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<h1 class="libtitle">IndPrinciples<span class="subtitle">Induction Principles</span></h1>
<div class="doc">
<div class="paragraph"> </div>
With the Curry-Howard correspondence and its realization in Coq in
mind, we can now take a deeper look at induction principles.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Set</span> <span class="id" type="var">Warnings</span> "-notation-overridden,-parsing".<br/>
<span class="id" type="var">From</span> <span class="id" type="var">LF</span> <span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">ProofObjects</span>.<br/>
</div>
<div class="doc">
<a name="lab281"></a><h1 class="section">Basics</h1>
<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 an <i>induction principle</i> for this type.
This induction principle is a theorem like any other: If <span class="inlinecode"><span class="id" type="var">t</span></span> is
defined inductively, the corresponding induction principle 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">
<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 <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop,<br/>
P 0 <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
(forall n : nat, P n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P (S n)) <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
forall n : nat, P n *)</span><br/>
</div>
<div class="doc">
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 with directly using <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">nat_ind</span></span>, 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
<a href="Basics.html"><span class="inlineref">Basics</span></a> chapter.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_r'</span> : <span style='font-size:120%;'>∀</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">(* n = O *)</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">intros</span> <span class="id" type="var">n'</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">rewrite</span> → <span class="id" type="var">IHn'</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<div class="doc">
This proof is basically the same as the earlier one, but a
few minor differences are worth noting.
<div class="paragraph"> </div>
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. By contrast, 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>
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 these bits 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="lab282"></a><h4 class="section">Exercise: 2 stars, standard, optional (plus_one_r')</h4>
Complete this proof without using the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_one_r'</span> : <span style='font-size:120%;'>∀</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>
<span class="proofbox">☐</span>
<div class="doc less-space">
<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 of
course 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">c<sub>1</sub></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> : <span style='font-size:120%;'>∀</span><span class="id" type="var">P</span> : <span class="id" type="var">t</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">c<sub>1</sub></span> ... →<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">c<sub>2</sub></span> ... → ...<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">cn</span> ... →<br/>
<span style='font-size:120%;'>∀</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">
<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><br/>
| <span class="id" type="var">no</span>.<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">yesno_ind</span>.<br/>
<span class="comment">(* ===> yesno_ind : forall P : yesno <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop,<br/>
P yes <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
P no <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
forall y : yesno, P y *)</span><br/>
</div>
<div class="doc">
<a name="lab283"></a><h4 class="section">Exercise: 1 star, standard, 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">
<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><br/>
| <span class="id" type="var">green</span><br/>
| <span class="id" type="var">blue</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">rgb_ind</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
Here's another example, this time with one of the constructors
taking some arguments.
</div>
<div class="code code-tight">
<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><br/>
| <span class="id" type="var">ncons</span> (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">l</span> : <span class="id" type="var">natlist</span>).<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">natlist_ind</span>.<br/>
<span class="comment">(* ===> (modulo a little variable renaming)<br/>
natlist_ind :<br/>
forall P : natlist <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop,<br/>
P nnil <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
(forall (n : nat) (l : natlist),<br/>
P l <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P (ncons n l)) <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
forall n : natlist, P n *)</span><br/>
</div>
<div class="doc">
<a name="lab284"></a><h4 class="section">Exercise: 1 star, standard, optional (natlist1)</h4>
Suppose we had written the above definition a little
differently:
</div>
<div class="code code-tight">
<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><br/>
| <span class="id" type="var">nsnoc1</span> (<span class="id" type="var">l</span> : <span class="id" type="var">natlist1</span>) (<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>).<br/>
</div>
<div class="doc">
Now what will the induction principle look like? <span class="proofbox">☐</span>
<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">a<sub>1</sub></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:
<div class="paragraph"> </div>
<ul class="doclist">
<li> "For all values <span class="inlinecode"><span class="id" type="var">x<sub>1</sub></span></span>...<span class="inlinecode"><span class="id" type="var">xn</span></span> of types <span class="inlinecode"><span class="id" type="var">a<sub>1</sub></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">x<sub>1</sub></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>
<a name="lab285"></a><h4 class="section">Exercise: 1 star, standard, optional (byntree_ind)</h4>
Write out the induction principle that Coq will generate for the
following datatype. (Again, 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">
<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><br/>
| <span class="id" type="var">bleaf</span> (<span class="id" type="var">yn</span> : <span class="id" type="var">yesno</span>)<br/>
| <span class="id" type="var">nbranch</span> (<span class="id" type="var">yn</span> : <span class="id" type="var">yesno</span>) (<span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></span> : <span class="id" type="var">byntree</span>).<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab286"></a><h4 class="section">Exercise: 1 star, standard, 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-size:120%;'>∀</span><span class="id" type="var">P</span> : <span class="id" type="var">ExSet</span> → <span class="id" type="keyword">Prop</span>,<br/>
(<span style='font-size:120%;'>∀</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>)) →<br/>
(<span style='font-size:120%;'>∀</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 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>)) →<br/>
<span style='font-size:120%;'>∀</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">
<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>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab287"></a><h1 class="section">Polymorphism</h1>
<div class="paragraph"> </div>
Next, 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 class="id" type="var">list</span> <span class="id" type="var">X</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-size:120%;'>∀</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 class="id" type="keyword">Prop</span>),<br/>
<span class="id" type="var">P</span> [] →<br/>
(<span style='font-size:120%;'>∀</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 class="id" type="var">P</span> (<span class="id" type="var">x</span> :: <span class="id" type="var">l</span>)) →<br/>
<span style='font-size:120%;'>∀</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 that 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="lab288"></a><h4 class="section">Exercise: 1 star, standard, 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">
<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 class="id" type="var">X</span>)<br/>
| <span class="id" type="var">node</span> (<span class="id" type="var">t<sub>1</sub></span> <span class="id" type="var">t<sub>2</sub></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>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab289"></a><h4 class="section">Exercise: 1 star, standard, 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-size:120%;'>∀</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 class="id" type="keyword">Prop</span>),<br/>
(<span style='font-size:120%;'>∀</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>)) →<br/>
(<span style='font-size:120%;'>∀</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>)) →<br/>
(<span style='font-size:120%;'>∀</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> →<br/>
<span style='font-size:120%;'>∀</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>)) →<br/>
<span style='font-size:120%;'>∀</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>
<span class="proofbox">☐</span>
<div class="paragraph"> </div>
<a name="lab290"></a><h4 class="section">Exercise: 1 star, standard, 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-size:120%;'>∀</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 class="id" type="keyword">Prop</span>),<br/>
(<span style='font-size:120%;'>∀</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>)) →<br/>
(<span style='font-size:120%;'>∀</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>)) →<br/>
(<span style='font-size:120%;'>∀</span><span class="id" type="var">f<sub>1</sub></span> : <span class="id" type="var">nat</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-size:120%;'>∀</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">f<sub>1</sub></span> <span class="id" type="var">n</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">f<sub>1</sub></span>)) →<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">f<sub>2</sub></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">f<sub>2</sub></span>
<div class="paragraph"> </div>
</div>
<span class="proofbox">☐</span>
<div class="paragraph"> </div>
<a name="lab291"></a><h4 class="section">Exercise: 1 star, standard, optional (foo')</h4>
Consider the following inductive definition:
</div>
<div class="code code-tight">
<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">C<sub>1</sub></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">C<sub>2</sub></span>.<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-size:120%;'>∀</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 class="id" type="keyword">Prop</span>),<br/>
(<span style='font-size:120%;'>∀</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> →<br/>
<span class="id" type="var">_______________________</span> ) →<br/>
<span class="id" type="var">___________________________________________</span> →<br/>
<span style='font-size:120%;'>∀</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>
<span class="proofbox">☐</span>
</div>
<div class="doc">
<a name="lab292"></a><h1 class="section">Induction Hypotheses</h1>
<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-size:120%;'>∀</span><span class="id" type="var">P</span> : <span class="id" type="var">nat</span> → <span class="id" type="keyword">Prop</span>,<br/>
<span class="id" type="var">P</span> 0 →<br/>
(<span style='font-size:120%;'>∀</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 class="id" type="var">P</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) →<br/>
<span style='font-size:120%;'>∀</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> (or rather, 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 class="id" type="keyword">Prop</span></span>.
<div class="paragraph"> </div>
We can make proofs by induction 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-size:120%;'>∀</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-size:120%;'>∀</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">
<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/>
</div>
<div class="doc">
... or equivalently:
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">P_m0r'</span> : <span class="id" type="var">nat</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/>
</div>
<div class="doc">
Now it is easier to see where <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> appears in the proof.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_r''</span> : <span style='font-size:120%;'>∀</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/>
</div>
<div class="doc">
This extra naming step isn't something that we 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-size:120%;'>∀</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-size:120%;'>∀</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> <span class="inlinecode"><span class="id" type="var">P_m0r</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="doc">
<a name="lab293"></a><h1 class="section">More on the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> Tactic</h1>
<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">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_assoc'</span> : <span style='font-size:120%;'>∀</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 class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<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">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_comm'</span> : <span style='font-size:120%;'>∀</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 class="id" type="var">plus_n_O</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 class="id" type="var">IHn'</span>.<br/>
<span class="id" type="tactic">rewrite</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/>
</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-size:120%;'>∀</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">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_comm''</span> : <span style='font-size:120%;'>∀</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 class="id" type="var">plus_n_O</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 class="id" type="var">IHm'</span>.<br/>
<span class="id" type="tactic">rewrite</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/>
</div>
<div class="doc">
<a name="lab294"></a><h4 class="section">Exercise: 1 star, standard, 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">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc">
<a name="lab295"></a><h1 class="section">Induction Principles in <span class="inlinecode"><span class="id" type="keyword">Prop</span></span></h1>
<div class="paragraph"> </div>
Earlier, we looked in detail at the induction principles that Coq
generates for inductively defined <i>sets</i>. The induction
principles for inductively defined <i>propositions</i> like <span class="inlinecode"><span class="id" type="var">even</span></span> are a
tiny bit more complicated. As with all induction principles, we
want to use the induction principle on <span class="inlinecode"><span class="id" type="var">even</span></span> to prove things by
inductively considering the possible shapes that something in <span class="inlinecode"><span class="id" type="var">even</span></span>
can have. Intuitively speaking, however, what we want to prove
are not statements about <i>evidence</i> but statements about
<i>numbers</i>: accordingly, we want an induction principle that lets
us prove properties of numbers by induction on evidence.
<div class="paragraph"> </div>
For example, from what we've said so far, you might expect the
inductive definition of <span class="inlinecode"><span class="id" type="var">even</span></span>...
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">even</span> : <span class="id" type="var">nat</span> → <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ev_0</span> : <span class="id" type="var">even</span> 0<br/>
| <span class="id" type="var">ev_SS</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">even</span> <span class="id" type="var">n</span> → <span class="id" type="var">even</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)).
<div class="paragraph"> </div>
</div>
...to give rise to an induction principle that looks like this...
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">ev_ind_max</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">P</span> : (<span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">even</span> <span class="id" type="var">n</span> → <span class="id" type="keyword">Prop</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">O</span> <span class="id" type="var">ev_0</span> →<br/>
(<span style='font-size:120%;'>∀</span>(<span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">E</span> : <span class="id" type="var">even</span> <span class="id" type="var">m</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">m</span> <span class="id" type="var">E</span> →<br/>
<span class="id" type="var">P</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>)) (<span class="id" type="var">ev_SS</span> <span class="id" type="var">m</span> <span class="id" type="var">E</span>)) →<br/>
<span style='font-size:120%;'>∀</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">even</span> <span class="id" type="var">n</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span>
<div class="paragraph"> </div>
</div>
... because:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Since <span class="inlinecode"><span class="id" type="var">even</span></span> is indexed by a number <span class="inlinecode"><span class="id" type="var">n</span></span> (every <span class="inlinecode"><span class="id" type="var">even</span></span> object <span class="inlinecode"><span class="id" type="var">E</span></span> is
a piece of evidence that some particular number <span class="inlinecode"><span class="id" type="var">n</span></span> is even),
the proposition <span class="inlinecode"><span class="id" type="var">P</span></span> is parameterized by both <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">E</span></span> —
that is, the induction principle can be used to prove
assertions involving both an even number and the evidence that
it is even.
<div class="paragraph"> </div>
</li>
<li> Since there are two ways of giving evidence of evenness (<span class="inlinecode"><span class="id" type="var">even</span></span>
has two constructors), applying the induction principle
generates two subgoals:
<div class="paragraph"> </div>
<ul class="doclist">
<li> We must prove that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">O</span></span> and <span class="inlinecode"><span class="id" type="var">ev_0</span></span>.
<div class="paragraph"> </div>
</li>
<li> We must prove that, whenever <span class="inlinecode"><span class="id" type="var">n</span></span> is an even number and <span class="inlinecode"><span class="id" type="var">E</span></span>
is an evidence of its evenness, if <span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">n</span></span> and
<span class="inlinecode"><span class="id" type="var">E</span></span>, then it also holds of <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> and <span class="inlinecode"><span class="id" type="var">ev_SS</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">E</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If these subgoals can be proved, then the induction principle
tells us that <span class="inlinecode"><span class="id" type="var">P</span></span> is true for <i>all</i> even numbers <span class="inlinecode"><span class="id" type="var">n</span></span> and
evidence <span class="inlinecode"><span class="id" type="var">E</span></span> of their evenness.
</li>
</ul>
<div class="paragraph"> </div>
This is more flexibility than we normally need or want: it is
giving us a way to prove logical assertions where the assertion
involves properties of some piece of <i>evidence</i> of evenness, while
all we really care about is proving properties of <i>numbers</i> that
are even — we are interested in assertions about numbers, not
about evidence. It would therefore be more convenient to have an
induction principle for proving propositions <span class="inlinecode"><span class="id" type="var">P</span></span> that are
parameterized just by <span class="inlinecode"><span class="id" type="var">n</span></span> and whose conclusion establishes <span class="inlinecode"><span class="id" type="var">P</span></span> for
all even numbers <span class="inlinecode"><span class="id" type="var">n</span></span>:
<div class="paragraph"> </div>
<div class="code code-tight">
<span style='font-size:120%;'>∀</span><span class="id" type="var">P</span> : <span class="id" type="var">nat</span> → <span class="id" type="keyword">Prop</span>,<br/>
... →<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">even</span> <span class="id" type="var">n</span> → <span class="id" type="var">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
For this reason, Coq actually generates the following simplified
induction principle for <span class="inlinecode"><span class="id" type="var">even</span></span>:
</div>
<div class="code code-tight">
<span class="id" type="keyword">Check</span> <span class="id" type="var">even_ind</span>.<br/>
<span class="comment">(* ===> ev_ind<br/>
: forall P : nat <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop,<br/>
P 0 <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
(forall n : nat, even n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P (S (S n))) <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
forall n : nat,<br/>
even n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P n *)</span><br/>
</div>
<div class="doc">
In particular, Coq has dropped the evidence term <span class="inlinecode"><span class="id" type="var">E</span></span> as a
parameter of the the proposition <span class="inlinecode"><span class="id" type="var">P</span></span>.
<div class="paragraph"> </div>
In English, <span class="inlinecode"><span class="id" type="var">ev_ind</span></span> says:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose, <span class="inlinecode"><span class="id" type="var">P</span></span> is a property of natural numbers (that is, <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is
a <span class="inlinecode"><span class="id" type="keyword">Prop</span></span> for every <span class="inlinecode"><span class="id" type="var">n</span></span>). To show that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> holds whenever <span class="inlinecode"><span class="id" type="var">n</span></span>
is even, it suffices to show:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode">0</span>,
<div class="paragraph"> </div>
</li>
<li> for any <span class="inlinecode"><span class="id" type="var">n</span></span>, if <span class="inlinecode"><span class="id" type="var">n</span></span> is even and <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">n</span></span>, then <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">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>.
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
As expected, we can apply <span class="inlinecode"><span class="id" type="var">ev_ind</span></span> directly instead of using
<span class="inlinecode"><span class="id" type="tactic">induction</span></span>. For example, we can use it to show that <span class="inlinecode"><span class="id" type="var">even'</span></span> (the
slightly awkward alternate definition of evenness that we saw in
an exercise in the \chap{IndProp} chapter) is equivalent to the
cleaner inductive definition <span class="inlinecode"><span class="id" type="var">even</span></span>:
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ev_ev'</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span>, <span class="id" type="var">even</span> <span class="id" type="var">n</span> → <span class="id" type="var">even'</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">even_ind</span>.<br/>
- <span class="comment">(* ev_0 *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">even'_0</span>.<br/>
- <span class="comment">(* ev_SS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">m</span> <span class="id" type="var">Hm</span> <span class="id" type="var">IH</span>.<br/>
<span class="id" type="tactic">apply</span> (<span class="id" type="var">even'_sum</span> 2 <span class="id" type="var">m</span>).<br/>
+ <span class="id" type="tactic">apply</span> <span class="id" type="var">even'_2</span>.<br/>
+ <span class="id" type="tactic">apply</span> <span class="id" type="var">IH</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The precise form of an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definition can affect the
induction principle Coq generates.
<div class="paragraph"> </div>
For example, in chapter <a href="IndProp.html"><span class="inlineref">IndProp</span></a>, we defined <span class="inlinecode">≤</span> as:
</div>
<div class="code code-tight">
<span class="comment">(* Inductive le : nat <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> nat <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop :=<br/>
| le_n : forall n, le n n<br/>
| le_S : forall n m, (le n m) <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> (le n (S m)). *)</span><br/>
</div>
<div class="doc">
This definition can be streamlined a little by observing that the
left-hand argument <span class="inlinecode"><span class="id" type="var">n</span></span> is the same everywhere in the definition,
so we can actually make it a "general parameter" to the whole
definition, rather than an argument to each constructor.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">le</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) : <span class="id" type="var">nat</span> → <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">le_n</span> : <span class="id" type="var">le</span> <span class="id" type="var">n</span> <span class="id" type="var">n</span><br/>
| <span class="id" type="var">le_S</span> <span class="id" type="var">m</span> (<span class="id" type="var">H</span> : <span class="id" type="var">le</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>) : <span class="id" type="var">le</span> <span class="id" type="var">n</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>).<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Notation</span> "m ≤ n" := (<span class="id" type="var">le</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span>).<br/>
</div>
<div class="doc">
The second one is better, even though it looks less symmetric.
Why? Because it gives us a simpler induction principle.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Check</span> <span class="id" type="var">le_ind</span>.<br/>
<span class="comment">(* ===> forall (n : nat) (P : nat <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop),<br/>
P n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
(forall m : nat, n <= m <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P m <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P (S m)) <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span><br/>
forall n<sub>0</sub> : nat, n <= n<sub>0</sub> <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> P n<sub>0</sub> *)</span><br/>
</div>
<div class="doc">
<a name="lab296"></a><h1 class="section">Formal vs. Informal Proofs by Induction</h1>
<div class="paragraph"> </div>
Question: 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>
Answer: The latter should <i>teach</i> the reader how to produce the
former.
<div class="paragraph"> </div>
Question: 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 will amount to just
transcribing the formal one into words). This may give the reader
the ability to reproduce the formal one for themselves, but it
probably doesn't <i>teach</i> them anything much.
<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 often
writing the proof requires one or more significant 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 work that we
went through to find the proof in the first place) plus high-level
suggestions for the more routine parts to save the reader from
spending too much time reconstructing these (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>
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. Although the ideal is somewhere in between, while one
is 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 — 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>).
<div class="paragraph"> </div>
<a name="lab297"></a><h2 class="section">Induction Over an Inductively Defined Set</h2>
<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">a<sub>1</sub></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">a<sub>1</sub></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...> <span class="proofbox">☐</span>
</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 <span class="inlinecode"><span class="id" type="var">index</span></span>.
<div class="paragraph"> </div>