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Hoare.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>Hoare: Hoare Logic, Part I</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">Hoare<span class="subtitle">Hoare Logic, Part I</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
</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">Imp</span>.<br/>
<br/>
</div>
<div class="doc">
In the past couple of chapters, we've begun applying the
mathematical tools developed in the first part of the course to
studying the theory of a small programming language, Imp.
<div class="paragraph"> </div>
<ul class="doclist">
<li> We defined a type of <i>abstract syntax trees</i> for Imp, together
with an <i>evaluation relation</i> (a partial function on states)
that specifies the <i>operational semantics</i> of programs.
<div class="paragraph"> </div>
The language we defined, though small, captures some of the key
features of full-blown languages like C, C++, and Java,
including the fundamental notion of mutable state and some
common control structures.
<div class="paragraph"> </div>
</li>
<li> We proved a number of <i>metatheoretic properties</i> — "meta" in
the sense that they are properties of the language as a whole,
rather than properties of particular programs in the language.
These included:
<div class="paragraph"> </div>
<ul class="doclist">
<li> determinism of evaluation
<div class="paragraph"> </div>
</li>
<li> equivalence of some different ways of writing down the
definitions (e.g. functional and relational definitions of
arithmetic expression evaluation)
<div class="paragraph"> </div>
</li>
<li> guaranteed termination of certain classes of programs
<div class="paragraph"> </div>
</li>
<li> correctness (in the sense of preserving meaning) of a number
of useful program transformations
<div class="paragraph"> </div>
</li>
<li> behavioral equivalence of programs (in the <span class="inlinecode"><span class="id" type="var">Equiv</span></span> chapter).
</li>
</ul>
</li>
</ul>
If we stopped here, we would already have something useful: a set
of tools for defining and discussing programming languages and
language features that are mathematically precise, flexible, and
easy to work with, applied to a set of key properties. All of
these properties are things that language designers, compiler
writers, and users might care about knowing. Indeed, many of them
are so fundamental to our understanding of the programming
languages we deal with that we might not consciously recognize
them as "theorems." But properties that seem intuitively obvious
can sometimes be quite subtle (in some cases, even subtly wrong!).
<div class="paragraph"> </div>
We'll return to the theme of metatheoretic properties of whole
languages later in the course when we discuss <i>types</i> and <i>type
soundness</i>. In this chapter, though, we'll turn to a different
set of issues.
<div class="paragraph"> </div>
Our goal is to see how to carry out some simple examples of
<i>program verification</i> — i.e., using the precise definition of
Imp to prove formally that particular programs satisfy particular
specifications of their behavior. We'll develop a reasoning system
called <i>Floyd-Hoare Logic</i> — often shortened to just <i>Hoare
Logic</i> — in which each of the syntactic constructs of Imp is
equipped with a single, generic "proof rule" that can be used to
reason compositionally about the correctness of programs involving
this construct.
<div class="paragraph"> </div>
Hoare Logic originates in the 1960s, and it continues to be the
subject of intensive research right up to the present day. It
lies at the core of a multitude of tools that are being used in
academia and industry to specify and verify real software
systems.
</div>
<div class="code code-tight">
<br/>
<br/>
</div>
<div class="doc">
<a name="lab512"></a><h1 class="section">Hoare Logic</h1>
<div class="paragraph"> </div>
Hoare Logic combines two beautiful ideas: a natural way of
writing down <i>specifications</i> of programs, and a <i>compositional
proof technique</i> for proving that programs are correct with
respect to such specifications — where by "compositional" we mean
that the structure of proofs directly mirrors the structure of the
programs that they are about.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab513"></a><h2 class="section">Assertions</h2>
<div class="paragraph"> </div>
To talk about specifications of programs, the first thing we
need is a way of making <i>assertions</i> about properties that hold at
particular points during a program's execution — i.e., claims
about the current state of the memory when program execution
reaches that point. Formally, an assertion is just a family of
propositions indexed by a <span class="inlinecode"><span class="id" type="var">state</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">Assertion</span> := <span class="id" type="var">state</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab514"></a><h4 class="section">Exercise: 1 star, optional (assertions)</h4>
</div>
<div class="code code-space">
<br/>
</div>
<div class="doc">
Paraphrase the following assertions in English.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as1</span> : <span class="id" type="var">Assertion</span> := <span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as2</span> : <span class="id" type="var">Assertion</span> := <span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> ≤ <span class="id" type="var">st</span> <span class="id" type="var">Y</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as3</span> : <span class="id" type="var">Assertion</span> :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3 <span style="font-family: arial;">∨</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> ≤ <span class="id" type="var">st</span> <span class="id" type="var">Y</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as4</span> : <span class="id" type="var">Assertion</span> :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">Z</span> × <span class="id" type="var">st</span> <span class="id" type="var">Z</span> ≤ <span class="id" type="var">st</span> <span class="id" type="var">X</span> <span style="font-family: arial;">∧</span><br/>
¬ (((<span class="id" type="var">S</span> (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>)) × (<span class="id" type="var">S</span> (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>))) ≤ <span class="id" type="var">st</span> <span class="id" type="var">X</span>).<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as5</span> : <span class="id" type="var">Assertion</span> := <span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">True</span>.<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">as6</span> : <span class="id" type="var">Assertion</span> := <span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">False</span>.<br/>
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
This way of writing assertions can be a little bit heavy,
for two reasons: (1) every single assertion that we ever write is
going to begin with <span class="inlinecode"><span class="id" type="keyword">fun</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode">⇒</span> <span class="inlinecode"></span>; and (2) this state <span class="inlinecode"><span class="id" type="var">st</span></span> is the
only one that we ever use to look up variables (we will never need
to talk about two different memory states at the same time). For
discussing examples informally, we'll adopt some simplifying
conventions: we'll drop the initial <span class="inlinecode"><span class="id" type="keyword">fun</span></span> <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode">⇒</span>, and we'll write
just <span class="inlinecode"><span class="id" type="var">X</span></span> to mean <span class="inlinecode"><span class="id" type="var">st</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>. Thus, instead of writing
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>) × (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>) ≤ <span class="id" type="var">m</span> <span style="font-family: arial;">∧</span><br/>
¬ ((<span class="id" type="var">S</span> (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>)) × (<span class="id" type="var">S</span> (<span class="id" type="var">st</span> <span class="id" type="var">Z</span>)) ≤ <span class="id" type="var">m</span>)
<div class="paragraph"> </div>
</div>
we'll write just
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">Z</span> × <span class="id" type="var">Z</span> ≤ <span class="id" type="var">m</span> <span style="font-family: arial;">∧</span> ~((<span class="id" type="var">S</span> <span class="id" type="var">Z</span>) × (<span class="id" type="var">S</span> <span class="id" type="var">Z</span>) ≤ <span class="id" type="var">m</span>).
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
Given two assertions <span class="inlinecode"><span class="id" type="var">P</span></span> and <span class="inlinecode"><span class="id" type="var">Q</span></span>, we say that <span class="inlinecode"><span class="id" type="var">P</span></span> <i>implies</i> <span class="inlinecode"><span class="id" type="var">Q</span></span>,
written <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span style="font-family: arial;">⇾</span></span> <span class="inlinecode"><span class="id" type="var">Q</span></span> (in ASCII, <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">-</span><span class="inlinecode">></span><span class="inlinecode">></span> <span class="inlinecode"><span class="id" type="var">Q</span></span>), if, whenever <span class="inlinecode"><span class="id" type="var">P</span></span>
holds in some state <span class="inlinecode"><span class="id" type="var">st</span></span>, <span class="inlinecode"><span class="id" type="var">Q</span></span> also holds.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">assert_implies</span> (<span class="id" type="var">P</span> <span class="id" type="var">Q</span> : <span class="id" type="var">Assertion</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Q</span> <span class="id" type="var">st</span>.<br/>
<br/>
<span class="id" type="keyword">Notation</span> "P <span style="font-family: arial;">⇾</span> Q" :=<br/>
(<span class="id" type="var">assert_implies</span> <span class="id" type="var">P</span> <span class="id" type="var">Q</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 80) : <span class="id" type="var">hoare_spec_scope</span>.<br/>
<span class="id" type="keyword">Open</span> <span class="id" type="keyword">Scope</span> <span class="id" type="var">hoare_spec_scope</span>.<br/>
<br/>
</div>
<div class="doc">
We'll also have occasion to use the "iff" variant of implication
between assertions:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "P <span style="font-family: arial;">⇿</span> Q" :=<br/>
(<span class="id" type="var">P</span> <span style="font-family: arial;">⇾</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">Q</span> <span style="font-family: arial;">⇾</span> <span class="id" type="var">P</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 80) : <span class="id" type="var">hoare_spec_scope</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab515"></a><h2 class="section">Hoare Triples</h2>
<div class="paragraph"> </div>
Next, we need a way of making formal claims about the
behavior of commands.
<div class="paragraph"> </div>
Since the behavior of a command is to transform one state to
another, it is natural to express claims about commands in terms
of assertions that are true before and after the command executes:
<div class="paragraph"> </div>
<ul class="doclist">
<li> "If command <span class="inlinecode"><span class="id" type="var">c</span></span> is started in a state satisfying assertion
<span class="inlinecode"><span class="id" type="var">P</span></span>, and if <span class="inlinecode"><span class="id" type="var">c</span></span> eventually terminates in some final state,
then this final state will satisfy the assertion <span class="inlinecode"><span class="id" type="var">Q</span></span>."
</li>
</ul>
<div class="paragraph"> </div>
Such a claim is called a <i>Hoare Triple</i>. The property <span class="inlinecode"><span class="id" type="var">P</span></span> is
called the <i>precondition</i> of <span class="inlinecode"><span class="id" type="var">c</span></span>, while <span class="inlinecode"><span class="id" type="var">Q</span></span> is the
<i>postcondition</i>. Formally:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">hoare_triple</span><br/>
(<span class="id" type="var">P</span>:<span class="id" type="var">Assertion</span>) (<span class="id" type="var">c</span>:<span class="id" type="var">com</span>) (<span class="id" type="var">Q</span>:<span class="id" type="var">Assertion</span>) : <span class="id" type="keyword">Prop</span> :=<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">st</span> <span class="id" type="var">st'</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">st'</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">Q</span> <span class="id" type="var">st'</span>.<br/>
<br/>
</div>
<div class="doc">
Since we'll be working a lot with Hoare triples, it's useful to
have a compact notation:
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.
<div class="paragraph"> </div>
</div>
(The traditional notation is <span class="inlinecode">{<span class="id" type="var">P</span>}</span> <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode">{<span class="id" type="var">Q</span>}</span>, but single braces
are already used for other things in Coq.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Notation</span> "<span style="letter-spacing:-.4em;">{</span>{ P <span style="letter-spacing:-.4em;">}</span>} c <span style="letter-spacing:-.4em;">{</span>{ Q <span style="letter-spacing:-.4em;">}</span>}" :=<br/>
(<span class="id" type="var">hoare_triple</span> <span class="id" type="var">P</span> <span class="id" type="var">c</span> <span class="id" type="var">Q</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 90, <span class="id" type="var">c</span> <span class="id" type="tactic">at</span> <span class="id" type="var">next</span> <span class="id" type="var">level</span>)<br/>
: <span class="id" type="var">hoare_spec_scope</span>.<br/>
<br/>
</div>
<div class="doc">
(The <span class="inlinecode"><span class="id" type="var">hoare_spec_scope</span></span> annotation here tells Coq that this
notation is not global but is intended to be used in particular
contexts. The <span class="inlinecode"><span class="id" type="keyword">Open</span></span> <span class="inlinecode"><span class="id" type="keyword">Scope</span></span> tells Coq that this file is one such
context.)
<div class="paragraph"> </div>
<a name="lab516"></a><h4 class="section">Exercise: 1 star, optional (triples)</h4>
Paraphrase the following Hoare triples in English.
<div class="paragraph"> </div>
<div class="code code-tight">
1) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 5<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
2) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = <span class="id" type="var">m</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = <span class="id" type="var">m</span> + 5)<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
3) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> ≤ <span class="id" type="var">Y</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Y</span> ≤ <span class="id" type="var">X</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
4) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
5) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = <span class="id" type="var">m</span><span style="letter-spacing:-.4em;">}</span>} <br/>
<span class="id" type="var">c</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Y</span> = <span class="id" type="var">real_fact</span> <span class="id" type="var">m</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<br/>
6) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <br/>
<span class="id" type="var">c</span> <br/>
<span style="letter-spacing:-.4em;">{</span>{(<span class="id" type="var">Z</span> × <span class="id" type="var">Z</span>) ≤ <span class="id" type="var">m</span> <span style="font-family: arial;">∧</span> ¬ (((<span class="id" type="var">S</span> <span class="id" type="var">Z</span>) × (<span class="id" type="var">S</span> <span class="id" type="var">Z</span>)) ≤ <span class="id" type="var">m</span>)<span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<div class="paragraph"> </div>
<font size=-2>☐</font>
<div class="paragraph"> </div>
<a name="lab517"></a><h4 class="section">Exercise: 1 star, optional (valid_triples)</h4>
Which of the following Hoare triples are <i>valid</i> — i.e., the
claimed relation between <span class="inlinecode"><span class="id" type="var">P</span></span>, <span class="inlinecode"><span class="id" type="var">c</span></span>, and <span class="inlinecode"><span class="id" type="var">Q</span></span> is true?
<div class="paragraph"> </div>
<div class="code code-tight">
1) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= 5 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 5<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
2) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 2<span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 3<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
3) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= 5; <span class="id" type="var">Y</span> ::= 0 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 5<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
4) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 2 <span style="font-family: arial;">∧</span> <span class="id" type="var">X</span> = 3<span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= 5 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 0<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
5) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">SKIP</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
6) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">SKIP</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
7) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">WHILE</span> <span class="id" type="var">True</span> <span class="id" type="var">DO</span> <span class="id" type="var">SKIP</span> <span class="id" type="var">END</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">False</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
8) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 0<span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> == 0 <span class="id" type="var">DO</span> <span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1 <span class="id" type="var">END</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 1<span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
9) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 1<span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">WHILE</span> <span class="id" type="var">X</span> ≠ 0 <span class="id" type="var">DO</span> <span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1 <span class="id" type="var">END</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 100<span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
</div>
<div class="code code-tight">
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2>☐</font>
<div class="paragraph"> </div>
(Note that we're using informal mathematical notations for
expressions inside of commands, for readability, rather than their
formal <span class="inlinecode"><span class="id" type="var">aexp</span></span> and <span class="inlinecode"><span class="id" type="var">bexp</span></span> encodings. We'll continue doing so
throughout the chapter.)
<div class="paragraph"> </div>
To get us warmed up for what's coming, here are two simple
facts about Hoare triples.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_post_true</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span> <span class="id" type="var">Q</span> : <span class="id" type="var">Assertion</span>) <span class="id" type="var">c</span>,<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, <span class="id" type="var">Q</span> <span class="id" type="var">st</span>) <span style="font-family: arial;">→</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">c</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">hoare_triple</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">Heval</span> <span class="id" type="var">HP</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">H</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_pre_false</span> : <span style="font-family: arial;">∀</span>(<span class="id" type="var">P</span> <span class="id" type="var">Q</span> : <span class="id" type="var">Assertion</span>) <span class="id" type="var">c</span>,<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">st</span>, ~(<span class="id" type="var">P</span> <span class="id" type="var">st</span>)) <span style="font-family: arial;">→</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">P</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">c</span> <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<div class="togglescript" id="proofcontrol2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')"><span class="show"></span></div>
<div class="proofscript" id="proof2" onclick="toggleDisplay('proof2');toggleDisplay('proofcontrol2')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">P</span> <span class="id" type="var">Q</span> <span class="id" type="var">c</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">unfold</span> <span class="id" type="var">hoare_triple</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">Heval</span> <span class="id" type="var">HP</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">not</span> <span class="id" type="keyword">in</span> <span class="id" type="var">H</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">H</span> <span class="id" type="keyword">in</span> <span class="id" type="var">HP</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HP</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
<a name="lab518"></a><h2 class="section">Proof Rules</h2>
<div class="paragraph"> </div>
The goal of Hoare logic is to provide a <i>compositional</i>
method for proving the validity of Hoare triples. That is, the
structure of a program's correctness proof should mirror the
structure of the program itself. To this end, in the sections
below, we'll introduce one rule for reasoning about each of the
different syntactic forms of commands in Imp — one for
assignment, one for sequencing, one for conditionals, etc. — plus
a couple of "structural" rules that are useful for gluing things
together. We will prove programs correct using these proof rules,
without ever unfolding the definition of <span class="inlinecode"><span class="id" type="var">hoare_triple</span></span>.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab519"></a><h3 class="section">Assignment</h3>
<div class="paragraph"> </div>
The rule for assignment is the most fundamental of the Hoare logic
proof rules. Here's how it works.
<div class="paragraph"> </div>
Consider this (valid) Hoare triple:
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">Y</span> = 1 <span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= <span class="id" type="var">Y</span> <span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> = 1 <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
In English: if we start out in a state where the value of <span class="inlinecode"><span class="id" type="var">Y</span></span>
is <span class="inlinecode">1</span> and we assign <span class="inlinecode"><span class="id" type="var">Y</span></span> to <span class="inlinecode"><span class="id" type="var">X</span></span>, then we'll finish in a
state where <span class="inlinecode"><span class="id" type="var">X</span></span> is <span class="inlinecode">1</span>. That is, the property of being equal
to <span class="inlinecode">1</span> gets transferred from <span class="inlinecode"><span class="id" type="var">Y</span></span> to <span class="inlinecode"><span class="id" type="var">X</span></span>.
<div class="paragraph"> </div>
Similarly, in
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">Y</span> + <span class="id" type="var">Z</span> = 1 <span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= <span class="id" type="var">Y</span> + <span class="id" type="var">Z</span> <span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> = 1 <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
the same property (being equal to one) gets transferred to
<span class="inlinecode"><span class="id" type="var">X</span></span> from the expression <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> on the right-hand side of
the assignment.
<div class="paragraph"> </div>
More generally, if <span class="inlinecode"><span class="id" type="var">a</span></span> is <i>any</i> arithmetic expression, then
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">a</span> = 1 <span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= <span class="id" type="var">a</span> <span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> = 1 <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
is a valid Hoare triple.
<div class="paragraph"> </div>
This can be made even more general. To conclude that an
<i>arbitrary</i> property <span class="inlinecode"><span class="id" type="var">Q</span></span> holds after <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode">::=</span> <span class="inlinecode"><span class="id" type="var">a</span></span>, we need to assume
that <span class="inlinecode"><span class="id" type="var">Q</span></span> holds before <span class="inlinecode"><span class="id" type="var">X</span></span> <span class="inlinecode">::=</span> <span class="inlinecode"><span class="id" type="var">a</span></span>, but <i>with all occurrences of</i> <span class="inlinecode"><span class="id" type="var">X</span></span>
replaced by <span class="inlinecode"><span class="id" type="var">a</span></span> in <span class="inlinecode"><span class="id" type="var">Q</span></span>. This leads to the Hoare rule for
assignment
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</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="var">a</span>] <span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= <span class="id" type="var">a</span> <span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">Q</span> <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
where "<span class="inlinecode"><span class="id" type="var">Q</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">a</span>]</span>" is pronounced "<span class="inlinecode"><span class="id" type="var">Q</span></span> where <span class="inlinecode"><span class="id" type="var">a</span></span> is substituted
for <span class="inlinecode"><span class="id" type="var">X</span></span>".
<div class="paragraph"> </div>
For example, these are valid applications of the assignment
rule:
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{ (<span class="id" type="var">X</span> ≤ 5) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> <span class="id" type="var">X</span> + 1]<br/>
<span class="id" type="var">i.e</span>., <span class="id" type="var">X</span> + 1 ≤ 5 <span style="letter-spacing:-.4em;">}</span>} <br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1 <br/>
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> ≤ 5 <span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
<span style="letter-spacing:-.4em;">{</span>{ (<span class="id" type="var">X</span> = 3) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> 3]<br/>
<span class="id" type="var">i.e</span>., 3 = 3<span style="letter-spacing:-.4em;">}</span>} <br/>
<span class="id" type="var">X</span> ::= 3 <br/>
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> = 3 <span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
<span style="letter-spacing:-.4em;">{</span>{ (0 ≤ <span class="id" type="var">X</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">X</span> ≤ 5) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> 3]<br/>
<span class="id" type="var">i.e</span>., (0 ≤ 3 <span style="font-family: arial;">∧</span> 3 ≤ 5)<span style="letter-spacing:-.4em;">}</span>} <br/>
<span class="id" type="var">X</span> ::= 3 <br/>
<span style="letter-spacing:-.4em;">{</span>{ 0 ≤ <span class="id" type="var">X</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">X</span> ≤ 5 <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
To formalize the rule, we must first formalize the idea of
"substituting an expression for an Imp variable in an assertion."
That is, given a proposition <span class="inlinecode"><span class="id" type="var">P</span></span>, a variable <span class="inlinecode"><span class="id" type="var">X</span></span>, and an
arithmetic expression <span class="inlinecode"><span class="id" type="var">a</span></span>, we want to derive another proposition
<span class="inlinecode"><span class="id" type="var">P'</span></span> that is just the same as <span class="inlinecode"><span class="id" type="var">P</span></span> except that, wherever <span class="inlinecode"><span class="id" type="var">P</span></span>
mentions <span class="inlinecode"><span class="id" type="var">X</span></span>, <span class="inlinecode"><span class="id" type="var">P'</span></span> should instead mention <span class="inlinecode"><span class="id" type="var">a</span></span>.
<div class="paragraph"> </div>
Since <span class="inlinecode"><span class="id" type="var">P</span></span> is an arbitrary Coq proposition, we can't directly
"edit" its text. Instead, we can achieve the effect we want by
evaluating <span class="inlinecode"><span class="id" type="var">P</span></span> in an updated state:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> <span class="id" type="var">a</span> <span class="id" type="var">P</span> : <span class="id" type="var">Assertion</span> :=<br/>
<span class="id" type="keyword">fun</span> (<span class="id" type="var">st</span> : <span class="id" type="var">state</span>) ⇒<br/>
<span class="id" type="var">P</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> <span class="id" type="var">a</span>)).<br/>
<br/>
<span class="id" type="keyword">Notation</span> "P [ X |-> a ]" := (<span class="id" type="var">assn_sub</span> <span class="id" type="var">X</span> <span class="id" type="var">a</span> <span class="id" type="var">P</span>) (<span class="id" type="tactic">at</span> <span class="id" type="var">level</span> 10).<br/>
<br/>
</div>
<div class="doc">
That is, <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 style="font-family: arial;">↦</span></span> <span class="inlinecode"><span class="id" type="var">a</span>]</span> is an assertion <span class="inlinecode"><span class="id" type="var">P'</span></span> that is just like <span class="inlinecode"><span class="id" type="var">P</span></span>
except that, wherever <span class="inlinecode"><span class="id" type="var">P</span></span> looks up the variable <span class="inlinecode"><span class="id" type="var">X</span></span> in the current
state, <span class="inlinecode"><span class="id" type="var">P'</span></span> instead uses the value of the expression <span class="inlinecode"><span class="id" type="var">a</span></span>.
<div class="paragraph"> </div>
To see how this works, let's calculate what happens with a couple
of examples. First, suppose <span class="inlinecode"><span class="id" type="var">P'</span></span> is <span class="inlinecode">(<span class="id" type="var">X</span></span> <span class="inlinecode">≤</span> <span class="inlinecode">5)</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">3]</span> — that
is, more formally, <span class="inlinecode"><span class="id" type="var">P'</span></span> is the Coq expression
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">st'</span> ⇒ <span class="id" type="var">st'</span> <span class="id" type="var">X</span> ≤ 5) <br/>
(<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> (<span class="id" type="var">ANum</span> 3))),
<div class="paragraph"> </div>
</div>
which simplifies to
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">st'</span> ⇒ <span class="id" type="var">st'</span> <span class="id" type="var">X</span> ≤ 5) <br/>
(<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> 3)
<div class="paragraph"> </div>
</div>
and further simplifies to
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
((<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> 3) <span class="id" type="var">X</span>) ≤ 5)
<div class="paragraph"> </div>
</div>
and by further simplification to
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(3 ≤ 5).
<div class="paragraph"> </div>
</div>
That is, <span class="inlinecode"><span class="id" type="var">P'</span></span> is the assertion that <span class="inlinecode">3</span> is less than or equal to
<span class="inlinecode">5</span> (as expected).
<div class="paragraph"> </div>
For a more interesting example, suppose <span class="inlinecode"><span class="id" type="var">P'</span></span> is <span class="inlinecode">(<span class="id" type="var">X</span></span> <span class="inlinecode">≤</span> <span class="inlinecode">5)</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">X</span>+1]</span>. Formally, <span class="inlinecode"><span class="id" type="var">P'</span></span> is the Coq expression
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(<span class="id" type="keyword">fun</span> <span class="id" type="var">st'</span> ⇒ <span class="id" type="var">st'</span> <span class="id" type="var">X</span> ≤ 5) <br/>
(<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 1)))),
<div class="paragraph"> </div>
</div>
which simplifies to
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(((<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> (<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 1))))) <span class="id" type="var">X</span>) ≤ 5
<div class="paragraph"> </div>
</div>
and further simplifies to
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <br/>
(<span class="id" type="var">aeval</span> <span class="id" type="var">st</span> (<span class="id" type="var">APlus</span> (<span class="id" type="var">AId</span> <span class="id" type="var">X</span>) (<span class="id" type="var">ANum</span> 1))) ≤ 5.
<div class="paragraph"> </div>
</div>
That is, <span class="inlinecode"><span class="id" type="var">P'</span></span> is the assertion that <span class="inlinecode"><span class="id" type="var">X</span>+1</span> is at most <span class="inlinecode">5</span>.
<div class="paragraph"> </div>
<div class="paragraph"> </div>
Now we can give the precise proof rule for assignment:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(hoare_asgn)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{Q [X <span style="font-family: arial;">↦</span> a]<span style="letter-spacing:-.4em;">}</span>} X ::= a <span style="letter-spacing:-.4em;">{</span>{Q<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
</table></center>
<div class="paragraph"> </div>
We can prove formally that this rule is indeed valid.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_asgn</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">Q</span> <span class="id" type="var">X</span> <span class="id" type="var">a</span>,<br/>
<span style="letter-spacing:-.4em;">{</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="var">a</span>]<span style="letter-spacing:-.4em;">}</span>} (<span class="id" type="var">X</span> ::= <span class="id" type="var">a</span>) <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">Q</span><span style="letter-spacing:-.4em;">}</span>}.<br/>
<div class="togglescript" id="proofcontrol3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')"><span class="show"></span></div>
<div class="proofscript" id="proof3" onclick="toggleDisplay('proof3');toggleDisplay('proofcontrol3')">
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">hoare_triple</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">Q</span> <span class="id" type="var">X</span> <span class="id" type="var">a</span> <span class="id" type="var">st</span> <span class="id" type="var">st'</span> <span class="id" type="var">HE</span> <span class="id" type="var">HQ</span>.<br/>
<span class="id" type="tactic">inversion</span> <span class="id" type="var">HE</span>. <span class="id" type="tactic">subst</span>.<br/>
<span class="id" type="tactic">unfold</span> <span class="id" type="var">assn_sub</span> <span class="id" type="keyword">in</span> <span class="id" type="var">HQ</span>. <span class="id" type="tactic">assumption</span>. <span class="id" type="keyword">Qed</span>.<br/>
</div>
<br/>
</div>
<div class="doc">
Here's a first formal proof using this rule.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Example</span> <span class="id" type="var">assn_sub_example</span> :<br/>
<span style="letter-spacing:-.4em;">{</span>{(<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> <span class="id" type="var">ANum</span> 3]<span style="letter-spacing:-.4em;">}</span>}<br/>
(<span class="id" type="var">X</span> ::= (<span class="id" type="var">ANum</span> 3))<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">st</span> <span class="id" type="var">X</span> = 3<span style="letter-spacing:-.4em;">}</span>}.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <span class="id" type="var">hoare_asgn</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab520"></a><h4 class="section">Exercise: 2 stars (hoare_asgn_examples)</h4>
Translate these informal Hoare triples...
<div class="paragraph"> </div>
<div class="code code-tight">
1) <span style="letter-spacing:-.4em;">{</span>{ (<span class="id" type="var">X</span> ≤ 5) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> <span class="id" type="var">X</span> + 1] <span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">X</span> + 1<br/>
<span style="letter-spacing:-.4em;">{</span>{ <span class="id" type="var">X</span> ≤ 5 <span style="letter-spacing:-.4em;">}</span>}<br/>
<br/>
2) <span style="letter-spacing:-.4em;">{</span>{ (0 ≤ <span class="id" type="var">X</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">X</span> ≤ 5) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> 3] <span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">X</span> ::= 3<br/>
<span style="letter-spacing:-.4em;">{</span>{ 0 ≤ <span class="id" type="var">X</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">X</span> ≤ 5 <span style="letter-spacing:-.4em;">}</span>}
<div class="paragraph"> </div>
</div>
...into formal statements and use <span class="inlinecode"><span class="id" type="var">hoare_asgn</span></span> to prove them.
</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="lab521"></a><h4 class="section">Exercise: 2 stars (hoare_asgn_wrong)</h4>
The assignment rule looks backward to almost everyone the first
time they see it. If it still seems backward to you, it may help
to think a little about alternative "forward" rules. Here is a
seemingly natural one:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(hoare_asgn_wrong)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{ True <span style="letter-spacing:-.4em;">}</span>} X ::= a <span style="letter-spacing:-.4em;">{</span>{ X = a <span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
</table></center> Give a counterexample showing that this rule is incorrect
(informally). Hint: The rule universally quantifies over the
arithmetic expression <span class="inlinecode"><span class="id" type="var">a</span></span>, and your counterexample needs to
exhibit an <span class="inlinecode"><span class="id" type="var">a</span></span> for which the rule doesn't work.
</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="lab522"></a><h4 class="section">Exercise: 3 stars, advanced (hoare_asgn_fwd)</h4>
However, using an auxiliary variable <span class="inlinecode"><span class="id" type="var">m</span></span> to remember the original
value of <span class="inlinecode"><span class="id" type="var">X</span></span> we can define a Hoare rule for assignment that does,
intuitively, "work forwards" rather than backwards.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(hoare_asgn_fwd)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{fun st ⇒ P st <span style="font-family: arial;">∧</span> st X = m<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">X ::= a</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{fun st ⇒ P st' <span style="font-family: arial;">∧</span> st X = aeval st' a <span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">(where st' = update st X m)</td>
<td></td>
</td>
</table></center> Note that we use the original value of <span class="inlinecode"><span class="id" type="var">X</span></span> to reconstruct the
state <span class="inlinecode"><span class="id" type="var">st'</span></span> before the assignment took place. Prove that this rule
is correct (the first hypothesis is the functional extensionality
axiom, which you will need at some point). Also note that this
rule is more complicated than <span class="inlinecode"><span class="id" type="var">hoare_asgn</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_asgn_fwd</span> :<br/>
(<span style="font-family: arial;">∀</span>{<span class="id" type="var">X</span> <span class="id" type="var">Y</span>: <span class="id" type="keyword">Type</span>} {<span class="id" type="var">f</span> <span class="id" type="var">g</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Y</span>},<br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">x</span>: <span class="id" type="var">X</span>), <span class="id" type="var">f</span> <span class="id" type="var">x</span> = <span class="id" type="var">g</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">f</span> = <span class="id" type="var">g</span>) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">m</span> <span class="id" type="var">a</span> <span class="id" type="var">P</span>,<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span> <span style="font-family: arial;">∧</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> = <span class="id" type="var">m</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">a</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span>) <span style="font-family: arial;">∧</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> = <span class="id" type="var">aeval</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span>) <span class="id" type="var">a</span> <span style="letter-spacing:-.4em;">}</span>}.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">functional_extensionality</span> <span class="id" type="var">m</span> <span class="id" type="var">a</span> <span class="id" type="var">P</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="lab523"></a><h4 class="section">Exercise: 2 stars, advanced (hoare_asgn_fwd_exists)</h4>
Another way to define a forward rule for assignment is to
existentially quantify over the previous value of the assigned
variable.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"> </td>
<td class="infrulenamecol" rowspan="3">
(hoare_asgn_fwd_exists)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{fun st ⇒ P st<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">X ::= a</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{fun st ⇒ <span style="font-family: arial;">∃</span>m, P (update st X m) <span style="font-family: arial;">∧</span></td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">st X = aeval (update st X m) a <span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
</table></center>
</div>
<div class="code code-tight">
<span class="comment">(* This rule was proposed by Nick Giannarakis and Zoe Paraskevopoulou. *)</span><br/>
<br/>
<span class="id" type="keyword">Theorem</span> <span class="id" type="var">hoare_asgn_fwd_exists</span> :<br/>
(<span style="font-family: arial;">∀</span>{<span class="id" type="var">X</span> <span class="id" type="var">Y</span>: <span class="id" type="keyword">Type</span>} {<span class="id" type="var">f</span> <span class="id" type="var">g</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">Y</span>},<br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">x</span>: <span class="id" type="var">X</span>), <span class="id" type="var">f</span> <span class="id" type="var">x</span> = <span class="id" type="var">g</span> <span class="id" type="var">x</span>) <span style="font-family: arial;">→</span> <span class="id" type="var">f</span> = <span class="id" type="var">g</span>) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">a</span> <span class="id" type="var">P</span>,<br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span class="id" type="var">P</span> <span class="id" type="var">st</span><span style="letter-spacing:-.4em;">}</span>}<br/>
<span class="id" type="var">X</span> ::= <span class="id" type="var">a</span><br/>
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="keyword">fun</span> <span class="id" type="var">st</span> ⇒ <span style="font-family: arial;">∃</span><span class="id" type="var">m</span>, <span class="id" type="var">P</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span>) <span style="font-family: arial;">∧</span><br/>
<span class="id" type="var">st</span> <span class="id" type="var">X</span> = <span class="id" type="var">aeval</span> (<span class="id" type="var">update</span> <span class="id" type="var">st</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span>) <span class="id" type="var">a</span> <span style="letter-spacing:-.4em;">}</span>}.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">functional_extensionality</span> <span class="id" type="var">a</span> <span class="id" type="var">P</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="lab524"></a><h3 class="section">Consequence</h3>
<div class="paragraph"> </div>
Sometimes the preconditions and postconditions we get from the
Hoare rules won't quite be the ones we want in the particular
situation at hand — they may be logically equivalent but have a
different syntactic form that fails to unify with the goal we are
trying to prove, or they actually may be logically weaker (for
preconditions) or stronger (for postconditions) than what we need.
<div class="paragraph"> </div>
For instance, while
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{(<span class="id" type="var">X</span> = 3) [<span class="id" type="var">X</span> <span style="font-family: arial;">↦</span> 3]<span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= 3 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 3<span style="letter-spacing:-.4em;">}</span>},
<div class="paragraph"> </div>
</div>
follows directly from the assignment rule,
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">True</span><span style="letter-spacing:-.4em;">}</span>} <span class="id" type="var">X</span> ::= 3 <span style="letter-spacing:-.4em;">{</span>{<span class="id" type="var">X</span> = 3<span style="letter-spacing:-.4em;">}</span>}.
<div class="paragraph"> </div>
</div>
does not. This triple is valid, but it is not an instance of
<span class="inlinecode"><span class="id" type="var">hoare_asgn</span></span> because <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">3)</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">3]</span> are not
syntactically equal assertions. However, they are logically
equivalent, so if one triple is valid, then the other must
certainly be as well. We might capture this observation with the
following rule:
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{P'<span style="letter-spacing:-.4em;">}</span>} c <span style="letter-spacing:-.4em;">{</span>{Q<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">P <span style="font-family: arial;">⇿</span> P'</td>
<td class="infrulenamecol" rowspan="3">
(hoare_consequence_pre_equiv)
</td></tr>
<tr class="infrulemiddle">
<td class="infrule"><hr /></td>
</tr>
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{P<span style="letter-spacing:-.4em;">}</span>} c <span style="letter-spacing:-.4em;">{</span>{Q<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
</table></center> Taking this line of thought a bit further, we can see that
strengthening the precondition or weakening the postcondition of a
valid triple always produces another valid triple. This
observation is captured by two <i>Rules of Consequence</i>.
<center><table class="infrule">
<tr class="infruleassumption">
<td class="infrule"><span style="letter-spacing:-.4em;">{</span>{P'<span style="letter-spacing:-.4em;">}</span>} c <span style="letter-spacing:-.4em;">{</span>{Q<span style="letter-spacing:-.4em;">}</span>}</td>
<td></td>
</td>
<tr class="infruleassumption">
<td class="infrule">P <span style="font-family: arial;">⇾</span> P'</td>
<td class="infrulenamecol" rowspan="3">
(hoare_consequence_pre)