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add some agda stuff
2022-06-23 22:12:24 +02:00
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<html><head><meta charset="utf-8"><title>Algebra.Definitions</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">-- Properties of functions, such as associativity and commutativity</a>
<a id="174" class="Comment">------------------------------------------------------------------------</a>
<a id="248" class="Comment">-- The contents of this module should be accessed via `Algebra`, unless</a>
<a id="320" class="Comment">-- you want to parameterise it via the equality relation.</a>
<a id="379" class="Symbol">{-#</a> <a id="383" class="Keyword">OPTIONS</a> <a id="391" class="Pragma">--without-K</a> <a id="403" class="Pragma">--safe</a> <a id="410" class="Symbol">#-}</a>
<a id="415" class="Keyword">open</a> <a id="420" class="Keyword">import</a> <a id="427" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a>
<a id="448" class="Keyword">open</a> <a id="453" class="Keyword">import</a> <a id="460" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="477" class="Keyword">using</a> <a id="483" class="Symbol">(</a><a id="484" href="Relation.Nullary.html#656" class="Function Operator">¬_</a><a id="486" class="Symbol">)</a>
<a id="489" class="Keyword">module</a> <a id="496" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a>
<a id="518" class="Symbol">{</a><a id="519" href="Algebra.Definitions.html#519" class="Bound">a</a> <a id="521" href="Algebra.Definitions.html#521" class="Bound">ℓ</a><a id="522" class="Symbol">}</a> <a id="524" class="Symbol">{</a><a id="525" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="527" class="Symbol">:</a> <a id="529" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="533" href="Algebra.Definitions.html#519" class="Bound">a</a><a id="534" class="Symbol">}</a> <a id="538" class="Comment">-- The underlying set</a>
<a id="562" class="Symbol">(</a><a id="563" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="567" class="Symbol">:</a> <a id="569" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="573" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="575" href="Algebra.Definitions.html#521" class="Bound">ℓ</a><a id="576" class="Symbol">)</a> <a id="582" class="Comment">-- The underlying equality</a>
<a id="611" class="Keyword">where</a>
<a id="618" class="Keyword">open</a> <a id="623" class="Keyword">import</a> <a id="630" href="Algebra.Core.html" class="Module">Algebra.Core</a>
<a id="643" class="Keyword">open</a> <a id="648" class="Keyword">import</a> <a id="655" href="Data.Product.html" class="Module">Data.Product</a>
<a id="668" class="Keyword">open</a> <a id="673" class="Keyword">import</a> <a id="680" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
<a id="695" class="Comment">------------------------------------------------------------------------</a>
<a id="768" class="Comment">-- Properties of operations</a>
<a id="Congruent₁"></a><a id="797" href="Algebra.Definitions.html#797" class="Function">Congruent₁</a> <a id="808" class="Symbol">:</a> <a id="810" href="Algebra.Core.html#475" class="Function">Op₁</a> <a id="814" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="816" class="Symbol">→</a> <a id="818" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="822" class="Symbol">_</a>
<a id="824" href="Algebra.Definitions.html#797" class="Function">Congruent₁</a> <a id="835" href="Algebra.Definitions.html#835" class="Bound">f</a> <a id="837" class="Symbol">=</a> <a id="839" href="Algebra.Definitions.html#835" class="Bound">f</a> <a id="841" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="851" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="855" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="857" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a>
<a id="Congruent₂"></a><a id="862" href="Algebra.Definitions.html#862" class="Function">Congruent₂</a> <a id="873" class="Symbol">:</a> <a id="875" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="879" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="881" class="Symbol">→</a> <a id="883" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="887" class="Symbol">_</a>
<a id="889" href="Algebra.Definitions.html#862" class="Function">Congruent₂</a> <a id="900" href="Algebra.Definitions.html#900" class="Bound">∙</a> <a id="902" class="Symbol">=</a> <a id="904" href="Algebra.Definitions.html#900" class="Bound">∙</a> <a id="906" href="Relation.Binary.Core.html#1689" class="Function Operator">Preserves₂</a> <a id="917" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="921" href="Relation.Binary.Core.html#1689" class="Function Operator">⟶</a> <a id="923" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="927" href="Relation.Binary.Core.html#1689" class="Function Operator">⟶</a> <a id="929" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a>
<a id="LeftCongruent"></a><a id="934" href="Algebra.Definitions.html#934" class="Function">LeftCongruent</a> <a id="948" class="Symbol">:</a> <a id="950" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="954" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="956" class="Symbol">→</a> <a id="958" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="962" class="Symbol">_</a>
<a id="964" href="Algebra.Definitions.html#934" class="Function">LeftCongruent</a> <a id="978" href="Algebra.Definitions.html#978" class="Bound Operator">_∙_</a> <a id="982" class="Symbol">=</a> <a id="984" class="Symbol">∀</a> <a id="986" class="Symbol">{</a><a id="987" href="Algebra.Definitions.html#987" class="Bound">x</a><a id="988" class="Symbol">}</a> <a id="990" class="Symbol">→</a> <a id="992" class="Symbol">(</a><a id="993" href="Algebra.Definitions.html#987" class="Bound">x</a> <a id="995" href="Algebra.Definitions.html#978" class="Bound Operator">∙_</a><a id="997" class="Symbol">)</a> <a id="999" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="1009" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="1013" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="1015" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a>
<a id="RightCongruent"></a><a id="1020" href="Algebra.Definitions.html#1020" class="Function">RightCongruent</a> <a id="1035" class="Symbol">:</a> <a id="1037" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1041" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1043" class="Symbol">→</a> <a id="1045" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1049" class="Symbol">_</a>
<a id="1051" href="Algebra.Definitions.html#1020" class="Function">RightCongruent</a> <a id="1066" href="Algebra.Definitions.html#1066" class="Bound Operator">_∙_</a> <a id="1070" class="Symbol">=</a> <a id="1072" class="Symbol">∀</a> <a id="1074" class="Symbol">{</a><a id="1075" href="Algebra.Definitions.html#1075" class="Bound">x</a><a id="1076" class="Symbol">}</a> <a id="1078" class="Symbol">→</a> <a id="1080" class="Symbol">(</a><a id="1081" href="Algebra.Definitions.html#1066" class="Bound Operator">_∙</a> <a id="1084" href="Algebra.Definitions.html#1075" class="Bound">x</a><a id="1085" class="Symbol">)</a> <a id="1087" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="1097" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a> <a id="1101" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="1103" href="Algebra.Definitions.html#563" class="Bound Operator">_≈_</a>
<a id="Associative"></a><a id="1108" href="Algebra.Definitions.html#1108" class="Function">Associative</a> <a id="1120" class="Symbol">:</a> <a id="1122" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1126" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1128" class="Symbol">→</a> <a id="1130" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1134" class="Symbol">_</a>
<a id="1136" href="Algebra.Definitions.html#1108" class="Function">Associative</a> <a id="1148" href="Algebra.Definitions.html#1148" class="Bound Operator">_∙_</a> <a id="1152" class="Symbol">=</a> <a id="1154" class="Symbol">∀</a> <a id="1156" href="Algebra.Definitions.html#1156" class="Bound">x</a> <a id="1158" href="Algebra.Definitions.html#1158" class="Bound">y</a> <a id="1160" href="Algebra.Definitions.html#1160" class="Bound">z</a> <a id="1162" class="Symbol">→</a> <a id="1164" class="Symbol">((</a><a id="1166" href="Algebra.Definitions.html#1156" class="Bound">x</a> <a id="1168" href="Algebra.Definitions.html#1148" class="Bound Operator">∙</a> <a id="1170" href="Algebra.Definitions.html#1158" class="Bound">y</a><a id="1171" class="Symbol">)</a> <a id="1173" href="Algebra.Definitions.html#1148" class="Bound Operator">∙</a> <a id="1175" href="Algebra.Definitions.html#1160" class="Bound">z</a><a id="1176" class="Symbol">)</a> <a id="1178" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1180" class="Symbol">(</a><a id="1181" href="Algebra.Definitions.html#1156" class="Bound">x</a> <a id="1183" href="Algebra.Definitions.html#1148" class="Bound Operator">∙</a> <a id="1185" class="Symbol">(</a><a id="1186" href="Algebra.Definitions.html#1158" class="Bound">y</a> <a id="1188" href="Algebra.Definitions.html#1148" class="Bound Operator">∙</a> <a id="1190" href="Algebra.Definitions.html#1160" class="Bound">z</a><a id="1191" class="Symbol">))</a>
<a id="Commutative"></a><a id="1195" href="Algebra.Definitions.html#1195" class="Function">Commutative</a> <a id="1207" class="Symbol">:</a> <a id="1209" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1213" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1215" class="Symbol">→</a> <a id="1217" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1221" class="Symbol">_</a>
<a id="1223" href="Algebra.Definitions.html#1195" class="Function">Commutative</a> <a id="1235" href="Algebra.Definitions.html#1235" class="Bound Operator">_∙_</a> <a id="1239" class="Symbol">=</a> <a id="1241" class="Symbol">∀</a> <a id="1243" href="Algebra.Definitions.html#1243" class="Bound">x</a> <a id="1245" href="Algebra.Definitions.html#1245" class="Bound">y</a> <a id="1247" class="Symbol">→</a> <a id="1249" class="Symbol">(</a><a id="1250" href="Algebra.Definitions.html#1243" class="Bound">x</a> <a id="1252" href="Algebra.Definitions.html#1235" class="Bound Operator">∙</a> <a id="1254" href="Algebra.Definitions.html#1245" class="Bound">y</a><a id="1255" class="Symbol">)</a> <a id="1257" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1259" class="Symbol">(</a><a id="1260" href="Algebra.Definitions.html#1245" class="Bound">y</a> <a id="1262" href="Algebra.Definitions.html#1235" class="Bound Operator">∙</a> <a id="1264" href="Algebra.Definitions.html#1243" class="Bound">x</a><a id="1265" class="Symbol">)</a>
<a id="LeftIdentity"></a><a id="1268" href="Algebra.Definitions.html#1268" class="Function">LeftIdentity</a> <a id="1281" class="Symbol">:</a> <a id="1283" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1285" class="Symbol">→</a> <a id="1287" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1291" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1293" class="Symbol">→</a> <a id="1295" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1299" class="Symbol">_</a>
<a id="1301" href="Algebra.Definitions.html#1268" class="Function">LeftIdentity</a> <a id="1314" href="Algebra.Definitions.html#1314" class="Bound">e</a> <a id="1316" href="Algebra.Definitions.html#1316" class="Bound Operator">_∙_</a> <a id="1320" class="Symbol">=</a> <a id="1322" class="Symbol">∀</a> <a id="1324" href="Algebra.Definitions.html#1324" class="Bound">x</a> <a id="1326" class="Symbol">→</a> <a id="1328" class="Symbol">(</a><a id="1329" href="Algebra.Definitions.html#1314" class="Bound">e</a> <a id="1331" href="Algebra.Definitions.html#1316" class="Bound Operator">∙</a> <a id="1333" href="Algebra.Definitions.html#1324" class="Bound">x</a><a id="1334" class="Symbol">)</a> <a id="1336" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1338" href="Algebra.Definitions.html#1324" class="Bound">x</a>
<a id="RightIdentity"></a><a id="1341" href="Algebra.Definitions.html#1341" class="Function">RightIdentity</a> <a id="1355" class="Symbol">:</a> <a id="1357" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1359" class="Symbol">→</a> <a id="1361" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1365" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1367" class="Symbol">→</a> <a id="1369" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1373" class="Symbol">_</a>
<a id="1375" href="Algebra.Definitions.html#1341" class="Function">RightIdentity</a> <a id="1389" href="Algebra.Definitions.html#1389" class="Bound">e</a> <a id="1391" href="Algebra.Definitions.html#1391" class="Bound Operator">_∙_</a> <a id="1395" class="Symbol">=</a> <a id="1397" class="Symbol">∀</a> <a id="1399" href="Algebra.Definitions.html#1399" class="Bound">x</a> <a id="1401" class="Symbol">→</a> <a id="1403" class="Symbol">(</a><a id="1404" href="Algebra.Definitions.html#1399" class="Bound">x</a> <a id="1406" href="Algebra.Definitions.html#1391" class="Bound Operator">∙</a> <a id="1408" href="Algebra.Definitions.html#1389" class="Bound">e</a><a id="1409" class="Symbol">)</a> <a id="1411" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1413" href="Algebra.Definitions.html#1399" class="Bound">x</a>
<a id="Identity"></a><a id="1416" href="Algebra.Definitions.html#1416" class="Function">Identity</a> <a id="1425" class="Symbol">:</a> <a id="1427" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1429" class="Symbol">→</a> <a id="1431" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1435" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1437" class="Symbol">→</a> <a id="1439" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1443" class="Symbol">_</a>
<a id="1445" href="Algebra.Definitions.html#1416" class="Function">Identity</a> <a id="1454" href="Algebra.Definitions.html#1454" class="Bound">e</a> <a id="1456" href="Algebra.Definitions.html#1456" class="Bound">∙</a> <a id="1458" class="Symbol">=</a> <a id="1460" class="Symbol">(</a><a id="1461" href="Algebra.Definitions.html#1268" class="Function">LeftIdentity</a> <a id="1474" href="Algebra.Definitions.html#1454" class="Bound">e</a> <a id="1476" href="Algebra.Definitions.html#1456" class="Bound">∙</a><a id="1477" class="Symbol">)</a> <a id="1479" href="Data.Product.html#1167" class="Function Operator">×</a> <a id="1481" class="Symbol">(</a><a id="1482" href="Algebra.Definitions.html#1341" class="Function">RightIdentity</a> <a id="1496" href="Algebra.Definitions.html#1454" class="Bound">e</a> <a id="1498" href="Algebra.Definitions.html#1456" class="Bound">∙</a><a id="1499" class="Symbol">)</a>
<a id="LeftZero"></a><a id="1502" href="Algebra.Definitions.html#1502" class="Function">LeftZero</a> <a id="1511" class="Symbol">:</a> <a id="1513" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1515" class="Symbol">→</a> <a id="1517" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1521" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1523" class="Symbol">→</a> <a id="1525" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1529" class="Symbol">_</a>
<a id="1531" href="Algebra.Definitions.html#1502" class="Function">LeftZero</a> <a id="1540" href="Algebra.Definitions.html#1540" class="Bound">z</a> <a id="1542" href="Algebra.Definitions.html#1542" class="Bound Operator">_∙_</a> <a id="1546" class="Symbol">=</a> <a id="1548" class="Symbol">∀</a> <a id="1550" href="Algebra.Definitions.html#1550" class="Bound">x</a> <a id="1552" class="Symbol">→</a> <a id="1554" class="Symbol">(</a><a id="1555" href="Algebra.Definitions.html#1540" class="Bound">z</a> <a id="1557" href="Algebra.Definitions.html#1542" class="Bound Operator">∙</a> <a id="1559" href="Algebra.Definitions.html#1550" class="Bound">x</a><a id="1560" class="Symbol">)</a> <a id="1562" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1564" href="Algebra.Definitions.html#1540" class="Bound">z</a>
<a id="RightZero"></a><a id="1567" href="Algebra.Definitions.html#1567" class="Function">RightZero</a> <a id="1577" class="Symbol">:</a> <a id="1579" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1581" class="Symbol">→</a> <a id="1583" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1587" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1589" class="Symbol">→</a> <a id="1591" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1595" class="Symbol">_</a>
<a id="1597" href="Algebra.Definitions.html#1567" class="Function">RightZero</a> <a id="1607" href="Algebra.Definitions.html#1607" class="Bound">z</a> <a id="1609" href="Algebra.Definitions.html#1609" class="Bound Operator">_∙_</a> <a id="1613" class="Symbol">=</a> <a id="1615" class="Symbol">∀</a> <a id="1617" href="Algebra.Definitions.html#1617" class="Bound">x</a> <a id="1619" class="Symbol">→</a> <a id="1621" class="Symbol">(</a><a id="1622" href="Algebra.Definitions.html#1617" class="Bound">x</a> <a id="1624" href="Algebra.Definitions.html#1609" class="Bound Operator">∙</a> <a id="1626" href="Algebra.Definitions.html#1607" class="Bound">z</a><a id="1627" class="Symbol">)</a> <a id="1629" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="1631" href="Algebra.Definitions.html#1607" class="Bound">z</a>
<a id="Zero"></a><a id="1634" href="Algebra.Definitions.html#1634" class="Function">Zero</a> <a id="1639" class="Symbol">:</a> <a id="1641" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1643" class="Symbol">→</a> <a id="1645" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="1649" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="1651" class="Symbol">→</a> <a id="1653" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="1657" class="Symbol">_</a>
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<a id="Selective"></a><a id="2782" href="Algebra.Definitions.html#2782" class="Function">Selective</a> <a id="2792" class="Symbol">:</a> <a id="2794" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="2798" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="2800" class="Symbol">→</a> <a id="2802" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="2806" class="Symbol">_</a>
<a id="2808" href="Algebra.Definitions.html#2782" class="Function">Selective</a> <a id="2818" href="Algebra.Definitions.html#2818" class="Bound Operator">_∙_</a> <a id="2822" class="Symbol">=</a> <a id="2824" class="Symbol">∀</a> <a id="2826" href="Algebra.Definitions.html#2826" class="Bound">x</a> <a id="2828" href="Algebra.Definitions.html#2828" class="Bound">y</a> <a id="2830" class="Symbol">→</a> <a id="2832" class="Symbol">(</a><a id="2833" href="Algebra.Definitions.html#2826" class="Bound">x</a> <a id="2835" href="Algebra.Definitions.html#2818" class="Bound Operator">∙</a> <a id="2837" href="Algebra.Definitions.html#2828" class="Bound">y</a><a id="2838" class="Symbol">)</a> <a id="2840" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="2842" href="Algebra.Definitions.html#2826" class="Bound">x</a> <a id="2844" href="Data.Sum.Base.html#734" class="Datatype Operator">⊎</a> <a id="2846" class="Symbol">(</a><a id="2847" href="Algebra.Definitions.html#2826" class="Bound">x</a> <a id="2849" href="Algebra.Definitions.html#2818" class="Bound Operator">∙</a> <a id="2851" href="Algebra.Definitions.html#2828" class="Bound">y</a><a id="2852" class="Symbol">)</a> <a id="2854" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="2856" href="Algebra.Definitions.html#2828" class="Bound">y</a>
<a id="_Absorbs_"></a><a id="2859" href="Algebra.Definitions.html#2859" class="Function Operator">_Absorbs_</a> <a id="2869" class="Symbol">:</a> <a id="2871" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="2875" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="2877" class="Symbol">→</a> <a id="2879" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="2883" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="2885" class="Symbol">→</a> <a id="2887" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="2891" class="Symbol">_</a>
<a id="2893" href="Algebra.Definitions.html#2893" class="Bound Operator">_∙_</a> <a id="2897" href="Algebra.Definitions.html#2859" class="Function Operator">Absorbs</a> <a id="2905" href="Algebra.Definitions.html#2905" class="Bound Operator">_∘_</a> <a id="2909" class="Symbol">=</a> <a id="2911" class="Symbol">∀</a> <a id="2913" href="Algebra.Definitions.html#2913" class="Bound">x</a> <a id="2915" href="Algebra.Definitions.html#2915" class="Bound">y</a> <a id="2917" class="Symbol">→</a> <a id="2919" class="Symbol">(</a><a id="2920" href="Algebra.Definitions.html#2913" class="Bound">x</a> <a id="2922" href="Algebra.Definitions.html#2893" class="Bound Operator">∙</a> <a id="2924" class="Symbol">(</a><a id="2925" href="Algebra.Definitions.html#2913" class="Bound">x</a> <a id="2927" href="Algebra.Definitions.html#2905" class="Bound Operator">∘</a> <a id="2929" href="Algebra.Definitions.html#2915" class="Bound">y</a><a id="2930" class="Symbol">))</a> <a id="2933" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="2935" href="Algebra.Definitions.html#2913" class="Bound">x</a>
<a id="Absorptive"></a><a id="2938" href="Algebra.Definitions.html#2938" class="Function">Absorptive</a> <a id="2949" class="Symbol">:</a> <a id="2951" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="2955" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="2957" class="Symbol">→</a> <a id="2959" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="2963" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="2965" class="Symbol">→</a> <a id="2967" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="2971" class="Symbol">_</a>
<a id="2973" href="Algebra.Definitions.html#2938" class="Function">Absorptive</a> <a id="2984" href="Algebra.Definitions.html#2984" class="Bound">∙</a> <a id="2986" href="Algebra.Definitions.html#2986" class="Bound">∘</a> <a id="2988" class="Symbol">=</a> <a id="2990" class="Symbol">(</a><a id="2991" href="Algebra.Definitions.html#2984" class="Bound">∙</a> <a id="2993" href="Algebra.Definitions.html#2859" class="Function Operator">Absorbs</a> <a id="3001" href="Algebra.Definitions.html#2986" class="Bound">∘</a><a id="3002" class="Symbol">)</a> <a id="3004" href="Data.Product.html#1167" class="Function Operator">×</a> <a id="3006" class="Symbol">(</a><a id="3007" href="Algebra.Definitions.html#2986" class="Bound">∘</a> <a id="3009" href="Algebra.Definitions.html#2859" class="Function Operator">Absorbs</a> <a id="3017" href="Algebra.Definitions.html#2984" class="Bound">∙</a><a id="3018" class="Symbol">)</a>
<a id="Involutive"></a><a id="3021" href="Algebra.Definitions.html#3021" class="Function">Involutive</a> <a id="3032" class="Symbol">:</a> <a id="3034" href="Algebra.Core.html#475" class="Function">Op₁</a> <a id="3038" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="3040" class="Symbol">→</a> <a id="3042" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="3046" class="Symbol">_</a>
<a id="3048" href="Algebra.Definitions.html#3021" class="Function">Involutive</a> <a id="3059" href="Algebra.Definitions.html#3059" class="Bound">f</a> <a id="3061" class="Symbol">=</a> <a id="3063" class="Symbol">∀</a> <a id="3065" href="Algebra.Definitions.html#3065" class="Bound">x</a> <a id="3067" class="Symbol">→</a> <a id="3069" href="Algebra.Definitions.html#3059" class="Bound">f</a> <a id="3071" class="Symbol">(</a><a id="3072" href="Algebra.Definitions.html#3059" class="Bound">f</a> <a id="3074" href="Algebra.Definitions.html#3065" class="Bound">x</a><a id="3075" class="Symbol">)</a> <a id="3077" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="3079" href="Algebra.Definitions.html#3065" class="Bound">x</a>
<a id="LeftCancellative"></a><a id="3082" href="Algebra.Definitions.html#3082" class="Function">LeftCancellative</a> <a id="3099" class="Symbol">:</a> <a id="3101" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="3105" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="3107" class="Symbol">→</a> <a id="3109" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="3113" class="Symbol">_</a>
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<a id="RightCancellative"></a><a id="3177" href="Algebra.Definitions.html#3177" class="Function">RightCancellative</a> <a id="3195" class="Symbol">:</a> <a id="3197" href="Algebra.Core.html#518" class="Function">Op₂</a> <a id="3201" href="Algebra.Definitions.html#525" class="Bound">A</a> <a id="3203" class="Symbol">→</a> <a id="3205" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="3209" class="Symbol">_</a>
<a id="3211" href="Algebra.Definitions.html#3177" class="Function">RightCancellative</a> <a id="3229" href="Algebra.Definitions.html#3229" class="Bound Operator">_•_</a> <a id="3233" class="Symbol">=</a> <a id="3235" class="Symbol">∀</a> <a id="3237" class="Symbol">{</a><a id="3238" href="Algebra.Definitions.html#3238" class="Bound">x</a><a id="3239" class="Symbol">}</a> <a id="3241" href="Algebra.Definitions.html#3241" class="Bound">y</a> <a id="3243" href="Algebra.Definitions.html#3243" class="Bound">z</a> <a id="3245" class="Symbol">→</a> <a id="3247" class="Symbol">(</a><a id="3248" href="Algebra.Definitions.html#3241" class="Bound">y</a> <a id="3250" href="Algebra.Definitions.html#3229" class="Bound Operator">•</a> <a id="3252" href="Algebra.Definitions.html#3238" class="Bound">x</a><a id="3253" class="Symbol">)</a> <a id="3255" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="3257" class="Symbol">(</a><a id="3258" href="Algebra.Definitions.html#3243" class="Bound">z</a> <a id="3260" href="Algebra.Definitions.html#3229" class="Bound Operator">•</a> <a id="3262" href="Algebra.Definitions.html#3238" class="Bound">x</a><a id="3263" class="Symbol">)</a> <a id="3265" class="Symbol">→</a> <a id="3267" href="Algebra.Definitions.html#3241" class="Bound">y</a> <a id="3269" href="Algebra.Definitions.html#563" class="Bound Operator">≈</a> <a id="3271" href="Algebra.Definitions.html#3243" class="Bound">z</a>
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