doc/guide/localhost/EqualityTypeVariable

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  26 <h1>EqualityTypeVariable</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p>An equality type variable is a type variable that starts with two or
  32 more primes, as in <span class="monospaced">''a</span> or <span class="monospaced">''b</span>.  The canonical use of equality type
  33 variables is in specifying the type of the <a href="PolymorphicEquality">PolymorphicEquality</a>
  34 function, which is <span class="monospaced">''a * ''a -&gt; bool</span>.  Equality type variables
  35 ensure that polymorphic equality is only used on
  36 <a href="EqualityType">equality types</a>, by requiring that at every use of a
  37 polymorphic value, equality type variables are instantiated by
  38 equality types.</p></div>
  39 <div class="paragraph"><p>For example, the following program is type correct because polymorphic
  40 equality is applied to variables of type <span class="monospaced">''a</span>.</p></div>
  41 <div class="listingblock">
  42 <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">&#39;&#39;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">&#39;&#39;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>
  43 </pre></div></div></div>
  44 <div class="paragraph"><p>On the other hand, the following program is not type correct, because
  45 polymorphic equality is applied to variables of type <span class="monospaced">'a</span>, which is
  46 not an equality type.</p></div>
  47 <div class="listingblock">
  48 <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">&#39;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">&#39;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>
  49 </pre></div></div></div>
  50 <div class="paragraph"><p>MLton reports the following error, indicating that polymorphic
  51 equality expects equality types, but didn&#8217;t get them.</p></div>
  52 <div class="listingblock">
  53 <div class="content monospaced">
  54 <pre>Error: z.sml 1.30-1.34.
  55   Function applied to incorrect argument.
  56     expects: [&lt;equality&gt;] * [&lt;equality&gt;]
  57     but got: ['a] * ['a]
  58     in: = (x, y)</pre>
  59 </div></div>
  60 <div class="paragraph"><p>As an example of using such a function that requires equality types,
  61 suppose that <span class="monospaced">f</span> has polymorphic type <span class="monospaced">''a -&gt; unit</span>.  Then, <span class="monospaced">f 13</span> is
  62 type correct because <span class="monospaced">int</span> is an equality type.  On the other hand,
  63 <span class="monospaced">f 13.0</span> and <span class="monospaced">f (fn x =&gt; x)</span> are not type correct, because <span class="monospaced">real</span> and
  64 arrow types are not equality types.  We can test these facts with the
  65 following short programs.  First, we verify that such an <span class="monospaced">f</span> can be
  66 applied to integers.</p></div>
  67 <div class="listingblock">
  68 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
  69 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  70 <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>
  71 <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>
  72 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  73 </pre></div></div></div>
  74 <div class="paragraph"><p>We can do better, and verify that such an <span class="monospaced">f</span> can be applied to
  75 any integer.</p></div>
  76 <div class="listingblock">
  77 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
  78 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  79 <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>
  80 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  81 </pre></div></div></div>
  82 <div class="paragraph"><p>Even better, we don&#8217;t need to introduce a dummy function name; we can
  83 use a type constraint.</p></div>
  84 <div class="listingblock">
  85 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
  86 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  87 <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">-&gt;</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
  88 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  89 </pre></div></div></div>
  90 <div class="paragraph"><p>Even better, we can use a signature constraint.</p></div>
  91 <div class="listingblock">
  92 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
  93 <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">-&gt;</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>
  94 </pre></div></div></div>
  95 <div class="paragraph"><p>This functor concisely verifies that a function of polymorphic type
  96 <span class="monospaced">''a -&gt; unit</span> can be safely used as a function of type <span class="monospaced">int -&gt; unit</span>.</p></div>
  97 <div class="paragraph"><p>As above, we can verify that such an <span class="monospaced">f</span> can not be used at
  98 non-equality types.</p></div>
  99 <div class="listingblock">
 100 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
 101 <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">-&gt;</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>
 102
 103 <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">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</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>
 104 <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">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</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>
 105 </pre></div></div></div>
 106 <div class="paragraph"><p>MLton reports the following errors.</p></div>
 107 <div class="listingblock">
 108 <div class="content monospaced">
 109 <pre>Error: z.sml 2.4-2.30.
 110   Variable in structure disagrees with signature (type): f.
 111     structure: val f: [&lt;equality&gt;] -&gt; _
 112     defn at: z.sml 1.25-1.25
 113     signature: val f: [real] -&gt; _
 114     spec at: z.sml 2.12-2.12
 115 Error: z.sml 5.4-5.36.
 116   Variable in structure disagrees with signature (type): f.
 117     structure: val f: [&lt;equality&gt;] -&gt; _
 118     defn at: z.sml 4.25-4.25
 119     signature: val f: [_ -&gt; _] -&gt; _
 120     spec at: z.sml 5.12-5.12</pre>
 121 </div></div>
 122 </div>
 123 </div>
 124 <div class="sect1">
 125 <h2 id="_equality_type_variables_in_type_and_datatype_declarations">Equality type variables in type and datatype declarations</h2>
 126 <div class="sectionbody">
 127 <div class="paragraph"><p>Equality type variables can be used in type and datatype declarations;
 128 however they play no special role.  For example,</p></div>
 129 <div class="listingblock">
 130 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="n">&#39;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">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">int</span><span class="w"></span>
 131 </pre></div></div></div>
 132 <div class="paragraph"><p>is completely identical to</p></div>
 133 <div class="listingblock">
 134 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="n">&#39;&#39;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">&#39;&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">int</span><span class="w"></span>
 135 </pre></div></div></div>
 136 <div class="paragraph"><p>In particular, such a definition does <em>not</em> require that <span class="monospaced">t</span> only be
 137 applied to equality types.</p></div>
 138 <div class="paragraph"><p>Similarly,</p></div>
 139 <div class="listingblock">
 140 <div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;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">&#39;a</span><span class="w"></span>
 141 </pre></div></div></div>
 142 <div class="paragraph"><p>is completely identical to</p></div>
 143 <div class="listingblock">
 144 <div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;&#39;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">&#39;&#39;a</span><span class="w"></span>
 145 </pre></div></div></div>
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