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<div class="header">
  <div class="summary">
<a href="#var-members">Variables</a>  </div>
  <div class="headertitle">
<div class="title">Applicative<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
</div><!--header-->
<div class="contents">
<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>The <code>Applicative</code> concept represents <code>Functor</code>s with the ability to lift values and combine computations. </p>
<p>A <code>Functor</code> can only take a normal function and map it over a structure containing values to obtain a new structure containing values. Intuitively, an <code>Applicative</code> can also take a value and lift it into the structure. In addition, an <code>Applicative</code> can take a structure containing functions and apply it to a structure containing values to obtain a new structure containing values. By currying the function(s) inside the structure, it is then also possible to apply n-ary functions to n structures containing values.</p>
<dl class="section note"><dt>Note</dt><dd>This documentation does not go into much details about the nature of applicatives. However, the <a href="https://wiki.haskell.org/Typeclassopedia#Applicative">Typeclassopedia</a> is a nice Haskell-oriented resource where such information can be found.</dd></dl>
<h2>Minimal complete definition </h2>
<p><code>lift</code> and <code>ap</code> satisfying the laws below. An <code>Applicative</code> must also be a <code>Functor</code>.</p>
<h2>Laws </h2>
<p>Given an <code>Applicative</code> <code>F</code>, the following laws must be satisfied:</p><ol type="1">
<li>Identity<br />
For all objects <code>xs</code> of tag <code>F(A)</code>, <div class="fragment"><div class="line"><a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(<span class="keywordtype">id</span>), xs) == xs</div></div><!-- fragment --></li>
<li>Composition<br />
For all objects <code>xs</code> of tag <code>F(A)</code> and functions-in-an-applicative \( fs : F(B \to C) \), \( gs : F(A \to B) \), <div class="fragment"><div class="line"><a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(<a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(<a class="code" href="group__group-functional.html#ga3b16146e53efcdf9ecbb9a7b21f8cd0b">compose</a>), fs, gs), xs) == <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(fs, <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(gs, xs))</div></div><!-- fragment --></li>
<li>Homomorphism<br />
For all objects <code>x</code> of tag <code>A</code> and functions \( f : A \to B \), <div class="fragment"><div class="line"><a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(f), lift&lt;F&gt;(x)) == lift&lt;F&gt;(f(x))</div></div><!-- fragment --></li>
<li>Interchange<br />
For all objects <code>x</code> of tag <code>A</code> and functions-in-an-applicative \( fs : F(A \to B) \), <div class="fragment"><div class="line"><a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(fs, lift&lt;F&gt;(x)) == <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(<a class="code" href="group__group-functional.html#ga30027c383676084be151ef3c6cf2829f">apply</a>(-, x)), fs)</div></div><!-- fragment --> where <code>apply(-, x)</code> denotes the partial application of the <code>apply</code> function from the <a class="el" href="group__group-functional.html">Functional</a> module to the <code>x</code> argument.</li>
</ol>
<p>As a consequence of these laws, the model of <code>Functor</code> for <code>F</code> will satisfy the following for all objects <code>xs</code> of tag <code>F(A)</code> and functions \( f : A \to B \): </p><div class="fragment"><div class="line"><a class="code" href="group__group-Functor.html#ga5a8975f6e55375d5b6038a9c36ee63e7">transform</a>(xs, f) == <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(f), xs)</div></div><!-- fragment --><h2>Refined concept </h2>
<ol type="1">
<li><code>Functor</code> (free model)<br />
As a consequence of the laws, any <code>Applicative F</code> can be made a <code>Functor</code> by setting <div class="fragment"><div class="line"><a class="code" href="group__group-Functor.html#ga5a8975f6e55375d5b6038a9c36ee63e7">transform</a>(xs, f) = <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(lift&lt;F&gt;(f), xs)</div></div><!-- fragment --></li>
</ol>
<h2>Concrete models </h2>
<p><code><a class="el" href="structboost_1_1hana_1_1lazy.html" title="hana::lazy implements superficial laziness via a monadic interface.">hana::lazy</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_1tuple.html" title="General purpose index-based heterogeneous sequence with a fixed length.">hana::tuple</a></code></p>
<p><a class="anchor" id="applicative-transformation"></a></p><h2>Structure-preserving functions </h2>
<p>An <em>applicative transformation</em> is a function \( t : F(X) \to G(X) \) between two Applicatives <code>F</code> and <code>G</code>, where <code>X</code> can be any tag, and which preserves the operations of an Applicative. In other words, for all objects <code>x</code> of tag <code>X</code>, functions-in-an-applicative \( fs : F(X \to Y) \) and objects <code>xs</code> of tag <code>F(X)</code>, </p><div class="fragment"><div class="line">t(lift&lt;F&gt;(x)) == lift&lt;G&gt;(x)</div><div class="line">t(<a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(fs, xs)) == <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a>(t(fs), t(xs))</div></div><!-- fragment --> <table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="var-members"></a>
Variables</h2></td></tr>
<tr class="memitem:ga6176a32953dbbdee96e3ab3c73e4c890"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">boost::hana::ap</a></td></tr>
<tr class="memdesc:ga6176a32953dbbdee96e3ab3c73e4c890"><td class="mdescLeft">&#160;</td><td class="mdescRight">Lifted application.Specifically, <code>ap</code> applies a structure containing functions to a structure containing values, and returns a new structure containing values. The exact way in which the functions are applied to the values depends on the <code>Applicative</code>.  <a href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">More...</a><br /></td></tr>
<tr class="separator:ga6176a32953dbbdee96e3ab3c73e4c890"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga712038d7abbc7159f8792788f7cd0c73"><td class="memTemplParams" colspan="2">template&lt;typename A &gt; </td></tr>
<tr class="memitem:ga712038d7abbc7159f8792788f7cd0c73"><td class="memTemplItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__group-Applicative.html#ga712038d7abbc7159f8792788f7cd0c73">boost::hana::lift</a></td></tr>
<tr class="memdesc:ga712038d7abbc7159f8792788f7cd0c73"><td class="mdescLeft">&#160;</td><td class="mdescRight">Lift a value into an <code>Applicative</code> structure.<code>lift&lt;A&gt;</code> takes a normal value and embeds it into a structure whose shape is represented by the <code>A</code> <code>Applicative</code>. Note that the value may be a function, in which case the created structure may be <code>ap</code>plied to another <code>Applicative</code> structure containing values.  <a href="group__group-Applicative.html#ga712038d7abbc7159f8792788f7cd0c73">More...</a><br /></td></tr>
<tr class="separator:ga712038d7abbc7159f8792788f7cd0c73"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
<a id="ga6176a32953dbbdee96e3ab3c73e4c890"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ga6176a32953dbbdee96e3ab3c73e4c890">&#9670;&nbsp;</a></span>ap</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2ap_8hpp.html">boost/hana/fwd/ap.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; f, <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 -->
<p>Lifted application.Specifically, <code>ap</code> applies a structure containing functions to a structure containing values, and returns a new structure containing values. The exact way in which the functions are applied to the values depends on the <code>Applicative</code>. </p>
<p><code>ap</code> can be called with two arguments or more; the functions in the <code>f</code> structure are curried and then applied to the values in each <code>x...</code> structure using the binary form of <code>ap</code>. Note that this requires the number of <code>x...</code> must match the arity of the functions in the <code>f</code> structure. In other words, <code>ap(f, x1, ..., xN)</code> is equivalent to </p><div class="fragment"><div class="line">((<a class="code" href="group__group-functional.html#ga49ea872ade5ac8f6c10052c495302e89">curry</a>(f) <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a> x1) <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a> x2) ... <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">ap</a> xN</div></div><!-- fragment --><p> where <code>x ap y</code> is just <code>ap(x, y)</code> written in infix notation to emphasize the left associativity.</p>
<h2>Signature </h2>
<p>Given an Applicative <code>A</code>, the signature is \( \mathtt{ap} : A(T_1 \times \cdots \times T_n \to U) \times A(T_1) \times \cdots \times A(T_n) \to A(U) \).</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">f</td><td>A structure containing function(s).</td></tr>
    <tr><td class="paramname">x...</td><td>Structure(s) containing value(s) and on which <code>f</code> is applied. The number of structures must match the arity of the functions in the <code>f</code> structure.</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="ap_8hpp.html">boost/hana/ap.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="config_8hpp.html">boost/hana/config.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="optional_8hpp.html">boost/hana/optional.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"></div><div class="line"><span class="preprocessor">#include &lt;functional&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">    <span class="comment">// with tuples</span></div><div class="line">    static_assert(</div><div class="line">        <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">hana::ap</a>(hana::make_tuple(std::plus&lt;&gt;{}), hana::make_tuple(1, 2),</div><div class="line">                                                  hana::make_tuple(3, 4, 5))</div><div class="line">            ==</div><div class="line">        hana::make_tuple(</div><div class="line">            1 + 3,      1 + 4,      1 + 5,</div><div class="line">            2 + 3,      2 + 4,      2 + 5</div><div class="line">        )</div><div class="line">    , <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line">    <span class="comment">// with optional values</span></div><div class="line">    BOOST_HANA_CONSTEXPR_LAMBDA <span class="keyword">auto</span> multiply = [](<span class="keyword">auto</span> a, <span class="keyword">auto</span> b, <span class="keyword">auto</span> c) {</div><div class="line">        <span class="keywordflow">return</span> a * b * c;</div><div class="line">    };</div><div class="line"></div><div class="line">    <a class="code" href="group__group-assertions.html#gac7aafc41e4dcc7d1f1929fb00f010d2a">BOOST_HANA_CONSTEXPR_CHECK</a>(</div><div class="line">        <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">hana::ap</a>(hana::just(multiply), hana::just(1),</div><div class="line">                                       hana::just(2),</div><div class="line">                                       hana::just(3))</div><div class="line">            ==</div><div class="line">        hana::just(1 * 2 * 3)</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>(</div><div class="line">        <a class="code" href="group__group-Applicative.html#ga6176a32953dbbdee96e3ab3c73e4c890">hana::ap</a>(hana::just(multiply), hana::just(1),</div><div class="line">                                       hana::nothing,</div><div class="line">                                       hana::just(3))</div><div class="line">            ==</div><div class="line">        hana::nothing</div><div class="line">    );</div><div class="line">}</div></div><!-- fragment --><dl class="todo"><dt><b><a class="el" href="todo.html#_todo000004">Todo:</a></b></dt><dd>Consider giving access to all the arguments to the tag-dispatched implementation for performance purposes. </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#ga712038d7abbc7159f8792788f7cd0c73">&#9670;&nbsp;</a></span>lift</h2>

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<p><code>#include &lt;<a class="el" href="fwd_2lift_8hpp.html">boost/hana/fwd/lift.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; x) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Lift a value into an <code>Applicative</code> structure.<code>lift&lt;A&gt;</code> takes a normal value and embeds it into a structure whose shape is represented by the <code>A</code> <code>Applicative</code>. Note that the value may be a function, in which case the created structure may be <code>ap</code>plied to another <code>Applicative</code> structure containing values. </p>
<h2>Signature </h2>
<p>Given an Applicative <code>A</code>, the signature is \( \mathtt{lift}_A : T \to A(T) \).</p>
<dl class="tparams"><dt>Template Parameters</dt><dd>
  <table class="tparams">
    <tr><td class="paramname">A</td><td>A tag representing the <code>Applicative</code> into which the value is lifted.</td></tr>
  </table>
  </dd>
</dl>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>The value to lift into the applicative.</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="equal_8hpp.html">boost/hana/equal.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="lift_8hpp.html">boost/hana/lift.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="optional_8hpp.html">boost/hana/optional.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(hana::lift&lt;hana::tuple_tag&gt;(<span class="charliteral">&#39;x&#39;</span>) == hana::make_tuple(<span class="charliteral">&#39;x&#39;</span>), <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(hana::lift&lt;hana::optional_tag&gt;(<span class="charliteral">&#39;x&#39;</span>) == hana::just(<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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