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<a href="./Home">MLton 20201002</a>
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<div id="header">
<h1>InfixingOperators</h1>
</div>
<div id="content">
<div id="preamble">
<div class="sectionbody">
<div class="paragraph"><p>Fixity specifications are not part of signatures in
<a href="StandardML">Standard ML</a>. When one wants to use a module that
provides functions designed to be used as infix operators there are
several obvious alternatives:</p></div>
<div class="ulist"><ul>
<li>
<p>
Use only prefix applications. Unfortunately there are situations
where infix applications lead to considerably more readable code.
</p>
</li>
<li>
<p>
Make the fixity declarations at the top-level. This may lead to
collisions and may be unsustainable in a large project. Pollution of
the top-level should be avoided.
</p>
</li>
<li>
<p>
Make the fixity declarations at each scope where you want to use
infix applications. The duplication becomes inconvenient if the
operators are widely used. Duplication of code should be avoided.
</p>
</li>
<li>
<p>
Use non-standard extensions, such as the <a href="MLBasis"> ML Basis system</a>
to control the scope of fixity declarations. This has the obvious
drawback of reduced portability.
</p>
</li>
<li>
<p>
Reuse existing infix operator symbols (<span class="monospaced">^</span>, <span class="monospaced">+</span>, <span class="monospaced">-</span>, &#8230;).  This
can be convenient when the standard operators aren&#8217;t needed in the
same scope with the new operators.  On the other hand, one is limited
to the standard operator symbols and the code may appear confusing.
</p>
</li>
</ul></div>
<div class="paragraph"><p>None of the obvious alternatives is best in every case. The following
describes a slightly less obvious alternative that can sometimes be
useful. The idea is to approximate Haskell&#8217;s special syntax for
treating any identifier enclosed in grave accents (backquotes) as an
infix operator. In Haskell, instead of writing the prefix application
<span class="monospaced">f x y</span> one can write the infix application <span class="monospaced">x &grave;f&grave; y</span>.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_infixing_operators">Infixing operators</h2>
<div class="sectionbody">
<div class="paragraph"><p>Let&#8217;s first take a look at the definitions of the operators:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">infix</span><span class="w">  </span><span class="mi">3</span><span class="w"> </span><span class="n">&lt;\</span><span class="w">     </span><span class="k">fun</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">&lt;\</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">y</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="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w">     </span><span class="cm">(* Left section      *)</span><span class="w"></span>
<span class="k">infix</span><span class="w">  </span><span class="mi">3</span><span class="w"> </span><span class="n">\&gt;</span><span class="w">     </span><span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">\&gt;</span><span class="w"> </span><span class="n">y</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">y</span><span class="w">                  </span><span class="cm">(* Left application  *)</span><span class="w"></span>
<span class="k">infixr</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="n">/&gt;</span><span class="w">     </span><span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">/&gt;</span><span class="w"> </span><span class="n">y</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">=&gt;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w">     </span><span class="cm">(* Right section     *)</span><span class="w"></span>
<span class="k">infixr</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="n">&lt;/</span><span class="w">     </span><span class="k">fun</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">&lt;/</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="n">x</span><span class="w">                  </span><span class="cm">(* Right application *)</span><span class="w"></span>

<span class="k">infix</span><span class="w">  </span><span class="mi">2</span><span class="w"> </span><span class="n">o</span><span class="w">  </span><span class="cm">(* See motivation below *)</span><span class="w"></span>
<span class="k">infix</span><span class="w">  </span><span class="mi">0</span><span class="w"> </span><span class="n">:=</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>The left and right sectioning operators, <span class="monospaced">&lt;\</span> and <span class="monospaced">/&gt;</span>, are useful in
SML for partial application of infix operators.
<a href="References#Paulson96"> ML For the Working Programmer</a> describes curried
functions <span class="monospaced">secl</span> and <span class="monospaced">secr</span> for the same purpose on pages 179-181.
For example,</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="n">List</span><span class="p">.</span><span class="n">map</span><span class="w"> </span><span class="p">(</span><span class="k">op</span><span class="n">-</span><span class="w"> </span><span class="n">/&gt;</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>is a function for subtracting <span class="monospaced">y</span> from a list of integers and</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="n">List</span><span class="p">.</span><span class="n">exists</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="n">&lt;\</span><span class="w"> </span><span class="k">op</span><span class="p">=)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>is a function for testing whether a list contains an <span class="monospaced">x</span>.</p></div>
<div class="paragraph"><p>Together with the left and right application operators, <span class="monospaced">\&gt;</span> and <span class="monospaced">&lt;/</span>,
the sectioning operators provide a way to treat any binary function
(i.e. a function whose domain is a pair) as an infix operator.  In
general,</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>x0 &lt;\f1\&gt; x1 &lt;\f2\&gt; x2 ... &lt;\fN\&gt; xN = fN (... f2 (f1 (x0, x1), x2) ..., xN)</pre>
</div></div>
<div class="paragraph"><p>and</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>xN &lt;/fN/&gt; ... x2 &lt;/f2/&gt; x1 &lt;/f1/&gt; x0  =  fN (xN, ... f2 (x2, f1 (x1, x0)) ...)</pre>
</div></div>
<div class="sect2">
<h3 id="_examples">Examples</h3>
<div class="paragraph"><p>As a fairly realistic example, consider providing a function for sequencing
comparisons:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">structure</span><span class="w"> </span><span class="n">Order</span><span class="w"> </span><span class="cm">(* ... *)</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">struct</span><span class="w"></span>
<span class="w">      </span><span class="cm">(* ... *)</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">orWhenEq</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="n">EQUAL</span><span class="p">,</span><span class="w"> </span><span class="n">th</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">th</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
<span class="w">                      </span><span class="p">|</span><span class="w"> </span><span class="p">(</span><span class="n">other</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">other</span><span class="w"></span>
<span class="w">      </span><span class="cm">(* ... *)</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>Using <span class="monospaced">orWhenEq</span> and the infixing operators, one can write a
<span class="monospaced">compare</span> function for triples as</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">fun</span><span class="w"> </span><span class="n">compare</span><span class="w"> </span><span class="p">(</span><span class="n">fad</span><span class="p">,</span><span class="w"> </span><span class="n">fbe</span><span class="p">,</span><span class="w"> </span><span class="n">fcf</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="p">(</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="p">=</span><span class="w"></span>
<span class="w">    </span><span class="n">fad</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">d</span><span class="p">)</span><span class="w"> </span><span class="n">&lt;\Order</span><span class="p">.</span><span class="n">orWhenEq\&gt;</span><span class="w"> </span><span class="n">`fbe</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">e</span><span class="p">)</span><span class="w"> </span><span class="n">&lt;\Order</span><span class="p">.</span><span class="n">orWhenEq\&gt;</span><span class="w"> </span><span class="n">`fcf</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>where <span class="monospaced">&grave;</span> is defined as</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">fun</span><span class="w"> </span><span class="n">`f</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="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">x</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>Although <span class="monospaced">orWhenEq</span> can be convenient (try rewriting the above without
it), it is probably not useful enough to be defined at the top level
as an infix operator. Fortunately we can use the infixing operators
and don&#8217;t have to.</p></div>
<div class="paragraph"><p>Another fairly realistic example would be to use the infixing operators with
the technique described on the <a href="Printf">Printf</a> page. Assuming that you would have
a <span class="monospaced">Printf</span> module binding <span class="monospaced">printf</span>, <span class="monospaced">&grave;</span>, and formatting combinators
named <span class="monospaced">int</span> and <span class="monospaced">string</span>, you could write</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">let</span><span class="w"> </span><span class="k">open</span><span class="w"> </span><span class="n">Printf</span><span class="w"> </span><span class="k">in</span><span class="w"></span>
<span class="w">  </span><span class="n">printf</span><span class="w"> </span><span class="p">(</span><span class="n">`</span><span class="s">&quot;Here&#39;s an int &quot;</span><span class="n">&lt;\int\&gt;</span><span class="s">&quot; and a string &quot;</span><span class="n">&lt;\string\&gt;</span><span class="s">&quot;.&quot;</span><span class="p">)</span><span class="w"> </span><span class="mi">13</span><span class="w"> </span><span class="s">&quot;foo&quot;</span><span class="w"> </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>without having to duplicate the fixity declarations. Alternatively, you could
write</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="n">P</span><span class="p">.</span><span class="n">printf</span><span class="w"> </span><span class="p">(</span><span class="n">P</span><span class="p">.</span><span class="n">`</span><span class="s">&quot;Here&#39;s an int &quot;</span><span class="n">&lt;\P</span><span class="p">.</span><span class="n">int\&gt;</span><span class="s">&quot; and a string &quot;</span><span class="n">&lt;\P</span><span class="p">.</span><span class="n">string\&gt;</span><span class="s">&quot;.&quot;</span><span class="p">)</span><span class="w"> </span><span class="mi">13</span><span class="w"> </span><span class="s">&quot;foo&quot;</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>assuming you have the made the binding</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">structure</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">Printf</span><span class="w"></span>
</pre></div></div></div>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_application_and_piping_operators">Application and piping operators</h2>
<div class="sectionbody">
<div class="paragraph"><p>The left and right application operators may also provide some notational
convenience on their own. In general,</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>f \&gt; x1 \&gt; ... \&gt; xN = f x1 ... xN</pre>
</div></div>
<div class="paragraph"><p>and</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>xN &lt;/ ... &lt;/ x1 &lt;/ f = f x1 ... xN</pre>
</div></div>
<div class="paragraph"><p>If nothing else, both of them can eliminate parentheses. For example,</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="n">foo</span><span class="w"> </span><span class="p">(</span><span class="mi">1</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="p">=</span><span class="w"> </span><span class="n">foo</span><span class="w"> </span><span class="n">\&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>The left and right application operators are related to operators
that could be described as the right and left piping operators:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">infix</span><span class="w">  </span><span class="mi">1</span><span class="w"> </span><span class="n">&gt;|</span><span class="w">     </span><span class="k">val</span><span class="w"> </span><span class="k">op</span><span class="n">&gt;|</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">op</span><span class="n">&lt;/</span><span class="w">      </span><span class="cm">(* Left pipe *)</span><span class="w"></span>
<span class="k">infixr</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="n">|&lt;</span><span class="w">     </span><span class="k">val</span><span class="w"> </span><span class="k">op</span><span class="n">|&lt;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">op</span><span class="n">\&gt;</span><span class="w">      </span><span class="cm">(* Right pipe *)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>As you can see, the left and right piping operators, <span class="monospaced">&gt;|</span> and <span class="monospaced">|&lt;</span>,
are the same as the right and left application operators,
respectively, except the associativities are reversed and the binding
strength is lower. They are useful for piping data through a sequence
of operations. In general,</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>x &gt;| f1 &gt;| ... &gt;| fN = fN (... (f1 x) ...) = (fN o ... o f1) x</pre>
</div></div>
<div class="paragraph"><p>and</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>fN |&lt; ... |&lt; f1 |&lt; x = fN (... (f1 x) ...) = (fN o ... o f1) x</pre>
</div></div>
<div class="paragraph"><p>The right piping operator, <span class="monospaced">|&lt;</span>, is provided by the Haskell prelude as
<span class="monospaced">$</span>. It can be convenient in CPS or continuation passing style.</p></div>
<div class="paragraph"><p>A use for the left piping operator is with parsing combinators. In a
strict language, like SML, eta-reduction is generally unsafe. Using
the left piping operator, parsing functions can be formatted
conveniently as</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="k">fun</span><span class="w"> </span><span class="n">parsingFunc</span><span class="w"> </span><span class="n">input</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="n">input</span><span class="w"> </span><span class="n">&gt;|</span><span class="w"> </span><span class="cm">(* ... *)</span><span class="w"></span>
<span class="w">         </span><span class="n">||</span><span class="w"> </span><span class="cm">(* ... *)</span><span class="w"></span>
<span class="w">         </span><span class="n">||</span><span class="w"> </span><span class="cm">(* ... *)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>where <span class="monospaced">||</span> is supposed to be a combinator provided by the parsing combinator
library.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_about_precedences">About precedences</h2>
<div class="sectionbody">
<div class="paragraph"><p>You probably noticed that we redefined the
<a href="OperatorPrecedence">precedences</a> of the function composition operator
<span class="monospaced">o</span> and the assignment operator <span class="monospaced">:=</span>. Doing so is not strictly
necessary, but can be convenient and should be relatively
safe. Consider the following motivating examples from
<a href="WesleyTerpstra"> Wesley W. Terpstra</a> relying on the redefined
precedences:</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span></span><span class="n">Word8</span><span class="p">.</span><span class="n">fromInt</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">Char</span><span class="p">.</span><span class="n">ord</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">&lt;\String</span><span class="p">.</span><span class="n">sub</span><span class="w"></span>
<span class="cm">(* Combining sectioning and composition *)</span><span class="w"></span>

<span class="n">x</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="n">&lt;\String</span><span class="p">.</span><span class="n">sub\&gt;</span><span class="w"> </span><span class="n">i</span><span class="w"></span>
<span class="cm">(* Assigning the result of an infixed application *)</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>In imperative languages, assignment usually has the lowest precedence
(ignoring statement separators). The precedence of <span class="monospaced">:=</span> in the
<a href="BasisLibrary"> Basis Library</a> is perhaps unnecessarily high, because
an expression of the form <span class="monospaced">r := x</span> always returns a unit, which makes
little sense to combine with anything. Dropping <span class="monospaced">:=</span> to the lowest
precedence level makes it behave more like in other imperative
languages.</p></div>
<div class="paragraph"><p>The case for <span class="monospaced">o</span> is different. With the exception of <span class="monospaced">before</span> and
<span class="monospaced">:=</span>, it doesn&#8217;t seem to make much sense to use <span class="monospaced">o</span> with any of the
operators defined by the <a href="BasisLibrary"> Basis Library</a> in an
unparenthesized expression. This is simply because none of the other
operators deal with functions. It would seem that the precedence of
<span class="monospaced">o</span> could be chosen completely arbitrarily from the set <span class="monospaced">{1, ..., 9}</span>
without having any adverse effects with respect to other infix
operators defined by the <a href="BasisLibrary"> Basis Library</a>.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_design_of_the_symbols">Design of the symbols</h2>
<div class="sectionbody">
<div class="paragraph"><p>The closest approximation of Haskell&#8217;s <span class="monospaced">x &grave;f&grave; y</span> syntax
achievable in Standard ML would probably be something like
<span class="monospaced">x &grave;f^ y</span>, but <span class="monospaced">^</span> is already used for string
concatenation by the <a href="BasisLibrary"> Basis Library</a>. Other
combinations of the characters <span class="monospaced">&grave;</span> and <span class="monospaced">^</span> would be
possible, but none seems clearly the best visually. The symbols <span class="monospaced">&lt;\</span>,
<span class="monospaced">\&gt;</span>, <span class="monospaced">&lt;/</span>, and <span class="monospaced">/&gt;</span> are reasonably concise and have a certain
self-documenting appearance and symmetry, which can help to remember
them.  As the names suggest, the symbols of the piping operators <span class="monospaced">&gt;|</span>
and <span class="monospaced">|&lt;</span> are inspired by Unix shell pipelines.</p></div>
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<h2 id="_also_see">Also see</h2>
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<li>
<p>
<a href="Utilities">Utilities</a>
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