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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>MoreInd: More on 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">MoreInd<span class="subtitle">More on Induction</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> "ProofObjects".<br/>
<br/>
</div>
<div class="doc">
<a name="lab326"></a><h1 class="section">Induction Principles</h1>
<div class="paragraph"> </div>
This is a good point to pause and take a deeper look at induction
principles.
<div class="paragraph"> </div>
Every time we declare a new <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> datatype, Coq
automatically generates and proves an <i>induction principle</i>
for this type.
<div class="paragraph"> </div>
The induction principle for a type <span class="inlinecode"><span class="id" type="var">t</span></span> is called <span class="inlinecode"><span class="id" type="var">t_ind</span></span>. Here is
the one for natural numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">nat_ind</span>.<br/>
<span class="comment">(* ===> nat_ind : <br/>
forall P : nat -> Prop,<br/>
P 0 -><br/>
(forall n : nat, P n -> P (S n)) -><br/>
forall n : nat, P n *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab327"></a><h3 class="section"> </h3>
The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic is a straightforward wrapper that, at
its core, simply performs <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">t_ind</span></span>. To see this more
clearly, let's experiment a little with using <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">nat_ind</span></span>
directly, instead of the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic, to carry out some
proofs. Here, for example, is an alternate proof of a theorem
that we saw in the <span class="inlinecode"><span class="id" type="var">Basics</span></span> chapter.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_r'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">n</span> × 0 = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">nat_ind</span>.<br/>
- <span class="comment">(* O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* S *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">IHn</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
This proof is basically the same as the earlier one, but a
few minor differences are worth noting. First, in the induction
step of the proof (the <span class="inlinecode">"<span class="id" type="var">S</span>"</span> case), we have to do a little
bookkeeping manually (the <span class="inlinecode"><span class="id" type="tactic">intros</span></span>) that <span class="inlinecode"><span class="id" type="tactic">induction</span></span> does
automatically.
<div class="paragraph"> </div>
Second, we do not introduce <span class="inlinecode"><span class="id" type="var">n</span></span> into the context before applying
<span class="inlinecode"><span class="id" type="var">nat_ind</span></span> — the conclusion of <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> is a quantified formula,
and <span class="inlinecode"><span class="id" type="tactic">apply</span></span> needs this conclusion to exactly match the shape of
the goal state, including the quantifier. The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic
works either with a variable in the context or a quantified
variable in the goal.
<div class="paragraph"> </div>
Third, the <span class="inlinecode"><span class="id" type="tactic">apply</span></span> tactic automatically chooses variable names for
us (in the second subgoal, here), whereas <span class="inlinecode"><span class="id" type="tactic">induction</span></span> lets us
specify (with the <span class="inlinecode"><span class="id" type="keyword">as</span>...</span> clause) what names should be used. The
automatic choice is actually a little unfortunate, since it
re-uses the name <span class="inlinecode"><span class="id" type="var">n</span></span> for a variable that is different from the <span class="inlinecode"><span class="id" type="var">n</span></span>
in the original theorem. This is we have labeled the second
subgoal just <span class="inlinecode"><span class="id" type="var">S</span></span> — if we tried to write it out in the more
explicit form that we've been using for most proofs, we'd have to
write <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>, which doesn't make a lot of sense! All of these
conveniences make <span class="inlinecode"><span class="id" type="tactic">induction</span></span> nicer to use in practice than
applying induction principles like <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> directly. But it is
important to realize that, modulo this little bit of bookkeeping,
applying <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> is what we are really doing.
<div class="paragraph"> </div>
<a name="lab328"></a><h4 class="section">Exercise: 2 stars, optional (plus_one_r')</h4>
Complete this proof as we did <span class="inlinecode"><span class="id" type="var">mult_0_r'</span></span> above, without using
the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_one_r'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">n</span> + 1 = <span class="id" type="var">S</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
Coq generates induction principles for every datatype defined with
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>, including those that aren't recursive. (Although
we don't need induction to prove properties of non-recursive
datatypes, the idea of an induction principle still makes sense
for them: it gives a way to prove that a property holds for all
values of the type.)
<div class="paragraph"> </div>
These generated principles follow a similar pattern. If we define a
type <span class="inlinecode"><span class="id" type="var">t</span></span> with constructors <span class="inlinecode"><span class="id" type="var">c1</span></span> ... <span class="inlinecode"><span class="id" type="var">cn</span></span>, Coq generates a theorem
with this shape:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">t_ind</span> :<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">c1</span> ... <span style="font-family: arial;">→</span><br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">c2</span> ... <span style="font-family: arial;">→</span><br/>
...<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">cn</span> ... <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">t</span>, <span class="id" type="var">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
The specific shape of each case depends on the arguments to the
corresponding constructor. Before trying to write down a general
rule, let's look at some more examples. First, an example where
the constructors take no arguments:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">yesno</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">yes</span> : <span class="id" type="var">yesno</span><br/>
| <span class="id" type="var">no</span> : <span class="id" type="var">yesno</span>.<br/>
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">yesno_ind</span>.<br/>
<span class="comment">(* ===> yesno_ind : forall P : yesno -> Prop, <br/>
P yes -><br/>
P no -><br/>
forall y : yesno, P y *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab329"></a><h4 class="section">Exercise: 1 star, optional (rgb)</h4>
Write out the induction principle that Coq will generate for the
following datatype. Write down your answer on paper or type it
into a comment, and then compare it with what Coq prints.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">rgb</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="tactic">red</span> : <span class="id" type="var">rgb</span><br/>
| <span class="id" type="var">green</span> : <span class="id" type="var">rgb</span><br/>
| <span class="id" type="var">blue</span> : <span class="id" type="var">rgb</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">rgb_ind</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
Here's another example, this time with one of the constructors
taking some arguments.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">natlist</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nnil</span> : <span class="id" type="var">natlist</span><br/>
| <span class="id" type="var">ncons</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">natlist</span> <span style="font-family: arial;">→</span> <span class="id" type="var">natlist</span>.<br/>
<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">natlist_ind</span>.<br/>
<span class="comment">(* ===> (modulo a little variable renaming for clarity)<br/>
natlist_ind :<br/>
forall P : natlist -> Prop,<br/>
P nnil -><br/>
(forall (n : nat) (l : natlist), P l -> P (ncons n l)) -><br/>
forall n : natlist, P n *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab330"></a><h4 class="section">Exercise: 1 star, optional (natlist1)</h4>
Suppose we had written the above definition a little
differently:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">natlist1</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nnil1</span> : <span class="id" type="var">natlist1</span><br/>
| <span class="id" type="var">nsnoc1</span> : <span class="id" type="var">natlist1</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">natlist1</span>.<br/>
<br/>
</div>
<div class="doc">
Now what will the induction principle look like? <font size=-2>☐</font>
<div class="paragraph"> </div>
From these examples, we can extract this general rule:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The type declaration gives several constructors; each
corresponds to one clause of the induction principle.
</li>
<li> Each constructor <span class="inlinecode"><span class="id" type="var">c</span></span> takes argument types <span class="inlinecode"><span class="id" type="var">a1</span></span>...<span class="inlinecode"><span class="id" type="var">an</span></span>.
</li>
<li> Each <span class="inlinecode"><span class="id" type="var">ai</span></span> can be either <span class="inlinecode"><span class="id" type="var">t</span></span> (the datatype we are defining) or
some other type <span class="inlinecode"><span class="id" type="var">s</span></span>.
</li>
<li> The corresponding case of the induction principle
says (in English):
<ul class="doclist">
<li> "for all values <span class="inlinecode"><span class="id" type="var">x1</span></span>...<span class="inlinecode"><span class="id" type="var">xn</span></span> of types <span class="inlinecode"><span class="id" type="var">a1</span></span>...<span class="inlinecode"><span class="id" type="var">an</span></span>, if <span class="inlinecode"><span class="id" type="var">P</span></span>
holds for each of the inductive arguments (each <span class="inlinecode"><span class="id" type="var">xi</span></span> of
type <span class="inlinecode"><span class="id" type="var">t</span></span>), then <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">x1</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">xn</span></span>".
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<a name="lab331"></a><h4 class="section">Exercise: 1 star, optional (byntree_ind)</h4>
Write out the induction principle that Coq will generate for the
following datatype. Write down your answer on paper or type it
into a comment, and then compare it with what Coq prints.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">byntree</span> : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">bempty</span> : <span class="id" type="var">byntree</span> <br/>
| <span class="id" type="var">bleaf</span> : <span class="id" type="var">yesno</span> <span style="font-family: arial;">→</span> <span class="id" type="var">byntree</span><br/>
| <span class="id" type="var">nbranch</span> : <span class="id" type="var">yesno</span> <span style="font-family: arial;">→</span> <span class="id" type="var">byntree</span> <span style="font-family: arial;">→</span> <span class="id" type="var">byntree</span> <span style="font-family: arial;">→</span> <span class="id" type="var">byntree</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab332"></a><h4 class="section">Exercise: 1 star, optional (ex_set)</h4>
Here is an induction principle for an inductively defined
set.
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">ExSet_ind</span> :<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">ExSet</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">b</span> : <span class="id" type="var">bool</span>, <span class="id" type="var">P</span> (<span class="id" type="var">con1</span> <span class="id" type="var">b</span>)) <span style="font-family: arial;">→</span><br/>
(<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">e</span> : <span class="id" type="var">ExSet</span>), <span class="id" type="var">P</span> <span class="id" type="var">e</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">con2</span> <span class="id" type="var">n</span> <span class="id" type="var">e</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">e</span> : <span class="id" type="var">ExSet</span>, <span class="id" type="var">P</span> <span class="id" type="var">e</span>
<div class="paragraph"> </div>
</div>
Give an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definition of <span class="inlinecode"><span class="id" type="var">ExSet</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ExSet</span> : <span class="id" type="keyword">Type</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
What about polymorphic datatypes?
<div class="paragraph"> </div>
The inductive definition of polymorphic lists
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">list</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nil</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">cons</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span>.
<div class="paragraph"> </div>
</div>
is very similar to that of <span class="inlinecode"><span class="id" type="var">natlist</span></span>. The main difference is
that, here, the whole definition is <i>parameterized</i> on a set <span class="inlinecode"><span class="id" type="var">X</span></span>:
that is, we are defining a <i>family</i> of inductive types <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>,
one for each <span class="inlinecode"><span class="id" type="var">X</span></span>. (Note that, wherever <span class="inlinecode"><span class="id" type="var">list</span></span> appears in the body
of the declaration, it is always applied to the parameter <span class="inlinecode"><span class="id" type="var">X</span></span>.)
The induction principle is likewise parameterized on <span class="inlinecode"><span class="id" type="var">X</span></span>:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">list_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
<span class="id" type="var">P</span> [] <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">x</span> : <span class="id" type="var">X</span>) (<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>), <span class="id" type="var">P</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">x</span> :: <span class="id" type="var">l</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">l</span>
<div class="paragraph"> </div>
</div>
Note the wording here (and, accordingly, the form of <span class="inlinecode"><span class="id" type="var">list_ind</span></span>):
The <i>whole</i> induction principle is parameterized on <span class="inlinecode"><span class="id" type="var">X</span></span>. That is,
<span class="inlinecode"><span class="id" type="var">list_ind</span></span> can be thought of as a polymorphic function that, when
applied to a type <span class="inlinecode"><span class="id" type="var">X</span></span>, gives us back an induction principle
specialized to the type <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>.
<div class="paragraph"> </div>
<a name="lab333"></a><h4 class="section">Exercise: 1 star, optional (tree)</h4>
Write out the induction principle that Coq will generate for
the following datatype. Compare your answer with what Coq
prints.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">tree</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">leaf</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tree</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">node</span> : <span class="id" type="var">tree</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tree</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">tree</span> <span class="id" type="var">X</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">tree_ind</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab334"></a><h4 class="section">Exercise: 1 star, optional (mytype)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">mytype_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span> : <span class="id" type="var">X</span>, <span class="id" type="var">P</span> (<span class="id" type="var">constr1</span> <span class="id" type="var">X</span> <span class="id" type="var">x</span>)) <span style="font-family: arial;">→</span><br/>
(<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">P</span> (<span class="id" type="var">constr2</span> <span class="id" type="var">X</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">m</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span> <br/>
<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">P</span> (<span class="id" type="var">constr3</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">m</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">m</span>
<div class="paragraph"> </div>
</div>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab335"></a><h4 class="section">Exercise: 1 star, optional (foo)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">foo_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span> : <span class="id" type="var">X</span>, <span class="id" type="var">P</span> (<span class="id" type="var">bar</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">x</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">y</span> : <span class="id" type="var">Y</span>, <span class="id" type="var">P</span> (<span class="id" type="var">baz</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">y</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">f1</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>,<br/>
(<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">P</span> (<span class="id" type="var">f1</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">quux</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">f1</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">f2</span> : <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>, <span class="id" type="var">P</span> <span class="id" type="var">f2</span>
<div class="paragraph"> </div>
</div>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab336"></a><h4 class="section">Exercise: 1 star, optional (foo')</h4>
Consider the following inductive definition:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">foo'</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">C1</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">foo'</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">foo'</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">C2</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span>.<br/>
<br/>
</div>
<div class="doc">
What induction principle will Coq generate for <span class="inlinecode"><span class="id" type="var">foo'</span></span>? Fill
in the blanks, then check your answer with Coq.)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">foo'_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">f</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">_______________________</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">_______________________</span> ) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">___________________________________________</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">f</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span>, <span class="id" type="var">________________________</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab337"></a><h2 class="section">Induction Hypotheses</h2>
<div class="paragraph"> </div>
Where does the phrase "induction hypothesis" fit into this story?
<div class="paragraph"> </div>
The induction principle for numbers
<div class="paragraph"> </div>
<div class="code code-tight">
<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">P</span> 0 <span style="font-family: arial;">→</span><br/>
(<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">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>)) <span style="font-family: arial;">→</span><br/>
<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">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
is a generic statement that holds for all propositions
<span class="inlinecode"><span class="id" type="var">P</span></span> (strictly speaking, for all families of propositions <span class="inlinecode"><span class="id" type="var">P</span></span>
indexed by a number <span class="inlinecode"><span class="id" type="var">n</span></span>). Each time we use this principle, we
are choosing <span class="inlinecode"><span class="id" type="var">P</span></span> to be a particular expression of type
<span class="inlinecode"><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span></span>.
<div class="paragraph"> </div>
We can make the proof more explicit by giving this expression a
name. For example, instead of stating the theorem <span class="inlinecode"><span class="id" type="var">mult_0_r</span></span> as
"<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</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">0</span>," we can write it as "<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>,</span> <span class="inlinecode"><span class="id" type="var">P_m0r</span></span>
<span class="inlinecode"><span class="id" type="var">n</span></span>", where <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> is defined as...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">P_m0r</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="keyword">Prop</span> := <br/>
<span class="id" type="var">n</span> × 0 = 0.<br/>
<br/>
</div>
<div class="doc">
... or equivalently...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">P_m0r'</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="keyword">fun</span> <span class="id" type="var">n</span> ⇒ <span class="id" type="var">n</span> × 0 = 0.<br/>
<br/>
</div>
<div class="doc">
Now when we do the proof it is easier to see where <span class="inlinecode"><span class="id" type="var">P_m0r</span></span>
appears.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_r''</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">P_m0r</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">nat_ind</span>.<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="comment">(* Note the proof state at this point! *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">IHn</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">P_m0r</span> <span class="id" type="keyword">in</span> <span class="id" type="var">IHn</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">P_m0r</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHn</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
This extra naming step isn't something that we'll do in
normal proofs, but it is useful to do it explicitly for an example
or two, because it allows us to see exactly what the induction
hypothesis is. If we prove <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>,</span> <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> by induction on
<span class="inlinecode"><span class="id" type="var">n</span></span> (using either <span class="inlinecode"><span class="id" type="tactic">induction</span></span> or <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">nat_ind</span></span>), we see that the
first subgoal requires us to prove <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode">0</span> ("<span class="inlinecode"><span class="id" type="var">P</span></span> holds for
zero"), while the second subgoal requires us to prove <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>,</span>
<span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><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_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> (that is "<span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> if it
holds of <span class="inlinecode"><span class="id" type="var">n'</span></span>" or, more elegantly, "<span class="inlinecode"><span class="id" type="var">P</span></span> is preserved by <span class="inlinecode"><span class="id" type="var">S</span></span>").
The <i>induction hypothesis</i> is the premise of this latter
implication — the assumption that <span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">n'</span></span>, which we are
allowed to use in proving that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <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/>
</div>
<div class="doc">
<a name="lab338"></a><h2 class="section">More on the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> Tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic actually does even more low-level
bookkeeping for us than we discussed above.
<div class="paragraph"> </div>
Recall the informal statement of the induction principle for
natural numbers:
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is some proposition involving a natural number n, and
we want to show that P holds for <i>all</i> numbers n, we can
reason like this:
<ul class="doclist">
<li> show that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span> holds
</li>
<li> show that, if <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> holds, then so does <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<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> <span class="inlinecode"><span class="id" type="var">n</span></span> holds for all n.
</li>
</ul>
</li>
</ul>
So, when we begin a proof with <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> and then <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>,
we are first telling Coq to consider a <i>particular</i> <span class="inlinecode"><span class="id" type="var">n</span></span> (by
introducing it into the context) and then telling it to prove
something about <i>all</i> numbers (by using induction).
<div class="paragraph"> </div>
What Coq actually does in this situation, internally, is to
"re-generalize" the variable we perform induction on. For
example, in our original proof that <span class="inlinecode"><span class="id" type="var">plus</span></span> is associative...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_assoc'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span> : <span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* ...we first introduce all 3 variables into the context,<br/>
which amounts to saying "Consider an arbitrary <span class="inlinecode"><span class="id" type="var">n</span></span>, <span class="inlinecode"><span class="id" type="var">m</span></span>, and<br/>
<span class="inlinecode"><span class="id" type="var">p</span></span>..." *)</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>.<br/>
<span class="comment">(* ...We now use the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic to prove <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> (that<br/>
is, <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">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>) for _all_ <span class="inlinecode"><span class="id" type="var">n</span></span>,<br/>
and hence also for the particular <span class="inlinecode"><span class="id" type="var">n</span></span> that is in the context<br/>
at the moment. *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="comment">(* In the second subgoal generated by <span class="inlinecode"><span class="id" type="tactic">induction</span></span> -- the<br/>
"inductive step" -- we must prove that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> implies <br/>
<span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> for all <span class="inlinecode"><span class="id" type="var">n'</span></span>. The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic <br/>
automatically introduces <span class="inlinecode"><span class="id" type="var">n'</span></span> and <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> into the context<br/>
for us, leaving just <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> as the goal. *)</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">
It also works to apply <span class="inlinecode"><span class="id" type="tactic">induction</span></span> to a variable that is
quantified in the goal.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_comm'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">n</span> + <span class="id" type="var">m</span> = <span class="id" type="var">m</span> + <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">plus_0_r</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">m</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>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">plus_n_Sm</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Note that <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> leaves <span class="inlinecode"><span class="id" type="var">m</span></span> still bound in the goal —
i.e., what we are proving inductively is a statement beginning
with <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>.
<div class="paragraph"> </div>
If we do <span class="inlinecode"><span class="id" type="tactic">induction</span></span> on a variable that is quantified in the goal
<i>after</i> some other quantifiers, the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic will
automatically introduce the variables bound by these quantifiers
into the context.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_comm''</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>, <br/>
<span class="id" type="var">n</span> + <span class="id" type="var">m</span> = <span class="id" type="var">m</span> + <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* Let's do induction on <span class="inlinecode"><span class="id" type="var">m</span></span> this time, instead of <span class="inlinecode"><span class="id" type="var">n</span></span>... *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
- <span class="comment">(* m = O *)</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">plus_0_r</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* m = S m' *)</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">IHm'</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">plus_n_Sm</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab339"></a><h4 class="section">Exercise: 1 star, optional (plus_explicit_prop)</h4>
Rewrite both <span class="inlinecode"><span class="id" type="var">plus_assoc'</span></span> and <span class="inlinecode"><span class="id" type="var">plus_comm'</span></span> and their proofs in
the same style as <span class="inlinecode"><span class="id" type="var">mult_0_r''</span></span> above — that is, for each theorem,
give an explicit <span class="inlinecode"><span class="id" type="keyword">Definition</span></span> of the proposition being proved by
induction, and state the theorem and proof in terms of this
defined proposition.
</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="lab340"></a><h2 class="section">Generalizing Inductions.</h2>
<div class="paragraph"> </div>
One potentially confusing feature of the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic is
that it happily lets you try to set up an induction over a term
that isn't sufficiently general. The net effect of this will be
to lose information (much as <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> can do), and leave
you unable to complete the proof. Here's an example:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">one_not_beautiful_FAILED</span>: ¬ <span class="id" type="var">beautiful</span> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">H</span>.<br/>
<span class="comment">(* Just doing an <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> on <span class="inlinecode"><span class="id" type="var">H</span></span> won't get us very far in the <span class="inlinecode"><span class="id" type="var">b_sum</span></span><br/>
case. (Try it!). So we'll need induction. A naive first attempt: *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>.<br/>
<span class="comment">(* But now, although we get four cases, as we would expect from<br/>
the definition of <span class="inlinecode"><span class="id" type="var">beautiful</span></span>, we lose all information about <span class="inlinecode"><span class="id" type="var">H</span></span> ! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
The problem is that <span class="inlinecode"><span class="id" type="tactic">induction</span></span> over a Prop only works properly over
completely general instances of the Prop, i.e. one in which all
the arguments are free (unconstrained) variables.
In this respect it behaves more
like <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> than like <span class="inlinecode"><span class="id" type="tactic">inversion</span></span>.
<div class="paragraph"> </div>
When you're tempted to do use <span class="inlinecode"><span class="id" type="tactic">induction</span></span> like this, it is generally
an indication that you need to be proving something more general.
But in some cases, it suffices to pull out any concrete arguments
into separate equations, like this:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">one_not_beautiful</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">n</span> = 1 <span style="font-family: arial;">→</span> ¬ <span class="id" type="var">beautiful</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> [| | | <span class="id" type="var">p</span> <span class="id" type="var">q</span> <span class="id" type="var">Hp</span> <span class="id" type="var">IHp</span> <span class="id" type="var">Hq</span> <span class="id" type="var">IHq</span>].<br/>
- <span class="comment">(* b_0 *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span>.<br/>
- <span class="comment">(* b_3 *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span>.<br/>
- <span class="comment">(* b_5 *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span>.<br/>
- <span class="comment">(* b_sum *)</span><br/>
<span class="comment">(* the rest is a tedious case analysis *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">p</span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">p'</span>].<br/>
+ <span class="comment">(* p = 0 *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">q</span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">q'</span>].<br/>
× <span class="comment">(* q = 0 *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span>.<br/>
× <span class="comment">(* q = S q' *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHq</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E</span>.<br/>
+ <span class="comment">(* p = S p' *)</span><br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">q</span> <span class="id" type="keyword">as</span> [|<span class="id" type="var">q'</span>].<br/>
× <span class="comment">(* q = 0 *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHp</span>. <span class="id" type="tactic">rewrite</span> <span class="id" type="var">plus_0_r</span> <span class="id" type="keyword">in</span> <span class="id" type="var">E</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">E</span>.<br/>
× <span class="comment">(* q = S q' *)</span><br/>
<span class="id" type="tactic">simpl</span> <span class="id" type="keyword">in</span> <span class="id" type="var">E</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">E</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">p'</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H0</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
There's a handy <span class="inlinecode"><span class="id" type="var">remember</span></span> tactic that can generate the second
proof state out of the original one.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">one_not_beautiful'</span>: ¬ <span class="id" type="var">beautiful</span> 1.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">remember</span> 1 <span class="id" type="keyword">as</span> <span class="id" type="var">n</span> <span class="id" type="var">eqn</span>:<span class="id" type="var">E</span>.<br/>
<span class="comment">(* now carry on as above *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab341"></a><h1 class="section">Informal Proofs (Advanced)</h1>
<div class="paragraph"> </div>
Q: What is the relation between a formal proof of a proposition
<span class="inlinecode"><span class="id" type="var">P</span></span> and an informal proof of the same proposition <span class="inlinecode"><span class="id" type="var">P</span></span>?
<div class="paragraph"> </div>
A: The latter should <i>teach</i> the reader how to produce the
former.
<div class="paragraph"> </div>
Q: How much detail is needed??
<div class="paragraph"> </div>
Unfortunately, There is no single right answer; rather, there is a
range of choices.
<div class="paragraph"> </div>
At one end of the spectrum, we can essentially give the reader the
whole formal proof (i.e., the informal proof amounts to just
transcribing the formal one into words). This gives the reader
the <i>ability</i> to reproduce the formal one for themselves, but it
doesn't <i>teach</i> them anything.
<div class="paragraph"> </div>
At the other end of the spectrum, we can say "The theorem is true
and you can figure out why for yourself if you think about it hard
enough." This is also not a good teaching strategy, because
usually writing the proof requires some deep insights into the
thing we're proving, and most readers will give up before they
rediscover all the same insights as we did.
<div class="paragraph"> </div>
In the middle is the golden mean — a proof that includes all of
the essential insights (saving the reader the hard part of work
that we went through to find the proof in the first place) and
clear high-level suggestions for the more routine parts to save the
reader from spending too much time reconstructing these
parts (e.g., what the IH says and what must be shown in each case
of an inductive proof), but not so much detail that the main ideas
are obscured.
<div class="paragraph"> </div>
Another key point: if we're comparing a formal proof of a
proposition <span class="inlinecode"><span class="id" type="var">P</span></span> and an informal proof of <span class="inlinecode"><span class="id" type="var">P</span></span>, the proposition <span class="inlinecode"><span class="id" type="var">P</span></span>
doesn't change. That is, formal and informal proofs are <i>talking
about the same world</i> and they <i>must play by the same rules</i>. <a name="lab342"></a><h2 class="section">Informal Proofs by Induction</h2>
<div class="paragraph"> </div>
Since we've spent much of this chapter looking "under the hood" at
formal proofs by induction, now is a good moment to talk a little
about <i>informal</i> proofs by induction.
<div class="paragraph"> </div>
In the real world of mathematical communication, written proofs
range from extremely longwinded and pedantic to extremely brief
and telegraphic. The ideal is somewhere in between, of course,
but while you are getting used to the style it is better to start
out at the pedantic end. Also, during the learning phase, it is
probably helpful to have a clear standard to compare against.
With this in mind, we offer two templates below — one for proofs
by induction over <i>data</i> (i.e., where the thing we're doing
induction on lives in <span class="inlinecode"><span class="id" type="keyword">Type</span></span>) and one for proofs by induction over
<i>evidence</i> (i.e., where the inductively defined thing lives in
<span class="inlinecode"><span class="id" type="keyword">Prop</span></span>). In the rest of this course, please follow one of the two
for <i>all</i> of your inductive proofs.
<div class="paragraph"> </div>
<a name="lab343"></a><h3 class="section">Induction Over an Inductively Defined Set</h3>
<div class="paragraph"> </div>
<i>Template</i>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>Theorem</i>: <Universally quantified proposition of the form
"For all <span class="inlinecode"><span class="id" type="var">n</span>:<span class="id" type="var">S</span></span>, <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span>," where <span class="inlinecode"><span class="id" type="var">S</span></span> is some inductively defined
set.>
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">n</span></span>.
<div class="paragraph"> </div>
<one case for each constructor <span class="inlinecode"><span class="id" type="var">c</span></span> of <span class="inlinecode"><span class="id" type="var">S</span></span>...>
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">a1</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">ak</span></span>, where <...and here we state
the IH for each of the <span class="inlinecode"><span class="id" type="var">a</span></span>'s that has type <span class="inlinecode"><span class="id" type="var">S</span></span>, if any>.
We must show <...and here we restate <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">a1</span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">ak</span>)</span>>.
<div class="paragraph"> </div>
<go on and prove <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> to finish the case...>
<div class="paragraph"> </div>
</li>
<li> <other cases similarly...> <font size=-2>☐</font>
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<i>Example</i>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>Theorem</i>: For all sets <span class="inlinecode"><span class="id" type="var">X</span></span>, lists <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>, and numbers
<span class="inlinecode"><span class="id" type="var">n</span></span>, if <span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span> then <span class="inlinecode"><span class="id" type="var">index</span></span> <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 class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">None</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">l</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode">[]</span>. We must show, for all numbers <span class="inlinecode"><span class="id" type="var">n</span></span>,
that, if <span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode">[]</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span>, then <span class="inlinecode"><span class="id" type="var">index</span></span> <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">=</span>
<span class="inlinecode"><span class="id" type="var">None</span></span>.
<div class="paragraph"> </div>
This follows immediately from the definition of index.
<div class="paragraph"> </div>
</li>
<li> Suppose <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">::</span> <span class="inlinecode"><span class="id" type="var">l'</span></span> for some <span class="inlinecode"><span class="id" type="var">x</span></span> and <span class="inlinecode"><span class="id" type="var">l'</span></span>, where
<span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode"><span class="id" type="var">l'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n'</span></span> implies <span class="inlinecode"><span class="id" type="var">index</span></span> <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 class="id" type="var">l'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">None</span></span>, for
any number <span class="inlinecode"><span class="id" type="var">n'</span></span>. We must show, for all <span class="inlinecode"><span class="id" type="var">n</span></span>, that, if
<span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode">(<span class="id" type="var">x</span>::<span class="id" type="var">l'</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span> then <span class="inlinecode"><span class="id" type="var">index</span></span> <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 class="id" type="var">x</span>::<span class="id" type="var">l'</span>)</span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" type="var">None</span></span>.
<div class="paragraph"> </div>
Let <span class="inlinecode"><span class="id" type="var">n</span></span> be a number with <span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode"><span class="id" type="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span>. Since
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">length</span> <span class="id" type="var">l</span> = <span class="id" type="var">length</span> (<span class="id" type="var">x</span>::<span class="id" type="var">l'</span>) = <span class="id" type="var">S</span> (<span class="id" type="var">length</span> <span class="id" type="var">l'</span>),
<div class="paragraph"> </div>
</div>
it suffices to show that
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">index</span> (<span class="id" type="var">S</span> (<span class="id" type="var">length</span> <span class="id" type="var">l'</span>)) <span class="id" type="var">l'</span> = <span class="id" type="var">None</span>.
<div class="paragraph"> </div>
</div>
But this follows directly from the induction hypothesis,
picking <span class="inlinecode"><span class="id" type="var">n'</span></span> to be <span class="inlinecode"><span class="id" type="var">length</span></span> <span class="inlinecode"><span class="id" type="var">l'</span></span>. <font size=-2>☐</font>
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab344"></a><h3 class="section">Induction Over an Inductively Defined Proposition</h3>
<div class="paragraph"> </div>
Since inductively defined proof objects are often called
"derivation trees," this form of proof is also known as <i>induction
on derivations</i>.
<div class="paragraph"> </div>
<i>Template</i>:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>Theorem</i>: <Proposition of the form "<span class="inlinecode"><span class="id" type="var">Q</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span>," where <span class="inlinecode"><span class="id" type="var">Q</span></span> is
some inductively defined proposition (more generally,
"For all <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span>, <span class="inlinecode"><span class="id" type="var">Q</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span>")>
<div class="paragraph"> </div>
<i>Proof</i>: By induction on a derivation of <span class="inlinecode"><span class="id" type="var">Q</span></span>. <Or, more
generally, "Suppose we are given <span class="inlinecode"><span class="id" type="var">x</span></span>, <span class="inlinecode"><span class="id" type="var">y</span></span>, and <span class="inlinecode"><span class="id" type="var">z</span></span>. We