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<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Relation.Binary.On</title><link rel="stylesheet" href="Agda.css"></head><body><pre><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="167" class="Comment">------------------------------------------------------------------------</a>

<a id="241" class="Keyword">open</a> <a id="246" class="Keyword">import</a> <a id="253" href="Relation.Binary.html" class="Module">Relation.Binary</a>

<a id="270" class="Keyword">module</a> <a id="277" href="Relation.Binary.On.html" class="Module">Relation.Binary.On</a> <a id="296" class="Keyword">where</a>

<a id="303" class="Keyword">open</a> <a id="308" class="Keyword">import</a> <a id="315" href="Function.html" class="Module">Function</a>
<a id="324" class="Keyword">open</a> <a id="329" class="Keyword">import</a> <a id="336" href="Data.Product.html" class="Module">Data.Product</a>

<a id="350" class="Keyword">module</a> <a id="357" href="Relation.Binary.On.html#357" class="Module">_</a> <a id="359" class="Symbol">{</a><a id="360" href="Relation.Binary.On.html#360" class="Bound">a</a> <a id="362" href="Relation.Binary.On.html#362" class="Bound">b</a><a id="363" class="Symbol">}</a> <a id="365" class="Symbol">{</a><a id="366" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="368" class="Symbol">:</a> <a id="370" class="PrimitiveType">Set</a> <a id="374" href="Relation.Binary.On.html#360" class="Bound">a</a><a id="375" class="Symbol">}</a> <a id="377" class="Symbol">{</a><a id="378" href="Relation.Binary.On.html#378" class="Bound">B</a> <a id="380" class="Symbol">:</a> <a id="382" class="PrimitiveType">Set</a> <a id="386" href="Relation.Binary.On.html#362" class="Bound">b</a><a id="387" class="Symbol">}</a> <a id="389" class="Symbol">(</a><a id="390" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="392" class="Symbol">:</a> <a id="394" href="Relation.Binary.On.html#378" class="Bound">B</a> <a id="396" class="Symbol"></a> <a id="398" href="Relation.Binary.On.html#366" class="Bound">A</a><a id="399" class="Symbol">)</a> <a id="401" class="Keyword">where</a>

  <a id="410" href="Relation.Binary.On.html#410" class="Function">implies</a> <a id="418" class="Symbol">:</a> <a id="420" class="Symbol"></a> <a id="422" class="Symbol">{</a><a id="423" href="Relation.Binary.On.html#423" class="Bound">ℓ₁</a> <a id="426" href="Relation.Binary.On.html#426" class="Bound">ℓ₂</a><a id="428" class="Symbol">}</a> <a id="430" class="Symbol">(</a><a id="431" href="Relation.Binary.On.html#431" class="Bound"></a> <a id="433" class="Symbol">:</a> <a id="435" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="439" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="441" href="Relation.Binary.On.html#423" class="Bound">ℓ₁</a><a id="443" class="Symbol">)</a> <a id="445" class="Symbol">(</a><a id="446" href="Relation.Binary.On.html#446" class="Bound"></a> <a id="448" class="Symbol">:</a> <a id="450" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="454" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="456" href="Relation.Binary.On.html#426" class="Bound">ℓ₂</a><a id="458" class="Symbol">)</a> <a id="460" class="Symbol"></a>
            <a id="474" href="Relation.Binary.On.html#431" class="Bound"></a> <a id="476" href="Relation.Binary.Core.html#_%E2%87%92_" class="Function Operator"></a> <a id="478" href="Relation.Binary.On.html#446" class="Bound"></a> <a id="480" class="Symbol"></a> <a id="482" class="Symbol">(</a><a id="483" href="Relation.Binary.On.html#431" class="Bound"></a> <a id="485" href="Function.html#_on_" class="Function Operator">on</a> <a id="488" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="489" class="Symbol">)</a> <a id="491" href="Relation.Binary.Core.html#_%E2%87%92_" class="Function Operator"></a> <a id="493" class="Symbol">(</a><a id="494" href="Relation.Binary.On.html#446" class="Bound"></a> <a id="496" href="Function.html#_on_" class="Function Operator">on</a> <a id="499" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="500" class="Symbol">)</a>
  <a id="504" href="Relation.Binary.On.html#410" class="Function">implies</a> <a id="512" class="Symbol">_</a> <a id="514" class="Symbol">_</a> <a id="516" href="Relation.Binary.On.html#516" class="Bound">impl</a> <a id="521" class="Symbol">=</a> <a id="523" href="Relation.Binary.On.html#516" class="Bound">impl</a>

  <a id="531" href="Relation.Binary.On.html#531" class="Function">reflexive</a> <a id="541" class="Symbol">:</a> <a id="543" class="Symbol"></a> <a id="545" class="Symbol">{</a><a id="546" href="Relation.Binary.On.html#546" class="Bound"></a><a id="547" class="Symbol">}</a> <a id="549" class="Symbol">(</a><a id="550" href="Relation.Binary.On.html#550" class="Bound"></a> <a id="552" class="Symbol">:</a> <a id="554" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="558" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="560" href="Relation.Binary.On.html#546" class="Bound"></a><a id="561" class="Symbol">)</a> <a id="563" class="Symbol"></a> <a id="565" href="Relation.Binary.Core.html#Reflexive" class="Function">Reflexive</a> <a id="575" href="Relation.Binary.On.html#550" class="Bound"></a> <a id="577" class="Symbol"></a> <a id="579" href="Relation.Binary.Core.html#Reflexive" class="Function">Reflexive</a> <a id="589" class="Symbol">(</a><a id="590" href="Relation.Binary.On.html#550" class="Bound"></a> <a id="592" href="Function.html#_on_" class="Function Operator">on</a> <a id="595" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="596" class="Symbol">)</a>
  <a id="600" href="Relation.Binary.On.html#531" class="Function">reflexive</a> <a id="610" class="Symbol">_</a> <a id="612" href="Relation.Binary.On.html#612" class="Bound">refl</a> <a id="617" class="Symbol">=</a> <a id="619" href="Relation.Binary.On.html#612" class="Bound">refl</a>

  <a id="627" href="Relation.Binary.On.html#627" class="Function">irreflexive</a> <a id="639" class="Symbol">:</a> <a id="641" class="Symbol"></a> <a id="643" class="Symbol">{</a><a id="644" href="Relation.Binary.On.html#644" class="Bound">ℓ₁</a> <a id="647" href="Relation.Binary.On.html#647" class="Bound">ℓ₂</a><a id="649" class="Symbol">}</a> <a id="651" class="Symbol">(</a><a id="652" href="Relation.Binary.On.html#652" class="Bound"></a> <a id="654" class="Symbol">:</a> <a id="656" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="660" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="662" href="Relation.Binary.On.html#644" class="Bound">ℓ₁</a><a id="664" class="Symbol">)</a> <a id="666" class="Symbol">(</a><a id="667" href="Relation.Binary.On.html#667" class="Bound"></a> <a id="669" class="Symbol">:</a> <a id="671" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="675" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="677" href="Relation.Binary.On.html#647" class="Bound">ℓ₂</a><a id="679" class="Symbol">)</a> <a id="681" class="Symbol"></a>
                <a id="699" href="Relation.Binary.Core.html#Irreflexive" class="Function">Irreflexive</a> <a id="711" href="Relation.Binary.On.html#652" class="Bound"></a> <a id="713" href="Relation.Binary.On.html#667" class="Bound"></a> <a id="715" class="Symbol"></a> <a id="717" href="Relation.Binary.Core.html#Irreflexive" class="Function">Irreflexive</a> <a id="729" class="Symbol">(</a><a id="730" href="Relation.Binary.On.html#652" class="Bound"></a> <a id="732" href="Function.html#_on_" class="Function Operator">on</a> <a id="735" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="736" class="Symbol">)</a> <a id="738" class="Symbol">(</a><a id="739" href="Relation.Binary.On.html#667" class="Bound"></a> <a id="741" href="Function.html#_on_" class="Function Operator">on</a> <a id="744" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="745" class="Symbol">)</a>
  <a id="749" href="Relation.Binary.On.html#627" class="Function">irreflexive</a> <a id="761" class="Symbol">_</a> <a id="763" class="Symbol">_</a> <a id="765" href="Relation.Binary.On.html#765" class="Bound">irrefl</a> <a id="772" class="Symbol">=</a> <a id="774" href="Relation.Binary.On.html#765" class="Bound">irrefl</a>

  <a id="784" href="Relation.Binary.On.html#784" class="Function">symmetric</a> <a id="794" class="Symbol">:</a> <a id="796" class="Symbol"></a> <a id="798" class="Symbol">{</a><a id="799" href="Relation.Binary.On.html#799" class="Bound"></a><a id="800" class="Symbol">}</a> <a id="802" class="Symbol">(</a><a id="803" href="Relation.Binary.On.html#803" class="Bound"></a> <a id="805" class="Symbol">:</a> <a id="807" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="811" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="813" href="Relation.Binary.On.html#799" class="Bound"></a><a id="814" class="Symbol">)</a> <a id="816" class="Symbol"></a> <a id="818" href="Relation.Binary.Core.html#Symmetric" class="Function">Symmetric</a> <a id="828" href="Relation.Binary.On.html#803" class="Bound"></a> <a id="830" class="Symbol"></a> <a id="832" href="Relation.Binary.Core.html#Symmetric" class="Function">Symmetric</a> <a id="842" class="Symbol">(</a><a id="843" href="Relation.Binary.On.html#803" class="Bound"></a> <a id="845" href="Function.html#_on_" class="Function Operator">on</a> <a id="848" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="849" class="Symbol">)</a>
  <a id="853" href="Relation.Binary.On.html#784" class="Function">symmetric</a> <a id="863" class="Symbol">_</a> <a id="865" href="Relation.Binary.On.html#865" class="Bound">sym</a> <a id="869" class="Symbol">=</a> <a id="871" href="Relation.Binary.On.html#865" class="Bound">sym</a>

  <a id="878" href="Relation.Binary.On.html#878" class="Function">transitive</a> <a id="889" class="Symbol">:</a> <a id="891" class="Symbol"></a> <a id="893" class="Symbol">{</a><a id="894" href="Relation.Binary.On.html#894" class="Bound"></a><a id="895" class="Symbol">}</a> <a id="897" class="Symbol">(</a><a id="898" href="Relation.Binary.On.html#898" class="Bound"></a> <a id="900" class="Symbol">:</a> <a id="902" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="906" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="908" href="Relation.Binary.On.html#894" class="Bound"></a><a id="909" class="Symbol">)</a> <a id="911" class="Symbol"></a> <a id="913" href="Relation.Binary.Core.html#Transitive" class="Function">Transitive</a> <a id="924" href="Relation.Binary.On.html#898" class="Bound"></a> <a id="926" class="Symbol"></a> <a id="928" href="Relation.Binary.Core.html#Transitive" class="Function">Transitive</a> <a id="939" class="Symbol">(</a><a id="940" href="Relation.Binary.On.html#898" class="Bound"></a> <a id="942" href="Function.html#_on_" class="Function Operator">on</a> <a id="945" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="946" class="Symbol">)</a>
  <a id="950" href="Relation.Binary.On.html#878" class="Function">transitive</a> <a id="961" class="Symbol">_</a> <a id="963" href="Relation.Binary.On.html#963" class="Bound">trans</a> <a id="969" class="Symbol">=</a> <a id="971" href="Relation.Binary.On.html#963" class="Bound">trans</a>

  <a id="980" href="Relation.Binary.On.html#980" class="Function">antisymmetric</a> <a id="994" class="Symbol">:</a> <a id="996" class="Symbol"></a> <a id="998" class="Symbol">{</a><a id="999" href="Relation.Binary.On.html#999" class="Bound">ℓ₁</a> <a id="1002" href="Relation.Binary.On.html#1002" class="Bound">ℓ₂</a><a id="1004" class="Symbol">}</a> <a id="1006" class="Symbol">(</a><a id="1007" href="Relation.Binary.On.html#1007" class="Bound"></a> <a id="1009" class="Symbol">:</a> <a id="1011" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1015" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1017" href="Relation.Binary.On.html#999" class="Bound">ℓ₁</a><a id="1019" class="Symbol">)</a> <a id="1021" class="Symbol">(</a><a id="1022" href="Relation.Binary.On.html#1022" class="Bound"></a> <a id="1024" class="Symbol">:</a> <a id="1026" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1030" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1032" href="Relation.Binary.On.html#1002" class="Bound">ℓ₂</a><a id="1034" class="Symbol">)</a> <a id="1036" class="Symbol"></a>
                  <a id="1056" href="Relation.Binary.Core.html#Antisymmetric" class="Function">Antisymmetric</a> <a id="1070" href="Relation.Binary.On.html#1007" class="Bound"></a> <a id="1072" href="Relation.Binary.On.html#1022" class="Bound"></a> <a id="1074" class="Symbol"></a> <a id="1076" href="Relation.Binary.Core.html#Antisymmetric" class="Function">Antisymmetric</a> <a id="1090" class="Symbol">(</a><a id="1091" href="Relation.Binary.On.html#1007" class="Bound"></a> <a id="1093" href="Function.html#_on_" class="Function Operator">on</a> <a id="1096" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1097" class="Symbol">)</a> <a id="1099" class="Symbol">(</a><a id="1100" href="Relation.Binary.On.html#1022" class="Bound"></a> <a id="1102" href="Function.html#_on_" class="Function Operator">on</a> <a id="1105" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1106" class="Symbol">)</a>
  <a id="1110" href="Relation.Binary.On.html#980" class="Function">antisymmetric</a> <a id="1124" class="Symbol">_</a> <a id="1126" class="Symbol">_</a> <a id="1128" href="Relation.Binary.On.html#1128" class="Bound">antisym</a> <a id="1136" class="Symbol">=</a> <a id="1138" href="Relation.Binary.On.html#1128" class="Bound">antisym</a>

  <a id="1149" href="Relation.Binary.On.html#1149" class="Function">asymmetric</a> <a id="1160" class="Symbol">:</a> <a id="1162" class="Symbol"></a> <a id="1164" class="Symbol">{</a><a id="1165" href="Relation.Binary.On.html#1165" class="Bound"></a><a id="1166" class="Symbol">}</a> <a id="1168" class="Symbol">(</a><a id="1169" href="Relation.Binary.On.html#1169" class="Bound">&lt;</a> <a id="1171" class="Symbol">:</a> <a id="1173" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1177" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1179" href="Relation.Binary.On.html#1165" class="Bound"></a><a id="1180" class="Symbol">)</a> <a id="1182" class="Symbol"></a> <a id="1184" href="Relation.Binary.Core.html#Asymmetric" class="Function">Asymmetric</a> <a id="1195" href="Relation.Binary.On.html#1169" class="Bound">&lt;</a> <a id="1197" class="Symbol"></a> <a id="1199" href="Relation.Binary.Core.html#Asymmetric" class="Function">Asymmetric</a> <a id="1210" class="Symbol">(</a><a id="1211" href="Relation.Binary.On.html#1169" class="Bound">&lt;</a> <a id="1213" href="Function.html#_on_" class="Function Operator">on</a> <a id="1216" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1217" class="Symbol">)</a>
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  <a id="1356" href="Relation.Binary.On.html#1249" class="Function">respects</a> <a id="1365" class="Symbol">_</a> <a id="1367" class="Symbol">_</a> <a id="1369" href="Relation.Binary.On.html#1369" class="Bound">resp</a> <a id="1374" class="Symbol">=</a> <a id="1376" href="Relation.Binary.On.html#1369" class="Bound">resp</a>

  <a id="1384" href="Relation.Binary.On.html#1384" class="Function">respects₂</a> <a id="1394" class="Symbol">:</a> <a id="1396" class="Symbol"></a> <a id="1398" class="Symbol">{</a><a id="1399" href="Relation.Binary.On.html#1399" class="Bound">ℓ₁</a> <a id="1402" href="Relation.Binary.On.html#1402" class="Bound">ℓ₂</a><a id="1404" class="Symbol">}</a> <a id="1406" class="Symbol">(</a><a id="1407" href="Relation.Binary.On.html#1407" class="Bound">∼₁</a> <a id="1410" class="Symbol">:</a> <a id="1412" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1416" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1418" href="Relation.Binary.On.html#1399" class="Bound">ℓ₁</a><a id="1420" class="Symbol">)</a> <a id="1422" class="Symbol">(</a><a id="1423" href="Relation.Binary.On.html#1423" class="Bound">∼₂</a> <a id="1426" class="Symbol">:</a> <a id="1428" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1432" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1434" href="Relation.Binary.On.html#1402" class="Bound">ℓ₂</a><a id="1436" class="Symbol">)</a> <a id="1438" class="Symbol"></a>
              <a id="1454" href="Relation.Binary.On.html#1407" class="Bound">∼₁</a> <a id="1457" href="Relation.Binary.Core.html#_Respects%E2%82%82_" class="Function Operator">Respects₂</a> <a id="1467" href="Relation.Binary.On.html#1423" class="Bound">∼₂</a> <a id="1470" class="Symbol"></a> <a id="1472" class="Symbol">(</a><a id="1473" href="Relation.Binary.On.html#1407" class="Bound">∼₁</a> <a id="1476" href="Function.html#_on_" class="Function Operator">on</a> <a id="1479" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1480" class="Symbol">)</a> <a id="1482" href="Relation.Binary.Core.html#_Respects%E2%82%82_" class="Function Operator">Respects₂</a> <a id="1492" class="Symbol">(</a><a id="1493" href="Relation.Binary.On.html#1423" class="Bound">∼₂</a> <a id="1496" href="Function.html#_on_" class="Function Operator">on</a> <a id="1499" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1500" class="Symbol">)</a>
  <a id="1504" href="Relation.Binary.On.html#1384" class="Function">respects₂</a> <a id="1514" class="Symbol">_</a> <a id="1516" class="Symbol">_</a> <a id="1518" class="Symbol">(</a><a id="1519" href="Relation.Binary.On.html#1519" class="Bound">resp₁</a> <a id="1525" href="Data.Product.html#%CE%A3._%2C_" class="InductiveConstructor Operator">,</a> <a id="1527" href="Relation.Binary.On.html#1527" class="Bound">resp₂</a><a id="1532" class="Symbol">)</a> <a id="1534" class="Symbol">=</a>
    <a id="1540" class="Symbol">((λ</a> <a id="1544" class="Symbol">{</a><a id="1545" href="Relation.Binary.On.html#1545" class="Bound">_</a><a id="1546" class="Symbol">}</a> <a id="1548" class="Symbol">{</a><a id="1549" href="Relation.Binary.On.html#1549" class="Bound">_</a><a id="1550" class="Symbol">}</a> <a id="1552" class="Symbol">{</a><a id="1553" href="Relation.Binary.On.html#1553" class="Bound">_</a><a id="1554" class="Symbol">}</a> <a id="1556" class="Symbol"></a> <a id="1558" href="Relation.Binary.On.html#1519" class="Bound">resp₁</a><a id="1563" class="Symbol">)</a> <a id="1565" href="Data.Product.html#%CE%A3._%2C_" class="InductiveConstructor Operator">,</a> <a id="1567" class="Symbol">λ</a> <a id="1569" class="Symbol">{</a><a id="1570" href="Relation.Binary.On.html#1570" class="Bound">_</a><a id="1571" class="Symbol">}</a> <a id="1573" class="Symbol">{</a><a id="1574" href="Relation.Binary.On.html#1574" class="Bound">_</a><a id="1575" class="Symbol">}</a> <a id="1577" class="Symbol">{</a><a id="1578" href="Relation.Binary.On.html#1578" class="Bound">_</a><a id="1579" class="Symbol">}</a> <a id="1581" class="Symbol"></a> <a id="1583" href="Relation.Binary.On.html#1527" class="Bound">resp₂</a><a id="1588" class="Symbol">)</a>

  <a id="1593" href="Relation.Binary.On.html#1593" class="Function">decidable</a> <a id="1603" class="Symbol">:</a> <a id="1605" class="Symbol"></a> <a id="1607" class="Symbol">{</a><a id="1608" href="Relation.Binary.On.html#1608" class="Bound"></a><a id="1609" class="Symbol">}</a> <a id="1611" class="Symbol">(</a><a id="1612" href="Relation.Binary.On.html#1612" class="Bound"></a> <a id="1614" class="Symbol">:</a> <a id="1616" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1620" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1622" href="Relation.Binary.On.html#1608" class="Bound"></a><a id="1623" class="Symbol">)</a> <a id="1625" class="Symbol"></a> <a id="1627" href="Relation.Binary.Core.html#Decidable" class="Function">Decidable</a> <a id="1637" href="Relation.Binary.On.html#1612" class="Bound"></a> <a id="1639" class="Symbol"></a> <a id="1641" href="Relation.Binary.Core.html#Decidable" class="Function">Decidable</a> <a id="1651" class="Symbol">(</a><a id="1652" href="Relation.Binary.On.html#1612" class="Bound"></a> <a id="1654" href="Function.html#_on_" class="Function Operator">on</a> <a id="1657" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1658" class="Symbol">)</a>
  <a id="1662" href="Relation.Binary.On.html#1593" class="Function">decidable</a> <a id="1672" class="Symbol">_</a> <a id="1674" href="Relation.Binary.On.html#1674" class="Bound">dec</a> <a id="1678" class="Symbol">=</a> <a id="1680" class="Symbol">λ</a> <a id="1682" href="Relation.Binary.On.html#1682" class="Bound">x</a> <a id="1684" href="Relation.Binary.On.html#1684" class="Bound">y</a> <a id="1686" class="Symbol"></a> <a id="1688" href="Relation.Binary.On.html#1674" class="Bound">dec</a> <a id="1692" class="Symbol">(</a><a id="1693" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1695" href="Relation.Binary.On.html#1682" class="Bound">x</a><a id="1696" class="Symbol">)</a> <a id="1698" class="Symbol">(</a><a id="1699" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1701" href="Relation.Binary.On.html#1684" class="Bound">y</a><a id="1702" class="Symbol">)</a>

  <a id="1707" href="Relation.Binary.On.html#1707" class="Function">total</a> <a id="1713" class="Symbol">:</a> <a id="1715" class="Symbol"></a> <a id="1717" class="Symbol">{</a><a id="1718" href="Relation.Binary.On.html#1718" class="Bound"></a><a id="1719" class="Symbol">}</a> <a id="1721" class="Symbol">(</a><a id="1722" href="Relation.Binary.On.html#1722" class="Bound"></a> <a id="1724" class="Symbol">:</a> <a id="1726" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1730" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1732" href="Relation.Binary.On.html#1718" class="Bound"></a><a id="1733" class="Symbol">)</a> <a id="1735" class="Symbol"></a> <a id="1737" href="Relation.Binary.Consequences.html#Total" class="Function">Total</a> <a id="1743" href="Relation.Binary.On.html#1722" class="Bound"></a> <a id="1745" class="Symbol"></a> <a id="1747" href="Relation.Binary.Consequences.html#Total" class="Function">Total</a> <a id="1753" class="Symbol">(</a><a id="1754" href="Relation.Binary.On.html#1722" class="Bound"></a> <a id="1756" href="Function.html#_on_" class="Function Operator">on</a> <a id="1759" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1760" class="Symbol">)</a>
  <a id="1764" href="Relation.Binary.On.html#1707" class="Function">total</a> <a id="1770" class="Symbol">_</a> <a id="1772" href="Relation.Binary.On.html#1772" class="Bound">tot</a> <a id="1776" class="Symbol">=</a> <a id="1778" class="Symbol">λ</a> <a id="1780" href="Relation.Binary.On.html#1780" class="Bound">x</a> <a id="1782" href="Relation.Binary.On.html#1782" class="Bound">y</a> <a id="1784" class="Symbol"></a> <a id="1786" href="Relation.Binary.On.html#1772" class="Bound">tot</a> <a id="1790" class="Symbol">(</a><a id="1791" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1793" href="Relation.Binary.On.html#1780" class="Bound">x</a><a id="1794" class="Symbol">)</a> <a id="1796" class="Symbol">(</a><a id="1797" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1799" href="Relation.Binary.On.html#1782" class="Bound">y</a><a id="1800" class="Symbol">)</a>

  <a id="1805" href="Relation.Binary.On.html#1805" class="Function">trichotomous</a> <a id="1818" class="Symbol">:</a> <a id="1820" class="Symbol"></a> <a id="1822" class="Symbol">{</a><a id="1823" href="Relation.Binary.On.html#1823" class="Bound">ℓ₁</a> <a id="1826" href="Relation.Binary.On.html#1826" class="Bound">ℓ₂</a><a id="1828" class="Symbol">}</a> <a id="1830" class="Symbol">(</a><a id="1831" href="Relation.Binary.On.html#1831" class="Bound"></a> <a id="1833" class="Symbol">:</a> <a id="1835" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1839" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1841" href="Relation.Binary.On.html#1823" class="Bound">ℓ₁</a><a id="1843" class="Symbol">)</a> <a id="1845" class="Symbol">(</a><a id="1846" href="Relation.Binary.On.html#1846" class="Bound">&lt;</a> <a id="1848" class="Symbol">:</a> <a id="1850" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="1854" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="1856" href="Relation.Binary.On.html#1826" class="Bound">ℓ₂</a><a id="1858" class="Symbol">)</a> <a id="1860" class="Symbol"></a>
                 <a id="1879" href="Relation.Binary.Core.html#Trichotomous" class="Function">Trichotomous</a> <a id="1892" href="Relation.Binary.On.html#1831" class="Bound"></a> <a id="1894" href="Relation.Binary.On.html#1846" class="Bound">&lt;</a> <a id="1896" class="Symbol"></a> <a id="1898" href="Relation.Binary.Core.html#Trichotomous" class="Function">Trichotomous</a> <a id="1911" class="Symbol">(</a><a id="1912" href="Relation.Binary.On.html#1831" class="Bound"></a> <a id="1914" href="Function.html#_on_" class="Function Operator">on</a> <a id="1917" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1918" class="Symbol">)</a> <a id="1920" class="Symbol">(</a><a id="1921" href="Relation.Binary.On.html#1846" class="Bound">&lt;</a> <a id="1923" href="Function.html#_on_" class="Function Operator">on</a> <a id="1926" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="1927" class="Symbol">)</a>
  <a id="1931" href="Relation.Binary.On.html#1805" class="Function">trichotomous</a> <a id="1944" class="Symbol">_</a> <a id="1946" class="Symbol">_</a> <a id="1948" href="Relation.Binary.On.html#1948" class="Bound">compare</a> <a id="1956" class="Symbol">=</a> <a id="1958" class="Symbol">λ</a> <a id="1960" href="Relation.Binary.On.html#1960" class="Bound">x</a> <a id="1962" href="Relation.Binary.On.html#1962" class="Bound">y</a> <a id="1964" class="Symbol"></a> <a id="1966" href="Relation.Binary.On.html#1948" class="Bound">compare</a> <a id="1974" class="Symbol">(</a><a id="1975" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1977" href="Relation.Binary.On.html#1960" class="Bound">x</a><a id="1978" class="Symbol">)</a> <a id="1980" class="Symbol">(</a><a id="1981" href="Relation.Binary.On.html#390" class="Bound">f</a> <a id="1983" href="Relation.Binary.On.html#1962" class="Bound">y</a><a id="1984" class="Symbol">)</a>

  <a id="1989" href="Relation.Binary.On.html#1989" class="Function">isEquivalence</a> <a id="2003" class="Symbol">:</a> <a id="2005" class="Symbol"></a> <a id="2007" class="Symbol">{</a><a id="2008" href="Relation.Binary.On.html#2008" class="Bound"></a><a id="2009" class="Symbol">}</a> <a id="2011" class="Symbol">{</a><a id="2012" href="Relation.Binary.On.html#2012" class="Bound"></a> <a id="2014" class="Symbol">:</a> <a id="2016" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="2020" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="2022" href="Relation.Binary.On.html#2008" class="Bound"></a><a id="2023" class="Symbol">}</a> <a id="2025" class="Symbol"></a>
                  <a id="2045" href="Relation.Binary.Core.html#IsEquivalence" class="Record">IsEquivalence</a> <a id="2059" href="Relation.Binary.On.html#2012" class="Bound"></a> <a id="2061" class="Symbol"></a> <a id="2063" href="Relation.Binary.Core.html#IsEquivalence" class="Record">IsEquivalence</a> <a id="2077" class="Symbol">(</a><a id="2078" href="Relation.Binary.On.html#2012" class="Bound"></a> <a id="2080" href="Function.html#_on_" class="Function Operator">on</a> <a id="2083" href="Relation.Binary.On.html#390" class="Bound">f</a><a id="2084" class="Symbol">)</a>
  <a id="2088" href="Relation.Binary.On.html#1989" class="Function">isEquivalence</a> <a id="2102" class="Symbol">{</a><a id="2103" class="Argument"></a> <a id="2105" class="Symbol">=</a> <a id="2107" href="Relation.Binary.On.html#2107" class="Bound"></a><a id="2108" class="Symbol">}</a> <a id="2110" href="Relation.Binary.On.html#2110" class="Bound">eq</a> <a id="2113" class="Symbol">=</a> <a id="2115" class="Keyword">record</a>
    <a id="2126" class="Symbol">{</a> <a id="2128" class="Field">refl</a>  <a id="2134" class="Symbol">=</a> <a id="2136" href="Relation.Binary.On.html#531" class="Function">reflexive</a>  <a id="2147" href="Relation.Binary.On.html#2107" class="Bound"></a> <a id="2149" href="Relation.Binary.Core.html#4934" class="Field">Eq.refl</a>
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    <a id="2231" class="Symbol">}</a>
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  <a id="2275" href="Relation.Binary.On.html#2275" class="Function">isPreorder</a> <a id="2286" class="Symbol">:</a> <a id="2288" class="Symbol"></a> <a id="2290" class="Symbol">{</a><a id="2291" href="Relation.Binary.On.html#2291" class="Bound">ℓ₁</a> <a id="2294" href="Relation.Binary.On.html#2294" class="Bound">ℓ₂</a><a id="2296" class="Symbol">}</a> <a id="2298" class="Symbol">{</a><a id="2299" href="Relation.Binary.On.html#2299" class="Bound"></a> <a id="2301" class="Symbol">:</a> <a id="2303" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="2307" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="2309" href="Relation.Binary.On.html#2291" class="Bound">ℓ₁</a><a id="2311" class="Symbol">}</a> <a id="2313" class="Symbol">{</a><a id="2314" href="Relation.Binary.On.html#2314" class="Bound"></a> <a id="2316" class="Symbol">:</a> <a id="2318" href="Relation.Binary.Core.html#Rel" class="Function">Rel</a> <a id="2322" href="Relation.Binary.On.html#366" class="Bound">A</a> <a id="2324" href="Relation.Binary.On.html#2294" class="Bound">ℓ₂</a><a id="2326" class="Symbol">}</a> <a id="2328" class="Symbol"></a>
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    <a id="2433" class="Symbol">{</a> <a id="2435" class="Field">isEquivalence</a> <a id="2449" class="Symbol">=</a> <a id="2451" href="Relation.Binary.On.html#1989" class="Function">isEquivalence</a> <a id="2465" href="Relation.Binary.html#1054" class="Field">Pre.isEquivalence</a>
    <a id="2487" class="Symbol">;</a> <a id="2489" class="Field">reflexive</a>     <a id="2503" class="Symbol">=</a> <a id="2505" href="Relation.Binary.On.html#410" class="Function">implies</a> <a id="2513" href="Relation.Binary.On.html#2409" class="Bound"></a> <a id="2515" href="Relation.Binary.On.html#2413" class="Bound"></a> <a id="2517" href="Relation.Binary.html#1160" class="Field">Pre.reflexive</a>
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    <a id="2580" class="Symbol">}</a>
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