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Induction.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>Induction</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">Induction</h1>
<div class="code code-tight">
</div>
<div class="doc">
<a name="lab41"></a><h1 class="section">归纳法: 用归纳法证明</h1>
<div class="paragraph"> </div>
下面这行代码会导入你在之前章节做的所有定义
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <span class="id" type="var">Basics</span>.<br/>
<br/>
</div>
<div class="doc">
为了让它起作用，你需要用 <span class="inlinecode"><span class="id" type="var">coqc</span></span> 把 <span class="inlinecode"><span class="id" type="var">Basics.v</span></span> 编译成 <span class="inlinecode"><span class="id" type="var">Basics.vo</span></span>。
（这就好像你把.java文件编译成.class文件，或者把.c文件编译成.o文件。）
有两种方式编译文件：
<div class="paragraph"> </div>
<ul class="doclist">
<li> CoqIDE:
打开 <span class="inlinecode"><span class="id" type="var">Basics.v</span></span>.
在 "Compile" 菜单, 点击 "Compile Buffer".
</li>
<li> 命令行:
运行 <span class="inlinecode"><span class="id" type="var">coqc</span></span> <span class="inlinecode"><span class="id" type="var">Basics.v</span></span>
</li>
</ul>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab42"></a><h1 class="section">为分类命名</h1>
<div class="paragraph"> </div>
现在有个问题是没有一个明确的命令让证明从一个分类跳到另一个分类，这让证明
脚本很难阅读。在一个极长的有很多分支的证明中，你甚至会迷失方向。（想象一下，
你试着记住一个内部分类的前五个子目标，这个分类的外边又有七个分类……）条理清晰
地使用缩进与注释可能会有点帮助，但是更好的方法还是使用 <span class="inlinecode"><span class="id" type="var">Case</span></span> 策略。
<div class="paragraph"> </div>
<span class="inlinecode"><span class="id" type="var">Case</span></span> 不是Coq内置的策略：我们需要自己定义。
现在还没必要理解它是怎么运作的 ———— 你可以跳过下面的示例定义 ———— 这里用了
一些我们还没提到的Coq的功能：字符串库 （用在引用字符串的具体语法上）和
<span class="inlinecode"><span class="id" type="keyword">Ltac</span></span> 命令 ———— 用来让我们自定义策略。感谢Aaron Bohannon做的这些工作！
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="var">String</span>. <span class="id" type="keyword">Open</span> <span class="id" type="keyword">Scope</span> <span class="id" type="var">string_scope</span>.<br/>
<br/>
<span class="id" type="keyword">Ltac</span> <span class="id" type="var">move_to_top</span> <span class="id" type="var">x</span> :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">reverse</span> <span class="id" type="var">goal</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">H</span> : <span class="id" type="var">_</span> <span style="font-family: arial;">⊢</span> <span class="id" type="var">_</span> ⇒ <span class="id" type="tactic">try</span> <span class="id" type="tactic">move</span> <span class="id" type="var">x</span> <span class="id" type="keyword">after</span> <span class="id" type="var">H</span><br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "assert_eq" <span class="id" type="var">ident</span>(<span class="id" type="var">x</span>) <span class="id" type="var">constr</span>(<span class="id" type="var">v</span>) :=<br/>
<span class="id" type="keyword">let</span> <span class="id" type="var">H</span> := <span class="id" type="tactic">fresh</span> <span class="id" type="keyword">in</span><br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">x</span> = <span class="id" type="var">v</span>) <span class="id" type="keyword">as</span> <span class="id" type="var">H</span> <span class="id" type="tactic">by</span> <span class="id" type="tactic">reflexivity</span>;<br/>
<span class="id" type="tactic">clear</span> <span class="id" type="var">H</span>.<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "Case_aux" <span class="id" type="var">ident</span>(<span class="id" type="var">x</span>) <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) :=<br/>
<span class="id" type="var">first</span> [<br/>
<span class="id" type="tactic">set</span> (<span class="id" type="var">x</span> := <span class="id" type="var">name</span>); <span class="id" type="var">move_to_top</span> <span class="id" type="var">x</span><br/>
| <span class="id" type="var">assert_eq</span> <span class="id" type="var">x</span> <span class="id" type="var">name</span>; <span class="id" type="var">move_to_top</span> <span class="id" type="var">x</span><br/>
| <span class="id" type="tactic">fail</span> 1 "because we are working on a different case" ].<br/>
<br/>
<span class="id" type="keyword">Tactic Notation</span> "Case" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">Case</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSSCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSSSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSSSCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSSSSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSSSSCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSSSSSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSSSSSCase</span> <span class="id" type="var">name</span>.<br/>
<span class="id" type="keyword">Tactic Notation</span> "SSSSSSSCase" <span class="id" type="var">constr</span>(<span class="id" type="var">name</span>) := <span class="id" type="var">Case_aux</span> <span class="id" type="var">SSSSSSSCase</span> <span class="id" type="var">name</span>.<br/>
</div>
<div class="doc">
这是一个 <span class="inlinecode"><span class="id" type="var">Case</span></span> 的例子。一步一步运行然后观察上下文的变化（CoqIde的右上角）
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">andb_true_elim1</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span> : <span class="id" type="var">bool</span>,<br/>
<span class="id" type="var">andb</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">b</span> = <span class="id" type="var">true</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">c</span> <span class="id" type="var">H</span>.<br/>
<span class="id" type="tactic">destruct</span> <span class="id" type="var">b</span>.<br/>
<span class="id" type="var">Case</span> "b = true". <span class="comment">(* <----- 看这 *)</span><br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "b = false". <span class="comment">(* <---- 再看这 *)</span><br/>
<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/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<span class="inlinecode"><span class="id" type="var">Case</span></span> 的作用非常直接：它简单地在当前目标的上下文里加上一个我们选择的字符串
（之前用标识符“Case”标注的）。当子目标生成的时候，这个字符串会被带进目标的上下文。
当最后一个子目标证明完毕、下一个顶级目标被激活，这个字符串就会在上下文中消失然后我
们会发现这个分类在我们引入的地方完成了。 另外，这还起到了使纠错更清晰的作用。比方说
我们试着运行一个新的<span class="inlinecode"><span class="id" type="var">Case</span></span>策略，然而上一个分类的字符串还在上下文里，显然我们有哪里
出错了。
<div class="paragraph"> </div>
对于嵌套的分类讨论（比方说，我们用了<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>之后在分支里又用了一个<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>，
这时我们应该用<span class="inlinecode"><span class="id" type="var">SCase</span></span>（subcase）策略来代替<span class="inlinecode"><span class="id" type="var">Case</span></span>。
<div class="paragraph"> </div>
<a name="lab43"></a><h4 class="section">练习: 2星 (andb_true_elim2)</h4>
证明 <span class="inlinecode"><span class="id" type="var">andb_true_elim2</span></span>, 当你使用<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>时用<span class="inlinecode"><span class="id" type="var">Case</span></span>（或<span class="inlinecode"><span class="id" type="var">SCase</span></span>）来标记
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">andb_true_elim2</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span> : <span class="id" type="var">bool</span>,<br/>
<span class="id" type="var">andb</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">c</span> = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* 请在这里填写代码 *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
There are no hard and fast rules for how proofs should be
formatted in Coq — in particular, where lines should be broken
and how sections of the proof should be indented to indicate their
nested structure. However, if the places where multiple subgoals
are generated are marked with explicit <span class="inlinecode"><span class="id" type="var">Case</span></span> tactics placed at
the beginning of lines, then the proof will be readable almost no
matter what choices are made about other aspects of layout.
<div class="paragraph"> </div>
This is a good place to mention one other piece of (possibly
obvious) advice about line lengths. Beginning Coq users sometimes
tend to the extremes, either writing each tactic on its own line
or entire proofs on one line. Good style lies somewhere in the
middle. In particular, one reasonable convention is to limit
yourself to 80-character lines. Lines longer than this are hard
to read and can be inconvenient to display and print. Many
editors have features that help enforce this.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab44"></a><h1 class="section">用归纳法来证明</h1>
<div class="paragraph"> </div>
我们在上一章简单地证明了<span class="inlinecode">0</span>对于<span class="inlinecode">+</span>是一个左单位元。实际上它也是个右单位元……
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_0_r_firsttry</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + 0 = <span class="id" type="var">n</span>.<br/>
<br/>
</div>
<div class="doc">
…… 它不能用同样简单的方式证明。只用<span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>不起作用：这里的<span class="inlinecode"><span class="id" type="var">n</span></span>在<span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode">0</span>
中是一个任意的未知数，所以<span class="inlinecode">+</span>的定义中的<span class="inlinecode"><span class="id" type="keyword">match</span></span>不能被化简。
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* 毫无变化！ *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab45"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
And reasoning by cases using <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> doesn't get us much
further: the branch of the case analysis where we assume <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>
goes through, but in the branch where <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> for some <span class="inlinecode"><span class="id" type="var">n'</span></span> we
get stuck in exactly the same way. We could use <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> to
get one step further, but since <span class="inlinecode"><span class="id" type="var">n</span></span> can be arbitrarily large, if we
try to keep on like this we'll never be done.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_0_r_secondtry</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + 0 = <span class="id" type="var">n</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="tactic">destruct</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "n = 0".<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="comment">(* 目前看上去还没什么问题…… *)</span><br/>
<span class="id" type="var">Case</span> "n = S n'".<br/>
<span class="id" type="tactic">simpl</span>. <span class="comment">(* ……但是在这里我们又卡住了 *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab46"></a><h3 class="section"> </h3>
<div class="paragraph"> </div>
To prove such facts — indeed, to prove most interesting
facts about numbers, lists, and other inductively defined sets —
we need a more powerful reasoning principle: <i>induction</i>.
<div class="paragraph"> </div>
Recall (from high school) the principle of induction over natural
numbers: If <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> is some proposition involving a natural number
<span class="inlinecode"><span class="id" type="var">n</span></span> and we want to show that P holds for <i>all</i> numbers <span class="inlinecode"><span class="id" type="var">n</span></span>, we can
reason like this:
<div class="paragraph"> </div>
<ul class="doclist">
<li> show that <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">O</span>)</span> holds;
</li>
<li> show that, for any <span class="inlinecode"><span class="id" type="var">n'</span></span>, if <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n'</span>)</span> holds, then so does
<span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span>;
</li>
<li> conclude that <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> holds for all <span class="inlinecode"><span class="id" type="var">n</span></span>.
</li>
</ul>
<div class="paragraph"> </div>
In Coq, the steps are the same but the order is backwards: we
begin with the goal of proving <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n</span>)</span> for all <span class="inlinecode"><span class="id" type="var">n</span></span> and break it
down (by applying the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic) into two separate
subgoals: first showing <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">O</span>)</span> and then showing <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">n'</span>)</span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P</span>(<span class="id" type="var">S</span></span>
<span class="inlinecode"><span class="id" type="var">n'</span>)</span>. Here's how this works for the theorem we are trying to
prove at the moment:
<div class="paragraph"> </div>
<a name="lab47"></a><h3 class="section"> </h3>
</div>
<div class="code code-space">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_0_r</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, <span class="id" type="var">n</span> + 0 = <span class="id" type="var">n</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="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "n = 0". <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "n = S n'". <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Like <span class="inlinecode"><span class="id" type="tactic">destruct</span></span>, the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic takes an <span class="inlinecode"><span class="id" type="keyword">as</span>...</span>
clause that specifies the names of the variables to be introduced
in the subgoals. In the first branch, <span class="inlinecode"><span class="id" type="var">n</span></span> is replaced by <span class="inlinecode">0</span> and
the goal becomes <span class="inlinecode">0</span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>, which follows by simplification. In
the second, <span class="inlinecode"><span class="id" type="var">n</span></span> is replaced by <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> and the assumption <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode">+</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span>
<span class="inlinecode"><span class="id" type="var">n'</span></span> is added to the context (with the name <span class="inlinecode"><span class="id" type="var">IHn'</span></span>, i.e., the
Induction Hypothesis for <span class="inlinecode"><span class="id" type="var">n'</span></span>). The goal in this case becomes <span class="inlinecode">(<span class="id" type="var">S</span></span>
<span class="inlinecode"><span class="id" type="var">n'</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode">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>, which simplifies to <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">n'</span></span> <span class="inlinecode">+</span> <span class="inlinecode">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>, which in
turn follows from the induction hypothesis.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">minus_diag</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>,<br/>
<span class="id" type="var">minus</span> <span class="id" type="var">n</span> <span class="id" type="var">n</span> = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* 课堂任务 *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "n = 0".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "n = S n'".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab48"></a><h4 class="section">Exercise: 2 stars (basic_induction)</h4>
<div class="paragraph"> </div>
Prove the following lemmas using induction. You might need
previously proven results.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_0_r</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> × 0 = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_n_Sm</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/>
<span class="id" type="var">S</span> (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) = <span class="id" type="var">n</span> + (<span class="id" type="var">S</span> <span class="id" type="var">m</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_comm</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/>
<span class="id" type="var">n</span> + <span class="id" type="var">m</span> = <span class="id" type="var">m</span> + <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_assoc</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab49"></a><h4 class="section">Exercise: 2 stars (double_plus)</h4>
<div class="paragraph"> </div>
Consider the following function, which doubles its argument:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Fixpoint</span> <span class="id" type="var">double</span> (<span class="id" type="var">n</span>:<span class="id" type="var">nat</span>) :=<br/>
<span class="id" type="keyword">match</span> <span class="id" type="var">n</span> <span class="id" type="keyword">with</span><br/>
| <span class="id" type="var">O</span> ⇒ <span class="id" type="var">O</span><br/>
| <span class="id" type="var">S</span> <span class="id" type="var">n'</span> ⇒ <span class="id" type="var">S</span> (<span class="id" type="var">S</span> (<span class="id" type="var">double</span> <span class="id" type="var">n'</span>))<br/>
<span class="id" type="keyword">end</span>.<br/>
<br/>
</div>
<div class="doc">
Use induction to prove this simple fact about <span class="inlinecode"><span class="id" type="var">double</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Lemma</span> <span class="id" type="var">double_plus</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <span class="id" type="var">double</span> <span class="id" type="var">n</span> = <span class="id" type="var">n</span> + <span class="id" type="var">n</span> .<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab50"></a><h4 class="section">Exercise: 1 star (destruct_induction)</h4>
Briefly explain the difference between the tactics
<span class="inlinecode"><span class="id" type="tactic">destruct</span></span> and <span class="inlinecode"><span class="id" type="tactic">induction</span></span>.
<div class="paragraph"> </div>
<span class="comment">(* FILL IN HERE *)</span><br/>
<div class="paragraph"> </div>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab51"></a><h1 class="section">证明里的证明</h1>
<div class="paragraph"> </div>
In Coq, as in informal mathematics, large proofs are very
often broken into a sequence of theorems, with later proofs
referring to earlier theorems. Occasionally, however, a proof
will need some miscellaneous fact that is too trivial (and of too
little general interest) to bother giving it its own top-level
name. In such cases, it is convenient to be able to simply state
and prove the needed "sub-theorem" right at the point where it is
used. The <span class="inlinecode"><span class="id" type="tactic">assert</span></span> tactic allows us to do this. For example, our
earlier proof of the <span class="inlinecode"><span class="id" type="var">mult_0_plus</span></span> theorem referred to a previous
theorem named <span class="inlinecode"><span class="id" type="var">plus_O_n</span></span>. We can also use <span class="inlinecode"><span class="id" type="tactic">assert</span></span> to state and
prove <span class="inlinecode"><span class="id" type="var">plus_O_n</span></span> in-line:
</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="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>).<br/>
<span class="id" type="var">Case</span> "Proof of assertion". <span class="id" type="tactic">reflexivity</span>.<br/>
<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>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">assert</span></span> tactic introduces two sub-goals. The first is
the assertion itself; by prefixing it with <span class="inlinecode"><span class="id" type="var">H</span>:</span> we name the
assertion <span class="inlinecode"><span class="id" type="var">H</span></span>. (Note that we could also name the assertion with
<span class="inlinecode"><span class="id" type="keyword">as</span></span> just as we did above with <span class="inlinecode"><span class="id" type="tactic">destruct</span></span> and <span class="inlinecode"><span class="id" type="tactic">induction</span></span>, i.e.,
<span class="inlinecode"><span class="id" type="tactic">assert</span></span> <span class="inlinecode">(0</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode"><span class="id" type="keyword">as</span></span> <span class="inlinecode"><span class="id" type="var">H</span></span>. Also note that we mark the proof of
this assertion with a <span class="inlinecode"><span class="id" type="var">Case</span></span>, both for readability and so that,
when using Coq interactively, we can see when we're finished
proving the assertion by observing when the <span class="inlinecode">"<span class="id" type="keyword">Proof</span></span> <span class="inlinecode"><span class="id" type="var">of</span></span> <span class="inlinecode"><span class="id" type="var">assertion</span>"</span>
string disappears from the context.) The second goal is the same
as the one at the point where we invoke <span class="inlinecode"><span class="id" type="tactic">assert</span></span>, except that, in
the context, we have the assumption <span class="inlinecode"><span class="id" type="var">H</span></span> that <span class="inlinecode">0</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">n</span></span>. That
is, <span class="inlinecode"><span class="id" type="tactic">assert</span></span> generates one subgoal where we must prove the
asserted fact and a second subgoal where we can use the asserted
fact to make progress on whatever we were trying to prove in the
first place.
<div class="paragraph"> </div>
Actually, <span class="inlinecode"><span class="id" type="tactic">assert</span></span> will turn out to be handy in many sorts of
situations. For example, suppose we want to prove that <span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">m</span>)</span>
<span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">p</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">q</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">(<span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">p</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">q</span>)</span>. The only difference between the
two sides of the <span class="inlinecode">=</span> is that the arguments <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">n</span></span> to the
first inner <span class="inlinecode">+</span> are swapped, so it seems we should be able to
use the commutativity of addition (<span class="inlinecode"><span class="id" type="var">plus_comm</span></span>) to rewrite one
into the other. However, the <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> tactic is a little stupid
about <i>where</i> it applies the rewrite. There are three uses of
<span class="inlinecode">+</span> here, and it turns out that doing <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">plus_comm</span></span>
will affect only the <i>outer</i> one.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_rearrange_firsttry</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">p</span> <span class="id" type="var">q</span> : <span class="id" type="var">nat</span>,<br/>
(<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + (<span class="id" type="var">p</span> + <span class="id" type="var">q</span>) = (<span class="id" type="var">m</span> + <span class="id" type="var">n</span>) + (<span class="id" type="var">p</span> + <span class="id" type="var">q</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> <span class="id" type="var">p</span> <span class="id" type="var">q</span>.<br/>
<span class="comment">(* We just need to swap (n + m) for (m + n)...<br/>
it seems like plus_comm should do the trick! *)</span><br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">plus_comm</span>.<br/>
<span class="comment">(* Doesn't work...Coq rewrote the wrong plus! *)</span><br/>
<span class="id" type="keyword">Abort</span>.<br/>
<br/>
</div>
<div class="doc">
To get <span class="inlinecode"><span class="id" type="var">plus_comm</span></span> to apply at the point where we want it, we can
introduce a local lemma stating that <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span></span> (for
the particular <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">n</span></span> that we are talking about here), prove
this lemma using <span class="inlinecode"><span class="id" type="var">plus_comm</span></span>, and then use this lemma to do the
desired rewrite.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_rearrange</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">p</span> <span class="id" type="var">q</span> : <span class="id" type="var">nat</span>,<br/>
(<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + (<span class="id" type="var">p</span> + <span class="id" type="var">q</span>) = (<span class="id" type="var">m</span> + <span class="id" type="var">n</span>) + (<span class="id" type="var">p</span> + <span class="id" type="var">q</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> <span class="id" type="var">p</span> <span class="id" type="var">q</span>.<br/>
<span class="id" type="tactic">assert</span> (<span class="id" type="var">H</span>: <span class="id" type="var">n</span> + <span class="id" type="var">m</span> = <span class="id" type="var">m</span> + <span class="id" type="var">n</span>).<br/>
<span class="id" type="var">Case</span> "Proof of assertion".<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">plus_comm</span>. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab52"></a><h4 class="section">Exercise: 4 stars (mult_comm)</h4>
Use <span class="inlinecode"><span class="id" type="tactic">assert</span></span> to help prove this theorem. You shouldn't need to
use induction.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_swap</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = <span class="id" type="var">m</span> + (<span class="id" type="var">n</span> + <span class="id" type="var">p</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
</div>
<div class="doc">
Now prove commutativity of multiplication. (You will probably
need to define and prove a separate subsidiary theorem to be used
in the proof of this one.) You may find that <span class="inlinecode"><span class="id" type="var">plus_swap</span></span> comes in
handy.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_comm</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">n</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">m</span> × <span class="id" type="var">n</span> = <span class="id" type="var">n</span> × <span class="id" type="var">m</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab53"></a><h4 class="section">Exercise: 2 stars, optional (evenb_n__oddb_Sn)</h4>
<div class="paragraph"> </div>
Prove the following simple fact:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">evenb_n__oddb_Sn</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">evenb</span> <span class="id" type="var">n</span> = <span class="id" type="var">negb</span> (<span class="id" type="var">evenb</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab54"></a><h1 class="section">更多习题</h1>
<div class="paragraph"> </div>
<a name="lab55"></a><h4 class="section">Exercise: 3 stars, optional (more_exercises)</h4>
Take a piece of paper. For each of the following theorems, first
<i>think</i> about whether (a) it can be proved using only
simplification and rewriting, (b) it also requires case
analysis (<span class="inlinecode"><span class="id" type="tactic">destruct</span></span>), or (c) it also requires induction. Write
down your prediction. Then fill in the proof. (There is no need
to turn in your piece of paper; this is just to encourage you to
reflect before hacking!)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">ble_nat_refl</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">true</span> = <span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">zero_nbeq_S</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">beq_nat</span> 0 (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">andb_false_r</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> : <span class="id" type="var">bool</span>,<br/>
<span class="id" type="var">andb</span> <span class="id" type="var">b</span> <span class="id" type="var">false</span> = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_ble_compat_l</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">ble_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> = <span class="id" type="var">true</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ble_nat</span> (<span class="id" type="var">p</span> + <span class="id" type="var">n</span>) (<span class="id" type="var">p</span> + <span class="id" type="var">m</span>) = <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">S_nbeq_0</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">beq_nat</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>) 0 = <span class="id" type="var">false</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_1_l</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<span class="id" type="var">nat</span>, 1 × <span class="id" type="var">n</span> = <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">all3_spec</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">b</span> <span class="id" type="var">c</span> : <span class="id" type="var">bool</span>,<br/>
<span class="id" type="var">orb</span><br/>
(<span class="id" type="var">andb</span> <span class="id" type="var">b</span> <span class="id" type="var">c</span>)<br/>
(<span class="id" type="var">orb</span> (<span class="id" type="var">negb</span> <span class="id" type="var">b</span>)<br/>
(<span class="id" type="var">negb</span> <span class="id" type="var">c</span>))<br/>
= <span class="id" type="var">true</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_plus_distr_r</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">p</span> : <span class="id" type="var">nat</span>,<br/>
(<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) × <span class="id" type="var">p</span> = (<span class="id" type="var">n</span> × <span class="id" type="var">p</span>) + (<span class="id" type="var">m</span> × <span class="id" type="var">p</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">mult_assoc</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> × (<span class="id" type="var">m</span> × <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> × <span class="id" type="var">m</span>) × <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab56"></a><h4 class="section">Exercise: 2 stars, optional (beq_nat_refl)</h4>
Prove the following theorem. Putting <span class="inlinecode"><span class="id" type="var">true</span></span> on the left-hand side
of the equality may seem odd, but this is how the theorem is stated in
the standard library, so we follow suit. Since rewriting
works equally well in either direction, we will have no
problem using the theorem no matter which way we state it.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">beq_nat_refl</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">true</span> = <span class="id" type="var">beq_nat</span> <span class="id" type="var">n</span> <span class="id" type="var">n</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab57"></a><h4 class="section">Exercise: 2 stars, optional (plus_swap')</h4>
The <span class="inlinecode"><span class="id" type="tactic">replace</span></span> tactic allows you to specify a particular subterm to
rewrite and what you want it rewritten to. More precisely,
<span class="inlinecode"><span class="id" type="tactic">replace</span></span> <span class="inlinecode">(<span class="id" type="var">t</span>)</span> <span class="inlinecode"><span class="id" type="keyword">with</span></span> <span class="inlinecode">(<span class="id" type="var">u</span>)</span> replaces (all copies of) expression <span class="inlinecode"><span class="id" type="var">t</span></span> in
the goal by expression <span class="inlinecode"><span class="id" type="var">u</span></span>, and generates <span class="inlinecode"><span class="id" type="var">t</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">u</span></span> as an additional
subgoal. This is often useful when a plain <span class="inlinecode"><span class="id" type="tactic">rewrite</span></span> acts on the wrong
part of the goal.
<div class="paragraph"> </div>
Use the <span class="inlinecode"><span class="id" type="tactic">replace</span></span> tactic to do a proof of <span class="inlinecode"><span class="id" type="var">plus_swap'</span></span>, just like
<span class="inlinecode"><span class="id" type="var">plus_swap</span></span> but without needing <span class="inlinecode"><span class="id" type="tactic">assert</span></span> <span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_swap'</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = <span class="id" type="var">m</span> + (<span class="id" type="var">n</span> + <span class="id" type="var">p</span>).<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab58"></a><h4 class="section">Exercise: 3 stars (binary_commute)</h4>
Recall the <span class="inlinecode"><span class="id" type="var">increment</span></span> and <span class="inlinecode"><span class="id" type="var">binary</span>-<span class="id" type="var">to</span>-<span class="id" type="var">unary</span></span> functions that you
wrote for the <span class="inlinecode"><span class="id" type="var">binary</span></span> exercise in the <span class="inlinecode"><span class="id" type="var">Basics</span></span> chapter. Prove
that these functions commute — that is, incrementing a binary
number and then converting it to unary yields the same result as
first converting it to unary and then incrementing.
Name your theorem <span class="inlinecode"><span class="id" type="var">bin_to_nat_pres_incr</span></span>.
<div class="paragraph"> </div>
(Before you start working on this exercise, please copy the
definitions from your solution to the <span class="inlinecode"><span class="id" type="var">binary</span></span> exercise here so
that this file can be graded on its own. If you find yourself
wanting to change your original definitions to make the property
easier to prove, feel free to do so.)
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab59"></a><h4 class="section">Exercise: 5 stars, advanced (binary_inverse)</h4>
This exercise is a continuation of the previous exercise about
binary numbers. You will need your definitions and theorems from
the previous exercise to complete this one.
<div class="paragraph"> </div>
(a) First, write a function to convert natural numbers to binary
numbers. Then prove that starting with any natural number,
converting to binary, then converting back yields the same
natural number you started with.
<div class="paragraph"> </div>
(b) You might naturally think that we should also prove the
opposite direction: that starting with a binary number,
converting to a natural, and then back to binary yields the
same number we started with. However, it is not true!
Explain what the problem is.
<div class="paragraph"> </div>
(c) Define a "direct" normalization function — i.e., a function
<span class="inlinecode"><span class="id" type="var">normalize</span></span> from binary numbers to binary numbers such that,
for any binary number b, converting to a natural and then back
to binary yields <span class="inlinecode">(<span class="id" type="var">normalize</span></span> <span class="inlinecode"><span class="id" type="var">b</span>)</span>. Prove it. (Warning: This
part is tricky!)
<div class="paragraph"> </div>
Again, feel free to change your earlier definitions if this helps
here.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab60"></a><h1 class="section">对比形式化和非形式化证明（高阶）</h1>
<div class="paragraph"> </div>
"Informal proofs are algorithms; formal proofs are code."
<div class="paragraph"> </div>
The question of what, exactly, constitutes a "proof" of a
mathematical claim has challenged philosophers for millennia. A
rough and ready definition, though, could be this: a proof of a
mathematical proposition <span class="inlinecode"><span class="id" type="var">P</span></span> is a written (or spoken) text that
instills in the reader or hearer the certainty that <span class="inlinecode"><span class="id" type="var">P</span></span> is true.
That is, a proof is an act of communication.
<div class="paragraph"> </div>
Now, acts of communication may involve different sorts of readers.
On one hand, the "reader" can be a program like Coq, in which case
the "belief" that is instilled is a simple mechanical check that
<span class="inlinecode"><span class="id" type="var">P</span></span> can be derived from a certain set of formal logical rules, and
the proof is a recipe that guides the program in performing this
check. Such recipes are <i>formal</i> proofs.
<div class="paragraph"> </div>
Alternatively, the reader can be a human being, in which case the
proof will be written in English or some other natural language,
thus necessarily <i>informal</i>. Here, the criteria for success are
less clearly specified. A "good" proof is one that makes the
reader believe <span class="inlinecode"><span class="id" type="var">P</span></span>. But the same proof may be read by many
different readers, some of whom may be convinced by a particular
way of phrasing the argument, while others may not be. One reader
may be particularly pedantic, inexperienced, or just plain
thick-headed; the only way to convince them will be to make the
argument in painstaking detail. But another reader, more familiar
in the area, may find all this detail so overwhelming that they
lose the overall thread. All they want is to be told the main
ideas, because it is easier to fill in the details for themselves.
Ultimately, there is no universal standard, because there is no
single way of writing an informal proof that is guaranteed to
convince every conceivable reader. In practice, however,
mathematicians have developed a rich set of conventions and idioms
for writing about complex mathematical objects that, within a
certain community, make communication fairly reliable. The
conventions of this stylized form of communication give a fairly
clear standard for judging proofs good or bad.
<div class="paragraph"> </div>
Because we are using Coq in this course, we will be working
heavily with formal proofs. But this doesn't mean we can ignore
the informal ones! Formal proofs are useful in many ways, but
they are <i>not</i> very efficient ways of communicating ideas between
human beings.
<div class="paragraph"> </div>
For example, here is a proof that addition is associative:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_assoc'</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>]. <span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Coq is perfectly happy with this as a proof. For a human,
however, it is difficult to make much sense of it. If you're used
to Coq you can probably step through the tactics one after the
other in your mind and imagine the state of the context and goal
stack at each point, but if the proof were even a little bit more
complicated this would be next to impossible. Instead, a
mathematician might write it something like this:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <i>Theorem</i>: For any <span class="inlinecode"><span class="id" type="var">n</span></span>, <span class="inlinecode"><span class="id" type="var">m</span></span> and <span class="inlinecode"><span class="id" type="var">p</span></span>,
n + (m + p) = (n + m) + p.
<i>Proof</i>: By induction on <span class="inlinecode"><span class="id" type="var">n</span></span>.
<ul class="doclist">
<li> First, suppose <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>. We must show
0 + (m + p) = (0 + m) + p.
This follows directly from the definition of <span class="inlinecode">+</span>.
</li>
<li> Next, suppose <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span>, where
n' + (m + p) = (n' + m) + p.
We must show
(S n') + (m + p) = ((S n') + m) + p.
By the definition of <span class="inlinecode">+</span>, this follows from
S (n' + (m + p)) = S ((n' + m) + p),
which is immediate from the induction hypothesis.
</li>
</ul>
</li>
</ul>
<i>Qed</i>
<div class="paragraph"> </div>
The overall form of the proof is basically similar. This is
no accident: Coq has been designed so that its <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic
generates the same sub-goals, in the same order, as the bullet
points that a mathematician would write. But there are
significant differences of detail: the formal proof is much more
explicit in some ways (e.g., the use of <span class="inlinecode"><span class="id" type="tactic">reflexivity</span></span>) but much
less explicit in others (in particular, the "proof state" at any
given point in the Coq proof is completely implicit, whereas the
informal proof reminds the reader several times where things
stand).
<div class="paragraph"> </div>
Here is a formal proof that shows the structure more
clearly:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">plus_assoc''</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">p</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">n</span> + (<span class="id" type="var">m</span> + <span class="id" type="var">p</span>) = (<span class="id" type="var">n</span> + <span class="id" type="var">m</span>) + <span class="id" type="var">p</span>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span>. <span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
<span class="id" type="var">Case</span> "n = 0".<br/>
<span class="id" type="tactic">reflexivity</span>.<br/>
<span class="id" type="var">Case</span> "n = S n'".<br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab61"></a><h4 class="section">Exercise: 2 stars, advanced (plus_comm_informal)</h4>
Translate your solution for <span class="inlinecode"><span class="id" type="var">plus_comm</span></span> into an informal proof.
<div class="paragraph"> </div>
Theorem: Addition is commutative.
Proof: <span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab62"></a><h4 class="section">Exercise: 2 stars, optional (beq_nat_refl_informal)</h4>
Write an informal proof of the following theorem, using the
informal proof of <span class="inlinecode"><span class="id" type="var">plus_assoc</span></span> as a model. Don't just
paraphrase the Coq tactics into English!
Theorem: <span class="inlinecode"><span class="id" type="var">true</span></span> <span class="inlinecode">=</span> <span class="inlinecode"><span class="id" type="var">beq_nat</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> for any <span class="inlinecode"><span class="id" type="var">n</span></span>.
Proof: <span class="comment">(* FILL IN HERE *)</span><br/>
<font size=-2>☐</font>
</div>
<div class="code code-tight">
</div>
</div>
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