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 123 <div class="header">
 124   <div class="summary">
 125 <a href="#var-members">Variables</a>  </div>
 126   <div class="headertitle">
 127 <div class="title">Group<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
 128 </div><!--header-->
 129 <div class="contents">
 130 <a name="details" id="details"></a><h2 class="groupheader">Description</h2>
 131 <p>The <code>Group</code> concept represents <code>Monoid</code>s where all objects have an inverse w.r.t. the <code>Monoid</code>'s binary operation. </p>
 132 <p>A <a href="http://en.wikipedia.org/wiki/Group_(mathematics)">Group</a> is an algebraic structure built on top of a <code>Monoid</code> which adds the ability to invert the action of the <code>Monoid</code>'s binary operation on any element of the set. Specifically, a <code>Group</code> is a <code>Monoid</code> <code>(S, +)</code> such that every element <code>s</code> in <code>S</code> has an inverse (say <code>s'</code>) which is such that </p><div class="fragment"><div class="line">s + s<span class="stringliteral">&#39; == s&#39;</span> + s == identity of the Monoid</div></div><!-- fragment --><p>There are many examples of <code>Group</code>s, one of which would be the additive <code>Monoid</code> on integers, where the inverse of any integer <code>n</code> is the integer <code>-n</code>. The method names used here refer to exactly this model.</p>
 133 <h2>Minimal complete definitions </h2>
 134 <ol type="1">
 135 <li><code>minus</code><br />
 136 When <code>minus</code> is specified, the <code>negate</code> method is defaulted by setting <div class="fragment"><div class="line"><a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x) = <a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(zero&lt;G&gt;(), x)</div></div><!-- fragment --></li>
 137 <li><code>negate</code><br />
 138 When <code>negate</code> is specified, the <code>minus</code> method is defaulted by setting <div class="fragment"><div class="line"><a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(y))</div></div><!-- fragment --></li>
 139 </ol>
 140 <h2>Laws </h2>
 141 <p>For all objects <code>x</code> of a <code>Group</code> <code>G</code>, the following laws must be satisfied: </p><div class="fragment"><div class="line"><a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x)) == zero&lt;G&gt;() <span class="comment">// right inverse</span></div><div class="line"><a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(<a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x), x) == zero&lt;G&gt;() <span class="comment">// left inverse</span></div></div><!-- fragment --><h2>Refined concept </h2>
 142 <p><code>Monoid</code></p>
 143 <h2>Concrete models </h2>
 144 <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></p>
 145 <h2>Free model for non-boolean arithmetic data types </h2>
 146 <p>A data type <code>T</code> is arithmetic if <code>std::is_arithmetic&lt;T&gt;::value</code> is true. For a non-boolean arithmetic data type <code>T</code>, a model of <code>Group</code> is automatically defined by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = (x - y)</div><div class="line"><a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x) = -x</div></div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>The rationale for not providing a Group model for <code>bool</code> is the same as for not providing a <code>Monoid</code> model.</dd></dl>
 147 <h2>Structure-preserving functions </h2>
 148 <p>Let <code>A</code> and <code>B</code> be two <code>Group</code>s. A function <code>f : A -&gt; B</code> is said to be a <a href="http://en.wikipedia.org/wiki/Group_homomorphism">Group morphism</a> if it preserves the group structure between <code>A</code> and <code>B</code>. Rigorously, for all objects <code>x, y</code> of data type <code>A</code>, </p><div class="fragment"><div class="line">f(<a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, y)) == <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(f(x), f(y))</div></div><!-- fragment --><p> Because of the <code>Group</code> structure, it is easy to prove that the following will then also be satisfied: </p><div class="fragment"><div class="line">f(<a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(x)) == <a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(f(x))</div><div class="line">f(<a class="code" href="group__group-Monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">zero</a>&lt;A&gt;()) == <a class="code" href="group__group-Monoid.html#gad459ac17b6bab8ead1cae7de0032f3c6">zero</a>&lt;B&gt;()</div></div><!-- fragment --><p> Functions with these properties interact nicely with <code>Group</code>s, which is why they are given such a special treatment.</p>
 149 <table class="memberdecls">
 150 <tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="var-members"></a>
 151 Variables</h2></td></tr>
 152 <tr class="memitem:ga2020c526324f361a2b990fe8d1b07c20"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">boost::hana::minus</a></td></tr>
 153 <tr class="memdesc:ga2020c526324f361a2b990fe8d1b07c20"><td class="mdescLeft">&#160;</td><td class="mdescRight">Subtract two elements of a group.Specifically, this performs the <code>Monoid</code> operation on the first argument and on the inverse of the second argument, thus being equivalent to:  <a href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">More...</a><br /></td></tr>
 154 <tr class="separator:ga2020c526324f361a2b990fe8d1b07c20"><td class="memSeparator" colspan="2">&#160;</td></tr>
 155 <tr class="memitem:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">boost::hana::negate</a></td></tr>
 156 <tr class="memdesc:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="mdescLeft">&#160;</td><td class="mdescRight">Return the inverse of an element of a group.  <a href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">More...</a><br /></td></tr>
 157 <tr class="separator:ga02e81002f40ba52eac4cf1974c7e0cdb"><td class="memSeparator" colspan="2">&#160;</td></tr>
 158 </table>
 159 <h2 class="groupheader">Variable Documentation</h2>
 160 <a class="anchor" id="ga2020c526324f361a2b990fe8d1b07c20"></a>
 161 <div class="memitem">
 162 <div class="memproto">
 163       <table class="memname">
 164         <tr>
 165           <td class="memname">constexpr auto boost::hana::minus</td>
 166         </tr>
 167       </table>
 168 </div><div class="memdoc">
 169
 170 <p><code>#include &lt;<a class="el" href="fwd_2minus_8hpp.html">boost/hana/fwd/minus.hpp</a>&gt;</code></p>
 171 <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 -->
 172 <p>Subtract two elements of a group.Specifically, this performs the <code>Monoid</code> operation on the first argument and on the inverse of the second argument, thus being equivalent to: </p>
 173 <div class="fragment"><div class="line"><a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) == <a class="code" href="group__group-Monoid.html#gaeb5d4a1e967e319712f9e4791948896c">plus</a>(x, <a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">negate</a>(y))</div></div><!-- fragment --><h2>Cross-type version of the method </h2>
 174 <p>The <code>minus</code> method is "overloaded" to handle distinct data types with certain properties. Specifically, <code>minus</code> is defined for <em>distinct</em> data types <code>A</code> and <code>B</code> such that</p><ol type="1">
 175 <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>
 176 <li><code>A</code>, <code>B</code> and <code>C</code> are all <code>Group</code>s when taken individually</li>
 177 <li><code>to&lt;C&gt; : A -&gt; B</code> and <code>to&lt;C&gt; : B -&gt; C</code> are <code>Group</code>-embeddings, as determined by the <code>is_embedding</code> metafunction.</li>
 178 </ol>
 179 <p>The definition of <code>minus</code> for data types satisfying the above properties is obtained by setting </p><div class="fragment"><div class="line"><a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(x, y) = <a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">minus</a>(to&lt;C&gt;(x), to&lt;C&gt;(y))</div></div><!-- fragment --><h2>Example </h2>
 180 <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="minus_8hpp.html">boost/hana/minus.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"><span class="keywordtype">int</span> main() {</div><div class="line">    <a class="code" href="group__group-assertions.html#ga4bf9e0c46c44e21fbe5c5fbb3ace8356">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">hana::minus</a>(hana::int_c&lt;3&gt;, hana::int_c&lt;5&gt;) == hana::int_c&lt;-2&gt;);</div><div class="line">    static_assert(<a class="code" href="group__group-Group.html#ga2020c526324f361a2b990fe8d1b07c20">hana::minus</a>(1, 2) == -1, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">}</div></div><!-- fragment -->
 181 </div>
 182 </div>
 183 <a class="anchor" id="ga02e81002f40ba52eac4cf1974c7e0cdb"></a>
 184 <div class="memitem">
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 186       <table class="memname">
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 188           <td class="memname">constexpr auto boost::hana::negate</td>
 189         </tr>
 190       </table>
 191 </div><div class="memdoc">
 192
 193 <p><code>#include &lt;<a class="el" href="fwd_2negate_8hpp.html">boost/hana/fwd/negate.hpp</a>&gt;</code></p>
 194 <b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x) -&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 -->
 195 <p>Return the inverse of an element of a group. </p>
 196 <h2>Example </h2>
 197 <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="negate_8hpp.html">boost/hana/negate.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"><span class="keywordtype">int</span> main() {</div><div class="line">    <a class="code" href="group__group-assertions.html#ga4bf9e0c46c44e21fbe5c5fbb3ace8356">BOOST_HANA_CONSTANT_CHECK</a>(<a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">hana::negate</a>(hana::int_c&lt;3&gt;) == hana::int_c&lt;-3&gt;);</div><div class="line">    static_assert(<a class="code" href="group__group-Group.html#ga02e81002f40ba52eac4cf1974c7e0cdb">hana::negate</a>(2) == -2, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">}</div></div><!-- fragment -->
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