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<div class="header">
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<a href="#var-members">Variables</a>  </div>
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<div class="title">Orderable<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
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<div class="contents">
<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>The <code>Orderable</code> concept represents totally ordered data types. </p>
<p>Intuitively, <code>Orderable</code> objects must define a binary predicate named <code>less</code> returning whether the first argument is to be considered less than the second argument. The word "total" means that <em>distinct</em> objects must always be ordered; if <code>a</code> and <code>b</code> are not equal, then exactly one of <code>less(a, b)</code> and <code>less(b, a)</code> must be true. This is a contrast with weaker kinds of orders that would allow some objects to be incomparable (neither less than nor greater than). Also note that a non-strict total order may always be obtained from a strict total order (and vice-versa) by setting </p><div class="fragment"><div class="line">a &lt;= b  =  !(b &lt; a)</div><div class="line">a &lt;  b  =  !(b &lt;= a)</div></div><!-- fragment --><p> The non-strict version is used in the description of the laws because it makes them easier to parse for humans, but they could be formulated equivalently using the strict order.</p>
<h2>Minimal complete definition </h2>
<p><code>less</code></p>
<p>When <code>less</code> is defined, the other methods are defined from it using the same definition as mandated in the laws below.</p>
<h2>Laws </h2>
<p>Rigorously speaking, a <a href="http://en.wikipedia.org/wiki/Total_order">total order</a> <code>&lt;=</code> on a set <code>S</code> is a binary predicate \( &lt;= \;: S \times S \to bool \) such that for all <code>a</code>, <code>b</code>, <code>c</code> in <code>S</code>, </p><div class="fragment"><div class="line"><span class="keywordflow">if</span>  a &lt;= b  and  b &lt;= a  <a class="code" href="group__group-Monad.html#gaaddd3789de43cf989babb10cdc0b447a">then</a>  a == b <span class="comment">// Antisymmetry</span></div><div class="line"><span class="keywordflow">if</span>  a &lt;= b  and  b &lt;= c  <a class="code" href="group__group-Monad.html#gaaddd3789de43cf989babb10cdc0b447a">then</a>  a &lt;= c <span class="comment">// Transitivity</span></div><div class="line">either  a &lt;= b  or  b &lt;= a            <span class="comment">// Totality</span></div></div><!-- fragment --><p> Additionally, the <code>less</code>, <code>greater</code> and <code>greater_equal</code> methods should have the following intuitive meanings: </p><div class="fragment"><div class="line">a &lt;  b  <span class="keywordflow">if</span> and only <span class="keywordflow">if</span>  !(b &lt;= a)</div><div class="line">a &gt;  b  <span class="keywordflow">if</span> and only <span class="keywordflow">if</span>    b &lt; a</div><div class="line">a &gt;= b  <span class="keywordflow">if</span> and only <span class="keywordflow">if</span>  !(a &lt; b)</div></div><!-- fragment --><h2>Refined concept </h2>
<ol type="1">
<li><code>Comparable</code> (free model)<br />
Since <code>Orderable</code> requires <code>less_equal</code> to be a total order, a model of <code>Comparable</code> may always be obtained by setting <div class="fragment"><div class="line"><a class="code" href="group__group-Comparable.html#gacaf1ebea6b3ab96ac9dcb82f0e64e547">equal</a>(x, y) = <a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(x, y) &amp;&amp; <a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(y, x)</div></div><!-- fragment --></li>
</ol>
<h2>Concrete models </h2>
<p><code><a class="el" href="structboost_1_1hana_1_1integral__constant.html" title="Compile-time value of an integral type.">hana::integral_constant</a></code>, <code><a class="el" href="structboost_1_1hana_1_1optional.html" title="Optional value whose optional-ness is known at compile-time.">hana::optional</a></code>, <code><a class="el" href="structboost_1_1hana_1_1pair.html" title="Generic container for two elements.">hana::pair</a></code>, <code><a class="el" href="structboost_1_1hana_1_1string.html" title="Compile-time string.">hana::string</a></code>, <code><a class="el" href="structboost_1_1hana_1_1tuple.html" title="General purpose index-based heterogeneous sequence with a fixed length.">hana::tuple</a></code></p>
<h2>Free model for <code>LessThanComparable</code> data types </h2>
<p>Two data types <code>T</code> and <code>U</code> that model the cross-type version of the usual <a href="http://en.cppreference.com/w/cpp/named_req/LessThanComparable">LessThanComparable</a> C++ concept are automatically a model of <code>Orderable</code> by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y) = (x &lt; y)</div></div><!-- fragment --><p> The cross-type version of the LessThanComparable concept is analogous to the cross-type version of the EqualityComparable concept presented in <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3351.pdf">N3351</a>, which is compatible with the usual single type definition. However, note that the LessThanComparable concept only requires <code>&lt;</code> to be a <a href="http://en.wikipedia.org/wiki/Strict_weak_ordering">strict weak ordering</a>, which is a weaker requirement than being a total order. Hence, if <code>less</code> is used with objects of a LessThanComparable data type that do not define a total order, some algorithms may have an unexpected behavior. It is the author's opinion that defining <code>operator&lt;</code> as a non-total order is a bad idea, but this is debatable and so the design choice of providing a model for LessThanComparable data types is open to debate. Waiting for some user input.</p>
<h2>Order-preserving functions </h2>
<p>Let <code>A</code> and <code>B</code> be two <code>Orderable</code> data types. A function \( f : A \to B\) is said to be order-preserving (also called monotone) if it preserves the structure of the <code>Orderable</code> concept, which can be rigorously stated as follows. For all objects <code>x</code>, <code>y</code> of data type <code>A</code>, </p><div class="fragment"><div class="line"><span class="keywordflow">if</span>  <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y)  <a class="code" href="group__group-Monad.html#gaaddd3789de43cf989babb10cdc0b447a">then</a>  <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(f(x), f(y))</div></div><!-- fragment --><p> Another important property is that of being order-reflecting, which can be stated as </p><div class="fragment"><div class="line"><span class="keywordflow">if</span>  <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(f(x), f(y))  <a class="code" href="group__group-Monad.html#gaaddd3789de43cf989babb10cdc0b447a">then</a>  <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y)</div></div><!-- fragment --><p> We say that a function is an order-embedding if it is both order-preserving and order-reflecting, i.e. if </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y)  <span class="keywordflow">if</span> and only <span class="keywordflow">if</span>  <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(f(x), f(y))</div></div><!-- fragment --><h2>Cross-type version of the methods </h2>
<p>The comparison methods (<code>less</code>, <code>less_equal</code>, <code>greater</code> and <code>greater_equal</code>) are "overloaded" to handle distinct data types with certain properties. Specifically, they are defined for <em>distinct</em> data types <code>A</code> and <code>B</code> such that</p><ol type="1">
<li><code>A</code> and <code>B</code> share a common data type <code>C</code>, as determined by the <code>common</code> metafunction</li>
<li><code>A</code>, <code>B</code> and <code>C</code> are all <code>Orderable</code> when taken individually</li>
<li>\(\mathrm{to&lt;C&gt;} : A \to C\) and \(\mathrm{to&lt;C&gt;} : B \to C\) are both order-embeddings as determined by the <code>is_embedding</code> metafunction.</li>
</ol>
<p>The method definitions for data types satisfying the above properties are </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y)          = <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div><div class="line"><a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(x, y)    = <a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div><div class="line"><a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">greater_equal</a>(x, y) = <a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">greater_equal</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div><div class="line"><a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">greater</a>(x, y)       = <a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">greater</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div></div><!-- fragment --><h2>Partial application of the methods </h2>
<p>The <code>less</code>, <code>greater</code>, <code>less_equal</code> and <code>greater_equal</code> methods can be called in two different ways. First, they can be called like normal functions: </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(x, y)</div><div class="line"><a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">greater</a>(x, y)</div><div class="line"></div><div class="line"><a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(x, y)</div><div class="line"><a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">greater_equal</a>(x, y)</div></div><!-- fragment --><p>However, they may also be partially applied to an argument as follows: </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>.than(x)(y)    == <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(y, x)</div><div class="line"><a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">greater</a>.than(x)(y) == <a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">greater</a>(y, x)</div><div class="line"></div><div class="line"><a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>.than(x)(y)    == <a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">less_equal</a>(y, x)</div><div class="line"><a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">greater_equal</a>.than(x)(y) == <a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">greater_equal</a>(y, x)</div></div><!-- fragment --><p>Take good note that the order of the arguments is reversed, so for example <code>less.than(x)(y)</code> is equivalent to <code>less(y, x)</code>, not <code>less(x, y)</code>. This is because those variants are meant to be used with higher order algorithms, where the chosen application order makes sense. </p>
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<tr class="memdesc:ga6023631e7d0a01e16dc3fa4221fbd703"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns a <code>Logical</code> representing whether <code>x</code> is greater than or equal to <code>y</code>.  <a href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">More...</a><br /></td></tr>
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<tr class="memdesc:gad510011602bdb14686f1c4ec145301c9"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns a <code>Logical</code> representing whether <code>x</code> is less than <code>y</code>.  <a href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">More...</a><br /></td></tr>
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<tr class="memdesc:ga9917dd82beb67151bf5657245d37b851"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns a <code>Logical</code> representing whether <code>x</code> is less than or equal to <code>y</code>.  <a href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">More...</a><br /></td></tr>
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<tr class="memdesc:ga999eee8ca8750f9b1afa0d7a1db28030"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns the greatest of its arguments according to the <code>less</code> ordering.  <a href="group__group-Orderable.html#ga999eee8ca8750f9b1afa0d7a1db28030">More...</a><br /></td></tr>
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<tr class="memitem:ga2d54f189ea6f57fb2c0d772169440c5c"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Orderable.html#ga2d54f189ea6f57fb2c0d772169440c5c">boost::hana::min</a></td></tr>
<tr class="memdesc:ga2d54f189ea6f57fb2c0d772169440c5c"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns the smallest of its arguments according to the <code>less</code> ordering.  <a href="group__group-Orderable.html#ga2d54f189ea6f57fb2c0d772169440c5c">More...</a><br /></td></tr>
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<tr class="memitem:gaf7e94ba859710cd6ba6152e5dc18977d"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Orderable.html#gaf7e94ba859710cd6ba6152e5dc18977d">boost::hana::ordering</a></td></tr>
<tr class="memdesc:gaf7e94ba859710cd6ba6152e5dc18977d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns a function performing <code>less</code> after applying a transformation to both arguments.<code>ordering</code> creates a total order based on the result of applying a function to some objects, which is especially useful in conjunction with algorithms that accept a custom predicate that must represent a total order.  <a href="group__group-Orderable.html#gaf7e94ba859710cd6ba6152e5dc18977d">More...</a><br /></td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
<a id="gaf9a073eafebbe514fb19dff82318f198"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gaf9a073eafebbe514fb19dff82318f198">&#9670;&nbsp;</a></span>greater</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2greater_8hpp.html">boost/hana/fwd/greater.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns a <code>Logical</code> representing whether <code>x</code> is greater than <code>y</code>. </p>
<h2>Signature </h2>
<p>Given a Logical <code>Bool</code> and two Orderables <code>A</code> and <code>B</code> with a common embedding, the signature is \( \mathrm{greater} : A \times B \to Bool \).</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two objects to compare.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="greater_8hpp.html">boost/hana/greater.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">hana::greater</a>(4, 1), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(!<a class="code" href="group__group-Orderable.html#gaf9a073eafebbe514fb19dff82318f198">hana::greater</a>(hana::int_c&lt;1&gt;, hana::int_c&lt;3&gt;));</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="ga6023631e7d0a01e16dc3fa4221fbd703"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga6023631e7d0a01e16dc3fa4221fbd703">&#9670;&nbsp;</a></span>greater_equal</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2greater__equal_8hpp.html">boost/hana/fwd/greater_equal.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns a <code>Logical</code> representing whether <code>x</code> is greater than or equal to <code>y</code>. </p>
<h2>Signature </h2>
<p>Given a Logical <code>Bool</code> and two Orderables <code>A</code> and <code>B</code> with a common embedding, the signature is \( \mathrm{greater\_equal} : A \times B \to Bool \).</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two objects to compare.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="greater__equal_8hpp.html">boost/hana/greater_equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">hana::greater_equal</a>(4, 1), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">hana::greater_equal</a>(1, 1), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(!<a class="code" href="group__group-Orderable.html#ga6023631e7d0a01e16dc3fa4221fbd703">hana::greater_equal</a>(hana::int_c&lt;1&gt;, hana::int_c&lt;2&gt;));</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="gad510011602bdb14686f1c4ec145301c9"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gad510011602bdb14686f1c4ec145301c9">&#9670;&nbsp;</a></span>less</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2less_8hpp.html">boost/hana/fwd/less.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns a <code>Logical</code> representing whether <code>x</code> is less than <code>y</code>. </p>
<h2>Signature </h2>
<p>Given a Logical <code>Bool</code> and two Orderables <code>A</code> and <code>B</code> with a common embedding, the signature is \( \mathrm{less} : A \times B \to Bool \).</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two objects to compare.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="all__of_8hpp.html">boost/hana/all_of.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="less_8hpp.html">boost/hana/less.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="not_8hpp.html">boost/hana/not.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="tuple_8hpp.html">boost/hana/tuple.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">hana::less</a>(1, 4), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(!<a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">hana::less</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;2&gt;));</div><div class="line"></div><div class="line"><span class="comment">// less.than is syntactic sugar</span></div><div class="line">static_assert(<a class="code" href="group__group-Searchable.html#ga3a168950082f38afd9edf256f336c8ba">hana::all_of</a>(hana::tuple_c&lt;int, 1, 2, 3, 4&gt;, <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">hana::less</a>.than(5)), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Searchable.html#ga3a168950082f38afd9edf256f336c8ba">hana::all_of</a>(hana::tuple_c&lt;int, 1, 2, 3, 4&gt;, <a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">hana::less_equal</a>.than(hana::int_c&lt;4&gt;)));</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="ga9917dd82beb67151bf5657245d37b851"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga9917dd82beb67151bf5657245d37b851">&#9670;&nbsp;</a></span>less_equal</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2less__equal_8hpp.html">boost/hana/fwd/less_equal.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns a <code>Logical</code> representing whether <code>x</code> is less than or equal to <code>y</code>. </p>
<h2>Signature </h2>
<p>Given a Logical <code>Bool</code> and two Orderables <code>A</code> and <code>B</code> with a common embedding, the signature is \( \mathrm{less\_equal} : A \times B \to Bool \).</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x,y</td><td>Two objects to compare.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="less__equal_8hpp.html">boost/hana/less_equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="not_8hpp.html">boost/hana/not.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">hana::less_equal</a>(1, 4), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">hana::less_equal</a>(1, 1), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(!<a class="code" href="group__group-Orderable.html#ga9917dd82beb67151bf5657245d37b851">hana::less_equal</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;2&gt;));</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="ga999eee8ca8750f9b1afa0d7a1db28030"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga999eee8ca8750f9b1afa0d7a1db28030">&#9670;&nbsp;</a></span>max</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2max_8hpp.html">boost/hana/fwd/max.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns the greatest of its arguments according to the <code>less</code> ordering. </p>
<dl class="todo"><dt><b><a class="el" href="todo.html#_todo000005">Todo:</a></b></dt><dd>Can't specify the signature here either. See <code>min</code> for details.</dd></dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="max_8hpp.html">boost/hana/max.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga999eee8ca8750f9b1afa0d7a1db28030">hana::max</a>(1, 4) == 4, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Orderable.html#ga999eee8ca8750f9b1afa0d7a1db28030">hana::max</a>(hana::int_c&lt;7&gt;, hana::int_c&lt;5&gt;) == hana::int_c&lt;7&gt;);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<a id="ga2d54f189ea6f57fb2c0d772169440c5c"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga2d54f189ea6f57fb2c0d772169440c5c">&#9670;&nbsp;</a></span>min</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2min_8hpp.html">boost/hana/fwd/min.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Returns the smallest of its arguments according to the <code>less</code> ordering. </p>
<dl class="todo"><dt><b><a class="el" href="todo.html#_todo000006">Todo:</a></b></dt><dd>We can't specify the signature right now, because the tag of the returned object depends on whether <code>x &lt; y</code> or not. If we wanted to be mathematically correct, we should probably ask that <code>if_(cond, x, y)</code> returns a common data type of <code>x</code> and <code>y</code>, and then the behavior of <code>min</code> would follow naturally. However, I'm unsure whether this is desirable because that's a big requirement.</dd></dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="integral__constant_8hpp.html">boost/hana/integral_constant.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="min_8hpp.html">boost/hana/min.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Orderable.html#ga2d54f189ea6f57fb2c0d772169440c5c">hana::min</a>(1, 4) == 1, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Orderable.html#ga2d54f189ea6f57fb2c0d772169440c5c">hana::min</a>(hana::int_c&lt;7&gt;, hana::int_c&lt;5&gt;) == hana::int_c&lt;5&gt;);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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<h2 class="memtitle"><span class="permalink"><a href="#gaf7e94ba859710cd6ba6152e5dc18977d">&#9670;&nbsp;</a></span>ordering</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2ordering_8hpp.html">boost/hana/fwd/ordering.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; f) {</div><div class="line">        <span class="keywordflow">return</span> [perfect-<a class="code" href="group__group-functional.html#ga41ada6b336e9d5bcb101ff0c737acbd0">capture</a>](<span class="keyword">auto</span>&amp;&amp; x, <span class="keyword">auto</span>&amp;&amp; y) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">            <span class="keywordflow">return</span> <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(f(forwarded(x)), f(forwarded(y)));</div><div class="line">        };</div><div class="line">    }</div><div class="ttc" id="group__group-Orderable_html_gad510011602bdb14686f1c4ec145301c9"><div class="ttname"><a href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">boost::hana::less</a></div><div class="ttdeci">constexpr auto less</div><div class="ttdoc">Returns a Logical representing whether x is less than y.</div><div class="ttdef"><b>Definition:</b> less.hpp:37</div></div>
<div class="ttc" id="group__group-functional_html_ga41ada6b336e9d5bcb101ff0c737acbd0"><div class="ttname"><a href="group__group-functional.html#ga41ada6b336e9d5bcb101ff0c737acbd0">boost::hana::capture</a></div><div class="ttdeci">constexpr auto capture</div><div class="ttdoc">Create a function capturing the given variables.</div><div class="ttdef"><b>Definition:</b> capture.hpp:45</div></div>
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<p>Returns a function performing <code>less</code> after applying a transformation to both arguments.<code>ordering</code> creates a total order based on the result of applying a function to some objects, which is especially useful in conjunction with algorithms that accept a custom predicate that must represent a total order. </p>
<p>Specifically, <code>ordering</code> is such that </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gaf7e94ba859710cd6ba6152e5dc18977d">ordering</a>(f) == <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a> ^<a class="code" href="group__group-functional.html#ga35c4fc3c5677b9f558150b90e74d3ab1">on</a>^ f</div></div><!-- fragment --><p> or, equivalently, </p><div class="fragment"><div class="line"><a class="code" href="group__group-Orderable.html#gaf7e94ba859710cd6ba6152e5dc18977d">ordering</a>(f)(x, y) == <a class="code" href="group__group-Orderable.html#gad510011602bdb14686f1c4ec145301c9">less</a>(f(x), f(y))</div></div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>This is not a tag-dispatched method (hence it can't be customized), but just a convenience function provided with the <code>Orderable</code> concept.</dd></dl>
<h2>Signature </h2>
<p>Given a Logical <code>Bool</code> and an Orderable <code>B</code>, the signature is \( \mathrm{ordering} : (A \to B) \to (A \times A \to Bool) \).</p>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2017</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="ordering_8hpp.html">boost/hana/ordering.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="sort_8hpp.html">boost/hana/sort.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="tuple_8hpp.html">boost/hana/tuple.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="type_8hpp.html">boost/hana/type.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">constexpr <span class="keyword">auto</span> sorted = <a class="code" href="group__group-Sequence.html#gac000a79eb7b9d44ecc8982c93daa40e5">hana::sort</a>.by(<a class="code" href="group__group-Orderable.html#gaf7e94ba859710cd6ba6152e5dc18977d">hana::ordering</a>(hana::sizeof_),</div><div class="line">    hana::tuple_t&lt;<span class="keywordtype">char</span>[3], <span class="keywordtype">char</span>[1], <span class="keywordtype">char</span>[2], <span class="keywordtype">char</span>[15]&gt;</div><div class="line">);</div><div class="line"></div><div class="line"><a class="code" href="group__group-assertions.html#ga2c5006540936d9f8880e3a39f4fcc035">BOOST_HANA_CONSTANT_CHECK</a>(sorted == hana::tuple_t&lt;<span class="keywordtype">char</span>[1], <span class="keywordtype">char</span>[2], <span class="keywordtype">char</span>[3], <span class="keywordtype">char</span>[15]&gt;);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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