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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="Comment">-- Note that very few of the element arguments are made implicit here,</a>
<a id="450" class="Comment">-- as we do not assume that the Agda can infer either the right or left</a>
<a id="522" class="Comment">-- argument of the binary operators. This is despite the fact that the</a>
<a id="593" class="Comment">-- library defines most of its concrete operators (e.g. in</a>
<a id="652" class="Comment">-- `Data.Nat.Base`) as being left-biased.</a>
<a id="695" class="Symbol">{-#</a> <a id="699" class="Keyword">OPTIONS</a> <a id="707" class="Pragma">--cubical-compatible</a> <a id="728" class="Pragma">--safe</a> <a id="735" class="Symbol">#-}</a>
<a id="740" class="Keyword">open</a> <a id="745" class="Keyword">import</a> <a id="752" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="773" class="Keyword">using</a> <a id="779" class="Symbol">(</a><a id="780" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="783" class="Symbol">;</a> <a id="785" href="Relation.Binary.Core.html#1577" class="Function Operator">_Preserves_⟶_</a><a id="798" class="Symbol">;</a> <a id="800" href="Relation.Binary.Core.html#1703" class="Function Operator">_Preserves₂_⟶_⟶_</a><a id="816" class="Symbol">)</a>
<a id="818" class="Keyword">open</a> <a id="823" class="Keyword">import</a> <a id="830" href="Relation.Nullary.Negation.Core.html" class="Module">Relation.Nullary.Negation.Core</a> <a id="861" class="Keyword">using</a> <a id="867" class="Symbol">(</a><a id="868" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬_</a><a id="870" class="Symbol">)</a>
<a id="873" class="Keyword">module</a> <a id="880" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a>
<a id="902" class="Symbol">{</a><a id="903" href="Algebra.Definitions.html#903" class="Bound">a</a> <a id="905" href="Algebra.Definitions.html#905" class="Bound">ℓ</a><a id="906" class="Symbol">}</a> <a id="908" class="Symbol">{</a><a id="909" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="911" class="Symbol">:</a> <a id="913" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="917" href="Algebra.Definitions.html#903" class="Bound">a</a><a id="918" class="Symbol">}</a> <a id="922" class="Comment">-- The underlying set</a>
<a id="946" class="Symbol">(</a><a id="947" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="951" class="Symbol">:</a> <a id="953" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="957" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="959" href="Algebra.Definitions.html#905" class="Bound">ℓ</a><a id="960" class="Symbol">)</a> <a id="966" class="Comment">-- The underlying equality</a>
<a id="995" class="Keyword">where</a>
<a id="1002" class="Keyword">open</a> <a id="1007" class="Keyword">import</a> <a id="1014" href="Algebra.Core.html" class="Module">Algebra.Core</a> <a id="1027" class="Keyword">using</a> <a id="1033" class="Symbol">(</a><a id="1034" href="Algebra.Core.html#484" class="Function">Op₁</a><a id="1037" class="Symbol">;</a> <a id="1039" href="Algebra.Core.html#527" class="Function">Op₂</a><a id="1042" class="Symbol">)</a>
<a id="1044" class="Keyword">open</a> <a id="1049" class="Keyword">import</a> <a id="1056" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="1074" class="Keyword">using</a> <a id="1080" class="Symbol">(</a><a id="1081" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="1084" class="Symbol">;</a> <a id="1086" href="Data.Product.Base.html#1371" class="Function">∃-syntax</a><a id="1094" class="Symbol">)</a>
<a id="1096" class="Keyword">open</a> <a id="1101" class="Keyword">import</a> <a id="1108" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="1122" class="Keyword">using</a> <a id="1128" class="Symbol">(</a><a id="1129" href="Data.Sum.Base.html#625" class="Datatype Operator">_⊎_</a><a id="1132" class="Symbol">)</a>
<a id="1135" class="Comment">------------------------------------------------------------------------</a>
<a id="1208" class="Comment">-- Properties of operations</a>
<a id="Congruent₁"></a><a id="1237" href="Algebra.Definitions.html#1237" class="Function">Congruent₁</a> <a id="1248" class="Symbol">:</a> <a id="1250" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="1254" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1256" class="Symbol">→</a> <a id="1258" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1262" class="Symbol">_</a>
<a id="1264" href="Algebra.Definitions.html#1237" class="Function">Congruent₁</a> <a id="1275" href="Algebra.Definitions.html#1275" class="Bound">f</a> <a id="1277" class="Symbol">=</a> <a id="1279" href="Algebra.Definitions.html#1275" class="Bound">f</a> <a id="1281" href="Relation.Binary.Core.html#1577" class="Function Operator">Preserves</a> <a id="1291" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="1295" href="Relation.Binary.Core.html#1577" class="Function Operator">⟶</a> <a id="1297" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a>
<a id="Congruent₂"></a><a id="1302" href="Algebra.Definitions.html#1302" class="Function">Congruent₂</a> <a id="1313" class="Symbol">:</a> <a id="1315" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1319" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1321" class="Symbol">→</a> <a id="1323" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1327" class="Symbol">_</a>
<a id="1329" href="Algebra.Definitions.html#1302" class="Function">Congruent₂</a> <a id="1340" href="Algebra.Definitions.html#1340" class="Bound">∙</a> <a id="1342" class="Symbol">=</a> <a id="1344" href="Algebra.Definitions.html#1340" class="Bound">∙</a> <a id="1346" href="Relation.Binary.Core.html#1703" class="Function Operator">Preserves₂</a> <a id="1357" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="1361" href="Relation.Binary.Core.html#1703" class="Function Operator">⟶</a> <a id="1363" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="1367" href="Relation.Binary.Core.html#1703" class="Function Operator">⟶</a> <a id="1369" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a>
<a id="LeftCongruent"></a><a id="1374" href="Algebra.Definitions.html#1374" class="Function">LeftCongruent</a> <a id="1388" class="Symbol">:</a> <a id="1390" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1394" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1396" class="Symbol">→</a> <a id="1398" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1402" class="Symbol">_</a>
<a id="1404" href="Algebra.Definitions.html#1374" class="Function">LeftCongruent</a> <a id="1418" href="Algebra.Definitions.html#1418" class="Bound Operator">_∙_</a> <a id="1422" class="Symbol">=</a> <a id="1424" class="Symbol">∀</a> <a id="1426" class="Symbol">{</a><a id="1427" href="Algebra.Definitions.html#1427" class="Bound">x</a><a id="1428" class="Symbol">}</a> <a id="1430" class="Symbol">→</a> <a id="1432" class="Symbol">(</a><a id="1433" href="Algebra.Definitions.html#1427" class="Bound">x</a> <a id="1435" href="Algebra.Definitions.html#1418" class="Bound Operator">∙_</a><a id="1437" class="Symbol">)</a> <a id="1439" href="Relation.Binary.Core.html#1577" class="Function Operator">Preserves</a> <a id="1449" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="1453" href="Relation.Binary.Core.html#1577" class="Function Operator">⟶</a> <a id="1455" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a>
<a id="RightCongruent"></a><a id="1460" href="Algebra.Definitions.html#1460" class="Function">RightCongruent</a> <a id="1475" class="Symbol">:</a> <a id="1477" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1481" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1483" class="Symbol">→</a> <a id="1485" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1489" class="Symbol">_</a>
<a id="1491" href="Algebra.Definitions.html#1460" class="Function">RightCongruent</a> <a id="1506" href="Algebra.Definitions.html#1506" class="Bound Operator">_∙_</a> <a id="1510" class="Symbol">=</a> <a id="1512" class="Symbol">∀</a> <a id="1514" class="Symbol">{</a><a id="1515" href="Algebra.Definitions.html#1515" class="Bound">x</a><a id="1516" class="Symbol">}</a> <a id="1518" class="Symbol">→</a> <a id="1520" class="Symbol">(</a><a id="1521" href="Algebra.Definitions.html#1506" class="Bound Operator">_∙</a> <a id="1524" href="Algebra.Definitions.html#1515" class="Bound">x</a><a id="1525" class="Symbol">)</a> <a id="1527" href="Relation.Binary.Core.html#1577" class="Function Operator">Preserves</a> <a id="1537" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a> <a id="1541" href="Relation.Binary.Core.html#1577" class="Function Operator">⟶</a> <a id="1543" href="Algebra.Definitions.html#947" class="Bound Operator">_≈_</a>
<a id="Associative"></a><a id="1548" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="1560" class="Symbol">:</a> <a id="1562" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1566" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1568" class="Symbol">→</a> <a id="1570" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1574" class="Symbol">_</a>
<a id="1576" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="1588" href="Algebra.Definitions.html#1588" class="Bound Operator">_∙_</a> <a id="1592" class="Symbol">=</a> <a id="1594" class="Symbol">∀</a> <a id="1596" href="Algebra.Definitions.html#1596" class="Bound">x</a> <a id="1598" href="Algebra.Definitions.html#1598" class="Bound">y</a> <a id="1600" href="Algebra.Definitions.html#1600" class="Bound">z</a> <a id="1602" class="Symbol">→</a> <a id="1604" class="Symbol">((</a><a id="1606" href="Algebra.Definitions.html#1596" class="Bound">x</a> <a id="1608" href="Algebra.Definitions.html#1588" class="Bound Operator">∙</a> <a id="1610" href="Algebra.Definitions.html#1598" class="Bound">y</a><a id="1611" class="Symbol">)</a> <a id="1613" href="Algebra.Definitions.html#1588" class="Bound Operator">∙</a> <a id="1615" href="Algebra.Definitions.html#1600" class="Bound">z</a><a id="1616" class="Symbol">)</a> <a id="1618" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="1620" class="Symbol">(</a><a id="1621" href="Algebra.Definitions.html#1596" class="Bound">x</a> <a id="1623" href="Algebra.Definitions.html#1588" class="Bound Operator">∙</a> <a id="1625" class="Symbol">(</a><a id="1626" href="Algebra.Definitions.html#1598" class="Bound">y</a> <a id="1628" href="Algebra.Definitions.html#1588" class="Bound Operator">∙</a> <a id="1630" href="Algebra.Definitions.html#1600" class="Bound">z</a><a id="1631" class="Symbol">))</a>
<a id="Commutative"></a><a id="1635" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="1647" class="Symbol">:</a> <a id="1649" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1653" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1655" class="Symbol">→</a> <a id="1657" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1661" class="Symbol">_</a>
<a id="1663" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="1675" href="Algebra.Definitions.html#1675" class="Bound Operator">_∙_</a> <a id="1679" class="Symbol">=</a> <a id="1681" class="Symbol">∀</a> <a id="1683" href="Algebra.Definitions.html#1683" class="Bound">x</a> <a id="1685" href="Algebra.Definitions.html#1685" class="Bound">y</a> <a id="1687" class="Symbol">→</a> <a id="1689" class="Symbol">(</a><a id="1690" href="Algebra.Definitions.html#1683" class="Bound">x</a> <a id="1692" href="Algebra.Definitions.html#1675" class="Bound Operator">∙</a> <a id="1694" href="Algebra.Definitions.html#1685" class="Bound">y</a><a id="1695" class="Symbol">)</a> <a id="1697" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="1699" class="Symbol">(</a><a id="1700" href="Algebra.Definitions.html#1685" class="Bound">y</a> <a id="1702" href="Algebra.Definitions.html#1675" class="Bound Operator">∙</a> <a id="1704" href="Algebra.Definitions.html#1683" class="Bound">x</a><a id="1705" class="Symbol">)</a>
<a id="LeftIdentity"></a><a id="1708" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="1721" class="Symbol">:</a> <a id="1723" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1725" class="Symbol">→</a> <a id="1727" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1731" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1733" class="Symbol">→</a> <a id="1735" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1739" class="Symbol">_</a>
<a id="1741" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="1754" href="Algebra.Definitions.html#1754" class="Bound">e</a> <a id="1756" href="Algebra.Definitions.html#1756" class="Bound Operator">_∙_</a> <a id="1760" class="Symbol">=</a> <a id="1762" class="Symbol">∀</a> <a id="1764" href="Algebra.Definitions.html#1764" class="Bound">x</a> <a id="1766" class="Symbol">→</a> <a id="1768" class="Symbol">(</a><a id="1769" href="Algebra.Definitions.html#1754" class="Bound">e</a> <a id="1771" href="Algebra.Definitions.html#1756" class="Bound Operator">∙</a> <a id="1773" href="Algebra.Definitions.html#1764" class="Bound">x</a><a id="1774" class="Symbol">)</a> <a id="1776" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="1778" href="Algebra.Definitions.html#1764" class="Bound">x</a>
<a id="RightIdentity"></a><a id="1781" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="1795" class="Symbol">:</a> <a id="1797" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1799" class="Symbol">→</a> <a id="1801" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1805" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1807" class="Symbol">→</a> <a id="1809" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1813" class="Symbol">_</a>
<a id="1815" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="1829" href="Algebra.Definitions.html#1829" class="Bound">e</a> <a id="1831" href="Algebra.Definitions.html#1831" class="Bound Operator">_∙_</a> <a id="1835" class="Symbol">=</a> <a id="1837" class="Symbol">∀</a> <a id="1839" href="Algebra.Definitions.html#1839" class="Bound">x</a> <a id="1841" class="Symbol">→</a> <a id="1843" class="Symbol">(</a><a id="1844" href="Algebra.Definitions.html#1839" class="Bound">x</a> <a id="1846" href="Algebra.Definitions.html#1831" class="Bound Operator">∙</a> <a id="1848" href="Algebra.Definitions.html#1829" class="Bound">e</a><a id="1849" class="Symbol">)</a> <a id="1851" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="1853" href="Algebra.Definitions.html#1839" class="Bound">x</a>
<a id="Identity"></a><a id="1856" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="1865" class="Symbol">:</a> <a id="1867" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1869" class="Symbol">→</a> <a id="1871" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1875" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1877" class="Symbol">→</a> <a id="1879" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1883" class="Symbol">_</a>
<a id="1885" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="1894" href="Algebra.Definitions.html#1894" class="Bound">e</a> <a id="1896" href="Algebra.Definitions.html#1896" class="Bound">∙</a> <a id="1898" class="Symbol">=</a> <a id="1900" class="Symbol">(</a><a id="1901" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="1914" href="Algebra.Definitions.html#1894" class="Bound">e</a> <a id="1916" href="Algebra.Definitions.html#1896" class="Bound">∙</a><a id="1917" class="Symbol">)</a> <a id="1919" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="1921" class="Symbol">(</a><a id="1922" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="1936" href="Algebra.Definitions.html#1894" class="Bound">e</a> <a id="1938" href="Algebra.Definitions.html#1896" class="Bound">∙</a><a id="1939" class="Symbol">)</a>
<a id="LeftZero"></a><a id="1942" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="1951" class="Symbol">:</a> <a id="1953" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1955" class="Symbol">→</a> <a id="1957" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1961" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="1963" class="Symbol">→</a> <a id="1965" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1969" class="Symbol">_</a>
<a id="1971" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="1980" href="Algebra.Definitions.html#1980" class="Bound">z</a> <a id="1982" href="Algebra.Definitions.html#1982" class="Bound Operator">_∙_</a> <a id="1986" class="Symbol">=</a> <a id="1988" class="Symbol">∀</a> <a id="1990" href="Algebra.Definitions.html#1990" class="Bound">x</a> <a id="1992" class="Symbol">→</a> <a id="1994" class="Symbol">(</a><a id="1995" href="Algebra.Definitions.html#1980" class="Bound">z</a> <a id="1997" href="Algebra.Definitions.html#1982" class="Bound Operator">∙</a> <a id="1999" href="Algebra.Definitions.html#1990" class="Bound">x</a><a id="2000" class="Symbol">)</a> <a id="2002" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2004" href="Algebra.Definitions.html#1980" class="Bound">z</a>
<a id="RightZero"></a><a id="2007" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="2017" class="Symbol">:</a> <a id="2019" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2021" class="Symbol">→</a> <a id="2023" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2027" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2029" class="Symbol">→</a> <a id="2031" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2035" class="Symbol">_</a>
<a id="2037" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="2047" href="Algebra.Definitions.html#2047" class="Bound">z</a> <a id="2049" href="Algebra.Definitions.html#2049" class="Bound Operator">_∙_</a> <a id="2053" class="Symbol">=</a> <a id="2055" class="Symbol">∀</a> <a id="2057" href="Algebra.Definitions.html#2057" class="Bound">x</a> <a id="2059" class="Symbol">→</a> <a id="2061" class="Symbol">(</a><a id="2062" href="Algebra.Definitions.html#2057" class="Bound">x</a> <a id="2064" href="Algebra.Definitions.html#2049" class="Bound Operator">∙</a> <a id="2066" href="Algebra.Definitions.html#2047" class="Bound">z</a><a id="2067" class="Symbol">)</a> <a id="2069" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2071" href="Algebra.Definitions.html#2047" class="Bound">z</a>
<a id="Zero"></a><a id="2074" href="Algebra.Definitions.html#2074" class="Function">Zero</a> <a id="2079" class="Symbol">:</a> <a id="2081" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2083" class="Symbol">→</a> <a id="2085" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2089" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2091" class="Symbol">→</a> <a id="2093" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2097" class="Symbol">_</a>
<a id="2099" href="Algebra.Definitions.html#2074" class="Function">Zero</a> <a id="2104" href="Algebra.Definitions.html#2104" class="Bound">z</a> <a id="2106" href="Algebra.Definitions.html#2106" class="Bound">∙</a> <a id="2108" class="Symbol">=</a> <a id="2110" class="Symbol">(</a><a id="2111" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="2120" href="Algebra.Definitions.html#2104" class="Bound">z</a> <a id="2122" href="Algebra.Definitions.html#2106" class="Bound">∙</a><a id="2123" class="Symbol">)</a> <a id="2125" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="2127" class="Symbol">(</a><a id="2128" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="2138" href="Algebra.Definitions.html#2104" class="Bound">z</a> <a id="2140" href="Algebra.Definitions.html#2106" class="Bound">∙</a><a id="2141" class="Symbol">)</a>
<a id="LeftInverse"></a><a id="2144" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="2156" class="Symbol">:</a> <a id="2158" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2160" class="Symbol">→</a> <a id="2162" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="2166" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2168" class="Symbol">→</a> <a id="2170" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2174" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2176" class="Symbol">→</a> <a id="2178" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2182" class="Symbol">_</a>
<a id="2184" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="2196" href="Algebra.Definitions.html#2196" class="Bound">e</a> <a id="2198" href="Algebra.Definitions.html#2198" class="Bound Operator">_⁻¹</a> <a id="2202" href="Algebra.Definitions.html#2202" class="Bound Operator">_∙_</a> <a id="2206" class="Symbol">=</a> <a id="2208" class="Symbol">∀</a> <a id="2210" href="Algebra.Definitions.html#2210" class="Bound">x</a> <a id="2212" class="Symbol">→</a> <a id="2214" class="Symbol">((</a><a id="2216" href="Algebra.Definitions.html#2210" class="Bound">x</a> <a id="2218" href="Algebra.Definitions.html#2198" class="Bound Operator">⁻¹</a><a id="2220" class="Symbol">)</a> <a id="2222" href="Algebra.Definitions.html#2202" class="Bound Operator">∙</a> <a id="2224" href="Algebra.Definitions.html#2210" class="Bound">x</a><a id="2225" class="Symbol">)</a> <a id="2227" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2229" href="Algebra.Definitions.html#2196" class="Bound">e</a>
<a id="RightInverse"></a><a id="2232" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="2245" class="Symbol">:</a> <a id="2247" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2249" class="Symbol">→</a> <a id="2251" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="2255" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2257" class="Symbol">→</a> <a id="2259" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2263" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2265" class="Symbol">→</a> <a id="2267" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2271" class="Symbol">_</a>
<a id="2273" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="2286" href="Algebra.Definitions.html#2286" class="Bound">e</a> <a id="2288" href="Algebra.Definitions.html#2288" class="Bound Operator">_⁻¹</a> <a id="2292" href="Algebra.Definitions.html#2292" class="Bound Operator">_∙_</a> <a id="2296" class="Symbol">=</a> <a id="2298" class="Symbol">∀</a> <a id="2300" href="Algebra.Definitions.html#2300" class="Bound">x</a> <a id="2302" class="Symbol">→</a> <a id="2304" class="Symbol">(</a><a id="2305" href="Algebra.Definitions.html#2300" class="Bound">x</a> <a id="2307" href="Algebra.Definitions.html#2292" class="Bound Operator">∙</a> <a id="2309" class="Symbol">(</a><a id="2310" href="Algebra.Definitions.html#2300" class="Bound">x</a> <a id="2312" href="Algebra.Definitions.html#2288" class="Bound Operator">⁻¹</a><a id="2314" class="Symbol">))</a> <a id="2317" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2319" href="Algebra.Definitions.html#2286" class="Bound">e</a>
<a id="Inverse"></a><a id="2322" href="Algebra.Definitions.html#2322" class="Function">Inverse</a> <a id="2330" class="Symbol">:</a> <a id="2332" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2334" class="Symbol">→</a> <a id="2336" href="Algebra.Core.html#484" class="Function">Op₁</a> <a id="2340" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2342" class="Symbol">→</a> <a id="2344" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2348" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2350" class="Symbol">→</a> <a id="2352" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2356" class="Symbol">_</a>
<a id="2358" href="Algebra.Definitions.html#2322" class="Function">Inverse</a> <a id="2366" href="Algebra.Definitions.html#2366" class="Bound">e</a> <a id="2368" href="Algebra.Definitions.html#2368" class="Bound">⁻¹</a> <a id="2371" href="Algebra.Definitions.html#2371" class="Bound">∙</a> <a id="2373" class="Symbol">=</a> <a id="2375" class="Symbol">(</a><a id="2376" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="2388" href="Algebra.Definitions.html#2366" class="Bound">e</a> <a id="2390" href="Algebra.Definitions.html#2368" class="Bound">⁻¹</a><a id="2392" class="Symbol">)</a> <a id="2394" href="Algebra.Definitions.html#2371" class="Bound">∙</a> <a id="2396" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="2398" class="Symbol">(</a><a id="2399" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="2412" href="Algebra.Definitions.html#2366" class="Bound">e</a> <a id="2414" href="Algebra.Definitions.html#2368" class="Bound">⁻¹</a> <a id="2417" href="Algebra.Definitions.html#2371" class="Bound">∙</a><a id="2418" class="Symbol">)</a>
<a id="2421" class="Comment">-- For structures in which not every element has an inverse (e.g. Fields)</a>
<a id="LeftInvertible"></a><a id="2495" href="Algebra.Definitions.html#2495" class="Function">LeftInvertible</a> <a id="2510" class="Symbol">:</a> <a id="2512" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2514" class="Symbol">→</a> <a id="2516" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2520" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2522" class="Symbol">→</a> <a id="2524" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2526" class="Symbol">→</a> <a id="2528" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2532" class="Symbol">_</a>
<a id="2534" href="Algebra.Definitions.html#2495" class="Function">LeftInvertible</a> <a id="2549" href="Algebra.Definitions.html#2549" class="Bound">e</a> <a id="2551" href="Algebra.Definitions.html#2551" class="Bound Operator">_∙_</a> <a id="2555" href="Algebra.Definitions.html#2555" class="Bound">x</a> <a id="2557" class="Symbol">=</a> <a id="2559" href="Data.Product.Base.html#1371" class="Function">∃[</a> <a id="2562" href="Algebra.Definitions.html#2562" class="Bound">x⁻¹</a> <a id="2566" href="Data.Product.Base.html#1371" class="Function">]</a> <a id="2568" class="Symbol">(</a><a id="2569" href="Algebra.Definitions.html#2562" class="Bound">x⁻¹</a> <a id="2573" href="Algebra.Definitions.html#2551" class="Bound Operator">∙</a> <a id="2575" href="Algebra.Definitions.html#2555" class="Bound">x</a><a id="2576" class="Symbol">)</a> <a id="2578" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2580" href="Algebra.Definitions.html#2549" class="Bound">e</a>
<a id="RightInvertible"></a><a id="2583" href="Algebra.Definitions.html#2583" class="Function">RightInvertible</a> <a id="2599" class="Symbol">:</a> <a id="2601" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2603" class="Symbol">→</a> <a id="2605" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2609" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2611" class="Symbol">→</a> <a id="2613" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2615" class="Symbol">→</a> <a id="2617" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2621" class="Symbol">_</a>
<a id="2623" href="Algebra.Definitions.html#2583" class="Function">RightInvertible</a> <a id="2639" href="Algebra.Definitions.html#2639" class="Bound">e</a> <a id="2641" href="Algebra.Definitions.html#2641" class="Bound Operator">_∙_</a> <a id="2645" href="Algebra.Definitions.html#2645" class="Bound">x</a> <a id="2647" class="Symbol">=</a> <a id="2649" href="Data.Product.Base.html#1371" class="Function">∃[</a> <a id="2652" href="Algebra.Definitions.html#2652" class="Bound">x⁻¹</a> <a id="2656" href="Data.Product.Base.html#1371" class="Function">]</a> <a id="2658" class="Symbol">(</a><a id="2659" href="Algebra.Definitions.html#2645" class="Bound">x</a> <a id="2661" href="Algebra.Definitions.html#2641" class="Bound Operator">∙</a> <a id="2663" href="Algebra.Definitions.html#2652" class="Bound">x⁻¹</a><a id="2666" class="Symbol">)</a> <a id="2668" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2670" href="Algebra.Definitions.html#2639" class="Bound">e</a>
<a id="2673" class="Comment">-- NB: this is not quite the same as</a>
<a id="2710" class="Comment">-- LeftInvertible e ∙ x × RightInvertible e ∙ x</a>
<a id="2758" class="Comment">-- since the left and right inverses have to coincide.</a>
<a id="Invertible"></a><a id="2813" href="Algebra.Definitions.html#2813" class="Function">Invertible</a> <a id="2824" class="Symbol">:</a> <a id="2826" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2828" class="Symbol">→</a> <a id="2830" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2834" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2836" class="Symbol">→</a> <a id="2838" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2840" class="Symbol">→</a> <a id="2842" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2846" class="Symbol">_</a>
<a id="2848" href="Algebra.Definitions.html#2813" class="Function">Invertible</a> <a id="2859" href="Algebra.Definitions.html#2859" class="Bound">e</a> <a id="2861" href="Algebra.Definitions.html#2861" class="Bound Operator">_∙_</a> <a id="2865" href="Algebra.Definitions.html#2865" class="Bound">x</a> <a id="2867" class="Symbol">=</a> <a id="2869" href="Data.Product.Base.html#1371" class="Function">∃[</a> <a id="2872" href="Algebra.Definitions.html#2872" class="Bound">x⁻¹</a> <a id="2876" href="Data.Product.Base.html#1371" class="Function">]</a> <a id="2878" class="Symbol">(</a><a id="2879" href="Algebra.Definitions.html#2872" class="Bound">x⁻¹</a> <a id="2883" href="Algebra.Definitions.html#2861" class="Bound Operator">∙</a> <a id="2885" href="Algebra.Definitions.html#2865" class="Bound">x</a><a id="2886" class="Symbol">)</a> <a id="2888" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2890" href="Algebra.Definitions.html#2859" class="Bound">e</a> <a id="2892" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="2894" class="Symbol">(</a><a id="2895" href="Algebra.Definitions.html#2865" class="Bound">x</a> <a id="2897" href="Algebra.Definitions.html#2861" class="Bound Operator">∙</a> <a id="2899" href="Algebra.Definitions.html#2872" class="Bound">x⁻¹</a><a id="2902" class="Symbol">)</a> <a id="2904" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2906" href="Algebra.Definitions.html#2859" class="Bound">e</a>
<a id="LeftConical"></a><a id="2909" href="Algebra.Definitions.html#2909" class="Function">LeftConical</a> <a id="2921" class="Symbol">:</a> <a id="2923" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2925" class="Symbol">→</a> <a id="2927" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="2931" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="2933" class="Symbol">→</a> <a id="2935" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2939" class="Symbol">_</a>
<a id="2941" href="Algebra.Definitions.html#2909" class="Function">LeftConical</a> <a id="2953" href="Algebra.Definitions.html#2953" class="Bound">e</a> <a id="2955" href="Algebra.Definitions.html#2955" class="Bound Operator">_∙_</a> <a id="2959" class="Symbol">=</a> <a id="2961" class="Symbol">∀</a> <a id="2963" href="Algebra.Definitions.html#2963" class="Bound">x</a> <a id="2965" href="Algebra.Definitions.html#2965" class="Bound">y</a> <a id="2967" class="Symbol">→</a> <a id="2969" class="Symbol">(</a><a id="2970" href="Algebra.Definitions.html#2963" class="Bound">x</a> <a id="2972" href="Algebra.Definitions.html#2955" class="Bound Operator">∙</a> <a id="2974" href="Algebra.Definitions.html#2965" class="Bound">y</a><a id="2975" class="Symbol">)</a> <a id="2977" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2979" href="Algebra.Definitions.html#2953" class="Bound">e</a> <a id="2981" class="Symbol">→</a> <a id="2983" href="Algebra.Definitions.html#2963" class="Bound">x</a> <a id="2985" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="2987" href="Algebra.Definitions.html#2953" class="Bound">e</a>
<a id="RightConical"></a><a id="2990" href="Algebra.Definitions.html#2990" class="Function">RightConical</a> <a id="3003" class="Symbol">:</a> <a id="3005" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3007" class="Symbol">→</a> <a id="3009" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3013" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3015" class="Symbol">→</a> <a id="3017" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="3021" class="Symbol">_</a>
<a id="3023" href="Algebra.Definitions.html#2990" class="Function">RightConical</a> <a id="3036" href="Algebra.Definitions.html#3036" class="Bound">e</a> <a id="3038" href="Algebra.Definitions.html#3038" class="Bound Operator">_∙_</a> <a id="3042" class="Symbol">=</a> <a id="3044" class="Symbol">∀</a> <a id="3046" href="Algebra.Definitions.html#3046" class="Bound">x</a> <a id="3048" href="Algebra.Definitions.html#3048" class="Bound">y</a> <a id="3050" class="Symbol">→</a> <a id="3052" class="Symbol">(</a><a id="3053" href="Algebra.Definitions.html#3046" class="Bound">x</a> <a id="3055" href="Algebra.Definitions.html#3038" class="Bound Operator">∙</a> <a id="3057" href="Algebra.Definitions.html#3048" class="Bound">y</a><a id="3058" class="Symbol">)</a> <a id="3060" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="3062" href="Algebra.Definitions.html#3036" class="Bound">e</a> <a id="3064" class="Symbol">→</a> <a id="3066" href="Algebra.Definitions.html#3048" class="Bound">y</a> <a id="3068" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="3070" href="Algebra.Definitions.html#3036" class="Bound">e</a>
<a id="Conical"></a><a id="3073" href="Algebra.Definitions.html#3073" class="Function">Conical</a> <a id="3081" class="Symbol">:</a> <a id="3083" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3085" class="Symbol">→</a> <a id="3087" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3091" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3093" class="Symbol">→</a> <a id="3095" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="3099" class="Symbol">_</a>
<a id="3101" href="Algebra.Definitions.html#3073" class="Function">Conical</a> <a id="3109" href="Algebra.Definitions.html#3109" class="Bound">e</a> <a id="3111" href="Algebra.Definitions.html#3111" class="Bound">∙</a> <a id="3113" class="Symbol">=</a> <a id="3115" class="Symbol">(</a><a id="3116" href="Algebra.Definitions.html#2909" class="Function">LeftConical</a> <a id="3128" href="Algebra.Definitions.html#3109" class="Bound">e</a> <a id="3130" href="Algebra.Definitions.html#3111" class="Bound">∙</a><a id="3131" class="Symbol">)</a> <a id="3133" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="3135" class="Symbol">(</a><a id="3136" href="Algebra.Definitions.html#2990" class="Function">RightConical</a> <a id="3149" href="Algebra.Definitions.html#3109" class="Bound">e</a> <a id="3151" href="Algebra.Definitions.html#3111" class="Bound">∙</a><a id="3152" class="Symbol">)</a>
<a id="_DistributesOverˡ_"></a><a id="3155" href="Algebra.Definitions.html#3155" class="Function Operator">_DistributesOverˡ_</a> <a id="3174" class="Symbol">:</a> <a id="3176" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3180" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3182" class="Symbol">→</a> <a id="3184" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="3188" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="3190" class="Symbol">→</a> <a id="3192" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="3196" class="Symbol">_</a>
<a id="3198" href="Algebra.Definitions.html#3198" class="Bound Operator">_*_</a> <a id="3202" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="3219" href="Algebra.Definitions.html#3219" class="Bound Operator">_+_</a> <a id="3223" class="Symbol">=</a>
<a id="3227" class="Symbol">∀</a> <a id="3229" href="Algebra.Definitions.html#3229" class="Bound">x</a> <a id="3231" href="Algebra.Definitions.html#3231" class="Bound">y</a> <a id="3233" href="Algebra.Definitions.html#3233" class="Bound">z</a> <a id="3235" class="Symbol">→</a> <a id="3237" class="Symbol">(</a><a id="3238" href="Algebra.Definitions.html#3229" class="Bound">x</a> <a id="3240" href="Algebra.Definitions.html#3198" class="Bound Operator">*</a> <a id="3242" class="Symbol">(</a><a id="3243" href="Algebra.Definitions.html#3231" class="Bound">y</a> <a id="3245" href="Algebra.Definitions.html#3219" class="Bound Operator">+</a> <a id="3247" href="Algebra.Definitions.html#3233" class="Bound">z</a><a id="3248" class="Symbol">))</a> <a id="3251" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="3253" class="Symbol">((</a><a id="3255" href="Algebra.Definitions.html#3229" class="Bound">x</a> <a id="3257" href="Algebra.Definitions.html#3198" class="Bound Operator">*</a> <a id="3259" href="Algebra.Definitions.html#3231" class="Bound">y</a><a id="3260" class="Symbol">)</a> <a id="3262" href="Algebra.Definitions.html#3219" class="Bound Operator">+</a> <a id="3264" class="Symbol">(</a><a id="3265" href="Algebra.Definitions.html#3229" class="Bound">x</a> <a id="3267" href="Algebra.Definitions.html#3198" class="Bound Operator">*</a> <a id="3269" href="Algebra.Definitions.html#3233" class="Bound">z</a><a id="3270" class="Symbol">))</a>
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<a id="4431" href="Algebra.Definitions.html#4402" class="Function">Cancellative</a> <a id="4444" href="Algebra.Definitions.html#4444" class="Bound Operator">_•_</a> <a id="4448" class="Symbol">=</a> <a id="4450" class="Symbol">(</a><a id="4451" href="Algebra.Definitions.html#4214" class="Function">LeftCancellative</a> <a id="4468" href="Algebra.Definitions.html#4444" class="Bound Operator">_•_</a><a id="4471" class="Symbol">)</a> <a id="4473" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="4475" class="Symbol">(</a><a id="4476" href="Algebra.Definitions.html#4307" class="Function">RightCancellative</a> <a id="4494" href="Algebra.Definitions.html#4444" class="Bound Operator">_•_</a><a id="4497" class="Symbol">)</a>
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<a id="4543" href="Algebra.Definitions.html#4500" class="Function">AlmostLeftCancellative</a> <a id="4566" href="Algebra.Definitions.html#4566" class="Bound">e</a> <a id="4568" href="Algebra.Definitions.html#4568" class="Bound Operator">_•_</a> <a id="4572" class="Symbol">=</a> <a id="4574" class="Symbol">∀</a> <a id="4576" href="Algebra.Definitions.html#4576" class="Bound">x</a> <a id="4578" href="Algebra.Definitions.html#4578" class="Bound">y</a> <a id="4580" href="Algebra.Definitions.html#4580" class="Bound">z</a> <a id="4582" class="Symbol">→</a> <a id="4584" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="4586" href="Algebra.Definitions.html#4576" class="Bound">x</a> <a id="4588" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4590" href="Algebra.Definitions.html#4566" class="Bound">e</a> <a id="4592" class="Symbol">→</a> <a id="4594" class="Symbol">(</a><a id="4595" href="Algebra.Definitions.html#4576" class="Bound">x</a> <a id="4597" href="Algebra.Definitions.html#4568" class="Bound Operator">•</a> <a id="4599" href="Algebra.Definitions.html#4578" class="Bound">y</a><a id="4600" class="Symbol">)</a> <a id="4602" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4604" class="Symbol">(</a><a id="4605" href="Algebra.Definitions.html#4576" class="Bound">x</a> <a id="4607" href="Algebra.Definitions.html#4568" class="Bound Operator">•</a> <a id="4609" href="Algebra.Definitions.html#4580" class="Bound">z</a><a id="4610" class="Symbol">)</a> <a id="4612" class="Symbol">→</a> <a id="4614" href="Algebra.Definitions.html#4578" class="Bound">y</a> <a id="4616" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4618" href="Algebra.Definitions.html#4580" class="Bound">z</a>
<a id="AlmostRightCancellative"></a><a id="4621" href="Algebra.Definitions.html#4621" class="Function">AlmostRightCancellative</a> <a id="4645" class="Symbol">:</a> <a id="4647" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="4649" class="Symbol">→</a> <a id="4651" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="4655" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="4657" class="Symbol">→</a> <a id="4659" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="4663" class="Symbol">_</a>
<a id="4665" href="Algebra.Definitions.html#4621" class="Function">AlmostRightCancellative</a> <a id="4689" href="Algebra.Definitions.html#4689" class="Bound">e</a> <a id="4691" href="Algebra.Definitions.html#4691" class="Bound Operator">_•_</a> <a id="4695" class="Symbol">=</a> <a id="4697" class="Symbol">∀</a> <a id="4699" href="Algebra.Definitions.html#4699" class="Bound">x</a> <a id="4701" href="Algebra.Definitions.html#4701" class="Bound">y</a> <a id="4703" href="Algebra.Definitions.html#4703" class="Bound">z</a> <a id="4705" class="Symbol">→</a> <a id="4707" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="4709" href="Algebra.Definitions.html#4699" class="Bound">x</a> <a id="4711" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4713" href="Algebra.Definitions.html#4689" class="Bound">e</a> <a id="4715" class="Symbol">→</a> <a id="4717" class="Symbol">(</a><a id="4718" href="Algebra.Definitions.html#4701" class="Bound">y</a> <a id="4720" href="Algebra.Definitions.html#4691" class="Bound Operator">•</a> <a id="4722" href="Algebra.Definitions.html#4699" class="Bound">x</a><a id="4723" class="Symbol">)</a> <a id="4725" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4727" class="Symbol">(</a><a id="4728" href="Algebra.Definitions.html#4703" class="Bound">z</a> <a id="4730" href="Algebra.Definitions.html#4691" class="Bound Operator">•</a> <a id="4732" href="Algebra.Definitions.html#4699" class="Bound">x</a><a id="4733" class="Symbol">)</a> <a id="4735" class="Symbol">→</a> <a id="4737" href="Algebra.Definitions.html#4701" class="Bound">y</a> <a id="4739" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="4741" href="Algebra.Definitions.html#4703" class="Bound">z</a>
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<a id="4783" href="Algebra.Definitions.html#4744" class="Function">AlmostCancellative</a> <a id="4802" href="Algebra.Definitions.html#4802" class="Bound">e</a> <a id="4804" href="Algebra.Definitions.html#4804" class="Bound Operator">_•_</a> <a id="4808" class="Symbol">=</a> <a id="4810" href="Algebra.Definitions.html#4500" class="Function">AlmostLeftCancellative</a> <a id="4833" href="Algebra.Definitions.html#4802" class="Bound">e</a> <a id="4835" href="Algebra.Definitions.html#4804" class="Bound Operator">_•_</a> <a id="4839" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="4841" href="Algebra.Definitions.html#4621" class="Function">AlmostRightCancellative</a> <a id="4865" href="Algebra.Definitions.html#4802" class="Bound">e</a> <a id="4867" href="Algebra.Definitions.html#4804" class="Bound Operator">_•_</a>
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<a id="7231" href="Algebra.Definitions.html#7206" class="Function">RightBol</a> <a id="7240" href="Algebra.Definitions.html#7240" class="Bound Operator">_∙_</a> <a id="7244" class="Symbol">=</a> <a id="7246" class="Symbol">∀</a> <a id="7248" href="Algebra.Definitions.html#7248" class="Bound">x</a> <a id="7250" href="Algebra.Definitions.html#7250" class="Bound">y</a> <a id="7252" href="Algebra.Definitions.html#7252" class="Bound">z</a> <a id="7254" class="Symbol">→</a> <a id="7256" class="Symbol">(((</a><a id="7259" href="Algebra.Definitions.html#7252" class="Bound">z</a> <a id="7261" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7263" href="Algebra.Definitions.html#7248" class="Bound">x</a><a id="7264" class="Symbol">)</a> <a id="7266" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7268" href="Algebra.Definitions.html#7250" class="Bound">y</a><a id="7269" class="Symbol">)</a> <a id="7271" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7273" href="Algebra.Definitions.html#7248" class="Bound">x</a><a id="7274" class="Symbol">)</a> <a id="7276" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="7278" class="Symbol">(</a><a id="7279" href="Algebra.Definitions.html#7252" class="Bound">z</a> <a id="7281" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7283" class="Symbol">((</a><a id="7285" href="Algebra.Definitions.html#7248" class="Bound">x</a> <a id="7287" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7289" href="Algebra.Definitions.html#7250" class="Bound">y</a><a id="7290" class="Symbol">)</a> <a id="7292" href="Algebra.Definitions.html#7240" class="Bound Operator">∙</a> <a id="7294" href="Algebra.Definitions.html#7248" class="Bound">x</a><a id="7295" class="Symbol">))</a>
<a id="MiddleBol"></a><a id="7299" href="Algebra.Definitions.html#7299" class="Function">MiddleBol</a> <a id="7309" class="Symbol">:</a> <a id="7311" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7315" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="7317" class="Symbol">→</a> <a id="7319" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7323" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="7326" class="Symbol">→</a> <a id="7328" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7332" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="7335" class="Symbol">→</a> <a id="7337" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="7341" class="Symbol">_</a>
<a id="7343" href="Algebra.Definitions.html#7299" class="Function">MiddleBol</a> <a id="7353" href="Algebra.Definitions.html#7353" class="Bound Operator">_∙_</a> <a id="7357" href="Algebra.Definitions.html#7357" class="Bound Operator">_\\_</a> <a id="7362" href="Algebra.Definitions.html#7362" class="Bound Operator">_//_</a> <a id="7367" class="Symbol">=</a> <a id="7369" class="Symbol">∀</a> <a id="7371" href="Algebra.Definitions.html#7371" class="Bound">x</a> <a id="7373" href="Algebra.Definitions.html#7373" class="Bound">y</a> <a id="7375" href="Algebra.Definitions.html#7375" class="Bound">z</a> <a id="7377" class="Symbol">→</a> <a id="7379" class="Symbol">(</a><a id="7380" href="Algebra.Definitions.html#7371" class="Bound">x</a> <a id="7382" href="Algebra.Definitions.html#7353" class="Bound Operator">∙</a> <a id="7384" class="Symbol">((</a><a id="7386" href="Algebra.Definitions.html#7373" class="Bound">y</a> <a id="7388" href="Algebra.Definitions.html#7353" class="Bound Operator">∙</a> <a id="7390" href="Algebra.Definitions.html#7375" class="Bound">z</a><a id="7391" class="Symbol">)</a> <a id="7393" href="Algebra.Definitions.html#7357" class="Bound Operator">\\</a> <a id="7396" href="Algebra.Definitions.html#7371" class="Bound">x</a><a id="7397" class="Symbol">))</a> <a id="7400" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="7402" class="Symbol">((</a><a id="7404" href="Algebra.Definitions.html#7371" class="Bound">x</a> <a id="7406" href="Algebra.Definitions.html#7362" class="Bound Operator">//</a> <a id="7409" href="Algebra.Definitions.html#7375" class="Bound">z</a><a id="7410" class="Symbol">)</a> <a id="7412" href="Algebra.Definitions.html#7353" class="Bound Operator">∙</a> <a id="7414" class="Symbol">(</a><a id="7415" href="Algebra.Definitions.html#7373" class="Bound">y</a> <a id="7417" href="Algebra.Definitions.html#7357" class="Bound Operator">\\</a> <a id="7420" href="Algebra.Definitions.html#7371" class="Bound">x</a><a id="7421" class="Symbol">))</a>
<a id="Identical"></a><a id="7425" href="Algebra.Definitions.html#7425" class="Function">Identical</a> <a id="7435" class="Symbol">:</a> <a id="7437" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7441" href="Algebra.Definitions.html#909" class="Bound">A</a> <a id="7443" class="Symbol">→</a> <a id="7445" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="7449" class="Symbol">_</a>
<a id="7451" href="Algebra.Definitions.html#7425" class="Function">Identical</a> <a id="7461" href="Algebra.Definitions.html#7461" class="Bound Operator">_∙_</a> <a id="7465" class="Symbol">=</a> <a id="7467" class="Symbol">∀</a> <a id="7469" href="Algebra.Definitions.html#7469" class="Bound">x</a> <a id="7471" href="Algebra.Definitions.html#7471" class="Bound">y</a> <a id="7473" href="Algebra.Definitions.html#7473" class="Bound">z</a> <a id="7475" class="Symbol">→</a> <a id="7477" class="Symbol">((</a><a id="7479" href="Algebra.Definitions.html#7473" class="Bound">z</a> <a id="7481" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7483" href="Algebra.Definitions.html#7469" class="Bound">x</a><a id="7484" class="Symbol">)</a> <a id="7486" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7488" class="Symbol">(</a><a id="7489" href="Algebra.Definitions.html#7471" class="Bound">y</a> <a id="7491" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7493" href="Algebra.Definitions.html#7473" class="Bound">z</a><a id="7494" class="Symbol">))</a> <a id="7497" href="Algebra.Definitions.html#947" class="Bound Operator">≈</a> <a id="7499" class="Symbol">(</a><a id="7500" href="Algebra.Definitions.html#7473" class="Bound">z</a> <a id="7502" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7504" class="Symbol">((</a><a id="7506" href="Algebra.Definitions.html#7469" class="Bound">x</a> <a id="7508" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7510" href="Algebra.Definitions.html#7471" class="Bound">y</a><a id="7511" class="Symbol">)</a> <a id="7513" href="Algebra.Definitions.html#7461" class="Bound Operator">∙</a> <a id="7515" href="Algebra.Definitions.html#7473" class="Bound">z</a><a id="7516" class="Symbol">))</a>
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