span class="id" type="tactic">assert</span></span> generates one subgoal where we must prove the asserted
fact and a second subgoal where we can use the asserted fact to
make progress on whatever we were trying to prove in the first
place.
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">assert</span></span> tactic is handy in many sorts of situations. For
example, suppose we want to prove that <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> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">p</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">q</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">(<span class="id" type="var">m</span></span> <span class="inlinecode">+</span>
<span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">p</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">q</span>)</span>. The only difference between the two sides of the
<span class="inlinecode">=</span> is that the arguments <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">n</span></span> to the first inner <span class="inlinecode">+</span> are
swapped, so it seems we should be able to use the commutativity of
addition (<span class="inlinecode"><span class="id" type="var">plus_comm</span></span>) to rewrite one into the other. However,
the <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> tactic is a little stupid about <i>where</i> it applies
the rewrite. There are three uses of <span class="inlinecode">+</span> here, and it turns out
that doing <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">plus_comm</span></span> will affect only the <i>outer</i>
one...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_rearrange_firsttry"><span class="id" type="lemma">plus_rearrange_firsttry</span></a> : <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> <span class="id" type="var">q</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
(<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) + (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#q"><span class="id" type="variable">q</span></a>) = (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) + (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#q"><span class="id" type="variable">q</span></a>).<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">m</span> <span class="id" type="var">p</span> <span class="id" type="var">q</span>.<br/>
<span class="comment">(* We just need to swap (n + m) for (m + n)...<br/>
it seems like plus_comm should do the trick! *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <a class="idref" href="Induction.html#plus_comm"><span class="id" type="axiom">plus_comm</span></a>.<br/>
<span class="comment">(* Doesn't work...Coq rewrote the wrong plus! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
To get <span class="inlinecode"><span class="id" type="var">plus_comm</span></span> to apply at the point where we want it to, we
can introduce a local lemma stating that <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> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> (for the
particular <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">n</span></span> that we are talking about here), prove this
lemma using <span class="inlinecode"><span class="id" type="var">plus_comm</span></span>, and then use it to do the desired
rewrite.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_rearrange"><span class="id" type="lemma">plus_rearrange</span></a> : <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> <span class="id" type="var">q</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
(<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) + (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#q"><span class="id" type="variable">q</span></a>) = (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) + (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#q"><span class="id" type="variable">q</span></a>).<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">m</span> <span class="id" type="var">p</span> <span class="id" type="var">q</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H</span>: <span class="id" type="var">n</span> + <span class="id" type="var">m</span> = <span class="id" type="var">m</span> + <span class="id" type="var">n</span>).<br/>
{ <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <a class="idref" href="Induction.html#plus_comm"><span class="id" type="axiom">plus_comm</span></a>. <span class="id" type="tactic">reflexivity</span>. }<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab47"></a><h1 class="section">More Exercises</h1>
<div class="paragraph"> </div>
<a name="lab48"></a><h4 class="section">Exercise: 3 stars, recommended (mult_comm)</h4>
Use <span class="inlinecode"><span class="id" type="tactic">assert</span></span> to help prove this theorem. You shouldn't need to
use induction on <span class="inlinecode"><span class="id" type="var">plus_swap</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_swap"><span class="id" type="lemma">plus_swap</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
Now prove commutativity of multiplication. (You will probably
need to define and prove a separate subsidiary theorem to be used
in the proof of this one. You may find that <span class="inlinecode"><span class="id" type="var">plus_swap</span></span> comes in
handy.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_comm"><span class="id" type="lemma">mult_comm</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> * <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab49"></a><h4 class="section">Exercise: 3 stars, optional (more_exercises)</h4>
Take a piece of paper. For each of the following theorems, first
<i>think</i> about whether (a) it can be proved using only
simplification and rewriting, (b) it also requires case
analysis (<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>), or (c) it also requires induction. Write
down your prediction. Then fill in the proof. (There is no need
to turn in your piece of paper; this is just to encourage you to
reflect before you hack!)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="leb_refl"><span class="id" type="lemma">leb_refl</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a> = <a class="idref" href="Basics.html#leb"><span class="id" type="definition">leb</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="zero_nbeq_S"><span class="id" type="lemma">zero_nbeq_S</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#beq_nat"><span class="id" type="definition">beq_nat</span></a> 0 (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) = <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="andb_false_r"><span class="id" type="lemma">andb_false_r</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> : <a class="idref" href="Basics.html#bool"><span class="id" type="inductive">bool</span></a>,<br/>
<a class="idref" href="Basics.html#andb"><span class="id" type="definition">andb</span></a> <a class="idref" href="Induction.html#b"><span class="id" type="variable">b</span></a> <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a> = <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_ble_compat_l"><span class="id" type="lemma">plus_ble_compat_l</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#leb"><span class="id" type="definition">leb</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> = <a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="Basics.html#leb"><span class="id" type="definition">leb</span></a> (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) (<a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) = <a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="S_nbeq_0"><span class="id" type="lemma">S_nbeq_0</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#beq_nat"><span class="id" type="definition">beq_nat</span></a> (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>) 0 = <a class="idref" href="Basics.html#false"><span class="id" type="constructor">false</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_1_l"><span class="id" type="lemma">mult_1_l</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>, 1 * <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> = <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="all3_spec"><span class="id" type="lemma">all3_spec</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span> : <a class="idref" href="Basics.html#bool"><span class="id" type="inductive">bool</span></a>,<br/>
<a class="idref" href="Basics.html#orb"><span class="id" type="definition">orb</span></a><br/>
(<a class="idref" href="Basics.html#andb"><span class="id" type="definition">andb</span></a> <a class="idref" href="Induction.html#b"><span class="id" type="variable">b</span></a> <a class="idref" href="Induction.html#c"><span class="id" type="variable">c</span></a>)<br/>
(<a class="idref" href="Basics.html#orb"><span class="id" type="definition">orb</span></a> (<a class="idref" href="Basics.html#negb"><span class="id" type="definition">negb</span></a> <a class="idref" href="Induction.html#b"><span class="id" type="variable">b</span></a>)<br/>
(<a class="idref" href="Basics.html#negb"><span class="id" type="definition">negb</span></a> <a class="idref" href="Induction.html#c"><span class="id" type="variable">c</span></a>))<br/>
= <a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_plus_distr_r"><span class="id" type="lemma">mult_plus_distr_r</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
(<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) * <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a> = (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> * <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_assoc"><span class="id" type="lemma">mult_assoc</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> * <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> * <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) * <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab50"></a><h4 class="section">Exercise: 2 stars, optional (beq_nat_refl)</h4>
Prove the following theorem. (Putting the <span class="inlinecode"><span class="id" type="var">true</span></span> on the left-hand
side of the equality may look odd, but this is how the theorem is
stated in the Coq standard library, so we follow suit. Rewriting
works equally well in either direction, so we will have no problem
using the theorem no matter which way we state it.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="beq_nat_refl"><span class="id" type="lemma">beq_nat_refl</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Basics.html#true"><span class="id" type="constructor">true</span></a> = <a class="idref" href="Basics.html#beq_nat"><span class="id" type="definition">beq_nat</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> <a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab51"></a><h4 class="section">Exercise: 2 stars, optional (plus_swap')</h4>
The <span class="inlinecode"><span class="id" type="tactic">replace</span></span> tactic allows you to specify a particular subterm to
rewrite and what you want it rewritten to: <span class="inlinecode"><span class="id" type="tactic">replace</span></span> <span class="inlinecode">(<span class="id" type="var">t</span>)</span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">(<span class="id" type="var">u</span>)</span>
replaces (all copies of) expression <span class="inlinecode"><span class="id" type="var">t</span></span> in the goal by expression
<span class="inlinecode"><span class="id" type="var">u</span></span>, and generates <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">u</span></span> as an additional subgoal. This is often
useful when a plain <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> acts on the wrong part of the goal.
<div class="paragraph"> </div>
Use the <span class="inlinecode"><span class="id" type="tactic">replace</span></span> tactic to do a proof of <span class="inlinecode"><span class="id" type="var">plus_swap'</span></span>, just like
<span class="inlinecode"><span class="id" type="var">plus_swap</span></span> but without needing <span class="inlinecode"><span class="id" type="tactic">assert</span></span> <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> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_swap'"><span class="id" type="lemma">plus_swap'</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab52"></a><h4 class="section">Exercise: 3 stars, recommended (binary_commute)</h4>
Recall the <span class="inlinecode"><span class="id" type="var">incr</span></span> and <span class="inlinecode"><span class="id" type="var">bin_to_nat</span></span> functions that you
wrote for the <span class="inlinecode"><span class="id" type="var">binary</span></span> exercise in the <a href="Basics.html"><span class="inlineref">Basics</span></a> chapter. Prove
that the following diagram commutes:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">bin</span> --------- <span class="id" type="var">incr</span> -------> <span class="id" type="var">bin</span><br/>
| |<br/>
<span class="id" type="var">bin_to_nat</span> <span class="id" type="var">bin_to_nat</span><br/>
| |<br/>
<span class="id" type="var">v</span> <span class="id" type="var">v</span><br/>
<span class="id" type="var">nat</span> ---------- <span class="id" type="var">S</span> ---------> <span class="id" type="var">nat</span>
<div class="paragraph"> </div>
</div>
That is, incrementing a binary number and then converting it to
a (unary) natural number yields the same result as first converting
it to a natural number and then incrementing.
Name your theorem <span class="inlinecode"><span class="id" type="var">bin_to_nat_pres_incr</span></span> ("pres" for "preserves").
<div class="paragraph"> </div>
Before you start working on this exercise, please copy the
definitions from your solution to the <span class="inlinecode"><span class="id" type="var">binary</span></span> exercise here so
that this file can be graded on its own. If you find yourself
wanting to change your original definitions to make the property
easier to prove, feel free to do so!
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab53"></a><h4 class="section">Exercise: 5 stars, advanced (binary_inverse)</h4>
This exercise is a continuation of the previous exercise about
binary numbers. You will need your definitions and theorems from
there to complete this one.
<div class="paragraph"> </div>
(a) First, write a function to convert natural numbers to binary
numbers. Then prove that starting with any natural number,
converting to binary, then converting back yields the same
natural number you started with.
<div class="paragraph"> </div>
(b) You might naturally think that we should also prove the
opposite direction: that starting with a binary number,
converting to a natural, and then back to binary yields the
same number we started with. However, this is not true!
Explain what the problem is.
<div class="paragraph"> </div>
(c) Define a "direct" normalization function — i.e., a function
<span class="inlinecode"><span class="id" type="var">normalize</span></span> from binary numbers to binary numbers such that,
for any binary number b, converting to a natural and then back
to binary yields <span class="inlinecode">(<span class="id" type="var">normalize</span></span> <span class="inlinecode"><span class="id" type="var">b</span>)</span>. Prove it. (Warning: This
part is tricky!)
<div class="paragraph"> </div>
Again, feel free to change your earlier definitions if this helps
here.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab54"></a><h1 class="section">Formal vs. Informal Proof (Optional)</h1>
<div class="paragraph"> </div>
<div class="quote">"<i>Informal proofs are algorithms; formal proofs are code</i>."</div>
<div class="paragraph"> </div>
The question of what constitutes a proof of a mathematical
claim has challenged philosophers for millennia, but a rough and
ready definition could be this: A proof of a mathematical
proposition <span class="inlinecode"><span class="id" type="var">P</span></span> is a written (or spoken) text that instills in the
reader or hearer the certainty that <span class="inlinecode"><span class="id" type="var">P</span></span> is true. That is, a proof
is an act of communication.
<div class="paragraph"> </div>
Acts of communication may involve different sorts of readers. On
one hand, the "reader" can be a program like Coq, in which case
the "belief" that is instilled is that <span class="inlinecode"><span class="id" type="var">P</span></span> can be mechanically
derived from a certain set of formal logical rules, and the proof
is a recipe that guides the program in checking this fact. Such
recipes are <i>formal</i> proofs.
<div class="paragraph"> </div>
Alternatively, the reader can be a human being, in which case the
proof will be written in English or some other natural language,
and will thus necessarily be <i>informal</i>. Here, the criteria for
success are less clearly specified. A "valid" proof is one that
makes the reader believe <span class="inlinecode"><span class="id" type="var">P</span></span>. But the same proof may be read by
many different readers, some of whom may be convinced by a
particular way of phrasing the argument, while others may not be.
Some readers may be particularly pedantic, inexperienced, or just
plain thick-headed; the only way to convince them will be to make
the argument in painstaking detail. But other readers, more
familiar in the area, may find all this detail so overwhelming
that they lose the overall thread; all they want is to be told the
main ideas, since it is easier for them to fill in the details for
themselves than to wade through a written presentation of them.
Ultimately, there is no universal standard, because there is no
single way of writing an informal proof that is guaranteed to
convince every conceivable reader.
<div class="paragraph"> </div>
In practice, however, mathematicians have developed a rich set of
conventions and idioms for writing about complex mathematical
objects that — at least within a certain community — make
communication fairly reliable. The conventions of this stylized
form of communication give a fairly clear standard for judging
proofs good or bad.
<div class="paragraph"> </div>
Because we are using Coq in this course, we will be working
heavily with formal proofs. But this doesn't mean we can
completely forget about informal ones! Formal proofs are useful
in many ways, but they are <i>not</i> very efficient ways of
communicating ideas between human beings.
<div class="paragraph"> </div>
For example, here is a proof that addition is associative:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_assoc'"><span class="id" type="lemma">plus_assoc'</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</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> <span class="id" type="var">IHn'</span>]. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Coq is perfectly happy with this. For a human, however, it
is difficult to make much sense of it. We can use comments and
bullets to show the structure a little more clearly...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_assoc''"><span class="id" type="lemma">plus_assoc''</span></a> : <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> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>) = (<a class="idref" href="Induction.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="Induction.html#m"><span class="id" type="variable">m</span></a>) + <a class="idref" href="Induction.html#p"><span class="id" type="variable">p</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</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> <span class="id" type="var">IHn'</span>].<br/>
- <span class="comment">(* n = 0 *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
... and if you're used to Coq you may be able to step
through the tactics one after the other in your mind and imagine
the state of the context and goal stack at each point, but if the
proof were even a little bit more complicated this would be next
to impossible.
<div class="paragraph"> </div>
A (pedantic) mathematician might write the proof something like
this:
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>Theorem</i>: For any <span class="inlinecode"><span class="id" type="var">n</span></span>, <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">p</span></span>,
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.
<div class="paragraph"> </div>
</div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">n</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> First, suppose <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>. We must show
<div class="paragraph"> </div>
<div class="code code-tight">
0 + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (0 + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.
<div class="paragraph"> </div>
</div>
This follows directly from the definition of <span class="inlinecode">+</span>.
<div class="paragraph"> </div>
</li>
<li> Next, suppose <span class="inlinecode"><span class="id" type="var">n</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">n'</span></span>, where
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">n'</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n'</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.
<div class="paragraph"> </div>
</div>
We must show
<div class="paragraph"> </div>
<div class="code code-tight">
(<span class="id" type="var">S</span> <span class="id" type="var">n'</span>) + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = ((<span class="id" type="var">S</span> <span class="id" type="var">n'</span>) + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.
<div class="paragraph"> </div>
</div>
By the definition of <span class="inlinecode">+</span>, this follows from
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">S</span> (<span class="id" type="var">n'</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>)) = <span class="id" type="var">S</span> ((<span class="id" type="var">n'</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>),
<div class="paragraph"> </div>
</div>
which is immediate from the induction hypothesis. <i>Qed</i>.
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
The overall form of the proof is basically similar, and of
course this is no accident: Coq has been designed so that its
<span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic generates the same sub-goals, in the same
order, as the bullet points that a mathematician would write. But
there are significant differences of detail: the formal proof is
much more explicit in some ways (e.g., the use of <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>)
but much less explicit in others (in particular, the "proof state"
at any given point in the Coq proof is completely implicit,
whereas the informal proof reminds the reader several times where
things stand).
<div class="paragraph"> </div>
<a name="lab55"></a><h4 class="section">Exercise: 2 stars, advanced, recommended (plus_comm_informal)</h4>
Translate your solution for <span class="inlinecode"><span class="id" type="var">plus_comm</span></span> into an informal proof:
<div class="paragraph"> </div>
Theorem: Addition is commutative.
<div class="paragraph"> </div>
Proof: <span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab56"></a><h4 class="section">Exercise: 2 stars, optional (beq_nat_refl_informal)</h4>
Write an informal proof of the following theorem, using the
informal proof of <span class="inlinecode"><span class="id" type="var">plus_assoc</span></span> as a model. Don't just
paraphrase the Coq tactics into English!
<div class="paragraph"> </div>
Theorem: <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">beq_nat</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> for any <span class="inlinecode"><span class="id" type="var">n</span></span>.
<div class="paragraph"> </div>
Proof: <span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
</div>
<div class="code code-tight">
</div>
</div>
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