<!DOCTYPE html> <html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta name="generator" content="AsciiDoc 8.6.8"> <title>EqualityTypeVariable</title> <link rel="stylesheet" href="./asciidoc.css" type="text/css"> <link rel="stylesheet" href="./pygments.css" type="text/css"> <script type="text/javascript" src="./asciidoc.js"></script> <script type="text/javascript"> /*<![CDATA[*/ asciidoc.install(); /*]]>*/ </script> <link rel="stylesheet" href="./mlton.css" type="text/css"/> </head> <body class="article"> <div id="banner"> <div id="banner-home"> <a href="./Home">MLton 20130715</a> </div> </div> <div id="header"> <h1>EqualityTypeVariable</h1> </div> <div id="content"> <div id="preamble"> <div class="sectionbody"> <div class="paragraph"><p>An equality type variable is a type variable that starts with two or more primes, as in <span class="monospaced">''a</span> or <span class="monospaced">''b</span>. The canonical use of equality type variables is in specifying the type of the <a href="PolymorphicEquality">PolymorphicEquality</a> function, which is <span class="monospaced">''a * ''a -> bool</span>. Equality type variables ensure that polymorphic equality is only used on <a href="EqualityType">equality types</a>, by requiring that at every use of a polymorphic value, equality type variables are instantiated by equality types.</p></div> <div class="paragraph"><p>For example, the following program is type correct because polymorphic equality is applied to variables of type <span class="monospaced">''a</span>.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="p">):</span><span class="w"> </span><span class="n">bool</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">y</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>On the other hand, the following program is not type correct, because polymorphic equality is applied to variables of type <span class="monospaced">'a</span>, which is not an equality type.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="p">):</span><span class="w"> </span><span class="n">bool</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">y</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>MLton reports the following error, indicating that polymorphic equality expects equality types, but didn’t get them.</p></div> <div class="listingblock"> <div class="content monospaced"> <pre>Error: z.sml 1.32. Function applied to incorrect argument. expects: [<equality>] * [<equality>] but got: [<non-equality>] * [<non-equality>] in: = (x, y)</pre> </div></div> <div class="paragraph"><p>As an example of using such a function that requires equality types, suppose that <span class="monospaced">f</span> has polymorphic type <span class="monospaced">''a -> unit</span>. Then, <span class="monospaced">f 13</span> is type correct because <span class="monospaced">int</span> is an equality type. On the other hand, <span class="monospaced">f 13.0</span> and <span class="monospaced">f (fn x => x)</span> are not type correct, because <span class="monospaced">real</span> and arrow types are not equality types. We can test these facts with the following short programs. First, we verify that such an <span class="monospaced">f</span> can be applied to integers.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Ok</span><span class="w"> </span><span class="p">(</span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="p">):</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"></span> <span class="w"> </span><span class="k">struct</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">14</span><span class="w"></span> <span class="w"> </span><span class="k">end</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>We can do better, and verify that such an <span class="monospaced">f</span> can be applied to any integer.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Ok</span><span class="w"> </span><span class="p">(</span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="p">):</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"></span> <span class="w"> </span><span class="k">struct</span><span class="w"></span> <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</span><span class="w"></span> <span class="w"> </span><span class="k">end</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>Even better, we don’t need to introduce a dummy function name; we can use a type constraint.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Ok</span><span class="w"> </span><span class="p">(</span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="p">):</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"></span> <span class="w"> </span><span class="k">struct</span><span class="w"></span> <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"></span> <span class="w"> </span><span class="k">end</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>Even better, we can use a signature constraint.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Ok</span><span class="w"> </span><span class="p">(</span><span class="n">S</span><span class="p">:</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="p">):</span><span class="w"></span> <span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">S</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>This functor concisely verifies that a function of polymorphic type <span class="monospaced">''a -> unit</span> can be safely used as a function of type <span class="monospaced">int -> unit</span>.</p></div> <div class="paragraph"><p>As above, we can verify that such an <span class="monospaced">f</span> can not be used at non equality types.</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Bad</span><span class="w"> </span><span class="p">(</span><span class="n">S</span><span class="p">:</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="p">):</span><span class="w"></span> <span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">S</span><span class="w"></span> </pre></div></div></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Bad</span><span class="w"> </span><span class="p">(</span><span class="n">S</span><span class="p">:</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="p">):</span><span class="w"></span> <span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">S</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>For each of these programs, MLton reports the following error.</p></div> <div class="listingblock"> <div class="content monospaced"> <pre>Error: z.sml 2.4. Variable type in structure disagrees with signature. variable: f structure: [<equality>] -> _ signature: [<non-equality>] -> _</pre> </div></div> </div> </div> <div class="sect1"> <h2 id="_equality_type_variables_in_type_and_datatype_declarations">Equality type variables in type and datatype declarations</h2> <div class="sectionbody"> <div class="paragraph"><p>Equality type variables can be used in type and datatype declarations; however they play no special role. For example,</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">int</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>is completely identical to</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">int</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>In particular, such a definition does <em>not</em> require that <span class="monospaced">t</span> only be applied to equality types.</p></div> <div class="paragraph"><p>Similarly,</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">'a</span><span class="w"></span> </pre></div></div></div> <div class="paragraph"><p>is completely identical to</p></div> <div class="listingblock"> <div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">''a</span><span class="w"></span> </pre></div></div></div> </div> </div> </div> <div id="footnotes"><hr></div> <div id="footer"> <div id="footer-text"> </div> <div id="footer-badges"> </div> </div> </body> </html>