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<title>src/Control/Category/Associative.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP #-}</span>
<a name="line-2"></a><span class='hs-cpp'>#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702</span>
<a name="line-3"></a><span class='hs-comment'>{-# LANGUAGE Trustworthy #-}</span>
<a name="line-4"></a><span class='hs-cpp'>#endif</span>
<a name="line-5"></a><span class='hs-comment'>{-# LANGUAGE MultiParamTypeClasses #-}</span>
<a name="line-6"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-7"></a><span class='hs-comment'>-- |</span>
<a name="line-8"></a><span class='hs-comment'>-- Module : Control.Category.Associative</span>
<a name="line-9"></a><span class='hs-comment'>-- Copyright : 2008 Edward Kmett</span>
<a name="line-10"></a><span class='hs-comment'>-- License : BSD</span>
<a name="line-11"></a><span class='hs-comment'>--</span>
<a name="line-12"></a><span class='hs-comment'>-- Maintainer : Edward Kmett <ekmett@gmail.com></span>
<a name="line-13"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-14"></a><span class='hs-comment'>-- Portability : portable</span>
<a name="line-15"></a><span class='hs-comment'>--</span>
<a name="line-16"></a><span class='hs-comment'>-- NB: this contradicts another common meaning for an 'Associative' 'Category', which is one</span>
<a name="line-17"></a><span class='hs-comment'>-- where the pentagonal condition does not hold, but for which there is an identity.</span>
<a name="line-18"></a><span class='hs-comment'>--</span>
<a name="line-19"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-20"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Category</span><span class='hs-varop'>.</span><span class='hs-conid'>Associative</span>
<a name="line-21"></a> <span class='hs-layout'>(</span> <span class='hs-conid'>Associative</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-22"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-23"></a>
<a name="line-24"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Categorical</span><span class='hs-varop'>.</span><span class='hs-conid'>Bifunctor</span>
<a name="line-25"></a>
<a name="line-26"></a><a name="Associative"></a><span class='hs-comment'>{- | A category with an associative bifunctor satisfying Mac Lane\'s pentagonal coherence identity law:
<a name="line-27"></a>
<a name="line-28"></a>> bimap id associate . associate . bimap associate id = associate . associate
<a name="line-29"></a>> bimap disassociate id . disassociate . bimap id disassociate = disassociate . disassociate
<a name="line-30"></a>-}</span>
<a name="line-31"></a><a name="Associative"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Bifunctor</span> <span class='hs-varid'>p</span> <span class='hs-varid'>k</span> <span class='hs-varid'>k</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Associative</span> <span class='hs-varid'>k</span> <span class='hs-varid'>p</span> <span class='hs-keyword'>where</span>
<a name="line-32"></a> <span class='hs-varid'>associate</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-33"></a> <span class='hs-varid'>disassociate</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>p</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span>
<a name="line-34"></a>
<a name="line-35"></a><span class='hs-comment'>{-- RULES
<a name="line-36"></a>"copentagonal coherence" first disassociate . disassociate . second disassociate = disassociate . disassociate
<a name="line-37"></a>"pentagonal coherence" second associate . associate . first associate = associate . associate
<a name="line-38"></a> --}</span>
<a name="line-39"></a>
<a name="line-40"></a><a name="instance%20Associative%20(-%3e)%20(,)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Associative</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-conid'>(,)</span> <span class='hs-keyword'>where</span>
<a name="line-41"></a> <span class='hs-varid'>associate</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span><span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-42"></a> <span class='hs-varid'>disassociate</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-layout'>(</span><span class='hs-varid'>b</span><span class='hs-layout'>,</span><span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span><span class='hs-varid'>c</span><span class='hs-layout'>)</span>
<a name="line-43"></a>
<a name="line-44"></a><a name="instance%20Associative%20(-%3e)%20Either"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Associative</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>-></span><span class='hs-layout'>)</span> <span class='hs-conid'>Either</span> <span class='hs-keyword'>where</span>
<a name="line-45"></a> <span class='hs-varid'>associate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>a</span>
<a name="line-46"></a> <span class='hs-varid'>associate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-47"></a> <span class='hs-varid'>associate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span>
<a name="line-48"></a> <span class='hs-varid'>disassociate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Left</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-49"></a> <span class='hs-varid'>disassociate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Left</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-50"></a> <span class='hs-varid'>disassociate</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Right</span> <span class='hs-varid'>c</span>
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