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<html><head><meta charset="utf-8"><title>Data.Sum.Relation.Binary.Pointwise</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">-- Pointwise sum</a>
<a id="123" class="Comment">------------------------------------------------------------------------</a>
<a id="197" class="Symbol">{-#</a> <a id="201" class="Keyword">OPTIONS</a> <a id="209" class="Pragma">--cubical-compatible</a> <a id="230" class="Pragma">--safe</a> <a id="237" class="Symbol">#-}</a>
<a id="242" class="Keyword">module</a> <a id="249" href="Data.Sum.Relation.Binary.Pointwise.html" class="Module">Data.Sum.Relation.Binary.Pointwise</a> <a id="284" class="Keyword">where</a>
<a id="291" class="Keyword">open</a> <a id="296" class="Keyword">import</a> <a id="303" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="321" class="Keyword">using</a> <a id="327" class="Symbol">(</a><a id="328" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="331" class="Symbol">)</a>
<a id="333" class="Keyword">open</a> <a id="338" class="Keyword">import</a> <a id="345" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="359" class="Symbol">as</a> <a id="362" class="Module">Sum</a> <a id="366" class="Keyword">using</a> <a id="372" class="Symbol">(</a><a id="373" href="Data.Sum.Base.html#625" class="Datatype Operator">_⊎_</a><a id="376" class="Symbol">;</a> <a id="378" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="382" class="Symbol">;</a> <a id="384" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="388" class="Symbol">)</a>
<a id="390" class="Keyword">open</a> <a id="395" class="Keyword">import</a> <a id="402" href="Data.Sum.Properties.html" class="Module">Data.Sum.Properties</a>
<a id="422" class="Keyword">open</a> <a id="427" class="Keyword">import</a> <a id="434" href="Level.html" class="Module">Level</a> <a id="440" class="Keyword">using</a> <a id="446" class="Symbol">(</a><a id="447" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="452" class="Symbol">;</a> <a id="454" href="Agda.Primitive.html#961" class="Primitive Operator">_⊔_</a><a id="457" class="Symbol">)</a>
<a id="459" class="Keyword">open</a> <a id="464" class="Keyword">import</a> <a id="471" href="Function.Base.html" class="Module">Function.Base</a> <a id="485" class="Keyword">using</a> <a id="491" class="Symbol">(</a><a id="492" href="Function.Base.html#725" class="Function">const</a><a id="497" class="Symbol">;</a> <a id="499" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="502" class="Symbol">;</a> <a id="504" href="Function.Base.html#704" class="Function">id</a><a id="506" class="Symbol">)</a>
<a id="508" class="Keyword">open</a> <a id="513" class="Keyword">import</a> <a id="520" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="537" class="Keyword">using</a> <a id="543" class="Symbol">(</a><a id="544" href="Function.Bundles.html#7340" class="Record">Inverse</a><a id="551" class="Symbol">;</a> <a id="553" href="Function.Bundles.html#14187" class="Function">mk↔</a><a id="556" class="Symbol">)</a>
<a id="558" class="Keyword">open</a> <a id="563" class="Keyword">import</a> <a id="570" href="Relation.Nullary.html" class="Module">Relation.Nullary</a>
<a id="587" class="Keyword">import</a> <a id="594" href="Relation.Nullary.Decidable.html" class="Module">Relation.Nullary.Decidable</a> <a id="621" class="Symbol">as</a> <a id="624" class="Module">Dec</a>
<a id="628" class="Keyword">open</a> <a id="633" class="Keyword">import</a> <a id="640" href="Relation.Binary.html" class="Module">Relation.Binary</a>
<a id="656" class="Keyword">open</a> <a id="661" class="Keyword">import</a> <a id="668" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a> <a id="711" class="Symbol">as</a> <a id="714" class="Module">P</a> <a id="716" class="Keyword">using</a> <a id="722" class="Symbol">(</a><a id="723" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="726" class="Symbol">)</a>
<a id="728" class="Keyword">import</a> <a id="735" href="Relation.Binary.PropositionalEquality.Properties.html" class="Module">Relation.Binary.PropositionalEquality.Properties</a> <a id="784" class="Symbol">as</a> <a id="787" class="Module">P</a>
<a id="790" class="Keyword">private</a>
<a id="800" class="Keyword">variable</a>
<a id="813" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="815" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="817" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="819" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a> <a id="821" href="Data.Sum.Relation.Binary.Pointwise.html#821" class="Generalizable">ℓ₁</a> <a id="824" href="Data.Sum.Relation.Binary.Pointwise.html#824" class="Generalizable">ℓ₂</a> <a id="827" href="Data.Sum.Relation.Binary.Pointwise.html#827" class="Generalizable">ℓ₃</a> <a id="830" href="Data.Sum.Relation.Binary.Pointwise.html#830" class="Generalizable">ℓ</a> <a id="832" class="Symbol">:</a> <a id="834" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="844" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="846" href="Data.Sum.Relation.Binary.Pointwise.html#846" class="Generalizable">B</a> <a id="848" href="Data.Sum.Relation.Binary.Pointwise.html#848" class="Generalizable">C</a> <a id="850" href="Data.Sum.Relation.Binary.Pointwise.html#850" class="Generalizable">D</a> <a id="852" class="Symbol">:</a> <a id="854" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="858" href="Data.Sum.Relation.Binary.Pointwise.html#830" class="Generalizable">ℓ</a>
<a id="864" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="866" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="868" href="Data.Sum.Relation.Binary.Pointwise.html#868" class="Generalizable">T</a> <a id="870" href="Data.Sum.Relation.Binary.Pointwise.html#870" class="Generalizable">U</a> <a id="872" class="Symbol">:</a> <a id="874" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="878" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="880" href="Data.Sum.Relation.Binary.Pointwise.html#846" class="Generalizable">B</a> <a id="882" href="Data.Sum.Relation.Binary.Pointwise.html#830" class="Generalizable">ℓ</a>
<a id="888" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="891" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="894" class="Symbol">:</a> <a id="896" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="900" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="902" href="Data.Sum.Relation.Binary.Pointwise.html#830" class="Generalizable">ℓ</a>
<a id="905" class="Comment">------------------------------------------------------------------------</a>
<a id="978" class="Comment">-- Definition</a>
<a id="993" class="Keyword">data</a> <a id="Pointwise"></a><a id="998" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1008" class="Symbol">{</a><a id="1009" href="Data.Sum.Relation.Binary.Pointwise.html#1009" class="Bound">A</a> <a id="1011" class="Symbol">:</a> <a id="1013" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1017" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a><a id="1018" class="Symbol">}</a> <a id="1020" class="Symbol">{</a><a id="1021" href="Data.Sum.Relation.Binary.Pointwise.html#1021" class="Bound">B</a> <a id="1023" class="Symbol">:</a> <a id="1025" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1029" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a><a id="1030" class="Symbol">}</a> <a id="1032" class="Symbol">{</a><a id="1033" href="Data.Sum.Relation.Binary.Pointwise.html#1033" class="Bound">C</a> <a id="1035" class="Symbol">:</a> <a id="1037" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1041" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a><a id="1042" class="Symbol">}</a> <a id="1044" class="Symbol">{</a><a id="1045" href="Data.Sum.Relation.Binary.Pointwise.html#1045" class="Bound">D</a> <a id="1047" class="Symbol">:</a> <a id="1049" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1053" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a><a id="1054" class="Symbol">}</a>
<a id="1071" class="Symbol">(</a><a id="1072" href="Data.Sum.Relation.Binary.Pointwise.html#1072" class="Bound">R</a> <a id="1074" class="Symbol">:</a> <a id="1076" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="1080" href="Data.Sum.Relation.Binary.Pointwise.html#1009" class="Bound">A</a> <a id="1082" href="Data.Sum.Relation.Binary.Pointwise.html#1033" class="Bound">C</a> <a id="1084" href="Data.Sum.Relation.Binary.Pointwise.html#821" class="Generalizable">ℓ₁</a><a id="1086" class="Symbol">)</a> <a id="1088" class="Symbol">(</a><a id="1089" href="Data.Sum.Relation.Binary.Pointwise.html#1089" class="Bound">S</a> <a id="1091" class="Symbol">:</a> <a id="1093" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="1097" href="Data.Sum.Relation.Binary.Pointwise.html#1021" class="Bound">B</a> <a id="1099" href="Data.Sum.Relation.Binary.Pointwise.html#1045" class="Bound">D</a> <a id="1101" href="Data.Sum.Relation.Binary.Pointwise.html#824" class="Generalizable">ℓ₂</a><a id="1103" class="Symbol">)</a>
<a id="1120" class="Symbol">:</a> <a id="1122" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="1126" class="Symbol">(</a><a id="1127" href="Data.Sum.Relation.Binary.Pointwise.html#1009" class="Bound">A</a> <a id="1129" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="1131" href="Data.Sum.Relation.Binary.Pointwise.html#1021" class="Bound">B</a><a id="1132" class="Symbol">)</a> <a id="1134" class="Symbol">(</a><a id="1135" href="Data.Sum.Relation.Binary.Pointwise.html#1033" class="Bound">C</a> <a id="1137" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="1139" href="Data.Sum.Relation.Binary.Pointwise.html#1045" class="Bound">D</a><a id="1140" class="Symbol">)</a> <a id="1142" class="Symbol">(</a><a id="1143" href="Data.Sum.Relation.Binary.Pointwise.html#1017" class="Bound">a</a> <a id="1145" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1147" href="Data.Sum.Relation.Binary.Pointwise.html#1029" class="Bound">b</a> <a id="1149" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1151" href="Data.Sum.Relation.Binary.Pointwise.html#1041" class="Bound">c</a> <a id="1153" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1155" href="Data.Sum.Relation.Binary.Pointwise.html#1053" class="Bound">d</a> <a id="1157" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1159" href="Data.Sum.Relation.Binary.Pointwise.html#1084" class="Bound">ℓ₁</a> <a id="1162" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1164" href="Data.Sum.Relation.Binary.Pointwise.html#1101" class="Bound">ℓ₂</a><a id="1166" class="Symbol">)</a> <a id="1168" class="Keyword">where</a>
<a id="Pointwise.inj₁"></a><a id="1176" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="1181" class="Symbol">:</a> <a id="1183" class="Symbol">∀</a> <a id="1185" class="Symbol">{</a><a id="1186" href="Data.Sum.Relation.Binary.Pointwise.html#1186" class="Bound">a</a> <a id="1188" href="Data.Sum.Relation.Binary.Pointwise.html#1188" class="Bound">c</a><a id="1189" class="Symbol">}</a> <a id="1191" class="Symbol">→</a> <a id="1193" href="Data.Sum.Relation.Binary.Pointwise.html#1072" class="Bound">R</a> <a id="1195" href="Data.Sum.Relation.Binary.Pointwise.html#1186" class="Bound">a</a> <a id="1197" href="Data.Sum.Relation.Binary.Pointwise.html#1188" class="Bound">c</a> <a id="1199" class="Symbol">→</a> <a id="1201" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1211" href="Data.Sum.Relation.Binary.Pointwise.html#1072" class="Bound">R</a> <a id="1213" href="Data.Sum.Relation.Binary.Pointwise.html#1089" class="Bound">S</a> <a id="1215" class="Symbol">(</a><a id="1216" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1221" href="Data.Sum.Relation.Binary.Pointwise.html#1186" class="Bound">a</a><a id="1222" class="Symbol">)</a> <a id="1224" class="Symbol">(</a><a id="1225" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1230" href="Data.Sum.Relation.Binary.Pointwise.html#1188" class="Bound">c</a><a id="1231" class="Symbol">)</a>
<a id="Pointwise.inj₂"></a><a id="1235" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="1240" class="Symbol">:</a> <a id="1242" class="Symbol">∀</a> <a id="1244" class="Symbol">{</a><a id="1245" href="Data.Sum.Relation.Binary.Pointwise.html#1245" class="Bound">b</a> <a id="1247" href="Data.Sum.Relation.Binary.Pointwise.html#1247" class="Bound">d</a><a id="1248" class="Symbol">}</a> <a id="1250" class="Symbol">→</a> <a id="1252" href="Data.Sum.Relation.Binary.Pointwise.html#1089" class="Bound">S</a> <a id="1254" href="Data.Sum.Relation.Binary.Pointwise.html#1245" class="Bound">b</a> <a id="1256" href="Data.Sum.Relation.Binary.Pointwise.html#1247" class="Bound">d</a> <a id="1258" class="Symbol">→</a> <a id="1260" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1270" href="Data.Sum.Relation.Binary.Pointwise.html#1072" class="Bound">R</a> <a id="1272" href="Data.Sum.Relation.Binary.Pointwise.html#1089" class="Bound">S</a> <a id="1274" class="Symbol">(</a><a id="1275" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1280" href="Data.Sum.Relation.Binary.Pointwise.html#1245" class="Bound">b</a><a id="1281" class="Symbol">)</a> <a id="1283" class="Symbol">(</a><a id="1284" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1289" href="Data.Sum.Relation.Binary.Pointwise.html#1247" class="Bound">d</a><a id="1290" class="Symbol">)</a>
<a id="1293" class="Comment">----------------------------------------------------------------------</a>
<a id="1364" class="Comment">-- Functions</a>
<a id="map"></a><a id="1378" href="Data.Sum.Relation.Binary.Pointwise.html#1378" class="Function">map</a> <a id="1382" class="Symbol">:</a> <a id="1384" class="Symbol">∀</a> <a id="1386" class="Symbol">{</a><a id="1387" href="Data.Sum.Relation.Binary.Pointwise.html#1387" class="Bound">f</a> <a id="1389" class="Symbol">:</a> <a id="1391" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="1393" class="Symbol">→</a> <a id="1395" href="Data.Sum.Relation.Binary.Pointwise.html#848" class="Generalizable">C</a><a id="1396" class="Symbol">}</a> <a id="1398" class="Symbol">{</a><a id="1399" href="Data.Sum.Relation.Binary.Pointwise.html#1399" class="Bound">g</a> <a id="1401" class="Symbol">:</a> <a id="1403" href="Data.Sum.Relation.Binary.Pointwise.html#846" class="Generalizable">B</a> <a id="1405" class="Symbol">→</a> <a id="1407" href="Data.Sum.Relation.Binary.Pointwise.html#850" class="Generalizable">D</a><a id="1408" class="Symbol">}</a> <a id="1410" class="Symbol">→</a>
<a id="1418" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1420" href="Relation.Binary.Core.html#1474" class="Function Operator">=[</a> <a id="1423" href="Data.Sum.Relation.Binary.Pointwise.html#1387" class="Bound">f</a> <a id="1425" href="Relation.Binary.Core.html#1474" class="Function Operator">]⇒</a> <a id="1428" href="Data.Sum.Relation.Binary.Pointwise.html#868" class="Generalizable">T</a> <a id="1430" class="Symbol">→</a> <a id="1432" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1434" href="Relation.Binary.Core.html#1474" class="Function Operator">=[</a> <a id="1437" href="Data.Sum.Relation.Binary.Pointwise.html#1399" class="Bound">g</a> <a id="1439" href="Relation.Binary.Core.html#1474" class="Function Operator">]⇒</a> <a id="1442" href="Data.Sum.Relation.Binary.Pointwise.html#870" class="Generalizable">U</a> <a id="1444" class="Symbol">→</a>
<a id="1452" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1462" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1464" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1466" href="Relation.Binary.Core.html#1474" class="Function Operator">=[</a> <a id="1469" href="Data.Sum.Base.html#1253" class="Function">Sum.map</a> <a id="1477" href="Data.Sum.Relation.Binary.Pointwise.html#1387" class="Bound">f</a> <a id="1479" href="Data.Sum.Relation.Binary.Pointwise.html#1399" class="Bound">g</a> <a id="1481" href="Relation.Binary.Core.html#1474" class="Function Operator">]⇒</a> <a id="1484" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1494" href="Data.Sum.Relation.Binary.Pointwise.html#868" class="Generalizable">T</a> <a id="1496" href="Data.Sum.Relation.Binary.Pointwise.html#870" class="Generalizable">U</a>
<a id="1498" href="Data.Sum.Relation.Binary.Pointwise.html#1378" class="Function">map</a> <a id="1502" href="Data.Sum.Relation.Binary.Pointwise.html#1502" class="Bound">R⇒T</a> <a id="1506" class="Symbol">_</a> <a id="1508" class="Symbol">(</a><a id="1509" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="1514" href="Data.Sum.Relation.Binary.Pointwise.html#1514" class="Bound">x</a><a id="1515" class="Symbol">)</a> <a id="1517" class="Symbol">=</a> <a id="1519" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="1524" class="Symbol">(</a><a id="1525" href="Data.Sum.Relation.Binary.Pointwise.html#1502" class="Bound">R⇒T</a> <a id="1529" href="Data.Sum.Relation.Binary.Pointwise.html#1514" class="Bound">x</a><a id="1530" class="Symbol">)</a>
<a id="1532" href="Data.Sum.Relation.Binary.Pointwise.html#1378" class="Function">map</a> <a id="1536" class="Symbol">_</a> <a id="1538" href="Data.Sum.Relation.Binary.Pointwise.html#1538" class="Bound">S⇒U</a> <a id="1542" class="Symbol">(</a><a id="1543" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="1548" href="Data.Sum.Relation.Binary.Pointwise.html#1548" class="Bound">x</a><a id="1549" class="Symbol">)</a> <a id="1551" class="Symbol">=</a> <a id="1553" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="1558" class="Symbol">(</a><a id="1559" href="Data.Sum.Relation.Binary.Pointwise.html#1538" class="Bound">S⇒U</a> <a id="1563" href="Data.Sum.Relation.Binary.Pointwise.html#1548" class="Bound">x</a><a id="1564" class="Symbol">)</a>
<a id="1567" class="Comment">------------------------------------------------------------------------</a>
<a id="1640" class="Comment">-- Relational properties</a>
<a id="drop-inj₁"></a><a id="1666" href="Data.Sum.Relation.Binary.Pointwise.html#1666" class="Function">drop-inj₁</a> <a id="1676" class="Symbol">:</a> <a id="1678" class="Symbol">∀</a> <a id="1680" class="Symbol">{</a><a id="1681" href="Data.Sum.Relation.Binary.Pointwise.html#1681" class="Bound">x</a> <a id="1683" href="Data.Sum.Relation.Binary.Pointwise.html#1683" class="Bound">y</a><a id="1684" class="Symbol">}</a> <a id="1686" class="Symbol">→</a> <a id="1688" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1698" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1700" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1702" class="Symbol">(</a><a id="1703" class="InductiveConstructor">inj₁</a> <a id="1708" href="Data.Sum.Relation.Binary.Pointwise.html#1681" class="Bound">x</a><a id="1709" class="Symbol">)</a> <a id="1711" class="Symbol">(</a><a id="1712" class="InductiveConstructor">inj₁</a> <a id="1717" href="Data.Sum.Relation.Binary.Pointwise.html#1683" class="Bound">y</a><a id="1718" class="Symbol">)</a> <a id="1720" class="Symbol">→</a> <a id="1722" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1724" href="Data.Sum.Relation.Binary.Pointwise.html#1681" class="Bound">x</a> <a id="1726" href="Data.Sum.Relation.Binary.Pointwise.html#1683" class="Bound">y</a>
<a id="1728" href="Data.Sum.Relation.Binary.Pointwise.html#1666" class="Function">drop-inj₁</a> <a id="1738" class="Symbol">(</a><a id="1739" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="1744" href="Data.Sum.Relation.Binary.Pointwise.html#1744" class="Bound">x</a><a id="1745" class="Symbol">)</a> <a id="1747" class="Symbol">=</a> <a id="1749" href="Data.Sum.Relation.Binary.Pointwise.html#1744" class="Bound">x</a>
<a id="drop-inj₂"></a><a id="1752" href="Data.Sum.Relation.Binary.Pointwise.html#1752" class="Function">drop-inj₂</a> <a id="1762" class="Symbol">:</a> <a id="1764" class="Symbol">∀</a> <a id="1766" class="Symbol">{</a><a id="1767" href="Data.Sum.Relation.Binary.Pointwise.html#1767" class="Bound">x</a> <a id="1769" href="Data.Sum.Relation.Binary.Pointwise.html#1769" class="Bound">y</a><a id="1770" class="Symbol">}</a> <a id="1772" class="Symbol">→</a> <a id="1774" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1784" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1786" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1788" class="Symbol">(</a><a id="1789" class="InductiveConstructor">inj₂</a> <a id="1794" href="Data.Sum.Relation.Binary.Pointwise.html#1767" class="Bound">x</a><a id="1795" class="Symbol">)</a> <a id="1797" class="Symbol">(</a><a id="1798" class="InductiveConstructor">inj₂</a> <a id="1803" href="Data.Sum.Relation.Binary.Pointwise.html#1769" class="Bound">y</a><a id="1804" class="Symbol">)</a> <a id="1806" class="Symbol">→</a> <a id="1808" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1810" href="Data.Sum.Relation.Binary.Pointwise.html#1767" class="Bound">x</a> <a id="1812" href="Data.Sum.Relation.Binary.Pointwise.html#1769" class="Bound">y</a>
<a id="1814" href="Data.Sum.Relation.Binary.Pointwise.html#1752" class="Function">drop-inj₂</a> <a id="1824" class="Symbol">(</a><a id="1825" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="1830" href="Data.Sum.Relation.Binary.Pointwise.html#1830" class="Bound">x</a><a id="1831" class="Symbol">)</a> <a id="1833" class="Symbol">=</a> <a id="1835" href="Data.Sum.Relation.Binary.Pointwise.html#1830" class="Bound">x</a>
<a id="⊎-refl"></a><a id="1838" href="Data.Sum.Relation.Binary.Pointwise.html#1838" class="Function">⊎-refl</a> <a id="1845" class="Symbol">:</a> <a id="1847" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1857" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1859" class="Symbol">→</a> <a id="1861" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1871" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="1873" class="Symbol">→</a> <a id="1875" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1885" class="Symbol">(</a><a id="1886" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="1896" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="1898" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="1899" class="Symbol">)</a>
<a id="1901" href="Data.Sum.Relation.Binary.Pointwise.html#1838" class="Function">⊎-refl</a> <a id="1908" href="Data.Sum.Relation.Binary.Pointwise.html#1908" class="Bound">refl₁</a> <a id="1914" href="Data.Sum.Relation.Binary.Pointwise.html#1914" class="Bound">refl₂</a> <a id="1920" class="Symbol">{</a><a id="1921" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1926" href="Data.Sum.Relation.Binary.Pointwise.html#1926" class="Bound">x</a><a id="1927" class="Symbol">}</a> <a id="1929" class="Symbol">=</a> <a id="1931" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="1936" href="Data.Sum.Relation.Binary.Pointwise.html#1908" class="Bound">refl₁</a>
<a id="1942" href="Data.Sum.Relation.Binary.Pointwise.html#1838" class="Function">⊎-refl</a> <a id="1949" href="Data.Sum.Relation.Binary.Pointwise.html#1949" class="Bound">refl₁</a> <a id="1955" href="Data.Sum.Relation.Binary.Pointwise.html#1955" class="Bound">refl₂</a> <a id="1961" class="Symbol">{</a><a id="1962" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1967" href="Data.Sum.Relation.Binary.Pointwise.html#1967" class="Bound">y</a><a id="1968" class="Symbol">}</a> <a id="1970" class="Symbol">=</a> <a id="1972" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="1977" href="Data.Sum.Relation.Binary.Pointwise.html#1955" class="Bound">refl₂</a>
<a id="⊎-symmetric"></a><a id="1984" href="Data.Sum.Relation.Binary.Pointwise.html#1984" class="Function">⊎-symmetric</a> <a id="1996" class="Symbol">:</a> <a id="1998" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2008" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2010" class="Symbol">→</a> <a id="2012" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2022" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="2024" class="Symbol">→</a>
<a id="2040" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2050" class="Symbol">(</a><a id="2051" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="2061" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2063" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="2064" class="Symbol">)</a>
<a id="2066" href="Data.Sum.Relation.Binary.Pointwise.html#1984" class="Function">⊎-symmetric</a> <a id="2078" href="Data.Sum.Relation.Binary.Pointwise.html#2078" class="Bound">sym₁</a> <a id="2083" href="Data.Sum.Relation.Binary.Pointwise.html#2083" class="Bound">sym₂</a> <a id="2088" class="Symbol">(</a><a id="2089" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2094" href="Data.Sum.Relation.Binary.Pointwise.html#2094" class="Bound">x</a><a id="2095" class="Symbol">)</a> <a id="2097" class="Symbol">=</a> <a id="2099" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2104" class="Symbol">(</a><a id="2105" href="Data.Sum.Relation.Binary.Pointwise.html#2078" class="Bound">sym₁</a> <a id="2110" href="Data.Sum.Relation.Binary.Pointwise.html#2094" class="Bound">x</a><a id="2111" class="Symbol">)</a>
<a id="2113" href="Data.Sum.Relation.Binary.Pointwise.html#1984" class="Function">⊎-symmetric</a> <a id="2125" href="Data.Sum.Relation.Binary.Pointwise.html#2125" class="Bound">sym₁</a> <a id="2130" href="Data.Sum.Relation.Binary.Pointwise.html#2130" class="Bound">sym₂</a> <a id="2135" class="Symbol">(</a><a id="2136" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2141" href="Data.Sum.Relation.Binary.Pointwise.html#2141" class="Bound">x</a><a id="2142" class="Symbol">)</a> <a id="2144" class="Symbol">=</a> <a id="2146" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2151" class="Symbol">(</a><a id="2152" href="Data.Sum.Relation.Binary.Pointwise.html#2130" class="Bound">sym₂</a> <a id="2157" href="Data.Sum.Relation.Binary.Pointwise.html#2141" class="Bound">x</a><a id="2158" class="Symbol">)</a>
<a id="⊎-transitive"></a><a id="2161" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="2174" class="Symbol">:</a> <a id="2176" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2187" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2189" class="Symbol">→</a> <a id="2191" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2202" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="2204" class="Symbol">→</a>
<a id="2221" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2232" class="Symbol">(</a><a id="2233" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="2243" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2245" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="2246" class="Symbol">)</a>
<a id="2248" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="2261" href="Data.Sum.Relation.Binary.Pointwise.html#2261" class="Bound">trans₁</a> <a id="2268" href="Data.Sum.Relation.Binary.Pointwise.html#2268" class="Bound">trans₂</a> <a id="2275" class="Symbol">(</a><a id="2276" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2281" href="Data.Sum.Relation.Binary.Pointwise.html#2281" class="Bound">x</a><a id="2282" class="Symbol">)</a> <a id="2284" class="Symbol">(</a><a id="2285" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2290" href="Data.Sum.Relation.Binary.Pointwise.html#2290" class="Bound">y</a><a id="2291" class="Symbol">)</a> <a id="2293" class="Symbol">=</a> <a id="2295" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2300" class="Symbol">(</a><a id="2301" href="Data.Sum.Relation.Binary.Pointwise.html#2261" class="Bound">trans₁</a> <a id="2308" href="Data.Sum.Relation.Binary.Pointwise.html#2281" class="Bound">x</a> <a id="2310" href="Data.Sum.Relation.Binary.Pointwise.html#2290" class="Bound">y</a><a id="2311" class="Symbol">)</a>
<a id="2313" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="2326" href="Data.Sum.Relation.Binary.Pointwise.html#2326" class="Bound">trans₁</a> <a id="2333" href="Data.Sum.Relation.Binary.Pointwise.html#2333" class="Bound">trans₂</a> <a id="2340" class="Symbol">(</a><a id="2341" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2346" href="Data.Sum.Relation.Binary.Pointwise.html#2346" class="Bound">x</a><a id="2347" class="Symbol">)</a> <a id="2349" class="Symbol">(</a><a id="2350" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2355" href="Data.Sum.Relation.Binary.Pointwise.html#2355" class="Bound">y</a><a id="2356" class="Symbol">)</a> <a id="2358" class="Symbol">=</a> <a id="2360" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2365" class="Symbol">(</a><a id="2366" href="Data.Sum.Relation.Binary.Pointwise.html#2333" class="Bound">trans₂</a> <a id="2373" href="Data.Sum.Relation.Binary.Pointwise.html#2346" class="Bound">x</a> <a id="2375" href="Data.Sum.Relation.Binary.Pointwise.html#2355" class="Bound">y</a><a id="2376" class="Symbol">)</a>
<a id="⊎-asymmetric"></a><a id="2379" href="Data.Sum.Relation.Binary.Pointwise.html#2379" class="Function">⊎-asymmetric</a> <a id="2392" class="Symbol">:</a> <a id="2394" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2405" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2407" class="Symbol">→</a> <a id="2409" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2420" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="2422" class="Symbol">→</a>
<a id="2439" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2450" class="Symbol">(</a><a id="2451" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="2461" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2463" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="2464" class="Symbol">)</a>
<a id="2466" href="Data.Sum.Relation.Binary.Pointwise.html#2379" class="Function">⊎-asymmetric</a> <a id="2479" href="Data.Sum.Relation.Binary.Pointwise.html#2479" class="Bound">asym₁</a> <a id="2485" href="Data.Sum.Relation.Binary.Pointwise.html#2485" class="Bound">asym₂</a> <a id="2491" class="Symbol">(</a><a id="2492" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2497" href="Data.Sum.Relation.Binary.Pointwise.html#2497" class="Bound">x</a><a id="2498" class="Symbol">)</a> <a id="2500" class="Symbol">=</a> <a id="2502" class="Symbol">λ</a> <a id="2504" class="Symbol">{</a> <a id="2506" class="Symbol">(</a><a id="2507" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2512" href="Data.Sum.Relation.Binary.Pointwise.html#2512" class="Bound">y</a><a id="2513" class="Symbol">)</a> <a id="2515" class="Symbol">→</a> <a id="2517" href="Data.Sum.Relation.Binary.Pointwise.html#2479" class="Bound">asym₁</a> <a id="2523" href="Data.Sum.Relation.Binary.Pointwise.html#2497" class="Bound">x</a> <a id="2525" href="Data.Sum.Relation.Binary.Pointwise.html#2512" class="Bound">y</a> <a id="2527" class="Symbol">}</a>
<a id="2529" href="Data.Sum.Relation.Binary.Pointwise.html#2379" class="Function">⊎-asymmetric</a> <a id="2542" href="Data.Sum.Relation.Binary.Pointwise.html#2542" class="Bound">asym₁</a> <a id="2548" href="Data.Sum.Relation.Binary.Pointwise.html#2548" class="Bound">asym₂</a> <a id="2554" class="Symbol">(</a><a id="2555" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2560" href="Data.Sum.Relation.Binary.Pointwise.html#2560" class="Bound">x</a><a id="2561" class="Symbol">)</a> <a id="2563" class="Symbol">=</a> <a id="2565" class="Symbol">λ</a> <a id="2567" class="Symbol">{</a> <a id="2569" class="Symbol">(</a><a id="2570" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2575" href="Data.Sum.Relation.Binary.Pointwise.html#2575" class="Bound">y</a><a id="2576" class="Symbol">)</a> <a id="2578" class="Symbol">→</a> <a id="2580" href="Data.Sum.Relation.Binary.Pointwise.html#2548" class="Bound">asym₂</a> <a id="2586" href="Data.Sum.Relation.Binary.Pointwise.html#2560" class="Bound">x</a> <a id="2588" href="Data.Sum.Relation.Binary.Pointwise.html#2575" class="Bound">y</a> <a id="2590" class="Symbol">}</a>
<a id="⊎-substitutive"></a><a id="2593" href="Data.Sum.Relation.Binary.Pointwise.html#2593" class="Function">⊎-substitutive</a> <a id="2608" class="Symbol">:</a> <a id="2610" href="Relation.Binary.Definitions.html#5955" class="Function">Substitutive</a> <a id="2623" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2625" href="Data.Sum.Relation.Binary.Pointwise.html#827" class="Generalizable">ℓ₃</a> <a id="2628" class="Symbol">→</a> <a id="2630" href="Relation.Binary.Definitions.html#5955" class="Function">Substitutive</a> <a id="2643" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="2645" href="Data.Sum.Relation.Binary.Pointwise.html#827" class="Generalizable">ℓ₃</a> <a id="2648" class="Symbol">→</a>
<a id="2667" href="Relation.Binary.Definitions.html#5955" class="Function">Substitutive</a> <a id="2680" class="Symbol">(</a><a id="2681" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="2691" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2693" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="2694" class="Symbol">)</a> <a id="2696" href="Data.Sum.Relation.Binary.Pointwise.html#827" class="Generalizable">ℓ₃</a>
<a id="2699" href="Data.Sum.Relation.Binary.Pointwise.html#2593" class="Function">⊎-substitutive</a> <a id="2714" href="Data.Sum.Relation.Binary.Pointwise.html#2714" class="Bound">subst₁</a> <a id="2721" href="Data.Sum.Relation.Binary.Pointwise.html#2721" class="Bound">subst₂</a> <a id="2728" href="Data.Sum.Relation.Binary.Pointwise.html#2728" class="Bound">P</a> <a id="2730" class="Symbol">(</a><a id="2731" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2736" href="Data.Sum.Relation.Binary.Pointwise.html#2736" class="Bound">x</a><a id="2737" class="Symbol">)</a> <a id="2739" class="Symbol">=</a> <a id="2741" href="Data.Sum.Relation.Binary.Pointwise.html#2714" class="Bound">subst₁</a> <a id="2748" class="Symbol">(</a><a id="2749" href="Data.Sum.Relation.Binary.Pointwise.html#2728" class="Bound">P</a> <a id="2751" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2753" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="2757" class="Symbol">)</a> <a id="2759" href="Data.Sum.Relation.Binary.Pointwise.html#2736" class="Bound">x</a>
<a id="2761" href="Data.Sum.Relation.Binary.Pointwise.html#2593" class="Function">⊎-substitutive</a> <a id="2776" href="Data.Sum.Relation.Binary.Pointwise.html#2776" class="Bound">subst₁</a> <a id="2783" href="Data.Sum.Relation.Binary.Pointwise.html#2783" class="Bound">subst₂</a> <a id="2790" href="Data.Sum.Relation.Binary.Pointwise.html#2790" class="Bound">P</a> <a id="2792" class="Symbol">(</a><a id="2793" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="2798" href="Data.Sum.Relation.Binary.Pointwise.html#2798" class="Bound">x</a><a id="2799" class="Symbol">)</a> <a id="2801" class="Symbol">=</a> <a id="2803" href="Data.Sum.Relation.Binary.Pointwise.html#2783" class="Bound">subst₂</a> <a id="2810" class="Symbol">(</a><a id="2811" href="Data.Sum.Relation.Binary.Pointwise.html#2790" class="Bound">P</a> <a id="2813" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2815" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="2819" class="Symbol">)</a> <a id="2821" href="Data.Sum.Relation.Binary.Pointwise.html#2798" class="Bound">x</a>
<a id="⊎-decidable"></a><a id="2824" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="2836" class="Symbol">:</a> <a id="2838" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2848" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2850" class="Symbol">→</a> <a id="2852" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2862" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="2864" class="Symbol">→</a> <a id="2866" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2876" class="Symbol">(</a><a id="2877" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="2887" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="2889" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="2890" class="Symbol">)</a>
<a id="2892" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="2904" href="Data.Sum.Relation.Binary.Pointwise.html#2904" class="Bound Operator">_≟₁_</a> <a id="2909" href="Data.Sum.Relation.Binary.Pointwise.html#2909" class="Bound Operator">_≟₂_</a> <a id="2914" class="Symbol">(</a><a id="2915" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2920" href="Data.Sum.Relation.Binary.Pointwise.html#2920" class="Bound">x</a><a id="2921" class="Symbol">)</a> <a id="2923" class="Symbol">(</a><a id="2924" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2929" href="Data.Sum.Relation.Binary.Pointwise.html#2929" class="Bound">y</a><a id="2930" class="Symbol">)</a> <a id="2932" class="Symbol">=</a> <a id="2934" href="Relation.Nullary.Decidable.Core.html#4901" class="Function">Dec.map′</a> <a id="2943" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="2948" href="Data.Sum.Relation.Binary.Pointwise.html#1666" class="Function">drop-inj₁</a> <a id="2958" class="Symbol">(</a><a id="2959" href="Data.Sum.Relation.Binary.Pointwise.html#2920" class="Bound">x</a> <a id="2961" href="Data.Sum.Relation.Binary.Pointwise.html#2904" class="Bound Operator">≟₁</a> <a id="2964" href="Data.Sum.Relation.Binary.Pointwise.html#2929" class="Bound">y</a><a id="2965" class="Symbol">)</a>
<a id="2967" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="2979" href="Data.Sum.Relation.Binary.Pointwise.html#2979" class="Bound Operator">_≟₁_</a> <a id="2984" href="Data.Sum.Relation.Binary.Pointwise.html#2984" class="Bound Operator">_≟₂_</a> <a id="2989" class="Symbol">(</a><a id="2990" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2995" href="Data.Sum.Relation.Binary.Pointwise.html#2995" class="Bound">x</a><a id="2996" class="Symbol">)</a> <a id="2998" class="Symbol">(</a><a id="2999" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3004" href="Data.Sum.Relation.Binary.Pointwise.html#3004" class="Bound">y</a><a id="3005" class="Symbol">)</a> <a id="3007" class="Symbol">=</a> <a id="3009" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="3012" class="Symbol">λ()</a>
<a id="3016" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="3028" href="Data.Sum.Relation.Binary.Pointwise.html#3028" class="Bound Operator">_≟₁_</a> <a id="3033" href="Data.Sum.Relation.Binary.Pointwise.html#3033" class="Bound Operator">_≟₂_</a> <a id="3038" class="Symbol">(</a><a id="3039" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3044" href="Data.Sum.Relation.Binary.Pointwise.html#3044" class="Bound">x</a><a id="3045" class="Symbol">)</a> <a id="3047" class="Symbol">(</a><a id="3048" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3053" href="Data.Sum.Relation.Binary.Pointwise.html#3053" class="Bound">y</a><a id="3054" class="Symbol">)</a> <a id="3056" class="Symbol">=</a> <a id="3058" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="3061" class="Symbol">λ()</a>
<a id="3065" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="3077" href="Data.Sum.Relation.Binary.Pointwise.html#3077" class="Bound Operator">_≟₁_</a> <a id="3082" href="Data.Sum.Relation.Binary.Pointwise.html#3082" class="Bound Operator">_≟₂_</a> <a id="3087" class="Symbol">(</a><a id="3088" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3093" href="Data.Sum.Relation.Binary.Pointwise.html#3093" class="Bound">x</a><a id="3094" class="Symbol">)</a> <a id="3096" class="Symbol">(</a><a id="3097" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3102" href="Data.Sum.Relation.Binary.Pointwise.html#3102" class="Bound">y</a><a id="3103" class="Symbol">)</a> <a id="3105" class="Symbol">=</a> <a id="3107" href="Relation.Nullary.Decidable.Core.html#4901" class="Function">Dec.map′</a> <a id="3116" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3121" href="Data.Sum.Relation.Binary.Pointwise.html#1752" class="Function">drop-inj₂</a> <a id="3131" class="Symbol">(</a><a id="3132" href="Data.Sum.Relation.Binary.Pointwise.html#3093" class="Bound">x</a> <a id="3134" href="Data.Sum.Relation.Binary.Pointwise.html#3082" class="Bound Operator">≟₂</a> <a id="3137" href="Data.Sum.Relation.Binary.Pointwise.html#3102" class="Bound">y</a><a id="3138" class="Symbol">)</a>
<a id="⊎-reflexive"></a><a id="3141" href="Data.Sum.Relation.Binary.Pointwise.html#3141" class="Function">⊎-reflexive</a> <a id="3153" class="Symbol">:</a> <a id="3155" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3158" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="3160" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3162" class="Symbol">→</a> <a id="3164" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="3167" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="3169" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="3171" class="Symbol">→</a>
<a id="3187" class="Symbol">(</a><a id="3188" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3198" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3201" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="3203" class="Symbol">)</a> <a id="3205" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="3207" class="Symbol">(</a><a id="3208" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3218" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3220" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="3221" class="Symbol">)</a>
<a id="3223" href="Data.Sum.Relation.Binary.Pointwise.html#3141" class="Function">⊎-reflexive</a> <a id="3235" href="Data.Sum.Relation.Binary.Pointwise.html#3235" class="Bound">refl₁</a> <a id="3241" href="Data.Sum.Relation.Binary.Pointwise.html#3241" class="Bound">refl₂</a> <a id="3247" class="Symbol">(</a><a id="3248" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3253" href="Data.Sum.Relation.Binary.Pointwise.html#3253" class="Bound">x</a><a id="3254" class="Symbol">)</a> <a id="3256" class="Symbol">=</a> <a id="3258" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3263" class="Symbol">(</a><a id="3264" href="Data.Sum.Relation.Binary.Pointwise.html#3235" class="Bound">refl₁</a> <a id="3270" href="Data.Sum.Relation.Binary.Pointwise.html#3253" class="Bound">x</a><a id="3271" class="Symbol">)</a>
<a id="3273" href="Data.Sum.Relation.Binary.Pointwise.html#3141" class="Function">⊎-reflexive</a> <a id="3285" href="Data.Sum.Relation.Binary.Pointwise.html#3285" class="Bound">refl₁</a> <a id="3291" href="Data.Sum.Relation.Binary.Pointwise.html#3291" class="Bound">refl₂</a> <a id="3297" class="Symbol">(</a><a id="3298" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3303" href="Data.Sum.Relation.Binary.Pointwise.html#3303" class="Bound">x</a><a id="3304" class="Symbol">)</a> <a id="3306" class="Symbol">=</a> <a id="3308" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3313" class="Symbol">(</a><a id="3314" href="Data.Sum.Relation.Binary.Pointwise.html#3291" class="Bound">refl₂</a> <a id="3320" href="Data.Sum.Relation.Binary.Pointwise.html#3303" class="Bound">x</a><a id="3321" class="Symbol">)</a>
<a id="⊎-irreflexive"></a><a id="3324" href="Data.Sum.Relation.Binary.Pointwise.html#3324" class="Function">⊎-irreflexive</a> <a id="3338" class="Symbol">:</a> <a id="3340" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="3352" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3355" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3357" class="Symbol">→</a> <a id="3359" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="3371" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="3374" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="3376" class="Symbol">→</a>
<a id="3394" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="3406" class="Symbol">(</a><a id="3407" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3417" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3420" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="3422" class="Symbol">)</a> <a id="3424" class="Symbol">(</a><a id="3425" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3435" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3437" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="3438" class="Symbol">)</a>
<a id="3440" href="Data.Sum.Relation.Binary.Pointwise.html#3324" class="Function">⊎-irreflexive</a> <a id="3454" href="Data.Sum.Relation.Binary.Pointwise.html#3454" class="Bound">irrefl₁</a> <a id="3462" href="Data.Sum.Relation.Binary.Pointwise.html#3462" class="Bound">irrefl₂</a> <a id="3470" class="Symbol">(</a><a id="3471" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3476" href="Data.Sum.Relation.Binary.Pointwise.html#3476" class="Bound">x</a><a id="3477" class="Symbol">)</a> <a id="3479" class="Symbol">(</a><a id="3480" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3485" href="Data.Sum.Relation.Binary.Pointwise.html#3485" class="Bound">y</a><a id="3486" class="Symbol">)</a> <a id="3488" class="Symbol">=</a> <a id="3490" href="Data.Sum.Relation.Binary.Pointwise.html#3454" class="Bound">irrefl₁</a> <a id="3498" href="Data.Sum.Relation.Binary.Pointwise.html#3476" class="Bound">x</a> <a id="3500" href="Data.Sum.Relation.Binary.Pointwise.html#3485" class="Bound">y</a>
<a id="3502" href="Data.Sum.Relation.Binary.Pointwise.html#3324" class="Function">⊎-irreflexive</a> <a id="3516" href="Data.Sum.Relation.Binary.Pointwise.html#3516" class="Bound">irrefl₁</a> <a id="3524" href="Data.Sum.Relation.Binary.Pointwise.html#3524" class="Bound">irrefl₂</a> <a id="3532" class="Symbol">(</a><a id="3533" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3538" href="Data.Sum.Relation.Binary.Pointwise.html#3538" class="Bound">x</a><a id="3539" class="Symbol">)</a> <a id="3541" class="Symbol">(</a><a id="3542" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3547" href="Data.Sum.Relation.Binary.Pointwise.html#3547" class="Bound">y</a><a id="3548" class="Symbol">)</a> <a id="3550" class="Symbol">=</a> <a id="3552" href="Data.Sum.Relation.Binary.Pointwise.html#3524" class="Bound">irrefl₂</a> <a id="3560" href="Data.Sum.Relation.Binary.Pointwise.html#3538" class="Bound">x</a> <a id="3562" href="Data.Sum.Relation.Binary.Pointwise.html#3547" class="Bound">y</a>
<a id="⊎-antisymmetric"></a><a id="3565" href="Data.Sum.Relation.Binary.Pointwise.html#3565" class="Function">⊎-antisymmetric</a> <a id="3581" class="Symbol">:</a> <a id="3583" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="3597" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3600" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3602" class="Symbol">→</a> <a id="3604" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="3618" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="3621" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="3623" class="Symbol">→</a>
<a id="3643" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="3657" class="Symbol">(</a><a id="3658" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3668" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3671" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="3673" class="Symbol">)</a> <a id="3675" class="Symbol">(</a><a id="3676" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3686" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3688" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="3689" class="Symbol">)</a>
<a id="3691" href="Data.Sum.Relation.Binary.Pointwise.html#3565" class="Function">⊎-antisymmetric</a> <a id="3707" href="Data.Sum.Relation.Binary.Pointwise.html#3707" class="Bound">antisym₁</a> <a id="3716" href="Data.Sum.Relation.Binary.Pointwise.html#3716" class="Bound">antisym₂</a> <a id="3725" class="Symbol">(</a><a id="3726" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3731" href="Data.Sum.Relation.Binary.Pointwise.html#3731" class="Bound">x</a><a id="3732" class="Symbol">)</a> <a id="3734" class="Symbol">(</a><a id="3735" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3740" href="Data.Sum.Relation.Binary.Pointwise.html#3740" class="Bound">y</a><a id="3741" class="Symbol">)</a> <a id="3743" class="Symbol">=</a> <a id="3745" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3750" class="Symbol">(</a><a id="3751" href="Data.Sum.Relation.Binary.Pointwise.html#3707" class="Bound">antisym₁</a> <a id="3760" href="Data.Sum.Relation.Binary.Pointwise.html#3731" class="Bound">x</a> <a id="3762" href="Data.Sum.Relation.Binary.Pointwise.html#3740" class="Bound">y</a><a id="3763" class="Symbol">)</a>
<a id="3765" href="Data.Sum.Relation.Binary.Pointwise.html#3565" class="Function">⊎-antisymmetric</a> <a id="3781" href="Data.Sum.Relation.Binary.Pointwise.html#3781" class="Bound">antisym₁</a> <a id="3790" href="Data.Sum.Relation.Binary.Pointwise.html#3790" class="Bound">antisym₂</a> <a id="3799" class="Symbol">(</a><a id="3800" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3805" href="Data.Sum.Relation.Binary.Pointwise.html#3805" class="Bound">x</a><a id="3806" class="Symbol">)</a> <a id="3808" class="Symbol">(</a><a id="3809" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3814" href="Data.Sum.Relation.Binary.Pointwise.html#3814" class="Bound">y</a><a id="3815" class="Symbol">)</a> <a id="3817" class="Symbol">=</a> <a id="3819" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="3824" class="Symbol">(</a><a id="3825" href="Data.Sum.Relation.Binary.Pointwise.html#3790" class="Bound">antisym₂</a> <a id="3834" href="Data.Sum.Relation.Binary.Pointwise.html#3805" class="Bound">x</a> <a id="3836" href="Data.Sum.Relation.Binary.Pointwise.html#3814" class="Bound">y</a><a id="3837" class="Symbol">)</a>
<a id="⊎-respectsˡ"></a><a id="3840" href="Data.Sum.Relation.Binary.Pointwise.html#3840" class="Function">⊎-respectsˡ</a> <a id="3852" class="Symbol">:</a> <a id="3854" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3856" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="3866" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3869" class="Symbol">→</a> <a id="3871" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="3873" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="3883" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="3886" class="Symbol">→</a>
<a id="3902" class="Symbol">(</a><a id="3903" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3913" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="3915" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="3916" class="Symbol">)</a> <a id="3918" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="3928" class="Symbol">(</a><a id="3929" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="3939" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="3942" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="3944" class="Symbol">)</a>
<a id="3946" href="Data.Sum.Relation.Binary.Pointwise.html#3840" class="Function">⊎-respectsˡ</a> <a id="3958" href="Data.Sum.Relation.Binary.Pointwise.html#3958" class="Bound">resp₁</a> <a id="3964" href="Data.Sum.Relation.Binary.Pointwise.html#3964" class="Bound">resp₂</a> <a id="3970" class="Symbol">(</a><a id="3971" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3976" href="Data.Sum.Relation.Binary.Pointwise.html#3976" class="Bound">x</a><a id="3977" class="Symbol">)</a> <a id="3979" class="Symbol">(</a><a id="3980" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3985" href="Data.Sum.Relation.Binary.Pointwise.html#3985" class="Bound">y</a><a id="3986" class="Symbol">)</a> <a id="3988" class="Symbol">=</a> <a id="3990" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="3995" class="Symbol">(</a><a id="3996" href="Data.Sum.Relation.Binary.Pointwise.html#3958" class="Bound">resp₁</a> <a id="4002" href="Data.Sum.Relation.Binary.Pointwise.html#3976" class="Bound">x</a> <a id="4004" href="Data.Sum.Relation.Binary.Pointwise.html#3985" class="Bound">y</a><a id="4005" class="Symbol">)</a>
<a id="4007" href="Data.Sum.Relation.Binary.Pointwise.html#3840" class="Function">⊎-respectsˡ</a> <a id="4019" href="Data.Sum.Relation.Binary.Pointwise.html#4019" class="Bound">resp₁</a> <a id="4025" href="Data.Sum.Relation.Binary.Pointwise.html#4025" class="Bound">resp₂</a> <a id="4031" class="Symbol">(</a><a id="4032" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4037" href="Data.Sum.Relation.Binary.Pointwise.html#4037" class="Bound">x</a><a id="4038" class="Symbol">)</a> <a id="4040" class="Symbol">(</a><a id="4041" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4046" href="Data.Sum.Relation.Binary.Pointwise.html#4046" class="Bound">y</a><a id="4047" class="Symbol">)</a> <a id="4049" class="Symbol">=</a> <a id="4051" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4056" class="Symbol">(</a><a id="4057" href="Data.Sum.Relation.Binary.Pointwise.html#4025" class="Bound">resp₂</a> <a id="4063" href="Data.Sum.Relation.Binary.Pointwise.html#4037" class="Bound">x</a> <a id="4065" href="Data.Sum.Relation.Binary.Pointwise.html#4046" class="Bound">y</a><a id="4066" class="Symbol">)</a>
<a id="⊎-respectsʳ"></a><a id="4069" href="Data.Sum.Relation.Binary.Pointwise.html#4069" class="Function">⊎-respectsʳ</a> <a id="4081" class="Symbol">:</a> <a id="4083" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="4085" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="4095" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4098" class="Symbol">→</a> <a id="4100" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="4102" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="4112" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="4115" class="Symbol">→</a>
<a id="4131" class="Symbol">(</a><a id="4132" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4142" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="4144" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="4145" class="Symbol">)</a> <a id="4147" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="4157" class="Symbol">(</a><a id="4158" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4168" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4171" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="4173" class="Symbol">)</a>
<a id="4175" href="Data.Sum.Relation.Binary.Pointwise.html#4069" class="Function">⊎-respectsʳ</a> <a id="4187" href="Data.Sum.Relation.Binary.Pointwise.html#4187" class="Bound">resp₁</a> <a id="4193" href="Data.Sum.Relation.Binary.Pointwise.html#4193" class="Bound">resp₂</a> <a id="4199" class="Symbol">(</a><a id="4200" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="4205" href="Data.Sum.Relation.Binary.Pointwise.html#4205" class="Bound">x</a><a id="4206" class="Symbol">)</a> <a id="4208" class="Symbol">(</a><a id="4209" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="4214" href="Data.Sum.Relation.Binary.Pointwise.html#4214" class="Bound">y</a><a id="4215" class="Symbol">)</a> <a id="4217" class="Symbol">=</a> <a id="4219" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="4224" class="Symbol">(</a><a id="4225" href="Data.Sum.Relation.Binary.Pointwise.html#4187" class="Bound">resp₁</a> <a id="4231" href="Data.Sum.Relation.Binary.Pointwise.html#4205" class="Bound">x</a> <a id="4233" href="Data.Sum.Relation.Binary.Pointwise.html#4214" class="Bound">y</a><a id="4234" class="Symbol">)</a>
<a id="4236" href="Data.Sum.Relation.Binary.Pointwise.html#4069" class="Function">⊎-respectsʳ</a> <a id="4248" href="Data.Sum.Relation.Binary.Pointwise.html#4248" class="Bound">resp₁</a> <a id="4254" href="Data.Sum.Relation.Binary.Pointwise.html#4254" class="Bound">resp₂</a> <a id="4260" class="Symbol">(</a><a id="4261" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4266" href="Data.Sum.Relation.Binary.Pointwise.html#4266" class="Bound">x</a><a id="4267" class="Symbol">)</a> <a id="4269" class="Symbol">(</a><a id="4270" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4275" href="Data.Sum.Relation.Binary.Pointwise.html#4275" class="Bound">y</a><a id="4276" class="Symbol">)</a> <a id="4278" class="Symbol">=</a> <a id="4280" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="4285" class="Symbol">(</a><a id="4286" href="Data.Sum.Relation.Binary.Pointwise.html#4254" class="Bound">resp₂</a> <a id="4292" href="Data.Sum.Relation.Binary.Pointwise.html#4266" class="Bound">x</a> <a id="4294" href="Data.Sum.Relation.Binary.Pointwise.html#4275" class="Bound">y</a><a id="4295" class="Symbol">)</a>
<a id="⊎-respects₂"></a><a id="4298" href="Data.Sum.Relation.Binary.Pointwise.html#4298" class="Function">⊎-respects₂</a> <a id="4310" class="Symbol">:</a> <a id="4312" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="4314" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="4324" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4327" class="Symbol">→</a> <a id="4329" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="4331" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="4341" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="4344" class="Symbol">→</a>
<a id="4360" class="Symbol">(</a><a id="4361" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4371" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="4373" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="4374" class="Symbol">)</a> <a id="4376" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="4386" class="Symbol">(</a><a id="4387" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4397" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4400" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="4402" class="Symbol">)</a>
<a id="4404" href="Data.Sum.Relation.Binary.Pointwise.html#4298" class="Function">⊎-respects₂</a> <a id="4416" class="Symbol">(</a><a id="4417" href="Data.Sum.Relation.Binary.Pointwise.html#4417" class="Bound">r₁</a> <a id="4420" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4422" href="Data.Sum.Relation.Binary.Pointwise.html#4422" class="Bound">l₁</a><a id="4424" class="Symbol">)</a> <a id="4426" class="Symbol">(</a><a id="4427" href="Data.Sum.Relation.Binary.Pointwise.html#4427" class="Bound">r₂</a> <a id="4430" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4432" href="Data.Sum.Relation.Binary.Pointwise.html#4432" class="Bound">l₂</a><a id="4434" class="Symbol">)</a> <a id="4436" class="Symbol">=</a> <a id="4438" href="Data.Sum.Relation.Binary.Pointwise.html#4069" class="Function">⊎-respectsʳ</a> <a id="4450" href="Data.Sum.Relation.Binary.Pointwise.html#4417" class="Bound">r₁</a> <a id="4453" href="Data.Sum.Relation.Binary.Pointwise.html#4427" class="Bound">r₂</a> <a id="4456" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4458" href="Data.Sum.Relation.Binary.Pointwise.html#3840" class="Function">⊎-respectsˡ</a> <a id="4470" href="Data.Sum.Relation.Binary.Pointwise.html#4422" class="Bound">l₁</a> <a id="4473" href="Data.Sum.Relation.Binary.Pointwise.html#4432" class="Bound">l₂</a>
<a id="4477" class="Comment">------------------------------------------------------------------------</a>
<a id="4550" class="Comment">-- Structures</a>
<a id="⊎-isEquivalence"></a><a id="4565" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="4581" class="Symbol">:</a> <a id="4583" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="4597" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4600" class="Symbol">→</a> <a id="4602" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="4616" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="4619" class="Symbol">→</a>
<a id="4639" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="4653" class="Symbol">(</a><a id="4654" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4664" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4667" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="4669" class="Symbol">)</a>
<a id="4671" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="4687" href="Data.Sum.Relation.Binary.Pointwise.html#4687" class="Bound">eq₁</a> <a id="4691" href="Data.Sum.Relation.Binary.Pointwise.html#4691" class="Bound">eq₂</a> <a id="4695" class="Symbol">=</a> <a id="4697" class="Keyword">record</a>
<a id="4706" class="Symbol">{</a> <a id="4708" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="4714" class="Symbol">=</a> <a id="4716" href="Data.Sum.Relation.Binary.Pointwise.html#1838" class="Function">⊎-refl</a> <a id="4729" class="Symbol">(</a><a id="4730" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="4736" href="Data.Sum.Relation.Binary.Pointwise.html#4687" class="Bound">eq₁</a><a id="4739" class="Symbol">)</a> <a id="4741" class="Symbol">(</a><a id="4742" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="4748" href="Data.Sum.Relation.Binary.Pointwise.html#4691" class="Bound">eq₂</a><a id="4751" class="Symbol">)</a>
<a id="4755" class="Symbol">;</a> <a id="4757" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="4763" class="Symbol">=</a> <a id="4765" href="Data.Sum.Relation.Binary.Pointwise.html#1984" class="Function">⊎-symmetric</a> <a id="4778" class="Symbol">(</a><a id="4779" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="4785" href="Data.Sum.Relation.Binary.Pointwise.html#4687" class="Bound">eq₁</a><a id="4788" class="Symbol">)</a> <a id="4790" class="Symbol">(</a><a id="4791" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="4797" href="Data.Sum.Relation.Binary.Pointwise.html#4691" class="Bound">eq₂</a><a id="4800" class="Symbol">)</a>
<a id="4804" class="Symbol">;</a> <a id="4806" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="4812" class="Symbol">=</a> <a id="4814" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="4827" class="Symbol">(</a><a id="4828" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="4834" href="Data.Sum.Relation.Binary.Pointwise.html#4687" class="Bound">eq₁</a><a id="4837" class="Symbol">)</a> <a id="4839" class="Symbol">(</a><a id="4840" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="4846" href="Data.Sum.Relation.Binary.Pointwise.html#4691" class="Bound">eq₂</a><a id="4849" class="Symbol">)</a>
<a id="4853" class="Symbol">}</a> <a id="4855" class="Keyword">where</a> <a id="4861" class="Keyword">open</a> <a id="4866" href="Relation.Binary.Structures.html#1550" class="Module">IsEquivalence</a>
<a id="⊎-isDecEquivalence"></a><a id="4881" href="Data.Sum.Relation.Binary.Pointwise.html#4881" class="Function">⊎-isDecEquivalence</a> <a id="4900" class="Symbol">:</a> <a id="4902" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="4919" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4922" class="Symbol">→</a> <a id="4924" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="4941" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="4944" class="Symbol">→</a>
<a id="4967" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="4984" class="Symbol">(</a><a id="4985" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="4995" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="4998" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="5000" class="Symbol">)</a>
<a id="5002" href="Data.Sum.Relation.Binary.Pointwise.html#4881" class="Function">⊎-isDecEquivalence</a> <a id="5021" href="Data.Sum.Relation.Binary.Pointwise.html#5021" class="Bound">eq₁</a> <a id="5025" href="Data.Sum.Relation.Binary.Pointwise.html#5025" class="Bound">eq₂</a> <a id="5029" class="Symbol">=</a> <a id="5031" class="Keyword">record</a>
<a id="5040" class="Symbol">{</a> <a id="5042" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="5056" class="Symbol">=</a>
<a id="5064" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="5080" class="Symbol">(</a><a id="5081" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="5095" href="Data.Sum.Relation.Binary.Pointwise.html#5021" class="Bound">eq₁</a><a id="5098" class="Symbol">)</a> <a id="5100" class="Symbol">(</a><a id="5101" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="5115" href="Data.Sum.Relation.Binary.Pointwise.html#5025" class="Bound">eq₂</a><a id="5118" class="Symbol">)</a>
<a id="5122" class="Symbol">;</a> <a id="5124" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="5138" class="Symbol">=</a> <a id="5140" href="Data.Sum.Relation.Binary.Pointwise.html#2824" class="Function">⊎-decidable</a> <a id="5152" class="Symbol">(</a><a id="5153" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="5157" href="Data.Sum.Relation.Binary.Pointwise.html#5021" class="Bound">eq₁</a><a id="5160" class="Symbol">)</a> <a id="5162" class="Symbol">(</a><a id="5163" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="5167" href="Data.Sum.Relation.Binary.Pointwise.html#5025" class="Bound">eq₂</a><a id="5170" class="Symbol">)</a>
<a id="5174" class="Symbol">}</a> <a id="5176" class="Keyword">where</a> <a id="5182" class="Keyword">open</a> <a id="5187" href="Relation.Binary.Structures.html#1852" class="Module">IsDecEquivalence</a>
<a id="⊎-isPreorder"></a><a id="5205" href="Data.Sum.Relation.Binary.Pointwise.html#5205" class="Function">⊎-isPreorder</a> <a id="5218" class="Symbol">:</a> <a id="5220" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="5231" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="5234" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="5236" class="Symbol">→</a> <a id="5238" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="5249" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="5252" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="5254" class="Symbol">→</a>
<a id="5271" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="5282" class="Symbol">(</a><a id="5283" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="5293" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="5296" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="5298" class="Symbol">)</a> <a id="5300" class="Symbol">(</a><a id="5301" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="5311" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="5313" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="5314" class="Symbol">)</a>
<a id="5316" href="Data.Sum.Relation.Binary.Pointwise.html#5205" class="Function">⊎-isPreorder</a> <a id="5329" href="Data.Sum.Relation.Binary.Pointwise.html#5329" class="Bound">pre₁</a> <a id="5334" href="Data.Sum.Relation.Binary.Pointwise.html#5334" class="Bound">pre₂</a> <a id="5339" class="Symbol">=</a> <a id="5341" class="Keyword">record</a>
<a id="5350" class="Symbol">{</a> <a id="5352" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="5366" class="Symbol">=</a>
<a id="5374" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="5390" class="Symbol">(</a><a id="5391" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="5405" href="Data.Sum.Relation.Binary.Pointwise.html#5329" class="Bound">pre₁</a><a id="5409" class="Symbol">)</a> <a id="5411" class="Symbol">(</a><a id="5412" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="5426" href="Data.Sum.Relation.Binary.Pointwise.html#5334" class="Bound">pre₂</a><a id="5430" class="Symbol">)</a>
<a id="5434" class="Symbol">;</a> <a id="5436" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="5450" class="Symbol">=</a> <a id="5452" href="Data.Sum.Relation.Binary.Pointwise.html#3141" class="Function">⊎-reflexive</a> <a id="5464" class="Symbol">(</a><a id="5465" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="5475" href="Data.Sum.Relation.Binary.Pointwise.html#5329" class="Bound">pre₁</a><a id="5479" class="Symbol">)</a> <a id="5481" class="Symbol">(</a><a id="5482" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="5492" href="Data.Sum.Relation.Binary.Pointwise.html#5334" class="Bound">pre₂</a><a id="5496" class="Symbol">)</a>
<a id="5500" class="Symbol">;</a> <a id="5502" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="5516" class="Symbol">=</a> <a id="5518" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="5531" class="Symbol">(</a><a id="5532" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="5538" href="Data.Sum.Relation.Binary.Pointwise.html#5329" class="Bound">pre₁</a><a id="5542" class="Symbol">)</a> <a id="5544" class="Symbol">(</a><a id="5545" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="5551" href="Data.Sum.Relation.Binary.Pointwise.html#5334" class="Bound">pre₂</a><a id="5555" class="Symbol">)</a>
<a id="5559" class="Symbol">}</a> <a id="5561" class="Keyword">where</a> <a id="5567" class="Keyword">open</a> <a id="5572" href="Relation.Binary.Structures.html#2191" class="Module">IsPreorder</a>
<a id="⊎-isPartialOrder"></a><a id="5584" href="Data.Sum.Relation.Binary.Pointwise.html#5584" class="Function">⊎-isPartialOrder</a> <a id="5601" class="Symbol">:</a> <a id="5603" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="5618" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="5621" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="5623" class="Symbol">→</a> <a id="5625" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="5640" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="5643" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="5645" class="Symbol">→</a>
<a id="5666" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a>
<a id="5702" class="Symbol">(</a><a id="5703" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="5713" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="5716" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="5718" class="Symbol">)</a> <a id="5720" class="Symbol">(</a><a id="5721" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="5731" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="5733" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="5734" class="Symbol">)</a>
<a id="5736" href="Data.Sum.Relation.Binary.Pointwise.html#5584" class="Function">⊎-isPartialOrder</a> <a id="5753" href="Data.Sum.Relation.Binary.Pointwise.html#5753" class="Bound">po₁</a> <a id="5757" href="Data.Sum.Relation.Binary.Pointwise.html#5757" class="Bound">po₂</a> <a id="5761" class="Symbol">=</a> <a id="5763" class="Keyword">record</a>
<a id="5772" class="Symbol">{</a> <a id="5774" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="5785" class="Symbol">=</a> <a id="5787" href="Data.Sum.Relation.Binary.Pointwise.html#5205" class="Function">⊎-isPreorder</a> <a id="5800" class="Symbol">(</a><a id="5801" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="5812" href="Data.Sum.Relation.Binary.Pointwise.html#5753" class="Bound">po₁</a><a id="5815" class="Symbol">)</a> <a id="5817" class="Symbol">(</a><a id="5818" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="5829" href="Data.Sum.Relation.Binary.Pointwise.html#5757" class="Bound">po₂</a><a id="5832" class="Symbol">)</a>
<a id="5836" class="Symbol">;</a> <a id="5838" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="5849" class="Symbol">=</a> <a id="5851" href="Data.Sum.Relation.Binary.Pointwise.html#3565" class="Function">⊎-antisymmetric</a> <a id="5867" class="Symbol">(</a><a id="5868" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="5876" href="Data.Sum.Relation.Binary.Pointwise.html#5753" class="Bound">po₁</a><a id="5879" class="Symbol">)</a> <a id="5881" class="Symbol">(</a><a id="5882" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="5893" href="Data.Sum.Relation.Binary.Pointwise.html#5757" class="Bound">po₂</a><a id="5896" class="Symbol">)</a>
<a id="5900" class="Symbol">}</a> <a id="5902" class="Keyword">where</a> <a id="5908" class="Keyword">open</a> <a id="5913" href="Relation.Binary.Structures.html#3522" class="Module">IsPartialOrder</a>
<a id="⊎-isStrictPartialOrder"></a><a id="5929" href="Data.Sum.Relation.Binary.Pointwise.html#5929" class="Function">⊎-isStrictPartialOrder</a> <a id="5952" class="Symbol">:</a> <a id="5954" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="5975" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="5978" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="5980" class="Symbol">→</a>
<a id="6007" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="6028" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a> <a id="6031" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a> <a id="6033" class="Symbol">→</a>
<a id="6060" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a>
<a id="6108" class="Symbol">(</a><a id="6109" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="6119" href="Data.Sum.Relation.Binary.Pointwise.html#888" class="Generalizable">≈₁</a> <a id="6122" href="Data.Sum.Relation.Binary.Pointwise.html#891" class="Generalizable">≈₂</a><a id="6124" class="Symbol">)</a> <a id="6126" class="Symbol">(</a><a id="6127" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="6137" href="Data.Sum.Relation.Binary.Pointwise.html#864" class="Generalizable">R</a> <a id="6139" href="Data.Sum.Relation.Binary.Pointwise.html#866" class="Generalizable">S</a><a id="6140" class="Symbol">)</a>
<a id="6142" href="Data.Sum.Relation.Binary.Pointwise.html#5929" class="Function">⊎-isStrictPartialOrder</a> <a id="6165" href="Data.Sum.Relation.Binary.Pointwise.html#6165" class="Bound">spo₁</a> <a id="6170" href="Data.Sum.Relation.Binary.Pointwise.html#6170" class="Bound">spo₂</a> <a id="6175" class="Symbol">=</a> <a id="6177" class="Keyword">record</a>
<a id="6186" class="Symbol">{</a> <a id="6188" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="6202" class="Symbol">=</a>
<a id="6210" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="6226" class="Symbol">(</a><a id="6227" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="6241" href="Data.Sum.Relation.Binary.Pointwise.html#6165" class="Bound">spo₁</a><a id="6245" class="Symbol">)</a> <a id="6247" class="Symbol">(</a><a id="6248" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="6262" href="Data.Sum.Relation.Binary.Pointwise.html#6170" class="Bound">spo₂</a><a id="6266" class="Symbol">)</a>
<a id="6270" class="Symbol">;</a> <a id="6272" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="6286" class="Symbol">=</a> <a id="6288" href="Data.Sum.Relation.Binary.Pointwise.html#3324" class="Function">⊎-irreflexive</a> <a id="6302" class="Symbol">(</a><a id="6303" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="6312" href="Data.Sum.Relation.Binary.Pointwise.html#6165" class="Bound">spo₁</a><a id="6316" class="Symbol">)</a> <a id="6318" class="Symbol">(</a><a id="6319" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="6328" href="Data.Sum.Relation.Binary.Pointwise.html#6170" class="Bound">spo₂</a><a id="6332" class="Symbol">)</a>
<a id="6336" class="Symbol">;</a> <a id="6338" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="6352" class="Symbol">=</a> <a id="6354" href="Data.Sum.Relation.Binary.Pointwise.html#2161" class="Function">⊎-transitive</a> <a id="6368" class="Symbol">(</a><a id="6369" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="6378" href="Data.Sum.Relation.Binary.Pointwise.html#6165" class="Bound">spo₁</a><a id="6382" class="Symbol">)</a> <a id="6384" class="Symbol">(</a><a id="6385" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="6394" href="Data.Sum.Relation.Binary.Pointwise.html#6170" class="Bound">spo₂</a><a id="6398" class="Symbol">)</a>
<a id="6402" class="Symbol">;</a> <a id="6404" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="6418" class="Symbol">=</a> <a id="6420" href="Data.Sum.Relation.Binary.Pointwise.html#4298" class="Function">⊎-respects₂</a> <a id="6434" class="Symbol">(</a><a id="6435" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="6444" href="Data.Sum.Relation.Binary.Pointwise.html#6165" class="Bound">spo₁</a><a id="6448" class="Symbol">)</a> <a id="6450" class="Symbol">(</a><a id="6451" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="6460" href="Data.Sum.Relation.Binary.Pointwise.html#6170" class="Bound">spo₂</a><a id="6464" class="Symbol">)</a>
<a id="6468" class="Symbol">}</a> <a id="6470" class="Keyword">where</a> <a id="6476" class="Keyword">open</a> <a id="6481" href="Relation.Binary.Structures.html#4298" class="Module">IsStrictPartialOrder</a>
<a id="6503" class="Comment">------------------------------------------------------------------------</a>
<a id="6576" class="Comment">-- Bundles</a>
<a id="⊎-setoid"></a><a id="6588" href="Data.Sum.Relation.Binary.Pointwise.html#6588" class="Function">⊎-setoid</a> <a id="6597" class="Symbol">:</a> <a id="6599" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="6606" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="6608" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="6610" class="Symbol">→</a> <a id="6612" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="6619" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="6621" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a> <a id="6623" class="Symbol">→</a> <a id="6625" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="6632" class="Symbol">_</a> <a id="6634" class="Symbol">_</a>
<a id="6636" href="Data.Sum.Relation.Binary.Pointwise.html#6588" class="Function">⊎-setoid</a> <a id="6645" href="Data.Sum.Relation.Binary.Pointwise.html#6645" class="Bound">s₁</a> <a id="6648" href="Data.Sum.Relation.Binary.Pointwise.html#6648" class="Bound">s₂</a> <a id="6651" class="Symbol">=</a> <a id="6653" class="Keyword">record</a>
<a id="6662" class="Symbol">{</a> <a id="6664" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="6678" class="Symbol">=</a>
<a id="6686" href="Data.Sum.Relation.Binary.Pointwise.html#4565" class="Function">⊎-isEquivalence</a> <a id="6702" class="Symbol">(</a><a id="6703" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="6717" href="Data.Sum.Relation.Binary.Pointwise.html#6645" class="Bound">s₁</a><a id="6719" class="Symbol">)</a> <a id="6721" class="Symbol">(</a><a id="6722" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="6736" href="Data.Sum.Relation.Binary.Pointwise.html#6648" class="Bound">s₂</a><a id="6738" class="Symbol">)</a>
<a id="6742" class="Symbol">}</a> <a id="6744" class="Keyword">where</a> <a id="6750" class="Keyword">open</a> <a id="6755" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a>
<a id="⊎-decSetoid"></a><a id="6763" href="Data.Sum.Relation.Binary.Pointwise.html#6763" class="Function">⊎-decSetoid</a> <a id="6775" class="Symbol">:</a> <a id="6777" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6787" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="6789" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="6791" class="Symbol">→</a> <a id="6793" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6803" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="6805" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a> <a id="6807" class="Symbol">→</a> <a id="6809" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6819" class="Symbol">_</a> <a id="6821" class="Symbol">_</a>
<a id="6823" href="Data.Sum.Relation.Binary.Pointwise.html#6763" class="Function">⊎-decSetoid</a> <a id="6835" href="Data.Sum.Relation.Binary.Pointwise.html#6835" class="Bound">ds₁</a> <a id="6839" href="Data.Sum.Relation.Binary.Pointwise.html#6839" class="Bound">ds₂</a> <a id="6843" class="Symbol">=</a> <a id="6845" class="Keyword">record</a>
<a id="6854" class="Symbol">{</a> <a id="6856" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6873" class="Symbol">=</a>
<a id="6881" href="Data.Sum.Relation.Binary.Pointwise.html#4881" class="Function">⊎-isDecEquivalence</a> <a id="6900" class="Symbol">(</a><a id="6901" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6918" href="Data.Sum.Relation.Binary.Pointwise.html#6835" class="Bound">ds₁</a><a id="6921" class="Symbol">)</a> <a id="6923" class="Symbol">(</a><a id="6924" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6941" href="Data.Sum.Relation.Binary.Pointwise.html#6839" class="Bound">ds₂</a><a id="6944" class="Symbol">)</a>
<a id="6948" class="Symbol">}</a> <a id="6950" class="Keyword">where</a> <a id="6956" class="Keyword">open</a> <a id="6961" href="Relation.Binary.Bundles.html#1548" class="Module">DecSetoid</a>
<a id="⊎-preorder"></a><a id="6972" href="Data.Sum.Relation.Binary.Pointwise.html#6972" class="Function">⊎-preorder</a> <a id="6983" class="Symbol">:</a> <a id="6985" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6994" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="6996" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="6998" href="Data.Sum.Relation.Binary.Pointwise.html#821" class="Generalizable">ℓ₁</a> <a id="7001" class="Symbol">→</a> <a id="7003" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="7012" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="7014" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a> <a id="7016" href="Data.Sum.Relation.Binary.Pointwise.html#824" class="Generalizable">ℓ₂</a> <a id="7019" class="Symbol">→</a> <a id="7021" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="7030" class="Symbol">_</a> <a id="7032" class="Symbol">_</a> <a id="7034" class="Symbol">_</a>
<a id="7036" href="Data.Sum.Relation.Binary.Pointwise.html#6972" class="Function">⊎-preorder</a> <a id="7047" href="Data.Sum.Relation.Binary.Pointwise.html#7047" class="Bound">p₁</a> <a id="7050" href="Data.Sum.Relation.Binary.Pointwise.html#7050" class="Bound">p₂</a> <a id="7053" class="Symbol">=</a> <a id="7055" class="Keyword">record</a>
<a id="7064" class="Symbol">{</a> <a id="7066" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="7079" class="Symbol">=</a>
<a id="7087" href="Data.Sum.Relation.Binary.Pointwise.html#5205" class="Function">⊎-isPreorder</a> <a id="7100" class="Symbol">(</a><a id="7101" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="7112" href="Data.Sum.Relation.Binary.Pointwise.html#7047" class="Bound">p₁</a><a id="7114" class="Symbol">)</a> <a id="7116" class="Symbol">(</a><a id="7117" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="7128" href="Data.Sum.Relation.Binary.Pointwise.html#7050" class="Bound">p₂</a><a id="7130" class="Symbol">)</a>
<a id="7134" class="Symbol">}</a> <a id="7136" class="Keyword">where</a> <a id="7142" class="Keyword">open</a> <a id="7147" href="Relation.Binary.Bundles.html#2121" class="Module">Preorder</a>
<a id="⊎-poset"></a><a id="7157" href="Data.Sum.Relation.Binary.Pointwise.html#7157" class="Function">⊎-poset</a> <a id="7165" class="Symbol">:</a> <a id="7167" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="7173" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="7175" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="7177" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="7179" class="Symbol">→</a> <a id="7181" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="7187" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="7189" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="7191" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="7193" class="Symbol">→</a> <a id="7195" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="7201" class="Symbol">_</a> <a id="7203" class="Symbol">_</a> <a id="7205" class="Symbol">_</a>
<a id="7207" href="Data.Sum.Relation.Binary.Pointwise.html#7157" class="Function">⊎-poset</a> <a id="7215" href="Data.Sum.Relation.Binary.Pointwise.html#7215" class="Bound">po₁</a> <a id="7219" href="Data.Sum.Relation.Binary.Pointwise.html#7219" class="Bound">po₂</a> <a id="7223" class="Symbol">=</a> <a id="7225" class="Keyword">record</a>
<a id="7234" class="Symbol">{</a> <a id="7236" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="7251" class="Symbol">=</a>
<a id="7259" href="Data.Sum.Relation.Binary.Pointwise.html#5584" class="Function">⊎-isPartialOrder</a> <a id="7276" class="Symbol">(</a><a id="7277" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="7292" href="Data.Sum.Relation.Binary.Pointwise.html#7215" class="Bound">po₁</a><a id="7295" class="Symbol">)</a> <a id="7297" class="Symbol">(</a><a id="7298" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="7313" href="Data.Sum.Relation.Binary.Pointwise.html#7219" class="Bound">po₂</a><a id="7316" class="Symbol">)</a>
<a id="7320" class="Symbol">}</a> <a id="7322" class="Keyword">where</a> <a id="7328" class="Keyword">open</a> <a id="7333" href="Relation.Binary.Bundles.html#3530" class="Module">Poset</a>
<a id="7340" class="Comment">------------------------------------------------------------------------</a>
<a id="7413" class="Comment">-- Additional notation</a>
<a id="7437" class="Comment">-- Infix combining setoids</a>
<a id="7464" class="Keyword">infix</a> <a id="7470" class="Number">4</a> <a id="7472" href="Data.Sum.Relation.Binary.Pointwise.html#7477" class="Function Operator">_⊎ₛ_</a>
<a id="_⊎ₛ_"></a><a id="7477" href="Data.Sum.Relation.Binary.Pointwise.html#7477" class="Function Operator">_⊎ₛ_</a> <a id="7482" class="Symbol">:</a> <a id="7484" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7491" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a> <a id="7493" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a> <a id="7495" class="Symbol">→</a> <a id="7497" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7504" href="Data.Sum.Relation.Binary.Pointwise.html#817" class="Generalizable">c</a> <a id="7506" href="Data.Sum.Relation.Binary.Pointwise.html#819" class="Generalizable">d</a> <a id="7508" class="Symbol">→</a> <a id="7510" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7517" class="Symbol">_</a> <a id="7519" class="Symbol">_</a>
<a id="7521" href="Data.Sum.Relation.Binary.Pointwise.html#7477" class="Function Operator">_⊎ₛ_</a> <a id="7526" class="Symbol">=</a> <a id="7528" href="Data.Sum.Relation.Binary.Pointwise.html#6588" class="Function">⊎-setoid</a>
<a id="7538" class="Comment">------------------------------------------------------------------------</a>
<a id="7611" class="Comment">-- The propositional equality setoid over products can be</a>
<a id="7669" class="Comment">-- decomposed using Pointwise</a>
<a id="Pointwise-≡⇒≡"></a><a id="7700" href="Data.Sum.Relation.Binary.Pointwise.html#7700" class="Function">Pointwise-≡⇒≡</a> <a id="7714" class="Symbol">:</a> <a id="7716" class="Symbol">(</a><a id="7717" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="7727" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7731" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="7734" class="Symbol">)</a> <a id="7736" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="7738" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7742" class="Symbol">{</a><a id="7743" class="Argument">A</a> <a id="7745" class="Symbol">=</a> <a id="7747" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="7749" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="7751" href="Data.Sum.Relation.Binary.Pointwise.html#846" class="Generalizable">B</a><a id="7752" class="Symbol">}</a>
<a id="7754" href="Data.Sum.Relation.Binary.Pointwise.html#7700" class="Function">Pointwise-≡⇒≡</a> <a id="7768" class="Symbol">(</a><a id="7769" href="Data.Sum.Relation.Binary.Pointwise.html#1176" class="InductiveConstructor">inj₁</a> <a id="7774" href="Data.Sum.Relation.Binary.Pointwise.html#7774" class="Bound">x</a><a id="7775" class="Symbol">)</a> <a id="7777" class="Symbol">=</a> <a id="7779" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">P.cong</a> <a id="7786" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="7791" href="Data.Sum.Relation.Binary.Pointwise.html#7774" class="Bound">x</a>
<a id="7793" href="Data.Sum.Relation.Binary.Pointwise.html#7700" class="Function">Pointwise-≡⇒≡</a> <a id="7807" class="Symbol">(</a><a id="7808" href="Data.Sum.Relation.Binary.Pointwise.html#1235" class="InductiveConstructor">inj₂</a> <a id="7813" href="Data.Sum.Relation.Binary.Pointwise.html#7813" class="Bound">x</a><a id="7814" class="Symbol">)</a> <a id="7816" class="Symbol">=</a> <a id="7818" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">P.cong</a> <a id="7825" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="7830" href="Data.Sum.Relation.Binary.Pointwise.html#7813" class="Bound">x</a>
<a id="≡⇒Pointwise-≡"></a><a id="7833" href="Data.Sum.Relation.Binary.Pointwise.html#7833" class="Function">≡⇒Pointwise-≡</a> <a id="7847" class="Symbol">:</a> <a id="7849" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7853" class="Symbol">{</a><a id="7854" class="Argument">A</a> <a id="7856" class="Symbol">=</a> <a id="7858" href="Data.Sum.Relation.Binary.Pointwise.html#844" class="Generalizable">A</a> <a id="7860" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="7862" href="Data.Sum.Relation.Binary.Pointwise.html#846" class="Generalizable">B</a><a id="7863" class="Symbol">}</a> <a id="7865" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="7867" class="Symbol">(</a><a id="7868" href="Data.Sum.Relation.Binary.Pointwise.html#998" class="Datatype">Pointwise</a> <a id="7878" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7882" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="7885" class="Symbol">)</a>
<a id="7887" href="Data.Sum.Relation.Binary.Pointwise.html#7833" class="Function">≡⇒Pointwise-≡</a> <a id="7901" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a> <a id="7908" class="Symbol">=</a> <a id="7910" href="Data.Sum.Relation.Binary.Pointwise.html#1838" class="Function">⊎-refl</a> <a id="7917" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a> <a id="7924" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a>
<a id="Pointwise-≡↔≡"></a><a id="7932" href="Data.Sum.Relation.Binary.Pointwise.html#7932" class="Function">Pointwise-≡↔≡</a> <a id="7946" class="Symbol">:</a> <a id="7948" class="Symbol">(</a><a id="7949" href="Data.Sum.Relation.Binary.Pointwise.html#7949" class="Bound">A</a> <a id="7951" class="Symbol">:</a> <a id="7953" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="7957" href="Data.Sum.Relation.Binary.Pointwise.html#813" class="Generalizable">a</a><a id="7958" class="Symbol">)</a> <a id="7960" class="Symbol">(</a><a id="7961" href="Data.Sum.Relation.Binary.Pointwise.html#7961" class="Bound">B</a> <a id="7963" class="Symbol">:</a> <a id="7965" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="7969" href="Data.Sum.Relation.Binary.Pointwise.html#815" class="Generalizable">b</a><a id="7970" class="Symbol">)</a> <a id="7972" class="Symbol">→</a>
<a id="7991" href="Function.Bundles.html#7340" class="Record">Inverse</a> <a id="7999" class="Symbol">(</a><a id="8000" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="8009" href="Data.Sum.Relation.Binary.Pointwise.html#7949" class="Bound">A</a> <a id="8011" href="Data.Sum.Relation.Binary.Pointwise.html#7477" class="Function Operator">⊎ₛ</a> <a id="8014" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="8023" href="Data.Sum.Relation.Binary.Pointwise.html#7961" class="Bound">B</a><a id="8024" class="Symbol">)</a> <a id="8026" class="Symbol">(</a><a id="8027" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="8036" class="Symbol">(</a><a id="8037" href="Data.Sum.Relation.Binary.Pointwise.html#7949" class="Bound">A</a> <a id="8039" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="8041" href="Data.Sum.Relation.Binary.Pointwise.html#7961" class="Bound">B</a><a id="8042" class="Symbol">))</a>
<a id="8045" href="Data.Sum.Relation.Binary.Pointwise.html#7932" class="Function">Pointwise-≡↔≡</a> <a id="8059" class="Symbol">_</a> <a id="8061" class="Symbol">_</a> <a id="8063" class="Symbol">=</a> <a id="8065" class="Keyword">record</a>
<a id="8074" class="Symbol">{</a> <a id="8076" href="Function.Bundles.html#7394" class="Field">to</a> <a id="8086" class="Symbol">=</a> <a id="8088" href="Function.Base.html#704" class="Function">id</a>
<a id="8093" class="Symbol">;</a> <a id="8095" href="Function.Bundles.html#7418" class="Field">from</a> <a id="8105" class="Symbol">=</a> <a id="8107" href="Function.Base.html#704" class="Function">id</a>
<a id="8112" class="Symbol">;</a> <a id="8114" href="Function.Bundles.html#7442" class="Field">to-cong</a> <a id="8124" class="Symbol">=</a> <a id="8126" href="Data.Sum.Relation.Binary.Pointwise.html#7700" class="Function">Pointwise-≡⇒≡</a>
<a id="8142" class="Symbol">;</a> <a id="8144" href="Function.Bundles.html#7483" class="Field">from-cong</a> <a id="8154" class="Symbol">=</a> <a id="8156" href="Data.Sum.Relation.Binary.Pointwise.html#7833" class="Function">≡⇒Pointwise-≡</a>
<a id="8172" class="Symbol">;</a> <a id="8174" href="Function.Bundles.html#7526" class="Field">inverse</a> <a id="8184" class="Symbol">=</a> <a id="8186" href="Data.Sum.Relation.Binary.Pointwise.html#7700" class="Function">Pointwise-≡⇒≡</a> <a id="8200" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="8202" href="Data.Sum.Relation.Binary.Pointwise.html#7833" class="Function">≡⇒Pointwise-≡</a>
<a id="8218" class="Symbol">}</a>
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