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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">--cubical-compatible</a> <a id="412" class="Pragma">--safe</a> <a id="419" class="Symbol">#-}</a>
<a id="424" class="Keyword">open</a> <a id="429" class="Keyword">import</a> <a id="436" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a>
<a id="457" class="Keyword">open</a> <a id="462" class="Keyword">import</a> <a id="469" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="486" class="Keyword">using</a> <a id="492" class="Symbol">(</a><a id="493" href="Relation.Nullary.html#665" class="Function Operator">¬_</a><a id="495" class="Symbol">)</a>
<a id="498" class="Keyword">module</a> <a id="505" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a>
<a id="527" class="Symbol">{</a><a id="528" href="Algebra.Definitions.html#528" class="Bound">a</a> <a id="530" href="Algebra.Definitions.html#530" class="Bound">ℓ</a><a id="531" class="Symbol">}</a> <a id="533" class="Symbol">{</a><a id="534" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="536" class="Symbol">:</a> <a id="538" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="542" href="Algebra.Definitions.html#528" class="Bound">a</a><a id="543" class="Symbol">}</a> <a id="547" class="Comment">-- The underlying set</a>
<a id="571" class="Symbol">(</a><a id="572" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="576" class="Symbol">:</a> <a id="578" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="582" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="584" href="Algebra.Definitions.html#530" class="Bound">ℓ</a><a id="585" class="Symbol">)</a> <a id="591" class="Comment">-- The underlying equality</a>
<a id="620" class="Keyword">where</a>
<a id="627" class="Keyword">open</a> <a id="632" class="Keyword">import</a> <a id="639" href="Algebra.Core.html" class="Module">Algebra.Core</a>
<a id="652" class="Keyword">open</a> <a id="657" class="Keyword">import</a> <a id="664" href="Data.Product.html" class="Module">Data.Product</a>
<a id="677" class="Keyword">open</a> <a id="682" class="Keyword">import</a> <a id="689" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
<a id="704" class="Comment">------------------------------------------------------------------------</a>
<a id="777" class="Comment">-- Properties of operations</a>
<a id="Congruent₁"></a><a id="806" href="Algebra.Definitions.html#806" class="Function">Congruent₁</a> <a id="817" class="Symbol">:</a> <a id="819" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="823" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="825" class="Symbol">→</a> <a id="827" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="831" class="Symbol">_</a>
<a id="833" href="Algebra.Definitions.html#806" class="Function">Congruent₁</a> <a id="844" href="Algebra.Definitions.html#844" class="Bound">f</a> <a id="846" class="Symbol">=</a> <a id="848" href="Algebra.Definitions.html#844" class="Bound">f</a> <a id="850" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="860" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="864" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="866" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a>
<a id="Congruent₂"></a><a id="871" href="Algebra.Definitions.html#871" class="Function">Congruent₂</a> <a id="882" class="Symbol">:</a> <a id="884" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="888" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="890" class="Symbol">→</a> <a id="892" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="896" class="Symbol">_</a>
<a id="898" href="Algebra.Definitions.html#871" class="Function">Congruent₂</a> <a id="909" href="Algebra.Definitions.html#909" class="Bound">∙</a> <a id="911" class="Symbol">=</a> <a id="913" href="Algebra.Definitions.html#909" class="Bound">∙</a> <a id="915" href="Relation.Binary.Core.html#1698" class="Function Operator">Preserves₂</a> <a id="926" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="930" href="Relation.Binary.Core.html#1698" class="Function Operator">⟶</a> <a id="932" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="936" href="Relation.Binary.Core.html#1698" class="Function Operator">⟶</a> <a id="938" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a>
<a id="LeftCongruent"></a><a id="943" href="Algebra.Definitions.html#943" class="Function">LeftCongruent</a> <a id="957" class="Symbol">:</a> <a id="959" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="963" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="965" class="Symbol">→</a> <a id="967" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="971" class="Symbol">_</a>
<a id="973" href="Algebra.Definitions.html#943" class="Function">LeftCongruent</a> <a id="987" href="Algebra.Definitions.html#987" class="Bound Operator">_∙_</a> <a id="991" class="Symbol">=</a> <a id="993" class="Symbol">∀</a> <a id="995" class="Symbol">{</a><a id="996" href="Algebra.Definitions.html#996" class="Bound">x</a><a id="997" class="Symbol">}</a> <a id="999" class="Symbol">→</a> <a id="1001" class="Symbol">(</a><a id="1002" href="Algebra.Definitions.html#996" class="Bound">x</a> <a id="1004" href="Algebra.Definitions.html#987" class="Bound Operator">∙_</a><a id="1006" class="Symbol">)</a> <a id="1008" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="1018" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="1022" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="1024" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a>
<a id="RightCongruent"></a><a id="1029" href="Algebra.Definitions.html#1029" class="Function">RightCongruent</a> <a id="1044" class="Symbol">:</a> <a id="1046" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1050" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1052" class="Symbol">→</a> <a id="1054" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1058" class="Symbol">_</a>
<a id="1060" href="Algebra.Definitions.html#1029" class="Function">RightCongruent</a> <a id="1075" href="Algebra.Definitions.html#1075" class="Bound Operator">_∙_</a> <a id="1079" class="Symbol">=</a> <a id="1081" class="Symbol">∀</a> <a id="1083" class="Symbol">{</a><a id="1084" href="Algebra.Definitions.html#1084" class="Bound">x</a><a id="1085" class="Symbol">}</a> <a id="1087" class="Symbol">→</a> <a id="1089" class="Symbol">(</a><a id="1090" href="Algebra.Definitions.html#1075" class="Bound Operator">_∙</a> <a id="1093" href="Algebra.Definitions.html#1084" class="Bound">x</a><a id="1094" class="Symbol">)</a> <a id="1096" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="1106" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a> <a id="1110" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="1112" href="Algebra.Definitions.html#572" class="Bound Operator">_≈_</a>
<a id="Associative"></a><a id="1117" href="Algebra.Definitions.html#1117" class="Function">Associative</a> <a id="1129" class="Symbol">:</a> <a id="1131" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1135" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1137" class="Symbol">→</a> <a id="1139" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1143" class="Symbol">_</a>
<a id="1145" href="Algebra.Definitions.html#1117" class="Function">Associative</a> <a id="1157" href="Algebra.Definitions.html#1157" class="Bound Operator">_∙_</a> <a id="1161" class="Symbol">=</a> <a id="1163" class="Symbol">∀</a> <a id="1165" href="Algebra.Definitions.html#1165" class="Bound">x</a> <a id="1167" href="Algebra.Definitions.html#1167" class="Bound">y</a> <a id="1169" href="Algebra.Definitions.html#1169" class="Bound">z</a> <a id="1171" class="Symbol">→</a> <a id="1173" class="Symbol">((</a><a id="1175" href="Algebra.Definitions.html#1165" class="Bound">x</a> <a id="1177" href="Algebra.Definitions.html#1157" class="Bound Operator">∙</a> <a id="1179" href="Algebra.Definitions.html#1167" class="Bound">y</a><a id="1180" class="Symbol">)</a> <a id="1182" href="Algebra.Definitions.html#1157" class="Bound Operator">∙</a> <a id="1184" href="Algebra.Definitions.html#1169" class="Bound">z</a><a id="1185" class="Symbol">)</a> <a id="1187" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1189" class="Symbol">(</a><a id="1190" href="Algebra.Definitions.html#1165" class="Bound">x</a> <a id="1192" href="Algebra.Definitions.html#1157" class="Bound Operator">∙</a> <a id="1194" class="Symbol">(</a><a id="1195" href="Algebra.Definitions.html#1167" class="Bound">y</a> <a id="1197" href="Algebra.Definitions.html#1157" class="Bound Operator">∙</a> <a id="1199" href="Algebra.Definitions.html#1169" class="Bound">z</a><a id="1200" class="Symbol">))</a>
<a id="Commutative"></a><a id="1204" href="Algebra.Definitions.html#1204" class="Function">Commutative</a> <a id="1216" class="Symbol">:</a> <a id="1218" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1222" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1224" class="Symbol">→</a> <a id="1226" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1230" class="Symbol">_</a>
<a id="1232" href="Algebra.Definitions.html#1204" class="Function">Commutative</a> <a id="1244" href="Algebra.Definitions.html#1244" class="Bound Operator">_∙_</a> <a id="1248" class="Symbol">=</a> <a id="1250" class="Symbol">∀</a> <a id="1252" href="Algebra.Definitions.html#1252" class="Bound">x</a> <a id="1254" href="Algebra.Definitions.html#1254" class="Bound">y</a> <a id="1256" class="Symbol">→</a> <a id="1258" class="Symbol">(</a><a id="1259" href="Algebra.Definitions.html#1252" class="Bound">x</a> <a id="1261" href="Algebra.Definitions.html#1244" class="Bound Operator">∙</a> <a id="1263" href="Algebra.Definitions.html#1254" class="Bound">y</a><a id="1264" class="Symbol">)</a> <a id="1266" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1268" class="Symbol">(</a><a id="1269" href="Algebra.Definitions.html#1254" class="Bound">y</a> <a id="1271" href="Algebra.Definitions.html#1244" class="Bound Operator">∙</a> <a id="1273" href="Algebra.Definitions.html#1252" class="Bound">x</a><a id="1274" class="Symbol">)</a>
<a id="LeftIdentity"></a><a id="1277" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="1290" class="Symbol">:</a> <a id="1292" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1294" class="Symbol">→</a> <a id="1296" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1300" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1302" class="Symbol">→</a> <a id="1304" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1308" class="Symbol">_</a>
<a id="1310" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="1323" href="Algebra.Definitions.html#1323" class="Bound">e</a> <a id="1325" href="Algebra.Definitions.html#1325" class="Bound Operator">_∙_</a> <a id="1329" class="Symbol">=</a> <a id="1331" class="Symbol">∀</a> <a id="1333" href="Algebra.Definitions.html#1333" class="Bound">x</a> <a id="1335" class="Symbol">→</a> <a id="1337" class="Symbol">(</a><a id="1338" href="Algebra.Definitions.html#1323" class="Bound">e</a> <a id="1340" href="Algebra.Definitions.html#1325" class="Bound Operator">∙</a> <a id="1342" href="Algebra.Definitions.html#1333" class="Bound">x</a><a id="1343" class="Symbol">)</a> <a id="1345" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1347" href="Algebra.Definitions.html#1333" class="Bound">x</a>
<a id="RightIdentity"></a><a id="1350" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="1364" class="Symbol">:</a> <a id="1366" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1368" class="Symbol">→</a> <a id="1370" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1374" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1376" class="Symbol">→</a> <a id="1378" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1382" class="Symbol">_</a>
<a id="1384" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="1398" href="Algebra.Definitions.html#1398" class="Bound">e</a> <a id="1400" href="Algebra.Definitions.html#1400" class="Bound Operator">_∙_</a> <a id="1404" class="Symbol">=</a> <a id="1406" class="Symbol">∀</a> <a id="1408" href="Algebra.Definitions.html#1408" class="Bound">x</a> <a id="1410" class="Symbol">→</a> <a id="1412" class="Symbol">(</a><a id="1413" href="Algebra.Definitions.html#1408" class="Bound">x</a> <a id="1415" href="Algebra.Definitions.html#1400" class="Bound Operator">∙</a> <a id="1417" href="Algebra.Definitions.html#1398" class="Bound">e</a><a id="1418" class="Symbol">)</a> <a id="1420" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1422" href="Algebra.Definitions.html#1408" class="Bound">x</a>
<a id="Identity"></a><a id="1425" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="1434" class="Symbol">:</a> <a id="1436" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1438" class="Symbol">→</a> <a id="1440" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1444" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1446" class="Symbol">→</a> <a id="1448" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1452" class="Symbol">_</a>
<a id="1454" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="1463" href="Algebra.Definitions.html#1463" class="Bound">e</a> <a id="1465" href="Algebra.Definitions.html#1465" class="Bound">∙</a> <a id="1467" class="Symbol">=</a> <a id="1469" class="Symbol">(</a><a id="1470" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="1483" href="Algebra.Definitions.html#1463" class="Bound">e</a> <a id="1485" href="Algebra.Definitions.html#1465" class="Bound">∙</a><a id="1486" class="Symbol">)</a> <a id="1488" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="1490" class="Symbol">(</a><a id="1491" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="1505" href="Algebra.Definitions.html#1463" class="Bound">e</a> <a id="1507" href="Algebra.Definitions.html#1465" class="Bound">∙</a><a id="1508" class="Symbol">)</a>
<a id="LeftZero"></a><a id="1511" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="1520" class="Symbol">:</a> <a id="1522" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1524" class="Symbol">→</a> <a id="1526" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1530" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1532" class="Symbol">→</a> <a id="1534" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1538" class="Symbol">_</a>
<a id="1540" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="1549" href="Algebra.Definitions.html#1549" class="Bound">z</a> <a id="1551" href="Algebra.Definitions.html#1551" class="Bound Operator">_∙_</a> <a id="1555" class="Symbol">=</a> <a id="1557" class="Symbol">∀</a> <a id="1559" href="Algebra.Definitions.html#1559" class="Bound">x</a> <a id="1561" class="Symbol">→</a> <a id="1563" class="Symbol">(</a><a id="1564" href="Algebra.Definitions.html#1549" class="Bound">z</a> <a id="1566" href="Algebra.Definitions.html#1551" class="Bound Operator">∙</a> <a id="1568" href="Algebra.Definitions.html#1559" class="Bound">x</a><a id="1569" class="Symbol">)</a> <a id="1571" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1573" href="Algebra.Definitions.html#1549" class="Bound">z</a>
<a id="RightZero"></a><a id="1576" href="Algebra.Definitions.html#1576" class="Function">RightZero</a> <a id="1586" class="Symbol">:</a> <a id="1588" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1590" class="Symbol">→</a> <a id="1592" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1596" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1598" class="Symbol">→</a> <a id="1600" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1604" class="Symbol">_</a>
<a id="1606" href="Algebra.Definitions.html#1576" class="Function">RightZero</a> <a id="1616" href="Algebra.Definitions.html#1616" class="Bound">z</a> <a id="1618" href="Algebra.Definitions.html#1618" class="Bound Operator">_∙_</a> <a id="1622" class="Symbol">=</a> <a id="1624" class="Symbol">∀</a> <a id="1626" href="Algebra.Definitions.html#1626" class="Bound">x</a> <a id="1628" class="Symbol">→</a> <a id="1630" class="Symbol">(</a><a id="1631" href="Algebra.Definitions.html#1626" class="Bound">x</a> <a id="1633" href="Algebra.Definitions.html#1618" class="Bound Operator">∙</a> <a id="1635" href="Algebra.Definitions.html#1616" class="Bound">z</a><a id="1636" class="Symbol">)</a> <a id="1638" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="1640" href="Algebra.Definitions.html#1616" class="Bound">z</a>
<a id="Zero"></a><a id="1643" href="Algebra.Definitions.html#1643" class="Function">Zero</a> <a id="1648" class="Symbol">:</a> <a id="1650" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1652" class="Symbol">→</a> <a id="1654" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1658" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="1660" class="Symbol">→</a> <a id="1662" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1666" class="Symbol">_</a>
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<a id="Selective"></a><a id="2791" href="Algebra.Definitions.html#2791" class="Function">Selective</a> <a id="2801" class="Symbol">:</a> <a id="2803" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2807" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="2809" class="Symbol">→</a> <a id="2811" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2815" class="Symbol">_</a>
<a id="2817" href="Algebra.Definitions.html#2791" class="Function">Selective</a> <a id="2827" href="Algebra.Definitions.html#2827" class="Bound Operator">_∙_</a> <a id="2831" class="Symbol">=</a> <a id="2833" class="Symbol">∀</a> <a id="2835" href="Algebra.Definitions.html#2835" class="Bound">x</a> <a id="2837" href="Algebra.Definitions.html#2837" class="Bound">y</a> <a id="2839" class="Symbol">→</a> <a id="2841" class="Symbol">(</a><a id="2842" href="Algebra.Definitions.html#2835" class="Bound">x</a> <a id="2844" href="Algebra.Definitions.html#2827" class="Bound Operator">∙</a> <a id="2846" href="Algebra.Definitions.html#2837" class="Bound">y</a><a id="2847" class="Symbol">)</a> <a id="2849" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="2851" href="Algebra.Definitions.html#2835" class="Bound">x</a> <a id="2853" href="Data.Sum.Base.html#743" class="Datatype Operator">⊎</a> <a id="2855" class="Symbol">(</a><a id="2856" href="Algebra.Definitions.html#2835" class="Bound">x</a> <a id="2858" href="Algebra.Definitions.html#2827" class="Bound Operator">∙</a> <a id="2860" href="Algebra.Definitions.html#2837" class="Bound">y</a><a id="2861" class="Symbol">)</a> <a id="2863" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="2865" href="Algebra.Definitions.html#2837" class="Bound">y</a>
<a id="_Absorbs_"></a><a id="2868" href="Algebra.Definitions.html#2868" class="Function Operator">_Absorbs_</a> <a id="2878" class="Symbol">:</a> <a id="2880" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2884" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="2886" class="Symbol">→</a> <a id="2888" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2892" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="2894" class="Symbol">→</a> <a id="2896" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2900" class="Symbol">_</a>
<a id="2902" href="Algebra.Definitions.html#2902" class="Bound Operator">_∙_</a> <a id="2906" href="Algebra.Definitions.html#2868" class="Function Operator">Absorbs</a> <a id="2914" href="Algebra.Definitions.html#2914" class="Bound Operator">_∘_</a> <a id="2918" class="Symbol">=</a> <a id="2920" class="Symbol">∀</a> <a id="2922" href="Algebra.Definitions.html#2922" class="Bound">x</a> <a id="2924" href="Algebra.Definitions.html#2924" class="Bound">y</a> <a id="2926" class="Symbol">→</a> <a id="2928" class="Symbol">(</a><a id="2929" href="Algebra.Definitions.html#2922" class="Bound">x</a> <a id="2931" href="Algebra.Definitions.html#2902" class="Bound Operator">∙</a> <a id="2933" class="Symbol">(</a><a id="2934" href="Algebra.Definitions.html#2922" class="Bound">x</a> <a id="2936" href="Algebra.Definitions.html#2914" class="Bound Operator">∘</a> <a id="2938" href="Algebra.Definitions.html#2924" class="Bound">y</a><a id="2939" class="Symbol">))</a> <a id="2942" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="2944" href="Algebra.Definitions.html#2922" class="Bound">x</a>
<a id="Absorptive"></a><a id="2947" href="Algebra.Definitions.html#2947" class="Function">Absorptive</a> <a id="2958" class="Symbol">:</a> <a id="2960" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2964" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="2966" class="Symbol">→</a> <a id="2968" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2972" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="2974" class="Symbol">→</a> <a id="2976" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2980" class="Symbol">_</a>
<a id="2982" href="Algebra.Definitions.html#2947" class="Function">Absorptive</a> <a id="2993" href="Algebra.Definitions.html#2993" class="Bound">∙</a> <a id="2995" href="Algebra.Definitions.html#2995" class="Bound">∘</a> <a id="2997" class="Symbol">=</a> <a id="2999" class="Symbol">(</a><a id="3000" href="Algebra.Definitions.html#2993" class="Bound">∙</a> <a id="3002" href="Algebra.Definitions.html#2868" class="Function Operator">Absorbs</a> <a id="3010" href="Algebra.Definitions.html#2995" class="Bound">∘</a><a id="3011" class="Symbol">)</a> <a id="3013" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="3015" class="Symbol">(</a><a id="3016" href="Algebra.Definitions.html#2995" class="Bound">∘</a> <a id="3018" href="Algebra.Definitions.html#2868" class="Function Operator">Absorbs</a> <a id="3026" href="Algebra.Definitions.html#2993" class="Bound">∙</a><a id="3027" class="Symbol">)</a>
<a id="Involutive"></a><a id="3030" href="Algebra.Definitions.html#3030" class="Function">Involutive</a> <a id="3041" class="Symbol">:</a> <a id="3043" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="3047" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3049" class="Symbol">→</a> <a id="3051" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3055" class="Symbol">_</a>
<a id="3057" href="Algebra.Definitions.html#3030" class="Function">Involutive</a> <a id="3068" href="Algebra.Definitions.html#3068" class="Bound">f</a> <a id="3070" class="Symbol">=</a> <a id="3072" class="Symbol">∀</a> <a id="3074" href="Algebra.Definitions.html#3074" class="Bound">x</a> <a id="3076" class="Symbol">→</a> <a id="3078" href="Algebra.Definitions.html#3068" class="Bound">f</a> <a id="3080" class="Symbol">(</a><a id="3081" href="Algebra.Definitions.html#3068" class="Bound">f</a> <a id="3083" href="Algebra.Definitions.html#3074" class="Bound">x</a><a id="3084" class="Symbol">)</a> <a id="3086" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3088" href="Algebra.Definitions.html#3074" class="Bound">x</a>
<a id="LeftCancellative"></a><a id="3091" href="Algebra.Definitions.html#3091" class="Function">LeftCancellative</a> <a id="3108" class="Symbol">:</a> <a id="3110" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3114" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3116" class="Symbol">→</a> <a id="3118" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3122" class="Symbol">_</a>
<a id="3124" href="Algebra.Definitions.html#3091" class="Function">LeftCancellative</a> <a id="3141" href="Algebra.Definitions.html#3141" class="Bound Operator">_•_</a> <a id="3145" class="Symbol">=</a> <a id="3147" class="Symbol">∀</a> <a id="3149" href="Algebra.Definitions.html#3149" class="Bound">x</a> <a id="3151" class="Symbol">{</a><a id="3152" href="Algebra.Definitions.html#3152" class="Bound">y</a> <a id="3154" href="Algebra.Definitions.html#3154" class="Bound">z</a><a id="3155" class="Symbol">}</a> <a id="3157" class="Symbol">→</a> <a id="3159" class="Symbol">(</a><a id="3160" href="Algebra.Definitions.html#3149" class="Bound">x</a> <a id="3162" href="Algebra.Definitions.html#3141" class="Bound Operator">•</a> <a id="3164" href="Algebra.Definitions.html#3152" class="Bound">y</a><a id="3165" class="Symbol">)</a> <a id="3167" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3169" class="Symbol">(</a><a id="3170" href="Algebra.Definitions.html#3149" class="Bound">x</a> <a id="3172" href="Algebra.Definitions.html#3141" class="Bound Operator">•</a> <a id="3174" href="Algebra.Definitions.html#3154" class="Bound">z</a><a id="3175" class="Symbol">)</a> <a id="3177" class="Symbol">→</a> <a id="3179" href="Algebra.Definitions.html#3152" class="Bound">y</a> <a id="3181" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3183" href="Algebra.Definitions.html#3154" class="Bound">z</a>
<a id="RightCancellative"></a><a id="3186" href="Algebra.Definitions.html#3186" class="Function">RightCancellative</a> <a id="3204" class="Symbol">:</a> <a id="3206" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3210" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3212" class="Symbol">→</a> <a id="3214" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3218" class="Symbol">_</a>
<a id="3220" href="Algebra.Definitions.html#3186" class="Function">RightCancellative</a> <a id="3238" href="Algebra.Definitions.html#3238" class="Bound Operator">_•_</a> <a id="3242" class="Symbol">=</a> <a id="3244" class="Symbol">∀</a> <a id="3246" class="Symbol">{</a><a id="3247" href="Algebra.Definitions.html#3247" class="Bound">x</a><a id="3248" class="Symbol">}</a> <a id="3250" href="Algebra.Definitions.html#3250" class="Bound">y</a> <a id="3252" href="Algebra.Definitions.html#3252" class="Bound">z</a> <a id="3254" class="Symbol">→</a> <a id="3256" class="Symbol">(</a><a id="3257" href="Algebra.Definitions.html#3250" class="Bound">y</a> <a id="3259" href="Algebra.Definitions.html#3238" class="Bound Operator">•</a> <a id="3261" href="Algebra.Definitions.html#3247" class="Bound">x</a><a id="3262" class="Symbol">)</a> <a id="3264" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3266" class="Symbol">(</a><a id="3267" href="Algebra.Definitions.html#3252" class="Bound">z</a> <a id="3269" href="Algebra.Definitions.html#3238" class="Bound Operator">•</a> <a id="3271" href="Algebra.Definitions.html#3247" class="Bound">x</a><a id="3272" class="Symbol">)</a> <a id="3274" class="Symbol">→</a> <a id="3276" href="Algebra.Definitions.html#3250" class="Bound">y</a> <a id="3278" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3280" href="Algebra.Definitions.html#3252" class="Bound">z</a>
<a id="Cancellative"></a><a id="3283" href="Algebra.Definitions.html#3283" class="Function">Cancellative</a> <a id="3296" class="Symbol">:</a> <a id="3298" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3302" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3304" class="Symbol">→</a> <a id="3306" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3310" class="Symbol">_</a>
<a id="3312" href="Algebra.Definitions.html#3283" class="Function">Cancellative</a> <a id="3325" href="Algebra.Definitions.html#3325" class="Bound Operator">_•_</a> <a id="3329" class="Symbol">=</a> <a id="3331" class="Symbol">(</a><a id="3332" href="Algebra.Definitions.html#3091" class="Function">LeftCancellative</a> <a id="3349" href="Algebra.Definitions.html#3325" class="Bound Operator">_•_</a><a id="3352" class="Symbol">)</a> <a id="3354" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="3356" class="Symbol">(</a><a id="3357" href="Algebra.Definitions.html#3186" class="Function">RightCancellative</a> <a id="3375" href="Algebra.Definitions.html#3325" class="Bound Operator">_•_</a><a id="3378" class="Symbol">)</a>
<a id="AlmostLeftCancellative"></a><a id="3381" href="Algebra.Definitions.html#3381" class="Function">AlmostLeftCancellative</a> <a id="3404" class="Symbol">:</a> <a id="3406" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3408" class="Symbol">→</a> <a id="3410" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3414" href="Algebra.Definitions.html#534" class="Bound">A</a> <a id="3416" class="Symbol">→</a> <a id="3418" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3422" class="Symbol">_</a>
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<a id="3548" href="Algebra.Definitions.html#3504" class="Function">AlmostRightCancellative</a> <a id="3572" href="Algebra.Definitions.html#3572" class="Bound">e</a> <a id="3574" href="Algebra.Definitions.html#3574" class="Bound Operator">_•_</a> <a id="3578" class="Symbol">=</a> <a id="3580" class="Symbol">∀</a> <a id="3582" class="Symbol">{</a><a id="3583" href="Algebra.Definitions.html#3583" class="Bound">x</a><a id="3584" class="Symbol">}</a> <a id="3586" href="Algebra.Definitions.html#3586" class="Bound">y</a> <a id="3588" href="Algebra.Definitions.html#3588" class="Bound">z</a> <a id="3590" class="Symbol">→</a> <a id="3592" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="3594" href="Algebra.Definitions.html#3583" class="Bound">x</a> <a id="3596" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3598" href="Algebra.Definitions.html#3572" class="Bound">e</a> <a id="3600" class="Symbol">→</a> <a id="3602" class="Symbol">(</a><a id="3603" href="Algebra.Definitions.html#3586" class="Bound">y</a> <a id="3605" href="Algebra.Definitions.html#3574" class="Bound Operator">•</a> <a id="3607" href="Algebra.Definitions.html#3583" class="Bound">x</a><a id="3608" class="Symbol">)</a> <a id="3610" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3612" class="Symbol">(</a><a id="3613" href="Algebra.Definitions.html#3588" class="Bound">z</a> <a id="3615" href="Algebra.Definitions.html#3574" class="Bound Operator">•</a> <a id="3617" href="Algebra.Definitions.html#3583" class="Bound">x</a><a id="3618" class="Symbol">)</a> <a id="3620" class="Symbol">→</a> <a id="3622" href="Algebra.Definitions.html#3586" class="Bound">y</a> <a id="3624" href="Algebra.Definitions.html#572" class="Bound Operator">≈</a> <a id="3626" href="Algebra.Definitions.html#3588" class="Bound">z</a>
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<a id="3668" href="Algebra.Definitions.html#3629" class="Function">AlmostCancellative</a> <a id="3687" href="Algebra.Definitions.html#3687" class="Bound">e</a> <a id="3689" href="Algebra.Definitions.html#3689" class="Bound Operator">_•_</a> <a id="3693" class="Symbol">=</a> <a id="3695" href="Algebra.Definitions.html#3381" class="Function">AlmostLeftCancellative</a> <a id="3718" href="Algebra.Definitions.html#3687" class="Bound">e</a> <a id="3720" href="Algebra.Definitions.html#3689" class="Bound Operator">_•_</a> <a id="3724" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="3726" href="Algebra.Definitions.html#3504" class="Function">AlmostRightCancellative</a> <a id="3750" href="Algebra.Definitions.html#3687" class="Bound">e</a> <a id="3752" href="Algebra.Definitions.html#3689" class="Bound Operator">_•_</a>
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