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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<title>Rel: Properties of Relations</title>
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</br><a href='index.html'> <span class='booktitleinheader'>Volume 1: Logical Foundations</span><br></br>
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<h1 class="libtitle">Rel<span class="subtitle">Properties of Relations</span></h1>
<div class="doc">
<div class="paragraph"> </div>
This short (and optional) chapter develops some basic definitions
and a few theorems about binary relations in Coq. The key
definitions are repeated where they are actually used (in the
<a href="https://www.cis.upenn.edu/~bcpierce/sf/plf-current/Smallstep.html"><span class="inlineref">Smallstep</span></a> chapter of <i>Programming Language Foundations</i>),
so readers who are already comfortable with these ideas can safely
skim or skip this chapter. However, relations are also a good
source of exercises for developing facility with Coq's basic
reasoning facilities, so it may be useful to look at this material
just after the <span class="inlinecode"><span class="id" type="var">IndProp</span></span> chapter.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Set</span> <span class="id" type="var">Warnings</span> "-notation-overridden,-parsing".<br/>
<span class="id" type="var">From</span> <span class="id" type="var">LF</span> <span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">IndProp</span>.<br/>
</div>
<div class="doc">
<a name="lab299"></a><h1 class="section">Relations</h1>
<div class="paragraph"> </div>
A binary <i>relation</i> on a set <span class="inlinecode"><span class="id" type="var">X</span></span> is a family of propositions
parameterized by two elements of <span class="inlinecode"><span class="id" type="var">X</span></span> — i.e., a proposition about
pairs of elements of <span class="inlinecode"><span class="id" type="var">X</span></span>.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">relation</span> (<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>) := <span class="id" type="var">X</span> → <span class="id" type="var">X</span> → <span class="id" type="keyword">Prop</span>.<br/>
</div>
<div class="doc">
Confusingly, the Coq standard library hijacks the generic term
"relation" for this specific instance of the idea. To maintain
consistency with the library, we will do the same. So, henceforth
the Coq identifier <span class="inlinecode"><span class="id" type="var">relation</span></span> will always refer to a binary
relation between some set and itself, whereas the English word
"relation" can refer either to the specific Coq concept or the
more general concept of a relation between any number of possibly
different sets. The context of the discussion should always make
clear which is meant.
<div class="paragraph"> </div>
An example relation on <span class="inlinecode"><span class="id" type="var">nat</span></span> is <span class="inlinecode"><span class="id" type="var">le</span></span>, the less-than-or-equal-to
relation, which we usually write <span class="inlinecode"><span class="id" type="var">n<sub>1</sub></span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" type="var">n<sub>2</sub></span></span>.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Print</span> <span class="id" type="var">le</span>.<br/>
<span class="comment">(* ====> Inductive le (n : nat) : nat <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> Prop :=<br/>
le_n : n <= n<br/>
| le_S : forall m : nat, n <= m <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> n <= S m *)</span><br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">le</span> : <span class="id" type="var">nat</span> → <span class="id" type="var">nat</span> → <span class="id" type="keyword">Prop</span>.<br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">le</span> : <span class="id" type="var">relation</span> <span class="id" type="var">nat</span>.<br/>
</div>
<div class="doc">
(Why did we write it this way instead of starting with <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>
<span class="inlinecode"><span class="id" type="var">le</span></span> <span class="inlinecode">:</span> <span class="inlinecode"><span class="id" type="var">relation</span></span> <span class="inlinecode"><span class="id" type="var">nat</span>...</span>? Because we wanted to put the first <span class="inlinecode"><span class="id" type="var">nat</span></span>
to the left of the <span class="inlinecode">:</span>, which makes Coq generate a somewhat nicer
induction principle for reasoning about <span class="inlinecode">≤</span>.)
</div>
<div class="doc">
<a name="lab300"></a><h1 class="section">Basic Properties</h1>
<div class="paragraph"> </div>
As anyone knows who has taken an undergraduate discrete math
course, there is a lot to be said about relations in general,
including ways of classifying relations (as reflexive, transitive,
etc.), theorems that can be proved generically about certain sorts
of relations, constructions that build one relation from another,
etc. For example...
<div class="paragraph"> </div>
<a name="lab301"></a><h3 class="section">Partial Functions</h3>
<div class="paragraph"> </div>
A relation <span class="inlinecode"><span class="id" type="var">R</span></span> on a set <span class="inlinecode"><span class="id" type="var">X</span></span> is a <i>partial function</i> if, for every
<span class="inlinecode"><span class="id" type="var">x</span></span>, there is at most one <span class="inlinecode"><span class="id" type="var">y</span></span> such that <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> — i.e., <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y<sub>1</sub></span></span>
and <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y<sub>2</sub></span></span> together imply <span class="inlinecode"><span class="id" type="var">y<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">y<sub>2</sub></span></span>.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">partial_function</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">x</span> <span class="id" type="var">y<sub>1</sub></span> <span class="id" type="var">y<sub>2</sub></span> : <span class="id" type="var">X</span>, <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y<sub>1</sub></span> → <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y<sub>2</sub></span> → <span class="id" type="var">y<sub>1</sub></span> = <span class="id" type="var">y<sub>2</sub></span>.<br/>
</div>
<div class="doc">
For example, the <span class="inlinecode"><span class="id" type="var">next_nat</span></span> relation defined earlier is a partial
function.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Print</span> <span class="id" type="var">next_nat</span>.<br/>
<span class="comment">(* ====> Inductive next_nat (n : nat) : 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/>
nn : next_nat n (S n) *)</span><br/>
<span class="id" type="keyword">Check</span> <span class="id" type="var">next_nat</span> : <span class="id" type="var">relation</span> <span class="id" type="var">nat</span>.<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">next_nat_partial_function</span> :<br/>
<span class="id" type="var">partial_function</span> <span class="id" type="var">next_nat</span>.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">partial_function</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">x</span> <span class="id" type="var">y<sub>1</sub></span> <span class="id" type="var">y<sub>2</sub></span> <span class="id" type="var">H<sub>1</sub></span> <span class="id" type="var">H<sub>2</sub></span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>1</sub></span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H<sub>2</sub></span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
However, the <span class="inlinecode">≤</span> relation on numbers is not a partial
function. (Assume, for a contradiction, that <span class="inlinecode">≤</span> is a partial
function. But then, since <span class="inlinecode">0</span> <span class="inlinecode">≤</span> <span class="inlinecode">0</span> and <span class="inlinecode">0</span> <span class="inlinecode">≤</span> <span class="inlinecode">1</span>, it follows that
<span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode">1</span>. This is nonsense, so our assumption was
contradictory.)
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_not_a_partial_function</span> :<br/>
¬(<span class="id" type="var">partial_function</span> <span class="id" type="var">le</span>).<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">not</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">partial_function</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">Hc</span>.<br/>
<span class="id" type="tactic">assert</span> (0 = 1) <span class="id" type="keyword">as</span> <span class="id" type="var">Nonsense</span>. {<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">Hc</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">x</span> := 0).<br/>
- <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>.<br/>
- <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>. }<br/>
<span class="id" type="tactic">discriminate</span> <span class="id" type="var">Nonsense</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab302"></a><h4 class="section">Exercise: 2 stars, standard, optional (total_relation_not_partial)</h4>
Show that the <span class="inlinecode"><span class="id" type="var">total_relation</span></span> defined in (an exercise in)
<a href="IndProp.html"><span class="inlineref">IndProp</span></a> is not a partial function.
</div>
<div class="code code-tight">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab303"></a><h4 class="section">Exercise: 2 stars, standard, optional (empty_relation_partial)</h4>
Show that the <span class="inlinecode"><span class="id" type="var">empty_relation</span></span> defined in (an exercise in)
<a href="IndProp.html"><span class="inlineref">IndProp</span></a> is a partial function.
</div>
<div class="code code-tight">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab304"></a><h3 class="section">Reflexive Relations</h3>
<div class="paragraph"> </div>
A <i>reflexive</i> relation on a set <span class="inlinecode"><span class="id" type="var">X</span></span> is one for which every element
of <span class="inlinecode"><span class="id" type="var">X</span></span> is related to itself.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">reflexive</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">a</span> : <span class="id" type="var">X</span>, <span class="id" type="var">R</span> <span class="id" type="var">a</span> <span class="id" type="var">a</span>.<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_reflexive</span> :<br/>
<span class="id" type="var">reflexive</span> <span class="id" type="var">le</span>.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">reflexive</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab305"></a><h3 class="section">Transitive Relations</h3>
<div class="paragraph"> </div>
A relation <span class="inlinecode"><span class="id" type="var">R</span></span> is <i>transitive</i> if <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> holds whenever <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span>
and <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> do.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">transitive</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> : <span class="id" type="var">X</span>, (<span class="id" type="var">R</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span>) → (<span class="id" type="var">R</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span>) → (<span class="id" type="var">R</span> <span class="id" type="var">a</span> <span class="id" type="var">c</span>).<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_trans</span> :<br/>
<span class="id" type="var">transitive</span> <span class="id" type="var">le</span>.<br/>
<div class="togglescript" id="proofcontrol4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')"><span class="show"></span></div>
<div class="proofscript" id="proof4" onclick="toggleDisplay('proof4');toggleDisplay('proofcontrol4')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">Hnm</span> <span class="id" type="var">Hmo</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Hmo</span>.<br/>
- <span class="comment">(* le_n *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">Hnm</span>.<br/>
- <span class="comment">(* le_S *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHHmo</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">lt_trans</span>:<br/>
<span class="id" type="var">transitive</span> <span class="id" type="var">lt</span>.<br/>
<div class="togglescript" id="proofcontrol5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')"><span class="show"></span></div>
<div class="proofscript" id="proof5" onclick="toggleDisplay('proof5');toggleDisplay('proofcontrol5')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">lt</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">transitive</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">Hnm</span> <span class="id" type="var">Hmo</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span> <span class="id" type="keyword">in</span> <span class="id" type="var">Hnm</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">le_trans</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">a</span> := (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) (<span class="id" type="var">b</span> := (<span class="id" type="var">S</span> <span class="id" type="var">m</span>)) (<span class="id" type="var">c</span> := <span class="id" type="var">o</span>).<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">Hnm</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">Hmo</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab306"></a><h4 class="section">Exercise: 2 stars, standard, optional (le_trans_hard_way)</h4>
We can also prove <span class="inlinecode"><span class="id" type="var">lt_trans</span></span> more laboriously by induction,
without using <span class="inlinecode"><span class="id" type="var">le_trans</span></span>. Do this.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">lt_trans'</span> :<br/>
<span class="id" type="var">transitive</span> <span class="id" type="var">lt</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* Prove this by induction on evidence that <span class="inlinecode"><span class="id" type="var">m</span></span> is less than <span class="inlinecode"><span class="id" type="var">o</span></span>. *)</span><br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">lt</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">transitive</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">Hnm</span> <span class="id" type="var">Hmo</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">Hmo</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span> <span class="id" type="var">Hm'o</span>].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab307"></a><h4 class="section">Exercise: 2 stars, standard, optional (lt_trans'')</h4>
Prove the same thing again by induction on <span class="inlinecode"><span class="id" type="var">o</span></span>.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">lt_trans''</span> :<br/>
<span class="id" type="var">transitive</span> <span class="id" type="var">lt</span>.<br/>
<div class="togglescript" id="proofcontrol6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')"><span class="show"></span></div>
<div class="proofscript" id="proof6" onclick="toggleDisplay('proof6');toggleDisplay('proofcontrol6')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">lt</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">transitive</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">o</span> <span class="id" type="var">Hnm</span> <span class="id" type="var">Hmo</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">o</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">o'</span>].<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
The transitivity of <span class="inlinecode"><span class="id" type="var">le</span></span>, in turn, can be used to prove some facts
that will be useful later (e.g., for the proof of antisymmetry
below)...
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_Sn_le</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>, <span class="id" type="var">S</span> <span class="id" type="var">n</span> ≤ <span class="id" type="var">m</span> → <span class="id" type="var">n</span> ≤ <span class="id" type="var">m</span>.<br/>
<div class="togglescript" id="proofcontrol7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')"><span class="show"></span></div>
<div class="proofscript" id="proof7" onclick="toggleDisplay('proof7');toggleDisplay('proofcontrol7')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_trans</span> <span class="id" type="keyword">with</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>).<br/>
- <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>.<br/>
- <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab308"></a><h4 class="section">Exercise: 1 star, standard, optional (le_S_n)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_S_n</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
(<span class="id" type="var">S</span> <span class="id" type="var">n</span> ≤ <span class="id" type="var">S</span> <span class="id" type="var">m</span>) → (<span class="id" type="var">n</span> ≤ <span class="id" type="var">m</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab309"></a><h4 class="section">Exercise: 2 stars, standard, optional (le_Sn_n_inf)</h4>
Provide an informal proof of the following theorem:
<div class="paragraph"> </div>
Theorem: For every <span class="inlinecode"><span class="id" type="var">n</span></span>, <span class="inlinecode">¬</span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">≤</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>
<div class="paragraph"> </div>
A formal proof of this is an optional exercise below, but try
writing an informal proof without doing the formal proof first.
<div class="paragraph"> </div>
Proof:
</div>
<div class="code code-tight">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab310"></a><h4 class="section">Exercise: 1 star, standard, optional (le_Sn_n)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_Sn_n</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span>,<br/>
¬(<span class="id" type="var">S</span> <span class="id" type="var">n</span> ≤ <span class="id" type="var">n</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
Reflexivity and transitivity are the main concepts we'll need for
later chapters, but, for a bit of additional practice working with
relations in Coq, let's look at a few other common ones...
<div class="paragraph"> </div>
<a name="lab311"></a><h3 class="section">Symmetric and Antisymmetric Relations</h3>
<div class="paragraph"> </div>
A relation <span class="inlinecode"><span class="id" type="var">R</span></span> is <i>symmetric</i> if <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> implies <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span>.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">symmetric</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">a</span> <span class="id" type="var">b</span> : <span class="id" type="var">X</span>, (<span class="id" type="var">R</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span>) → (<span class="id" type="var">R</span> <span class="id" type="var">b</span> <span class="id" type="var">a</span>).<br/>
</div>
<div class="doc">
<a name="lab312"></a><h4 class="section">Exercise: 2 stars, standard, optional (le_not_symmetric)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_not_symmetric</span> :<br/>
¬(<span class="id" type="var">symmetric</span> <span class="id" type="var">le</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
A relation <span class="inlinecode"><span class="id" type="var">R</span></span> is <i>antisymmetric</i> if <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> and <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> together
imply <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">b</span></span> — that is, if the only "cycles" in <span class="inlinecode"><span class="id" type="var">R</span></span> are trivial
ones.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">antisymmetric</span> {<span class="id" type="var">X</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
<span style='font-size:120%;'>∀</span><span class="id" type="var">a</span> <span class="id" type="var">b</span> : <span class="id" type="var">X</span>, (<span class="id" type="var">R</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span>) → (<span class="id" type="var">R</span> <span class="id" type="var">b</span> <span class="id" type="var">a</span>) → <span class="id" type="var">a</span> = <span class="id" type="var">b</span>.<br/>
</div>
<div class="doc">
<a name="lab313"></a><h4 class="section">Exercise: 2 stars, standard, optional (le_antisymmetric)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_antisymmetric</span> :<br/>
<span class="id" type="var">antisymmetric</span> <span class="id" type="var">le</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab314"></a><h4 class="section">Exercise: 2 stars, standard, optional (le_step)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_step</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span>,<br/>
<span class="id" type="var">n</span> < <span class="id" type="var">m</span> →<br/>
<span class="id" type="var">m</span> ≤ <span class="id" type="var">S</span> <span class="id" type="var">p</span> →<br/>
<span class="id" type="var">n</span> ≤ <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
<a name="lab315"></a><h3 class="section">Equivalence Relations</h3>
<div class="paragraph"> </div>
A relation is an <i>equivalence</i> if it's reflexive, symmetric, and
transitive.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">equivalence</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
(<span class="id" type="var">reflexive</span> <span class="id" type="var">R</span>) ∧ (<span class="id" type="var">symmetric</span> <span class="id" type="var">R</span>) ∧ (<span class="id" type="var">transitive</span> <span class="id" type="var">R</span>).<br/>
</div>
<div class="doc">
<a name="lab316"></a><h3 class="section">Partial Orders and Preorders</h3>
<div class="paragraph"> </div>
A relation is a <i>partial order</i> when it's reflexive,
<i>anti</i>-symmetric, and transitive. In the Coq standard library
it's called just "order" for short.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">order</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
(<span class="id" type="var">reflexive</span> <span class="id" type="var">R</span>) ∧ (<span class="id" type="var">antisymmetric</span> <span class="id" type="var">R</span>) ∧ (<span class="id" type="var">transitive</span> <span class="id" type="var">R</span>).<br/>
</div>
<div class="doc">
A preorder is almost like a partial order, but doesn't have to be
antisymmetric.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Definition</span> <span class="id" type="var">preorder</span> {<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) :=<br/>
(<span class="id" type="var">reflexive</span> <span class="id" type="var">R</span>) ∧ (<span class="id" type="var">transitive</span> <span class="id" type="var">R</span>).<br/><hr class='doublespaceincode'/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">le_order</span> :<br/>
<span class="id" type="var">order</span> <span class="id" type="var">le</span>.<br/>
<div class="togglescript" id="proofcontrol8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')"><span class="show"></span></div>
<div class="proofscript" id="proof8" onclick="toggleDisplay('proof8');toggleDisplay('proofcontrol8')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">order</span>. <span class="id" type="tactic">split</span>.<br/>
- <span class="comment">(* refl *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">le_reflexive</span>.<br/>
- <span class="id" type="tactic">split</span>.<br/>
+ <span class="comment">(* antisym *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">le_antisymmetric</span>.<br/>
+ <span class="comment">(* transitive. *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">le_trans</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab317"></a><h1 class="section">Reflexive, Transitive Closure</h1>
<div class="paragraph"> </div>
The <i>reflexive, transitive closure</i> of a relation <span class="inlinecode"><span class="id" type="var">R</span></span> is the
smallest relation that contains <span class="inlinecode"><span class="id" type="var">R</span></span> and that is both reflexive and
transitive. Formally, it is defined like this in the Relations
module of the Coq standard library:
</div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">clos_refl_trans</span> {<span class="id" type="var">A</span>: <span class="id" type="keyword">Type</span>} (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">A</span>) : <span class="id" type="var">relation</span> <span class="id" type="var">A</span> :=<br/>
| <span class="id" type="var">rt_step</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> (<span class="id" type="var">H</span> : <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>) : <span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span><br/>
| <span class="id" type="var">rt_refl</span> <span class="id" type="var">x</span> : <span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">x</span><br/>
| <span class="id" type="var">rt_trans</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span><br/>
(<span class="id" type="var">Hxy</span> : <span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>)<br/>
(<span class="id" type="var">Hyz</span> : <span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>) :<br/>
<span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">z</span>.<br/>
</div>
<div class="doc">
For example, the reflexive and transitive closure of the
<span class="inlinecode"><span class="id" type="var">next_nat</span></span> relation coincides with the <span class="inlinecode"><span class="id" type="var">le</span></span> relation.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">next_nat_closure_is_le</span> : <span style='font-size:120%;'>∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
(<span class="id" type="var">n</span> ≤ <span class="id" type="var">m</span>) ↔ ((<span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">next_nat</span>) <span class="id" type="var">n</span> <span class="id" type="var">m</span>).<br/>
<div class="togglescript" id="proofcontrol9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')"><span class="show"></span></div>
<div class="proofscript" id="proof9" onclick="toggleDisplay('proof9');toggleDisplay('proofcontrol9')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">split</span>.<br/>
- <span class="comment">(* <span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>></span></span> *)</span><br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>.<br/>
+ <span class="comment">(* le_n *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">rt_refl</span>.<br/>
+ <span class="comment">(* le_S *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">rt_trans</span> <span class="id" type="keyword">with</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHle</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">rt_step</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">nn</span>.<br/>
- <span class="comment">(* <- *)</span><br/>
<span class="id" type="tactic">intro</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">H</span>.<br/>
+ <span class="comment">(* rt_step *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_S</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>.<br/>
+ <span class="comment">(* rt_refl *)</span> <span class="id" type="tactic">apply</span> <span class="id" type="var">le_n</span>.<br/>
+ <span class="comment">(* rt_trans *)</span><br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">le_trans</span> <span class="id" type="keyword">with</span> <span class="id" type="var">y</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHclos_refl_trans1</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHclos_refl_trans2</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
The above definition of reflexive, transitive closure is natural:
it says, explicitly, that the reflexive and transitive closure of
<span class="inlinecode"><span class="id" type="var">R</span></span> is the least relation that includes <span class="inlinecode"><span class="id" type="var">R</span></span> and that is closed
under rules of reflexivity and transitivity. But it turns out
that this definition is not very convenient for doing proofs,
since the "nondeterminism" of the <span class="inlinecode"><span class="id" type="var">rt_trans</span></span> rule can sometimes
lead to tricky inductions. Here is a more useful definition:
</div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">clos_refl_trans_1n</span> {<span class="id" type="var">A</span> : <span class="id" type="keyword">Type</span>}<br/>
(<span class="id" type="var">R</span> : <span class="id" type="var">relation</span> <span class="id" type="var">A</span>) (<span class="id" type="var">x</span> : <span class="id" type="var">A</span>)<br/>
: <span class="id" type="var">A</span> → <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">rt1n_refl</span> : <span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">x</span><br/>
| <span class="id" type="var">rt1n_trans</span> (<span class="id" type="var">y</span> <span class="id" type="var">z</span> : <span class="id" type="var">A</span>)<br/>
(<span class="id" type="var">Hxy</span> : <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>) (<span class="id" type="var">Hrest</span> : <span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>) :<br/>
<span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">z</span>.<br/>
</div>
<div class="doc">
Our new definition of reflexive, transitive closure "bundles"
the <span class="inlinecode"><span class="id" type="var">rt_step</span></span> and <span class="inlinecode"><span class="id" type="var">rt_trans</span></span> rules into the single rule step.
The left-hand premise of this step is a single use of <span class="inlinecode"><span class="id" type="var">R</span></span>,
leading to a much simpler induction principle.
<div class="paragraph"> </div>
Before we go on, we should check that the two definitions do
indeed define the same relation...
<div class="paragraph"> </div>
First, we prove two lemmas showing that <span class="inlinecode"><span class="id" type="var">clos_refl_trans_1n</span></span> mimics
the behavior of the two "missing" <span class="inlinecode"><span class="id" type="var">clos_refl_trans</span></span>
constructors.
</div>
<div class="code code-tight">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">rsc_R</span> : <span style='font-size:120%;'>∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">R</span>:<span class="id" type="var">relation</span> <span class="id" type="var">X</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span> : <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> → <span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>.<br/>
<div class="togglescript" id="proofcontrol10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')"><span class="show"></span></div>
<div class="proofscript" id="proof10" onclick="toggleDisplay('proof10');toggleDisplay('proofcontrol10')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">X</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">rt1n_trans</span> <span class="id" type="keyword">with</span> <span class="id" type="var">y</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">rt1n_refl</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
</div>
<div class="doc">
<a name="lab318"></a><h4 class="section">Exercise: 2 stars, standard, optional (rsc_trans)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">rsc_trans</span> :<br/>
<span style='font-size:120%;'>∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span> : <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> →<br/>
<span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span> →<br/>
<span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">z</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="doc less-space">
<div class="paragraph"> </div>
Then we use these facts to prove that the two definitions of
reflexive, transitive closure do indeed define the same
relation.
<div class="paragraph"> </div>
<a name="lab319"></a><h4 class="section">Exercise: 3 stars, standard, optional (rtc_rsc_coincide)</h4>
</div>
<div class="code code-space">
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">rtc_rsc_coincide</span> :<br/>
<span style='font-size:120%;'>∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">R</span>: <span class="id" type="var">relation</span> <span class="id" type="var">X</span>) (<span class="id" type="var">x</span> <span class="id" type="var">y</span> : <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">clos_refl_trans</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> ↔ <span class="id" type="var">clos_refl_trans_1n</span> <span class="id" type="var">R</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<span class="proofbox">☐</span>
<div class="code code-tight">
<span class="comment">(* Wed Jan 9 12:02:46 EST 2019 *)</span><br/>
</div>
</div>
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