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<html><head><meta charset="utf-8"><title>Relation.Binary.Construct.On</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">-- Many properties which hold for `_∼_` also hold for `_∼_ on f`</a>
<a id="171" class="Comment">------------------------------------------------------------------------</a>
<a id="245" class="Symbol">{-#</a> <a id="249" class="Keyword">OPTIONS</a> <a id="257" class="Pragma">--cubical-compatible</a> <a id="278" class="Pragma">--safe</a> <a id="285" class="Symbol">#-}</a>
<a id="290" class="Keyword">module</a> <a id="297" href="Relation.Binary.Construct.On.html" class="Module">Relation.Binary.Construct.On</a> <a id="326" class="Keyword">where</a>
<a id="333" class="Keyword">open</a> <a id="338" class="Keyword">import</a> <a id="345" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="363" class="Keyword">using</a> <a id="369" class="Symbol">(</a><a id="370" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="373" class="Symbol">)</a>
<a id="375" class="Keyword">open</a> <a id="380" class="Keyword">import</a> <a id="387" href="Function.Base.html" class="Module">Function.Base</a> <a id="401" class="Keyword">using</a> <a id="407" class="Symbol">(</a><a id="408" href="Function.Base.html#6209" class="Function Operator">_on_</a><a id="412" class="Symbol">;</a> <a id="414" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="417" class="Symbol">)</a>
<a id="419" class="Keyword">open</a> <a id="424" class="Keyword">import</a> <a id="431" href="Induction.WellFounded.html" class="Module">Induction.WellFounded</a> <a id="453" class="Keyword">using</a> <a id="459" class="Symbol">(</a><a id="460" href="Induction.WellFounded.html#1604" class="Function">WellFounded</a><a id="471" class="Symbol">;</a> <a id="473" href="Induction.WellFounded.html#1356" class="Datatype">Acc</a><a id="476" class="Symbol">;</a> <a id="478" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a><a id="481" class="Symbol">)</a>
<a id="483" class="Keyword">open</a> <a id="488" class="Keyword">import</a> <a id="495" href="Level.html" class="Module">Level</a> <a id="501" class="Keyword">using</a> <a id="507" class="Symbol">(</a><a id="508" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="513" class="Symbol">)</a>
<a id="515" class="Keyword">open</a> <a id="520" class="Keyword">import</a> <a id="527" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="548" class="Keyword">using</a> <a id="554" class="Symbol">(</a><a id="555" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="558" class="Symbol">;</a> <a id="560" href="Relation.Binary.Core.html#1268" class="Function Operator">_⇒_</a><a id="563" class="Symbol">)</a>
<a id="565" class="Keyword">open</a> <a id="570" class="Keyword">import</a> <a id="577" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a>
<a id="603" class="Keyword">using</a> <a id="609" class="Symbol">(</a><a id="610" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a><a id="618" class="Symbol">;</a> <a id="620" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="626" class="Symbol">;</a> <a id="628" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a><a id="637" class="Symbol">;</a> <a id="639" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="644" class="Symbol">;</a> <a id="646" href="Relation.Binary.Bundles.html#4140" class="Record">DecPoset</a><a id="654" class="Symbol">;</a> <a id="656" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a><a id="674" class="Symbol">;</a> <a id="676" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a><a id="686" class="Symbol">;</a> <a id="688" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a><a id="701" class="Symbol">;</a> <a id="703" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a><a id="719" class="Symbol">)</a>
<a id="721" class="Keyword">open</a> <a id="726" class="Keyword">import</a> <a id="733" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
<a id="762" class="Keyword">using</a> <a id="768" class="Symbol">(</a><a id="769" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="782" class="Symbol">;</a> <a id="784" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a><a id="800" class="Symbol">;</a> <a id="802" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="812" class="Symbol">;</a> <a id="814" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="828" class="Symbol">;</a> <a id="830" href="Relation.Binary.Structures.html#3809" class="Record">IsDecPartialOrder</a><a id="847" class="Symbol">;</a> <a id="849" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a><a id="869" class="Symbol">;</a> <a id="871" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a><a id="883" class="Symbol">;</a> <a id="885" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a><a id="900" class="Symbol">;</a> <a id="902" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a><a id="920" class="Symbol">)</a>
<a id="922" class="Keyword">open</a> <a id="927" class="Keyword">import</a> <a id="934" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
<a id="964" class="Keyword">using</a> <a id="970" class="Symbol">(</a><a id="971" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a><a id="980" class="Symbol">;</a> <a id="982" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a><a id="993" class="Symbol">;</a> <a id="995" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a><a id="1004" class="Symbol">;</a> <a id="1006" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a><a id="1016" class="Symbol">;</a> <a id="1018" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a><a id="1031" class="Symbol">;</a> <a id="1033" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a><a id="1043" class="Symbol">;</a> <a id="1045" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="1054" class="Symbol">;</a> <a id="1056" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="1061" class="Symbol">;</a> <a id="1063" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a><a id="1075" class="Symbol">;</a> <a id="1077" href="Relation.Binary.Definitions.html#5179" class="Function Operator">_Respects_</a><a id="1087" class="Symbol">;</a> <a id="1089" href="Relation.Binary.Definitions.html#5663" class="Function Operator">_Respects₂_</a><a id="1100" class="Symbol">)</a>
<a id="1103" class="Keyword">private</a>
<a id="1113" class="Keyword">variable</a>
<a id="1126" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="1128" href="Relation.Binary.Construct.On.html#1128" class="Generalizable">b</a> <a id="1130" href="Relation.Binary.Construct.On.html#1130" class="Generalizable">p</a> <a id="1132" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a> <a id="1134" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a> <a id="1137" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a> <a id="1140" class="Symbol">:</a> <a id="1142" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="1152" href="Relation.Binary.Construct.On.html#1152" class="Generalizable">A</a> <a id="1154" class="Symbol">:</a> <a id="1156" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1160" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a>
<a id="1166" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="1168" class="Symbol">:</a> <a id="1170" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1174" href="Relation.Binary.Construct.On.html#1128" class="Generalizable">b</a>
<a id="1177" class="Comment">------------------------------------------------------------------------</a>
<a id="1250" class="Comment">-- Definitions</a>
<a id="1266" class="Keyword">module</a> <a id="1273" href="Relation.Binary.Construct.On.html#1273" class="Module">_</a> <a id="1275" class="Symbol">(</a><a id="1276" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="1278" class="Symbol">:</a> <a id="1280" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="1282" class="Symbol">→</a> <a id="1284" href="Relation.Binary.Construct.On.html#1152" class="Generalizable">A</a><a id="1285" class="Symbol">)</a> <a id="1287" class="Keyword">where</a>
<a id="1296" href="Relation.Binary.Construct.On.html#1296" class="Function">implies</a> <a id="1304" class="Symbol">:</a> <a id="1306" class="Symbol">(</a><a id="1307" href="Relation.Binary.Construct.On.html#1307" class="Bound">≈</a> <a id="1309" class="Symbol">:</a> <a id="1311" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1315" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1317" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="1319" class="Symbol">)</a> <a id="1321" class="Symbol">(</a><a id="1322" href="Relation.Binary.Construct.On.html#1322" class="Bound">∼</a> <a id="1324" class="Symbol">:</a> <a id="1326" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1330" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1332" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="1334" class="Symbol">)</a> <a id="1336" class="Symbol">→</a>
<a id="1350" href="Relation.Binary.Construct.On.html#1307" class="Bound">≈</a> <a id="1352" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1354" href="Relation.Binary.Construct.On.html#1322" class="Bound">∼</a> <a id="1356" class="Symbol">→</a> <a id="1358" class="Symbol">(</a><a id="1359" href="Relation.Binary.Construct.On.html#1307" class="Bound">≈</a> <a id="1361" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1364" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1365" class="Symbol">)</a> <a id="1367" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1369" class="Symbol">(</a><a id="1370" href="Relation.Binary.Construct.On.html#1322" class="Bound">∼</a> <a id="1372" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1375" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1376" class="Symbol">)</a>
<a id="1380" href="Relation.Binary.Construct.On.html#1296" class="Function">implies</a> <a id="1388" class="Symbol">_</a> <a id="1390" class="Symbol">_</a> <a id="1392" href="Relation.Binary.Construct.On.html#1392" class="Bound">impl</a> <a id="1397" class="Symbol">=</a> <a id="1399" href="Relation.Binary.Construct.On.html#1392" class="Bound">impl</a>
<a id="1407" href="Relation.Binary.Construct.On.html#1407" class="Function">reflexive</a> <a id="1417" class="Symbol">:</a> <a id="1419" class="Symbol">(</a><a id="1420" href="Relation.Binary.Construct.On.html#1420" class="Bound">∼</a> <a id="1422" class="Symbol">:</a> <a id="1424" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1428" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1430" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="1431" class="Symbol">)</a> <a id="1433" class="Symbol">→</a> <a id="1435" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1445" href="Relation.Binary.Construct.On.html#1420" class="Bound">∼</a> <a id="1447" class="Symbol">→</a> <a id="1449" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1459" class="Symbol">(</a><a id="1460" href="Relation.Binary.Construct.On.html#1420" class="Bound">∼</a> <a id="1462" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1465" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1466" class="Symbol">)</a>
<a id="1470" href="Relation.Binary.Construct.On.html#1407" class="Function">reflexive</a> <a id="1480" class="Symbol">_</a> <a id="1482" href="Relation.Binary.Construct.On.html#1482" class="Bound">refl</a> <a id="1487" class="Symbol">=</a> <a id="1489" href="Relation.Binary.Construct.On.html#1482" class="Bound">refl</a>
<a id="1497" href="Relation.Binary.Construct.On.html#1497" class="Function">irreflexive</a> <a id="1509" class="Symbol">:</a> <a id="1511" class="Symbol">(</a><a id="1512" href="Relation.Binary.Construct.On.html#1512" class="Bound">≈</a> <a id="1514" class="Symbol">:</a> <a id="1516" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1520" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1522" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="1524" class="Symbol">)</a> <a id="1526" class="Symbol">(</a><a id="1527" href="Relation.Binary.Construct.On.html#1527" class="Bound">∼</a> <a id="1529" class="Symbol">:</a> <a id="1531" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1535" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1537" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="1539" class="Symbol">)</a> <a id="1541" class="Symbol">→</a>
<a id="1559" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1571" href="Relation.Binary.Construct.On.html#1512" class="Bound">≈</a> <a id="1573" href="Relation.Binary.Construct.On.html#1527" class="Bound">∼</a> <a id="1575" class="Symbol">→</a> <a id="1577" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1589" class="Symbol">(</a><a id="1590" href="Relation.Binary.Construct.On.html#1512" class="Bound">≈</a> <a id="1592" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1595" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1596" class="Symbol">)</a> <a id="1598" class="Symbol">(</a><a id="1599" href="Relation.Binary.Construct.On.html#1527" class="Bound">∼</a> <a id="1601" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1604" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1605" class="Symbol">)</a>
<a id="1609" href="Relation.Binary.Construct.On.html#1497" class="Function">irreflexive</a> <a id="1621" class="Symbol">_</a> <a id="1623" class="Symbol">_</a> <a id="1625" href="Relation.Binary.Construct.On.html#1625" class="Bound">irrefl</a> <a id="1632" class="Symbol">=</a> <a id="1634" href="Relation.Binary.Construct.On.html#1625" class="Bound">irrefl</a>
<a id="1644" href="Relation.Binary.Construct.On.html#1644" class="Function">symmetric</a> <a id="1654" class="Symbol">:</a> <a id="1656" class="Symbol">(</a><a id="1657" href="Relation.Binary.Construct.On.html#1657" class="Bound">∼</a> <a id="1659" class="Symbol">:</a> <a id="1661" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1665" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1667" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="1668" class="Symbol">)</a> <a id="1670" class="Symbol">→</a> <a id="1672" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1682" href="Relation.Binary.Construct.On.html#1657" class="Bound">∼</a> <a id="1684" class="Symbol">→</a> <a id="1686" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1696" class="Symbol">(</a><a id="1697" href="Relation.Binary.Construct.On.html#1657" class="Bound">∼</a> <a id="1699" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1702" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1703" class="Symbol">)</a>
<a id="1707" href="Relation.Binary.Construct.On.html#1644" class="Function">symmetric</a> <a id="1717" class="Symbol">_</a> <a id="1719" href="Relation.Binary.Construct.On.html#1719" class="Bound">sym</a> <a id="1723" class="Symbol">=</a> <a id="1725" href="Relation.Binary.Construct.On.html#1719" class="Bound">sym</a>
<a id="1732" href="Relation.Binary.Construct.On.html#1732" class="Function">transitive</a> <a id="1743" class="Symbol">:</a> <a id="1745" class="Symbol">(</a><a id="1746" href="Relation.Binary.Construct.On.html#1746" class="Bound">∼</a> <a id="1748" class="Symbol">:</a> <a id="1750" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1754" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1756" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="1757" class="Symbol">)</a> <a id="1759" class="Symbol">→</a> <a id="1761" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1772" href="Relation.Binary.Construct.On.html#1746" class="Bound">∼</a> <a id="1774" class="Symbol">→</a> <a id="1776" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1787" class="Symbol">(</a><a id="1788" href="Relation.Binary.Construct.On.html#1746" class="Bound">∼</a> <a id="1790" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1793" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1794" class="Symbol">)</a>
<a id="1798" href="Relation.Binary.Construct.On.html#1732" class="Function">transitive</a> <a id="1809" class="Symbol">_</a> <a id="1811" href="Relation.Binary.Construct.On.html#1811" class="Bound">trans</a> <a id="1817" class="Symbol">=</a> <a id="1819" href="Relation.Binary.Construct.On.html#1811" class="Bound">trans</a>
<a id="1828" href="Relation.Binary.Construct.On.html#1828" class="Function">antisymmetric</a> <a id="1842" class="Symbol">:</a> <a id="1844" class="Symbol">(</a><a id="1845" href="Relation.Binary.Construct.On.html#1845" class="Bound">≈</a> <a id="1847" class="Symbol">:</a> <a id="1849" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1853" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1855" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="1857" class="Symbol">)</a> <a id="1859" class="Symbol">(</a><a id="1860" href="Relation.Binary.Construct.On.html#1860" class="Bound">≤</a> <a id="1862" class="Symbol">:</a> <a id="1864" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1868" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="1870" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="1872" class="Symbol">)</a> <a id="1874" class="Symbol">→</a>
<a id="1894" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1908" href="Relation.Binary.Construct.On.html#1845" class="Bound">≈</a> <a id="1910" href="Relation.Binary.Construct.On.html#1860" class="Bound">≤</a> <a id="1912" class="Symbol">→</a> <a id="1914" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1928" class="Symbol">(</a><a id="1929" href="Relation.Binary.Construct.On.html#1845" class="Bound">≈</a> <a id="1931" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1934" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1935" class="Symbol">)</a> <a id="1937" class="Symbol">(</a><a id="1938" href="Relation.Binary.Construct.On.html#1860" class="Bound">≤</a> <a id="1940" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1943" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="1944" class="Symbol">)</a>
<a id="1948" href="Relation.Binary.Construct.On.html#1828" class="Function">antisymmetric</a> <a id="1962" class="Symbol">_</a> <a id="1964" class="Symbol">_</a> <a id="1966" href="Relation.Binary.Construct.On.html#1966" class="Bound">antisym</a> <a id="1974" class="Symbol">=</a> <a id="1976" href="Relation.Binary.Construct.On.html#1966" class="Bound">antisym</a>
<a id="1987" href="Relation.Binary.Construct.On.html#1987" class="Function">asymmetric</a> <a id="1998" class="Symbol">:</a> <a id="2000" class="Symbol">(</a><a id="2001" href="Relation.Binary.Construct.On.html#2001" class="Bound">&lt;</a> <a id="2003" class="Symbol">:</a> <a id="2005" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2009" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2011" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2012" class="Symbol">)</a> <a id="2014" class="Symbol">→</a> <a id="2016" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2027" href="Relation.Binary.Construct.On.html#2001" class="Bound">&lt;</a> <a id="2029" class="Symbol">→</a> <a id="2031" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2042" class="Symbol">(</a><a id="2043" href="Relation.Binary.Construct.On.html#2001" class="Bound">&lt;</a> <a id="2045" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2048" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2049" class="Symbol">)</a>
<a id="2053" href="Relation.Binary.Construct.On.html#1987" class="Function">asymmetric</a> <a id="2064" class="Symbol">_</a> <a id="2066" href="Relation.Binary.Construct.On.html#2066" class="Bound">asym</a> <a id="2071" class="Symbol">=</a> <a id="2073" href="Relation.Binary.Construct.On.html#2066" class="Bound">asym</a>
<a id="2081" href="Relation.Binary.Construct.On.html#2081" class="Function">respects</a> <a id="2090" class="Symbol">:</a> <a id="2092" class="Symbol">(</a><a id="2093" href="Relation.Binary.Construct.On.html#2093" class="Bound">∼</a> <a id="2095" class="Symbol">:</a> <a id="2097" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2101" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2103" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2104" class="Symbol">)</a> <a id="2106" class="Symbol">(</a><a id="2107" href="Relation.Binary.Construct.On.html#2107" class="Bound">P</a> <a id="2109" class="Symbol">:</a> <a id="2111" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2113" class="Symbol">→</a> <a id="2115" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2119" href="Relation.Binary.Construct.On.html#1130" class="Generalizable">p</a><a id="2120" class="Symbol">)</a> <a id="2122" class="Symbol">→</a>
<a id="2137" href="Relation.Binary.Construct.On.html#2107" class="Bound">P</a> <a id="2139" href="Relation.Binary.Definitions.html#5179" class="Function Operator">Respects</a> <a id="2148" href="Relation.Binary.Construct.On.html#2093" class="Bound">∼</a> <a id="2150" class="Symbol">→</a> <a id="2152" class="Symbol">(</a><a id="2153" href="Relation.Binary.Construct.On.html#2107" class="Bound">P</a> <a id="2155" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2157" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2158" class="Symbol">)</a> <a id="2160" href="Relation.Binary.Definitions.html#5179" class="Function Operator">Respects</a> <a id="2169" class="Symbol">(</a><a id="2170" href="Relation.Binary.Construct.On.html#2093" class="Bound">∼</a> <a id="2172" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2175" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2176" class="Symbol">)</a>
<a id="2180" href="Relation.Binary.Construct.On.html#2081" class="Function">respects</a> <a id="2189" class="Symbol">_</a> <a id="2191" class="Symbol">_</a> <a id="2193" href="Relation.Binary.Construct.On.html#2193" class="Bound">resp</a> <a id="2198" class="Symbol">=</a> <a id="2200" href="Relation.Binary.Construct.On.html#2193" class="Bound">resp</a>
<a id="2208" href="Relation.Binary.Construct.On.html#2208" class="Function">respects₂</a> <a id="2218" class="Symbol">:</a> <a id="2220" class="Symbol">(</a><a id="2221" href="Relation.Binary.Construct.On.html#2221" class="Bound">∼₁</a> <a id="2224" class="Symbol">:</a> <a id="2226" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2230" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2232" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="2234" class="Symbol">)</a> <a id="2236" class="Symbol">(</a><a id="2237" href="Relation.Binary.Construct.On.html#2237" class="Bound">∼₂</a> <a id="2240" class="Symbol">:</a> <a id="2242" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2246" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2248" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="2250" class="Symbol">)</a> <a id="2252" class="Symbol">→</a>
<a id="2268" href="Relation.Binary.Construct.On.html#2221" class="Bound">∼₁</a> <a id="2271" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2281" href="Relation.Binary.Construct.On.html#2237" class="Bound">∼₂</a> <a id="2284" class="Symbol">→</a> <a id="2286" class="Symbol">(</a><a id="2287" href="Relation.Binary.Construct.On.html#2221" class="Bound">∼₁</a> <a id="2290" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2293" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2294" class="Symbol">)</a> <a id="2296" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2306" class="Symbol">(</a><a id="2307" href="Relation.Binary.Construct.On.html#2237" class="Bound">∼₂</a> <a id="2310" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2313" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2314" class="Symbol">)</a>
<a id="2318" href="Relation.Binary.Construct.On.html#2208" class="Function">respects₂</a> <a id="2328" class="Symbol">_</a> <a id="2330" class="Symbol">_</a> <a id="2332" class="Symbol">(</a><a id="2333" href="Relation.Binary.Construct.On.html#2333" class="Bound">resp₁</a> <a id="2339" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2341" href="Relation.Binary.Construct.On.html#2341" class="Bound">resp₂</a><a id="2346" class="Symbol">)</a> <a id="2348" class="Symbol">=</a> <a id="2350" class="Symbol">(</a><a id="2351" href="Relation.Binary.Construct.On.html#2333" class="Bound">resp₁</a> <a id="2357" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2359" href="Relation.Binary.Construct.On.html#2341" class="Bound">resp₂</a><a id="2364" class="Symbol">)</a>
<a id="2369" href="Relation.Binary.Construct.On.html#2369" class="Function">decidable</a> <a id="2379" class="Symbol">:</a> <a id="2381" class="Symbol">(</a><a id="2382" href="Relation.Binary.Construct.On.html#2382" class="Bound">∼</a> <a id="2384" class="Symbol">:</a> <a id="2386" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2390" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2392" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2393" class="Symbol">)</a> <a id="2395" class="Symbol">→</a> <a id="2397" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2407" href="Relation.Binary.Construct.On.html#2382" class="Bound">∼</a> <a id="2409" class="Symbol">→</a> <a id="2411" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2421" class="Symbol">(</a><a id="2422" href="Relation.Binary.Construct.On.html#2382" class="Bound">∼</a> <a id="2424" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2427" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2428" class="Symbol">)</a>
<a id="2432" href="Relation.Binary.Construct.On.html#2369" class="Function">decidable</a> <a id="2442" class="Symbol">_</a> <a id="2444" href="Relation.Binary.Construct.On.html#2444" class="Bound">dec</a> <a id="2448" href="Relation.Binary.Construct.On.html#2448" class="Bound">x</a> <a id="2450" href="Relation.Binary.Construct.On.html#2450" class="Bound">y</a> <a id="2452" class="Symbol">=</a> <a id="2454" href="Relation.Binary.Construct.On.html#2444" class="Bound">dec</a> <a id="2458" class="Symbol">(</a><a id="2459" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2461" href="Relation.Binary.Construct.On.html#2448" class="Bound">x</a><a id="2462" class="Symbol">)</a> <a id="2464" class="Symbol">(</a><a id="2465" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2467" href="Relation.Binary.Construct.On.html#2450" class="Bound">y</a><a id="2468" class="Symbol">)</a>
<a id="2473" href="Relation.Binary.Construct.On.html#2473" class="Function">total</a> <a id="2479" class="Symbol">:</a> <a id="2481" class="Symbol">(</a><a id="2482" href="Relation.Binary.Construct.On.html#2482" class="Bound">∼</a> <a id="2484" class="Symbol">:</a> <a id="2486" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2490" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2492" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2493" class="Symbol">)</a> <a id="2495" class="Symbol">→</a> <a id="2497" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2503" href="Relation.Binary.Construct.On.html#2482" class="Bound">∼</a> <a id="2505" class="Symbol">→</a> <a id="2507" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2513" class="Symbol">(</a><a id="2514" href="Relation.Binary.Construct.On.html#2482" class="Bound">∼</a> <a id="2516" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2519" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2520" class="Symbol">)</a>
<a id="2524" href="Relation.Binary.Construct.On.html#2473" class="Function">total</a> <a id="2530" class="Symbol">_</a> <a id="2532" href="Relation.Binary.Construct.On.html#2532" class="Bound">tot</a> <a id="2536" href="Relation.Binary.Construct.On.html#2536" class="Bound">x</a> <a id="2538" href="Relation.Binary.Construct.On.html#2538" class="Bound">y</a> <a id="2540" class="Symbol">=</a> <a id="2542" href="Relation.Binary.Construct.On.html#2532" class="Bound">tot</a> <a id="2546" class="Symbol">(</a><a id="2547" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2549" href="Relation.Binary.Construct.On.html#2536" class="Bound">x</a><a id="2550" class="Symbol">)</a> <a id="2552" class="Symbol">(</a><a id="2553" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2555" href="Relation.Binary.Construct.On.html#2538" class="Bound">y</a><a id="2556" class="Symbol">)</a>
<a id="2561" href="Relation.Binary.Construct.On.html#2561" class="Function">trichotomous</a> <a id="2574" class="Symbol">:</a> <a id="2576" class="Symbol">(</a><a id="2577" href="Relation.Binary.Construct.On.html#2577" class="Bound">≈</a> <a id="2579" class="Symbol">:</a> <a id="2581" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2585" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2587" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="2589" class="Symbol">)</a> <a id="2591" class="Symbol">(</a><a id="2592" href="Relation.Binary.Construct.On.html#2592" class="Bound">&lt;</a> <a id="2594" class="Symbol">:</a> <a id="2596" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2600" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2602" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="2604" class="Symbol">)</a> <a id="2606" class="Symbol">→</a>
<a id="2625" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="2638" href="Relation.Binary.Construct.On.html#2577" class="Bound">≈</a> <a id="2640" href="Relation.Binary.Construct.On.html#2592" class="Bound">&lt;</a> <a id="2642" class="Symbol">→</a> <a id="2644" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="2657" class="Symbol">(</a><a id="2658" href="Relation.Binary.Construct.On.html#2577" class="Bound">≈</a> <a id="2660" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2663" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2664" class="Symbol">)</a> <a id="2666" class="Symbol">(</a><a id="2667" href="Relation.Binary.Construct.On.html#2592" class="Bound">&lt;</a> <a id="2669" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2672" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2673" class="Symbol">)</a>
<a id="2677" href="Relation.Binary.Construct.On.html#2561" class="Function">trichotomous</a> <a id="2690" class="Symbol">_</a> <a id="2692" class="Symbol">_</a> <a id="2694" href="Relation.Binary.Construct.On.html#2694" class="Bound">compare</a> <a id="2702" href="Relation.Binary.Construct.On.html#2702" class="Bound">x</a> <a id="2704" href="Relation.Binary.Construct.On.html#2704" class="Bound">y</a> <a id="2706" class="Symbol">=</a> <a id="2708" href="Relation.Binary.Construct.On.html#2694" class="Bound">compare</a> <a id="2716" class="Symbol">(</a><a id="2717" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2719" href="Relation.Binary.Construct.On.html#2702" class="Bound">x</a><a id="2720" class="Symbol">)</a> <a id="2722" class="Symbol">(</a><a id="2723" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2725" href="Relation.Binary.Construct.On.html#2704" class="Bound">y</a><a id="2726" class="Symbol">)</a>
<a id="2731" href="Relation.Binary.Construct.On.html#2731" class="Function">accessible</a> <a id="2742" class="Symbol">:</a> <a id="2744" class="Symbol">∀</a> <a id="2746" class="Symbol">{</a><a id="2747" href="Relation.Binary.Construct.On.html#2747" class="Bound">∼</a> <a id="2749" class="Symbol">:</a> <a id="2751" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2755" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2757" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2758" class="Symbol">}</a> <a id="2760" class="Symbol">{</a><a id="2761" href="Relation.Binary.Construct.On.html#2761" class="Bound">x</a><a id="2762" class="Symbol">}</a> <a id="2764" class="Symbol">→</a> <a id="2766" href="Induction.WellFounded.html#1356" class="Datatype">Acc</a> <a id="2770" href="Relation.Binary.Construct.On.html#2747" class="Bound">∼</a> <a id="2772" class="Symbol">(</a><a id="2773" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2775" href="Relation.Binary.Construct.On.html#2761" class="Bound">x</a><a id="2776" class="Symbol">)</a> <a id="2778" class="Symbol">→</a> <a id="2780" href="Induction.WellFounded.html#1356" class="Datatype">Acc</a> <a id="2784" class="Symbol">(</a><a id="2785" href="Relation.Binary.Construct.On.html#2747" class="Bound">∼</a> <a id="2787" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2790" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2791" class="Symbol">)</a> <a id="2793" href="Relation.Binary.Construct.On.html#2761" class="Bound">x</a>
<a id="2797" href="Relation.Binary.Construct.On.html#2731" class="Function">accessible</a> <a id="2808" class="Symbol">(</a><a id="2809" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a> <a id="2813" href="Relation.Binary.Construct.On.html#2813" class="Bound">rs</a><a id="2815" class="Symbol">)</a> <a id="2817" class="Symbol">=</a> <a id="2819" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a> <a id="2823" class="Symbol">(λ</a> <a id="2826" href="Relation.Binary.Construct.On.html#2826" class="Bound">fy&lt;fx</a> <a id="2832" class="Symbol">→</a> <a id="2834" href="Relation.Binary.Construct.On.html#2731" class="Function">accessible</a> <a id="2845" class="Symbol">(</a><a id="2846" href="Relation.Binary.Construct.On.html#2813" class="Bound">rs</a> <a id="2849" href="Relation.Binary.Construct.On.html#2826" class="Bound">fy&lt;fx</a><a id="2854" class="Symbol">))</a>
<a id="2860" href="Relation.Binary.Construct.On.html#2860" class="Function">wellFounded</a> <a id="2872" class="Symbol">:</a> <a id="2874" class="Symbol">{</a><a id="2875" href="Relation.Binary.Construct.On.html#2875" class="Bound">∼</a> <a id="2877" class="Symbol">:</a> <a id="2879" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2883" href="Relation.Binary.Construct.On.html#1284" class="Bound">A</a> <a id="2885" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="2886" class="Symbol">}</a> <a id="2888" class="Symbol">→</a> <a id="2890" href="Induction.WellFounded.html#1604" class="Function">WellFounded</a> <a id="2902" href="Relation.Binary.Construct.On.html#2875" class="Bound">∼</a> <a id="2904" class="Symbol">→</a> <a id="2906" href="Induction.WellFounded.html#1604" class="Function">WellFounded</a> <a id="2918" class="Symbol">(</a><a id="2919" href="Relation.Binary.Construct.On.html#2875" class="Bound">∼</a> <a id="2921" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="2924" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a><a id="2925" class="Symbol">)</a>
<a id="2929" href="Relation.Binary.Construct.On.html#2860" class="Function">wellFounded</a> <a id="2941" href="Relation.Binary.Construct.On.html#2941" class="Bound">wf</a> <a id="2944" href="Relation.Binary.Construct.On.html#2944" class="Bound">x</a> <a id="2946" class="Symbol">=</a> <a id="2948" href="Relation.Binary.Construct.On.html#2731" class="Function">accessible</a> <a id="2959" class="Symbol">(</a><a id="2960" href="Relation.Binary.Construct.On.html#2941" class="Bound">wf</a> <a id="2963" class="Symbol">(</a><a id="2964" href="Relation.Binary.Construct.On.html#1276" class="Bound">f</a> <a id="2966" href="Relation.Binary.Construct.On.html#2944" class="Bound">x</a><a id="2967" class="Symbol">))</a>
<a id="2971" class="Comment">------------------------------------------------------------------------</a>
<a id="3044" class="Comment">-- Structures</a>
<a id="3059" class="Keyword">module</a> <a id="3066" href="Relation.Binary.Construct.On.html#3066" class="Module">_</a> <a id="3068" class="Symbol">(</a><a id="3069" href="Relation.Binary.Construct.On.html#3069" class="Bound">f</a> <a id="3071" class="Symbol">:</a> <a id="3073" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="3075" class="Symbol">→</a> <a id="3077" href="Relation.Binary.Construct.On.html#1152" class="Generalizable">A</a><a id="3078" class="Symbol">)</a> <a id="3080" class="Symbol">{</a><a id="3081" href="Relation.Binary.Construct.On.html#3081" class="Bound">≈</a> <a id="3083" class="Symbol">:</a> <a id="3085" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="3089" href="Relation.Binary.Construct.On.html#1152" class="Generalizable">A</a> <a id="3091" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a><a id="3093" class="Symbol">}</a> <a id="3095" class="Keyword">where</a>
<a id="3104" href="Relation.Binary.Construct.On.html#3104" class="Function">isEquivalence</a> <a id="3118" class="Symbol">:</a> <a id="3120" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="3134" href="Relation.Binary.Construct.On.html#3081" class="Bound">≈</a> <a id="3136" class="Symbol">→</a>
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<a id="4917" class="Symbol">}</a> <a id="4919" class="Keyword">where</a> <a id="4925" class="Keyword">module</a> <a id="4932" href="Relation.Binary.Construct.On.html#4932" class="Module">Spo</a> <a id="4936" class="Symbol">=</a> <a id="4938" href="Relation.Binary.Structures.html#4298" class="Module">IsStrictPartialOrder</a> <a id="4959" href="Relation.Binary.Construct.On.html#4695" class="Bound">spo</a>
<a id="4966" href="Relation.Binary.Construct.On.html#4966" class="Function">isTotalOrder</a> <a id="4979" class="Symbol">:</a> <a id="4981" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="4994" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="4996" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="4998" class="Symbol">→</a>
<a id="5017" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="5030" class="Symbol">(</a><a id="5031" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5033" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5036" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5037" class="Symbol">)</a> <a id="5039" class="Symbol">(</a><a id="5040" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5042" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5045" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5046" class="Symbol">)</a>
<a id="5050" href="Relation.Binary.Construct.On.html#4966" class="Function">isTotalOrder</a> <a id="5063" href="Relation.Binary.Construct.On.html#5063" class="Bound">to</a> <a id="5066" class="Symbol">=</a> <a id="5068" class="Keyword">record</a>
<a id="5079" class="Symbol">{</a> <a id="5081" href="Relation.Binary.Structures.html#5575" class="Field">isPartialOrder</a> <a id="5096" class="Symbol">=</a> <a id="5098" href="Relation.Binary.Construct.On.html#3967" class="Function">isPartialOrder</a> <a id="5113" href="Relation.Binary.Structures.html#5575" class="Field">To.isPartialOrder</a>
<a id="5135" class="Symbol">;</a> <a id="5137" href="Relation.Binary.Structures.html#5615" class="Field">total</a> <a id="5152" class="Symbol">=</a> <a id="5154" href="Relation.Binary.Construct.On.html#2473" class="Function">total</a> <a id="5160" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a> <a id="5162" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5164" href="Relation.Binary.Structures.html#5615" class="Field">To.total</a>
<a id="5177" class="Symbol">}</a> <a id="5179" class="Keyword">where</a> <a id="5185" class="Keyword">module</a> <a id="5192" href="Relation.Binary.Construct.On.html#5192" class="Module">To</a> <a id="5195" class="Symbol">=</a> <a id="5197" href="Relation.Binary.Structures.html#5508" class="Module">IsTotalOrder</a> <a id="5210" href="Relation.Binary.Construct.On.html#5063" class="Bound">to</a>
<a id="5216" href="Relation.Binary.Construct.On.html#5216" class="Function">isDecTotalOrder</a> <a id="5232" class="Symbol">:</a> <a id="5234" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="5250" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5252" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5254" class="Symbol">→</a>
<a id="5276" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="5292" class="Symbol">(</a><a id="5293" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5295" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5298" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5299" class="Symbol">)</a> <a id="5301" class="Symbol">(</a><a id="5302" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5304" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5307" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5308" class="Symbol">)</a>
<a id="5312" href="Relation.Binary.Construct.On.html#5216" class="Function">isDecTotalOrder</a> <a id="5328" href="Relation.Binary.Construct.On.html#5328" class="Bound">dec</a> <a id="5332" class="Symbol">=</a> <a id="5334" class="Keyword">record</a>
<a id="5345" class="Symbol">{</a> <a id="5347" href="Relation.Binary.Structures.html#5914" class="Field">isTotalOrder</a> <a id="5360" class="Symbol">=</a> <a id="5362" href="Relation.Binary.Construct.On.html#4966" class="Function">isTotalOrder</a> <a id="5375" href="Relation.Binary.Structures.html#5914" class="Field">Dec.isTotalOrder</a>
<a id="5396" class="Symbol">;</a> <a id="5398" href="Relation.Binary.Structures.html#5950" class="Field Operator">_≟_</a> <a id="5411" class="Symbol">=</a> <a id="5413" href="Relation.Binary.Construct.On.html#2369" class="Function">decidable</a> <a id="5423" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a> <a id="5425" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5427" href="Relation.Binary.Structures.html#5950" class="Field Operator">Dec._≟_</a>
<a id="5439" class="Symbol">;</a> <a id="5441" href="Relation.Binary.Structures.html#5983" class="Field Operator">_≤?_</a> <a id="5454" class="Symbol">=</a> <a id="5456" href="Relation.Binary.Construct.On.html#2369" class="Function">decidable</a> <a id="5466" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a> <a id="5468" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5470" href="Relation.Binary.Structures.html#5983" class="Field Operator">Dec._≤?_</a>
<a id="5483" class="Symbol">}</a> <a id="5485" class="Keyword">where</a> <a id="5491" class="Keyword">module</a> <a id="5498" href="Relation.Binary.Construct.On.html#5498" class="Module">Dec</a> <a id="5502" class="Symbol">=</a> <a id="5504" href="Relation.Binary.Structures.html#5825" class="Module">IsDecTotalOrder</a> <a id="5520" href="Relation.Binary.Construct.On.html#5328" class="Bound">dec</a>
<a id="5527" href="Relation.Binary.Construct.On.html#5527" class="Function">isStrictTotalOrder</a> <a id="5546" class="Symbol">:</a> <a id="5548" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="5567" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5569" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5571" class="Symbol">→</a>
<a id="5596" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="5615" class="Symbol">(</a><a id="5616" href="Relation.Binary.Construct.On.html#3648" class="Bound">≈</a> <a id="5618" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5621" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5622" class="Symbol">)</a> <a id="5624" class="Symbol">(</a><a id="5625" href="Relation.Binary.Construct.On.html#3663" class="Bound">∼</a> <a id="5627" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="5630" href="Relation.Binary.Construct.On.html#3636" class="Bound">f</a><a id="5631" class="Symbol">)</a>
<a id="5635" href="Relation.Binary.Construct.On.html#5527" class="Function">isStrictTotalOrder</a> <a id="5654" href="Relation.Binary.Construct.On.html#5654" class="Bound">sto</a> <a id="5658" class="Symbol">=</a> <a id="5660" class="Keyword">record</a>
<a id="5671" class="Symbol">{</a> <a id="5673" href="Relation.Binary.Structures.html#6805" class="Field">isStrictPartialOrder</a> <a id="5694" class="Symbol">=</a> <a id="5696" href="Relation.Binary.Construct.On.html#4558" class="Function">isStrictPartialOrder</a> <a id="5717" href="Relation.Binary.Structures.html#6805" class="Field">Sto.isStrictPartialOrder</a>
<a id="5746" class="Symbol">;</a> <a id="5748" href="Relation.Binary.Structures.html#6857" class="Field">compare</a> <a id="5769" class="Symbol">=</a> <a id="5771" href="Relation.Binary.Construct.On.html#2561" class="Function">trichotomous</a> <a id="5784" class="Symbol">_</a> <a id="5786" class="Symbol">_</a> <a id="5788" class="Symbol">_</a> <a id="5790" href="Relation.Binary.Structures.html#6857" class="Field">Sto.compare</a>
<a id="5806" class="Symbol">}</a> <a id="5808" class="Keyword">where</a> <a id="5814" class="Keyword">module</a> <a id="5821" href="Relation.Binary.Construct.On.html#5821" class="Module">Sto</a> <a id="5825" class="Symbol">=</a> <a id="5827" href="Relation.Binary.Structures.html#6732" class="Module">IsStrictTotalOrder</a> <a id="5846" href="Relation.Binary.Construct.On.html#5654" class="Bound">sto</a>
<a id="5851" class="Comment">------------------------------------------------------------------------</a>
<a id="5924" class="Comment">-- Bundles</a>
<a id="preorder"></a><a id="5936" href="Relation.Binary.Construct.On.html#5936" class="Function">preorder</a> <a id="5945" class="Symbol">:</a> <a id="5947" class="Symbol">(</a><a id="5948" href="Relation.Binary.Construct.On.html#5948" class="Bound">P</a> <a id="5950" class="Symbol">:</a> <a id="5952" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="5961" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="5963" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a> <a id="5966" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="5968" class="Symbol">)</a> <a id="5970" class="Symbol">→</a>
<a id="5983" class="Symbol">(</a><a id="5984" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="5986" class="Symbol">→</a> <a id="5988" href="Relation.Binary.Bundles.html#2200" class="Field">Preorder.Carrier</a> <a id="6005" href="Relation.Binary.Construct.On.html#5948" class="Bound">P</a><a id="6006" class="Symbol">)</a> <a id="6008" class="Symbol">→</a>
<a id="6021" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6030" class="Symbol">_</a> <a id="6032" class="Symbol">_</a> <a id="6034" class="Symbol">_</a>
<a id="6036" href="Relation.Binary.Construct.On.html#5936" class="Function">preorder</a> <a id="6045" href="Relation.Binary.Construct.On.html#6045" class="Bound">P</a> <a id="6047" href="Relation.Binary.Construct.On.html#6047" class="Bound">f</a> <a id="6049" class="Symbol">=</a> <a id="6051" class="Keyword">record</a>
<a id="6060" class="Symbol">{</a> <a id="6062" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="6073" class="Symbol">=</a> <a id="6075" href="Relation.Binary.Construct.On.html#3686" class="Function">isPreorder</a> <a id="6086" href="Relation.Binary.Construct.On.html#6047" class="Bound">f</a> <a id="6088" class="Symbol">(</a><a id="6089" href="Relation.Binary.Bundles.html#2334" class="Field">Preorder.isPreorder</a> <a id="6109" href="Relation.Binary.Construct.On.html#6045" class="Bound">P</a><a id="6110" class="Symbol">)</a>
<a id="6114" class="Symbol">}</a>
<a id="setoid"></a><a id="6117" href="Relation.Binary.Construct.On.html#6117" class="Function">setoid</a> <a id="6124" class="Symbol">:</a> <a id="6126" class="Symbol">(</a><a id="6127" href="Relation.Binary.Construct.On.html#6127" class="Bound">S</a> <a id="6129" class="Symbol">:</a> <a id="6131" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="6138" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="6140" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="6141" class="Symbol">)</a> <a id="6143" class="Symbol">→</a>
<a id="6154" class="Symbol">(</a><a id="6155" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="6157" class="Symbol">→</a> <a id="6159" href="Relation.Binary.Bundles.html#1143" class="Field">Setoid.Carrier</a> <a id="6174" href="Relation.Binary.Construct.On.html#6127" class="Bound">S</a><a id="6175" class="Symbol">)</a> <a id="6177" class="Symbol">→</a>
<a id="6188" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="6195" class="Symbol">_</a> <a id="6197" class="Symbol">_</a>
<a id="6199" href="Relation.Binary.Construct.On.html#6117" class="Function">setoid</a> <a id="6206" href="Relation.Binary.Construct.On.html#6206" class="Bound">S</a> <a id="6208" href="Relation.Binary.Construct.On.html#6208" class="Bound">f</a> <a id="6210" class="Symbol">=</a> <a id="6212" class="Keyword">record</a>
<a id="6221" class="Symbol">{</a> <a id="6223" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="6237" class="Symbol">=</a> <a id="6239" href="Relation.Binary.Construct.On.html#3104" class="Function">isEquivalence</a> <a id="6253" href="Relation.Binary.Construct.On.html#6208" class="Bound">f</a> <a id="6255" class="Symbol">(</a><a id="6256" href="Relation.Binary.Bundles.html#1203" class="Field">Setoid.isEquivalence</a> <a id="6277" href="Relation.Binary.Construct.On.html#6206" class="Bound">S</a><a id="6278" class="Symbol">)</a>
<a id="6282" class="Symbol">}</a>
<a id="decSetoid"></a><a id="6285" href="Relation.Binary.Construct.On.html#6285" class="Function">decSetoid</a> <a id="6295" class="Symbol">:</a> <a id="6297" class="Symbol">(</a><a id="6298" href="Relation.Binary.Construct.On.html#6298" class="Bound">D</a> <a id="6300" class="Symbol">:</a> <a id="6302" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6312" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="6314" href="Relation.Binary.Construct.On.html#1132" class="Generalizable">ℓ</a><a id="6315" class="Symbol">)</a> <a id="6317" class="Symbol">→</a>
<a id="6331" class="Symbol">(</a><a id="6332" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="6334" class="Symbol">→</a> <a id="6336" href="Relation.Binary.Bundles.html#1614" class="Field">DecSetoid.Carrier</a> <a id="6354" href="Relation.Binary.Construct.On.html#6298" class="Bound">D</a><a id="6355" class="Symbol">)</a> <a id="6357" class="Symbol">→</a>
<a id="6371" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6381" class="Symbol">_</a> <a id="6383" class="Symbol">_</a>
<a id="6385" href="Relation.Binary.Construct.On.html#6285" class="Function">decSetoid</a> <a id="6395" href="Relation.Binary.Construct.On.html#6395" class="Bound">D</a> <a id="6397" href="Relation.Binary.Construct.On.html#6397" class="Bound">f</a> <a id="6399" class="Symbol">=</a> <a id="6401" class="Keyword">record</a>
<a id="6410" class="Symbol">{</a> <a id="6412" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6429" class="Symbol">=</a> <a id="6431" href="Relation.Binary.Construct.On.html#3362" class="Function">isDecEquivalence</a> <a id="6448" href="Relation.Binary.Construct.On.html#6397" class="Bound">f</a> <a id="6450" class="Symbol">(</a><a id="6451" href="Relation.Binary.Bundles.html#1680" class="Field">DecSetoid.isDecEquivalence</a> <a id="6478" href="Relation.Binary.Construct.On.html#6395" class="Bound">D</a><a id="6479" class="Symbol">)</a>
<a id="6483" class="Symbol">}</a>
<a id="poset"></a><a id="6486" href="Relation.Binary.Construct.On.html#6486" class="Function">poset</a> <a id="6492" class="Symbol">:</a> <a id="6494" class="Symbol">∀</a> <a id="6496" class="Symbol">(</a><a id="6497" href="Relation.Binary.Construct.On.html#6497" class="Bound">P</a> <a id="6499" class="Symbol">:</a> <a id="6501" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6507" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="6509" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a> <a id="6512" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="6514" class="Symbol">)</a> <a id="6516" class="Symbol">→</a>
<a id="6526" class="Symbol">(</a><a id="6527" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="6529" class="Symbol">→</a> <a id="6531" href="Relation.Binary.Bundles.html#3606" class="Field">Poset.Carrier</a> <a id="6545" href="Relation.Binary.Construct.On.html#6497" class="Bound">P</a><a id="6546" class="Symbol">)</a> <a id="6548" class="Symbol">→</a>
<a id="6558" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6564" class="Symbol">_</a> <a id="6566" class="Symbol">_</a> <a id="6568" class="Symbol">_</a>
<a id="6570" href="Relation.Binary.Construct.On.html#6486" class="Function">poset</a> <a id="6576" href="Relation.Binary.Construct.On.html#6576" class="Bound">P</a> <a id="6578" href="Relation.Binary.Construct.On.html#6578" class="Bound">f</a> <a id="6580" class="Symbol">=</a> <a id="6582" class="Keyword">record</a>
<a id="6591" class="Symbol">{</a> <a id="6593" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="6608" class="Symbol">=</a> <a id="6610" href="Relation.Binary.Construct.On.html#3967" class="Function">isPartialOrder</a> <a id="6625" href="Relation.Binary.Construct.On.html#6578" class="Bound">f</a> <a id="6627" class="Symbol">(</a><a id="6628" href="Relation.Binary.Bundles.html#3705" class="Field">Poset.isPartialOrder</a> <a id="6649" href="Relation.Binary.Construct.On.html#6576" class="Bound">P</a><a id="6650" class="Symbol">)</a>
<a id="6654" class="Symbol">}</a>
<a id="decPoset"></a><a id="6657" href="Relation.Binary.Construct.On.html#6657" class="Function">decPoset</a> <a id="6666" class="Symbol">:</a> <a id="6668" class="Symbol">(</a><a id="6669" href="Relation.Binary.Construct.On.html#6669" class="Bound">D</a> <a id="6671" class="Symbol">:</a> <a id="6673" href="Relation.Binary.Bundles.html#4140" class="Record">DecPoset</a> <a id="6682" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="6684" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a> <a id="6687" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="6689" class="Symbol">)</a> <a id="6691" class="Symbol">→</a>
<a id="6704" class="Symbol">(</a><a id="6705" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="6707" class="Symbol">→</a> <a id="6709" href="Relation.Binary.Bundles.html#4219" class="Field">DecPoset.Carrier</a> <a id="6726" href="Relation.Binary.Construct.On.html#6669" class="Bound">D</a><a id="6727" class="Symbol">)</a> <a id="6729" class="Symbol">→</a>
<a id="6742" href="Relation.Binary.Bundles.html#4140" class="Record">DecPoset</a> <a id="6751" class="Symbol">_</a> <a id="6753" class="Symbol">_</a> <a id="6755" class="Symbol">_</a>
<a id="6757" href="Relation.Binary.Construct.On.html#6657" class="Function">decPoset</a> <a id="6766" href="Relation.Binary.Construct.On.html#6766" class="Bound">D</a> <a id="6768" href="Relation.Binary.Construct.On.html#6768" class="Bound">f</a> <a id="6770" class="Symbol">=</a> <a id="6772" class="Keyword">record</a>
<a id="6781" class="Symbol">{</a> <a id="6783" href="Relation.Binary.Bundles.html#4327" class="Field">isDecPartialOrder</a> <a id="6801" class="Symbol">=</a>
<a id="6809" href="Relation.Binary.Construct.On.html#4225" class="Function">isDecPartialOrder</a> <a id="6827" href="Relation.Binary.Construct.On.html#6768" class="Bound">f</a> <a id="6829" class="Symbol">(</a><a id="6830" href="Relation.Binary.Bundles.html#4327" class="Field">DecPoset.isDecPartialOrder</a> <a id="6857" href="Relation.Binary.Construct.On.html#6766" class="Bound">D</a><a id="6858" class="Symbol">)</a>
<a id="6862" class="Symbol">}</a>
<a id="strictPartialOrder"></a><a id="6865" href="Relation.Binary.Construct.On.html#6865" class="Function">strictPartialOrder</a> <a id="6884" class="Symbol">:</a> <a id="6886" class="Symbol">(</a><a id="6887" href="Relation.Binary.Construct.On.html#6887" class="Bound">S</a> <a id="6889" class="Symbol">:</a> <a id="6891" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6910" href="Relation.Binary.Construct.On.html#1126" class="Generalizable">a</a> <a id="6912" href="Relation.Binary.Construct.On.html#1134" class="Generalizable">ℓ₁</a> <a id="6915" href="Relation.Binary.Construct.On.html#1137" class="Generalizable">ℓ₂</a><a id="6917" class="Symbol">)</a> <a id="6919" class="Symbol">→</a>
<a id="6942" class="Symbol">(</a><a id="6943" href="Relation.Binary.Construct.On.html#1166" class="Generalizable">B</a> <a id="6945" class="Symbol">→</a> <a id="6947" href="Relation.Binary.Bundles.html#4928" class="Field">StrictPartialOrder.Carrier</a> <a id="6974" href="Relation.Binary.Construct.On.html#6887" class="Bound">S</a><a id="6975" class="Symbol">)</a> <a id="6977" class="Symbol">→</a>
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<a id="7090" href="Relation.Binary.Construct.On.html#4558" class="Function">isStrictPartialOrder</a> <a id="7111" href="Relation.Binary.Construct.On.html#7046" class="Bound">f</a> <a id="7113" class="Symbol">(</a><a id="7114" href="Relation.Binary.Bundles.html#5045" class="Field">StrictPartialOrder.isStrictPartialOrder</a> <a id="7154" href="Relation.Binary.Construct.On.html#7044" class="Bound">S</a><a id="7155" class="Symbol">)</a>
<a id="7159" class="Symbol">}</a>
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<a id="7300" class="Symbol">{</a> <a id="7302" href="Relation.Binary.Bundles.html#6632" class="Field">isTotalOrder</a> <a id="7315" class="Symbol">=</a> <a id="7317" href="Relation.Binary.Construct.On.html#4966" class="Function">isTotalOrder</a> <a id="7330" href="Relation.Binary.Construct.On.html#7287" class="Bound">f</a> <a id="7332" class="Symbol">(</a><a id="7333" href="Relation.Binary.Bundles.html#6632" class="Field">TotalOrder.isTotalOrder</a> <a id="7357" href="Relation.Binary.Construct.On.html#7285" class="Bound">T</a><a id="7358" class="Symbol">)</a>
<a id="7362" class="Symbol">}</a>
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<a id="7495" href="Relation.Binary.Construct.On.html#7365" class="Function">decTotalOrder</a> <a id="7509" href="Relation.Binary.Construct.On.html#7509" class="Bound">D</a> <a id="7511" href="Relation.Binary.Construct.On.html#7511" class="Bound">f</a> <a id="7513" class="Symbol">=</a> <a id="7515" class="Keyword">record</a>
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<a id="7873" class="Symbol">}</a>
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