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<h1 class="libtitle">IndPrinciples<span class="subtitle">Induction Principles</span></h1>
<div class="doc">
<div class="paragraph"> </div>
Every time we declare a new <span class="inlinecode"><span class="id" title="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" title="var">t</span></span> is
defined inductively, the corresponding induction principle is
called <span class="inlinecode"><span class="id" title="var">t_ind</span></span>.
</div>
<div class="doc">
<a id="lab324"></a><h1 class="section">Basics</h1>
<div class="paragraph"> </div>
Here is the induction principle for natural numbers:
</div>
<div class="code">
<span class="id" title="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat_ind"><span class="id" title="definition">nat_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:1" class="idref" href="#P:1"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="IndPrinciples.html#P:1"><span class="id" title="variable">P</span></a> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:2" class="idref" href="#n:2"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:2"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:1"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#n:2"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="n:3" class="idref" href="#n:3"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#P:1"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:3"><span class="id" title="variable">n</span></a>.<br/>
</div>
<div class="doc">
In English: Suppose <span class="inlinecode"><span class="id" title="var">P</span></span> is a property of natural numbers (that is,
<span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> is a <span class="inlinecode"><span class="id" title="keyword">Prop</span></span> for every <span class="inlinecode"><span class="id" title="var">n</span></span>). To show that <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> holds of all
<span class="inlinecode"><span class="id" title="var">n</span></span>, it suffices to show:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="var">P</span></span> holds of <span class="inlinecode">0</span>
</li>
<li> for any <span class="inlinecode"><span class="id" title="var">n</span></span>, if <span class="inlinecode"><span class="id" title="var">P</span></span> holds of <span class="inlinecode"><span class="id" title="var">n</span></span>, then <span class="inlinecode"><span class="id" title="var">P</span></span> holds of <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="tactic">induction</span></span> tactic is a straightforward wrapper that, at its
core, simply performs <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">t_ind</span></span>. To see this more clearly,
let's experiment with directly using <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">nat_ind</span></span>, instead of
the <span class="inlinecode"><span class="id" title="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="Induction.html"><span class="inlineref">Induction</span></a> chapter.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="mul_0_r'" class="idref" href="#mul_0_r'"><span class="id" title="lemma">mul_0_r'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:4" class="idref" href="#n:4"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n:4"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#ea2ff3d561159081cea6fb2e8113cc<sub>54</sub>"><span class="id" title="notation">×</span></a> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat_ind"><span class="id" title="definition">nat_ind</span></a>.<br/>
- <span class="comment">(* n = O *)</span> <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">n'</span> <span class="id" title="var">IHn'</span>. <span class="id" title="tactic">rewrite</span> → <span class="id" title="var">IHn'</span>.<br/>
<span class="id" title="tactic">reflexivity</span>. <span class="id" title="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" title="var">S</span></span> case), we
have to do a little bookkeeping manually (the <span class="inlinecode"><span class="id" title="tactic">intros</span></span>) that
<span class="inlinecode"><span class="id" title="tactic">induction</span></span> does automatically.
<div class="paragraph"> </div>
Second, we do not introduce <span class="inlinecode"><span class="id" title="var">n</span></span> into the context before applying
<span class="inlinecode"><span class="id" title="var">nat_ind</span></span> -- the conclusion of <span class="inlinecode"><span class="id" title="var">nat_ind</span></span> is a quantified formula,
and <span class="inlinecode"><span class="id" title="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" title="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, we had to manually supply the name of the induction principle
with <span class="inlinecode"><span class="id" title="tactic">apply</span></span>, but <span class="inlinecode"><span class="id" title="tactic">induction</span></span> figures that out itself.
<div class="paragraph"> </div>
These conveniences make <span class="inlinecode"><span class="id" title="tactic">induction</span></span> nicer to use in practice than
applying induction principles like <span class="inlinecode"><span class="id" title="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" title="var">nat_ind</span></span> is what we are really doing.
<div class="paragraph"> </div>
<a id="lab325"></a><h4 class="section">Exercise: 2 stars, standard (plus_one_r')</h4>
Complete this proof without using the <span class="inlinecode"><span class="id" title="tactic">induction</span></span> tactic.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="plus_one_r'" class="idref" href="#plus_one_r'"><span class="id" title="lemma">plus_one_r'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:5" class="idref" href="#n:5"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n:5"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#n:5"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
Coq generates induction principles for every datatype
defined with <span class="inlinecode"><span class="id" title="keyword">Inductive</span></span>, including those that aren't recursive.
Although of course we don't need the proof technique of 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" title="var">t</span></span> with constructors <span class="inlinecode"><span class="id" title="var">c<sub>1</sub></span></span> ... <span class="inlinecode"><span class="id" title="var">cn</span></span>, Coq generates a
theorem with this shape:
<br/>
<span class="inlinecode"> <span class="id" title="var">t_ind</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : <span class="id" title="var">t</span> → <span class="id" title="keyword">Prop</span>,<br/>
... <span class="id" title="tactic">case</span> <span class="id" title="keyword">for</span> <span class="id" title="var">c<sub>1</sub></span> ... →<br/>
... <span class="id" title="tactic">case</span> <span class="id" title="keyword">for</span> <span class="id" title="var">c<sub>2</sub></span> ... → ...<br/>
... <span class="id" title="tactic">case</span> <span class="id" title="keyword">for</span> <span class="id" title="var">cn</span> ... →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">t</span>, <span class="id" title="var">P</span> <span class="id" title="var">n</span>
</span> The specific shape of each case depends on the arguments to the
corresponding constructor.
<div class="paragraph"> </div>
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">
<span class="id" title="keyword">Inductive</span> <a id="time" class="idref" href="#time"><span class="id" title="inductive">time</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="day" class="idref" href="#day"><span class="id" title="constructor">day</span></a><br/>
| <a id="night" class="idref" href="#night"><span class="id" title="constructor">night</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#time_ind"><span class="id" title="definition">time_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:8" class="idref" href="#P:8"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#time"><span class="id" title="inductive">time</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="IndPrinciples.html#P:8"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#day"><span class="id" title="constructor">day</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="IndPrinciples.html#P:8"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#night"><span class="id" title="constructor">night</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="t:9" class="idref" href="#t:9"><span class="id" title="binder">t</span></a> : <a class="idref" href="IndPrinciples.html#time"><span class="id" title="inductive">time</span></a>, <a class="idref" href="IndPrinciples.html#P:8"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#t:9"><span class="id" title="variable">t</span></a>.<br/>
</div>
<div class="doc">
<a id="lab326"></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">
<span class="id" title="keyword">Inductive</span> <a id="rgb" class="idref" href="#rgb"><span class="id" title="inductive">rgb</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="red" class="idref" href="#red"><span class="id" title="constructor">red</span></a><br/>
| <a id="green" class="idref" href="#green"><span class="id" title="constructor">green</span></a><br/>
| <a id="blue" class="idref" href="#blue"><span class="id" title="constructor">blue</span></a>.<br/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#rgb_ind"><span class="id" title="definition">rgb_ind</span></a>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
Here's another example, this time with one of the constructors
taking some arguments.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="natlist" class="idref" href="#natlist"><span class="id" title="inductive">natlist</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="nnil" class="idref" href="#nnil"><span class="id" title="constructor">nnil</span></a><br/>
| <a id="ncons" class="idref" href="#ncons"><span class="id" title="constructor">ncons</span></a> (<a id="n:14" class="idref" href="#n:14"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:15" class="idref" href="#l:15"><span class="id" title="binder">l</span></a> : <a class="idref" href="IndPrinciples.html#natlist:12"><span class="id" title="inductive">natlist</span></a>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#natlist_ind"><span class="id" title="definition">natlist_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:16" class="idref" href="#P:16"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#natlist"><span class="id" title="inductive">natlist</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="IndPrinciples.html#P:16"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#nnil"><span class="id" title="constructor">nnil</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="n:17" class="idref" href="#n:17"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="l:18" class="idref" href="#l:18"><span class="id" title="binder">l</span></a> : <a class="idref" href="IndPrinciples.html#natlist"><span class="id" title="inductive">natlist</span></a>),<br/>
<a class="idref" href="IndPrinciples.html#P:16"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#l:18"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:16"><span class="id" title="variable">P</span></a> (<a class="idref" href="IndPrinciples.html#ncons"><span class="id" title="constructor">ncons</span></a> <a class="idref" href="IndPrinciples.html#n:17"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#l:18"><span class="id" title="variable">l</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="l:19" class="idref" href="#l:19"><span class="id" title="binder">l</span></a> : <a class="idref" href="IndPrinciples.html#natlist"><span class="id" title="inductive">natlist</span></a>, <a class="idref" href="IndPrinciples.html#P:16"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#l:19"><span class="id" title="variable">l</span></a>.<br/>
</div>
<div class="doc">
In general, the automatically generated induction principle for
inductive type <span class="inlinecode"><span class="id" title="var">t</span></span> is formed as follows:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Each constructor <span class="inlinecode"><span class="id" title="var">c</span></span> generates one case of the principle.
</li>
<li> If <span class="inlinecode"><span class="id" title="var">c</span></span> takes no arguments, that case is:
<br/>
<span class="inlinecode"> "P holds of c"
</span>
</li>
<li> If <span class="inlinecode"><span class="id" title="var">c</span></span> takes arguments <span class="inlinecode"><span class="id" title="var">x<sub>1</sub></span>:<span class="id" title="var">a<sub>1</sub></span></span> ... <span class="inlinecode"><span class="id" title="var">xn</span>:<span class="id" title="var">an</span></span>, that case is:
<br/>
<span class="inlinecode"> "For all x<sub>1</sub>:a<sub>1</sub> ... xn:an,
if [P] holds of each of the arguments of type [t],
then [P] holds of [c x<sub>1</sub> ... xn]"
</span> But that oversimplifies a little. An assumption about <span class="inlinecode"><span class="id" title="var">P</span></span>
holding of an argument <span class="inlinecode"><span class="id" title="var">x</span></span> of type <span class="inlinecode"><span class="id" title="var">t</span></span> actually occurs
immediately after the quantification of <span class="inlinecode"><span class="id" title="var">x</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
For example, suppose we had written the definition of <span class="inlinecode"><span class="id" title="var">natlist</span></span> a little
differently:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="natlist'" class="idref" href="#natlist'"><span class="id" title="inductive">natlist'</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="nnil'" class="idref" href="#nnil'"><span class="id" title="constructor">nnil'</span></a><br/>
| <a id="nsnoc" class="idref" href="#nsnoc"><span class="id" title="constructor">nsnoc</span></a> (<a id="l:22" class="idref" href="#l:22"><span class="id" title="binder">l</span></a> : <a class="idref" href="IndPrinciples.html#natlist':20"><span class="id" title="inductive">natlist'</span></a>) (<a id="n:23" class="idref" href="#n:23"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).<br/>
</div>
<div class="doc">
Now the induction principle case for <span class="inlinecode"><span class="id" title="var">nsnoc1</span></span> is a bit different
than the earlier case for <span class="inlinecode"><span class="id" title="var">ncons</span></span>:
</div>
<div class="code">
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#natlist'_ind"><span class="id" title="definition">natlist'_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:24" class="idref" href="#P:24"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#natlist'"><span class="id" title="inductive">natlist'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="IndPrinciples.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#nnil'"><span class="id" title="constructor">nnil'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="l:25" class="idref" href="#l:25"><span class="id" title="binder">l</span></a> : <a class="idref" href="IndPrinciples.html#natlist'"><span class="id" title="inductive">natlist'</span></a>, <a class="idref" href="IndPrinciples.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#l:25"><span class="id" title="variable">l</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">∀</span> <a id="n:26" class="idref" href="#n:26"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#P:24"><span class="id" title="variable">P</span></a> (<a class="idref" href="IndPrinciples.html#nsnoc"><span class="id" title="constructor">nsnoc</span></a> <a class="idref" href="IndPrinciples.html#l:25"><span class="id" title="variable">l</span></a> <a class="idref" href="IndPrinciples.html#n:26"><span class="id" title="variable">n</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="n:27" class="idref" href="#n:27"><span class="id" title="binder">n</span></a> : <a class="idref" href="IndPrinciples.html#natlist'"><span class="id" title="inductive">natlist'</span></a>, <a class="idref" href="IndPrinciples.html#P:24"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:27"><span class="id" title="variable">n</span></a>.<br/>
</div>
<div class="doc">
<a id="lab327"></a><h4 class="section">Exercise: 2 stars, standard (booltree_ind)</h4>
Here is a type for trees that contain a boolean value at each leaf
and branch.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="booltree" class="idref" href="#booltree"><span class="id" title="inductive">booltree</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="bt_empty" class="idref" href="#bt_empty"><span class="id" title="constructor">bt_empty</span></a><br/>
| <a id="bt_leaf" class="idref" href="#bt_leaf"><span class="id" title="constructor">bt_leaf</span></a> (<a id="b:30" class="idref" href="#b:30"><span class="id" title="binder">b</span></a> : <a class="idref" href="Basics.html#bool"><span class="id" title="inductive">bool</span></a>)<br/>
| <a id="bt_branch" class="idref" href="#bt_branch"><span class="id" title="constructor">bt_branch</span></a> (<a id="b:31" class="idref" href="#b:31"><span class="id" title="binder">b</span></a> : <a class="idref" href="Basics.html#bool"><span class="id" title="inductive">bool</span></a>) (<a id="t<sub>1</sub>:32" class="idref" href="#t<sub>1</sub>:32"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>2</sub>:33" class="idref" href="#t<sub>2</sub>:33"><span class="id" title="binder">t<sub>2</sub></span></a> : <a class="idref" href="IndPrinciples.html#booltree:28"><span class="id" title="inductive">booltree</span></a>).<br/><hr class='doublespaceincode'/>
<span class="comment">(* What is the induction principle for <span class="inlinecode"><span class="id" title="var">booltree</span></span>? Of course you could<br/>
ask Coq, but try not to do that. Instead, write it down yourself on<br/>
paper. Then look at the definition of <span class="inlinecode"><span class="id" title="var">booltree_ind_type</span></span>, below.<br/>
It has three missing pieces, which are provided by the definitions<br/>
in between here and there. Fill in those definitions based on what<br/>
you wrote on paper. *)</span><br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="booltree_property_type" class="idref" href="#booltree_property_type"><span class="id" title="definition">booltree_property_type</span></a> : <span class="id" title="keyword">Type</span> := <a class="idref" href="IndPrinciples.html#booltree"><span class="id" title="inductive">booltree</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="base_case" class="idref" href="#base_case"><span class="id" title="definition">base_case</span></a> (<a id="P:34" class="idref" href="#P:34"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#booltree_property_type"><span class="id" title="definition">booltree_property_type</span></a>) : <span class="id" title="keyword">Prop</span><br/>
<span class="comment">(* REPLACE THIS LINE WITH ":= _your_definition_ ." *)</span>. <span class="id" title="var">Admitted</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="leaf_case" class="idref" href="#leaf_case"><span class="id" title="definition">leaf_case</span></a> (<a id="P:35" class="idref" href="#P:35"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#booltree_property_type"><span class="id" title="definition">booltree_property_type</span></a>) : <span class="id" title="keyword">Prop</span><br/>
<span class="comment">(* REPLACE THIS LINE WITH ":= _your_definition_ ." *)</span>. <span class="id" title="var">Admitted</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="branch_case" class="idref" href="#branch_case"><span class="id" title="definition">branch_case</span></a> (<a id="P:36" class="idref" href="#P:36"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#booltree_property_type"><span class="id" title="definition">booltree_property_type</span></a>) : <span class="id" title="keyword">Prop</span><br/>
<span class="comment">(* REPLACE THIS LINE WITH ":= _your_definition_ ." *)</span>. <span class="id" title="var">Admitted</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="booltree_ind_type" class="idref" href="#booltree_ind_type"><span class="id" title="definition">booltree_ind_type</span></a> :=<br/>
<span class="id" title="keyword">∀</span> (<a id="P:37" class="idref" href="#P:37"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#booltree_property_type"><span class="id" title="definition">booltree_property_type</span></a>),<br/>
<a class="idref" href="IndPrinciples.html#base_case"><span class="id" title="axiom">base_case</span></a> <a class="idref" href="IndPrinciples.html#P:37"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="IndPrinciples.html#leaf_case"><span class="id" title="axiom">leaf_case</span></a> <a class="idref" href="IndPrinciples.html#P:37"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="IndPrinciples.html#branch_case"><span class="id" title="axiom">branch_case</span></a> <a class="idref" href="IndPrinciples.html#P:37"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> (<a id="b:38" class="idref" href="#b:38"><span class="id" title="binder">b</span></a> : <a class="idref" href="IndPrinciples.html#booltree"><span class="id" title="inductive">booltree</span></a>), <a class="idref" href="IndPrinciples.html#P:37"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#b:38"><span class="id" title="variable">b</span></a>.<br/>
</div>
<div class="doc">
Now check the correctness of your answers by proving the following
theorem. If you have them right, you can complete the proof with
just one tactic: <span class="inlinecode"><span class="id" title="tactic">exact</span></span> <span class="inlinecode"><span class="id" title="var">booltree_ind</span></span>. That will work because the
automatically generated induction principle <span class="inlinecode"><span class="id" title="var">booltree_ind</span></span> has the
same type as what you just defined.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="booltree_ind_type_correct" class="idref" href="#booltree_ind_type_correct"><span class="id" title="lemma">booltree_ind_type_correct</span></a> : <a class="idref" href="IndPrinciples.html#booltree_ind_type"><span class="id" title="definition">booltree_ind_type</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab328"></a><h4 class="section">Exercise: 2 stars, standard (toy_ind)</h4>
Here is an induction principle for a toy type:
<br/>
<span class="inlinecode"> <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : <span class="id" title="var">Toy</span> → <span class="id" title="keyword">Prop</span>,<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">b</span> : <span class="id" title="var">bool</span>, <span class="id" title="var">P</span> (<span class="id" title="var">con1</span> <span class="id" title="var">b</span>)) →<br/>
(<span class="id" title="keyword">∀</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">t</span> : <span class="id" title="var">Toy</span>), <span class="id" title="var">P</span> <span class="id" title="var">t</span> → <span class="id" title="var">P</span> (<span class="id" title="var">con2</span> <span class="id" title="var">n</span> <span class="id" title="var">t</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">t</span> : <span class="id" title="var">Toy</span>, <span class="id" title="var">P</span> <span class="id" title="var">t</span>
</span> Give an <span class="inlinecode"><span class="id" title="keyword">Inductive</span></span> definition of <span class="inlinecode"><span class="id" title="var">Toy</span></span>, such that the induction
principle Coq generates is that given above:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="Toy" class="idref" href="#Toy"><span class="id" title="inductive">Toy</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
</div>
<div class="doc">
Show that your definition is correct by proving the following theorem.
You should be able to instantiate <span class="inlinecode"><span class="id" title="var">f</span></span> and <span class="inlinecode"><span class="id" title="var">g</span></span> with your two constructors,
then immediately finish the proof with <span class="inlinecode"><span class="id" title="tactic">exact</span></span> <span class="inlinecode"><span class="id" title="var">Toy_ind</span></span>. As in the previous
exercise, that will work because the automatically generated induction
principle <span class="inlinecode"><span class="id" title="var">Toy_ind</span></span> will have the same type.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="Toy_correct" class="idref" href="#Toy_correct"><span class="id" title="lemma">Toy_correct</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="f:41" class="idref" href="#f:41"><span class="id" title="binder">f</span></a> <a id="g:42" class="idref" href="#g:42"><span class="id" title="binder">g</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="P:43" class="idref" href="#P:43"><span class="id" title="binder">P</span></a> : <a class="idref" href="IndPrinciples.html#Toy"><span class="id" title="inductive">Toy</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="b:44" class="idref" href="#b:44"><span class="id" title="binder">b</span></a> : <a class="idref" href="Basics.html#bool"><span class="id" title="inductive">bool</span></a>, <a class="idref" href="IndPrinciples.html#P:43"><span class="id" title="variable">P</span></a> (<a class="idref" href="IndPrinciples.html#f:41"><span class="id" title="variable">f</span></a> <a class="idref" href="IndPrinciples.html#b:44"><span class="id" title="variable">b</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> (<a id="n:45" class="idref" href="#n:45"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="t:46" class="idref" href="#t:46"><span class="id" title="binder">t</span></a> : <a class="idref" href="IndPrinciples.html#Toy"><span class="id" title="inductive">Toy</span></a>), <a class="idref" href="IndPrinciples.html#P:43"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#t:46"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:43"><span class="id" title="variable">P</span></a> (<a class="idref" href="IndPrinciples.html#g:42"><span class="id" title="variable">g</span></a> <a class="idref" href="IndPrinciples.html#n:45"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#t:46"><span class="id" title="variable">t</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="t:47" class="idref" href="#t:47"><span class="id" title="binder">t</span></a> : <a class="idref" href="IndPrinciples.html#Toy"><span class="id" title="inductive">Toy</span></a>, <a class="idref" href="IndPrinciples.html#P:43"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#t:47"><span class="id" title="variable">t</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="comment">(* FILL IN HERE *)</span> <span class="id" title="var">Admitted</span>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab329"></a><h1 class="section">Polymorphism</h1>
<div class="paragraph"> </div>
What about polymorphic datatypes?
<div class="paragraph"> </div>
The inductive definition of polymorphic lists
<br/>
<span class="inlinecode"> <span class="id" title="keyword">Inductive</span> <span class="id" title="var">list</span> (<span class="id" title="var">X</span>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
| <span class="id" title="var">nil</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span><br/>
| <span class="id" title="var">cons</span> : <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span> → <span class="id" title="var">list</span> <span class="id" title="var">X</span>.
</span> is very similar to that of <span class="inlinecode"><span class="id" title="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" title="var">X</span></span>:
that is, we are defining a <i>family</i> of inductive types <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>,
one for each <span class="inlinecode"><span class="id" title="var">X</span></span>. (Note that, wherever <span class="inlinecode"><span class="id" title="var">list</span></span> appears in the body
of the declaration, it is always applied to the parameter <span class="inlinecode"><span class="id" title="var">X</span></span>.)
<div class="paragraph"> </div>
The induction principle is likewise parameterized on <span class="inlinecode"><span class="id" title="var">X</span></span>:
<br/>
<span class="inlinecode"> <span class="id" title="var">list_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span> → <span class="id" title="keyword">Prop</span>),<br/>
<span class="id" title="var">P</span> [] →<br/>
(<span class="id" title="keyword">∀</span> (<span class="id" title="var">x</span> : <span class="id" title="var">X</span>) (<span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span>), <span class="id" title="var">P</span> <span class="id" title="var">l</span> → <span class="id" title="var">P</span> (<span class="id" title="var">x</span> :: <span class="id" title="var">l</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span>, <span class="id" title="var">P</span> <span class="id" title="var">l</span>
</span> Note that the <i>whole</i> induction principle is parameterized on
<span class="inlinecode"><span class="id" title="var">X</span></span>. That is, <span class="inlinecode"><span class="id" title="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" title="var">X</span></span>, gives us back an
induction principle specialized to the type <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>.
<div class="paragraph"> </div>
<a id="lab330"></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">
<span class="id" title="keyword">Inductive</span> <a id="tree" class="idref" href="#tree"><span class="id" title="inductive">tree</span></a> (<a id="X:48" class="idref" href="#X:48"><span class="id" title="binder">X</span></a>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="leaf" class="idref" href="#leaf"><span class="id" title="constructor">leaf</span></a> (<a id="x:51" class="idref" href="#x:51"><span class="id" title="binder">x</span></a> : <a class="idref" href="IndPrinciples.html#X:48"><span class="id" title="variable">X</span></a>)<br/>
| <a id="node" class="idref" href="#node"><span class="id" title="constructor">node</span></a> (<a id="t<sub>1</sub>:52" class="idref" href="#t<sub>1</sub>:52"><span class="id" title="binder">t<sub>1</sub></span></a> <a id="t<sub>2</sub>:53" class="idref" href="#t<sub>2</sub>:53"><span class="id" title="binder">t<sub>2</sub></span></a> : <a class="idref" href="IndPrinciples.html#tree:49"><span class="id" title="inductive">tree</span></a> <a class="idref" href="IndPrinciples.html#X:48"><span class="id" title="variable">X</span></a>).<br/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#tree_ind"><span class="id" title="definition">tree_ind</span></a>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<div class="paragraph"> </div>
<a id="lab331"></a><h4 class="section">Exercise: 1 star, standard, optional (mytype)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<br/>
<span class="inlinecode"> <span class="id" title="var">mytype_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">mytype</span> <span class="id" title="var">X</span> → <span class="id" title="keyword">Prop</span>),<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">X</span>, <span class="id" title="var">P</span> (<span class="id" title="var">constr1</span> <span class="id" title="var">X</span> <span class="id" title="var">x</span>)) →<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">P</span> (<span class="id" title="var">constr2</span> <span class="id" title="var">X</span> <span class="id" title="var">n</span>)) →<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> : <span class="id" title="var">mytype</span> <span class="id" title="var">X</span>, <span class="id" title="var">P</span> <span class="id" title="var">m</span> →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">P</span> (<span class="id" title="var">constr3</span> <span class="id" title="var">X</span> <span class="id" title="var">m</span> <span class="id" title="var">n</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> : <span class="id" title="var">mytype</span> <span class="id" title="var">X</span>, <span class="id" title="var">P</span> <span class="id" title="var">m</span>
</span> <font size=-2>☐</font>
<div class="paragraph"> </div>
<a id="lab332"></a><h4 class="section">Exercise: 1 star, standard, optional (foo)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<br/>
<span class="inlinecode"> <span class="id" title="var">foo_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">X</span> <span class="id" title="var">Y</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">foo</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> → <span class="id" title="keyword">Prop</span>),<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <span class="id" title="var">X</span>, <span class="id" title="var">P</span> (<span class="id" title="var">bar</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">x</span>)) →<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">y</span> : <span class="id" title="var">Y</span>, <span class="id" title="var">P</span> (<span class="id" title="var">baz</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">y</span>)) →<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">f<sub>1</sub></span> : <span class="id" title="var">nat</span> → <span class="id" title="var">foo</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span>,<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">P</span> (<span class="id" title="var">f<sub>1</sub></span> <span class="id" title="var">n</span>)) → <span class="id" title="var">P</span> (<span class="id" title="var">quux</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span> <span class="id" title="var">f<sub>1</sub></span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">f<sub>2</sub></span> : <span class="id" title="var">foo</span> <span class="id" title="var">X</span> <span class="id" title="var">Y</span>, <span class="id" title="var">P</span> <span class="id" title="var">f<sub>2</sub></span>
</span> <font size=-2>☐</font>
<div class="paragraph"> </div>
<a id="lab333"></a><h4 class="section">Exercise: 1 star, standard, optional (foo')</h4>
Consider the following inductive definition:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="foo'" class="idref" href="#foo'"><span class="id" title="inductive">foo'</span></a> (<a id="X:54" class="idref" href="#X:54"><span class="id" title="binder">X</span></a>:<span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> :=<br/>
| <a id="C<sub>1</sub>" class="idref" href="#C<sub>1</sub>"><span class="id" title="constructor">C<sub>1</sub></span></a> (<a id="l:57" class="idref" href="#l:57"><span class="id" title="binder">l</span></a> : <a class="idref" href="Poly.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="IndPrinciples.html#X:54"><span class="id" title="variable">X</span></a>) (<a id="f:58" class="idref" href="#f:58"><span class="id" title="binder">f</span></a> : <a class="idref" href="IndPrinciples.html#foo':55"><span class="id" title="inductive">foo'</span></a> <a class="idref" href="IndPrinciples.html#X:54"><span class="id" title="variable">X</span></a>)<br/>
| <a id="C<sub>2</sub>" class="idref" href="#C<sub>2</sub>"><span class="id" title="constructor">C<sub>2</sub></span></a>.<br/>
</div>
<div class="doc">
What induction principle will Coq generate for <span class="inlinecode"><span class="id" title="var">foo'</span></span>? (Fill
in the blanks, then check your answer with Coq.)
<br/>
<span class="inlinecode"> <span class="id" title="var">foo'_ind</span> :<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">X</span> : <span class="id" title="keyword">Type</span>) (<span class="id" title="var">P</span> : <span class="id" title="var">foo'</span> <span class="id" title="var">X</span> → <span class="id" title="keyword">Prop</span>),<br/>
(<span class="id" title="keyword">∀</span> (<span class="id" title="var">l</span> : <span class="id" title="var">list</span> <span class="id" title="var">X</span>) (<span class="id" title="var">f</span> : <span class="id" title="var">foo'</span> <span class="id" title="var">X</span>),<br/>
<span class="id" title="var">_______________________</span> →<br/>
<span class="id" title="var">_______________________</span> ) →<br/>
<span class="id" title="var">___________________________________________</span> →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">f</span> : <span class="id" title="var">foo'</span> <span class="id" title="var">X</span>, <span class="id" title="var">________________________</span>
</span>
<div class="paragraph"> </div>
<font size=-2>☐</font>
</div>
<div class="doc">
<a id="lab334"></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
<br/>
<span class="inlinecode"> <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : <span class="id" title="var">nat</span> → <span class="id" title="keyword">Prop</span>,<br/>
<span class="id" title="var">P</span> 0 →<br/>
(<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">P</span> <span class="id" title="var">n</span> → <span class="id" title="var">P</span> (<span class="id" title="var">S</span> <span class="id" title="var">n</span>)) →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">P</span> <span class="id" title="var">n</span>
</span> is a generic statement that holds for all propositions
<span class="inlinecode"><span class="id" title="var">P</span></span> (or rather, strictly speaking, for all families of
propositions <span class="inlinecode"><span class="id" title="var">P</span></span> indexed by a number <span class="inlinecode"><span class="id" title="var">n</span></span>). Each time we
use this principle, we are choosing <span class="inlinecode"><span class="id" title="var">P</span></span> to be a particular
expression of type <span class="inlinecode"><span class="id" title="var">nat</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="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" title="var">mul_0_r</span></span> as "<span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">n</span>,</span> <span class="inlinecode"><span class="id" title="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 class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">n</span>,</span> <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>", where <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> is defined
as...
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="P_m0r" class="idref" href="#P_m0r"><span class="id" title="definition">P_m0r</span></a> (<a id="n:59" class="idref" href="#n:59"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <span class="id" title="keyword">Prop</span> :=<br/>
<a class="idref" href="IndPrinciples.html#n:59"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#ea2ff3d561159081cea6fb2e8113cc<sub>54</sub>"><span class="id" title="notation">×</span></a> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
</div>
<div class="doc">
... or equivalently:
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a id="P_m0r'" class="idref" href="#P_m0r'"><span class="id" title="definition">P_m0r'</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
<span class="id" title="keyword">fun</span> <a id="n:60" class="idref" href="#n:60"><span class="id" title="binder">n</span></a> ⇒ <a class="idref" href="IndPrinciples.html#n:60"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#ea2ff3d561159081cea6fb2e8113cc<sub>54</sub>"><span class="id" title="notation">×</span></a> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 0.<br/>
</div>
<div class="doc">
Now it is easier to see where <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> appears in the proof.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="mul_0_r''" class="idref" href="#mul_0_r''"><span class="id" title="lemma">mul_0_r''</span></a> : <span class="id" title="keyword">∀</span> <a id="n:61" class="idref" href="#n:61"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" title="definition">P_m0r</span></a> <a class="idref" href="IndPrinciples.html#n:61"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat_ind"><span class="id" title="definition">nat_ind</span></a>.<br/>
- <span class="comment">(* n = O *)</span> <span class="id" title="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" title="tactic">intros</span> <span class="id" title="var">n</span> <span class="id" title="var">IHn</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" title="definition">P_m0r</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">IHn</span>. <span class="id" title="tactic">unfold</span> <a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" title="definition">P_m0r</span></a>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IHn</span>. <span class="id" title="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 class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">n</span>,</span> <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> by induction on
<span class="inlinecode"><span class="id" title="var">n</span></span> (using either <span class="inlinecode"><span class="id" title="tactic">induction</span></span> or <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">nat_ind</span></span>), we see that the
first subgoal requires us to prove <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> <span class="inlinecode">0</span> ("<span class="inlinecode"><span class="id" title="var">P</span></span> holds for
zero"), while the second subgoal requires us to prove <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">n'</span>,</span>
<span class="inlinecode"><span class="id" title="var">P_m0r</span></span> <span class="inlinecode"><span class="id" title="var">n'</span></span> <span class="inlinecode">→</span> <span class="inlinecode"><span class="id" title="var">P_m0r</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n'</span>)</span> (that is "<span class="inlinecode"><span class="id" title="var">P</span></span> holds of <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n'</span></span> if it
holds of <span class="inlinecode"><span class="id" title="var">n'</span></span>" or, more elegantly, "<span class="inlinecode"><span class="id" title="var">P</span></span> is preserved by <span class="inlinecode"><span class="id" title="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" title="var">P</span></span> holds of <span class="inlinecode"><span class="id" title="var">n'</span></span>, which we are
allowed to use in proving that <span class="inlinecode"><span class="id" title="var">P</span></span> holds for <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n'</span></span>.
</div>
<div class="doc">
<a id="lab335"></a><h1 class="section">More on the <span class="inlinecode"><span class="id" title="tactic">induction</span></span> Tactic</h1>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="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:
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="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" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">O</span></span> holds
</li>
<li> show that, if <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">n'</span></span> holds, then so does <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n'</span>)</span>
</li>
<li> conclude that <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="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" title="tactic">intros</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> and then <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span>,
we are first telling Coq to consider a <i>particular</i> <span class="inlinecode"><span class="id" title="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 it
"re-generalizes" the variable we perform induction on. For
example, in our original proof that <span class="inlinecode"><span class="id" title="var">plus</span></span> is associative...
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="add_assoc'" class="idref" href="#add_assoc'"><span class="id" title="lemma">add_assoc'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:62" class="idref" href="#n:62"><span class="id" title="binder">n</span></a> <a id="m:63" class="idref" href="#m:63"><span class="id" title="binder">m</span></a> <a id="p:64" class="idref" href="#p:64"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="IndPrinciples.html#m:63"><span class="id" title="variable">m</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#p:64"><span class="id" title="variable">p</span></a><a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="IndPrinciples.html#n:62"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#m:63"><span class="id" title="variable">m</span></a><a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#p:64"><span class="id" title="variable">p</span></a>.<br/>
<span class="id" title="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" title="var">n</span></span>, <span class="inlinecode"><span class="id" title="var">m</span></span>, and<br/>
<span class="inlinecode"><span class="id" title="var">p</span></span>..." *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span> <span class="id" title="var">p</span>.<br/>
<span class="comment">(* ...We now use the <span class="inlinecode"><span class="id" title="tactic">induction</span></span> tactic to prove <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> (that<br/>
is, <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" title="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">p</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">(<span class="id" title="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">m</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">p</span></span>) for _all_ <span class="inlinecode"><span class="id" title="var">n</span></span>,<br/>
and hence also for the particular <span class="inlinecode"><span class="id" title="var">n</span></span> that is in the context<br/>
at the moment. *)</span><br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> → <span class="id" title="var">IHn'</span>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
It also works to apply <span class="inlinecode"><span class="id" title="tactic">induction</span></span> to a variable that is
quantified in the goal.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="add_comm'" class="idref" href="#add_comm'"><span class="id" title="lemma">add_comm'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:65" class="idref" href="#n:65"><span class="id" title="binder">n</span></a> <a id="m:66" class="idref" href="#m:66"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n:65"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#m:66"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="IndPrinciples.html#m:66"><span class="id" title="variable">m</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#n:65"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">n</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" title="tactic">intros</span> <span class="id" title="var">m</span>. <span class="id" title="tactic">rewrite</span> → <a class="idref" href="Induction.html#add_0_r"><span class="id" title="lemma">add_0_r</span></a>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" title="tactic">intros</span> <span class="id" title="var">m</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> → <span class="id" title="var">IHn'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="Induction.html#plus_n_Sm"><span class="id" title="axiom">plus_n_Sm</span></a>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Note that <span class="inlinecode"><span class="id" title="tactic">induction</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> leaves <span class="inlinecode"><span class="id" title="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 class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span>.
<div class="paragraph"> </div>
If we do <span class="inlinecode"><span class="id" title="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" title="tactic">induction</span></span> tactic will
automatically introduce the variables bound by these quantifiers
into the context.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="add_comm''" class="idref" href="#add_comm''"><span class="id" title="lemma">add_comm''</span></a> : <span class="id" title="keyword">∀</span> <a id="n:67" class="idref" href="#n:67"><span class="id" title="binder">n</span></a> <a id="m:68" class="idref" href="#m:68"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n:67"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#m:68"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="IndPrinciples.html#m:68"><span class="id" title="variable">m</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#n:67"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="comment">(* Let's do induction on <span class="inlinecode"><span class="id" title="var">m</span></span> this time, instead of <span class="inlinecode"><span class="id" title="var">n</span></span>... *)</span><br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">m</span> <span class="id" title="keyword">as</span> [| <span class="id" title="var">m'</span>]. <span class="comment">(* <span class="inlinecode"><span class="id" title="var">n</span></span> is already introduced into the context *)</span><br/>
- <span class="comment">(* m = O *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> → <a class="idref" href="Induction.html#add_0_r"><span class="id" title="lemma">add_0_r</span></a>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* m = S m' *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">IHm'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="Induction.html#plus_n_Sm"><span class="id" title="axiom">plus_n_Sm</span></a>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab336"></a><h4 class="section">Exercise: 1 star, standard, optional (plus_explicit_prop)</h4>
Rewrite both <span class="inlinecode"><span class="id" title="var">add_assoc'</span></span> and <span class="inlinecode"><span class="id" title="var">add_comm'</span></span> and their proofs in
the same style as <span class="inlinecode"><span class="id" title="var">mul_0_r''</span></span> above -- that is, for each theorem,
give an explicit <span class="inlinecode"><span class="id" title="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">
<span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<a id="lab337"></a><h1 class="section">Induction Principles for Propositions</h1>
<div class="paragraph"> </div>
Inductive definitions of propositions also cause Coq to generate
induction priniciples. For example, recall our proposition <span class="inlinecode"><span class="id" title="var">ev</span></span>
from <a href="IndProp.html"><span class="inlineref">IndProp</span></a>:
</div>
<div class="code">
<span class="id" title="keyword">Print</span> <a class="idref" href="ProofObjects.html#ev"><span class="id" title="inductive">ev</span></a>.<br/><hr class='doublespaceincode'/>
<span class="comment">(* ===><br/>
<br/>
Inductive ev : 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/>
| ev_0 : ev 0<br/>
| ev_SS : forall n : nat, ev n <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> ev (S (S n)))<br/>
<br/>
*)</span><br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <a class="idref" href="ProofObjects.html#ev_ind"><span class="id" title="definition">ev_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:69" class="idref" href="#P:69"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="IndPrinciples.html#P:69"><span class="id" title="variable">P</span></a> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:70" class="idref" href="#n:70"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="ProofObjects.html#ev"><span class="id" title="inductive">ev</span></a> <a class="idref" href="IndPrinciples.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:69"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:70"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:69"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#n:70"><span class="id" title="variable">n</span></a>))<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="n:71" class="idref" href="#n:71"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="ProofObjects.html#ev"><span class="id" title="inductive">ev</span></a> <a class="idref" href="IndPrinciples.html#n:71"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:69"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:71"><span class="id" title="variable">n</span></a>.<br/>
</div>
<div class="doc">
In English, <span class="inlinecode"><span class="id" title="var">ev_ind</span></span> says: Suppose <span class="inlinecode"><span class="id" title="var">P</span></span> is a property of natural
numbers. To show that <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">n</span></span> holds whenever <span class="inlinecode"><span class="id" title="var">n</span></span> is even, it suffices
to show:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="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" title="var">n</span></span>, if <span class="inlinecode"><span class="id" title="var">n</span></span> is even and <span class="inlinecode"><span class="id" title="var">P</span></span> holds for <span class="inlinecode"><span class="id" title="var">n</span></span>, then <span class="inlinecode"><span class="id" title="var">P</span></span>
holds for <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span>.
</li>
</ul>
<div class="paragraph"> </div>
As expected, we can apply <span class="inlinecode"><span class="id" title="var">ev_ind</span></span> directly instead of using
<span class="inlinecode"><span class="id" title="tactic">induction</span></span>. For example, we can use it to show that <span class="inlinecode"><span class="id" title="var">ev'</span></span> (the
slightly awkward alternate definition of evenness that we saw in
an exercise in the <a href="IndProp.html"><span class="inlineref">IndProp</span></a> chapter) is equivalent to the
cleaner inductive definition <span class="inlinecode"><span class="id" title="var">ev</span></span>:
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="ev'" class="idref" href="#ev'"><span class="id" title="inductive">ev'</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="ev'_0" class="idref" href="#ev'_0"><span class="id" title="constructor">ev'_0</span></a> : <a class="idref" href="IndPrinciples.html#ev':72"><span class="id" title="inductive">ev'</span></a> 0<br/>
| <a id="ev'_2" class="idref" href="#ev'_2"><span class="id" title="constructor">ev'_2</span></a> : <a class="idref" href="IndPrinciples.html#ev':72"><span class="id" title="inductive">ev'</span></a> 2<br/>
| <a id="ev'_sum" class="idref" href="#ev'_sum"><span class="id" title="constructor">ev'_sum</span></a> <a id="n:74" class="idref" href="#n:74"><span class="id" title="binder">n</span></a> <a id="m:75" class="idref" href="#m:75"><span class="id" title="binder">m</span></a> (<a id="Hn:76" class="idref" href="#Hn:76"><span class="id" title="binder">Hn</span></a> : <a class="idref" href="IndPrinciples.html#ev':72"><span class="id" title="inductive">ev'</span></a> <a class="idref" href="IndPrinciples.html#n:74"><span class="id" title="variable">n</span></a>) (<a id="Hm:77" class="idref" href="#Hm:77"><span class="id" title="binder">Hm</span></a> : <a class="idref" href="IndPrinciples.html#ev':72"><span class="id" title="inductive">ev'</span></a> <a class="idref" href="IndPrinciples.html#m:75"><span class="id" title="variable">m</span></a>) : <a class="idref" href="IndPrinciples.html#ev':72"><span class="id" title="inductive">ev'</span></a> (<a class="idref" href="IndPrinciples.html#n:74"><span class="id" title="variable">n</span></a> <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="IndPrinciples.html#m:75"><span class="id" title="variable">m</span></a>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="ev_ev'" class="idref" href="#ev_ev'"><span class="id" title="lemma">ev_ev'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:78" class="idref" href="#n:78"><span class="id" title="binder">n</span></a>, <a class="idref" href="ProofObjects.html#ev"><span class="id" title="inductive">ev</span></a> <a class="idref" href="IndPrinciples.html#n:78"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#ev'"><span class="id" title="inductive">ev'</span></a> <a class="idref" href="IndPrinciples.html#n:78"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="ProofObjects.html#ev_ind"><span class="id" title="definition">ev_ind</span></a>.<br/>
- <span class="comment">(* ev_0 *)</span><br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="IndPrinciples.html#ev'_0"><span class="id" title="constructor">ev'_0</span></a>.<br/>
- <span class="comment">(* ev_SS *)</span><br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">m</span> <span class="id" title="var">Hm</span> <span class="id" title="var">IH</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="IndPrinciples.html#ev'_sum"><span class="id" title="constructor">ev'_sum</span></a> 2 <span class="id" title="var">m</span>).<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndPrinciples.html#ev'_2"><span class="id" title="constructor">ev'_2</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The precise form of an <span class="inlinecode"><span class="id" title="keyword">Inductive</span></span> definition can affect the
induction principle Coq generates.
</div>
<div class="code">
<span class="id" title="keyword">Inductive</span> <a id="le<sub>1</sub>" class="idref" href="#le<sub>1</sub>"><span class="id" title="inductive">le<sub>1</sub></span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="le1_n" class="idref" href="#le1_n"><span class="id" title="constructor">le1_n</span></a> : <span class="id" title="keyword">∀</span> <a id="n:81" class="idref" href="#n:81"><span class="id" title="binder">n</span></a>, <a class="idref" href="IndPrinciples.html#le<sub>1</sub>:79"><span class="id" title="inductive">le<sub>1</sub></span></a> <a class="idref" href="IndPrinciples.html#n:81"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#n:81"><span class="id" title="variable">n</span></a><br/>
| <a id="le1_S" class="idref" href="#le1_S"><span class="id" title="constructor">le1_S</span></a> : <span class="id" title="keyword">∀</span> <a id="n:82" class="idref" href="#n:82"><span class="id" title="binder">n</span></a> <a id="m:83" class="idref" href="#m:83"><span class="id" title="binder">m</span></a>, <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="IndPrinciples.html#le<sub>1</sub>:79"><span class="id" title="inductive">le<sub>1</sub></span></a> <a class="idref" href="IndPrinciples.html#n:82"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#m:83"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><a class="idref" href="IndPrinciples.html#le<sub>1</sub>:79"><span class="id" title="inductive">le<sub>1</sub></span></a> <a class="idref" href="IndPrinciples.html#n:82"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#m:83"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="dcb49d48dea0c82eb271ef74f5398ffa" class="idref" href="#dcb49d48dea0c82eb271ef74f5398ffa"><span class="id" title="notation">"</span></a>m <=1 n" := (<a class="idref" href="IndPrinciples.html#le<sub>1</sub>"><span class="id" title="inductive">le<sub>1</sub></span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<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" title="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">
<span class="id" title="keyword">Inductive</span> <a id="le<sub>2</sub>" class="idref" href="#le<sub>2</sub>"><span class="id" title="inductive">le<sub>2</sub></span></a> (<a id="n:84" class="idref" href="#n:84"><span class="id" title="binder">n</span></a>:<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
| <a id="le2_n" class="idref" href="#le2_n"><span class="id" title="constructor">le2_n</span></a> : <a class="idref" href="IndPrinciples.html#le<sub>2</sub>:85"><span class="id" title="inductive">le<sub>2</sub></span></a> <a class="idref" href="IndPrinciples.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#n:84"><span class="id" title="variable">n</span></a><br/>
| <a id="le2_S" class="idref" href="#le2_S"><span class="id" title="constructor">le2_S</span></a> <a id="m:87" class="idref" href="#m:87"><span class="id" title="binder">m</span></a> (<a id="H:88" class="idref" href="#H:88"><span class="id" title="binder">H</span></a> : <a class="idref" href="IndPrinciples.html#le<sub>2</sub>:85"><span class="id" title="inductive">le<sub>2</sub></span></a> <a class="idref" href="IndPrinciples.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#m:87"><span class="id" title="variable">m</span></a>) : <a class="idref" href="IndPrinciples.html#le<sub>2</sub>:85"><span class="id" title="inductive">le<sub>2</sub></span></a> <a class="idref" href="IndPrinciples.html#n:84"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#m:87"><span class="id" title="variable">m</span></a>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Notation</span> <a id="b22f9d37eb3ba29bb4f11d08927ef059" class="idref" href="#b22f9d37eb3ba29bb4f11d08927ef059"><span class="id" title="notation">"</span></a>m <=2 n" := (<a class="idref" href="IndPrinciples.html#le<sub>2</sub>"><span class="id" title="inductive">le<sub>2</sub></span></a> <span class="id" title="var">m</span> <span class="id" title="var">n</span>) (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<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">
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#le1_ind"><span class="id" title="definition">le1_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> <a id="P:89" class="idref" href="#P:89"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>,<br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:90" class="idref" href="#n:90"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:90"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#n:90"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="n:91" class="idref" href="#n:91"><span class="id" title="binder">n</span></a> <a id="m:92" class="idref" href="#m:92"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#n:91"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#dcb49d48dea0c82eb271ef74f5398ffa"><span class="id" title="notation"><=1</span></a> <a class="idref" href="IndPrinciples.html#m:92"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:91"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#m:92"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:91"><span class="id" title="variable">n</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#m:92"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="n:93" class="idref" href="#n:93"><span class="id" title="binder">n</span></a> <a id="n<sub>0</sub>:94" class="idref" href="#n<sub>0</sub>:94"><span class="id" title="binder">n<sub>0</sub></span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#n:93"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#dcb49d48dea0c82eb271ef74f5398ffa"><span class="id" title="notation"><=1</span></a> <a class="idref" href="IndPrinciples.html#n<sub>0</sub>:94"><span class="id" title="variable">n<sub>0</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:89"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:93"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#n<sub>0</sub>:94"><span class="id" title="variable">n<sub>0</sub></span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndPrinciples.html#le2_ind"><span class="id" title="definition">le2_ind</span></a> :<br/>
<span class="id" title="keyword">∀</span> (<a id="n:95" class="idref" href="#n:95"><span class="id" title="binder">n</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) (<a id="P:96" class="idref" href="#P:96"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
<a class="idref" href="IndPrinciples.html#P:96"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n:95"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="m:97" class="idref" href="#m:97"><span class="id" title="binder">m</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#n:95"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#b22f9d37eb3ba29bb4f11d08927ef059"><span class="id" title="notation"><=2</span></a> <a class="idref" href="IndPrinciples.html#m:97"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:96"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#m:97"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:96"><span class="id" title="variable">P</span></a> (<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#m:97"><span class="id" title="variable">m</span></a>)<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <a id="n<sub>0</sub>:98" class="idref" href="#n<sub>0</sub>:98"><span class="id" title="binder">n<sub>0</sub></span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="IndPrinciples.html#n:95"><span class="id" title="variable">n</span></a> <a class="idref" href="IndPrinciples.html#b22f9d37eb3ba29bb4f11d08927ef059"><span class="id" title="notation"><=2</span></a> <a class="idref" href="IndPrinciples.html#n<sub>0</sub>:98"><span class="id" title="variable">n<sub>0</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'->'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="IndPrinciples.html#P:96"><span class="id" title="variable">P</span></a> <a class="idref" href="IndPrinciples.html#n<sub>0</sub>:98"><span class="id" title="variable">n<sub>0</sub></span></a>.<br/>
</div>
<div class="doc">
<a id="lab338"></a><h1 class="section">Another Form of Induction Principles on Propositions (Optional)</h1>
<div class="paragraph"> </div>
The induction principle that Coq generated for <span class="inlinecode"><span class="id" title="var">ev</span></span> was parameterized
on a natural number <span class="inlinecode"><span class="id" title="var">n</span></span>. It could have additionally been parameterized
on the evidence that <span class="inlinecode"><span class="id" title="var">n</span></span> was even, which would have led to this
induction principle:
<br/>
<span class="inlinecode"> <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : (<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>, <span class="id" title="var">ev''</span> <span class="id" title="var">n</span> → <span class="id" title="keyword">Prop</span>),<br/>
<span class="id" title="var">P</span> <span class="id" title="var">O</span> <span class="id" title="var">ev_0</span> →<br/>
(<span class="id" title="keyword">∀</span> (<span class="id" title="var">m</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">E</span> : <span class="id" title="var">ev''</span> <span class="id" title="var">m</span>),<br/>
<span class="id" title="var">P</span> <span class="id" title="var">m</span> <span class="id" title="var">E</span> → <span class="id" title="var">P</span> (<span class="id" title="var">S</span> (<span class="id" title="var">S</span> <span class="id" title="var">m</span>)) (<span class="id" title="var">ev_SS</span> <span class="id" title="var">m</span> <span class="id" title="var">E</span>)) →<br/>
<span class="id" title="keyword">∀</span> (<span class="id" title="var">n</span> : <span class="id" title="var">nat</span>) (<span class="id" title="var">E</span> : <span class="id" title="var">ev''</span> <span class="id" title="var">n</span>), <span class="id" title="var">P</span> <span class="id" title="var">n</span> <span class="id" title="var">E</span>
</span>
<div class="paragraph"> </div>
... because:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Since <span class="inlinecode"><span class="id" title="var">ev</span></span> is indexed by a number <span class="inlinecode"><span class="id" title="var">n</span></span> (every <span class="inlinecode"><span class="id" title="var">ev</span></span> object <span class="inlinecode"><span class="id" title="var">E</span></span> is
a piece of evidence that some particular number <span class="inlinecode"><span class="id" title="var">n</span></span> is even),
the proposition <span class="inlinecode"><span class="id" title="var">P</span></span> is parameterized by both <span class="inlinecode"><span class="id" title="var">n</span></span> and <span class="inlinecode"><span class="id" title="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" title="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" title="var">P</span></span> holds for <span class="inlinecode"><span class="id" title="var">O</span></span> and <span class="inlinecode"><span class="id" title="var">ev_0</span></span>.
<div class="paragraph"> </div>
</li>
<li> We must prove that, whenever <span class="inlinecode"><span class="id" title="var">m</span></span> is an even number and <span class="inlinecode"><span class="id" title="var">E</span></span>
is an evidence of its evenness, if <span class="inlinecode"><span class="id" title="var">P</span></span> holds of <span class="inlinecode"><span class="id" title="var">m</span></span> and
<span class="inlinecode"><span class="id" title="var">E</span></span>, then it also holds of <span class="inlinecode"><span class="id" title="var">S</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">m</span>)</span> and <span class="inlinecode"><span class="id" title="var">ev_SS</span></span> <span class="inlinecode"><span class="id" title="var">m</span></span> <span class="inlinecode"><span class="id" title="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" title="var">P</span></span> is true for <i>all</i> even numbers <span class="inlinecode"><span class="id" title="var">n</span></span> and
evidence <span class="inlinecode"><span class="id" title="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" title="var">P</span></span> that are
parameterized just by <span class="inlinecode"><span class="id" title="var">n</span></span> and whose conclusion establishes <span class="inlinecode"><span class="id" title="var">P</span></span> for
all even numbers <span class="inlinecode"><span class="id" title="var">n</span></span>:
<br/>
<span class="inlinecode"> <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : <span class="id" title="var">nat</span> → <span class="id" title="keyword">Prop</span>,<br/>
... →<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">n</span> : <span class="id" title="var">nat</span>,<br/>
<span class="id" title="var">even</span> <span class="id" title="var">n</span> → <span class="id" title="var">P</span> <span class="id" title="var">n</span>
</span> That is why Coq actually generates the induction principle
<span class="inlinecode"><span class="id" title="var">ev_ind</span></span> that we saw before.
</div>
<div class="doc">
<a id="lab339"></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" title="var">P</span></span> and an informal proof of the same proposition <span class="inlinecode"><span class="id" title="var">P</span></span>?
<div class="paragraph"> </div>
Answer: The latter should <i>teach</i> the reader everything they would
need to understand to be able to produce the former.
<div class="paragraph"> </div>
Question: How much detail does that require?
<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" title="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" title="keyword">Prop</span></span>).
<div class="paragraph"> </div>
<a id="lab340"></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" title="var">n</span>:<span class="id" title="var">S</span></span>, <span class="inlinecode"><span class="id" title="var">P</span>(<span class="id" title="var">n</span>)</span>," where <span class="inlinecode"><span class="id" title="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" title="var">n</span></span>.
<div class="paragraph"> </div>
<one case for each constructor <span class="inlinecode"><span class="id" title="var">c</span></span> of <span class="inlinecode"><span class="id" title="var">S</span></span>...>
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" title="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">a<sub>1</sub></span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" title="var">ak</span></span>, where <...and here we state
the IH for each of the <span class="inlinecode"><span class="id" title="var">a</span></span>'s that has type <span class="inlinecode"><span class="id" title="var">S</span></span>, if any>.
We must show <...and here we restate <span class="inlinecode"><span class="id" title="var">P</span>(<span class="id" title="var">c</span></span> <span class="inlinecode"><span class="id" title="var">a<sub>1</sub></span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" title="var">ak</span>)</span>>.
<div class="paragraph"> </div>
<go on and prove <span class="inlinecode"><span class="id" title="var">P</span>(<span class="id" title="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" title="var">X</span></span>, lists <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" title="var">list</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>, and numbers
<span class="inlinecode"><span class="id" title="var">n</span></span>, if <span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">n</span></span> then <span class="inlinecode"><span class="id" title="var">index</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span> <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">None</span></span>.
<div class="paragraph"> </div>
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" title="var">l</span></span>.
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode">[]</span>. We must show, for all numbers <span class="inlinecode"><span class="id" title="var">n</span></span>,
that, if <span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode">[]</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">n</span></span>, then <span class="inlinecode"><span class="id" title="var">index</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span> <span class="inlinecode">[]</span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" title="var">None</span></span>.
<div class="paragraph"> </div>
This follows immediately from the definition of <span class="inlinecode"><span class="id" title="var">index</span></span>.
<div class="paragraph"> </div>
</li>
<li> Suppose <span class="inlinecode"><span class="id" title="var">l</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">::</span> <span class="inlinecode"><span class="id" title="var">l'</span></span> for some <span class="inlinecode"><span class="id" title="var">x</span></span> and <span class="inlinecode"><span class="id" title="var">l'</span></span>, where
<span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode"><span class="id" title="var">l'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">n'</span></span> implies <span class="inlinecode"><span class="id" title="var">index</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n'</span>)</span> <span class="inlinecode"><span class="id" title="var">l'</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">None</span></span>, for
any number <span class="inlinecode"><span class="id" title="var">n'</span></span>. We must show, for all <span class="inlinecode"><span class="id" title="var">n</span></span>, that, if
<span class="inlinecode"><span class="id" title="var">length</span></span> <span class="inlinecode">(<span class="id" title="var">x</span>::<span class="id" title="var">l'</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">n</span></span> then <span class="inlinecode"><span class="id" title="var">index</span></span> <span class="inlinecode">(<span class="id" title="var">S</span></span> <span class="inlinecode"><span class="id" title="var">n</span>)</span> <span class="inlinecode">(<span class="id" title="var">x</span>::<span class="id" title="var">l'</span>)</span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" title="var">None</span></span>.
<div class="paragraph"> </div>