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Leon Vatthauer 2bf4655eae
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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>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Some theory for Semigroup</a>
<a id="135" class="Comment">------------------------------------------------------------------------</a>
<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>
<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#4139" class="Record">Semigroup</a><a id="290" class="Symbol">)</a>
<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#4139" 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>
<a id="362" class="Keyword">open</a> <a id="367" href="Algebra.Bundles.html#4139" class="Module">Semigroup</a> <a id="377" href="Algebra.Properties.Semigroup.html#336" class="Bound">S</a>
<a id="379" class="Keyword">open</a> <a id="384" class="Keyword">import</a> <a id="391" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="411" href="Algebra.Bundles.html#4245" class="Field Operator">_≈_</a>
<a id="415" class="Keyword">open</a> <a id="420" class="Keyword">import</a> <a id="427" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="445" class="Keyword">using</a> <a id="451" class="Symbol">(</a><a id="452" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="455" class="Symbol">)</a>
<a id="x∙yz≈xy∙z"></a><a id="458" href="Algebra.Properties.Semigroup.html#458" class="Function">x∙yz≈xy∙z</a> <a id="468" class="Symbol">:</a> <a id="470" class="Symbol">∀</a> <a id="472" href="Algebra.Properties.Semigroup.html#472" class="Bound">x</a> <a id="474" href="Algebra.Properties.Semigroup.html#474" class="Bound">y</a> <a id="476" href="Algebra.Properties.Semigroup.html#476" class="Bound">z</a> <a id="478" class="Symbol">→</a> <a id="480" href="Algebra.Properties.Semigroup.html#472" class="Bound">x</a> <a id="482" href="Algebra.Bundles.html#4277" class="Field Operator">∙</a> <a id="484" class="Symbol">(</a><a id="485" href="Algebra.Properties.Semigroup.html#474" class="Bound">y</a> <a id="487" href="Algebra.Bundles.html#4277" class="Field Operator">∙</a> <a id="489" href="Algebra.Properties.Semigroup.html#476" class="Bound">z</a><a id="490" class="Symbol">)</a> <a id="492" href="Algebra.Bundles.html#4245" class="Field Operator">≈</a> <a id="494" class="Symbol">(</a><a id="495" href="Algebra.Properties.Semigroup.html#472" class="Bound">x</a> <a id="497" href="Algebra.Bundles.html#4277" class="Field Operator">∙</a> <a id="499" href="Algebra.Properties.Semigroup.html#474" class="Bound">y</a><a id="500" class="Symbol">)</a> <a id="502" href="Algebra.Bundles.html#4277" class="Field Operator">∙</a> <a id="504" href="Algebra.Properties.Semigroup.html#476" class="Bound">z</a>
<a id="506" href="Algebra.Properties.Semigroup.html#458" class="Function">x∙yz≈xy∙z</a> <a id="516" href="Algebra.Properties.Semigroup.html#516" class="Bound">x</a> <a id="518" href="Algebra.Properties.Semigroup.html#518" class="Bound">y</a> <a id="520" href="Algebra.Properties.Semigroup.html#520" class="Bound">z</a> <a id="522" class="Symbol">=</a> <a id="524" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="528" class="Symbol">(</a><a id="529" href="Algebra.Structures.html#2977" class="Function">assoc</a> <a id="535" href="Algebra.Properties.Semigroup.html#516" class="Bound">x</a> <a id="537" href="Algebra.Properties.Semigroup.html#518" class="Bound">y</a> <a id="539" href="Algebra.Properties.Semigroup.html#520" class="Bound">z</a><a id="540" class="Symbol">)</a>
<a id="alternativeˡ"></a><a id="543" href="Algebra.Properties.Semigroup.html#543" class="Function">alternativeˡ</a> <a id="556" class="Symbol">:</a> <a id="558" href="Algebra.Definitions.html#6357" class="Function">LeftAlternative</a> <a id="574" href="Algebra.Bundles.html#4277" class="Field Operator">_∙_</a>
<a id="578" href="Algebra.Properties.Semigroup.html#543" class="Function">alternativeˡ</a> <a id="591" href="Algebra.Properties.Semigroup.html#591" class="Bound">x</a> <a id="593" href="Algebra.Properties.Semigroup.html#593" class="Bound">y</a> <a id="595" class="Symbol">=</a> <a id="597" href="Algebra.Structures.html#2977" class="Function">assoc</a> <a id="603" href="Algebra.Properties.Semigroup.html#591" class="Bound">x</a> <a id="605" href="Algebra.Properties.Semigroup.html#591" class="Bound">x</a> <a id="607" href="Algebra.Properties.Semigroup.html#593" class="Bound">y</a>
<a id="alternativeʳ"></a><a id="610" href="Algebra.Properties.Semigroup.html#610" class="Function">alternativeʳ</a> <a id="623" class="Symbol">:</a> <a id="625" href="Algebra.Definitions.html#6452" class="Function">RightAlternative</a> <a id="642" href="Algebra.Bundles.html#4277" class="Field Operator">_∙_</a>
<a id="646" href="Algebra.Properties.Semigroup.html#610" class="Function">alternativeʳ</a> <a id="659" href="Algebra.Properties.Semigroup.html#659" class="Bound">x</a> <a id="661" href="Algebra.Properties.Semigroup.html#661" class="Bound">y</a> <a id="663" class="Symbol">=</a> <a id="665" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="669" class="Symbol">(</a><a id="670" href="Algebra.Structures.html#2977" class="Function">assoc</a> <a id="676" href="Algebra.Properties.Semigroup.html#659" class="Bound">x</a> <a id="678" href="Algebra.Properties.Semigroup.html#661" class="Bound">y</a> <a id="680" href="Algebra.Properties.Semigroup.html#661" class="Bound">y</a><a id="681" class="Symbol">)</a>
<a id="alternative"></a><a id="684" href="Algebra.Properties.Semigroup.html#684" class="Function">alternative</a> <a id="696" class="Symbol">:</a> <a id="698" href="Algebra.Definitions.html#6547" class="Function">Alternative</a> <a id="710" href="Algebra.Bundles.html#4277" class="Field Operator">_∙_</a>
<a id="714" href="Algebra.Properties.Semigroup.html#684" class="Function">alternative</a> <a id="726" class="Symbol">=</a> <a id="728" href="Algebra.Properties.Semigroup.html#543" class="Function">alternativeˡ</a> <a id="741" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="743" href="Algebra.Properties.Semigroup.html#610" class="Function">alternativeʳ</a>
<a id="flexible"></a><a id="757" href="Algebra.Properties.Semigroup.html#757" class="Function">flexible</a> <a id="766" class="Symbol">:</a> <a id="768" href="Algebra.Definitions.html#6642" class="Function">Flexible</a> <a id="777" href="Algebra.Bundles.html#4277" class="Field Operator">_∙_</a>
<a id="781" href="Algebra.Properties.Semigroup.html#757" class="Function">flexible</a> <a id="790" href="Algebra.Properties.Semigroup.html#790" class="Bound">x</a> <a id="792" href="Algebra.Properties.Semigroup.html#792" class="Bound">y</a> <a id="794" class="Symbol">=</a> <a id="796" href="Algebra.Structures.html#2977" class="Function">assoc</a> <a id="802" href="Algebra.Properties.Semigroup.html#790" class="Bound">x</a> <a id="804" href="Algebra.Properties.Semigroup.html#792" class="Bound">y</a> <a id="806" href="Algebra.Properties.Semigroup.html#790" class="Bound">x</a>
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