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| 6 | <title>EqualityTypeVariable</title>\r | |
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| 22 | <a href="./Home">MLton 20180207</a>\r | |
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| 25 | <div id="header">\r | |
| 26 | <h1>EqualityTypeVariable</h1>\r | |
| 27 | </div>\r | |
| 28 | <div id="content">\r | |
| 29 | <div id="preamble">\r | |
| 30 | <div class="sectionbody">\r | |
| 31 | <div class="paragraph"><p>An equality type variable is a type variable that starts with two or\r | |
| 32 | more primes, as in <span class="monospaced">''a</span> or <span class="monospaced">''b</span>. The canonical use of equality type\r | |
| 33 | variables is in specifying the type of the <a href="PolymorphicEquality">PolymorphicEquality</a>\r | |
| 34 | function, which is <span class="monospaced">''a * ''a -> bool</span>. Equality type variables\r | |
| 35 | ensure that polymorphic equality is only used on\r | |
| 36 | <a href="EqualityType">equality types</a>, by requiring that at every use of a\r | |
| 37 | polymorphic value, equality type variables are instantiated by\r | |
| 38 | equality types.</p></div>\r | |
| 39 | <div class="paragraph"><p>For example, the following program is type correct because polymorphic\r | |
| 40 | equality is applied to variables of type <span class="monospaced">''a</span>.</p></div>\r | |
| 41 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 43 | </pre></div></div></div>\r | |
| 44 | <div class="paragraph"><p>On the other hand, the following program is not type correct, because\r | |
| 45 | polymorphic equality is applied to variables of type <span class="monospaced">'a</span>, which is\r | |
| 46 | not an equality type.</p></div>\r | |
| 47 | <div class="listingblock">\r | |
| 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">'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>\r | |
| 49 | </pre></div></div></div>\r | |
| 50 | <div class="paragraph"><p>MLton reports the following error, indicating that polymorphic\r | |
| 51 | equality expects equality types, but didn’t get them.</p></div>\r | |
| 52 | <div class="listingblock">\r | |
| 53 | <div class="content monospaced">\r | |
| 54 | <pre>Error: z.sml 1.30-1.34.\r | |
| 55 | Function applied to incorrect argument.\r | |
| 56 | expects: [<equality>] * [<equality>]\r | |
| 57 | but got: ['a] * ['a]\r | |
| 58 | in: = (x, y)</pre>\r | |
| 59 | </div></div>\r | |
| 60 | <div class="paragraph"><p>As an example of using such a function that requires equality types,\r | |
| 61 | suppose that <span class="monospaced">f</span> has polymorphic type <span class="monospaced">''a -> unit</span>. Then, <span class="monospaced">f 13</span> is\r | |
| 62 | type correct because <span class="monospaced">int</span> is an equality type. On the other hand,\r | |
| 63 | <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\r | |
| 64 | arrow types are not equality types. We can test these facts with the\r | |
| 65 | following short programs. First, we verify that such an <span class="monospaced">f</span> can be\r | |
| 66 | applied to integers.</p></div>\r | |
| 67 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 69 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r | |
| 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>\r | |
| 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>\r | |
| 72 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
| 73 | </pre></div></div></div>\r | |
| 74 | <div class="paragraph"><p>We can do better, and verify that such an <span class="monospaced">f</span> can be applied to\r | |
| 75 | any integer.</p></div>\r | |
| 76 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 78 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r | |
| 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>\r | |
| 80 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
| 81 | </pre></div></div></div>\r | |
| 82 | <div class="paragraph"><p>Even better, we don’t need to introduce a dummy function name; we can\r | |
| 83 | use a type constraint.</p></div>\r | |
| 84 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 86 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r | |
| 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">-></span><span class="w"> </span><span class="n">unit</span><span class="w"></span>\r | |
| 88 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
| 89 | </pre></div></div></div>\r | |
| 90 | <div class="paragraph"><p>Even better, we can use a signature constraint.</p></div>\r | |
| 91 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 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">-></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>\r | |
| 94 | </pre></div></div></div>\r | |
| 95 | <div class="paragraph"><p>This functor concisely verifies that a function of polymorphic type\r | |
| 96 | <span class="monospaced">''a -> unit</span> can be safely used as a function of type <span class="monospaced">int -> unit</span>.</p></div>\r | |
| 97 | <div class="paragraph"><p>As above, we can verify that such an <span class="monospaced">f</span> can not be used at\r | |
| 98 | non-equality types.</p></div>\r | |
| 99 | <div class="listingblock">\r | |
| 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">''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>\r | |
| 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">-></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>\r | |
| 102 | \r | |
| 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">''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>\r | |
| 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">'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>\r | |
| 105 | </pre></div></div></div>\r | |
| 106 | <div class="paragraph"><p>MLton reports the following errors.</p></div>\r | |
| 107 | <div class="listingblock">\r | |
| 108 | <div class="content monospaced">\r | |
| 109 | <pre>Error: z.sml 2.4-2.30.\r | |
| 110 | Variable in structure disagrees with signature (type): f.\r | |
| 111 | structure: val f: [<equality>] -> _\r | |
| 112 | defn at: z.sml 1.25-1.25\r | |
| 113 | signature: val f: [real] -> _\r | |
| 114 | spec at: z.sml 2.12-2.12\r | |
| 115 | Error: z.sml 5.4-5.36.\r | |
| 116 | Variable in structure disagrees with signature (type): f.\r | |
| 117 | structure: val f: [<equality>] -> _\r | |
| 118 | defn at: z.sml 4.25-4.25\r | |
| 119 | signature: val f: [_ -> _] -> _\r | |
| 120 | spec at: z.sml 5.12-5.12</pre>\r | |
| 121 | </div></div>\r | |
| 122 | </div>\r | |
| 123 | </div>\r | |
| 124 | <div class="sect1">\r | |
| 125 | <h2 id="_equality_type_variables_in_type_and_datatype_declarations">Equality type variables in type and datatype declarations</h2>\r | |
| 126 | <div class="sectionbody">\r | |
| 127 | <div class="paragraph"><p>Equality type variables can be used in type and datatype declarations;\r | |
| 128 | however they play no special role. For example,</p></div>\r | |
| 129 | <div class="listingblock">\r | |
| 130 | <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>\r | |
| 131 | </pre></div></div></div>\r | |
| 132 | <div class="paragraph"><p>is completely identical to</p></div>\r | |
| 133 | <div class="listingblock">\r | |
| 134 | <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>\r | |
| 135 | </pre></div></div></div>\r | |
| 136 | <div class="paragraph"><p>In particular, such a definition does <em>not</em> require that <span class="monospaced">t</span> only be\r | |
| 137 | applied to equality types.</p></div>\r | |
| 138 | <div class="paragraph"><p>Similarly,</p></div>\r | |
| 139 | <div class="listingblock">\r | |
| 140 | <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>\r | |
| 141 | </pre></div></div></div>\r | |
| 142 | <div class="paragraph"><p>is completely identical to</p></div>\r | |
| 143 | <div class="listingblock">\r | |
| 144 | <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>\r | |
| 145 | </pre></div></div></div>\r | |
| 146 | </div>\r | |
| 147 | </div>\r | |
| 148 | </div>\r | |
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