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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="common/css/sf.css" rel="stylesheet" type="text/css" />
<title>AltAuto: A Streamlined Treatment of Automation</title>
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<script src="common/jquery-ui/external/jquery/jquery.js"></script>
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<script src="common/toggleproofs.js"></script>
<link href="common/css/lf.css" rel="stylesheet" type="text/css"/>
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<body>
<div id="page">
<div id="header">
<div id='logoinheader'><a href='https://softwarefoundations.cis.upenn.edu'>
<img src='common/media/image/sf_logo_sm.png' alt='Software Foundations Logo'></a></div>
<div class='booktitleinheader'><a href='index.html'>Volume 1: Logical Foundations</a></div>
<ul id='menu'>
<li class='section_name'><a href='toc.html'>Table of Contents</a></li>
<li class='section_name'><a href='coqindex.html'>Index</a></li>
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</ul>
</div>
<div id="main">
<h1 class="libtitle">AltAuto<span class="subtitle">A Streamlined Treatment of Automation</span></h1>
<div class="doc">
<div class="paragraph"> </div>
So far, we've been doing all our proofs using just a small
handful of Coq's tactics and completely ignoring its powerful
facilities for constructing parts of proofs automatically. Getting
used to them will take some work -- Coq's automation is a power
tool -- but it will allow us to scale up our efforts to more
complex definitions and more interesting properties without
becoming overwhelmed by boring, repetitive, low-level details.
<div class="paragraph"> </div>
In this chapter, we'll learn about
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>tacticals</i>, which allow tactics to be combined;
<div class="paragraph"> </div>
</li>
<li> new tactics that make dealing with hypothesis names less fussy
and more maintainable;
<div class="paragraph"> </div>
</li>
<li> <i>automatic solvers</i> that can prove limited classes of theorems
without any human assistance;
<div class="paragraph"> </div>
</li>
<li> <i>proof search</i> with the <span class="inlinecode"><span class="id" title="tactic">auto</span></span> tactic; and
<div class="paragraph"> </div>
</li>
<li> the <i>Ltac</i> language for writing tactics.
</li>
</ul>
<div class="paragraph"> </div>
These features enable startlingly short proofs. Used properly,
they can also make proofs more maintainable and robust to changes
in underlying definitions.
<div class="paragraph"> </div>
This chapter is an alternative to the combination of <a href="Imp.html"><span class="inlineref">Imp</span></a>
and <a href="Auto.html"><span class="inlineref">Auto</span></a>, which cover roughly the same material about
automation, but in the context of programming language metatheory.
A deeper treatment of <span class="inlinecode"><span class="id" title="tactic">auto</span></span> can be found in the <a href="https://softwarefoundations.cis.upenn.edu/plf-current/UseAuto.html"><span class="inlineref">UseAuto</span></a>
chapter in <i>Programming Language Foundations</i>.
</div>
<div class="code">
<span class="id" title="keyword">Set</span> <span class="id" title="var">Warnings</span> "-notation-overridden,-parsing,-deprecated-hint-without-locality,-deprecated-syntactic-definition,-deprecated]".<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Arith.Arith.html#"><span class="id" title="library">Arith</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Lists.List.html#"><span class="id" title="library">List</span></a>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">LF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="IndProp.html#"><span class="id" title="library">IndProp</span></a>.<br/>
</div>
<div class="doc">
As a simple illustration of the benefits of automation,
let's consider another problem on regular expressions, which we
formalized in <a href="IndProp.html"><span class="inlineref">IndProp</span></a>. A given set of strings can be
denoted by many different regular expressions. For example, <span class="inlinecode"><span class="id" title="var">App</span></span>
<span class="inlinecode"><span class="id" title="var">EmptyString</span></span> <span class="inlinecode"><span class="id" title="var">re</span></span> matches exactly the same strings as <span class="inlinecode"><span class="id" title="var">re</span></span>. We can
write a function that "optimizes" any regular expression into a
potentially simpler one by applying this fact throughout the r.e.
(Note that, for simplicity, the function does not optimize
expressions that arise as the result of other optimizations.)
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <a id="re_opt_e" class="idref" href="#re_opt_e"><span class="id" title="definition">re_opt_e</span></a> {<a id="T:1" class="idref" href="#T:1"><span class="id" title="binder">T</span></a>:<span class="id" title="keyword">Type</span>} (<a id="re:2" class="idref" href="#re:2"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:1"><span class="id" title="variable">T</span></a>) : <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:1"><span class="id" title="variable">T</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="AltAuto.html#re:2"><span class="id" title="variable">re</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <span class="id" title="var">re<sub>2</sub></span><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> (<a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <span class="id" title="var">re<sub>1</sub></span>) (<a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <span class="id" title="var">re<sub>2</sub></span>)<br/>
| <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> (<a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <span class="id" title="var">re<sub>1</sub></span>) (<a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <span class="id" title="var">re<sub>2</sub></span>)<br/>
| <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> <span class="id" title="var">re</span> ⇒ <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> (<a class="idref" href="AltAuto.html#re_opt_e:3"><span class="id" title="definition">re_opt_e</span></a> <a class="idref" href="AltAuto.html#re:2"><span class="id" title="variable">re</span></a>)<br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="AltAuto.html#re:2"><span class="id" title="variable">re</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
</div>
<div class="doc">
We would like to show the equivalence of re's with their
"optimized" form. One direction of this equivalence looks like
this (the other is similar).
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="re_opt_e_match" class="idref" href="#re_opt_e_match"><span class="id" title="lemma">re_opt_e_match</span></a> : <span class="id" title="keyword">∀</span> <a id="T:5" class="idref" href="#T:5"><span class="id" title="binder">T</span></a> (<a id="re:6" class="idref" href="#re:6"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:5"><span class="id" title="variable">T</span></a>) <a id="s:7" class="idref" href="#s:7"><span class="id" title="binder">s</span></a>,<br/>
<a class="idref" href="AltAuto.html#s:7"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re:6"><span class="id" title="variable">re</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="AltAuto.html#s:7"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re_opt_e"><span class="id" title="definition">re_opt_e</span></a> <a class="idref" href="AltAuto.html#re:6"><span class="id" title="variable">re</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">T</span> <span class="id" title="var">re</span> <span class="id" title="var">s</span> <span class="id" title="var">M</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">M</span><br/>
<span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span> | <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span><br/>
| <span class="id" title="var">re</span> | <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re</span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span>].<br/>
- <span class="comment">(* MEmpty *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
- <span class="comment">(* MChar *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MChar"><span class="id" title="constructor">MChar</span></a>.<br/>
- <span class="comment">(* MApp *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">Hmatch1</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
- <span class="comment">(* MUnionL *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MUnionR *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MStar0 *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
- <span class="comment">(* MStarApp *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStarApp"><span class="id" title="constructor">MStarApp</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The amount of repetition in that proof is annoying. And if
we wanted to extend the optimization function to handle other,
similar, rewriting opportunities, it would start to be a real
problem. We can streamline the proof with <i>tacticals</i>, which we
turn to, next.
</div>
<div class="doc">
<a id="lab444"></a><h1 class="section">Tacticals</h1>
<div class="paragraph"> </div>
<i>Tacticals</i> are tactics that take other tactics as arguments --
"higher-order tactics," if you will.
</div>
<div class="doc">
<a id="lab445"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="tactic">try</span></span> Tactical</h2>
<div class="paragraph"> </div>
If <span class="inlinecode"><span class="id" title="var">T</span></span> is a tactic, then <span class="inlinecode"><span class="id" title="tactic">try</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span> is a tactic that is just like <span class="inlinecode"><span class="id" title="var">T</span></span>
except that, if <span class="inlinecode"><span class="id" title="var">T</span></span> fails, <span class="inlinecode"><span class="id" title="tactic">try</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span> <i>successfully</i> does nothing at
all instead of failing.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="silly1" class="idref" href="#silly1"><span class="id" title="lemma">silly1</span></a> : <span class="id" title="keyword">∀</span> <a id="n:8" class="idref" href="#n:8"><span class="id" title="binder">n</span></a>, 1 <a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">+</span></a> <a class="idref" href="AltAuto.html#n:8"><span class="id" title="variable">n</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="http://coq.inria.fr/library//Coq.Init.Datatypes.html#S"><span class="id" title="constructor">S</span></a> <a class="idref" href="AltAuto.html#n:8"><span class="id" title="variable">n</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">try</span> <span class="id" title="tactic">reflexivity</span>. <span class="comment">(* this just does <span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span> *)</span> <span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="silly2" class="idref" href="#silly2"><span class="id" title="lemma">silly2</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:9" class="idref" href="#P:9"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>), <a class="idref" href="AltAuto.html#P:9"><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> <a class="idref" href="AltAuto.html#P:9"><span class="id" title="variable">P</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">HP</span>.<br/>
<span class="id" title="var">Fail</span> <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">reflexivity</span>. <span class="comment">(* proof state is unchanged *)</span><br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">HP</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
There is no real reason to use <span class="inlinecode"><span class="id" title="tactic">try</span></span> in completely manual
proofs like these, but it is very useful for doing automated
proofs in conjunction with the <span class="inlinecode">;</span> tactical, which we show
next.
</div>
<div class="doc">
<a id="lab446"></a><h2 class="section">The Sequence Tactical <span class="inlinecode">;</span> (Simple Form)</h2>
<div class="paragraph"> </div>
In its most common form, the sequence tactical, written with
semicolon <span class="inlinecode">;</span>, takes two tactics as arguments. The compound
tactic <span class="inlinecode"><span class="id" title="var">T</span>;</span> <span class="inlinecode"><span class="id" title="var">T'</span></span> first performs <span class="inlinecode"><span class="id" title="var">T</span></span> and then performs <span class="inlinecode"><span class="id" title="var">T'</span></span> on <i>each
subgoal</i> generated by <span class="inlinecode"><span class="id" title="var">T</span></span>.
<div class="paragraph"> </div>
For example, consider the following trivial lemma:
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="simple_semi" class="idref" href="#simple_semi"><span class="id" title="lemma">simple_semi</span></a> : <span class="id" title="keyword">∀</span> <a id="n:10" class="idref" href="#n:10"><span class="id" title="binder">n</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="AltAuto.html#n:10"><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="Basics.html#ad2ec4e405f68c46c0a176e3e94ae2e<sub>3</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> <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#false"><span class="id" title="constructor">false</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">n</span> <span class="id" title="var">eqn</span>:<span class="id" title="var">E</span>.<br/>
<span class="comment">(* Leaves two subgoals, which are discharged identically... *)</span><br/>
- <span class="comment">(* n=0 *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n=Sn' *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
We can simplify this proof using the <span class="inlinecode">;</span> tactical:
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="simple_semi'" class="idref" href="#simple_semi'"><span class="id" title="lemma">simple_semi'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:11" class="idref" href="#n:11"><span class="id" title="binder">n</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="AltAuto.html#n:11"><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="Basics.html#ad2ec4e405f68c46c0a176e3e94ae2e<sub>3</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> <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#false"><span class="id" title="constructor">false</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">n</span>.<br/>
<span class="comment">(* <span class="inlinecode"><span class="id" title="tactic">destruct</span></span> the current goal *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">n</span>;<br/>
<span class="comment">(* then <span class="inlinecode"><span class="id" title="tactic">simpl</span></span> each resulting subgoal *)</span><br/>
<span class="id" title="tactic">simpl</span>;<br/>
<span class="comment">(* and do <span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span> on each resulting subgoal *)</span><br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Or even more tersely, <span class="inlinecode"><span class="id" title="tactic">destruct</span></span> can do the <span class="inlinecode"><span class="id" title="tactic">intro</span></span>, and <span class="inlinecode"><span class="id" title="tactic">simpl</span></span>
can be omitted:
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="simple_semi''" class="idref" href="#simple_semi''"><span class="id" title="lemma">simple_semi''</span></a> : <span class="id" title="keyword">∀</span> <a id="n:12" class="idref" href="#n:12"><span class="id" title="binder">n</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="AltAuto.html#n:12"><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="Basics.html#ad2ec4e405f68c46c0a176e3e94ae2e<sub>3</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> <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#false"><span class="id" title="constructor">false</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">n</span>; <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab447"></a><h4 class="section">Exercise: 3 stars, standard (try_sequence)</h4>
<div class="paragraph"> </div>
Prove the following theorems using <span class="inlinecode"><span class="id" title="tactic">try</span></span> and <span class="inlinecode">;</span>. Like
<span class="inlinecode"><span class="id" title="var">simple_semi''</span></span> above, each proof script should be a sequence <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span>;</span>
<span class="inlinecode">...;</span> <span class="inlinecode"><span class="id" title="var">tn</span>.</span> of tactics, and there should be only one period in
between <span class="inlinecode"><span class="id" title="keyword">Proof</span>.</span> and <span class="inlinecode"><span class="id" title="keyword">Qed</span>.</span>. Let's call that a "one shot"
proof.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="andb_eq_orb" class="idref" href="#andb_eq_orb"><span class="id" title="lemma">andb_eq_orb</span></a> :<br/>
<span class="id" title="keyword">∀</span> (<a id="b:13" class="idref" href="#b:13"><span class="id" title="binder">b</span></a> <a id="c:14" class="idref" href="#c:14"><span class="id" title="binder">c</span></a> : <a class="idref" href="Basics.html#bool"><span class="id" title="inductive">bool</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><a class="idref" href="Basics.html#andb"><span class="id" title="definition">andb</span></a> <a class="idref" href="AltAuto.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="AltAuto.html#c:14"><span class="id" title="variable">c</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#orb"><span class="id" title="definition">orb</span></a> <a class="idref" href="AltAuto.html#b:13"><span class="id" title="variable">b</span></a> <a class="idref" href="AltAuto.html#c:14"><span class="id" title="variable">c</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="AltAuto.html#b:13"><span class="id" title="variable">b</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="AltAuto.html#c:14"><span class="id" title="variable">c</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/><hr class='doublespaceincode'/>
<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:15" class="idref" href="#n:15"><span class="id" title="binder">n</span></a> <a id="m:16" class="idref" href="#m:16"><span class="id" title="binder">m</span></a> <a id="p:17" class="idref" href="#p:17"><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="AltAuto.html#n:15"><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="AltAuto.html#m:16"><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="AltAuto.html#p:17"><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="AltAuto.html#n:15"><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="AltAuto.html#m:16"><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="AltAuto.html#p:17"><span class="id" title="variable">p</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/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Fixpoint</span> <a id="nonzeros" class="idref" href="#nonzeros"><span class="id" title="definition">nonzeros</span></a> (<a id="lst:18" class="idref" href="#lst:18"><span class="id" title="binder">lst</span></a> : <a class="idref" href="Poly.html#list"><span class="id" title="inductive">list</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/>
<span class="id" title="keyword">match</span> <a class="idref" href="AltAuto.html#lst:18"><span class="id" title="variable">lst</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="Poly.html#2c60282cbb04e070c60ae01e76f3865a"><span class="id" title="notation">[]</span></a> ⇒ <a class="idref" href="Poly.html#2c60282cbb04e070c60ae01e76f3865a"><span class="id" title="notation">[]</span></a><br/>
| 0 <a class="idref" href="Poly.html#:::x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">t</span> ⇒ <a class="idref" href="AltAuto.html#nonzeros:19"><span class="id" title="definition">nonzeros</span></a> <span class="id" title="var">t</span><br/>
| <span class="id" title="var">h</span> <a class="idref" href="Poly.html#:::x_'::'_x"><span class="id" title="notation">::</span></a> <span class="id" title="var">t</span> ⇒ <span class="id" title="var">h</span> <a class="idref" href="Poly.html#:::x_'::'_x"><span class="id" title="notation">::</span></a> <a class="idref" href="AltAuto.html#nonzeros:19"><span class="id" title="definition">nonzeros</span></a> <span class="id" title="var">t</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Lemma</span> <a id="nonzeros_app" class="idref" href="#nonzeros_app"><span class="id" title="lemma">nonzeros_app</span></a> : <span class="id" title="keyword">∀</span> <a id="lst1:21" class="idref" href="#lst1:21"><span class="id" title="binder">lst1</span></a> <a id="lst2:22" class="idref" href="#lst2:22"><span class="id" title="binder">lst2</span></a> : <a class="idref" href="Poly.html#list"><span class="id" title="inductive">list</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="AltAuto.html#nonzeros"><span class="id" title="definition">nonzeros</span></a> (<a class="idref" href="AltAuto.html#lst1:21"><span class="id" title="variable">lst1</span></a> <a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">++</span></a> <a class="idref" href="AltAuto.html#lst2:22"><span class="id" title="variable">lst2</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="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">(</span></a><a class="idref" href="AltAuto.html#nonzeros"><span class="id" title="definition">nonzeros</span></a> <a class="idref" href="AltAuto.html#lst1:21"><span class="id" title="variable">lst1</span></a><a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">)</span></a> <a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">++</span></a> <a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">(</span></a><a class="idref" href="AltAuto.html#nonzeros"><span class="id" title="definition">nonzeros</span></a> <a class="idref" href="AltAuto.html#lst2:22"><span class="id" title="variable">lst2</span></a><a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">)</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>
Using <span class="inlinecode"><span class="id" title="tactic">try</span></span> and <span class="inlinecode">;</span> together, we can improve the proof about
regular expression optimization.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="re_opt_e_match'" class="idref" href="#re_opt_e_match'"><span class="id" title="lemma">re_opt_e_match'</span></a> : <span class="id" title="keyword">∀</span> <a id="T:23" class="idref" href="#T:23"><span class="id" title="binder">T</span></a> (<a id="re:24" class="idref" href="#re:24"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:23"><span class="id" title="variable">T</span></a>) <a id="s:25" class="idref" href="#s:25"><span class="id" title="binder">s</span></a>,<br/>
<a class="idref" href="AltAuto.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re:24"><span class="id" title="variable">re</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="AltAuto.html#s:25"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re_opt_e"><span class="id" title="definition">re_opt_e</span></a> <a class="idref" href="AltAuto.html#re:24"><span class="id" title="variable">re</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">T</span> <span class="id" title="var">re</span> <span class="id" title="var">s</span> <span class="id" title="var">M</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">M</span><br/>
<span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span> | <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span><br/>
| <span class="id" title="var">re</span> | <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re</span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span>];<br/>
<span class="comment">(* Do the <span class="inlinecode"><span class="id" title="tactic">simpl</span></span> for every case here: *)</span><br/>
<span class="id" title="tactic">simpl</span>.<br/>
- <span class="comment">(* MEmpty *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
- <span class="comment">(* MChar *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MChar"><span class="id" title="constructor">MChar</span></a>.<br/>
- <span class="comment">(* MApp *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>;<br/>
<span class="comment">(* Most cases follow by the same formula. Notice that <span class="inlinecode"><span class="id" title="tactic">apply</span></span>
<span class="inlinecode"><span class="id" title="var">MApp</span></span> gives two subgoals: <span class="inlinecode"><span class="id" title="tactic">try</span></span> <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">IH<sub>1</sub></span></span> is run on _both_ of<br/>
them and succeeds on the first but not the second; <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">IH<sub>2</sub></span></span><br/>
is then run on this remaining goal. *)</span><br/>
<span class="id" title="tactic">try</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>; <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>; <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>).<br/>
<span class="comment">(* The interesting case, on which <span class="inlinecode"><span class="id" title="tactic">try</span>...</span> does nothing, is when<br/>
<span class="inlinecode"><span class="id" title="var">re<sub>1</sub></span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">EmptyStr</span></span>. In this case, we have to appeal to the fact<br/>
that <span class="inlinecode"><span class="id" title="var">re<sub>1</sub></span></span> matches only the empty string: *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Hmatch1</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
- <span class="comment">(* MUnionL *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MUnionR *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MStar0 *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
- <span class="comment">(* MStarApp *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStarApp"><span class="id" title="constructor">MStarApp</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab448"></a><h2 class="section">The Sequence Tactical <span class="inlinecode">;</span> (Local Form)</h2>
<div class="paragraph"> </div>
The sequence tactical <span class="inlinecode">;</span> also has a more general form than the
simple <span class="inlinecode"><span class="id" title="var">T</span>;</span> <span class="inlinecode"><span class="id" title="var">T'</span></span> we saw above. If <span class="inlinecode"><span class="id" title="var">T</span></span>, <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span>, ..., <span class="inlinecode"><span class="id" title="var">Tn</span></span> are tactics,
then
<div class="paragraph"> </div>
<span class="inlinecode">[</span> <span class="inlinecode"><span class="id" title="var">T</span>;</span> <span class="inlinecode">[<span class="id" title="var">T<sub>1</sub></span></span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" title="var">T<sub>2</sub></span></span> <span class="inlinecode">|</span> <span class="inlinecode">...</span> <span class="inlinecode">|</span> <span class="inlinecode"><span class="id" title="var">Tn</span>]</span> <span class="inlinecode">]</span>
<div class="paragraph"> </div>
is a tactic that first performs <span class="inlinecode"><span class="id" title="var">T</span></span> and then locally performs <span class="inlinecode"><span class="id" title="var">T<sub>1</sub></span></span>
on the first subgoal generated by <span class="inlinecode"><span class="id" title="var">T</span></span>, locally performs <span class="inlinecode"><span class="id" title="var">T<sub>2</sub></span></span> on
the second subgoal, etc.
<div class="paragraph"> </div>
So <span class="inlinecode"><span class="id" title="var">T</span>;</span> <span class="inlinecode"><span class="id" title="var">T'</span></span> is just special notation for the case when all of the
<span class="inlinecode"><span class="id" title="var">Ti</span></span>'s are the same tactic; i.e., <span class="inlinecode"><span class="id" title="var">T</span>;</span> <span class="inlinecode"><span class="id" title="var">T'</span></span> is shorthand for:
<div class="paragraph"> </div>
<br/>
<span class="inlinecode"> <span class="id" title="var">T</span>; [<span class="id" title="var">T'</span> | <span class="id" title="var">T'</span> | ... | <span class="id" title="var">T'</span>]
</span>
<div class="paragraph"> </div>
For example, the following proof makes it clear which tactics are
used to solve the base case vs. the inductive case.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="app_length" class="idref" href="#app_length"><span class="id" title="lemma">app_length</span></a> : <span class="id" title="keyword">∀</span> (<a id="X:26" class="idref" href="#X:26"><span class="id" title="binder">X</span></a> : <span class="id" title="keyword">Type</span>) (<a id="lst1:27" class="idref" href="#lst1:27"><span class="id" title="binder">lst1</span></a> <a id="lst2:28" class="idref" href="#lst2:28"><span class="id" title="binder">lst2</span></a> : <a class="idref" href="Poly.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="AltAuto.html#X:26"><span class="id" title="variable">X</span></a>),<br/>
<a class="idref" href="Poly.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="AltAuto.html#lst1:27"><span class="id" title="variable">lst1</span></a> <a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">++</span></a> <a class="idref" href="AltAuto.html#lst2:28"><span class="id" title="variable">lst2</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="Poly.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="AltAuto.html#lst1:27"><span class="id" title="variable">lst1</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="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="Poly.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="AltAuto.html#lst2:28"><span class="id" title="variable">lst2</span></a><a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">induction</span> <span class="id" title="var">lst1</span>;<br/>
[<span class="id" title="tactic">reflexivity</span> | <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHlst1</span>; <span class="id" title="tactic">reflexivity</span>].<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The identity tactic <span class="inlinecode"><span class="id" title="tactic">idtac</span></span> always succeeds without changing the
proof state. We can use it to factor out <span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span> in the
previous proof.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="app_length'" class="idref" href="#app_length'"><span class="id" title="lemma">app_length'</span></a> : <span class="id" title="keyword">∀</span> (<a id="X:29" class="idref" href="#X:29"><span class="id" title="binder">X</span></a> : <span class="id" title="keyword">Type</span>) (<a id="lst1:30" class="idref" href="#lst1:30"><span class="id" title="binder">lst1</span></a> <a id="lst2:31" class="idref" href="#lst2:31"><span class="id" title="binder">lst2</span></a> : <a class="idref" href="Poly.html#list"><span class="id" title="inductive">list</span></a> <a class="idref" href="AltAuto.html#X:29"><span class="id" title="variable">X</span></a>),<br/>
<a class="idref" href="Poly.html#length"><span class="id" title="definition">length</span></a> (<a class="idref" href="AltAuto.html#lst1:30"><span class="id" title="variable">lst1</span></a> <a class="idref" href="Poly.html#f03f7a04ef75ff3ac66ca5c23554e52e"><span class="id" title="notation">++</span></a> <a class="idref" href="AltAuto.html#lst2:31"><span class="id" title="variable">lst2</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="Poly.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="AltAuto.html#lst1:30"><span class="id" title="variable">lst1</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="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">(</span></a><a class="idref" href="Poly.html#length"><span class="id" title="definition">length</span></a> <a class="idref" href="AltAuto.html#lst2:31"><span class="id" title="variable">lst2</span></a><a class="idref" href="Basics.html#0dacc1786c5ba797d47dd85006231633"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">induction</span> <span class="id" title="var">lst1</span>;<br/>
[<span class="id" title="tactic">idtac</span> | <span class="id" title="tactic">simpl</span>; <span class="id" title="tactic">rewrite</span> <span class="id" title="var">IHlst1</span>];<br/>
<span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab449"></a><h4 class="section">Exercise: 1 star, standard (notry_sequence)</h4>
<div class="paragraph"> </div>
Prove the following theorem with a one-shot proof, but this
time, do not use <span class="inlinecode"><span class="id" title="tactic">try</span></span>.
</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:32" class="idref" href="#n:32"><span class="id" title="binder">n</span></a> <a id="m:33" class="idref" href="#m:33"><span class="id" title="binder">m</span></a> <a id="p:34" class="idref" href="#p:34"><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="AltAuto.html#n:32"><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="AltAuto.html#m:33"><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="AltAuto.html#p:34"><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="AltAuto.html#n:32"><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="AltAuto.html#m:33"><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="AltAuto.html#p:34"><span class="id" title="variable">p</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>
We can use the local form of the sequence tactical to give a
slightly neater version of our optimization proof. Two lines
change, as shown below with <span class="inlinecode"><===</span>.
</div>
<div class="code">
<span class="id" title="keyword">Lemma</span> <a id="re_opt_e_match''" class="idref" href="#re_opt_e_match''"><span class="id" title="lemma">re_opt_e_match''</span></a> : <span class="id" title="keyword">∀</span> <a id="T:35" class="idref" href="#T:35"><span class="id" title="binder">T</span></a> (<a id="re:36" class="idref" href="#re:36"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:35"><span class="id" title="variable">T</span></a>) <a id="s:37" class="idref" href="#s:37"><span class="id" title="binder">s</span></a>,<br/>
<a class="idref" href="AltAuto.html#s:37"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re:36"><span class="id" title="variable">re</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="AltAuto.html#s:37"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re_opt_e"><span class="id" title="definition">re_opt_e</span></a> <a class="idref" href="AltAuto.html#re:36"><span class="id" title="variable">re</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">T</span> <span class="id" title="var">re</span> <span class="id" title="var">s</span> <span class="id" title="var">M</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">M</span><br/>
<span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span> | <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span><br/>
| <span class="id" title="var">re</span> | <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re</span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span>];<br/>
<span class="comment">(* Do the <span class="inlinecode"><span class="id" title="tactic">simpl</span></span> for every case here: *)</span><br/>
<span class="id" title="tactic">simpl</span>.<br/>
- <span class="comment">(* MEmpty *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
- <span class="comment">(* MChar *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MChar"><span class="id" title="constructor">MChar</span></a>.<br/>
- <span class="comment">(* MApp *)</span><br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>;<br/>
<span class="id" title="tactic">try</span> (<span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>; [<span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span> | <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>]). <span class="comment">(* <=== *)</span><br/>
<span class="id" title="tactic">inversion</span> <span class="id" title="var">Hmatch1</span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
- <span class="comment">(* MUnionL *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MUnionR *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MStar0 *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
- <span class="comment">(* MStarApp *)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStarApp"><span class="id" title="constructor">MStarApp</span></a>; [<span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span> | <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>]. <span class="comment">(* <=== *)</span><br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab450"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> Tactical</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> tactical takes another tactic and keeps
applying this tactic until it fails or stops making progress. Here
is an example showing that <span class="inlinecode">10</span> is in a long list:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="In<sub>10</sub>" class="idref" href="#In<sub>10</sub>"><span class="id" title="lemma">In<sub>10</sub></span></a> : <a class="idref" href="Logic.html#In"><span class="id" title="definition">In</span></a> 10 <a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a>1<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>2<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>3<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>4<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>5<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>6<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>7<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>8<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>9<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>10<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">try</span> (<span class="id" title="tactic">left</span>; <span class="id" title="tactic">reflexivity</span>); <span class="id" title="tactic">right</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span> never fails: if the tactic <span class="inlinecode"><span class="id" title="var">T</span></span> doesn't apply
to the original goal, then <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> still succeeds without
changing the original goal (i.e., it repeats zero times).
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="In<sub>10</sub>'" class="idref" href="#In<sub>10</sub>'"><span class="id" title="lemma">In<sub>10</sub>'</span></a> : <a class="idref" href="Logic.html#In"><span class="id" title="definition">In</span></a> 10 <a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a>1<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>2<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>3<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>4<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>5<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>6<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>7<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>8<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>9<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a>10<a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">left</span>; <span class="id" title="tactic">reflexivity</span>).<br/>
<span class="id" title="tactic">repeat</span> (<span class="id" title="tactic">right</span>; <span class="id" title="tactic">try</span> (<span class="id" title="tactic">left</span>; <span class="id" title="tactic">reflexivity</span>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="var">T</span></span> also does not have any upper bound on the
number of times it applies <span class="inlinecode"><span class="id" title="var">T</span></span>. If <span class="inlinecode"><span class="id" title="var">T</span></span> is a tactic that always
succeeds, then repeat <span class="inlinecode"><span class="id" title="var">T</span></span> will loop forever (e.g., <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> <span class="inlinecode"><span class="id" title="tactic">simpl</span></span>
loops, since <span class="inlinecode"><span class="id" title="tactic">simpl</span></span> always succeeds). Evaluation in Coq's term
language, Gallina, is guaranteed to terminate, but tactic
evaluation is not. This does not affect Coq's logical consistency,
however, since the job of <span class="inlinecode"><span class="id" title="tactic">repeat</span></span> and other tactics is to guide
Coq in constructing proofs. If the construction process diverges,
it simply means that we have failed to construct a proof, not that
we have constructed an incorrect proof.
<div class="paragraph"> </div>
<a id="lab451"></a><h4 class="section">Exercise: 1 star, standard (ev100)</h4>
Prove that 100 is even. Your proof script should be quite short.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="ev100" class="idref" href="#ev100"><span class="id" title="lemma">ev100</span></a>: <a class="idref" href="IndProp.html#ev"><span class="id" title="inductive">ev</span></a> 100.<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">
<a id="lab452"></a><h2 class="section">An Optimization Exercise</h2>
<a id="lab453"></a><h4 class="section">Exercise: 4 stars, standard (re_opt)</h4>
<div class="paragraph"> </div>
Consider this more powerful version of the regular expression
optimizer.
</div>
<div class="code">
<span class="id" title="keyword">Fixpoint</span> <a id="re_opt" class="idref" href="#re_opt"><span class="id" title="definition">re_opt</span></a> {<a id="T:38" class="idref" href="#T:38"><span class="id" title="binder">T</span></a>:<span class="id" title="keyword">Type</span>} (<a id="re:39" class="idref" href="#re:39"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:38"><span class="id" title="variable">T</span></a>) : <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:38"><span class="id" title="variable">T</span></a> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="AltAuto.html#re:39"><span class="id" title="variable">re</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <span class="id" title="var">_</span> <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a> ⇒ <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>2</sub></span><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <span class="id" title="var">re<sub>1</sub></span> <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a> ⇒ <a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>1</sub></span><br/>
| <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="IndProp.html#App"><span class="id" title="constructor">App</span></a> (<a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>1</sub></span>) (<a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>2</sub></span>)<br/>
| <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>2</sub></span><br/>
| <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> <span class="id" title="var">re<sub>1</sub></span> <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a> ⇒ <a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>1</sub></span><br/>
| <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> ⇒ <a class="idref" href="IndProp.html#Union"><span class="id" title="constructor">Union</span></a> (<a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>1</sub></span>) (<a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <span class="id" title="var">re<sub>2</sub></span>)<br/>
| <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a> ⇒ <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a><br/>
| <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a> ⇒ <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a><br/>
| <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> <span class="id" title="var">re</span> ⇒ <a class="idref" href="IndProp.html#Star"><span class="id" title="constructor">Star</span></a> (<a class="idref" href="AltAuto.html#re_opt:40"><span class="id" title="definition">re_opt</span></a> <a class="idref" href="AltAuto.html#re:39"><span class="id" title="variable">re</span></a>)<br/>
| <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a> ⇒ <a class="idref" href="IndProp.html#EmptySet"><span class="id" title="constructor">EmptySet</span></a><br/>
| <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a> ⇒ <a class="idref" href="IndProp.html#EmptyStr"><span class="id" title="constructor">EmptyStr</span></a><br/>
| <a class="idref" href="IndProp.html#Char"><span class="id" title="constructor">Char</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="IndProp.html#Char"><span class="id" title="constructor">Char</span></a> <span class="id" title="var">x</span><br/>
<span class="id" title="keyword">end</span>.<br/><hr class='doublespaceincode'/>
<span class="comment">(* Here is an incredibly tedious manual proof of (one direction of)<br/>
its correctness: *)</span><br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Lemma</span> <a id="re_opt_match" class="idref" href="#re_opt_match"><span class="id" title="lemma">re_opt_match</span></a> : <span class="id" title="keyword">∀</span> <a id="T:42" class="idref" href="#T:42"><span class="id" title="binder">T</span></a> (<a id="re:43" class="idref" href="#re:43"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:42"><span class="id" title="variable">T</span></a>) <a id="s:44" class="idref" href="#s:44"><span class="id" title="binder">s</span></a>,<br/>
<a class="idref" href="AltAuto.html#s:44"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re:43"><span class="id" title="variable">re</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="AltAuto.html#s:44"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re_opt"><span class="id" title="definition">re_opt</span></a> <a class="idref" href="AltAuto.html#re:43"><span class="id" title="variable">re</span></a>.<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" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">T</span> <span class="id" title="var">re</span> <span class="id" title="var">s</span> <span class="id" title="var">M</span>.<br/>
<span class="id" title="tactic">induction</span> <span class="id" title="var">M</span><br/>
<span class="id" title="keyword">as</span> [| <span class="id" title="var">x'</span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span><br/>
| <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span> | <span class="id" title="var">re<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re<sub>2</sub></span> <span class="id" title="var">Hmatch</span> <span class="id" title="var">IH</span><br/>
| <span class="id" title="var">re</span> | <span class="id" title="var">s<sub>1</sub></span> <span class="id" title="var">s<sub>2</sub></span> <span class="id" title="var">re</span> <span class="id" title="var">Hmatch1</span> <span class="id" title="var">IH<sub>1</sub></span> <span class="id" title="var">Hmatch2</span> <span class="id" title="var">IH<sub>2</sub></span>].<br/>
- <span class="comment">(* MEmpty *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
- <span class="comment">(* MChar *)</span> <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MChar"><span class="id" title="constructor">MChar</span></a>.<br/>
- <span class="comment">(* MApp *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>1</sub></span>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="Poly.html#app_nil_r"><span class="id" title="axiom">app_nil_r</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="Poly.html#app_nil_r"><span class="id" title="axiom">app_nil_r</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="Poly.html#app_nil_r"><span class="id" title="axiom">app_nil_r</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="Poly.html#app_nil_r"><span class="id" title="axiom">app_nil_r</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MApp"><span class="id" title="constructor">MApp</span></a>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
-- <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
- <span class="comment">(* MUnionL *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionL"><span class="id" title="constructor">MUnionL</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MUnionR *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
+ <span class="id" title="tactic">destruct</span> <span class="id" title="var">re<sub>2</sub></span>.<br/>
× <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MUnionR"><span class="id" title="constructor">MUnionR</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH</span>.<br/>
- <span class="comment">(* MStar0 *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re</span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
+ <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re</span>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar0"><span class="id" title="constructor">MStar0</span></a>.<br/>
- <span class="comment">(* MStarApp *)</span> <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">re</span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
+ <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>1</sub></span>. <span class="id" title="tactic">inversion</span> <span class="id" title="var">IH<sub>2</sub></span>. <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MEmpty"><span class="id" title="constructor">MEmpty</span></a>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#star_app"><span class="id" title="lemma">star_app</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar1"><span class="id" title="lemma">MStar1</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#star_app"><span class="id" title="lemma">star_app</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar1"><span class="id" title="lemma">MStar1</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#star_app"><span class="id" title="lemma">star_app</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar1"><span class="id" title="lemma">MStar1</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#star_app"><span class="id" title="lemma">star_app</span></a>.<br/>
× <span class="id" title="tactic">apply</span> <a class="idref" href="IndProp.html#MStar1"><span class="id" title="lemma">MStar1</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>1</sub></span>.<br/>
× <span class="id" title="tactic">apply</span> <span class="id" title="var">IH<sub>2</sub></span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="comment">(* Use the tacticals described so far to shorten the proof. The proof<br/>
above is about 200 lines. Reduce it to 50 or fewer lines of similar<br/>
density. Solve each of the seven top-level bullets with a one-shot<br/>
proof.<br/>
<br/>
Hint: use a bottom-up approach. First copy-paste the entire proof<br/>
below. Then automate the innermost bullets first, proceeding<br/>
outwards. Frequently double-check that the entire proof still<br/>
compiles. If it doesn't, undo the most recent changes you made<br/>
until you get back to a compiling proof. *)</span><br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Lemma</span> <a id="re_opt_match'" class="idref" href="#re_opt_match'"><span class="id" title="lemma">re_opt_match'</span></a> : <span class="id" title="keyword">∀</span> <a id="T:45" class="idref" href="#T:45"><span class="id" title="binder">T</span></a> (<a id="re:46" class="idref" href="#re:46"><span class="id" title="binder">re</span></a>: <a class="idref" href="IndProp.html#reg_exp"><span class="id" title="inductive">reg_exp</span></a> <a class="idref" href="AltAuto.html#T:45"><span class="id" title="variable">T</span></a>) <a id="s:47" class="idref" href="#s:47"><span class="id" title="binder">s</span></a>,<br/>
<a class="idref" href="AltAuto.html#s:47"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re:46"><span class="id" title="variable">re</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="AltAuto.html#s:47"><span class="id" title="variable">s</span></a> <a class="idref" href="IndProp.html#70ea788eca33f3ac1bb7ed2e8169c791"><span class="id" title="notation">=~</span></a> <a class="idref" href="AltAuto.html#re_opt"><span class="id" title="definition">re_opt</span></a> <a class="idref" href="AltAuto.html#re:46"><span class="id" title="variable">re</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/>
<span class="comment">(* Do not modify the following line: *)</span><br/>
<span class="id" title="keyword">Definition</span> <a id="manual_grade_for_re_opt" class="idref" href="#manual_grade_for_re_opt"><span class="id" title="definition">manual_grade_for_re_opt</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</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="Poly.html#11c698c8685bb8ab1cf725545c085ac<sub>4</sub>"><span class="id" title="notation">×</span></a><a class="idref" href="IndProp.html#string"><span class="id" title="definition">string</span></a>) := <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<font size=-2>☐</font>
</div>
<div class="doc">
<a id="lab454"></a><h1 class="section">Tactics that Make Mentioning Names Unnecessary</h1>
<div class="paragraph"> </div>
So far we have been dependent on knowing the names of
hypotheses. For example, to prove the following simple theorem,
we hardcode the name <span class="inlinecode"><span class="id" title="var">HP</span></span>:
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="hyp_name" class="idref" href="#hyp_name"><span class="id" title="lemma">hyp_name</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:48" class="idref" href="#P:48"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>), <a class="idref" href="AltAuto.html#P:48"><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> <a class="idref" href="AltAuto.html#P:48"><span class="id" title="variable">P</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">HP</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">HP</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
We took the trouble to invent a name for <span class="inlinecode"><span class="id" title="var">HP</span></span>, then we had
to remember that name. If we later change the name in one place,
we have to change it everywhere. Likewise, if we were to add new
arguments to the theorem, we would have to adjust the <span class="inlinecode"><span class="id" title="tactic">intros</span></span>
list. That makes it challenging to maintain large proofs. So, Coq
provides several tactics that make it possible to write proof
scripts that do not hardcode names.
</div>
<div class="doc">
<a id="lab455"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="tactic">assumption</span></span> tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="tactic">assumption</span></span> tactic is useful to streamline the proof
above. It looks through the hypotheses and, if it finds the goal
as one them, it uses that to finish the proof.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="no_hyp_name" class="idref" href="#no_hyp_name"><span class="id" title="lemma">no_hyp_name</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:49" class="idref" href="#P:49"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>), <a class="idref" href="AltAuto.html#P:49"><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> <a class="idref" href="AltAuto.html#P:49"><span class="id" title="variable">P</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Some might argue to the contrary that hypothesis names
improve self-documention of proof scripts. Maybe they do,
sometimes. But in the case of the two proofs above, the first
mentions unnecessary detail, whereas the second could be
paraphrased simply as "the conclusion follows from the
assumptions."
<div class="paragraph"> </div>
Anyway, unlike informal (good) mathematical proofs, Coq proof
scripts are generally not that illuminating to readers. Worries
about rich, self-documenting names for hypotheses might be
misplaced.
</div>
<div class="doc">
<a id="lab456"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="var">contradiction</span></span> tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="var">contradiction</span></span> tactic handles some ad hoc situations where a
hypothesis contains <span class="inlinecode"><span class="id" title="var">False</span></span>, or two hypotheses derive <span class="inlinecode"><span class="id" title="var">False</span></span>.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="false_assumed" class="idref" href="#false_assumed"><span class="id" title="lemma">false_assumed</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</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> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="false_assumed'" class="idref" href="#false_assumed'"><span class="id" title="lemma">false_assumed'</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#False"><span class="id" title="inductive">False</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> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="var">contradiction</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="contras" class="idref" href="#contras"><span class="id" title="lemma">contras</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:50" class="idref" href="#P:50"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>), <a class="idref" href="AltAuto.html#P:50"><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> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="AltAuto.html#P:50"><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> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">HP</span> <span class="id" title="var">HNP</span>. <span class="id" title="var">exfalso</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">HNP</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">HP</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="contras'" class="idref" href="#contras'"><span class="id" title="lemma">contras'</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:51" class="idref" href="#P:51"><span class="id" title="binder">P</span></a> : <span class="id" title="keyword">Prop</span>), <a class="idref" href="AltAuto.html#P:51"><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> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#63a68285c81db8f9bc456233bb9ed181"><span class="id" title="notation">¬</span></a><a class="idref" href="AltAuto.html#P:51"><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> 0 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 1.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="var">contradiction</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
<a id="lab457"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="tactic">subst</span></span> tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="tactic">subst</span></span> tactic substitutes away an identifier, replacing
it everywhere and eliminating it from the context. That helps
us to avoid naming hypotheses in <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span>s.
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a id="many_eq" class="idref" href="#many_eq"><span class="id" title="lemma">many_eq</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:52" class="idref" href="#n:52"><span class="id" title="binder">n</span></a> <a id="m:53" class="idref" href="#m:53"><span class="id" title="binder">m</span></a> <a id="o:54" class="idref" href="#o:54"><span class="id" title="binder">o</span></a> <a id="p:55" class="idref" href="#p:55"><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="AltAuto.html#n:52"><span class="id" title="variable">n</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="AltAuto.html#m:53"><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/>
<a class="idref" href="AltAuto.html#o:54"><span class="id" title="variable">o</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="AltAuto.html#p:55"><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="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="AltAuto.html#n:52"><span class="id" title="variable">n</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="AltAuto.html#o:54"><span class="id" title="variable">o</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><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="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="AltAuto.html#m:53"><span class="id" title="variable">m</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="AltAuto.html#p:55"><span class="id" title="variable">p</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Proof</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">o</span> <span class="id" title="var">p</span> <span class="id" title="var">Hnm</span> <span class="id" title="var">Hop</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hnm</span>. <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hop</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="many_eq'" class="idref" href="#many_eq'"><span class="id" title="lemma">many_eq'</span></a> : <span class="id" title="keyword">∀</span> (<a id="n:56" class="idref" href="#n:56"><span class="id" title="binder">n</span></a> <a id="m:57" class="idref" href="#m:57"><span class="id" title="binder">m</span></a> <a id="o:58" class="idref" href="#o:58"><span class="id" title="binder">o</span></a> <a id="p:59" class="idref" href="#p:59"><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="AltAuto.html#n:56"><span class="id" title="variable">n</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="AltAuto.html#m:57"><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/>
<a class="idref" href="AltAuto.html#o:58"><span class="id" title="variable">o</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="AltAuto.html#p:59"><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="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="AltAuto.html#n:56"><span class="id" title="variable">n</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="AltAuto.html#o:58"><span class="id" title="variable">o</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><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="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">[</span></a><a class="idref" href="AltAuto.html#m:57"><span class="id" title="variable">m</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="AltAuto.html#p:59"><span class="id" title="variable">p</span></a><a class="idref" href="Poly.html#fa57d319973f6d58544a8887d0d48ea<sub>6</sub>"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="tactic">subst</span>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Actually there are two forms of this tactic.
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="tactic">subst</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> finds an assumption <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">e</span></span> or <span class="inlinecode"><span class="id" title="var">e</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">x</span></span> in the
context, replaces <span class="inlinecode"><span class="id" title="var">x</span></span> with <span class="inlinecode"><span class="id" title="var">e</span></span> throughout the context and
current goal, and removes the assumption from the context.
<div class="paragraph"> </div>
</li>
<li> <span class="inlinecode"><span class="id" title="tactic">subst</span></span> substitutes away <i>all</i> assumptions of the form <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">e</span></span>
or <span class="inlinecode"><span class="id" title="var">e</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" title="var">x</span></span>.
</li>
</ul>
</div>
<div class="doc">
<a id="lab458"></a><h2 class="section">The <span class="inlinecode"><span class="id" title="tactic">constructor</span></span> tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="tactic">constructor</span></span> tactic tries to find a constructor <span class="inlinecode"><span class="id" title="var">c</span></span> (from the
appropriate <span class="inlinecode"><span class="id" title="keyword">Inductive</span></span> definition in the current environment)
that can be applied to solve the current goal.
</div>
<div class="code">
<span class="id" title="keyword">Check</span> <a class="idref" href="IndProp.html#ev_0"><span class="id" title="constructor">ev_0</span></a> : <a class="idref" href="IndProp.html#ev"><span class="id" title="inductive">ev</span></a> 0.<br/>
<span class="id" title="keyword">Check</span> <a class="idref" href="IndProp.html#ev_SS"><span class="id" title="constructor">ev_SS</span></a> : <span class="id" title="keyword">∀</span> <a id="n:60" class="idref" href="#n:60"><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="IndProp.html#ev"><span class="id" title="inductive">ev</span></a> <a class="idref" href="AltAuto.html#n:60"><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="IndProp.html#ev"><span class="id" title="inductive">ev</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="AltAuto.html#n:60"><span class="id" title="variable">n</span></a>)).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="constructor_example" class="idref" href="#constructor_example"><span class="id" title="definition">constructor_example</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="IndProp.html#ev"><span class="id" title="inductive">ev</span></a> (<a class="idref" href="AltAuto.html#n:61"><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="AltAuto.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">induction</span> <span class="id" title="var">n</span>; <span class="id" title="tactic">simpl</span>.<br/>
- <span class="id" title="tactic">constructor</span>. <span class="comment">(* applies ev_0 *)</span><br/>
- <span class="id" title="tactic">rewrite</span> <a class="idref" href="Induction.html#add_comm"><span class="id" title="axiom">add_comm</span></a>. <span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">constructor</span>. <span class="comment">(* applies ev_SS *)</span><br/>
<span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
<div class="doc">
Warning: if more than one constructor can apply,
<span class="inlinecode"><span class="id" title="tactic">constructor</span></span> picks the first one, in the order in which they were
defined in the <span class="inlinecode"><span class="id" title="keyword">Inductive</span></span> definition. That might not be the one
you want.
</div>
<div class="doc">
<a id="lab459"></a><h1 class="section">Automatic Solvers</h1>
<div class="paragraph"> </div>
Coq has several special-purpose tactics that can solve
certain kinds of goals in a completely automated way. These
tactics are based on sophisticated algorithms developed for
verification in specific mathematical or logical domains.
<div class="paragraph"> </div>
Some automatic solvers are <i>decision procedures</i>, which are
algorithms that always terminate, and always give a correct
answer. Here, that means that they always find a correct proof, or
correctly determine that the goal is invalid. Other automatic
solvers are <i>incomplete</i>: they might fail to find a proof of a
valid goal.
</div>
<div class="doc">
<a id="lab460"></a><h2 class="section">Linear Integer Arithmetic: The <span class="inlinecode"><span class="id" title="var">lia</span></span> Tactic</h2>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" title="var">lia</span></span> tactic implements a decision procedure for integer
linear arithmetic, a subset of propositional logic and arithmetic.
As input it accepts goals constructed as follows:
<div class="paragraph"> </div>
<ul class="doclist">
<li> variables and constants of type <span class="inlinecode"><span class="id" title="var">nat</span></span>, <span class="inlinecode"><span class="id" title="var">Z</span></span>, and other integer
types;
<div class="paragraph"> </div>
</li>
<li> arithmetic operators <span class="inlinecode">+</span>, <span class="inlinecode">-</span>, <span class="inlinecode">×</span>, and <span class="inlinecode">^</span>;
<div class="paragraph"> </div>
</li>
<li> equality <span class="inlinecode">=</span> and ordering <span class="inlinecode"><</span>, <span class="inlinecode">></span>, <span class="inlinecode">≤</span>, <span class="inlinecode">≥</span>; and
<div class="paragraph"> </div>
</li>
<li> the logical connectives <span class="inlinecode">∧</span>, <span class="inlinecode">∨</span>, <span class="inlinecode">¬</span>, <span class="inlinecode">→</span>, and <span class="inlinecode">↔</span>; and
constants <span class="inlinecode"><span class="id" title="var">True</span></span> and <span class="inlinecode"><span class="id" title="var">False</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
<i>Linear</i> goals involve (in)equalities over expressions of the form
<span class="inlinecode"><span class="id" title="var">c<sub>1</sub></span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">x<sub>1</sub></span></span> <span class="inlinecode">+</span> <span class="inlinecode">...</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" title="var">cn</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" title="var">xn</span></span>, where <span class="inlinecode"><span class="id" title="var">ci</span></span> are constants and <span class="inlinecode"><span class="id" title="var">xi</span></span> are
variables.
<div class="paragraph"> </div>
<ul class="doclist">
<li> For linear goals, <span class="inlinecode"><span class="id" title="var">lia</span></span> will either solve the goal or fail,
meaning that the goal is actually invalid.
<div class="paragraph"> </div>
</li>
<li> For non-linear goals, <span class="inlinecode"><span class="id" title="var">lia</span></span> will also either solve the goal or
fail. But in this case, the failure does not necessarily mean
that the goal is invalid -- it might just be beyond <span class="inlinecode"><span class="id" title="var">lia</span></span>'s
reach to prove because of non-linearity.
</li>
</ul>