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UseTactics.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>UseTactics: Tactic Library for Coq: A Gentle Introduction</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">UseTactics<span class="subtitle">Tactic Library for Coq: A Gentle Introduction</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Chapter maintained by Arthur Chargueraud *)</span><br/>
<br/>
</div>
<div class="doc">
Coq comes with a set of builtin tactics, such as <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>,
<span class="inlinecode"><span class="id" type="tactic">intros</span></span>, <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> and so on. While it is possible to conduct
proofs using only those tactics, you can significantly increase
your productivity by working with a set of more powerful tactics.
This chapter describes a number of such very useful tactics, which,
for various reasons, are not yet available by default in Coq.
These tactics are defined in the <span class="inlinecode"><span class="id" type="var">LibTactics.v</span></span> file.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">LibTactics</span>.<br/>
<br/>
</div>
<div class="doc">
Remark: SSReflect is another package providing powerful tactics.
The library "LibTactics" differs from "SSReflect" in two respects:
<div class="paragraph"> </div>
<ul class="doclist">
<li> "SSReflect" was primarily developed for proving mathematical
theorems, whereas "LibTactics" was primarily developed for proving
theorems on programming languages. In particular, "LibTactics"
provides a number of useful tactics that have no counterpart in the
"SSReflect" package.
</li>
<li> "SSReflect" entirely rethinks the presentation of tactics,
whereas "LibTactics" mostly stick to the traditional
presentation of Coq tactics, simply providing a number of
additional tactics. For this reason, "LibTactics" is
probably easier to get started with than "SSReflect".
</li>
</ul>
<div class="paragraph"> </div>
This chapter is a tutorial focusing on the most useful features
from the "LibTactics" library. It does not aim at presenting all
the features of "LibTactics". The detailed specification of tactics
can be found in the source file <span class="inlinecode"><span class="id" type="var">LibTactics.v</span></span>. Further documentation
as well as demos can be found at http://www.chargueraud.org/softs/tlc/ .
<div class="paragraph"> </div>
In this tutorial, tactics are presented using examples taken from
the core chapters of the "Software Foundations" course. To illustrate
the various ways in which a given tactic can be used, we use a
tactic that duplicates a given goal. More precisely, <span class="inlinecode"><span class="id" type="var">dup</span></span> produces
two copies of the current goal, and <span class="inlinecode"><span class="id" type="var">dup</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> produces <span class="inlinecode"><span class="id" type="var">n</span></span> copies of it.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab971"></a><h1 class="section">Tactics for introduction and case analysis</h1>
<div class="paragraph"> </div>
This section presents the following tactics:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">introv</span></span>, for naming hypotheses more efficiently,
</li>
<li> <span class="inlinecode"><span class="id" type="var">inverts</span></span>, for improving the <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic,
</li>
<li> <span class="inlinecode"><span class="id" type="var">cases</span></span>, for performing a case analysis without losing information,
</li>
<li> <span class="inlinecode"><span class="id" type="var">cases_if</span></span>, for automating case analysis on the argument of <span class="inlinecode"><span class="id" type="keyword">if</span></span>.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab972"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">introv</span></span></h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">IntrovExamples</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Stlc</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">Imp</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">introv</span></span> allows to automatically introduce the
variables of a theorem and explicitly name the hypotheses
involved. In the example shown next, the variables <span class="inlinecode"><span class="id" type="var">c</span></span>,
<span class="inlinecode"><span class="id" type="var">st</span></span>, <span class="inlinecode"><span class="id" type="var">st1</span></span> and <span class="inlinecode"><span class="id" type="var">st2</span></span> involved in the statement of determinism
need not be named explicitly, because their name where already
given in the statement of the lemma. On the contrary, it is
useful to provide names for the two hypotheses, which we
name <span class="inlinecode"><span class="id" type="var">E1</span></span> and <span class="inlinecode"><span class="id" type="var">E2</span></span>, respectively.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ceval_deterministic</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st1</span> <span class="id" type="var">st2</span>,<br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st1</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">st1</span> = <span class="id" type="var">st2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">E1</span> <span class="id" type="var">E2</span>. <span class="comment">(* was <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">st1</span></span> <span class="inlinecode"><span class="id" type="var">st2</span></span> <span class="inlinecode"><span class="id" type="var">E1</span></span> <span class="inlinecode"><span class="id" type="var">E2</span></span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
When there is no hypothesis to be named, one can call
<span class="inlinecode"><span class="id" type="var">introv</span></span> without any argument.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">dist_exists_or</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> <span class="id" type="var">Q</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">Q</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">↔</span> (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">∨</span> (<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">Q</span> <span class="id" type="var">x</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span>. <span class="comment">(* was <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">Q</span></span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">introv</span></span> also applies to statements in which
<span class="inlinecode"><span style="font-family: arial;">∀</span></span> and <span class="inlinecode"><span style="font-family: arial;">→</span></span> are interleaved.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ceval_deterministic'</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st1</span>,<br/>
(<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st1</span>) <span style="font-family: arial;">→</span> <span style="font-family: arial;">∀</span><span class="id" type="var">st2</span>, (<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st2</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">st1</span> = <span class="id" type="var">st2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">E1</span> <span class="id" type="var">E2</span>. <span class="comment">(* was <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">st1</span></span> <span class="inlinecode"><span class="id" type="var">E1</span></span> <span class="inlinecode"><span class="id" type="var">st2</span></span> <span class="inlinecode"><span class="id" type="var">E2</span></span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
Like the arguments of <span class="inlinecode"><span class="id" type="tactic">intros</span></span>, the arguments of <span class="inlinecode"><span class="id" type="var">introv</span></span>
can be structured patterns.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">exists_impl</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">X</span> (<span class="id" type="var">P</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>) (<span class="id" type="var">Q</span> : <span class="id" type="keyword">Prop</span>) (<span class="id" type="var">R</span> : <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span>) <span style="font-family: arial;">→</span><br/>
((<span style="font-family: arial;">∃</span><span class="id" type="var">x</span>, <span class="id" type="var">P</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> [<span class="id" type="var">x</span> <span class="id" type="var">H2</span>]. <span class="id" type="tactic">eauto</span>.<br/>
<span class="comment">(* same as <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">Q</span></span> <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">H1</span></span> <span class="inlinecode">[<span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">H2</span>].</span>, which is itself short <br/>
for <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">Q</span></span> <span class="inlinecode"><span class="id" type="var">R</span></span> <span class="inlinecode"><span class="id" type="var">H1</span></span> <span class="inlinecode"><span class="id" type="var">H2</span>.</span> <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">H2</span></span> <span class="inlinecode"><span class="id" type="keyword">as</span></span> <span class="inlinecode">[<span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">H2</span>].</span> *)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Remark: the tactic <span class="inlinecode"><span class="id" type="var">introv</span></span> works even when definitions
need to be unfolded in order to reveal hypotheses.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">IntrovExamples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab973"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span></h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">InvertsExamples</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Stlc</span> <span class="id" type="var">Equiv</span> <span class="id" type="var">Imp</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> tactic of Coq is not very satisfying for
three reasons. First, it produces a bunch of equalities
which one typically wants to substitute away, using <span class="inlinecode"><span class="id" type="tactic">subst</span></span>.
Second, it introduces meaningless names for hypotheses.
Third, a call to <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> does not remove <span class="inlinecode"><span class="id" type="var">H</span></span> from the
context, even though in most cases an hypothesis is no longer
needed after being inverted. The tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> address all
of these three issues. It is intented to be used in place of
the tactic <span class="inlinecode"><span class="id" type="tactic">inversion</span></span>.
<div class="paragraph"> </div>
The following example illustrates how the tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span>
behaves mostly like <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> except that it performs
some substitutions in order to eliminate the trivial equalities
that are being produced by <span class="inlinecode"><span class="id" type="tactic">inversion</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">skip_left</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span>,<br/>
<span class="id" type="var">cequiv</span> (<span class="id" type="var">SKIP</span>;; <span class="id" type="var">c</span>) <span class="id" type="var">c</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span>. <span class="id" type="tactic">split</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">dup</span>. <span class="comment">(* duplicate the goal for comparison *)</span><br/>
<span class="comment">(* was: *)</span><br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="comment">(* now: *)</span><br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span>. <span class="id" type="var">inverts</span> <span class="id" type="var">H2</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
A slightly more interesting example appears next.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ceval_deterministic</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st1</span> <span class="id" type="var">st2</span>,<br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st1</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">st1</span> = <span class="id" type="var">st2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">E1</span> <span class="id" type="var">E2</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">st2</span>.<br/>
(<span class="id" type="var">ceval_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">E1</span>) <span class="id" type="var">Case</span>); <span class="id" type="tactic">intros</span> <span class="id" type="var">st2</span> <span class="id" type="var">E2</span>.<br/>
<span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>. <span class="comment">(* skip some basic cases *)</span><br/>
<span class="id" type="var">dup</span>. <span class="comment">(* duplicate the goal for comparison *)</span><br/>
<span class="comment">(* was: *)</span> <span class="id" type="tactic">inversion</span> <span class="id" type="var">E2</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="var">admit</span>.<br/>
<span class="comment">(* now: *)</span> <span class="id" type="var">inverts</span> <span class="id" type="var">E2</span>. <span class="id" type="var">admit</span>.<br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode"><span class="id" type="keyword">as</span>.</span> is like <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> except that the
variables and hypotheses being produced are placed in the goal
rather than in the context. This strategy allows naming those
new variables and hypotheses explicitly, using either <span class="inlinecode"><span class="id" type="tactic">intros</span></span>
or <span class="inlinecode"><span class="id" type="var">introv</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ceval_deterministic'</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span> <span class="id" type="var">st</span> <span class="id" type="var">st1</span> <span class="id" type="var">st2</span>,<br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st1</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">c</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇓</span> <span class="id" type="var">st2</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">st1</span> = <span class="id" type="var">st2</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">E1</span> <span class="id" type="var">E2</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">st2</span>.<br/>
(<span class="id" type="var">ceval_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">E1</span>) <span class="id" type="var">Case</span>); <span class="id" type="tactic">intros</span> <span class="id" type="var">st2</span> <span class="id" type="var">E2</span>; <br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">E2</span> <span class="id" type="keyword">as</span>.<br/>
<span class="id" type="var">Case</span> "E_Skip". <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "E_Ass".<br/>
<span class="comment">(* Observe that the variable <span class="inlinecode"><span class="id" type="var">n</span></span> is not automatically <br/>
substituted because, contrary to <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> <span class="inlinecode"><span class="id" type="var">E2</span>;</span> <span class="inlinecode"><span class="id" type="tactic">subst</span></span>,<br/>
the tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">E2</span></span> does not substitute the equalities<br/>
that exist before running the inversion. *)</span><br/>
<span class="comment">(* new: *)</span> <span class="id" type="tactic">subst</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "E_Seq".<br/>
<span class="comment">(* Here, the newly created variables can be introduced<br/>
using intros, so they can be assigned meaningful names,<br/>
for example <span class="inlinecode"><span class="id" type="var">st3</span></span> instead of <span class="inlinecode"><span class="id" type="var">st'0</span></span>. *)</span><br/>
<span class="comment">(* new: *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">st3</span> <span class="id" type="var">Red1</span> <span class="id" type="var">Red2</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">st'</span> = <span class="id" type="var">st3</span>) <span class="id" type="keyword">as</span> <span class="id" type="var">EQ1</span>.<br/>
<span class="id" type="var">SCase</span> "Proof of assertion". <span class="id" type="tactic">apply</span> <span class="id" type="var">IHE1_1</span>; <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="tactic">subst</span> <span class="id" type="var">st3</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHE1_2</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">Case</span> "E_IfTrue".<br/>
<span class="id" type="var">SCase</span> "b1 evaluates to true".<br/>
<span class="comment">(* In an easy case like this one, there is no need to<br/>
provide meaningful names, so we can just use <span class="inlinecode"><span class="id" type="tactic">intros</span></span> *)</span><br/>
<span class="comment">(* new: *)</span> <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">IHE1</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="var">SCase</span> "b1 evaluates to false (contradiction)".<br/>
<span class="comment">(* new: *)</span> <span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H5</span>. <span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>.<br/>
<span class="comment">(* The other cases are similiar *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
In the particular case where a call to <span class="inlinecode"><span class="id" type="tactic">inversion</span></span> produces
a single subgoal, one can use the syntax <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode"><span class="id" type="keyword">as</span></span> <span class="inlinecode"><span class="id" type="var">H1</span></span> <span class="inlinecode"><span class="id" type="var">H2</span></span> <span class="inlinecode"><span class="id" type="var">H3</span></span>
for calling <span class="inlinecode"><span class="id" type="var">inverts</span></span> and naming the new hypotheses <span class="inlinecode"><span class="id" type="var">H1</span></span>, <span class="inlinecode"><span class="id" type="var">H2</span></span>
and <span class="inlinecode"><span class="id" type="var">H3</span></span>. In other words, the tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode"><span class="id" type="keyword">as</span></span> <span class="inlinecode"><span class="id" type="var">H1</span></span> <span class="inlinecode"><span class="id" type="var">H2</span></span> <span class="inlinecode"><span class="id" type="var">H3</span></span> is
equivalent to <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode"><span class="id" type="keyword">as</span>;</span> <span class="inlinecode"><span class="id" type="var">introv</span></span> <span class="inlinecode"><span class="id" type="var">H1</span></span> <span class="inlinecode"><span class="id" type="var">H2</span></span> <span class="inlinecode"><span class="id" type="var">H3</span></span>. An example follows.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">skip_left'</span>: <span style="font-family: arial;">∀</span><span class="id" type="var">c</span>,<br/>
<span class="id" type="var">cequiv</span> (<span class="id" type="var">SKIP</span>;; <span class="id" type="var">c</span>) <span class="id" type="var">c</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span>. <span class="id" type="tactic">split</span>; <span class="id" type="tactic">intros</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">U</span> <span class="id" type="var">V</span>. <span class="comment">(* new hypotheses are named <span class="inlinecode"><span class="id" type="var">U</span></span> and <span class="inlinecode"><span class="id" type="var">V</span></span> *)</span><br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">U</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
A more involved example appears next. In particular, this example
shows that the name of the hypothesis being inverted can be reused.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">typing_nonexample_1</span> :<br/>
¬ <span style="font-family: arial;">∃</span><span class="id" type="var">T</span>,<br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">TBool</span><br/>
(<span class="id" type="var">tabs</span> <span class="id" type="var">y</span> <span class="id" type="var">TBool</span><br/>
(<span class="id" type="var">tapp</span> (<span class="id" type="var">tvar</span> <span class="id" type="var">x</span>) (<span class="id" type="var">tvar</span> <span class="id" type="var">y</span>))))<br/>
<span class="id" type="var">T</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">dup</span> 3.<br/>
<br/>
<span class="comment">(* The old proof: *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">C</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">C</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H4</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H4</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H2</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H5</span>. <span class="id" type="tactic">subst</span>. <span class="id" type="tactic">clear</span> <span class="id" type="var">H5</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">H1</span>.<br/>
<br/>
<span class="comment">(* The new proof: *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">C</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">C</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H1</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H2</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H3</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H3</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H4</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H4</span>.<br/>
<br/>
<span class="comment">(* The new proof, alternative: *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">C</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">C</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span> <span class="id" type="keyword">as</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="var">inverts</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">InvertsExamples</span>.<br/>
<br/>
</div>
<div class="doc">
Note: in the rare cases where one needs to perform an inversion
on an hypothesis <span class="inlinecode"><span class="id" type="var">H</span></span> without clearing <span class="inlinecode"><span class="id" type="var">H</span></span> from the context,
one can use the tactic <span class="inlinecode"><span class="id" type="var">inverts</span></span> <span class="inlinecode"><span class="id" type="var">keep</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span>, where the keyword <span class="inlinecode"><span class="id" type="var">keep</span></span>
indicates that the hypothesis should be kept in the context.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab974"></a><h1 class="section">Tactics for n-ary connectives</h1>
<div class="paragraph"> </div>
Because Coq encodes conjunctions and disjunctions using binary
constructors <span class="inlinecode"><span style="font-family: arial;">∧</span></span> and <span class="inlinecode"><span style="font-family: arial;">∨</span></span>, working with a conjunction or a
disjunction of <span class="inlinecode"><span class="id" type="var">N</span></span> facts can sometimes be quite cumbursome.
For this reason, "LibTactics" provides tactics offering direct
support for n-ary conjunctions and disjunctions. It also provides
direct support for n-ary existententials.
<div class="paragraph"> </div>
This section presents the following tactics:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">splits</span></span> for decomposing n-ary conjunctions,
</li>
<li> <span class="inlinecode"><span class="id" type="var">branch</span></span> for decomposing n-ary disjunctions,
</li>
<li> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> for proving n-ary existentials.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">NaryExamples</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">References</span> <span class="id" type="var">SfLib</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">STLCRef</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab975"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">splits</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">splits</span></span> applies to a goal made of a conjunction
of <span class="inlinecode"><span class="id" type="var">n</span></span> propositions and it produces <span class="inlinecode"><span class="id" type="var">n</span></span> subgoals. For example,
it decomposes the goal <span class="inlinecode"><span class="id" type="var">G1</span></span> <span class="inlinecode"><span style="font-family: arial;">∧</span></span> <span class="inlinecode"><span class="id" type="var">G2</span></span> <span class="inlinecode"><span style="font-family: arial;">∧</span></span> <span class="inlinecode"><span class="id" type="var">G3</span></span> into the three subgoals
<span class="inlinecode"><span class="id" type="var">G1</span></span>, <span class="inlinecode"><span class="id" type="var">G2</span></span> and <span class="inlinecode"><span class="id" type="var">G3</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_splits</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">n</span> > 0 <span style="font-family: arial;">∧</span> <span class="id" type="var">n</span> < <span class="id" type="var">m</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">m</span> < <span class="id" type="var">n</span>+10 <span style="font-family: arial;">∧</span> <span class="id" type="var">m</span> ≠ 3.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="var">splits</span>.<br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab976"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">branch</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">branch</span></span> <span class="inlinecode"><span class="id" type="var">k</span></span> can be used to prove a n-ary disjunction.
For example, if the goal takes the form <span class="inlinecode"><span class="id" type="var">G1</span></span> <span class="inlinecode"><span style="font-family: arial;">∨</span></span> <span class="inlinecode"><span class="id" type="var">G2</span></span> <span class="inlinecode"><span style="font-family: arial;">∨</span></span> <span class="inlinecode"><span class="id" type="var">G3</span></span>,
the tactic <span class="inlinecode"><span class="id" type="var">branch</span></span> <span class="inlinecode">2</span> leaves only <span class="inlinecode"><span class="id" type="var">G2</span></span> as subgoal. The following
example illustrates the behavior of the <span class="inlinecode"><span class="id" type="var">branch</span></span> tactic.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_branch</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span>,<br/>
<span class="id" type="var">n</span> < <span class="id" type="var">m</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">n</span> = <span class="id" type="var">m</span> <span style="font-family: arial;">∨</span> <span class="id" type="var">m</span> < <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>.<br/>
<span class="id" type="tactic">destruct</span> (<span class="id" type="var">lt_eq_lt_dec</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>) <span class="id" type="keyword">as</span> [[<span class="id" type="var">H1</span>|<span class="id" type="var">H2</span>]|<span class="id" type="var">H3</span>].<br/>
<span class="id" type="var">branch</span> 1. <span class="id" type="tactic">apply</span> <span class="id" type="var">H1</span>.<br/>
<span class="id" type="var">branch</span> 2. <span class="id" type="tactic">apply</span> <span class="id" type="var">H2</span>.<br/>
<span class="id" type="var">branch</span> 3. <span class="id" type="tactic">apply</span> <span class="id" type="var">H3</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab977"></a><h2 class="section">The tactic <span class="inlinecode"><span style="font-family: arial;">∃</span></span></h2>
<div class="paragraph"> </div>
The library "LibTactics" introduces a notation for n-ary
existentials. For example, one can write <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="var">z</span>,</span> <span class="inlinecode"><span class="id" type="var">H</span></span>
instead of <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">x</span>,</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">y</span>,</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">z</span>,</span> <span class="inlinecode"><span class="id" type="var">H</span></span>. Similarly,
the library provides a n-ary tactic <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span>, which is a
shorthand for <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">a</span>;</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">b</span>;</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">c</span></span>. The following
example illustrates both the notation and the tactic for
dealing with n-ary existentials.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="tactic">progress</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">ST</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">st</span>,<br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <span class="id" type="var">ST</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">store_well_typed</span> <span class="id" type="var">ST</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">value</span> <span class="id" type="var">t</span> <span style="font-family: arial;">∨</span> <span style="font-family: arial;">∃</span><span class="id" type="var">t'</span> <span class="id" type="var">st'</span>, <span class="id" type="var">t</span> / <span class="id" type="var">st</span> <span style="font-family: arial;">⇒</span> <span class="id" type="var">t'</span> / <span class="id" type="var">st'</span>.<br/>
<span class="comment">(* was: <span class="inlinecode"><span class="id" type="var">value</span></span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode"><span style="font-family: arial;">∨</span></span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">t'</span>,</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">st'</span>,</span> <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span style="font-family: arial;">⇒</span></span> <span class="inlinecode"><span class="id" type="var">t'</span></span> <span class="inlinecode">/</span> <span class="inlinecode"><span class="id" type="var">st'</span></span> *)</span><br/>
<span class="id" type="keyword">Proof</span> <span class="id" type="keyword">with</span> <span class="id" type="tactic">eauto</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">ST</span> <span class="id" type="var">t</span> <span class="id" type="var">T</span> <span class="id" type="var">st</span> <span class="id" type="var">Ht</span> <span class="id" type="var">HST</span>. <span class="id" type="var">remember</span> (@<span class="id" type="var">empty</span> <span class="id" type="var">ty</span>) <span class="id" type="keyword">as</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
(<span class="id" type="var">has_type_cases</span> (<span class="id" type="tactic">induction</span> <span class="id" type="var">Ht</span>) <span class="id" type="var">Case</span>); <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>...<br/>
<span class="id" type="var">Case</span> "T_App".<br/>
<span class="id" type="var">right</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt1</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Ht1p</span> | <span class="id" type="var">Ht1p</span>]...<br/>
<span class="id" type="var">SCase</span> "t<sub>1</sub> is a value".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Ht1p</span>; <span class="id" type="tactic">subst</span>; <span class="id" type="tactic">try</span> <span class="id" type="var">solve</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">inversion</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">IHHt2</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">Ht2p</span> | <span class="id" type="var">Ht2p</span>]...<br/>
<span class="id" type="var">SSCase</span> "t<sub>2</sub> steps".<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">Ht2p</span> <span class="id" type="keyword">as</span> [<span class="id" type="var">t<sub>2</sub>'</span> [<span class="id" type="var">st'</span> <span class="id" type="var">Hstep</span>]].<br/>
<span style="font-family: arial;">∃</span>(<span class="id" type="var">tapp</span> (<span class="id" type="var">tabs</span> <span class="id" type="var">x</span> <span class="id" type="var">T</span> <span class="id" type="var">t</span>) <span class="id" type="var">t<sub>2</sub>'</span>) <span class="id" type="var">st'</span>...<br/>
<span class="comment">(* was: <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode">(<span class="id" type="var">tapp</span></span> <span class="inlinecode">(<span class="id" type="var">tabs</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">T</span></span> <span class="inlinecode"><span class="id" type="var">t</span>)</span> <span class="inlinecode"><span class="id" type="var">t<sub>2</sub>'</span>).</span> <span class="inlinecode"><span style="font-family: arial;">∃</span></span> <span class="inlinecode"><span class="id" type="var">st'</span>...</span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
Remark: a similar facility for n-ary existentials is provided
by the module <span class="inlinecode"><span class="id" type="var">Coq.Program.Syntax</span></span> from the standard library.
(<span class="inlinecode"><span class="id" type="var">Coq.Program.Syntax</span></span> supports existentials up to arity 4;
<span class="inlinecode"><span class="id" type="var">LibTactics</span></span> supports them up to arity 10.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">NaryExamples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab978"></a><h1 class="section">Tactics for working with equality</h1>
<div class="paragraph"> </div>
One of the major weakness of Coq compared with other interactive
proof assistants is its relatively poor support for reasoning
with equalities. The tactics described next aims at simplifying
pieces of proof scripts manipulating equalities.
<div class="paragraph"> </div>
This section presents the following tactics:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> for introducing an equality to rewrite with,
</li>
<li> <span class="inlinecode"><span class="id" type="var">cuts_rewrite</span></span>, which is similar except that its subgoals are swapped,
</li>
<li> <span class="inlinecode"><span class="id" type="var">substs</span></span> for improving the <span class="inlinecode"><span class="id" type="tactic">subst</span></span> tactic,
</li>
<li> <span class="inlinecode"><span class="id" type="var">fequals</span></span> for improving the <span class="inlinecode"><span class="id" type="tactic">f_equal</span></span> tactic,
</li>
<li> <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> for proving <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> using an hypothesis <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span>,
automatically producing an equality <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">z</span></span> as subgoal.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">EqualityExamples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab979"></a><h2 class="section">The tactics <span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> and <span class="inlinecode"><span class="id" type="var">cuts_rewrite</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> <span class="inlinecode">(<span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span>)</span> replaces <span class="inlinecode"><span class="id" type="var">E1</span></span> with <span class="inlinecode"><span class="id" type="var">E2</span></span> in
the goal, and produces the goal <span class="inlinecode"><span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_plus</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>,<br/>
(0 + <span class="id" type="var">n</span>) × <span class="id" type="var">m</span> = <span class="id" type="var">n</span> × <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">dup</span>.<br/>
<span class="comment">(* The old proof: *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H</span>: 0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>). <span class="id" type="tactic">reflexivity</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<br/>
<span class="comment">(* The new proof: *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>.<br/>
<span class="id" type="var">asserts_rewrite</span> (0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>).<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* subgoal <span class="inlinecode">0+<span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span> *)</span><br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* subgoal <span class="inlinecode"><span class="id" type="var">n</span>×<span class="id" type="var">m</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span>×<span class="id" type="var">n</span></span> *)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
<span class="comment">(*** Remark: the syntax <span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> <span class="inlinecode">(<span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span>)</span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> allows<br/>
rewriting in the hypothesis <span class="inlinecode"><span class="id" type="var">H</span></span> rather than in the goal. *)</span><br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">cuts_rewrite</span></span> <span class="inlinecode">(<span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span>)</span> is like
<span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> <span class="inlinecode">(<span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span>)</span>, except that the equality <span class="inlinecode"><span class="id" type="var">E1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">E2</span></span>
appears as first subgoal.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_plus'</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <span class="id" type="var">nat</span>,<br/>
(0 + <span class="id" type="var">n</span>) × <span class="id" type="var">m</span> = <span class="id" type="var">n</span> × <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span>.<br/>
<span class="id" type="var">cuts_rewrite</span> (0 + <span class="id" type="var">n</span> = <span class="id" type="var">n</span>).<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* subgoal <span class="inlinecode"><span class="id" type="var">n</span>×<span class="id" type="var">m</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span>×<span class="id" type="var">n</span></span> *)</span><br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* subgoal <span class="inlinecode">0+<span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span> *)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
More generally, the tactics <span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> and <span class="inlinecode"><span class="id" type="var">cuts_rewrite</span></span>
can be provided a lemma as argument. For example, one can write
<span class="inlinecode"><span class="id" type="var">asserts_rewrite</span></span> <span class="inlinecode">(<span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span>,</span> <span class="inlinecode"><span class="id" type="var">a</span>*(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">b</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">a</span>×<span class="id" type="var">b</span>+<span class="id" type="var">a</span>)</span>.
This formulation is useful when <span class="inlinecode"><span class="id" type="var">a</span></span> and <span class="inlinecode"><span class="id" type="var">b</span></span> are big terms,
since there is no need to repeat their statements.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_plus''</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">u</span> <span class="id" type="var">v</span> <span class="id" type="var">w</span> <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>: <span class="id" type="var">nat</span>,<br/>
(<span class="id" type="var">u</span> + <span class="id" type="var">v</span>) × (<span class="id" type="var">S</span> (<span class="id" type="var">w</span> × <span class="id" type="var">x</span> + <span class="id" type="var">y</span>)) = <span class="id" type="var">z</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="var">asserts_rewrite</span> (<span style="font-family: arial;">∀</span><span class="id" type="var">a</span> <span class="id" type="var">b</span>, <span class="id" type="var">a</span>*(<span class="id" type="var">S</span> <span class="id" type="var">b</span>) = <span class="id" type="var">a</span>×<span class="id" type="var">b</span>+<span class="id" type="var">a</span>).<br/>
<span class="comment">(* first subgoal: <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode"><span class="id" type="var">b</span>,</span> <span class="inlinecode"><span class="id" type="var">a</span>*(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">b</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">a</span>×<span class="id" type="var">b</span>+<span class="id" type="var">a</span></span> *)</span><br/>
<span class="comment">(* second subgoal: <span class="inlinecode">(<span class="id" type="var">u</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">v</span>)</span> <span class="inlinecode">×</span> <span class="inlinecode">(<span class="id" type="var">w</span></span> <span class="inlinecode">×</span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">y</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">u</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">v</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">z</span></span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab980"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">substs</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">substs</span></span> is similar to <span class="inlinecode"><span class="id" type="tactic">subst</span></span> except that it
does not fail when the goal contains "circular equalities",
such as <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">f</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_substs</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">x</span> <span class="id" type="var">y</span> (<span class="id" type="var">f</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>), <br/>
<span class="id" type="var">x</span> = <span class="id" type="var">f</span> <span class="id" type="var">x</span> <span style="font-family: arial;">→</span> <span class="id" type="var">y</span> = <span class="id" type="var">x</span> <span style="font-family: arial;">→</span> <span class="id" type="var">y</span> = <span class="id" type="var">f</span> <span class="id" type="var">x</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="var">substs</span>. <span class="comment">(* the tactic <span class="inlinecode"><span class="id" type="tactic">subst</span></span> would fail here *)</span><br/>
<span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab981"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">fequals</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">fequals</span></span> is similar to <span class="inlinecode"><span class="id" type="tactic">f_equal</span></span> except that it
directly discharges all the trivial subgoals produced. Moreover,
the tactic <span class="inlinecode"><span class="id" type="var">fequals</span></span> features an enhanced treatment of equalities
between tuples.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_fequals</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> <span class="id" type="var">e</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">f</span> : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>),<br/>
<span class="id" type="var">a</span> = 1 <span style="font-family: arial;">→</span> <span class="id" type="var">b</span> = <span class="id" type="var">e</span> <span style="font-family: arial;">→</span> <span class="id" type="var">e</span> = 2 <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">f</span> <span class="id" type="var">a</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> <span class="id" type="var">d</span> = <span class="id" type="var">f</span> 1 2 <span class="id" type="var">c</span> 4.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="var">fequals</span>.<br/>
<span class="comment">(* subgoals <span class="inlinecode"><span class="id" type="var">a</span></span> <span class="inlinecode">=</span> <span class="inlinecode">1</span>, <span class="inlinecode"><span class="id" type="var">b</span></span> <span class="inlinecode">=</span> <span class="inlinecode">2</span> and <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">c</span></span> are proved, <span class="inlinecode"><span class="id" type="var">d</span></span> <span class="inlinecode">=</span> <span class="inlinecode">4</span> remains *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab982"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">applys_eq</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> is a variant of <span class="inlinecode"><span class="id" type="tactic">eapply</span></span> that introduces
equalities for subterms that do not unify. For example, assume
the goal is the proposition <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> and assume we have the
assumption <span class="inlinecode"><span class="id" type="var">H</span></span> asserting that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span> holds. We know that we can
prove <span class="inlinecode"><span class="id" type="var">y</span></span> to be equal to <span class="inlinecode"><span class="id" type="var">z</span></span>. So, we could call the tactic
<span class="inlinecode"><span class="id" type="var">assert_rewrite</span></span> <span class="inlinecode">(<span class="id" type="var">y</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">z</span>)</span> and change the goal to <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span>, but
this would require copy-pasting the values of <span class="inlinecode"><span class="id" type="var">y</span></span> and <span class="inlinecode"><span class="id" type="var">z</span></span>.
With the tactic <span class="inlinecode"><span class="id" type="var">applys_eq</span></span>, we can call <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">1</span>, which
proves the goal and leaves only the subgoal <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">z</span></span>. The value <span class="inlinecode">1</span>
given as argument to <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> indicates that we want an equality
to be introduced for the first argument of <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> counting from
the right. The three following examples illustrate the behavior
of a call to <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">1</span>, a call to <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">2</span>, and a
call to <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">1</span> <span class="inlinecode">2</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Axiom</span> <span class="id" type="var">big_expression_using</span> : <span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span>. <span class="comment">(* Used in the example *)</span><br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_applys_eq_1</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>, <br/>
<span class="id" type="var">P</span> <span class="id" type="var">x</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">z</span>) <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">P</span> <span class="id" type="var">x</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">y</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">H</span>. <span class="id" type="var">dup</span>.<br/>
<br/>
<span class="comment">(* The old proof: *)</span><br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">Eq</span>: <span class="id" type="var">big_expression_using</span> <span class="id" type="var">y</span> = <span class="id" type="var">big_expression_using</span> <span class="id" type="var">z</span>).<br/>
<span class="id" type="var">admit</span>. <span class="comment">(* Assume we can prove this equality somehow. *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span class="id" type="var">Eq</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>.<br/>
<br/>
<span class="comment">(* The new proof: *)</span><br/>
<span class="id" type="var">applys_eq</span> <span class="id" type="var">H</span> 1.<br/>
<span class="id" type="var">admit</span>. <span class="comment">(* Assume we can prove this equality somehow. *)</span><br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
If the mismatch was on the first argument of <span class="inlinecode"><span class="id" type="var">P</span></span> instead of
the second, we would have written <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">2</span>. Recall
that the occurences are counted from the right.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_applys_eq_2</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) <span class="id" type="var">x</span> <span class="id" type="var">y</span> <span class="id" type="var">z</span>, <br/>
<span class="id" type="var">P</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">z</span>) <span class="id" type="var">x</span> <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">P</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">y</span>) <span class="id" type="var">x</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">H</span>. <span class="id" type="var">applys_eq</span> <span class="id" type="var">H</span> 2.<br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
When we have a mismatch on two arguments, we want to produce
two equalities. To achieve this, we may call <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> <span class="inlinecode">1</span> <span class="inlinecode">2</span>.
More generally, the tactic <span class="inlinecode"><span class="id" type="var">applys_eq</span></span> expects a lemma and a
sequence of natural numbers as arguments.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_applys_eq_3</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span>:<span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span>) <span class="id" type="var">x1</span> <span class="id" type="var">x2</span> <span class="id" type="var">y1</span> <span class="id" type="var">y2</span>, <br/>
<span class="id" type="var">P</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">x2</span>) (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">y2</span>) <span style="font-family: arial;">→</span> <br/>
<span class="id" type="var">P</span> (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">x1</span>) (<span class="id" type="var">big_expression_using</span> <span class="id" type="var">y1</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">introv</span> <span class="id" type="var">H</span>. <span class="id" type="var">applys_eq</span> <span class="id" type="var">H</span> 1 2.<br/>
<span class="comment">(* produces two subgoals:<br/>
<span class="inlinecode"><span class="id" type="var">big_expression_using</span></span> <span class="inlinecode"><span class="id" type="var">x1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">big_expression_using</span></span> <span class="inlinecode"><span class="id" type="var">x2</span></span><br/>
<span class="inlinecode"><span class="id" type="var">big_expression_using</span></span> <span class="inlinecode"><span class="id" type="var">y1</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">big_expression_using</span></span> <span class="inlinecode"><span class="id" type="var">y2</span></span> *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">EqualityExamples</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab983"></a><h1 class="section">Some convenient shorthands</h1>
<div class="paragraph"> </div>
This section of the tutorial introduces a few tactics
that help make proof scripts shorter and more readable:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">unfolds</span></span> (without argument) for unfolding the head definition,
</li>
<li> <span class="inlinecode"><span class="id" type="var">false</span></span> for replacing the goal with <span class="inlinecode"><span class="id" type="var">False</span></span>,
</li>
<li> <span class="inlinecode"><span class="id" type="var">gen</span></span> as a shorthand for <span class="inlinecode"><span class="id" type="tactic">dependent</span></span> <span class="inlinecode"><span class="id" type="tactic">generalize</span></span>,
</li>
<li> <span class="inlinecode"><span class="id" type="var">skip</span></span> for skipping a subgoal even if it contains existential variables,
</li>
<li> <span class="inlinecode"><span class="id" type="var">sort</span></span> for re-ordering the proof context by moving moving all
propositions at the bottom.
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab984"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">unfolds</span></span></h2>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Module</span> <span class="id" type="var">UnfoldsExample</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Hoare</span>.<br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">unfolds</span></span> (without any argument) unfolds the
head constant of the goal. This tactic saves the need to
name the constant explicitly.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">bexp_eval_true</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">st</span>,<br/>
<span class="id" type="var">beval</span> <span class="id" type="var">st</span> <span class="id" type="var">b</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> (<span class="id" type="var">bassn</span> <span class="id" type="var">b</span>) <span class="id" type="var">st</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">b</span> <span class="id" type="var">st</span> <span class="id" type="var">Hbe</span>. <span class="id" type="var">dup</span>.<br/>
<br/>
<span class="comment">(* The old proof: *)</span><br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">bassn</span>. <span class="id" type="tactic">assumption</span>.<br/>
<br/>
<span class="comment">(* The new proof: *)</span><br/>
<span class="id" type="var">unfolds</span>. <span class="id" type="tactic">assumption</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Remark: contrary to the tactic <span class="inlinecode"><span class="id" type="var">hnf</span></span>, which may unfold several
constants, <span class="inlinecode"><span class="id" type="var">unfolds</span></span> performs only a single step of unfolding.
<div class="paragraph"> </div>
Remark: the tactic <span class="inlinecode"><span class="id" type="var">unfolds</span></span> <span class="inlinecode"><span class="id" type="keyword">in</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> can be used to unfold the
head definition of the hypothesis <span class="inlinecode"><span class="id" type="var">H</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">End</span> <span class="id" type="var">UnfoldsExample</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab985"></a><h2 class="section">The tactics <span class="inlinecode"><span class="id" type="var">false</span></span> and <span class="inlinecode"><span class="id" type="var">tryfalse</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">false</span></span> can be used to replace any goal with <span class="inlinecode"><span class="id" type="var">False</span></span>.
In short, it is a shorthand for <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">ex_falso_quodlibet</span></span>.
Moreover, <span class="inlinecode"><span class="id" type="var">false</span></span> proves the goal if it contains an absurd
assumption, such as <span class="inlinecode"><span class="id" type="var">False</span></span> or <span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>, or if it contains
contradictory assumptions, such as <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">true</span></span> and <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">false</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_false</span> : <br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">S</span> <span class="id" type="var">n</span> = 1 <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">false</span></span> can be given an argument: <span class="inlinecode"><span class="id" type="var">false</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span> replace
the goals with <span class="inlinecode"><span class="id" type="var">False</span></span> and then applies <span class="inlinecode"><span class="id" type="var">H</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_false_arg</span> : <br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">n</span> < 0 <span style="font-family: arial;">→</span> <span class="id" type="var">False</span>) <span style="font-family: arial;">→</span> (3 < 0) <span style="font-family: arial;">→</span> 4 < 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">H</span> <span class="id" type="var">L</span>. <span class="id" type="var">false</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">L</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
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</div>
<div class="doc">
The tactic <span class="inlinecode"><span class="id" type="var">tryfalse</span></span> is a shorthand for <span class="inlinecode"><span class="id" type="tactic">try</span></span> <span class="inlinecode"><span class="id" type="var">solve</span></span> <span class="inlinecode">[<span class="id" type="var">false</span>]</span>:
it tries to find a contradiction in the goal. The tactic
<span class="inlinecode"><span class="id" type="var">tryfalse</span></span> is generally called after a case analysis.
</div>
<div class="code code-tight">
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<span class="id" type="keyword">Lemma</span> <span class="id" type="var">demo_tryfalse</span> : <br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">S</span> <span class="id" type="var">n</span> = 1 <span style="font-family: arial;">→</span> <span class="id" type="var">n</span> = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span>. <span class="id" type="tactic">destruct</span> <span class="id" type="var">n</span>; <span class="id" type="var">tryfalse</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
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</div>
<div class="doc">
<a name="lab986"></a><h2 class="section">The tactic <span class="inlinecode"><span class="id" type="var">gen</span></span></h2>
<div class="paragraph"> </div>
The tactic <span class="inlinecode"><span class="id" type="var">gen</span></span> is a shortand for <span class="inlinecode"><span class="id" type="tactic">generalize</span></span> <span class="inlinecode"><span class="id" type="tactic">dependent</span></span>
that accepts several arguments at once. An invokation of
this tactic takes the form <span class="inlinecode"><span class="id" type="var">gen</span></span> <span class="inlinecode"><span class="id" type="var">x</span></span> <span class="inlinecode"><span class="id" type="var">y</span></span> <span class="inlinecode"><span class="id" type="var">z</span></span>.
</div>
<div class="code code-tight">
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<span class="id" type="keyword">Module</span> <span class="id" type="var">GenExample</span>.<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Import</span> <span class="id" type="var">Stlc</span>.<br/>
<span class="id" type="keyword">Import</span> <span class="id" type="var">STLC</span>.<br/>
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">substitution_preserves_typing</span> : <span style="font-family: arial;">∀</span><span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span> <span class="id" type="var">v</span> <span class="id" type="var">t</span> <span class="id" type="var">S</span>,<br/>
<span class="id" type="var">has_type</span> (<span class="id" type="var">extend</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span>) <span class="id" type="var">t</span> <span class="id" type="var">S</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span class="id" type="var">empty</span> <span class="id" type="var">v</span> <span class="id" type="var">U</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">has_type</span> <span style="font-family: serif; font-size:85%;">Γ</span> ([<span class="id" type="var">x</span>:=<span class="id" type="var">v</span>]<span class="id" type="var">t</span>) <span class="id" type="var">S</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="var">dup</span>.<br/>
<br/>
<span class="comment">(* The old proof: *)</span><br/>
<span class="id" type="tactic">intros</span> <span style="font-family: serif; font-size:85%;">Γ</span> <span class="id" type="var">x</span> <span class="id" type="var">U</span> <span class="id" type="var">v</span> <span class="id" type="var">t</span> <span class="id" type="var">S</span> <span class="id" type="var">Htypt</span> <span class="id" type="var">Htypv</span>.<br/>
<span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span class="id" type="var">S</span>. <span class="id" type="tactic">generalize</span> <span class="id" type="tactic">dependent</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">t</span>; <span class="id" type="tactic">intros</span>; <span class="id" type="tactic">simpl</span>.<br/>
<span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>. <span class="id" type="var">admit</span>.<br/>
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<span class="comment">(* The new proof: *)</span><br/>
<span class="id" type="var">introv</span> <span class="id" type="var">Htypt</span> <span class="id" type="var">Htypv</span>. <span class="id" type="var">gen</span> <span class="id" type="var">S</span> <span style="font-family: serif; font-size:85%;">Γ</span>.<br/>