doc/guide/localhost/Fold01N

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  26 <h1>Fold01N</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p>A common use pattern of <a href="Fold">Fold</a> is to define a variable-arity
  32 function that combines multiple arguments together using a binary
  33 function.  It is slightly tricky to do this directly using fold,
  34 because of the special treatment required for the case of zero or one
  35 argument.  Here is a structure, <span class="monospaced">Fold01N</span>, that solves the problem
  36 once and for all, and eases the definition of such functions.</p></div>
  37 <div class="listingblock">
  38 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Fold01N</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  39 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  40 <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">finish</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">zero</span><span class="p">}</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  41 <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="n">finish</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">=&gt;</span><span class="w"> </span><span class="n">zero</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">),</span><span class="w"></span>
  42 <span class="w">                    </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">finish</span><span class="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="p">_)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="p">()))</span><span class="w"></span>
  43
  44 <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">combine</span><span class="p">,</span><span class="w"> </span><span class="n">input</span><span class="p">}</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  45 <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="n">finish</span><span class="p">,</span><span class="w"> </span><span class="p">_,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
  46 <span class="w">                     </span><span class="p">(</span><span class="n">finish</span><span class="p">,</span><span class="w"></span>
  47 <span class="w">                      </span><span class="n">finish</span><span class="p">,</span><span class="w"></span>
  48 <span class="w">                      </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">input</span><span class="p">,</span><span class="w"></span>
  49 <span class="w">                      </span><span class="k">fn</span><span class="w"> </span><span class="n">x&#39;</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">combine</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">input</span><span class="p">,</span><span class="w"> </span><span class="n">x&#39;</span><span class="p">)))</span><span class="w"></span>
  50
  51 <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">combine</span><span class="p">}</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="n">input</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  52 <span class="w">         </span><span class="n">step0</span><span class="w"> </span><span class="p">{</span><span class="n">combine</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">combine</span><span class="p">,</span><span class="w"> </span><span class="n">input</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">input</span><span class="p">}</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
  53 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  54 </pre></div></div></div>
  55 <div class="paragraph"><p>If one has a value <span class="monospaced">zero</span>, and functions <span class="monospaced">start</span>, <span class="monospaced">c</span>, and <span class="monospaced">finish</span>,
  56 then one can define a variable-arity function <span class="monospaced">f</span> and stepper
  57 <span class="monospaced">&grave;</span> as follows.</p></div>
  58 <div class="listingblock">
  59 <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">=&gt;</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">fold</span><span class="w"> </span><span class="p">{</span><span class="n">finish</span><span class="w"> </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">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">zero</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">zero</span><span class="p">}</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
  60 <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">=&gt;</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"> </span><span class="p">{</span><span class="n">combine</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="n">z</span><span class="w"></span>
  61 </pre></div></div></div>
  62 <div class="paragraph"><p>One can then use the fold equation to prove the following equations.</p></div>
  63 <div class="listingblock">
  64 <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">zero</span><span class="w"></span>
  65 <span class="n">f</span><span class="w"> </span><span class="n">`a1</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">start</span><span class="w"> </span><span class="n">a1</span><span class="p">)</span><span class="w"></span>
  66 <span class="n">f</span><span class="w"> </span><span class="n">`a1</span><span class="w"> </span><span class="n">`a2</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="p">(</span><span class="n">start</span><span class="w"> </span><span class="n">a1</span><span class="p">,</span><span class="w"> </span><span class="n">a2</span><span class="p">))</span><span class="w"></span>
  67 <span class="n">f</span><span class="w"> </span><span class="n">`a1</span><span class="w"> </span><span class="n">`a2</span><span class="w"> </span><span class="n">`a3</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="p">(</span><span class="n">start</span><span class="w"> </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">a3</span><span class="p">))</span><span class="w"></span>
  68 <span class="p">...</span><span class="w"></span>
  69 </pre></div></div></div>
  70 <div class="paragraph"><p>For an example of <span class="monospaced">Fold01N</span>, see <a href="VariableArityPolymorphism">VariableArityPolymorphism</a>.</p></div>
  71 </div>
  72 </div>
  73 <div class="sect1">
  74 <h2 id="_typing_fold01n">Typing Fold01N</h2>
  75 <div class="sectionbody">
  76 <div class="paragraph"><p>Here is the signature for <span class="monospaced">Fold01N</span>.  We use a trick to avoid having
  77 to duplicate the definition of some rather complex types in both the
  78 signature and the structure.  We first define the types in a
  79 structure.  Then, we define them via type re-definitions in the
  80 signature, and via <span class="monospaced">open</span> in the full structure.</p></div>
  81 <div class="listingblock">
  82 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Fold01N</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  83 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  84 <span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;answer</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;zero</span><span class="p">,</span><span class="w"></span>
  85 <span class="w">            </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</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>
  86 <span class="w">         </span><span class="p">((</span><span class="n">&#39;zero</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;zero</span><span class="p">)</span><span class="w"></span>
  87 <span class="w">          </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;accum2</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;answer</span><span class="p">)</span><span class="w"></span>
  88 <span class="w">          </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;zero</span><span class="p">)</span><span class="w"></span>
  89 <span class="w">          </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">),</span><span class="w"></span>
  90 <span class="w">          </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"></span>
  91 <span class="w">          </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"></span>
  92 <span class="w">          </span><span class="n">&#39;e</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>
  93
  94 <span class="w">       </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;input2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"></span>
  95 <span class="w">            </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</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>
  96 <span class="w">         </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input1</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">),</span><span class="w"></span>
  97 <span class="w">          </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input2</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">),</span><span class="w"></span>
  98 <span class="w">          </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</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>
  99
 100 <span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;input</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"></span>
 101 <span class="w">            </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</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>
 102 <span class="w">         </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"></span>
 103 <span class="w">          </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;d</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">),</span><span class="w"></span>
 104 <span class="w">          </span><span class="n">&#39;c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">),</span><span class="w"></span>
 105 <span class="w">          </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</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>
 106 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 107
 108 <span class="k">signature</span><span class="w"> </span><span class="n">FOLD_01N</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 109 <span class="w">   </span><span class="k">sig</span><span class="w"></span>
 110 <span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</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>
 111 <span class="w">         </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 112 <span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</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>
 113 <span class="w">         </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step0</span><span class="w"></span>
 114 <span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</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>
 115 <span class="w">         </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;h</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;i</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;j</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>
 116
 117 <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>
 118 <span class="w">         </span><span class="p">{</span><span class="n">finish</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;answer</span><span class="p">,</span><span class="w"></span>
 119 <span class="w">          </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;input</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"></span>
 120 <span class="w">          </span><span class="n">zero</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;zero</span><span class="p">}</span><span class="w"></span>
 121 <span class="w">         </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;answer</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;zero</span><span class="p">,</span><span class="w"></span>
 122 <span class="w">             </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 123
 124 <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>
 125 <span class="w">         </span><span class="p">{</span><span class="n">combine</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;input2</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"></span>
 126 <span class="w">          </span><span class="n">input</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;input1</span><span class="p">}</span><span class="w"></span>
 127 <span class="w">         </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;input1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;input2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"></span>
 128 <span class="w">             </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">)</span><span class="w"> </span><span class="n">step0</span><span class="w"></span>
 129
 130 <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>
 131 <span class="w">         </span><span class="p">{</span><span class="n">combine</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;accum1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;input</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">}</span><span class="w"></span>
 132 <span class="w">         </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;accum1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;input</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;accum2</span><span class="p">,</span><span class="w"></span>
 133 <span class="w">             </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;d</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;e</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;f</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;g</span><span class="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"></span>
 134 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 135
 136 <span class="k">structure</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">:</span><span class="w"> </span><span class="n">FOLD_01N</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 137 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
 138 <span class="w">      </span><span class="k">open</span><span class="w"> </span><span class="n">Fold01N</span><span class="w"></span>
 139
 140 <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">finish</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">zero</span><span class="p">}</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 141 <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="n">finish</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">=&gt;</span><span class="w"> </span><span class="n">zero</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">),</span><span class="w"></span>
 142 <span class="w">                    </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">finish</span><span class="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="p">_)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="p">()))</span><span class="w"></span>
 143
 144 <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">combine</span><span class="p">,</span><span class="w"> </span><span class="n">input</span><span class="p">}</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 145 <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="n">finish</span><span class="p">,</span><span class="w"> </span><span class="p">_,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 146 <span class="w">                     </span><span class="p">(</span><span class="n">finish</span><span class="p">,</span><span class="w"></span>
 147 <span class="w">                      </span><span class="n">finish</span><span class="p">,</span><span class="w"></span>
 148 <span class="w">                      </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">input</span><span class="p">,</span><span class="w"></span>
 149 <span class="w">                      </span><span class="k">fn</span><span class="w"> </span><span class="n">x&#39;</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">combine</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="n">input</span><span class="p">,</span><span class="w"> </span><span class="n">x&#39;</span><span class="p">)))</span><span class="w"></span>
 150
 151 <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">combine</span><span class="p">}</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="n">input</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 152 <span class="w">         </span><span class="n">step0</span><span class="w"> </span><span class="p">{</span><span class="n">combine</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">combine</span><span class="p">,</span><span class="w"> </span><span class="n">input</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">input</span><span class="p">}</span><span class="w"> </span><span class="n">z</span><span class="w"></span>
 153 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 154 </pre></div></div></div>
 155 </div>
 156 </div>
 157 </div>
 158 <div id="footnotes"><hr></div>
 159 <div id="footer">
 160 <div id="footer-text">
 161 </div>
 162 <div id="footer-badges">
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