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30 <div class="sectionbody">
31 <div class="paragraph"><p>This page describes a technique that enables convenient syntax for a
32 number of language features that are not explicitly supported by
33 <a href="StandardML">Standard ML</a>, including: variable number of arguments,
34 <a href="OptionalArguments">optional arguments and labeled arguments</a>,
35 <a href="ArrayLiteral">array and vector literals</a>,
36 <a href="FunctionalRecordUpdate">functional record update</a>,
37 and (seemingly) dependently typed functions like <a href="Printf">printf</a> and scanf.</p></div>
38 <div class="paragraph"><p>The key idea to <em>fold</em> is to define functions <span class="monospaced">fold</span>, <span class="monospaced">step0</span>,
39 and <span class="monospaced">$</span> such that the following equation holds.</p></div>
40 <div class="listingblock">
41 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
42 <span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">))))</span><span class="w"></span>
43 </pre></div></div></div>
44 <div class="paragraph"><p>The name <span class="monospaced">fold</span> comes because this is like a traditional list fold,
45 where <span class="monospaced">a</span> is the <em>base element</em>, and each <em>step function</em>,
46 <span class="monospaced">step0 hi</span>, corresponds to one element of the list and does one
47 step of the fold. The name <span class="monospaced">$</span> is chosen to mean "end of
48 arguments" from its common use in regular-expression syntax.</p></div>
49 <div class="paragraph"><p>Unlike the usual list fold in which the same function is used to step
50 over each element in the list, this fold allows the step functions to
51 be different from each other, and even to be of different types. Also
52 unlike the usual list fold, this fold includes a "finishing
53 function", <span class="monospaced">f</span>, that is applied to the result of the fold. The
54 presence of the finishing function may seem odd because there is no
55 analogy in list fold. However, the finishing function is essential;
56 without it, there would be no way for the folder to perform an
57 arbitrary computation after processing all the arguments. The
58 examples below will make this clear.</p></div>
59 <div class="paragraph"><p>The functions <span class="monospaced">fold</span>, <span class="monospaced">step0</span>, and <span class="monospaced">$</span> are easy to
61 <div class="listingblock">
62 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</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">a</span><span class="w"></span>
63 <span class="k">fun</span><span class="w"> </span><span class="n">id</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">x</span><span class="w"></span>
64 <span class="k">structure</span><span class="w"> </span><span class="n">Fold</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
65 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
66 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">g</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
67 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
68 <span class="w"> </span><span class="k">end</span><span class="w"></span>
69 </pre></div></div></div>
70 <div class="paragraph"><p>We’ve placed <span class="monospaced">fold</span> and <span class="monospaced">step0</span> in the <span class="monospaced">Fold</span> structure
71 but left <span class="monospaced">$</span> at the toplevel because it is convenient in code to
72 always have <span class="monospaced">$</span> in scope. We’ve also defined the identity
73 function, <span class="monospaced">id</span>, at the toplevel since we use it so frequently.</p></div>
74 <div class="paragraph"><p>Plugging in the definitions, it is easy to verify the equation from
76 <div class="listingblock">
77 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
78 <span class="p">=</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
79 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
80 <span class="p">=</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
81 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
82 <span class="p">...</span><span class="w"></span>
83 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</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">$</span><span class="w"></span>
84 <span class="p">=</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">))),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
85 <span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">))))</span><span class="w"></span>
86 </pre></div></div></div>
90 <h2 id="_example_variable_number_of_arguments">Example: variable number of arguments</h2>
91 <div class="sectionbody">
92 <div class="paragraph"><p>The simplest example of fold is accepting a variable number of
93 (curried) arguments. We’ll define a function <span class="monospaced">f</span> and argument
94 <span class="monospaced">a</span> such that all of the following expressions are valid.</p></div>
95 <div class="listingblock">
96 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
97 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
98 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
99 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
100 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</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">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="cm">(* as many a's as we want *)</span><span class="w"></span>
101 </pre></div></div></div>
102 <div class="paragraph"><p>Off-hand it may appear impossible that all of the above expressions
103 are type correct SML — how can a function <span class="monospaced">f</span> accept a variable
104 number of curried arguments? What could the type of <span class="monospaced">f</span> be?
105 We’ll have more to say later on how type checking works. For now,
106 once we have supplied the definitions below, you can check that the
107 expressions are type correct by feeding them to your favorite SML
108 implementation.</p></div>
109 <div class="paragraph"><p>It is simple to define <span class="monospaced">f</span> and <span class="monospaced">a</span>. We define <span class="monospaced">f</span> as a
110 folder whose base element is <span class="monospaced">()</span> and whose finish function does
111 nothing. We define <span class="monospaced">a</span> as the step function that does nothing.
112 The only trickiness is that we must <a href="EtaExpansion">eta expand</a> the
113 definition of <span class="monospaced">f</span> and <span class="monospaced">a</span> to work around the ValueRestriction;
114 we frequently use eta expansion for this purpose without mention.</p></div>
115 <div class="listingblock">
116 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
117 <span class="k">fun</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
118 <span class="k">fun</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
119 <span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">base</span><span class="p">,</span><span class="w"> </span><span class="n">finish</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
120 <span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
121 </pre></div></div></div>
122 <div class="paragraph"><p>One can easily apply the fold equation to verify by hand that <span class="monospaced">f</span>
123 applied to any number of <span class="monospaced">a</span>'s evaluates to <span class="monospaced">()</span>.</p></div>
124 <div class="listingblock">
125 <div class="content"><div class="highlight"><pre><span class="n">f</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">a</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
126 <span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">step</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">step</span><span class="w"> </span><span class="n">base</span><span class="p">)))</span><span class="w"></span>
127 <span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">step</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">()))</span><span class="w"></span>
128 <span class="p">...</span><span class="w"></span>
129 <span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
130 <span class="p">=</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
131 </pre></div></div></div>
135 <h2 id="_example_variable_argument_sum">Example: variable-argument sum</h2>
136 <div class="sectionbody">
137 <div class="paragraph"><p>Let’s look at an example that computes something: a variable-argument
138 function <span class="monospaced">sum</span> and a stepper <span class="monospaced">a</span> such that</p></div>
139 <div class="listingblock">
140 <div class="content"><div class="highlight"><pre><span class="n">sum</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">i1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">i2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">im</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i1</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">i2</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">im</span><span class="w"></span>
141 </pre></div></div></div>
142 <div class="paragraph"><p>The idea is simple — the folder starts with a base accumulator of
143 <span class="monospaced">0</span> and the stepper adds each element to the accumulator, <span class="monospaced">s</span>,
144 which the folder simply returns at the end.</p></div>
145 <div class="listingblock">
146 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">sum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
147 <span class="k">fun</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">s</span><span class="p">)</span><span class="w"></span>
148 </pre></div></div></div>
149 <div class="paragraph"><p>Using the fold equation, one can verify the following.</p></div>
150 <div class="listingblock">
151 <div class="content"><div class="highlight"><pre><span class="n">sum</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">6</span><span class="w"></span>
152 </pre></div></div></div>
156 <h2 id="_step1">Step1</h2>
157 <div class="sectionbody">
158 <div class="paragraph"><p>It is sometimes syntactically convenient to omit the parentheses
159 around the steps in a fold. This is easily done by defining a new
160 function, <span class="monospaced">step1</span>, as follows.</p></div>
161 <div class="listingblock">
162 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Fold</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
163 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
164 <span class="w"> </span><span class="k">open</span><span class="w"> </span><span class="n">Fold</span><span class="w"></span>
165 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">b</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">f</span><span class="p">)</span><span class="w"></span>
166 <span class="w"> </span><span class="k">end</span><span class="w"></span>
167 </pre></div></div></div>
168 <div class="paragraph"><p>From the definition of <span class="monospaced">step1</span>, we have the following
169 equivalence.</p></div>
170 <div class="listingblock">
171 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step1</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="n">b</span><span class="w"></span>
172 <span class="p">=</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">b</span><span class="w"></span>
173 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">b</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">f</span><span class="p">)</span><span class="w"></span>
174 </pre></div></div></div>
175 <div class="paragraph"><p>Using the above equivalence, we can compute the following equation for
176 <span class="monospaced">step1</span>.</p></div>
177 <div class="listingblock">
178 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step1</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="n">b1</span><span class="w"> </span><span class="p">(</span><span class="n">step1</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="n">b2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step1</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">bn</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
179 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step1</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="n">b2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step1</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">bn</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
180 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">b2</span><span class="p">,</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">b1</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">f</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step1</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">bn</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
181 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(</span><span class="n">bn</span><span class="p">,</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">b2</span><span class="p">,</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)))),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
182 <span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(</span><span class="n">bn</span><span class="p">,</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(</span><span class="n">b2</span><span class="p">,</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)))))</span><span class="w"></span>
183 </pre></div></div></div>
184 <div class="paragraph"><p>Here is an example using <span class="monospaced">step1</span> to define a variable-argument
185 product function, <span class="monospaced">prod</span>, with a convenient syntax.</p></div>
186 <div class="listingblock">
187 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">prod</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
188 <span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
189 </pre></div></div></div>
190 <div class="paragraph"><p>The functions <span class="monospaced">prod</span> and <span class="monospaced">`</span> satisfy the following equation.</p></div>
191 <div class="listingblock">
192 <div class="content"><div class="highlight"><pre><span class="n">prod</span><span class="w"> </span><span class="n">`i1</span><span class="w"> </span><span class="n">`i2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">`im</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">i2</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">im</span><span class="w"></span>
193 </pre></div></div></div>
194 <div class="paragraph"><p>Note that in SML, <span class="monospaced">`i1</span> is two different tokens, <span class="monospaced">`</span> and
195 <span class="monospaced">i1</span>. We often use <span class="monospaced">`</span> for an instance of a <span class="monospaced">step1</span> function
196 because of its syntactic unobtrusiveness and because no space is
197 required to separate it from an alphanumeric token.</p></div>
198 <div class="paragraph"><p>Also note that there are no parenthesis around the steps. That is,
199 the following expression is not the same as the above one (in fact, it
200 is not type correct).</p></div>
201 <div class="listingblock">
202 <div class="content"><div class="highlight"><pre><span class="n">prod</span><span class="w"> </span><span class="p">(</span><span class="n">`i1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">`i2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">`im</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
203 </pre></div></div></div>
207 <h2 id="_example_list_literals">Example: list literals</h2>
208 <div class="sectionbody">
209 <div class="paragraph"><p>SML already has a syntax for list literals, e.g. <span class="monospaced">[w, x, y, z]</span>.
210 However, using fold, we can define our own syntax.</p></div>
211 <div class="listingblock">
212 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">([],</span><span class="w"> </span><span class="n">rev</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
213 <span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">::</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
214 </pre></div></div></div>
215 <div class="paragraph"><p>The idea is that the folder starts out with the empty list, the steps
216 accumulate the elements into a list, and then the finishing function
217 reverses the list at the end.</p></div>
218 <div class="paragraph"><p>With these definitions one can write a list like:</p></div>
219 <div class="listingblock">
220 <div class="content"><div class="highlight"><pre><span class="n">list</span><span class="w"> </span><span class="n">`w</span><span class="w"> </span><span class="n">`x</span><span class="w"> </span><span class="n">`y</span><span class="w"> </span><span class="n">`z</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
221 </pre></div></div></div>
222 <div class="paragraph"><p>While the example is not practically useful, it does demonstrate the
223 need for the finishing function to be incorporated in <span class="monospaced">fold</span>.
224 Without a finishing function, every use of <span class="monospaced">list</span> would need to be
225 wrapped in <span class="monospaced">rev</span>, as follows.</p></div>
226 <div class="listingblock">
227 <div class="content"><div class="highlight"><pre><span class="n">rev</span><span class="w"> </span><span class="p">(</span><span class="n">list</span><span class="w"> </span><span class="n">`w</span><span class="w"> </span><span class="n">`x</span><span class="w"> </span><span class="n">`y</span><span class="w"> </span><span class="n">`z</span><span class="w"> </span><span class="n">$</span><span class="p">)</span><span class="w"></span>
228 </pre></div></div></div>
229 <div class="paragraph"><p>The finishing function allows us to incorporate the reversal into the
230 definition of <span class="monospaced">list</span>, and to treat <span class="monospaced">list</span> as a truly variable
231 argument function, performing an arbitrary computation after receiving
232 all of its arguments.</p></div>
233 <div class="paragraph"><p>See <a href="ArrayLiteral">ArrayLiteral</a> for a similar use of <span class="monospaced">fold</span> that provides a
234 syntax for array and vector literals, which are not built in to SML.</p></div>
238 <h2 id="_fold_right">Fold right</h2>
239 <div class="sectionbody">
240 <div class="paragraph"><p>Just as <span class="monospaced">fold</span> is analogous to a fold left, in which the functions
241 are applied to the accumulator left-to-right, we can define a variant
242 of <span class="monospaced">fold</span> that is analogous to a fold right, in which the
243 functions are applied to the accumulator right-to-left. That is, we
244 can define functions <span class="monospaced">foldr</span> and <span class="monospaced">step0</span> such that the
245 following equation holds.</p></div>
246 <div class="listingblock">
247 <div class="content"><div class="highlight"><pre><span class="n">foldr</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step0</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
248 <span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="n">a</span><span class="p">))))</span><span class="w"></span>
249 </pre></div></div></div>
250 <div class="paragraph"><p>The implementation of fold right is easy, using fold. The idea is for
251 the fold to start with <span class="monospaced">f</span> and for each step to precompose the
252 next <span class="monospaced">hi</span>. Then, the finisher applies the composed function to
253 the base value, <span class="monospaced">a</span>. Here is the code.</p></div>
254 <div class="listingblock">
255 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Foldr</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
256 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
257 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">foldr</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
258 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"></span>
259 <span class="w"> </span><span class="k">end</span><span class="w"></span>
260 </pre></div></div></div>
261 <div class="paragraph"><p>Verifying the fold-right equation is straightforward, using the
262 fold-left equation.</p></div>
263 <div class="listingblock">
264 <div class="content"><div class="highlight"><pre><span class="n">foldr</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">Foldr</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">Foldr</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="p">(</span><span class="n">Foldr</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
265 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
266 <span class="w"> </span><span class="p">(</span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="p">))</span><span class="w"></span>
267 <span class="w"> </span><span class="p">(</span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="p">))</span><span class="w"></span>
268 <span class="w"> </span><span class="p">...</span><span class="w"></span>
269 <span class="w"> </span><span class="p">(</span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">))</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
270 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
271 <span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="p">)</span><span class="w"> </span><span class="n">f</span><span class="p">))))</span><span class="w"></span>
272 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
273 <span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="p">))))</span><span class="w"></span>
274 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">((</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="p">)))</span><span class="w"></span>
275 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"></span>
276 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h1</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h2</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">hn</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="w"></span>
277 <span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="p">(</span><span class="n">h2</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="n">a</span><span class="p">))))</span><span class="w"></span>
278 </pre></div></div></div>
279 <div class="paragraph"><p>One can also define the fold-right analogue of <span class="monospaced">step1</span>.</p></div>
280 <div class="listingblock">
281 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Foldr</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
282 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
283 <span class="w"> </span><span class="k">open</span><span class="w"> </span><span class="n">Foldr</span><span class="w"></span>
284 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="p">(</span><span class="k">fn</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">h</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)))</span><span class="w"></span>
285 <span class="w"> </span><span class="k">end</span><span class="w"></span>
286 </pre></div></div></div>
290 <h2 id="_example_list_literals_via_fold_right">Example: list literals via fold right</h2>
291 <div class="sectionbody">
292 <div class="paragraph"><p>Revisiting the list literal example from earlier, we can use fold
293 right to define a syntax for list literals that doesn’t do a reversal.</p></div>
294 <div class="listingblock">
295 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Foldr</span><span class="p">.</span><span class="n">foldr</span><span class="w"> </span><span class="p">([],</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
296 <span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Foldr</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="w"> </span><span class="n">::</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
297 </pre></div></div></div>
298 <div class="paragraph"><p>As before, with these definitions, one can write a list like:</p></div>
299 <div class="listingblock">
300 <div class="content"><div class="highlight"><pre><span class="n">list</span><span class="w"> </span><span class="n">`w</span><span class="w"> </span><span class="n">`x</span><span class="w"> </span><span class="n">`y</span><span class="w"> </span><span class="n">`z</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
301 </pre></div></div></div>
302 <div class="paragraph"><p>The difference between the fold-left and fold-right approaches is that
303 the fold-right approach does not have to reverse the list at the end,
304 since it accumulates the elements in the correct order. In practice,
305 MLton will simplify away all of the intermediate function composition,
306 so the the fold-right approach will be more efficient.</p></div>
310 <h2 id="_mixing_steppers">Mixing steppers</h2>
311 <div class="sectionbody">
312 <div class="paragraph"><p>All of the examples so far have used the same step function throughout
313 a fold. This need not be the case. For example, consider the
315 <div class="listingblock">
316 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
317 <span class="k">val</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
318 <span class="k">val</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
319 </pre></div></div></div>
320 <div class="paragraph"><p>Here we have one folder, <span class="monospaced">n</span>, that can be used with two different
321 steppers, <span class="monospaced">I</span> and <span class="monospaced">O</span>. By using the fold equation, one can
322 verify the following equations.</p></div>
323 <div class="listingblock">
324 <div class="content"><div class="highlight"><pre><span class="n">n</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"></span>
325 <span class="n">n</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
326 <span class="n">n</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
327 <span class="n">n</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">5</span><span class="w"></span>
328 <span class="n">n</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">14</span><span class="w"></span>
329 </pre></div></div></div>
330 <div class="paragraph"><p>That is, we’ve defined a syntax for writing binary integer constants.</p></div>
331 <div class="paragraph"><p>Not only can one use different instances of <span class="monospaced">step0</span> in the same
332 fold, one can also intermix uses of <span class="monospaced">step0</span> and <span class="monospaced">step1</span>. For
333 example, consider the following.</p></div>
334 <div class="listingblock">
335 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
336 <span class="k">val</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="mi">8</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
337 <span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="mi">8</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
338 </pre></div></div></div>
339 <div class="paragraph"><p>Using the straightforward generalization of the fold equation to mixed
340 steppers, one can verify the following equations.</p></div>
341 <div class="listingblock">
342 <div class="content"><div class="highlight"><pre><span class="n">n</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"></span>
343 <span class="n">n</span><span class="w"> </span><span class="n">`</span><span class="mi">3</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">24</span><span class="w"></span>
344 <span class="n">n</span><span class="w"> </span><span class="n">`</span><span class="mi">1</span><span class="w"> </span><span class="n">O</span><span class="w"> </span><span class="n">`</span><span class="mi">7</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">71</span><span class="w"></span>
345 </pre></div></div></div>
346 <div class="paragraph"><p>That is, we’ve defined a syntax for writing octal integer constants,
347 with a special syntax, <span class="monospaced">O</span>, for the zero digit (admittedly
348 contrived, since one could just write <span class="monospaced">`0</span> instead of <span class="monospaced">O</span>).</p></div>
349 <div class="paragraph"><p>See <a href="NumericLiteral">NumericLiteral</a> for a practical extension of this approach that
350 supports numeric constants in any base and of any type.</p></div>
354 <h2 id="_seemingly_dependent_types">(Seemingly) dependent types</h2>
355 <div class="sectionbody">
356 <div class="paragraph"><p>A normal list fold always returns the same type no matter what
357 elements are in the list or how long the list is. Variable-argument
358 fold is more powerful, because the result type can vary based both on
359 the arguments that are passed and on their number. This can provide
360 the illusion of dependent types.</p></div>
361 <div class="paragraph"><p>For example, consider the following.</p></div>
362 <div class="listingblock">
363 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">((),</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
364 <span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="s">"hello"</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
365 <span class="k">val</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="mi">13</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
366 <span class="k">val</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">))</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
367 </pre></div></div></div>
368 <div class="paragraph"><p>Using the fold equation, one can verify the following equations.</p></div>
369 <div class="listingblock">
370 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"hello"</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"></span>
371 <span class="n">f</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">13</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"></span>
372 <span class="n">f</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">):</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">int</span><span class="w"></span>
373 </pre></div></div></div>
374 <div class="paragraph"><p>That is, <span class="monospaced">f</span> returns a value of a different type depending on
375 whether it is applied to argument <span class="monospaced">a</span>, argument <span class="monospaced">b</span>, or
376 argument <span class="monospaced">c</span>.</p></div>
377 <div class="paragraph"><p>The following example shows how the type of a fold can depend on the
378 number of arguments.</p></div>
379 <div class="listingblock">
380 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">grow</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">([],</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
381 <span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=></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">z</span><span class="w"></span>
382 </pre></div></div></div>
383 <div class="paragraph"><p>Using the fold equation, one can verify the following equations.</p></div>
384 <div class="listingblock">
385 <div class="content"><div class="highlight"><pre><span class="n">grow</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</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">list</span><span class="w"></span>
386 <span class="n">grow</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[[]]:</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="n">list</span><span class="w"></span>
387 <span class="n">grow</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</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">list</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="n">list</span><span class="w"></span>
388 </pre></div></div></div>
389 <div class="paragraph"><p>Clearly, the result type of a call to the variable argument <span class="monospaced">grow</span>
390 function depends on the number of arguments that are passed.</p></div>
391 <div class="paragraph"><p>As a reminder, this is well-typed SML. You can check it out in any
392 implementation.</p></div>
396 <h2 id="_seemingly_dependently_typed_functional_results">(Seemingly) dependently-typed functional results</h2>
397 <div class="sectionbody">
398 <div class="paragraph"><p>Fold is especially useful when it returns a curried function whose
399 arity depends on the number of arguments. For example, consider the
401 <div class="listingblock">
402 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">makeSum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">id</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">f</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">0</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
403 <span class="k">val</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</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">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
404 </pre></div></div></div>
405 <div class="paragraph"><p>The <span class="monospaced">makeSum</span> folder constructs a function whose arity depends on
406 the number of <span class="monospaced">I</span> arguments and that adds together all of its
407 arguments. For example,
408 <span class="monospaced">makeSum I $</span> is of type <span class="monospaced">int -> int</span> and
409 <span class="monospaced">makeSum I I $</span> is of type <span class="monospaced">int -> int -> int</span>.</p></div>
410 <div class="paragraph"><p>One can use the fold equation to verify that the <span class="monospaced">makeSum</span> works
411 correctly. For example, one can easily check by hand the following
413 <div class="listingblock">
414 <div class="content"><div class="highlight"><pre><span class="n">makeSum</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
415 <span class="n">makeSum</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">3</span><span class="w"></span>
416 <span class="n">makeSum</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">6</span><span class="w"></span>
417 </pre></div></div></div>
418 <div class="paragraph"><p>Returning a function becomes especially interesting when there are
419 steppers of different types. For example, the following <span class="monospaced">makeSum</span>
420 folder constructs functions that sum integers and reals.</p></div>
421 <div class="listingblock">
422 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">makeSum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Foldr</span><span class="p">.</span><span class="n">foldr</span><span class="w"> </span><span class="p">(</span><span class="n">id</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mf">0.0</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
423 <span class="k">val</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Foldr</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=></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="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">i</span><span class="p">))</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
424 <span class="k">val</span><span class="w"> </span><span class="n">R</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Foldr</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">x</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="k">fn</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=></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="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">r</span><span class="p">))</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
425 </pre></div></div></div>
426 <div class="paragraph"><p>With these definitions, <span class="monospaced">makeSum I R $</span> is of type
427 <span class="monospaced">int -> real -> real</span> and <span class="monospaced">makeSum R I I $</span> is of type
428 <span class="monospaced">real -> int -> int -> real</span>. One can use the foldr equation to
429 check the following equations.</p></div>
430 <div class="listingblock">
431 <div class="content"><div class="highlight"><pre><span class="n">makeSum</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">1.0</span><span class="w"></span>
432 <span class="n">makeSum</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">R</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="mf">2.5</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">3.5</span><span class="w"></span>
433 <span class="n">makeSum</span><span class="w"> </span><span class="n">R</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mf">1.5</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">6.5</span><span class="w"></span>
434 </pre></div></div></div>
435 <div class="paragraph"><p>We used <span class="monospaced">foldr</span> instead of <span class="monospaced">fold</span> for this so that the order
436 in which the specifiers <span class="monospaced">I</span> and <span class="monospaced">R</span> appear is the same as the
437 order in which the arguments appear. Had we used <span class="monospaced">fold</span>, things
438 would have been reversed.</p></div>
439 <div class="paragraph"><p>An extension of this idea is sufficient to define <a href="Printf">Printf</a>-like
440 functions in SML.</p></div>
444 <h2 id="_an_idiom_for_combining_steps">An idiom for combining steps</h2>
445 <div class="sectionbody">
446 <div class="paragraph"><p>It is sometimes useful to combine a number of steps together and name
447 them as a single step. As a simple example, suppose that one often
448 sees an integer follower by a real in the <span class="monospaced">makeSum</span> example above.
449 One can define a new <em>compound step</em> <span class="monospaced">IR</span> as follows.</p></div>
450 <div class="listingblock">
451 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">IR</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">I</span><span class="w"> </span><span class="n">R</span><span class="w"></span>
452 </pre></div></div></div>
453 <div class="paragraph"><p>With this definition in place, one can verify the following.</p></div>
454 <div class="listingblock">
455 <div class="content"><div class="highlight"><pre><span class="n">makeSum</span><span class="w"> </span><span class="n">IR</span><span class="w"> </span><span class="n">IR</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="mf">2.2</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="mf">4.4</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">10.6</span><span class="w"></span>
456 </pre></div></div></div>
457 <div class="paragraph"><p>In general, one can combine steps <span class="monospaced">s1</span>, <span class="monospaced">s2</span>, … <span class="monospaced">sn</span> as</p></div>
458 <div class="listingblock">
459 <div class="content"><div class="highlight"><pre><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"></span>
460 </pre></div></div></div>
461 <div class="paragraph"><p>The following calculation shows why a compound step behaves as the
462 composition of its constituent steps.</p></div>
463 <div class="listingblock">
464 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="p">)</span><span class="w"></span>
465 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="p">)</span><span class="w"> </span><span class="n">u</span><span class="w"></span>
466 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"></span>
467 </pre></div></div></div>
471 <h2 id="_post_composition">Post composition</h2>
472 <div class="sectionbody">
473 <div class="paragraph"><p>Suppose we already have a function defined via fold,
474 <span class="monospaced">w = fold (a, f)</span>, and we would like to construct a new fold
475 function that is like <span class="monospaced">w</span>, but applies <span class="monospaced">g</span> to the result
476 produced by <span class="monospaced">w</span>. This is similar to function composition, but we
477 can’t just do <span class="monospaced">g o w</span>, because we don’t want to use <span class="monospaced">g</span> until
478 <span class="monospaced">w</span> has been applied to all of its arguments and received the
479 end-of-arguments terminator <span class="monospaced">$</span>.</p></div>
480 <div class="paragraph"><p>More precisely, we want to define a post-composition function
481 <span class="monospaced">post</span> that satisfies the following equation.</p></div>
482 <div class="listingblock">
483 <div class="content"><div class="highlight"><pre><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="p">)</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="p">)</span><span class="w"></span>
484 </pre></div></div></div>
485 <div class="paragraph"><p>Here is the definition of <span class="monospaced">post</span>.</p></div>
486 <div class="listingblock">
487 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Fold</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
488 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
489 <span class="w"> </span><span class="k">open</span><span class="w"> </span><span class="n">Fold</span><span class="w"></span>
490 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
491 <span class="w"> </span><span class="k">end</span><span class="w"></span>
492 </pre></div></div></div>
493 <div class="paragraph"><p>The following calculations show that <span class="monospaced">post</span> satisfies the desired
494 equation, where <span class="monospaced">w = fold (a, f)</span>.</p></div>
495 <div class="listingblock">
496 <div class="content"><div class="highlight"><pre><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
497 <span class="p">=</span><span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
498 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
499 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
500 <span class="p">=</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
501 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
502 </pre></div></div></div>
503 <div class="paragraph"><p>Now, suppose <span class="monospaced">si = step0 hi</span> for <span class="monospaced">i</span> from <span class="monospaced">1</span> to <span class="monospaced">n</span>.</p></div>
504 <div class="listingblock">
505 <div class="content"><div class="highlight"><pre><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="p">)</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
506 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
507 <span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">g</span><span class="w"> </span><span class="n">o</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">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">)))</span><span class="w"></span>
508 <span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">hn</span><span class="w"> </span><span class="p">(...</span><span class="w"> </span><span class="p">(</span><span class="n">h1</span><span class="w"> </span><span class="n">a</span><span class="p">))))</span><span class="w"></span>
509 <span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">s1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="p">)</span><span class="w"></span>
510 <span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">sn</span><span class="w"> </span><span class="n">$</span><span class="p">)</span><span class="w"></span>
511 </pre></div></div></div>
512 <div class="paragraph"><p>For a practical example of post composition, see <a href="ArrayLiteral">ArrayLiteral</a>.</p></div>
516 <h2 id="_lift">Lift</h2>
517 <div class="sectionbody">
518 <div class="paragraph"><p>We now define a peculiar-looking function, <span class="monospaced">lift0</span>, that is,
519 equationally speaking, equivalent to the identity function on a step
521 <div class="listingblock">
522 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">lift0</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
523 </pre></div></div></div>
524 <div class="paragraph"><p>Using the definitions, we can prove the following equation.</p></div>
525 <div class="listingblock">
526 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">lift0</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step0</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"></span>
527 </pre></div></div></div>
528 <div class="paragraph"><p>Here is the proof.</p></div>
529 <div class="listingblock">
530 <div class="content"><div class="highlight"><pre><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">lift0</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
531 <span class="p">=</span><span class="w"> </span><span class="n">lift0</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
532 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
533 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
534 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
535 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
536 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">id</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
537 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
538 <span class="p">=</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
539 <span class="p">=</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</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">step0</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"></span>
540 </pre></div></div></div>
541 <div class="paragraph"><p>If <span class="monospaced">lift0</span> is the identity, then why even define it? The answer
542 lies in the typing of fold expressions, which we have, until now, left
543 unexplained.</p></div>
547 <h2 id="_typing">Typing</h2>
548 <div class="sectionbody">
549 <div class="paragraph"><p>Perhaps the most surprising aspect of fold is that it can be checked
550 by the SML type system. The types involved in fold expressions are
551 complex; fortunately type inference is able to deduce them.
552 Nevertheless, it is instructive to study the types of fold functions
553 and steppers. More importantly, it is essential to understand the
554 typing aspects of fold in order to write down signatures of functions
555 defined using fold and step.</p></div>
556 <div class="paragraph"><p>Here is the <span class="monospaced">FOLD</span> signature, and a recapitulation of the entire
557 <span class="monospaced">Fold</span> structure, with additional type annotations.</p></div>
558 <div class="listingblock">
559 <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">FOLD</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
560 <span class="w"> </span><span class="k">sig</span><span class="w"></span>
561 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</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="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
562 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
563 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
564 <span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
565 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="p">,</span><span class="w"> </span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
566 <span class="w"> </span><span class="p">(</span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'a11</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
568 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">fold</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="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
569 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">lift0</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
570 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
571 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">post</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c1</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'c1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c2</span><span class="p">)</span><span class="w"></span>
572 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c2</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
573 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">step0</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
574 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">step1</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a12</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"></span>
575 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="p">,</span><span class="w"> </span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"></span>
576 <span class="w"> </span><span class="k">end</span><span class="w"></span>
578 <span class="k">structure</span><span class="w"> </span><span class="n">Fold</span><span class="p">:></span><span class="w"> </span><span class="n">FOLD</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
579 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
580 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</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="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
582 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
584 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
585 <span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
587 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="p">,</span><span class="w"> </span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
588 <span class="w"> </span><span class="p">(</span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'a11</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
590 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">f</span><span class="p">:</span><span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"></span>
591 <span class="w"> </span><span class="p">(</span><span class="n">g</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="p">):</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
592 <span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
594 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"></span>
595 <span class="w"> </span><span class="p">(</span><span class="n">a1</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
596 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a1</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
598 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="p">:</span><span class="w"> </span><span class="n">'a11</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a12</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"></span>
599 <span class="w"> </span><span class="p">(</span><span class="n">a12</span><span class="p">:</span><span class="w"> </span><span class="n">'a12</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"></span>
600 <span class="w"> </span><span class="p">(</span><span class="n">a11</span><span class="p">:</span><span class="w"> </span><span class="n">'a11</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
601 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a11</span><span class="p">,</span><span class="w"> </span><span class="n">a12</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
603 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">lift0</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="p">)</span><span class="w"></span>
604 <span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
605 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
607 <span class="w"> </span><span class="k">fun</span><span class="w"> </span><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c1</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"></span>
608 <span class="w"> </span><span class="n">g</span><span class="p">:</span><span class="w"> </span><span class="n">'c1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c2</span><span class="p">)</span><span class="w"></span>
609 <span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c2</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="p">):</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
610 <span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
611 <span class="w"> </span><span class="k">end</span><span class="w"></span>
612 </pre></div></div></div>
613 <div class="paragraph"><p>That’s a lot to swallow, so let’s walk through it one step at a time.
614 First, we have the definition of type <span class="monospaced">Fold.step</span>.</p></div>
615 <div class="listingblock">
616 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</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="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
617 </pre></div></div></div>
618 <div class="paragraph"><p>As a fold proceeds over its arguments, it maintains two things: the
619 accumulator, of type <span class="monospaced">'a</span>, and the finishing function, of type
620 <span class="monospaced">'b -> 'c</span>. Each step in the fold is a function that takes those
621 two pieces (i.e. <span class="monospaced">'a * ('b -> 'c)</span> and does something to them
622 (i.e. produces <span class="monospaced">'d</span>). The result type of the step is completely
623 left open to be filled in by type inference, as it is an arrow type
624 that is capable of consuming the rest of the arguments to the fold.</p></div>
625 <div class="paragraph"><p>A folder, of type <span class="monospaced">Fold.t</span>, is a function that consumes a single
627 <div class="listingblock">
628 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
629 </pre></div></div></div>
630 <div class="paragraph"><p>Expanding out the type, we have:</p></div>
631 <div class="listingblock">
632 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
633 </pre></div></div></div>
634 <div class="paragraph"><p>This shows that the only thing a folder does is to hand its
635 accumulator (<span class="monospaced">'a</span>) and finisher (<span class="monospaced">'b -> 'c</span>) to the next step
636 (<span class="monospaced">'a * ('b -> 'c) -> 'd</span>). If SML had <a href="FirstClassPolymorphism">first-class polymorphism</a>,
637 we would write the fold type as follows.</p></div>
638 <div class="listingblock">
639 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">)</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">Forall</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="err">. ('a, 'b, 'c, 'd) step -> 'd</span>
640 </pre></div></div></div>
641 <div class="paragraph"><p>This type definition shows that a folder had nothing to do with
642 the rest of the fold, it only deals with the next step.</p></div>
643 <div class="paragraph"><p>We now can understand the type of <span class="monospaced">fold</span>, which takes the initial
644 value of the accumulator and the finishing function, and constructs a
645 folder, i.e. a function awaiting the next step.</p></div>
646 <div class="listingblock">
647 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">fold</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="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
648 <span class="k">fun</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</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">f</span><span class="p">:</span><span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"></span>
649 <span class="w"> </span><span class="p">(</span><span class="n">g</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="p">):</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
650 <span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
651 </pre></div></div></div>
652 <div class="paragraph"><p>Continuing on, we have the type of step functions.</p></div>
653 <div class="listingblock">
654 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
655 <span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
656 </pre></div></div></div>
657 <div class="paragraph"><p>Expanding out the type a bit gives:</p></div>
658 <div class="listingblock">
659 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
660 <span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
661 </pre></div></div></div>
662 <div class="paragraph"><p>So, a step function takes the accumulator (<span class="monospaced">'a1</span>) and finishing
663 function (<span class="monospaced">'b -> 'c</span>), which will be passed to it by the previous
664 folder, and transforms them to a new folder. This new folder has a
665 new accumulator (<span class="monospaced">'a2</span>) and the same finishing function.</p></div>
666 <div class="paragraph"><p>Again, imagining that SML had <a href="FirstClassPolymorphism">first-class polymorphism</a> makes the type
668 <div class="listingblock">
669 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
670 <span class="w"> </span><span class="n">Forall</span><span class="w"> </span><span class="p">(</span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="err">. ('a1, 'b, 'c, ('a2, 'b, 'c) t) step</span>
671 </pre></div></div></div>
672 <div class="paragraph"><p>Thus, in essence, a <span class="monospaced">step0</span> function is a wrapper around a
673 function of type <span class="monospaced">'a1 -> 'a2</span>, which is exactly what the
674 definition of <span class="monospaced">step0</span> does.</p></div>
675 <div class="listingblock">
676 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">step0</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
677 <span class="k">fun</span><span class="w"> </span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"></span>
678 <span class="w"> </span><span class="p">(</span><span class="n">a1</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
679 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="n">a1</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
680 </pre></div></div></div>
681 <div class="paragraph"><p>It is not much beyond <span class="monospaced">step0</span> to understand <span class="monospaced">step1</span>.</p></div>
682 <div class="listingblock">
683 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="p">,</span><span class="w"> </span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
684 <span class="w"> </span><span class="p">(</span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'a11</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
685 </pre></div></div></div>
686 <div class="paragraph"><p>A <span class="monospaced">step1</span> function takes the accumulator (<span class="monospaced">'a12</span>) and finisher
687 (<span class="monospaced">'b -> 'c</span>) passed to it by the previous folder and transforms
688 them into a function that consumes the next argument (<span class="monospaced">'a11</span>) and
689 produces a folder that will continue the fold with a new accumulator
690 (<span class="monospaced">'a2</span>) and the same finisher.</p></div>
691 <div class="listingblock">
692 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="p">:</span><span class="w"> </span><span class="n">'a11</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a12</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"></span>
693 <span class="w"> </span><span class="p">(</span><span class="n">a12</span><span class="p">:</span><span class="w"> </span><span class="n">'a12</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"></span>
694 <span class="w"> </span><span class="p">(</span><span class="n">a11</span><span class="p">:</span><span class="w"> </span><span class="n">'a11</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
695 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">h</span><span class="w"> </span><span class="p">(</span><span class="n">a11</span><span class="p">,</span><span class="w"> </span><span class="n">a12</span><span class="p">),</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
696 </pre></div></div></div>
697 <div class="paragraph"><p>With <a href="FirstClassPolymorphism">first-class polymorphism</a>, a <span class="monospaced">step1</span> function is more clearly
698 seen as a wrapper around a binary function of type
699 <span class="monospaced">'a11 * 'a12 -> 'a2</span>.</p></div>
700 <div class="listingblock">
701 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'a11</span><span class="p">,</span><span class="w"> </span><span class="n">'a12</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
702 <span class="w"> </span><span class="n">Forall</span><span class="w"> </span><span class="p">(</span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">)</span><span class="w"> </span><span class="err">. ('a12, 'b, 'c, 'a11 -> ('a2, 'b, 'c) t) step</span>
703 </pre></div></div></div>
704 <div class="paragraph"><p>The type of <span class="monospaced">post</span> is clear: it takes a folder with a finishing
705 function that produces type <span class="monospaced">'c1</span>, and a function of type
706 <span class="monospaced">'c1 -> 'c2</span> to postcompose onto the folder. It returns a new
707 folder with a finishing function that produces type <span class="monospaced">'c2</span>.</p></div>
708 <div class="listingblock">
709 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">post</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c1</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'c1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c2</span><span class="p">)</span><span class="w"></span>
710 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c2</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
711 <span class="k">fun</span><span class="w"> </span><span class="n">post</span><span class="w"> </span><span class="p">(</span><span class="n">w</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c1</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"></span>
712 <span class="w"> </span><span class="n">g</span><span class="p">:</span><span class="w"> </span><span class="n">'c1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c2</span><span class="p">)</span><span class="w"></span>
713 <span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c2</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step</span><span class="p">):</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
714 <span class="w"> </span><span class="n">w</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">h</span><span class="p">))</span><span class="w"></span>
715 </pre></div></div></div>
716 <div class="paragraph"><p>We will return to <span class="monospaced">lift0</span> after an example.</p></div>
720 <h2 id="_an_example_typing">An example typing</h2>
721 <div class="sectionbody">
722 <div class="paragraph"><p>Let’s type check our simplest example, a variable-argument fold.
723 Recall that we have a folder <span class="monospaced">f</span> and a stepper <span class="monospaced">a</span> defined as
725 <div class="listingblock">
726 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">((),</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
727 <span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
728 </pre></div></div></div>
729 <div class="paragraph"><p>Since the accumulator and finisher are uninteresting, we’ll use some
730 abbreviations to simplify things.</p></div>
731 <div class="listingblock">
732 <div class="content"><div class="highlight"><pre><span class="k">type</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="p">,</span><span class="w"> </span><span class="n">unit</span><span class="p">,</span><span class="w"> </span><span class="n">unit</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step</span><span class="w"></span>
733 <span class="k">type</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">step</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'d</span><span class="w"></span>
734 </pre></div></div></div>
735 <div class="paragraph"><p>With these abbreviations, <span class="monospaced">f</span> and <span class="monospaced">a</span> have the following polymorphic
737 <div class="listingblock">
738 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">fold</span><span class="w"></span>
739 <span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
740 </pre></div></div></div>
741 <div class="paragraph"><p>Suppose we want to type check</p></div>
742 <div class="listingblock">
743 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$:</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
744 </pre></div></div></div>
745 <div class="paragraph"><p>As a reminder, the fully parenthesized expression is</p></div>
746 <div class="listingblock">
747 <div class="content"><div class="highlight"><pre><span class="p">((((</span><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
748 </pre></div></div></div>
749 <div class="paragraph"><p>The observation that we will use repeatedly is that for any type
750 <span class="monospaced">z</span>, if <span class="monospaced">f: z fold</span> and <span class="monospaced">s: z step</span>, then <span class="monospaced">f s: z</span>.
751 So, if we want</p></div>
752 <div class="listingblock">
753 <div class="content"><div class="highlight"><pre><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="n">$:</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
754 </pre></div></div></div>
755 <div class="paragraph"><p>then we must have</p></div>
756 <div class="listingblock">
757 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"></span>
758 <span class="n">$:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
759 </pre></div></div></div>
760 <div class="paragraph"><p>Applying the observation again, we must have</p></div>
761 <div class="listingblock">
762 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"></span>
763 <span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
764 </pre></div></div></div>
765 <div class="paragraph"><p>Applying the observation two more times leads to the following type
766 derivation.</p></div>
767 <div class="listingblock">
768 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
769 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
770 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
771 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">$:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
772 <span class="n">f</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">$:</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
773 </pre></div></div></div>
774 <div class="paragraph"><p>So, each application is a fold that consumes the next step, producing
775 a fold of one smaller type.</p></div>
776 <div class="paragraph"><p>One can expand some of the type definitions in <span class="monospaced">f</span> to see that it is
777 indeed a function that takes four curried arguments, each one a step
779 <div class="listingblock">
780 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
781 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
782 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
783 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">step</span><span class="w"></span>
784 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
785 </pre></div></div></div>
786 <div class="paragraph"><p>This example shows why we must eta expand uses of <span class="monospaced">fold</span> and <span class="monospaced">step0</span>
787 to work around the value restriction and make folders and steppers
788 polymorphic. The type of a fold function like <span class="monospaced">f</span> depends on the
789 number of arguments, and so will vary from use to use. Similarly,
790 each occurrence of an argument like <span class="monospaced">a</span> has a different type,
791 depending on the number of remaining arguments.</p></div>
792 <div class="paragraph"><p>This example also shows that the type of a folder, when fully
793 expanded, is exponential in the number of arguments: there are as many
794 nested occurrences of the <span class="monospaced">fold</span> type constructor as there are
795 arguments, and each occurrence duplicates its type argument. One can
796 observe this exponential behavior in a type checker that doesn’t share
797 enough of the representation of types (e.g. one that represents types
798 as trees rather than directed acyclic graphs).</p></div>
799 <div class="paragraph"><p>Generalizing this type derivation to uses of fold where the
800 accumulator and finisher are more interesting is straightforward. One
801 simply includes the type of the accumulator, which may change, for
802 each step, and the type of the finisher, which doesn’t change from
803 step to step.</p></div>
807 <h2 id="_typing_lift">Typing lift</h2>
808 <div class="sectionbody">
809 <div class="paragraph"><p>The lack of <a href="FirstClassPolymorphism">first-class polymorphism</a> in SML
810 causes problems if one wants to use a step in a first-class way.
811 Consider the following <span class="monospaced">double</span> function, which takes a step, <span class="monospaced">s</span>, and
812 produces a composite step that does <span class="monospaced">s</span> twice.</p></div>
813 <div class="listingblock">
814 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">double</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
815 </pre></div></div></div>
816 <div class="paragraph"><p>The definition of <span class="monospaced">double</span> is not type correct. The problem is that
817 the type of a step depends on the number of remaining arguments but
818 that the parameter <span class="monospaced">s</span> is not polymorphic, and so can not be used in
819 two different positions.</p></div>
820 <div class="paragraph"><p>Fortunately, we can define a function, <span class="monospaced">lift0</span>, that takes a monotyped
821 step function and <em>lifts</em> it into a polymorphic step function. This
822 is apparent in the type of <span class="monospaced">lift0</span>.</p></div>
823 <div class="listingblock">
824 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">lift0</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
825 <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
826 <span class="k">fun</span><span class="w"> </span><span class="n">lift0</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="p">)</span><span class="w"></span>
827 <span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</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">'b</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'c</span><span class="p">):</span><span class="w"> </span><span class="p">(</span><span class="n">'a2</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
828 <span class="w"> </span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">fold</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">$</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
829 </pre></div></div></div>
830 <div class="paragraph"><p>The following definition of <span class="monospaced">double</span> uses <span class="monospaced">lift0</span>, appropriately eta
831 wrapped, to fix the problem.</p></div>
832 <div class="listingblock">
833 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">double</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
834 <span class="w"> </span><span class="k">let</span><span class="w"></span>
835 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">lift0</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
836 <span class="w"> </span><span class="k">in</span><span class="w"></span>
837 <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="n">u</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
838 <span class="w"> </span><span class="k">end</span><span class="w"></span>
839 </pre></div></div></div>
840 <div class="paragraph"><p>With that definition of <span class="monospaced">double</span> in place, we can use it as in the
841 following example.</p></div>
842 <div class="listingblock">
843 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">((),</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
844 <span class="k">val</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
845 <span class="k">val</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">double</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
846 <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="n">a</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">$</span><span class="w"></span>
847 </pre></div></div></div>
848 <div class="paragraph"><p>Of course, we must eta wrap the call <span class="monospaced">double</span> in order to use its
849 result, which is a step function, polymorphically.</p></div>
853 <h2 id="_hiding_the_type_of_the_accumulator">Hiding the type of the accumulator</h2>
854 <div class="sectionbody">
855 <div class="paragraph"><p>For clarity and to avoid mistakes, it can be useful to hide the type
856 of the accumulator in a fold. Reworking the simple variable-argument
857 example to do this leads to the following.</p></div>
858 <div class="listingblock">
859 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">S</span><span class="p">:></span><span class="w"></span>
860 <span class="w"> </span><span class="k">sig</span><span class="w"></span>
861 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">ac</span><span class="w"></span>
862 <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">ac</span><span class="p">,</span><span class="w"> </span><span class="n">ac</span><span class="p">,</span><span class="w"> </span><span class="n">unit</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
863 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">ac</span><span class="p">,</span><span class="w"> </span><span class="n">ac</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"> </span><span class="n">'c</span><span class="p">,</span><span class="w"> </span><span class="n">'d</span><span class="p">)</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"></span>
864 <span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
865 <span class="w"> </span><span class="k">struct</span><span class="w"></span>
866 <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">ac</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>
867 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">((),</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
868 <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">())</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
869 <span class="w"> </span><span class="k">end</span><span class="w"></span>
870 </pre></div></div></div>
871 <div class="paragraph"><p>The idea is to name the accumulator type and use opaque signature
872 matching to make it abstract. This can prevent improper manipulation
873 of the accumulator by client code and ensure invariants that the
874 folder and stepper would like to maintain.</p></div>
875 <div class="paragraph"><p>For a practical example of this technique, see <a href="ArrayLiteral">ArrayLiteral</a>.</p></div>
879 <h2 id="_also_see">Also see</h2>
880 <div class="sectionbody">
881 <div class="paragraph"><p>Fold has a number of practical applications. Here are some of them.</p></div>
882 <div class="ulist"><ul>
885 <a href="ArrayLiteral">ArrayLiteral</a>
890 <a href="Fold01N">Fold01N</a>
895 <a href="FunctionalRecordUpdate">FunctionalRecordUpdate</a>
900 <a href="NumericLiteral">NumericLiteral</a>
905 <a href="OptionalArguments">OptionalArguments</a>
910 <a href="Printf">Printf</a>
915 <a href="VariableArityPolymorphism">VariableArityPolymorphism</a>
919 <div class="paragraph"><p>There are a number of related techniques. Here are some of them.</p></div>
920 <div class="ulist"><ul>
923 <a href="StaticSum">StaticSum</a>
928 <a href="TypeIndexedValues">TypeIndexedValues</a>
935 <div id="footnotes"><hr></div>
937 <div id="footer-text">
939 <div id="footer-badges">
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