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<html><head><meta charset="utf-8"><title>Algebra.Properties.Semigroup</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
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<a id="74" class="Comment">-- The Agda standard library</a>
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<a id="103" class="Comment">--</a>
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<a id="106" class="Comment">-- Some theory for Semigroup</a>
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<a id="135" class="Comment">------------------------------------------------------------------------</a>
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<a id="209" class="Symbol">{-#</a> <a id="213" class="Keyword">OPTIONS</a> <a id="221" class="Pragma">--cubical-compatible</a> <a id="242" class="Pragma">--safe</a> <a id="249" class="Symbol">#-}</a>
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<a id="254" class="Keyword">open</a> <a id="259" class="Keyword">import</a> <a id="266" href="Algebra.html" class="Module">Algebra</a> <a id="274" class="Keyword">using</a> <a id="280" class="Symbol">(</a><a id="281" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a><a id="290" class="Symbol">)</a>
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<a id="293" class="Keyword">module</a> <a id="300" href="Algebra.Properties.Semigroup.html" class="Module">Algebra.Properties.Semigroup</a> <a id="329" class="Symbol">{</a><a id="330" href="Algebra.Properties.Semigroup.html#330" class="Bound">a</a> <a id="332" href="Algebra.Properties.Semigroup.html#332" class="Bound">ℓ</a><a id="333" class="Symbol">}</a> <a id="335" class="Symbol">(</a><a id="336" href="Algebra.Properties.Semigroup.html#336" class="Bound">S</a> <a id="338" class="Symbol">:</a> <a id="340" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a> <a id="350" href="Algebra.Properties.Semigroup.html#330" class="Bound">a</a> <a id="352" href="Algebra.Properties.Semigroup.html#332" class="Bound">ℓ</a><a id="353" class="Symbol">)</a> <a id="355" class="Keyword">where</a>
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<a id="362" class="Keyword">open</a> <a id="367" href="Algebra.Bundles.html#2152" class="Module">Semigroup</a> <a id="377" href="Algebra.Properties.Semigroup.html#336" class="Bound">S</a>
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<a id="x∙yz≈xy∙z"></a><a id="380" href="Algebra.Properties.Semigroup.html#380" class="Function">x∙yz≈xy∙z</a> <a id="390" class="Symbol">:</a> <a id="392" class="Symbol">∀</a> <a id="394" href="Algebra.Properties.Semigroup.html#394" class="Bound">x</a> <a id="396" href="Algebra.Properties.Semigroup.html#396" class="Bound">y</a> <a id="398" href="Algebra.Properties.Semigroup.html#398" class="Bound">z</a> <a id="400" class="Symbol">→</a> <a id="402" href="Algebra.Properties.Semigroup.html#394" class="Bound">x</a> <a id="404" href="Algebra.Bundles.html#2290" class="Field Operator">∙</a> <a id="406" class="Symbol">(</a><a id="407" href="Algebra.Properties.Semigroup.html#396" class="Bound">y</a> <a id="409" href="Algebra.Bundles.html#2290" class="Field Operator">∙</a> <a id="411" href="Algebra.Properties.Semigroup.html#398" class="Bound">z</a><a id="412" class="Symbol">)</a> <a id="414" href="Algebra.Bundles.html#2258" class="Field Operator">≈</a> <a id="416" class="Symbol">(</a><a id="417" href="Algebra.Properties.Semigroup.html#394" class="Bound">x</a> <a id="419" href="Algebra.Bundles.html#2290" class="Field Operator">∙</a> <a id="421" href="Algebra.Properties.Semigroup.html#396" class="Bound">y</a><a id="422" class="Symbol">)</a> <a id="424" href="Algebra.Bundles.html#2290" class="Field Operator">∙</a> <a id="426" href="Algebra.Properties.Semigroup.html#398" class="Bound">z</a>
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<a id="428" href="Algebra.Properties.Semigroup.html#380" class="Function">x∙yz≈xy∙z</a> <a id="438" href="Algebra.Properties.Semigroup.html#438" class="Bound">x</a> <a id="440" href="Algebra.Properties.Semigroup.html#440" class="Bound">y</a> <a id="442" href="Algebra.Properties.Semigroup.html#442" class="Bound">z</a> <a id="444" class="Symbol">=</a> <a id="446" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="450" class="Symbol">(</a><a id="451" href="Algebra.Structures.html#1876" class="Function">assoc</a> <a id="457" href="Algebra.Properties.Semigroup.html#438" class="Bound">x</a> <a id="459" href="Algebra.Properties.Semigroup.html#440" class="Bound">y</a> <a id="461" href="Algebra.Properties.Semigroup.html#442" class="Bound">z</a><a id="462" class="Symbol">)</a>
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