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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><meta name="viewport" content="width=device-width, initial-scale=0.8"/><title>7.5 Contracts on Structures</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-style.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../doc-site.css" title="default"/><script type="text/javascript" src="../scribble-common.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../doc-site.js"></script><script type="text/javascript" src="../local-redirect/local-redirect.js"></script><script type="text/javascript" src="../local-redirect/local-user-redirect.js"></script><!--[if IE 6]><style type="text/css">.SIEHidden { overflow: hidden; }</style><![endif]--></head><body id="doc-racket-lang-org"><div class="tocset"><div class="tocview"><div class="tocviewlist tocviewlisttopspace"><div class="tocviewtitle"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_0");">►</a></td><td></td><td><a href="index.html" class="tocviewlink" data-pltdoc="x">The Racket Guide</a></td></tr></table></div><div class="tocviewsublisttop" style="display: none;" id="tocview_0"><table cellspacing="0" cellpadding="0"><tr><td align="right">1 </td><td><a href="intro.html" class="tocviewlink" data-pltdoc="x">Welcome to Racket</a></td></tr><tr><td align="right">2 </td><td><a href="to-scheme.html" class="tocviewlink" data-pltdoc="x">Racket Essentials</a></td></tr><tr><td align="right">3 </td><td><a href="datatypes.html" class="tocviewlink" data-pltdoc="x">Built-<wbr></wbr>In Datatypes</a></td></tr><tr><td align="right">4 </td><td><a href="scheme-forms.html" class="tocviewlink" data-pltdoc="x">Expressions and Definitions</a></td></tr><tr><td align="right">5 </td><td><a href="define-struct.html" class="tocviewlink" data-pltdoc="x">Programmer-<wbr></wbr>Defined Datatypes</a></td></tr><tr><td align="right">6 </td><td><a href="modules.html" class="tocviewlink" data-pltdoc="x">Modules</a></td></tr><tr><td align="right">7 </td><td><a href="contracts.html" class="tocviewselflink" data-pltdoc="x">Contracts</a></td></tr><tr><td align="right">8 </td><td><a href="i_o.html" class="tocviewlink" data-pltdoc="x">Input and Output</a></td></tr><tr><td align="right">9 </td><td><a href="regexp.html" class="tocviewlink" data-pltdoc="x">Regular Expressions</a></td></tr><tr><td align="right">10 </td><td><a href="control.html" class="tocviewlink" data-pltdoc="x">Exceptions and Control</a></td></tr><tr><td align="right">11 </td><td><a href="for.html" class="tocviewlink" data-pltdoc="x">Iterations and Comprehensions</a></td></tr><tr><td align="right">12 </td><td><a href="match.html" class="tocviewlink" data-pltdoc="x">Pattern Matching</a></td></tr><tr><td align="right">13 </td><td><a href="classes.html" class="tocviewlink" data-pltdoc="x">Classes and Objects</a></td></tr><tr><td align="right">14 </td><td><a href="units.html" class="tocviewlink" data-pltdoc="x">Units</a></td></tr><tr><td align="right">15 </td><td><a href="reflection.html" class="tocviewlink" data-pltdoc="x">Reflection and Dynamic Evaluation</a></td></tr><tr><td align="right">16 </td><td><a href="macros.html" class="tocviewlink" data-pltdoc="x">Macros</a></td></tr><tr><td align="right">17 </td><td><a href="languages.html" class="tocviewlink" data-pltdoc="x">Creating Languages</a></td></tr><tr><td align="right">18 </td><td><a href="concurrency.html" class="to
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<span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span> definitions, i.e., the ability to create
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structs of a certain kind, to access their fields, to modify
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them, and to distinguish structs of this kind against every
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other kind of value in the world. Second, on occasion a
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module exports a specific struct and wishes to promise that
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its fields contain values of a certain kind. This section
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explains how to protect structs with contracts for both
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uses.</p><h5 x-source-module="(lib "scribblings/guide/guide.scrbl")" x-source-pkg="racket-doc" x-part-tag=""contracts-single-struct"">7.5.1<tt> </tt><a name="(part._contracts-single-struct)"></a>Guarantees for a Specific Value</h5><p>If your module defines a variable to be a structure value, then you can
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specify the structure’s shape using <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span>.</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><a href="Module_Syntax.html#%28part._hash-lang%29" class="RktModLink" data-pltdoc="x"><span class="RktMod">#lang</span></a><span class="hspace"> </span><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=index.html&version=8.6" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">racket</span></a></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span><span class="hspace"> </span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">x</span><span class="hspace"> </span><span class="RktSym">y</span><span class="RktPn">]</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">origin</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._provide%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">provide</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=attaching-contracts-to-values.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fbase..rkt%2529._contract-out%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">contract-out</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">origin</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span><span class="hspace"> </span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._zero%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">zero?</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.or
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2-dimensional positions and then creates one specific instance:
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<span class="RktSym">origin</span>. The export of this instance guarantees that its fields
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are <span class="RktVal">0</span>, i.e., they represent <span class="stt">(0,0)</span>, on the Cartesian grid.</p><blockquote class="refpara"><blockquote class="refcolumn"><blockquote class="refcontent"><p>See also <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528def._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fvector..rkt%2529._vector%252Fc%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">vector/c</a></span> and similar contract
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combinators for (flat) compound data.</p></blockquote></blockquote></blockquote><h5 x-source-module="(lib "scribblings/guide/guide.scrbl")" x-source-pkg="racket-doc" x-part-tag=""contracts-define-struct"">7.5.2<tt> </tt><a name="(part._contracts-define-struct)"></a>Guarantees for All Values</h5><p>The book <span style="font-style: italic"><a href="https://htdp.org">How to Design Programs</a></span> teaches that <span class="RktSym">posn</span>s should contain only
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numbers in their two fields. With contracts we would enforce this
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informal data definition as follows:</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><a href="Module_Syntax.html#%28part._hash-lang%29" class="RktModLink" data-pltdoc="x"><span class="RktMod">#lang</span></a><span class="hspace"> </span><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=index.html&version=8.6" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">racket</span></a></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span><span class="hspace"> </span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">x</span><span class="hspace"> </span><span class="RktSym">y</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._provide%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">provide</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=attaching-contracts-to-values.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fbase..rkt%2529._contract-out%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">contract-out</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span><span class="hspace"> </span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktPn">(</span><span class="RktSym">x</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._number%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">number?</a></span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">y</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._number%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">number?</a></span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">]</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">p-okay</span><span class="hspace"> </span><span class="RktSym">posn?</span><span class="RktPn">]</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">p-sick</span><span class="hspace"> </span><span class="RktSym">posn?</span><span class="RktPn">]</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.htm
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<span class="RktSym">posn?</span>, <span class="RktSym">posn-x</span>, <span class="RktSym">posn-y</span>,
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<span class="RktSym">set-posn-x!</span>, and <span class="RktSym">set-posn-y!</span>. Each function enforces
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or promises that the two fields of a <span class="RktSym">posn</span> structure are
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numbers —<wbr></wbr> when the values flow across the module boundary. Thus, if
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a client calls <span class="RktSym">posn</span> on <span class="RktVal">10</span> and <span class="RktVal">'</span><span class="RktVal">a</span>, the
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contract system signals a contract violation.</p><p>The creation of <span class="RktSym">p-sick</span> inside of the <span class="RktSym">posn</span> module,
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however, does not violate the contracts. The function <span class="RktSym">posn</span> is
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used internally, so <span class="RktVal">'</span><span class="RktVal">a</span> and <span class="RktVal">'</span><span class="RktVal">b</span> don’t cross the module
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boundary. Similarly, when <span class="RktSym">p-sick</span> crosses the boundary of
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<span class="RktSym">posn</span>, the contract promises a <span class="RktSym">posn?</span> and nothing
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else. In particular, this check does <span style="font-style: italic">not</span> require that the
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fields of <span class="RktSym">p-sick</span> are numbers.</p><p>The association of contract checking with module boundaries implies that
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<span class="RktSym">p-okay</span> and <span class="RktSym">p-sick</span> look alike from a client’s
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perspective until the client extracts the pieces:</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><a href="Module_Syntax.html#%28part._hash-lang%29" class="RktModLink" data-pltdoc="x"><span class="RktMod">#lang</span></a><span class="hspace"> </span><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=index.html&version=8.6" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">racket</span></a></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._require%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">require</a></span><span class="hspace"> </span><span class="RktSym">lang/posn</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529._......%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">...</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">posn-x</span><span class="hspace"> </span><span class="RktSym">p-sick</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529._......%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">...</a></span></td></tr></table></blockquote><p>Using <span class="RktSym">posn-x</span> is the only way the client can find out what
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a <span class="RktSym">posn</span> contains in the <span class="RktSym">x</span> field. The application of
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<span class="RktSym">posn-x</span> sends <span class="RktSym">p-sick</span> back into the
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<span class="RktSym">posn</span> module and the result value – <span class="RktVal">'</span><span class="RktVal">a</span> here – back to
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the client, again across the module boundary. At this very point, the contract
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system discovers that a promise is broken. Specifically, <span class="RktSym">posn-x</span>
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doesn’t return a number but a symbol and is therefore blamed.</p><p>This specific example shows that the explanation for a contract violation
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doesn’t always pinpoint the source of the error. The good news is that the
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error is located in the <span class="RktSym">posn</span> module. The bad news is that the
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explanation is misleading. Although it is true that <span class="RktSym">posn-x</span>
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produced a symbol instead of a number, it is the fault of the programmer who
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created a <span class="RktSym">posn</span> from symbols, i.e., the programmer who added</p><blockquote class="SCodeFlow"><p><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">p-sick</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">a</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">b</span><span class="RktPn">)</span><span class="RktPn">)</span></p></blockquote><p>to the module. So, when you are looking for bugs based on contract
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violations, keep this example in mind.</p><p>If we want to fix the contract for <span class="RktSym">p-sick</span> so that the error
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is caught when <span class="RktSym">sick</span> is exported, a single change suffices:</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._provide%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">provide</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=attaching-contracts-to-values.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fbase..rkt%2529._contract-out%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">contract-out</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529._......%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">...</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">p-sick</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span><span class="hspace"> </span><span class="RktSym">posn</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._number%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">number?</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=number-types.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._number%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">number?</a></span><span class="RktPn">)</span><span class="RktPn">]</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></blockquote><p>That is, instead of exporting <span class="RktSym">p-sick</span> as a plain
|
||
|
|
<span class="RktSym">posn?</span>, we use a <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span> contract to enforce
|
||
|
|
constraints on its components.</p><h5 x-source-module="(lib "scribblings/guide/guide.scrbl")" x-source-pkg="racket-doc" x-part-tag=""contracts-lazy-contracts"">7.5.3<tt> </tt><a name="(part._contracts-lazy-contracts)"></a>Checking Properties of Data Structures</h5><p>Contracts written using <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span> immediately
|
||
|
|
check the fields of the data structure, but sometimes this
|
||
|
|
can have disastrous effects on the performance of a program
|
||
|
|
that does not, itself, inspect the entire data structure.</p><p>As an example, consider the binary search tree
|
||
|
|
search algorithm. A binary search tree is like a binary
|
||
|
|
tree, except that the numbers are organized in the tree to
|
||
|
|
make searching the tree fast. In particular, for each
|
||
|
|
interior node in the tree, all of the numbers in the left
|
||
|
|
subtree are smaller than the number in the node, and all of
|
||
|
|
the numbers in the right subtree are larger than the number
|
||
|
|
in the node.</p><p><div class="SIntrapara">We can implement a search function <span class="RktSym">in?</span> that takes
|
||
|
|
advantage of the structure of the binary search tree.
|
||
|
|
</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><a href="Module_Syntax.html#%28part._hash-lang%29" class="RktModLink" data-pltdoc="x"><span class="RktMod">#lang</span></a><span class="hspace"> </span><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=index.html&version=8.6" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">racket</span></a></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span><span class="hspace"> </span><span class="RktSym">node</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">val</span><span class="hspace"> </span><span class="RktSym">left</span><span class="hspace"> </span><span class="RktSym">right</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">determines if `n' is in the binary search tree `b',</span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">exploiting the binary search tree invariant</span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">in?</span><span class="hspace"> </span><span class="RktSym">n</span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fletstx-scheme..rkt%2529._cond%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">cond</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._null%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">null?</a></span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="RktPn">]</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fletstx-scheme..rkt%2529._else%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">else</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fletstx-scheme..rkt%2529._cond%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">cond</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktPn">(</span><
|
||
|
|
the <span class="RktSym">in?</span> function only has to explore a
|
||
|
|
logarithmic number of nodes.</p><p>The contract on <span class="RktSym">in?</span> guarantees that its input
|
||
|
|
is a binary search tree. But a little careful thought
|
||
|
|
reveals that this contract defeats the purpose of the binary
|
||
|
|
search tree algorithm. In particular, consider the
|
||
|
|
inner <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fletstx-scheme..rkt%2529._cond%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">cond</a></span> in the <span class="RktSym">in?</span>
|
||
|
|
function. This is where the <span class="RktSym">in?</span> function gets
|
||
|
|
its speed: it avoids searching an entire subtree at each
|
||
|
|
recursive call. Now compare that to the <span class="RktSym">bst-between?</span>
|
||
|
|
function. In the case that it returns <span class="RktVal">#t</span>, it
|
||
|
|
traverses the entire tree, meaning that the speedup
|
||
|
|
of <span class="RktSym">in?</span> is lost.</p><p>In order to fix that, we can employ a new strategy for
|
||
|
|
checking the binary search tree contract. In particular, if
|
||
|
|
we only checked the contract on the nodes
|
||
|
|
that <span class="RktSym">in?</span> looks at, we can still guarantee that
|
||
|
|
the tree is at least partially well-formed, but without
|
||
|
|
changing the complexity.</p><p>To do that, we need to use <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fdc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/dc</a></span> to define
|
||
|
|
<span class="RktSym">bst-between?</span>. Like <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span>, <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fdc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/dc</a></span> defines a
|
||
|
|
contract for a structure. Unlike
|
||
|
|
<span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/c</a></span>, it allows fields to be marked as lazy, so that
|
||
|
|
the contracts are only checked when the matching selector is called.
|
||
|
|
Also, it does not allow mutable fields to be marked as lazy.</p><p>The <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fdc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/dc</a></span> form accepts a contract for each
|
||
|
|
field of the struct and returns a contract on the
|
||
|
|
struct. More interestingly, <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fdc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/dc</a></span> allows us to write dependent
|
||
|
|
contracts, i.e., contracts where some of the contracts on
|
||
|
|
the fields depend on the values of other fields. We can use
|
||
|
|
this to define the binary search tree contract:</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><a href="Module_Syntax.html#%28part._hash-lang%29" class="RktModLink" data-pltdoc="x"><span class="RktMod">#lang</span></a><span class="hspace"> </span><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=index.html&version=8.6" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">racket</span></a></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define-struct.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._struct%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct</a></span><span class="hspace"> </span><span class="RktSym">node</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">val</span><span class="hspace"> </span><span class="RktSym">left</span><span class="hspace"> </span><span class="RktSym">right</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">determines if `n' is in the binary search tree `b'</span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">in?</span><span class="hspace"> </span><span class="RktSym">n</span><span class="hspace"> </span><span class="RktSym">b</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529._......%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">...</a></span><span class="hspace"> </span><span class="RktSym">as</span><span class="hspace"> </span><span class="RktSym">before</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529._......%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">...</a></span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">bst-between : number number -> contract</span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">builds a contract for binary search trees</span></td></tr><tr><td><span class="RktCmt">;</span><span class="RktCmt"> </span><span class="RktCmt">whose values are between low and high</span></td></tr><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">bst-between/c</span><span class="hspace"> </span><span class="RktSym">low</span><span class="hspace"> </span><span class="RktSym">high</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span>
|
||
|
|
fields and then specify contracts for each field. In the
|
||
|
|
above, the <span class="RktSym">val</span> field is a contract that accepts
|
||
|
|
values between <span class="RktSym">low</span> and <span class="RktSym">high</span>.
|
||
|
|
The <span class="RktSym">left</span> and <span class="RktSym">right</span> fields are
|
||
|
|
dependent on the value of the <span class="RktSym">val</span> field,
|
||
|
|
indicated by their second sub-expressions. They are
|
||
|
|
also marked with the <span class="RktPn">#:lazy</span> keyword to indicate
|
||
|
|
that they should be checked only when the appropriate
|
||
|
|
accessor is called on the struct instance. Their contracts
|
||
|
|
are built by recursive calls to
|
||
|
|
the <span class="RktSym">bst-between/c</span> function. Taken together,
|
||
|
|
this contract ensures the same thing that
|
||
|
|
the <span class="RktSym">bst-between?</span> function checked in the
|
||
|
|
original example, but here the checking only happens
|
||
|
|
as <span class="RktSym">in?</span> explores the tree.</p><p>Although this contract improves the performance
|
||
|
|
of <span class="RktSym">in?</span>, restoring it to the logarithmic
|
||
|
|
behavior that the contract-less version had, it is still
|
||
|
|
imposes a fairly large constant overhead. So, the contract
|
||
|
|
library also provides <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=contract-utilities.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fopt..rkt%2529._define-opt%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define-opt/c</a></span> that brings
|
||
|
|
down that constant factor by optimizing its body. Its shape
|
||
|
|
is just like the <span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define</a></span> above. It expects its
|
||
|
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body to be a contract and then optimizes that contract.</p><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=contract-utilities.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fopt..rkt%2529._define-opt%252Fc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">define-opt/c</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">bst-between/c</span><span class="hspace"> </span><span class="RktSym">low</span><span class="hspace"> </span><span class="RktSym">high</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528def._%2528%2528lib._racket%252Fcontract%252Fbase..rkt%2529._or%252Fc%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">or/c</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._null%7E3f%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">null?</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528form._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fstruct-dc..rkt%2529._struct%252Fdc%2529%2529&version=8.6" class="RktStxLink Sq" data-pltdoc="x">struct/dc</a></span><span class="hspace"> </span><span class="RktSym">node</span><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">val</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/releases/8.6/doc/local-redirect/index.html?doc=reference&rel=data-structure-contracts.html%23%2528def._%2528%2528lib._racket%252Fcontract%252Fprivate%252Fmisc..rkt%2529._between%252Fc%2529%2529&version=8.6" class="RktValLink Sq" data-pltdoc="x">between/c</a></span><span class="hspace"> </span><span class="RktSym">low</span><span class="hspace"> </span><span class="RktSym">high</span><span class="RktPn">)</span><span class="RktPn">]</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">left</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">val</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:lazy</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">bst-between/c</span><span class="hspace"> </span><span class="RktSym">low</span><span class="hspace"> </span><span class="RktSym">val</span><span class="RktPn">)</span><span class="RktPn">]</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">[</span><span class="RktSym">right</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">val</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:lazy</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">bst-between/c</span><span class="hspace"> </span><span class="RktSym">val</span><span class="hspace"> </span><span class="RktSym">high<
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