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   &#160;<span id="projectnumber">1.6.0</span>
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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">Product<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
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<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>Represents types that are generic containers of two elements. </p>
<p>This concept basically represents types that are like <code><a class="el" href="structstd_1_1pair.html" title="Adaptation of std::pair for Hana.">std::pair</a></code>. The motivation for making such a precise concept is similar to the motivation behind the <code>Sequence</code> concept; there are many different implementations of <code><a class="el" href="structstd_1_1pair.html" title="Adaptation of std::pair for Hana.">std::pair</a></code> in different libraries, and we would like to manipulate any of them generically.</p>
<p>Since a <code>Product</code> is basically a pair, it is unsurprising that the operations provided by this concept are getting the first and second element of a pair, creating a pair from two elements and other simmilar operations.</p>
<dl class="section note"><dt>Note</dt><dd>Mathematically, this concept represents types that are category theoretical <a href="http://en.wikipedia.org/wiki/Product_(category_theory)">products</a>. This is also where the name comes from.</dd></dl>
<h2>Minimal complete definition </h2>
<p><code>first</code>, <code>second</code> and <code>make</code></p>
<p><code>first</code> and <code>second</code> must obviously return the first and the second element of the pair, respectively. <code>make</code> must take two arguments <code>x</code> and <code>y</code> representing the first and the second element of the pair, and return a pair <code>p</code> such that <code>first(p) == x</code> and <code>second(p) == y</code>. </p><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="first_8hpp.html">boost/hana/first.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="pair_8hpp.html">boost/hana/pair.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="second_8hpp.html">boost/hana/second.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-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4">hana::first</a>(hana::make&lt;hana::pair_tag&gt;(1, <span class="charliteral">&#39;x&#39;</span>)) == 1, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(<a class="code" href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077">hana::second</a>(hana::make&lt;hana::pair_tag&gt;(1, <span class="charliteral">&#39;x&#39;</span>)) == <span class="charliteral">&#39;x&#39;</span>, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --><h2>Laws </h2>
<p>For a model <code>P</code> of <code>Product</code>, the following laws must be satisfied. For every data types <code>X</code> and <code>Y</code>, there must be a unique function \( \mathtt{make} : X \times Y \to P \) such that for every <code>x</code>, <code>y</code>, </p><div class="fragment"><div class="line">x == <a class="code" href="group__group-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4">first</a>(make&lt;P&gt;(x, y))</div><div class="line">y == <a class="code" href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077">second</a>(make&lt;P&gt;(x, y))</div></div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>This law is less general than the universal property typically used to define category theoretical products, but it is vastly enough for what we need.</dd></dl>
<p>This is basically saying that a <code>Product</code> must be the most general object able to contain a pair of objects <code>(P1, P2)</code>, but nothing more. Since the categorical product is defined by a universal property, all the models of this concept are isomorphic, and the isomorphism is unique. In other words, there is one and only one way to convert one <code>Product</code> to another.</p>
<p>Another property that must be satisfied by <code>first</code> and <code>second</code> is that of <a class="el" href="group__group-Struct.html#move-independence">move-independence</a>, which ensures that we can optimally decompose a <code>Product</code> into its two members without making redundant copies.</p>
<h2>Refined concepts </h2>
<ol type="1">
<li><code>Comparable</code> (free model)<br />
Two products <code>x</code> and <code>y</code> are equal iff they are equal element-wise, by comparing the first element before the second element. <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="not__equal_8hpp.html">boost/hana/not_equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="pair_8hpp.html">boost/hana/pair.hpp</a>&gt;</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;string&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"><span class="keyword">using namespace </span>std::literals;</div><div class="line"></div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() {</div><div class="line">    <a class="code" href="group__group-assertions.html#ga4796ae107d58b67e0bbccd5ae6f70101">BOOST_HANA_RUNTIME_CHECK</a>(hana::make_pair(1, <span class="stringliteral">&quot;234&quot;</span>s) == hana::make_pair(1ll, <span class="stringliteral">&quot;234&quot;</span>s));</div><div class="line">    static_assert(hana::make_pair(<span class="charliteral">&#39;x&#39;</span>, 2) != hana::make_pair(<span class="charliteral">&#39;y&#39;</span>, 2), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">}</div></div><!-- fragment --></li>
<li><code>Orderable</code> (free model)<br />
Products are ordered using a lexicographical ordering as-if they were 2-element tuples.</li>
<li><code>Foldable</code> (free model)<br />
Folding a <code>Product</code> <code>p</code> is equivalent to folding a list containing <code>first(p)</code> and <code>second(p)</code>, in that order.</li>
</ol>
<h2>Concrete models </h2>
<p><code><a class="el" href="structboost_1_1hana_1_1pair.html" title="Generic container for two elements.">hana::pair</a></code> </p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="var-members"></a>
Variables</h2></td></tr>
<tr class="memitem:ga34bbf4281de06dc3540441e8b2bd24f4"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4">boost::hana::first</a></td></tr>
<tr class="memdesc:ga34bbf4281de06dc3540441e8b2bd24f4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns the first element of a pair.Note that if the <code>Product</code> actually stores the elements it contains, <code><a class="el" href="group__group-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4" title="Returns the first element of a pair.Note that if the Product actually stores the elements it contains...">hana::first</a></code> is required to return a lvalue reference, a lvalue reference to const or a rvalue reference to the first element, where the type of reference must match that of the pair passed to <code>first</code>. If the <code>Product</code> does not store the elements it contains (i.e. it generates them on demand), this requirement is dropped.  <a href="group__group-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4">More...</a><br /></td></tr>
<tr class="separator:ga34bbf4281de06dc3540441e8b2bd24f4"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga7bb979d59ffc3ab862cb7d9dc7730077"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077">boost::hana::second</a></td></tr>
<tr class="memdesc:ga7bb979d59ffc3ab862cb7d9dc7730077"><td class="mdescLeft">&#160;</td><td class="mdescRight">Returns the second element of a pair.Note that if the <code>Product</code> actually stores the elements it contains, <code><a class="el" href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077" title="Returns the second element of a pair.Note that if the Product actually stores the elements it contain...">hana::second</a></code> is required to return a lvalue reference, a lvalue reference to const or a rvalue reference to the second element, where the type of reference must match that of the pair passed to <code>second</code>. If the <code>Product</code> does not store the elements it contains (i.e. it generates them on demand), this requirement is dropped.  <a href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077">More...</a><br /></td></tr>
<tr class="separator:ga7bb979d59ffc3ab862cb7d9dc7730077"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
<a id="ga34bbf4281de06dc3540441e8b2bd24f4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga34bbf4281de06dc3540441e8b2bd24f4">&#9670;&nbsp;</a></span>first</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2first_8hpp.html">boost/hana/fwd/first.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; <a class="code" href="group__group-Foldable.html#ga17fe9c1982c882807f3358b4138c5744">product</a>) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div><div class="ttc" id="group__group-Foldable_html_ga17fe9c1982c882807f3358b4138c5744"><div class="ttname"><a href="group__group-Foldable.html#ga17fe9c1982c882807f3358b4138c5744">boost::hana::product</a></div><div class="ttdeci">constexpr auto product</div><div class="ttdoc">Compute the product of the numbers of a structure.More generally, product will take any foldable stru...</div><div class="ttdef"><b>Definition:</b> product.hpp:57</div></div>
</div><!-- fragment -->
<p>Returns the first element of a pair.Note that if the <code>Product</code> actually stores the elements it contains, <code><a class="el" href="group__group-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4" title="Returns the first element of a pair.Note that if the Product actually stores the elements it contains...">hana::first</a></code> is required to return a lvalue reference, a lvalue reference to const or a rvalue reference to the first element, where the type of reference must match that of the pair passed to <code>first</code>. If the <code>Product</code> does not store the elements it contains (i.e. it generates them on demand), this requirement is dropped. </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="first_8hpp.html">boost/hana/first.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="pair_8hpp.html">boost/hana/pair.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-Product.html#ga34bbf4281de06dc3540441e8b2bd24f4">hana::first</a>(hana::make_pair(1, <span class="charliteral">&#39;x&#39;</span>)) == 1, <span class="stringliteral">&quot;&quot;</span>);</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="#ga7bb979d59ffc3ab862cb7d9dc7730077">&#9670;&nbsp;</a></span>second</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2second_8hpp.html">boost/hana/fwd/second.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; <a class="code" href="group__group-Foldable.html#ga17fe9c1982c882807f3358b4138c5744">product</a>) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div><div class="ttc" id="group__group-Foldable_html_ga17fe9c1982c882807f3358b4138c5744"><div class="ttname"><a href="group__group-Foldable.html#ga17fe9c1982c882807f3358b4138c5744">boost::hana::product</a></div><div class="ttdeci">constexpr auto product</div><div class="ttdoc">Compute the product of the numbers of a structure.More generally, product will take any foldable stru...</div><div class="ttdef"><b>Definition:</b> product.hpp:57</div></div>
</div><!-- fragment -->
<p>Returns the second element of a pair.Note that if the <code>Product</code> actually stores the elements it contains, <code><a class="el" href="group__group-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077" title="Returns the second element of a pair.Note that if the Product actually stores the elements it contain...">hana::second</a></code> is required to return a lvalue reference, a lvalue reference to const or a rvalue reference to the second element, where the type of reference must match that of the pair passed to <code>second</code>. If the <code>Product</code> does not store the elements it contains (i.e. it generates them on demand), this requirement is dropped. </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="second_8hpp.html">boost/hana/second.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="pair_8hpp.html">boost/hana/pair.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-Product.html#ga7bb979d59ffc3ab862cb7d9dc7730077">hana::second</a>(hana::make_pair(1, <span class="charliteral">&#39;x&#39;</span>)) == <span class="charliteral">&#39;x&#39;</span>, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment --> 
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