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| 22 | <a href="./Home">MLton 20180207</a>\r |
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| 25 | <div id="header">\r |
| 26 | <h1>Fold01N</h1>\r |
| 27 | </div>\r |
| 28 | <div id="content">\r |
| 29 | <div id="preamble">\r |
| 30 | <div class="sectionbody">\r |
| 31 | <div class="paragraph"><p>A common use pattern of <a href="Fold">Fold</a> is to define a variable-arity\r |
| 32 | function that combines multiple arguments together using a binary\r |
| 33 | function. It is slightly tricky to do this directly using fold,\r |
| 34 | because of the special treatment required for the case of zero or one\r |
| 35 | argument. Here is a structure, <span class="monospaced">Fold01N</span>, that solves the problem\r |
| 36 | once and for all, and eases the definition of such functions.</p></div>\r |
| 37 | <div class="listingblock">\r |
| 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>\r |
| 39 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r |
| 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>\r |
| 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">=></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>\r |
| 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">=></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>\r |
| 43 | \r |
| 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>\r |
| 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">=></span><span class="w"></span>\r |
| 46 | <span class="w"> </span><span class="p">(</span><span class="n">finish</span><span class="p">,</span><span class="w"></span>\r |
| 47 | <span class="w"> </span><span class="n">finish</span><span class="p">,</span><span class="w"></span>\r |
| 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">=></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>\r |
| 49 | <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">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'</span><span class="p">)))</span><span class="w"></span>\r |
| 50 | \r |
| 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>\r |
| 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>\r |
| 53 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 54 | </pre></div></div></div>\r |
| 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>,\r |
| 56 | then one can define a variable-arity function <span class="monospaced">f</span> and stepper\r |
| 57 | <span class="monospaced">`</span> as follows.</p></div>\r |
| 58 | <div class="listingblock">\r |
| 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">=></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>\r |
| 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">=></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>\r |
| 61 | </pre></div></div></div>\r |
| 62 | <div class="paragraph"><p>One can then use the fold equation to prove the following equations.</p></div>\r |
| 63 | <div class="listingblock">\r |
| 64 | <div class="content"><div class="highlight"><pre><span class="n">f</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">zero</span><span class="w"></span>\r |
| 65 | <span class="n">f</span><span class="w"> </span><span class="n">`a1</span><span class="w"> </span><span class="n">$</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>\r |
| 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">$</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">finish</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="p">(</span><span class="n">start</span><span class="w"> </span><span class="n">a1</span><span class="p">,</span><span class="w"> </span><span class="n">a2</span><span class="p">))</span><span class="w"></span>\r |
| 67 | <span class="n">f</span><span class="w"> </span><span class="n">`a1</span><span class="w"> </span><span class="n">`a2</span><span class="w"> </span><span class="n">`a3</span><span class="w"> </span><span class="n">$</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>\r |
| 68 | <span class="p">...</span><span class="w"></span>\r |
| 69 | </pre></div></div></div>\r |
| 70 | <div class="paragraph"><p>For an example of <span class="monospaced">Fold01N</span>, see <a href="VariableArityPolymorphism">VariableArityPolymorphism</a>.</p></div>\r |
| 71 | </div>\r |
| 72 | </div>\r |
| 73 | <div class="sect1">\r |
| 74 | <h2 id="_typing_fold01n">Typing Fold01N</h2>\r |
| 75 | <div class="sectionbody">\r |
| 76 | <div class="paragraph"><p>Here is the signature for <span class="monospaced">Fold01N</span>. We use a trick to avoid having\r |
| 77 | to duplicate the definition of some rather complex types in both the\r |
| 78 | signature and the structure. We first define the types in a\r |
| 79 | structure. Then, we define them via type re-definitions in the\r |
| 80 | signature, and via <span class="monospaced">open</span> in the full structure.</p></div>\r |
| 81 | <div class="listingblock">\r |
| 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>\r |
| 83 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r |
| 84 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'input</span><span class="p">,</span><span class="w"> </span><span class="n">'accum1</span><span class="p">,</span><span class="w"> </span><span class="n">'accum2</span><span class="p">,</span><span class="w"> </span><span class="n">'answer</span><span class="p">,</span><span class="w"> </span><span class="n">'zero</span><span class="p">,</span><span class="w"></span>\r |
| 85 | <span class="w"> </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">'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>\r |
| 86 | <span class="w"> </span><span class="p">((</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>\r |
| 87 | <span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'accum2</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'answer</span><span class="p">)</span><span class="w"></span>\r |
| 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">-></span><span class="w"> </span><span class="n">'zero</span><span class="p">)</span><span class="w"></span>\r |
| 89 | <span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'input</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum1</span><span class="p">),</span><span class="w"></span>\r |
| 90 | <span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'b</span><span class="p">)</span><span class="w"> </span><span class="n">*</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="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'d</span><span class="p">,</span><span class="w"></span>\r |
| 91 | <span class="w"> </span><span class="n">'b</span><span class="p">,</span><span class="w"></span>\r |
| 92 | <span class="w"> </span><span class="n">'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>\r |
| 93 | \r |
| 94 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'input1</span><span class="p">,</span><span class="w"> </span><span class="n">'accum1</span><span class="p">,</span><span class="w"> </span><span class="n">'input2</span><span class="p">,</span><span class="w"> </span><span class="n">'accum2</span><span class="p">,</span><span class="w"></span>\r |
| 95 | <span class="w"> </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">'e</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">step0</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r |
| 96 | <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="n">'b</span><span class="w"> </span><span class="n">*</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="n">'input1</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum1</span><span class="p">),</span><span class="w"></span>\r |
| 97 | <span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="n">*</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="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'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">'input2</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum2</span><span class="p">),</span><span class="w"></span>\r |
| 98 | <span class="w"> </span><span class="n">'d</span><span class="p">,</span><span class="w"> </span><span class="n">'e</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">Fold</span><span class="p">.</span><span class="n">step0</span><span class="w"></span>\r |
| 99 | \r |
| 100 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">'accum1</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">'accum2</span><span class="p">,</span><span class="w"></span>\r |
| 101 | <span class="w"> </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">'e</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="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r |
| 102 | <span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"></span>\r |
| 103 | <span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'c</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'d</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum1</span><span class="p">),</span><span class="w"></span>\r |
| 104 | <span class="w"> </span><span class="n">'c</span><span class="w"> </span><span class="n">*</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="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'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">'input</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum2</span><span class="p">),</span><span class="w"></span>\r |
| 105 | <span class="w"> </span><span class="n">'e</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="p">)</span><span class="w"> </span><span class="n">Fold</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>\r |
| 106 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 107 | \r |
| 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>\r |
| 109 | <span class="w"> </span><span class="k">sig</span><span class="w"></span>\r |
| 110 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 111 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 112 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 113 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 114 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 115 | <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">'e</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="p">,</span><span class="w"> </span><span class="n">'h</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">'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>\r |
| 116 | \r |
| 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>\r |
| 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">'accum2</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'answer</span><span class="p">,</span><span class="w"></span>\r |
| 119 | <span class="w"> </span><span class="n">start</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">'accum1</span><span class="p">,</span><span class="w"></span>\r |
| 120 | <span class="w"> </span><span class="n">zero</span><span class="p">:</span><span class="w"> </span><span class="n">'zero</span><span class="p">}</span><span class="w"></span>\r |
| 121 | <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'input</span><span class="p">,</span><span class="w"> </span><span class="n">'accum1</span><span class="p">,</span><span class="w"> </span><span class="n">'accum2</span><span class="p">,</span><span class="w"> </span><span class="n">'answer</span><span class="p">,</span><span class="w"> </span><span class="n">'zero</span><span class="p">,</span><span class="w"></span>\r |
| 122 | <span class="w"> </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">'e</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r |
| 123 | \r |
| 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>\r |
| 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">'accum1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'input2</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'accum2</span><span class="p">,</span><span class="w"></span>\r |
| 126 | <span class="w"> </span><span class="n">input</span><span class="p">:</span><span class="w"> </span><span class="n">'input1</span><span class="p">}</span><span class="w"></span>\r |
| 127 | <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'input1</span><span class="p">,</span><span class="w"> </span><span class="n">'accum1</span><span class="p">,</span><span class="w"> </span><span class="n">'input2</span><span class="p">,</span><span class="w"> </span><span class="n">'accum2</span><span class="p">,</span><span class="w"></span>\r |
| 128 | <span class="w"> </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">'e</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">step0</span><span class="w"></span>\r |
| 129 | \r |
| 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>\r |
| 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">'accum1</span><span class="w"> </span><span class="n">*</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">'accum2</span><span class="p">}</span><span class="w"></span>\r |
| 132 | <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'accum1</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">'accum2</span><span class="p">,</span><span class="w"></span>\r |
| 133 | <span class="w"> </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">'e</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="p">)</span><span class="w"> </span><span class="n">step1</span><span class="w"></span>\r |
| 134 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 135 | \r |
| 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>\r |
| 137 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r |
| 138 | <span class="w"> </span><span class="k">open</span><span class="w"> </span><span class="n">Fold01N</span><span class="w"></span>\r |
| 139 | \r |
| 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>\r |
| 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">=></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>\r |
| 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">=></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>\r |
| 143 | \r |
| 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>\r |
| 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">=></span><span class="w"></span>\r |
| 146 | <span class="w"> </span><span class="p">(</span><span class="n">finish</span><span class="p">,</span><span class="w"></span>\r |
| 147 | <span class="w"> </span><span class="n">finish</span><span class="p">,</span><span class="w"></span>\r |
| 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">=></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>\r |
| 149 | <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">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'</span><span class="p">)))</span><span class="w"></span>\r |
| 150 | \r |
| 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>\r |
| 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>\r |
| 153 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 154 | </pre></div></div></div>\r |
| 155 | </div>\r |
| 156 | </div>\r |
| 157 | </div>\r |
| 158 | <div id="footnotes"><hr></div>\r |
| 159 | <div id="footer">\r |
| 160 | <div id="footer-text">\r |
| 161 | </div>\r |
| 162 | <div id="footer-badges">\r |
| 163 | </div>\r |
| 164 | </div>\r |
| 165 | </body>\r |
| 166 | </html>\r |