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<html><head><meta charset="utf-8"><title>Relation.Binary.Construct.Converse</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike</a>
<a id="175" class="Comment">-- the module `Relation.Binary.Construct.Flip` this module does not</a>
<a id="243" class="Comment">-- flip the underlying equality.</a>
<a id="276" class="Comment">------------------------------------------------------------------------</a>
<a id="350" class="Symbol">{-#</a> <a id="354" class="Keyword">OPTIONS</a> <a id="362" class="Pragma">--cubical-compatible</a> <a id="383" class="Pragma">--safe</a> <a id="390" class="Symbol">#-}</a>
<a id="395" class="Keyword">open</a> <a id="400" class="Keyword">import</a> <a id="407" href="Relation.Binary.html" class="Module">Relation.Binary</a>
<a id="424" class="Keyword">module</a> <a id="431" href="Relation.Binary.Construct.Converse.html" class="Module">Relation.Binary.Construct.Converse</a> <a id="466" class="Keyword">where</a>
<a id="473" class="Keyword">open</a> <a id="478" class="Keyword">import</a> <a id="485" href="Function.Base.html" class="Module">Function.Base</a> <a id="499" class="Keyword">using</a> <a id="505" class="Symbol">(</a><a id="506" href="Function.Base.html#1563" class="Function">flip</a><a id="510" class="Symbol">;</a> <a id="512" href="Function.Base.html#1040" class="Function Operator">_∘_</a><a id="515" class="Symbol">)</a>
<a id="517" class="Keyword">open</a> <a id="522" class="Keyword">import</a> <a id="529" href="Data.Product.html" class="Module">Data.Product</a>
<a id="543" class="Comment">------------------------------------------------------------------------</a>
<a id="616" class="Comment">-- Properties</a>
<a id="631" class="Keyword">module</a> <a id="638" href="Relation.Binary.Construct.Converse.html#638" class="Module">_</a> <a id="640" class="Symbol">{</a><a id="641" href="Relation.Binary.Construct.Converse.html#641" class="Bound">a</a> <a id="643" href="Relation.Binary.Construct.Converse.html#643" class="Bound">ℓ</a><a id="644" class="Symbol">}</a> <a id="646" class="Symbol">{</a><a id="647" href="Relation.Binary.Construct.Converse.html#647" class="Bound">A</a> <a id="649" class="Symbol">:</a> <a id="651" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="655" href="Relation.Binary.Construct.Converse.html#641" class="Bound">a</a><a id="656" class="Symbol">}</a> <a id="658" class="Symbol">(</a><a id="659" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="661" class="Symbol">:</a> <a id="663" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="667" href="Relation.Binary.Construct.Converse.html#647" class="Bound">A</a> <a id="669" href="Relation.Binary.Construct.Converse.html#643" class="Bound">ℓ</a><a id="670" class="Symbol">)</a> <a id="672" class="Keyword">where</a>
<a id="681" href="Relation.Binary.Construct.Converse.html#681" class="Function">refl</a> <a id="686" class="Symbol">:</a> <a id="688" href="Relation.Binary.Definitions.html#1348" class="Function">Reflexive</a> <a id="698" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="700" class="Symbol">→</a> <a id="702" href="Relation.Binary.Definitions.html#1348" class="Function">Reflexive</a> <a id="712" class="Symbol">(</a><a id="713" href="Function.Base.html#1563" class="Function">flip</a> <a id="718" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="719" class="Symbol">)</a>
<a id="723" href="Relation.Binary.Construct.Converse.html#681" class="Function">refl</a> <a id="728" href="Relation.Binary.Construct.Converse.html#728" class="Bound">refl</a> <a id="733" class="Symbol">=</a> <a id="735" href="Relation.Binary.Construct.Converse.html#728" class="Bound">refl</a>
<a id="743" href="Relation.Binary.Construct.Converse.html#743" class="Function">sym</a> <a id="747" class="Symbol">:</a> <a id="749" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="759" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="761" class="Symbol">→</a> <a id="763" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="773" class="Symbol">(</a><a id="774" href="Function.Base.html#1563" class="Function">flip</a> <a id="779" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="780" class="Symbol">)</a>
<a id="784" href="Relation.Binary.Construct.Converse.html#743" class="Function">sym</a> <a id="788" href="Relation.Binary.Construct.Converse.html#788" class="Bound">sym</a> <a id="792" class="Symbol">=</a> <a id="794" href="Relation.Binary.Construct.Converse.html#788" class="Bound">sym</a>
<a id="801" href="Relation.Binary.Construct.Converse.html#801" class="Function">trans</a> <a id="807" class="Symbol">:</a> <a id="809" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="820" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="822" class="Symbol">→</a> <a id="824" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="835" class="Symbol">(</a><a id="836" href="Function.Base.html#1563" class="Function">flip</a> <a id="841" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="842" class="Symbol">)</a>
<a id="846" href="Relation.Binary.Construct.Converse.html#801" class="Function">trans</a> <a id="852" href="Relation.Binary.Construct.Converse.html#852" class="Bound">trans</a> <a id="858" class="Symbol">=</a> <a id="860" href="Function.Base.html#1563" class="Function">flip</a> <a id="865" href="Relation.Binary.Construct.Converse.html#852" class="Bound">trans</a>
<a id="874" href="Relation.Binary.Construct.Converse.html#874" class="Function">asym</a> <a id="879" class="Symbol">:</a> <a id="881" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="892" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="894" class="Symbol">→</a> <a id="896" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="907" class="Symbol">(</a><a id="908" href="Function.Base.html#1563" class="Function">flip</a> <a id="913" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="914" class="Symbol">)</a>
<a id="918" href="Relation.Binary.Construct.Converse.html#874" class="Function">asym</a> <a id="923" href="Relation.Binary.Construct.Converse.html#923" class="Bound">asym</a> <a id="928" class="Symbol">=</a> <a id="930" href="Relation.Binary.Construct.Converse.html#923" class="Bound">asym</a>
<a id="938" href="Relation.Binary.Construct.Converse.html#938" class="Function">total</a> <a id="944" class="Symbol">:</a> <a id="946" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="952" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="954" class="Symbol">→</a> <a id="956" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="962" class="Symbol">(</a><a id="963" href="Function.Base.html#1563" class="Function">flip</a> <a id="968" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="969" class="Symbol">)</a>
<a id="973" href="Relation.Binary.Construct.Converse.html#938" class="Function">total</a> <a id="979" href="Relation.Binary.Construct.Converse.html#979" class="Bound">total</a> <a id="985" href="Relation.Binary.Construct.Converse.html#985" class="Bound">x</a> <a id="987" href="Relation.Binary.Construct.Converse.html#987" class="Bound">y</a> <a id="989" class="Symbol">=</a> <a id="991" href="Relation.Binary.Construct.Converse.html#979" class="Bound">total</a> <a id="997" href="Relation.Binary.Construct.Converse.html#987" class="Bound">y</a> <a id="999" href="Relation.Binary.Construct.Converse.html#985" class="Bound">x</a>
<a id="1004" href="Relation.Binary.Construct.Converse.html#1004" class="Function">resp</a> <a id="1009" class="Symbol">:</a> <a id="1011" class="Symbol">∀</a> <a id="1013" class="Symbol">{</a><a id="1014" href="Relation.Binary.Construct.Converse.html#1014" class="Bound">p</a><a id="1015" class="Symbol">}</a> <a id="1017" class="Symbol">(</a><a id="1018" href="Relation.Binary.Construct.Converse.html#1018" class="Bound">P</a> <a id="1020" class="Symbol">:</a> <a id="1022" href="Relation.Binary.Construct.Converse.html#647" class="Bound">A</a> <a id="1024" class="Symbol">→</a> <a id="1026" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1030" href="Relation.Binary.Construct.Converse.html#1014" class="Bound">p</a><a id="1031" class="Symbol">)</a> <a id="1033" class="Symbol">→</a> <a id="1035" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1045" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="1047" class="Symbol">→</a>
<a id="1062" href="Relation.Binary.Construct.Converse.html#1018" class="Bound">P</a> <a id="1064" href="Relation.Binary.Definitions.html#3602" class="Function Operator">Respects</a> <a id="1073" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="1075" class="Symbol">→</a> <a id="1077" href="Relation.Binary.Construct.Converse.html#1018" class="Bound">P</a> <a id="1079" href="Relation.Binary.Definitions.html#3602" class="Function Operator">Respects</a> <a id="1088" class="Symbol">(</a><a id="1089" href="Function.Base.html#1563" class="Function">flip</a> <a id="1094" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="1095" class="Symbol">)</a>
<a id="1099" href="Relation.Binary.Construct.Converse.html#1004" class="Function">resp</a> <a id="1104" class="Symbol">_</a> <a id="1106" href="Relation.Binary.Construct.Converse.html#1106" class="Bound">sym</a> <a id="1110" href="Relation.Binary.Construct.Converse.html#1110" class="Bound">resp</a> <a id="1115" href="Relation.Binary.Construct.Converse.html#1115" class="Bound">∼</a> <a id="1117" class="Symbol">=</a> <a id="1119" href="Relation.Binary.Construct.Converse.html#1110" class="Bound">resp</a> <a id="1124" class="Symbol">(</a><a id="1125" href="Relation.Binary.Construct.Converse.html#1106" class="Bound">sym</a> <a id="1129" href="Relation.Binary.Construct.Converse.html#1115" class="Bound">∼</a><a id="1130" class="Symbol">)</a>
<a id="1135" href="Relation.Binary.Construct.Converse.html#1135" class="Function">max</a> <a id="1139" class="Symbol">:</a> <a id="1141" class="Symbol">∀</a> <a id="1143" class="Symbol">{</a><a id="1144" href="Relation.Binary.Construct.Converse.html#1144" class="Bound">⊥</a><a id="1145" class="Symbol">}</a> <a id="1147" class="Symbol">→</a> <a id="1149" href="Relation.Binary.Definitions.html#3346" class="Function">Minimum</a> <a id="1157" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="1159" href="Relation.Binary.Construct.Converse.html#1144" class="Bound">⊥</a> <a id="1161" class="Symbol">→</a> <a id="1163" href="Relation.Binary.Definitions.html#3197" class="Function">Maximum</a> <a id="1171" class="Symbol">(</a><a id="1172" href="Function.Base.html#1563" class="Function">flip</a> <a id="1177" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="1178" class="Symbol">)</a> <a id="1180" href="Relation.Binary.Construct.Converse.html#1144" class="Bound">⊥</a>
<a id="1184" href="Relation.Binary.Construct.Converse.html#1135" class="Function">max</a> <a id="1188" href="Relation.Binary.Construct.Converse.html#1188" class="Bound">min</a> <a id="1192" class="Symbol">=</a> <a id="1194" href="Relation.Binary.Construct.Converse.html#1188" class="Bound">min</a>
<a id="1201" href="Relation.Binary.Construct.Converse.html#1201" class="Function">min</a> <a id="1205" class="Symbol">:</a> <a id="1207" class="Symbol">∀</a> <a id="1209" class="Symbol">{</a><a id="1210" href="Relation.Binary.Construct.Converse.html#1210" class="Bound">⊤</a><a id="1211" class="Symbol">}</a> <a id="1213" class="Symbol">→</a> <a id="1215" href="Relation.Binary.Definitions.html#3197" class="Function">Maximum</a> <a id="1223" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a> <a id="1225" href="Relation.Binary.Construct.Converse.html#1210" class="Bound">⊤</a> <a id="1227" class="Symbol">→</a> <a id="1229" href="Relation.Binary.Definitions.html#3346" class="Function">Minimum</a> <a id="1237" class="Symbol">(</a><a id="1238" href="Function.Base.html#1563" class="Function">flip</a> <a id="1243" href="Relation.Binary.Construct.Converse.html#659" class="Bound">∼</a><a id="1244" class="Symbol">)</a> <a id="1246" href="Relation.Binary.Construct.Converse.html#1210" class="Bound">⊤</a>
<a id="1250" href="Relation.Binary.Construct.Converse.html#1201" class="Function">min</a> <a id="1254" href="Relation.Binary.Construct.Converse.html#1254" class="Bound">max</a> <a id="1258" class="Symbol">=</a> <a id="1260" href="Relation.Binary.Construct.Converse.html#1254" class="Bound">max</a>
<a id="1265" class="Keyword">module</a> <a id="1272" href="Relation.Binary.Construct.Converse.html#1272" class="Module">_</a> <a id="1274" class="Symbol">{</a><a id="1275" href="Relation.Binary.Construct.Converse.html#1275" class="Bound">a</a> <a id="1277" href="Relation.Binary.Construct.Converse.html#1277" class="Bound">ℓ₁</a> <a id="1280" href="Relation.Binary.Construct.Converse.html#1280" class="Bound">ℓ₂</a><a id="1282" class="Symbol">}</a> <a id="1284" class="Symbol">{</a><a id="1285" href="Relation.Binary.Construct.Converse.html#1285" class="Bound">A</a> <a id="1287" class="Symbol">:</a> <a id="1289" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1293" href="Relation.Binary.Construct.Converse.html#1275" class="Bound">a</a><a id="1294" class="Symbol">}</a> <a id="1296" class="Symbol">{</a><a id="1297" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1299" class="Symbol">:</a> <a id="1301" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1305" href="Relation.Binary.Construct.Converse.html#1285" class="Bound">A</a> <a id="1307" href="Relation.Binary.Construct.Converse.html#1277" class="Bound">ℓ₁</a><a id="1309" class="Symbol">}</a> <a id="1311" class="Symbol">(</a><a id="1312" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a> <a id="1314" class="Symbol">:</a> <a id="1316" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1320" href="Relation.Binary.Construct.Converse.html#1285" class="Bound">A</a> <a id="1322" href="Relation.Binary.Construct.Converse.html#1280" class="Bound">ℓ₂</a><a id="1324" class="Symbol">)</a> <a id="1326" class="Keyword">where</a>
<a id="1335" href="Relation.Binary.Construct.Converse.html#1335" class="Function">reflexive</a> <a id="1345" class="Symbol">:</a> <a id="1347" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1357" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1359" class="Symbol">→</a> <a id="1361" class="Symbol">(</a><a id="1362" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1364" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1366" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a><a id="1367" class="Symbol">)</a> <a id="1369" class="Symbol">→</a> <a id="1371" class="Symbol">(</a><a id="1372" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1374" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1376" href="Function.Base.html#1563" class="Function">flip</a> <a id="1381" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a><a id="1382" class="Symbol">)</a>
<a id="1386" href="Relation.Binary.Construct.Converse.html#1335" class="Function">reflexive</a> <a id="1396" href="Relation.Binary.Construct.Converse.html#1396" class="Bound">sym</a> <a id="1400" href="Relation.Binary.Construct.Converse.html#1400" class="Bound">impl</a> <a id="1405" class="Symbol">=</a> <a id="1407" href="Relation.Binary.Construct.Converse.html#1400" class="Bound">impl</a> <a id="1412" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="1414" href="Relation.Binary.Construct.Converse.html#1396" class="Bound">sym</a>
<a id="1421" href="Relation.Binary.Construct.Converse.html#1421" class="Function">irrefl</a> <a id="1428" class="Symbol">:</a> <a id="1430" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1440" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1442" class="Symbol">→</a> <a id="1444" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="1456" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1458" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a> <a id="1460" class="Symbol">→</a> <a id="1462" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="1474" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1476" class="Symbol">(</a><a id="1477" href="Function.Base.html#1563" class="Function">flip</a> <a id="1482" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a><a id="1483" class="Symbol">)</a>
<a id="1487" href="Relation.Binary.Construct.Converse.html#1421" class="Function">irrefl</a> <a id="1494" href="Relation.Binary.Construct.Converse.html#1494" class="Bound">sym</a> <a id="1498" href="Relation.Binary.Construct.Converse.html#1498" class="Bound">irrefl</a> <a id="1505" href="Relation.Binary.Construct.Converse.html#1505" class="Bound">x≈y</a> <a id="1509" href="Relation.Binary.Construct.Converse.html#1509" class="Bound">y∼x</a> <a id="1513" class="Symbol">=</a> <a id="1515" href="Relation.Binary.Construct.Converse.html#1498" class="Bound">irrefl</a> <a id="1522" class="Symbol">(</a><a id="1523" href="Relation.Binary.Construct.Converse.html#1494" class="Bound">sym</a> <a id="1527" href="Relation.Binary.Construct.Converse.html#1505" class="Bound">x≈y</a><a id="1530" class="Symbol">)</a> <a id="1532" href="Relation.Binary.Construct.Converse.html#1509" class="Bound">y∼x</a>
<a id="1539" href="Relation.Binary.Construct.Converse.html#1539" class="Function">antisym</a> <a id="1547" class="Symbol">:</a> <a id="1549" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1563" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1565" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a> <a id="1567" class="Symbol">→</a> <a id="1569" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1583" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1585" class="Symbol">(</a><a id="1586" href="Function.Base.html#1563" class="Function">flip</a> <a id="1591" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a><a id="1592" class="Symbol">)</a>
<a id="1596" href="Relation.Binary.Construct.Converse.html#1539" class="Function">antisym</a> <a id="1604" href="Relation.Binary.Construct.Converse.html#1604" class="Bound">antisym</a> <a id="1612" class="Symbol">=</a> <a id="1614" href="Function.Base.html#1563" class="Function">flip</a> <a id="1619" href="Relation.Binary.Construct.Converse.html#1604" class="Bound">antisym</a>
<a id="1630" href="Relation.Binary.Construct.Converse.html#1630" class="Function">compare</a> <a id="1638" class="Symbol">:</a> <a id="1640" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="1653" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1655" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a> <a id="1657" class="Symbol">→</a> <a id="1659" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="1672" href="Relation.Binary.Construct.Converse.html#1297" class="Bound">≈</a> <a id="1674" class="Symbol">(</a><a id="1675" href="Function.Base.html#1563" class="Function">flip</a> <a id="1680" href="Relation.Binary.Construct.Converse.html#1312" class="Bound">∼</a><a id="1681" class="Symbol">)</a>
<a id="1685" href="Relation.Binary.Construct.Converse.html#1630" class="Function">compare</a> <a id="1693" href="Relation.Binary.Construct.Converse.html#1693" class="Bound">cmp</a> <a id="1697" href="Relation.Binary.Construct.Converse.html#1697" class="Bound">x</a> <a id="1699" href="Relation.Binary.Construct.Converse.html#1699" class="Bound">y</a> <a id="1701" class="Keyword">with</a> <a id="1706" href="Relation.Binary.Construct.Converse.html#1693" class="Bound">cmp</a> <a id="1710" href="Relation.Binary.Construct.Converse.html#1697" class="Bound">x</a> <a id="1712" href="Relation.Binary.Construct.Converse.html#1699" class="Bound">y</a>
<a id="1716" class="Symbol">...</a> <a id="1720" class="Symbol">|</a> <a id="1722" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri&lt;</a> <a id="1727" href="Relation.Binary.Construct.Converse.html#1727" class="Bound">x&lt;y</a> <a id="1731" href="Relation.Binary.Construct.Converse.html#1731" class="Bound">x≉y</a> <a id="1735" href="Relation.Binary.Construct.Converse.html#1735" class="Bound">y≮x</a> <a id="1739" class="Symbol">=</a> <a id="1741" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri&gt;</a> <a id="1746" href="Relation.Binary.Construct.Converse.html#1735" class="Bound">y≮x</a> <a id="1750" href="Relation.Binary.Construct.Converse.html#1731" class="Bound">x≉y</a> <a id="1754" href="Relation.Binary.Construct.Converse.html#1727" class="Bound">x&lt;y</a>
<a id="1760" class="Symbol">...</a> <a id="1764" class="Symbol">|</a> <a id="1766" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="1771" href="Relation.Binary.Construct.Converse.html#1771" class="Bound">x≮y</a> <a id="1775" href="Relation.Binary.Construct.Converse.html#1775" class="Bound">x≈y</a> <a id="1779" href="Relation.Binary.Construct.Converse.html#1779" class="Bound">y≮x</a> <a id="1783" class="Symbol">=</a> <a id="1785" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="1790" href="Relation.Binary.Construct.Converse.html#1779" class="Bound">y≮x</a> <a id="1794" href="Relation.Binary.Construct.Converse.html#1775" class="Bound">x≈y</a> <a id="1798" href="Relation.Binary.Construct.Converse.html#1771" class="Bound">x≮y</a>
<a id="1804" class="Symbol">...</a> <a id="1808" class="Symbol">|</a> <a id="1810" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri&gt;</a> <a id="1815" href="Relation.Binary.Construct.Converse.html#1815" class="Bound">x≮y</a> <a id="1819" href="Relation.Binary.Construct.Converse.html#1819" class="Bound">x≉y</a> <a id="1823" href="Relation.Binary.Construct.Converse.html#1823" class="Bound">y&lt;x</a> <a id="1827" class="Symbol">=</a> <a id="1829" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri&lt;</a> <a id="1834" href="Relation.Binary.Construct.Converse.html#1823" class="Bound">y&lt;x</a> <a id="1838" href="Relation.Binary.Construct.Converse.html#1819" class="Bound">x≉y</a> <a id="1842" href="Relation.Binary.Construct.Converse.html#1815" class="Bound">x≮y</a>
<a id="1847" class="Keyword">module</a> <a id="1854" href="Relation.Binary.Construct.Converse.html#1854" class="Module">_</a> <a id="1856" class="Symbol">{</a><a id="1857" href="Relation.Binary.Construct.Converse.html#1857" class="Bound">a</a> <a id="1859" href="Relation.Binary.Construct.Converse.html#1859" class="Bound">ℓ₁</a> <a id="1862" href="Relation.Binary.Construct.Converse.html#1862" class="Bound">ℓ₂</a><a id="1864" class="Symbol">}</a> <a id="1866" class="Symbol">{</a><a id="1867" href="Relation.Binary.Construct.Converse.html#1867" class="Bound">A</a> <a id="1869" class="Symbol">:</a> <a id="1871" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1875" href="Relation.Binary.Construct.Converse.html#1857" class="Bound">a</a><a id="1876" class="Symbol">}</a> <a id="1878" class="Symbol">(</a><a id="1879" href="Relation.Binary.Construct.Converse.html#1879" class="Bound">∼₁</a> <a id="1882" class="Symbol">:</a> <a id="1884" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1888" href="Relation.Binary.Construct.Converse.html#1867" class="Bound">A</a> <a id="1890" href="Relation.Binary.Construct.Converse.html#1859" class="Bound">ℓ₁</a><a id="1892" class="Symbol">)</a> <a id="1894" class="Symbol">(</a><a id="1895" href="Relation.Binary.Construct.Converse.html#1895" class="Bound">∼₂</a> <a id="1898" class="Symbol">:</a> <a id="1900" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1904" href="Relation.Binary.Construct.Converse.html#1867" class="Bound">A</a> <a id="1906" href="Relation.Binary.Construct.Converse.html#1862" class="Bound">ℓ₂</a><a id="1908" class="Symbol">)</a> <a id="1910" class="Keyword">where</a>
<a id="1919" href="Relation.Binary.Construct.Converse.html#1919" class="Function">resp₂</a> <a id="1925" class="Symbol">:</a> <a id="1927" href="Relation.Binary.Construct.Converse.html#1879" class="Bound">∼₁</a> <a id="1930" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="1940" href="Relation.Binary.Construct.Converse.html#1895" class="Bound">∼₂</a> <a id="1943" class="Symbol">→</a> <a id="1945" class="Symbol">(</a><a id="1946" href="Function.Base.html#1563" class="Function">flip</a> <a id="1951" href="Relation.Binary.Construct.Converse.html#1879" class="Bound">∼₁</a><a id="1953" class="Symbol">)</a> <a id="1955" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="1965" href="Relation.Binary.Construct.Converse.html#1895" class="Bound">∼₂</a>
<a id="1970" href="Relation.Binary.Construct.Converse.html#1919" class="Function">resp₂</a> <a id="1976" class="Symbol">(</a><a id="1977" href="Relation.Binary.Construct.Converse.html#1977" class="Bound">resp₁</a> <a id="1983" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1985" href="Relation.Binary.Construct.Converse.html#1985" class="Bound">resp₂</a><a id="1990" class="Symbol">)</a> <a id="1992" class="Symbol">=</a> <a id="1994" href="Relation.Binary.Construct.Converse.html#1985" class="Bound">resp₂</a> <a id="2000" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2002" href="Relation.Binary.Construct.Converse.html#1977" class="Bound">resp₁</a>
<a id="2009" class="Keyword">module</a> <a id="2016" href="Relation.Binary.Construct.Converse.html#2016" class="Module">_</a> <a id="2018" class="Symbol">{</a><a id="2019" href="Relation.Binary.Construct.Converse.html#2019" class="Bound">a</a> <a id="2021" href="Relation.Binary.Construct.Converse.html#2021" class="Bound">b</a> <a id="2023" href="Relation.Binary.Construct.Converse.html#2023" class="Bound">ℓ</a><a id="2024" class="Symbol">}</a> <a id="2026" class="Symbol">{</a><a id="2027" href="Relation.Binary.Construct.Converse.html#2027" class="Bound">A</a> <a id="2029" class="Symbol">:</a> <a id="2031" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2035" href="Relation.Binary.Construct.Converse.html#2019" class="Bound">a</a><a id="2036" class="Symbol">}</a> <a id="2038" class="Symbol">{</a><a id="2039" href="Relation.Binary.Construct.Converse.html#2039" class="Bound">B</a> <a id="2041" class="Symbol">:</a> <a id="2043" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2047" href="Relation.Binary.Construct.Converse.html#2021" class="Bound">b</a><a id="2048" class="Symbol">}</a> <a id="2050" class="Symbol">(</a><a id="2051" href="Relation.Binary.Construct.Converse.html#2051" class="Bound">∼</a> <a id="2053" class="Symbol">:</a> <a id="2055" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="2059" href="Relation.Binary.Construct.Converse.html#2027" class="Bound">A</a> <a id="2061" href="Relation.Binary.Construct.Converse.html#2039" class="Bound">B</a> <a id="2063" href="Relation.Binary.Construct.Converse.html#2023" class="Bound">ℓ</a><a id="2064" class="Symbol">)</a> <a id="2066" class="Keyword">where</a>
<a id="2075" href="Relation.Binary.Construct.Converse.html#2075" class="Function">dec</a> <a id="2079" class="Symbol">:</a> <a id="2081" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="2091" href="Relation.Binary.Construct.Converse.html#2051" class="Bound">∼</a> <a id="2093" class="Symbol">→</a> <a id="2095" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="2105" class="Symbol">(</a><a id="2106" href="Function.Base.html#1563" class="Function">flip</a> <a id="2111" href="Relation.Binary.Construct.Converse.html#2051" class="Bound">∼</a><a id="2112" class="Symbol">)</a>
<a id="2116" href="Relation.Binary.Construct.Converse.html#2075" class="Function">dec</a> <a id="2120" href="Relation.Binary.Construct.Converse.html#2120" class="Bound">dec</a> <a id="2124" class="Symbol">=</a> <a id="2126" href="Function.Base.html#1563" class="Function">flip</a> <a id="2131" href="Relation.Binary.Construct.Converse.html#2120" class="Bound">dec</a>
<a id="2136" class="Comment">------------------------------------------------------------------------</a>
<a id="2209" class="Comment">-- Structures</a>
<a id="2224" class="Keyword">module</a> <a id="2231" href="Relation.Binary.Construct.Converse.html#2231" class="Module">_</a> <a id="2233" class="Symbol">{</a><a id="2234" href="Relation.Binary.Construct.Converse.html#2234" class="Bound">a</a> <a id="2236" href="Relation.Binary.Construct.Converse.html#2236" class="Bound">ℓ</a><a id="2237" class="Symbol">}</a> <a id="2239" class="Symbol">{</a><a id="2240" href="Relation.Binary.Construct.Converse.html#2240" class="Bound">A</a> <a id="2242" class="Symbol">:</a> <a id="2244" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2248" href="Relation.Binary.Construct.Converse.html#2234" class="Bound">a</a><a id="2249" class="Symbol">}</a> <a id="2251" class="Symbol">{</a><a id="2252" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2254" class="Symbol">:</a> <a id="2256" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2260" href="Relation.Binary.Construct.Converse.html#2240" class="Bound">A</a> <a id="2262" href="Relation.Binary.Construct.Converse.html#2236" class="Bound">ℓ</a><a id="2263" class="Symbol">}</a> <a id="2265" class="Keyword">where</a>
<a id="2274" href="Relation.Binary.Construct.Converse.html#2274" class="Function">isEquivalence</a> <a id="2288" class="Symbol">:</a> <a id="2290" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a> <a id="2304" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2306" class="Symbol">→</a> <a id="2308" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a> <a id="2322" class="Symbol">(</a><a id="2323" href="Function.Base.html#1563" class="Function">flip</a> <a id="2328" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a><a id="2329" class="Symbol">)</a>
<a id="2333" href="Relation.Binary.Construct.Converse.html#2274" class="Function">isEquivalence</a> <a id="2347" href="Relation.Binary.Construct.Converse.html#2347" class="Bound">eq</a> <a id="2350" class="Symbol">=</a> <a id="2352" class="Keyword">record</a>
<a id="2363" class="Symbol">{</a> <a id="2365" href="Relation.Binary.Structures.html#1577" class="Field">refl</a> <a id="2371" class="Symbol">=</a> <a id="2373" href="Relation.Binary.Construct.Converse.html#681" class="Function">refl</a> <a id="2379" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2381" href="Relation.Binary.Structures.html#1577" class="Field">Eq.refl</a>
<a id="2393" class="Symbol">;</a> <a id="2395" href="Relation.Binary.Structures.html#1603" class="Field">sym</a> <a id="2401" class="Symbol">=</a> <a id="2403" href="Relation.Binary.Construct.Converse.html#743" class="Function">sym</a> <a id="2409" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2411" href="Relation.Binary.Structures.html#1603" class="Field">Eq.sym</a>
<a id="2422" class="Symbol">;</a> <a id="2424" href="Relation.Binary.Structures.html#1629" class="Field">trans</a> <a id="2430" class="Symbol">=</a> <a id="2432" href="Relation.Binary.Construct.Converse.html#801" class="Function">trans</a> <a id="2438" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2440" href="Relation.Binary.Structures.html#1629" class="Field">Eq.trans</a>
<a id="2453" class="Symbol">}</a>
<a id="2459" class="Keyword">where</a> <a id="2465" class="Keyword">module</a> <a id="2472" href="Relation.Binary.Construct.Converse.html#2472" class="Module">Eq</a> <a id="2475" class="Symbol">=</a> <a id="2477" href="Relation.Binary.Structures.html#1531" class="Module">IsEquivalence</a> <a id="2491" href="Relation.Binary.Construct.Converse.html#2347" class="Bound">eq</a>
<a id="2497" href="Relation.Binary.Construct.Converse.html#2497" class="Function">isDecEquivalence</a> <a id="2514" class="Symbol">:</a> <a id="2516" href="Relation.Binary.Structures.html#1833" class="Record">IsDecEquivalence</a> <a id="2533" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2535" class="Symbol">→</a> <a id="2537" href="Relation.Binary.Structures.html#1833" class="Record">IsDecEquivalence</a> <a id="2554" class="Symbol">(</a><a id="2555" href="Function.Base.html#1563" class="Function">flip</a> <a id="2560" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a><a id="2561" class="Symbol">)</a>
<a id="2565" href="Relation.Binary.Construct.Converse.html#2497" class="Function">isDecEquivalence</a> <a id="2582" href="Relation.Binary.Construct.Converse.html#2582" class="Bound">eq</a> <a id="2585" class="Symbol">=</a> <a id="2587" class="Keyword">record</a>
<a id="2598" class="Symbol">{</a> <a id="2600" href="Relation.Binary.Structures.html#1896" class="Field">isEquivalence</a> <a id="2614" class="Symbol">=</a> <a id="2616" href="Relation.Binary.Construct.Converse.html#2274" class="Function">isEquivalence</a> <a id="2630" href="Relation.Binary.Structures.html#1896" class="Field">Dec.isEquivalence</a>
<a id="2652" class="Symbol">;</a> <a id="2654" href="Relation.Binary.Structures.html#1930" class="Field Operator">_≟_</a> <a id="2668" class="Symbol">=</a> <a id="2670" href="Relation.Binary.Construct.Converse.html#2075" class="Function">dec</a> <a id="2674" href="Relation.Binary.Construct.Converse.html#2252" class="Bound">≈</a> <a id="2676" href="Relation.Binary.Structures.html#1930" class="Field Operator">Dec._≟_</a>
<a id="2688" class="Symbol">}</a>
<a id="2694" class="Keyword">where</a> <a id="2700" class="Keyword">module</a> <a id="2707" href="Relation.Binary.Construct.Converse.html#2707" class="Module">Dec</a> <a id="2711" class="Symbol">=</a> <a id="2713" href="Relation.Binary.Structures.html#1833" class="Module">IsDecEquivalence</a> <a id="2730" href="Relation.Binary.Construct.Converse.html#2582" class="Bound">eq</a>
<a id="2734" class="Keyword">module</a> <a id="2741" href="Relation.Binary.Construct.Converse.html#2741" class="Module">_</a> <a id="2743" class="Symbol">{</a><a id="2744" href="Relation.Binary.Construct.Converse.html#2744" class="Bound">a</a> <a id="2746" href="Relation.Binary.Construct.Converse.html#2746" class="Bound">ℓ₁</a> <a id="2749" href="Relation.Binary.Construct.Converse.html#2749" class="Bound">ℓ₂</a><a id="2751" class="Symbol">}</a> <a id="2753" class="Symbol">{</a><a id="2754" href="Relation.Binary.Construct.Converse.html#2754" class="Bound">A</a> <a id="2756" class="Symbol">:</a> <a id="2758" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2762" href="Relation.Binary.Construct.Converse.html#2744" class="Bound">a</a><a id="2763" class="Symbol">}</a> <a id="2765" class="Symbol">{</a><a id="2766" href="Relation.Binary.Construct.Converse.html#2766" class="Bound">≈</a> <a id="2768" class="Symbol">:</a> <a id="2770" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2774" href="Relation.Binary.Construct.Converse.html#2754" class="Bound">A</a> <a id="2776" href="Relation.Binary.Construct.Converse.html#2746" class="Bound">ℓ₁</a><a id="2778" class="Symbol">}</a> <a id="2780" class="Symbol">{</a><a id="2781" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a> <a id="2783" class="Symbol">:</a> <a id="2785" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2789" href="Relation.Binary.Construct.Converse.html#2754" class="Bound">A</a> <a id="2791" href="Relation.Binary.Construct.Converse.html#2749" class="Bound">ℓ₂</a><a id="2793" class="Symbol">}</a> <a id="2795" class="Keyword">where</a>
<a id="2804" href="Relation.Binary.Construct.Converse.html#2804" class="Function">isPreorder</a> <a id="2815" class="Symbol">:</a> <a id="2817" href="Relation.Binary.Structures.html#2172" class="Record">IsPreorder</a> <a id="2828" href="Relation.Binary.Construct.Converse.html#2766" class="Bound">≈</a> <a id="2830" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a> <a id="2832" class="Symbol">→</a> <a id="2834" href="Relation.Binary.Structures.html#2172" class="Record">IsPreorder</a> <a id="2845" href="Relation.Binary.Construct.Converse.html#2766" class="Bound">≈</a> <a id="2847" class="Symbol">(</a><a id="2848" href="Function.Base.html#1563" class="Function">flip</a> <a id="2853" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a><a id="2854" class="Symbol">)</a>
<a id="2858" href="Relation.Binary.Construct.Converse.html#2804" class="Function">isPreorder</a> <a id="2869" href="Relation.Binary.Construct.Converse.html#2869" class="Bound">O</a> <a id="2871" class="Symbol">=</a> <a id="2873" class="Keyword">record</a>
<a id="2884" class="Symbol">{</a> <a id="2886" href="Relation.Binary.Structures.html#2237" class="Field">isEquivalence</a> <a id="2900" class="Symbol">=</a> <a id="2902" href="Relation.Binary.Structures.html#2237" class="Field">O.isEquivalence</a>
<a id="2922" class="Symbol">;</a> <a id="2924" href="Relation.Binary.Structures.html#2340" class="Field">reflexive</a> <a id="2938" class="Symbol">=</a> <a id="2940" href="Relation.Binary.Construct.Converse.html#1335" class="Function">reflexive</a> <a id="2950" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a> <a id="2952" href="Relation.Binary.Structures.html#1603" class="Function">O.Eq.sym</a> <a id="2961" href="Relation.Binary.Structures.html#2340" class="Field">O.reflexive</a>
<a id="2977" class="Symbol">;</a> <a id="2979" href="Relation.Binary.Structures.html#2370" class="Field">trans</a> <a id="2993" class="Symbol">=</a> <a id="2995" href="Relation.Binary.Construct.Converse.html#801" class="Function">trans</a> <a id="3001" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a> <a id="3003" href="Relation.Binary.Structures.html#2370" class="Field">O.trans</a>
<a id="3015" class="Symbol">}</a>
<a id="3021" class="Keyword">where</a> <a id="3027" class="Keyword">module</a> <a id="3034" href="Relation.Binary.Construct.Converse.html#3034" class="Module">O</a> <a id="3036" class="Symbol">=</a> <a id="3038" href="Relation.Binary.Structures.html#2172" class="Module">IsPreorder</a> <a id="3049" href="Relation.Binary.Construct.Converse.html#2869" class="Bound">O</a>
<a id="3054" href="Relation.Binary.Construct.Converse.html#3054" class="Function">isTotalPreorder</a> <a id="3070" class="Symbol">:</a> <a id="3072" href="Relation.Binary.Structures.html#2742" class="Record">IsTotalPreorder</a> <a id="3088" href="Relation.Binary.Construct.Converse.html#2766" class="Bound">≈</a> <a id="3090" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a> <a id="3092" class="Symbol">→</a> <a id="3094" href="Relation.Binary.Structures.html#2742" class="Record">IsTotalPreorder</a> <a id="3110" href="Relation.Binary.Construct.Converse.html#2766" class="Bound">≈</a> <a id="3112" class="Symbol">(</a><a id="3113" href="Function.Base.html#1563" class="Function">flip</a> <a id="3118" href="Relation.Binary.Construct.Converse.html#2781" class="Bound">∼</a><a id="3119" class="Symbol">)</a>
<a id="3123" href="Relation.Binary.Construct.Converse.html#3054" class="Function">isTotalPreorder</a> <a id="3139" href="Relation.Binary.Construct.Converse.html#3139" class="Bound">O</a> <a id="3141" class="Symbol">=</a> <a id="3143" class="Keyword">record</a>
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<a id="4115" class="Symbol">{</a> <a id="4117" href="Relation.Binary.Structures.html#4034" class="Field">isEquivalence</a> <a id="4131" class="Symbol">=</a> <a id="4133" href="Relation.Binary.Structures.html#4034" class="Field">O.isEquivalence</a>
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<a id="4928" class="Symbol">}</a>
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<a id="5108" class="Symbol">}</a>
<a id="5114" class="Keyword">where</a> <a id="5120" class="Keyword">module</a> <a id="5127" href="Relation.Binary.Construct.Converse.html#5127" class="Module">O</a> <a id="5129" class="Symbol">=</a> <a id="5131" href="Relation.Binary.Bundles.html#1929" class="Module">Preorder</a> <a id="5140" href="Relation.Binary.Construct.Converse.html#5050" class="Bound">O</a>
<a id="5145" href="Relation.Binary.Construct.Converse.html#5145" class="Function">totalPreorder</a> <a id="5159" class="Symbol">:</a> <a id="5161" href="Relation.Binary.Bundles.html#2384" class="Record">TotalPreorder</a> <a id="5175" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5177" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5180" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="5183" class="Symbol">→</a> <a id="5185" href="Relation.Binary.Bundles.html#2384" class="Record">TotalPreorder</a> <a id="5199" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5201" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5204" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="5209" href="Relation.Binary.Construct.Converse.html#5145" class="Function">totalPreorder</a> <a id="5223" href="Relation.Binary.Construct.Converse.html#5223" class="Bound">O</a> <a id="5225" class="Symbol">=</a> <a id="5227" class="Keyword">record</a>
<a id="5238" class="Symbol">{</a> <a id="5240" href="Relation.Binary.Bundles.html#2617" class="Field">isTotalPreorder</a> <a id="5256" class="Symbol">=</a> <a id="5258" href="Relation.Binary.Construct.Converse.html#3054" class="Function">isTotalPreorder</a> <a id="5274" href="Relation.Binary.Bundles.html#2617" class="Field">O.isTotalPreorder</a>
<a id="5296" class="Symbol">}</a> <a id="5298" class="Keyword">where</a> <a id="5304" class="Keyword">module</a> <a id="5311" href="Relation.Binary.Construct.Converse.html#5311" class="Module">O</a> <a id="5313" class="Symbol">=</a> <a id="5315" href="Relation.Binary.Bundles.html#2384" class="Module">TotalPreorder</a> <a id="5329" href="Relation.Binary.Construct.Converse.html#5223" class="Bound">O</a>
<a id="5334" href="Relation.Binary.Construct.Converse.html#5334" class="Function">poset</a> <a id="5340" class="Symbol">:</a> <a id="5342" href="Relation.Binary.Bundles.html#3037" class="Record">Poset</a> <a id="5348" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5350" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5353" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="5356" class="Symbol">→</a> <a id="5358" href="Relation.Binary.Bundles.html#3037" class="Record">Poset</a> <a id="5364" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5366" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5369" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="5374" href="Relation.Binary.Construct.Converse.html#5334" class="Function">poset</a> <a id="5380" href="Relation.Binary.Construct.Converse.html#5380" class="Bound">O</a> <a id="5382" class="Symbol">=</a> <a id="5384" class="Keyword">record</a>
<a id="5395" class="Symbol">{</a> <a id="5397" href="Relation.Binary.Bundles.html#3212" class="Field">isPartialOrder</a> <a id="5412" class="Symbol">=</a> <a id="5414" href="Relation.Binary.Construct.Converse.html#3272" class="Function">isPartialOrder</a> <a id="5429" href="Relation.Binary.Bundles.html#3212" class="Field">O.isPartialOrder</a>
<a id="5450" class="Symbol">}</a>
<a id="5456" class="Keyword">where</a> <a id="5462" class="Keyword">module</a> <a id="5469" href="Relation.Binary.Construct.Converse.html#5469" class="Module">O</a> <a id="5471" class="Symbol">=</a> <a id="5473" href="Relation.Binary.Bundles.html#3037" class="Module">Poset</a> <a id="5479" href="Relation.Binary.Construct.Converse.html#5380" class="Bound">O</a>
<a id="5484" href="Relation.Binary.Construct.Converse.html#5484" class="Function">totalOrder</a> <a id="5495" class="Symbol">:</a> <a id="5497" href="Relation.Binary.Bundles.html#5476" class="Record">TotalOrder</a> <a id="5508" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5510" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5513" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="5516" class="Symbol">→</a> <a id="5518" href="Relation.Binary.Bundles.html#5476" class="Record">TotalOrder</a> <a id="5529" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5531" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5534" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="5539" href="Relation.Binary.Construct.Converse.html#5484" class="Function">totalOrder</a> <a id="5550" href="Relation.Binary.Construct.Converse.html#5550" class="Bound">O</a> <a id="5552" class="Symbol">=</a> <a id="5554" class="Keyword">record</a>
<a id="5565" class="Symbol">{</a> <a id="5567" href="Relation.Binary.Bundles.html#5650" class="Field">isTotalOrder</a> <a id="5580" class="Symbol">=</a> <a id="5582" href="Relation.Binary.Construct.Converse.html#3493" class="Function">isTotalOrder</a> <a id="5595" href="Relation.Binary.Bundles.html#5650" class="Field">O.isTotalOrder</a>
<a id="5614" class="Symbol">}</a>
<a id="5620" class="Keyword">where</a> <a id="5626" class="Keyword">module</a> <a id="5633" href="Relation.Binary.Construct.Converse.html#5633" class="Module">O</a> <a id="5635" class="Symbol">=</a> <a id="5637" href="Relation.Binary.Bundles.html#5476" class="Module">TotalOrder</a> <a id="5648" href="Relation.Binary.Construct.Converse.html#5550" class="Bound">O</a>
<a id="5653" href="Relation.Binary.Construct.Converse.html#5653" class="Function">decTotalOrder</a> <a id="5667" class="Symbol">:</a> <a id="5669" href="Relation.Binary.Bundles.html#6016" class="Record">DecTotalOrder</a> <a id="5683" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5685" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5688" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="5691" class="Symbol">→</a> <a id="5693" href="Relation.Binary.Bundles.html#6016" class="Record">DecTotalOrder</a> <a id="5707" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5709" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5712" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="5717" href="Relation.Binary.Construct.Converse.html#5653" class="Function">decTotalOrder</a> <a id="5731" href="Relation.Binary.Construct.Converse.html#5731" class="Bound">O</a> <a id="5733" class="Symbol">=</a> <a id="5735" class="Keyword">record</a>
<a id="5746" class="Symbol">{</a> <a id="5748" href="Relation.Binary.Bundles.html#6202" class="Field">isDecTotalOrder</a> <a id="5764" class="Symbol">=</a> <a id="5766" href="Relation.Binary.Construct.Converse.html#3716" class="Function">isDecTotalOrder</a> <a id="5782" href="Relation.Binary.Bundles.html#6202" class="Field">O.isDecTotalOrder</a>
<a id="5804" class="Symbol">}</a>
<a id="5810" class="Keyword">where</a> <a id="5816" class="Keyword">module</a> <a id="5823" href="Relation.Binary.Construct.Converse.html#5823" class="Module">O</a> <a id="5825" class="Symbol">=</a> <a id="5827" href="Relation.Binary.Bundles.html#6016" class="Module">DecTotalOrder</a> <a id="5841" href="Relation.Binary.Construct.Converse.html#5731" class="Bound">O</a>
<a id="5846" href="Relation.Binary.Construct.Converse.html#5846" class="Function">strictPartialOrder</a> <a id="5865" class="Symbol">:</a> <a id="5867" href="Relation.Binary.Bundles.html#4117" class="Record">StrictPartialOrder</a> <a id="5886" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5888" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5891" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="5894" class="Symbol">→</a>
<a id="5919" href="Relation.Binary.Bundles.html#4117" class="Record">StrictPartialOrder</a> <a id="5938" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="5940" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="5943" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="5948" href="Relation.Binary.Construct.Converse.html#5846" class="Function">strictPartialOrder</a> <a id="5967" href="Relation.Binary.Construct.Converse.html#5967" class="Bound">O</a> <a id="5969" class="Symbol">=</a> <a id="5971" class="Keyword">record</a>
<a id="5982" class="Symbol">{</a> <a id="5984" href="Relation.Binary.Bundles.html#4323" class="Field">isStrictPartialOrder</a> <a id="6005" class="Symbol">=</a> <a id="6007" href="Relation.Binary.Construct.Converse.html#3970" class="Function">isStrictPartialOrder</a> <a id="6028" href="Relation.Binary.Bundles.html#4323" class="Field">O.isStrictPartialOrder</a>
<a id="6055" class="Symbol">}</a>
<a id="6061" class="Keyword">where</a> <a id="6067" class="Keyword">module</a> <a id="6074" href="Relation.Binary.Construct.Converse.html#6074" class="Module">O</a> <a id="6076" class="Symbol">=</a> <a id="6078" href="Relation.Binary.Bundles.html#4117" class="Module">StrictPartialOrder</a> <a id="6097" href="Relation.Binary.Construct.Converse.html#5967" class="Bound">O</a>
<a id="6102" href="Relation.Binary.Construct.Converse.html#6102" class="Function">strictTotalOrder</a> <a id="6119" class="Symbol">:</a> <a id="6121" href="Relation.Binary.Bundles.html#6937" class="Record">StrictTotalOrder</a> <a id="6138" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="6140" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="6143" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a> <a id="6146" class="Symbol">→</a>
<a id="6169" href="Relation.Binary.Bundles.html#6937" class="Record">StrictTotalOrder</a> <a id="6186" href="Relation.Binary.Construct.Converse.html#4974" class="Bound">a</a> <a id="6188" href="Relation.Binary.Construct.Converse.html#4976" class="Bound">ℓ₁</a> <a id="6191" href="Relation.Binary.Construct.Converse.html#4979" class="Bound">ℓ₂</a>
<a id="6196" href="Relation.Binary.Construct.Converse.html#6102" class="Function">strictTotalOrder</a> <a id="6213" href="Relation.Binary.Construct.Converse.html#6213" class="Bound">O</a> <a id="6215" class="Symbol">=</a> <a id="6217" class="Keyword">record</a>
<a id="6228" class="Symbol">{</a> <a id="6230" href="Relation.Binary.Bundles.html#7135" class="Field">isStrictTotalOrder</a> <a id="6249" class="Symbol">=</a> <a id="6251" href="Relation.Binary.Construct.Converse.html#4332" class="Function">isStrictTotalOrder</a> <a id="6270" href="Relation.Binary.Bundles.html#7135" class="Field">O.isStrictTotalOrder</a>
<a id="6295" class="Symbol">}</a>
<a id="6301" class="Keyword">where</a> <a id="6307" class="Keyword">module</a> <a id="6314" href="Relation.Binary.Construct.Converse.html#6314" class="Module">O</a> <a id="6316" class="Symbol">=</a> <a id="6318" href="Relation.Binary.Bundles.html#6937" class="Module">StrictTotalOrder</a> <a id="6335" href="Relation.Binary.Construct.Converse.html#6213" class="Bound">O</a>
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