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<html><head><meta charset="utf-8"><title>Relation.Binary.Construct.Flip.EqAndOrd</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
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<a id="74" class="Comment">-- The Agda standard library</a>
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<a id="103" class="Comment">--</a>
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<a id="106" class="Comment">-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike</a>
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<a id="175" class="Comment">-- the module `Relation.Binary.Construct.Flip.Ord` this module does not</a>
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<a id="247" class="Comment">-- flip the underlying equality.</a>
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<a id="280" class="Comment">------------------------------------------------------------------------</a>
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<a id="354" class="Symbol">{-#</a> <a id="358" class="Keyword">OPTIONS</a> <a id="366" class="Pragma">--cubical-compatible</a> <a id="387" class="Pragma">--safe</a> <a id="394" class="Symbol">#-}</a>
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<a id="399" class="Keyword">open</a> <a id="404" class="Keyword">import</a> <a id="411" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="432" class="Keyword">using</a> <a id="438" class="Symbol">(</a><a id="439" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="442" class="Symbol">;</a> <a id="444" href="Relation.Binary.Core.html#780" class="Function">REL</a><a id="447" class="Symbol">;</a> <a id="449" href="Relation.Binary.Core.html#1268" class="Function Operator">_⇒_</a><a id="452" class="Symbol">)</a>
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<a id="454" class="Keyword">open</a> <a id="459" class="Keyword">import</a> <a id="466" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a>
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<a id="492" class="Keyword">using</a> <a id="498" class="Symbol">(</a><a id="499" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="505" class="Symbol">;</a> <a id="507" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a><a id="516" class="Symbol">;</a> <a id="518" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a><a id="526" class="Symbol">;</a> <a id="528" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="533" class="Symbol">;</a> <a id="535" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a><a id="545" class="Symbol">;</a> <a id="547" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a><a id="560" class="Symbol">;</a> <a id="562" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a><a id="580" class="Symbol">;</a> <a id="582" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a><a id="598" class="Symbol">;</a> <a id="600" href="Relation.Binary.Bundles.html#2842" class="Record">TotalPreorder</a><a id="613" class="Symbol">)</a>
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<a id="615" class="Keyword">open</a> <a id="620" class="Keyword">import</a> <a id="627" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
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<a id="656" class="Keyword">using</a> <a id="662" class="Symbol">(</a><a id="663" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="676" class="Symbol">;</a> <a id="678" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a><a id="694" class="Symbol">;</a> <a id="696" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="706" class="Symbol">;</a> <a id="708" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="722" class="Symbol">;</a> <a id="724" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a><a id="736" class="Symbol">;</a> <a id="738" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a><a id="753" class="Symbol">;</a> <a id="755" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a><a id="775" class="Symbol">;</a> <a id="777" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a><a id="795" class="Symbol">;</a> <a id="797" href="Relation.Binary.Structures.html#3186" class="Record">IsTotalPreorder</a><a id="812" class="Symbol">)</a>
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<a id="814" class="Keyword">open</a> <a id="819" class="Keyword">import</a> <a id="826" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
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<a id="856" class="Keyword">using</a> <a id="862" class="Symbol">(</a><a id="863" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a><a id="872" class="Symbol">;</a> <a id="874" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a><a id="883" class="Symbol">;</a> <a id="885" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a><a id="895" class="Symbol">;</a> <a id="897" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a><a id="907" class="Symbol">;</a> <a id="909" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="914" class="Symbol">;</a> <a id="916" href="Relation.Binary.Definitions.html#5179" class="Function Operator">_Respects_</a><a id="926" class="Symbol">;</a> <a id="928" href="Relation.Binary.Definitions.html#5663" class="Function Operator">_Respects₂_</a><a id="939" class="Symbol">;</a> <a id="941" href="Relation.Binary.Definitions.html#3663" class="Function">Minimum</a><a id="948" class="Symbol">;</a> <a id="950" href="Relation.Binary.Definitions.html#3514" class="Function">Maximum</a><a id="957" class="Symbol">;</a> <a id="959" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a><a id="970" class="Symbol">;</a> <a id="972" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a><a id="985" class="Symbol">;</a> <a id="987" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a><a id="999" class="Symbol">;</a> <a id="1001" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="1010" class="Symbol">;</a> <a id="1012" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri<</a><a id="1016" class="Symbol">;</a> <a id="1018" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri></a><a id="1022" class="Symbol">;</a> <a id="1024" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a><a id="1028" class="Symbol">)</a>
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<a id="1031" class="Keyword">module</a> <a id="1038" href="Relation.Binary.Construct.Flip.EqAndOrd.html" class="Module">Relation.Binary.Construct.Flip.EqAndOrd</a> <a id="1078" class="Keyword">where</a>
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<a id="1085" class="Keyword">open</a> <a id="1090" class="Keyword">import</a> <a id="1097" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="1115" class="Keyword">using</a> <a id="1121" class="Symbol">(</a><a id="1122" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="1125" class="Symbol">)</a>
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<a id="1127" class="Keyword">open</a> <a id="1132" class="Keyword">import</a> <a id="1139" href="Function.Base.html" class="Module">Function.Base</a> <a id="1153" class="Keyword">using</a> <a id="1159" class="Symbol">(</a><a id="1160" href="Function.Base.html#1638" class="Function">flip</a><a id="1164" class="Symbol">;</a> <a id="1166" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="1169" class="Symbol">)</a>
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<a id="1171" class="Keyword">open</a> <a id="1176" class="Keyword">import</a> <a id="1183" href="Level.html" class="Module">Level</a> <a id="1189" class="Keyword">using</a> <a id="1195" class="Symbol">(</a><a id="1196" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="1201" class="Symbol">)</a>
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<a id="1204" class="Keyword">private</a>
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<a id="1214" class="Keyword">variable</a>
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<a id="1227" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="1229" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1229" class="Generalizable">b</a> <a id="1231" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1231" class="Generalizable">p</a> <a id="1233" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a> <a id="1235" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="1238" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="1241" class="Symbol">:</a> <a id="1243" href="Agda.Primitive.html#742" class="Postulate">Level</a>
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<a id="1253" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="1255" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1255" class="Generalizable">B</a> <a id="1257" class="Symbol">:</a> <a id="1259" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1263" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a>
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<a id="1269" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="1271" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1271" class="Generalizable">∼</a> <a id="1273" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a> <a id="1275" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1275" class="Generalizable"><</a> <a id="1277" class="Symbol">:</a> <a id="1279" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1283" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="1285" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a>
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<a id="1288" class="Comment">------------------------------------------------------------------------</a>
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<a id="1361" class="Comment">-- Properties</a>
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<a id="1376" class="Keyword">module</a> <a id="1383" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1383" class="Module">_</a> <a id="1385" class="Symbol">(</a><a id="1386" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1388" class="Symbol">:</a> <a id="1390" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1394" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="1396" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a><a id="1397" class="Symbol">)</a> <a id="1399" class="Keyword">where</a>
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<a id="1408" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1408" class="Function">refl</a> <a id="1413" class="Symbol">:</a> <a id="1415" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1425" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1427" class="Symbol">→</a> <a id="1429" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1439" class="Symbol">(</a><a id="1440" href="Function.Base.html#1638" class="Function">flip</a> <a id="1445" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1446" class="Symbol">)</a>
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<a id="1450" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1408" class="Function">refl</a> <a id="1455" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1455" class="Bound">refl</a> <a id="1460" class="Symbol">=</a> <a id="1462" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1455" class="Bound">refl</a>
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<a id="1470" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1470" class="Function">sym</a> <a id="1474" class="Symbol">:</a> <a id="1476" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1486" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1488" class="Symbol">→</a> <a id="1490" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1500" class="Symbol">(</a><a id="1501" href="Function.Base.html#1638" class="Function">flip</a> <a id="1506" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1507" class="Symbol">)</a>
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<a id="1511" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1470" class="Function">sym</a> <a id="1515" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1515" class="Bound">sym</a> <a id="1519" class="Symbol">=</a> <a id="1521" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1515" class="Bound">sym</a>
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<a id="1528" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1528" class="Function">trans</a> <a id="1534" class="Symbol">:</a> <a id="1536" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1547" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1549" class="Symbol">→</a> <a id="1551" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1562" class="Symbol">(</a><a id="1563" href="Function.Base.html#1638" class="Function">flip</a> <a id="1568" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1569" class="Symbol">)</a>
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<a id="1573" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1528" class="Function">trans</a> <a id="1579" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1579" class="Bound">trans</a> <a id="1585" class="Symbol">=</a> <a id="1587" href="Function.Base.html#1638" class="Function">flip</a> <a id="1592" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1579" class="Bound">trans</a>
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<a id="1601" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1601" class="Function">asym</a> <a id="1606" class="Symbol">:</a> <a id="1608" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="1619" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1621" class="Symbol">→</a> <a id="1623" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="1634" class="Symbol">(</a><a id="1635" href="Function.Base.html#1638" class="Function">flip</a> <a id="1640" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1641" class="Symbol">)</a>
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<a id="1645" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1601" class="Function">asym</a> <a id="1650" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1650" class="Bound">asym</a> <a id="1655" class="Symbol">=</a> <a id="1657" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1650" class="Bound">asym</a>
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<a id="1665" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1665" class="Function">total</a> <a id="1671" class="Symbol">:</a> <a id="1673" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="1679" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1681" class="Symbol">→</a> <a id="1683" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="1689" class="Symbol">(</a><a id="1690" href="Function.Base.html#1638" class="Function">flip</a> <a id="1695" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1696" class="Symbol">)</a>
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<a id="1700" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1665" class="Function">total</a> <a id="1706" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1706" class="Bound">total</a> <a id="1712" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1712" class="Bound">x</a> <a id="1714" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1714" class="Bound">y</a> <a id="1716" class="Symbol">=</a> <a id="1718" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1706" class="Bound">total</a> <a id="1724" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1714" class="Bound">y</a> <a id="1726" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1712" class="Bound">x</a>
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<a id="1731" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1731" class="Function">resp</a> <a id="1736" class="Symbol">:</a> <a id="1738" class="Symbol">∀</a> <a id="1740" class="Symbol">{</a><a id="1741" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1741" class="Bound">p</a><a id="1742" class="Symbol">}</a> <a id="1744" class="Symbol">(</a><a id="1745" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1745" class="Bound">P</a> <a id="1747" class="Symbol">:</a> <a id="1749" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1394" class="Bound">A</a> <a id="1751" class="Symbol">→</a> <a id="1753" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1757" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1741" class="Bound">p</a><a id="1758" class="Symbol">)</a> <a id="1760" class="Symbol">→</a> <a id="1762" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1772" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1774" class="Symbol">→</a>
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<a id="1789" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1745" class="Bound">P</a> <a id="1791" href="Relation.Binary.Definitions.html#5179" class="Function Operator">Respects</a> <a id="1800" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1802" class="Symbol">→</a> <a id="1804" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1745" class="Bound">P</a> <a id="1806" href="Relation.Binary.Definitions.html#5179" class="Function Operator">Respects</a> <a id="1815" class="Symbol">(</a><a id="1816" href="Function.Base.html#1638" class="Function">flip</a> <a id="1821" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1822" class="Symbol">)</a>
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<a id="1826" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1731" class="Function">resp</a> <a id="1831" class="Symbol">_</a> <a id="1833" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1833" class="Bound">sym</a> <a id="1837" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1837" class="Bound">resp</a> <a id="1842" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1842" class="Bound">∼</a> <a id="1844" class="Symbol">=</a> <a id="1846" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1837" class="Bound">resp</a> <a id="1851" class="Symbol">(</a><a id="1852" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1833" class="Bound">sym</a> <a id="1856" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1842" class="Bound">∼</a><a id="1857" class="Symbol">)</a>
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<a id="1862" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1862" class="Function">max</a> <a id="1866" class="Symbol">:</a> <a id="1868" class="Symbol">∀</a> <a id="1870" class="Symbol">{</a><a id="1871" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1871" class="Bound">⊥</a><a id="1872" class="Symbol">}</a> <a id="1874" class="Symbol">→</a> <a id="1876" href="Relation.Binary.Definitions.html#3663" class="Function">Minimum</a> <a id="1884" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1886" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1871" class="Bound">⊥</a> <a id="1888" class="Symbol">→</a> <a id="1890" href="Relation.Binary.Definitions.html#3514" class="Function">Maximum</a> <a id="1898" class="Symbol">(</a><a id="1899" href="Function.Base.html#1638" class="Function">flip</a> <a id="1904" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1905" class="Symbol">)</a> <a id="1907" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1871" class="Bound">⊥</a>
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<a id="1911" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1862" class="Function">max</a> <a id="1915" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1915" class="Bound">min</a> <a id="1919" class="Symbol">=</a> <a id="1921" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1915" class="Bound">min</a>
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<a id="1928" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1928" class="Function">min</a> <a id="1932" class="Symbol">:</a> <a id="1934" class="Symbol">∀</a> <a id="1936" class="Symbol">{</a><a id="1937" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1937" class="Bound">⊤</a><a id="1938" class="Symbol">}</a> <a id="1940" class="Symbol">→</a> <a id="1942" href="Relation.Binary.Definitions.html#3514" class="Function">Maximum</a> <a id="1950" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a> <a id="1952" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1937" class="Bound">⊤</a> <a id="1954" class="Symbol">→</a> <a id="1956" href="Relation.Binary.Definitions.html#3663" class="Function">Minimum</a> <a id="1964" class="Symbol">(</a><a id="1965" href="Function.Base.html#1638" class="Function">flip</a> <a id="1970" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1386" class="Bound">∼</a><a id="1971" class="Symbol">)</a> <a id="1973" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1937" class="Bound">⊤</a>
|
||
<a id="1977" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1928" class="Function">min</a> <a id="1981" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1981" class="Bound">max</a> <a id="1985" class="Symbol">=</a> <a id="1987" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1981" class="Bound">max</a>
|
||
|
||
<a id="1992" class="Keyword">module</a> <a id="1999" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1999" class="Module">_</a> <a id="2001" class="Symbol">{</a><a id="2002" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2004" class="Symbol">:</a> <a id="2006" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2010" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="2012" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a><a id="2014" class="Symbol">}</a> <a id="2016" class="Symbol">(</a><a id="2017" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a> <a id="2019" class="Symbol">:</a> <a id="2021" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2025" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="2027" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a><a id="2029" class="Symbol">)</a> <a id="2031" class="Keyword">where</a>
|
||
|
||
<a id="2040" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2040" class="Function">reflexive</a> <a id="2050" class="Symbol">:</a> <a id="2052" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2062" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2064" class="Symbol">→</a> <a id="2066" class="Symbol">(</a><a id="2067" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2069" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="2071" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a><a id="2072" class="Symbol">)</a> <a id="2074" class="Symbol">→</a> <a id="2076" class="Symbol">(</a><a id="2077" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2079" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="2081" href="Function.Base.html#1638" class="Function">flip</a> <a id="2086" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a><a id="2087" class="Symbol">)</a>
|
||
<a id="2091" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2040" class="Function">reflexive</a> <a id="2101" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2101" class="Bound">sym</a> <a id="2105" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2105" class="Bound">impl</a> <a id="2110" class="Symbol">=</a> <a id="2112" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2105" class="Bound">impl</a> <a id="2117" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2119" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2101" class="Bound">sym</a>
|
||
|
||
<a id="2126" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2126" class="Function">irrefl</a> <a id="2133" class="Symbol">:</a> <a id="2135" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2145" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2147" class="Symbol">→</a> <a id="2149" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="2161" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2163" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a> <a id="2165" class="Symbol">→</a> <a id="2167" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="2179" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2181" class="Symbol">(</a><a id="2182" href="Function.Base.html#1638" class="Function">flip</a> <a id="2187" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a><a id="2188" class="Symbol">)</a>
|
||
<a id="2192" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2126" class="Function">irrefl</a> <a id="2199" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2199" class="Bound">sym</a> <a id="2203" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2203" class="Bound">irrefl</a> <a id="2210" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2210" class="Bound">x≈y</a> <a id="2214" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2214" class="Bound">y∼x</a> <a id="2218" class="Symbol">=</a> <a id="2220" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2203" class="Bound">irrefl</a> <a id="2227" class="Symbol">(</a><a id="2228" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2199" class="Bound">sym</a> <a id="2232" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2210" class="Bound">x≈y</a><a id="2235" class="Symbol">)</a> <a id="2237" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2214" class="Bound">y∼x</a>
|
||
|
||
<a id="2244" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2244" class="Function">antisym</a> <a id="2252" class="Symbol">:</a> <a id="2254" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2268" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2270" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a> <a id="2272" class="Symbol">→</a> <a id="2274" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2288" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2290" class="Symbol">(</a><a id="2291" href="Function.Base.html#1638" class="Function">flip</a> <a id="2296" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a><a id="2297" class="Symbol">)</a>
|
||
<a id="2301" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2244" class="Function">antisym</a> <a id="2309" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2309" class="Bound">antisym</a> <a id="2317" class="Symbol">=</a> <a id="2319" href="Function.Base.html#1638" class="Function">flip</a> <a id="2324" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2309" class="Bound">antisym</a>
|
||
|
||
<a id="2335" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2335" class="Function">compare</a> <a id="2343" class="Symbol">:</a> <a id="2345" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="2358" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2360" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a> <a id="2362" class="Symbol">→</a> <a id="2364" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="2377" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2002" class="Bound">≈</a> <a id="2379" class="Symbol">(</a><a id="2380" href="Function.Base.html#1638" class="Function">flip</a> <a id="2385" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2017" class="Bound">∼</a><a id="2386" class="Symbol">)</a>
|
||
<a id="2390" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2335" class="Function">compare</a> <a id="2398" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2398" class="Bound">cmp</a> <a id="2402" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2402" class="Bound">x</a> <a id="2404" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2404" class="Bound">y</a> <a id="2406" class="Keyword">with</a> <a id="2411" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2398" class="Bound">cmp</a> <a id="2415" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2402" class="Bound">x</a> <a id="2417" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2404" class="Bound">y</a>
|
||
<a id="2421" class="Symbol">...</a> <a id="2425" class="Symbol">|</a> <a id="2427" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri<</a> <a id="2432" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2432" class="Bound">x<y</a> <a id="2436" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2436" class="Bound">x≉y</a> <a id="2440" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2440" class="Bound">y≮x</a> <a id="2444" class="Symbol">=</a> <a id="2446" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri></a> <a id="2451" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2440" class="Bound">y≮x</a> <a id="2455" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2436" class="Bound">x≉y</a> <a id="2459" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2432" class="Bound">x<y</a>
|
||
<a id="2465" class="Symbol">...</a> <a id="2469" class="Symbol">|</a> <a id="2471" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a> <a id="2476" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2476" class="Bound">x≮y</a> <a id="2480" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2480" class="Bound">x≈y</a> <a id="2484" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2484" class="Bound">y≮x</a> <a id="2488" class="Symbol">=</a> <a id="2490" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a> <a id="2495" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2484" class="Bound">y≮x</a> <a id="2499" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2480" class="Bound">x≈y</a> <a id="2503" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2476" class="Bound">x≮y</a>
|
||
<a id="2509" class="Symbol">...</a> <a id="2513" class="Symbol">|</a> <a id="2515" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri></a> <a id="2520" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2520" class="Bound">x≮y</a> <a id="2524" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2524" class="Bound">x≉y</a> <a id="2528" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2528" class="Bound">y<x</a> <a id="2532" class="Symbol">=</a> <a id="2534" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri<</a> <a id="2539" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2528" class="Bound">y<x</a> <a id="2543" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2524" class="Bound">x≉y</a> <a id="2547" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2520" class="Bound">x≮y</a>
|
||
|
||
<a id="2552" class="Keyword">module</a> <a id="2559" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2559" class="Module">_</a> <a id="2561" class="Symbol">(</a><a id="2562" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2562" class="Bound">∼₁</a> <a id="2565" class="Symbol">:</a> <a id="2567" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2571" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="2573" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a><a id="2575" class="Symbol">)</a> <a id="2577" class="Symbol">(</a><a id="2578" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2578" class="Bound">∼₂</a> <a id="2581" class="Symbol">:</a> <a id="2583" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2587" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="2589" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a><a id="2591" class="Symbol">)</a> <a id="2593" class="Keyword">where</a>
|
||
|
||
<a id="2602" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2602" class="Function">resp₂</a> <a id="2608" class="Symbol">:</a> <a id="2610" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2562" class="Bound">∼₁</a> <a id="2613" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2623" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2578" class="Bound">∼₂</a> <a id="2626" class="Symbol">→</a> <a id="2628" class="Symbol">(</a><a id="2629" href="Function.Base.html#1638" class="Function">flip</a> <a id="2634" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2562" class="Bound">∼₁</a><a id="2636" class="Symbol">)</a> <a id="2638" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2648" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2578" class="Bound">∼₂</a>
|
||
<a id="2653" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2602" class="Function">resp₂</a> <a id="2659" class="Symbol">(</a><a id="2660" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2660" class="Bound">resp₁</a> <a id="2666" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2668" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2668" class="Bound">resp₂</a><a id="2673" class="Symbol">)</a> <a id="2675" class="Symbol">=</a> <a id="2677" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2668" class="Bound">resp₂</a> <a id="2683" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2685" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2660" class="Bound">resp₁</a>
|
||
|
||
<a id="2692" class="Keyword">module</a> <a id="2699" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2699" class="Module">_</a> <a id="2701" class="Symbol">(</a><a id="2702" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2702" class="Bound">∼</a> <a id="2704" class="Symbol">:</a> <a id="2706" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="2710" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1253" class="Generalizable">A</a> <a id="2712" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1255" class="Generalizable">B</a> <a id="2714" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a><a id="2715" class="Symbol">)</a> <a id="2717" class="Keyword">where</a>
|
||
|
||
<a id="2726" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2726" class="Function">dec</a> <a id="2730" class="Symbol">:</a> <a id="2732" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2742" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2702" class="Bound">∼</a> <a id="2744" class="Symbol">→</a> <a id="2746" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2756" class="Symbol">(</a><a id="2757" href="Function.Base.html#1638" class="Function">flip</a> <a id="2762" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2702" class="Bound">∼</a><a id="2763" class="Symbol">)</a>
|
||
<a id="2767" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2726" class="Function">dec</a> <a id="2771" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2771" class="Bound">dec</a> <a id="2775" class="Symbol">=</a> <a id="2777" href="Function.Base.html#1638" class="Function">flip</a> <a id="2782" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2771" class="Bound">dec</a>
|
||
|
||
<a id="2787" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="2860" class="Comment">-- Structures</a>
|
||
|
||
<a id="isEquivalence"></a><a id="2875" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2875" class="Function">isEquivalence</a> <a id="2889" class="Symbol">:</a> <a id="2891" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="2905" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="2907" class="Symbol">→</a> <a id="2909" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="2923" class="Symbol">(</a><a id="2924" href="Function.Base.html#1638" class="Function">flip</a> <a id="2929" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a><a id="2930" class="Symbol">)</a>
|
||
<a id="2932" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2875" class="Function">isEquivalence</a> <a id="2946" class="Symbol">{</a><a id="2947" class="Argument">≈</a> <a id="2949" class="Symbol">=</a> <a id="2951" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2951" class="Bound">≈</a><a id="2952" class="Symbol">}</a> <a id="2954" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2954" class="Bound">eq</a> <a id="2957" class="Symbol">=</a> <a id="2959" class="Keyword">record</a>
|
||
<a id="2968" class="Symbol">{</a> <a id="2970" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="2976" class="Symbol">=</a> <a id="2978" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1408" class="Function">refl</a> <a id="2984" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2951" class="Bound">≈</a> <a id="2986" href="Relation.Binary.Structures.html#1596" class="Field">Eq.refl</a>
|
||
<a id="2996" class="Symbol">;</a> <a id="2998" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="3004" class="Symbol">=</a> <a id="3006" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1470" class="Function">sym</a> <a id="3012" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2951" class="Bound">≈</a> <a id="3014" href="Relation.Binary.Structures.html#1622" class="Field">Eq.sym</a>
|
||
<a id="3023" class="Symbol">;</a> <a id="3025" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="3031" class="Symbol">=</a> <a id="3033" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1528" class="Function">trans</a> <a id="3039" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2951" class="Bound">≈</a> <a id="3041" href="Relation.Binary.Structures.html#1648" class="Field">Eq.trans</a>
|
||
<a id="3052" class="Symbol">}</a> <a id="3054" class="Keyword">where</a> <a id="3060" class="Keyword">module</a> <a id="3067" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3067" class="Module">Eq</a> <a id="3070" class="Symbol">=</a> <a id="3072" href="Relation.Binary.Structures.html#1550" class="Module">IsEquivalence</a> <a id="3086" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2954" class="Bound">eq</a>
|
||
|
||
<a id="isDecEquivalence"></a><a id="3090" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3090" class="Function">isDecEquivalence</a> <a id="3107" class="Symbol">:</a> <a id="3109" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="3126" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3128" class="Symbol">→</a> <a id="3130" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="3147" class="Symbol">(</a><a id="3148" href="Function.Base.html#1638" class="Function">flip</a> <a id="3153" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a><a id="3154" class="Symbol">)</a>
|
||
<a id="3156" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3090" class="Function">isDecEquivalence</a> <a id="3173" class="Symbol">{</a><a id="3174" class="Argument">≈</a> <a id="3176" class="Symbol">=</a> <a id="3178" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3178" class="Bound">≈</a><a id="3179" class="Symbol">}</a> <a id="3181" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3181" class="Bound">eq</a> <a id="3184" class="Symbol">=</a> <a id="3186" class="Keyword">record</a>
|
||
<a id="3195" class="Symbol">{</a> <a id="3197" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="3211" class="Symbol">=</a> <a id="3213" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2875" class="Function">isEquivalence</a> <a id="3227" href="Relation.Binary.Structures.html#1915" class="Field">Dec.isEquivalence</a>
|
||
<a id="3247" class="Symbol">;</a> <a id="3249" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="3263" class="Symbol">=</a> <a id="3265" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2726" class="Function">dec</a> <a id="3269" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3178" class="Bound">≈</a> <a id="3271" href="Relation.Binary.Structures.html#1949" class="Field Operator">Dec._≟_</a>
|
||
<a id="3281" class="Symbol">}</a> <a id="3283" class="Keyword">where</a> <a id="3289" class="Keyword">module</a> <a id="3296" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3296" class="Module">Dec</a> <a id="3300" class="Symbol">=</a> <a id="3302" href="Relation.Binary.Structures.html#1852" class="Module">IsDecEquivalence</a> <a id="3319" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3181" class="Bound">eq</a>
|
||
|
||
<a id="isPreorder"></a><a id="3323" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3323" class="Function">isPreorder</a> <a id="3334" class="Symbol">:</a> <a id="3336" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="3347" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3349" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1271" class="Generalizable">∼</a> <a id="3351" class="Symbol">→</a> <a id="3353" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="3364" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3366" class="Symbol">(</a><a id="3367" href="Function.Base.html#1638" class="Function">flip</a> <a id="3372" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1271" class="Generalizable">∼</a><a id="3373" class="Symbol">)</a>
|
||
<a id="3375" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3323" class="Function">isPreorder</a> <a id="3386" class="Symbol">{</a><a id="3387" class="Argument">≈</a> <a id="3389" class="Symbol">=</a> <a id="3391" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3391" class="Bound">≈</a><a id="3392" class="Symbol">}</a> <a id="3394" class="Symbol">{</a><a id="3395" class="Argument">∼</a> <a id="3397" class="Symbol">=</a> <a id="3399" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3399" class="Bound">∼</a><a id="3400" class="Symbol">}</a> <a id="3402" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3402" class="Bound">O</a> <a id="3404" class="Symbol">=</a> <a id="3406" class="Keyword">record</a>
|
||
<a id="3415" class="Symbol">{</a> <a id="3417" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="3431" class="Symbol">=</a> <a id="3433" href="Relation.Binary.Structures.html#2256" class="Field">O.isEquivalence</a>
|
||
<a id="3451" class="Symbol">;</a> <a id="3453" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="3467" class="Symbol">=</a> <a id="3469" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2040" class="Function">reflexive</a> <a id="3479" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3399" class="Bound">∼</a> <a id="3481" href="Relation.Binary.Structures.html#1622" class="Function">O.Eq.sym</a> <a id="3490" href="Relation.Binary.Structures.html#2359" class="Field">O.reflexive</a>
|
||
<a id="3504" class="Symbol">;</a> <a id="3506" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="3520" class="Symbol">=</a> <a id="3522" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1528" class="Function">trans</a> <a id="3528" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3399" class="Bound">∼</a> <a id="3530" href="Relation.Binary.Structures.html#2389" class="Field">O.trans</a>
|
||
<a id="3540" class="Symbol">}</a> <a id="3542" class="Keyword">where</a> <a id="3548" class="Keyword">module</a> <a id="3555" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3555" class="Module">O</a> <a id="3557" class="Symbol">=</a> <a id="3559" href="Relation.Binary.Structures.html#2191" class="Module">IsPreorder</a> <a id="3570" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3402" class="Bound">O</a>
|
||
|
||
<a id="isTotalPreorder"></a><a id="3573" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3573" class="Function">isTotalPreorder</a> <a id="3589" class="Symbol">:</a> <a id="3591" href="Relation.Binary.Structures.html#3186" class="Record">IsTotalPreorder</a> <a id="3607" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3609" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1271" class="Generalizable">∼</a> <a id="3611" class="Symbol">→</a> <a id="3613" href="Relation.Binary.Structures.html#3186" class="Record">IsTotalPreorder</a> <a id="3629" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3631" class="Symbol">(</a><a id="3632" href="Function.Base.html#1638" class="Function">flip</a> <a id="3637" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1271" class="Generalizable">∼</a><a id="3638" class="Symbol">)</a>
|
||
<a id="3640" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3573" class="Function">isTotalPreorder</a> <a id="3656" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3656" class="Bound">O</a> <a id="3658" class="Symbol">=</a> <a id="3660" class="Keyword">record</a>
|
||
<a id="3669" class="Symbol">{</a> <a id="3671" href="Relation.Binary.Structures.html#3256" class="Field">isPreorder</a> <a id="3682" class="Symbol">=</a> <a id="3684" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3323" class="Function">isPreorder</a> <a id="3695" href="Relation.Binary.Structures.html#3256" class="Field">O.isPreorder</a>
|
||
<a id="3710" class="Symbol">;</a> <a id="3712" href="Relation.Binary.Structures.html#3288" class="Field">total</a> <a id="3723" class="Symbol">=</a> <a id="3725" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1665" class="Function">total</a> <a id="3731" class="Symbol">_</a> <a id="3733" href="Relation.Binary.Structures.html#3288" class="Field">O.total</a>
|
||
<a id="3743" class="Symbol">}</a> <a id="3745" class="Keyword">where</a> <a id="3751" class="Keyword">module</a> <a id="3758" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3758" class="Module">O</a> <a id="3760" class="Symbol">=</a> <a id="3762" href="Relation.Binary.Structures.html#3186" class="Module">IsTotalPreorder</a> <a id="3778" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3656" class="Bound">O</a>
|
||
|
||
<a id="isPartialOrder"></a><a id="3781" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3781" class="Function">isPartialOrder</a> <a id="3796" class="Symbol">:</a> <a id="3798" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="3813" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3815" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a> <a id="3817" class="Symbol">→</a> <a id="3819" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="3834" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="3836" class="Symbol">(</a><a id="3837" href="Function.Base.html#1638" class="Function">flip</a> <a id="3842" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a><a id="3843" class="Symbol">)</a>
|
||
<a id="3845" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3781" class="Function">isPartialOrder</a> <a id="3860" class="Symbol">{</a><a id="3861" class="Argument">≤</a> <a id="3863" class="Symbol">=</a> <a id="3865" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3865" class="Bound">≤</a><a id="3866" class="Symbol">}</a> <a id="3868" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3868" class="Bound">O</a> <a id="3870" class="Symbol">=</a> <a id="3872" class="Keyword">record</a>
|
||
<a id="3881" class="Symbol">{</a> <a id="3883" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="3894" class="Symbol">=</a> <a id="3896" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3323" class="Function">isPreorder</a> <a id="3907" href="Relation.Binary.Structures.html#3591" class="Field">O.isPreorder</a>
|
||
<a id="3922" class="Symbol">;</a> <a id="3924" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="3935" class="Symbol">=</a> <a id="3937" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2244" class="Function">antisym</a> <a id="3945" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3865" class="Bound">≤</a> <a id="3947" href="Relation.Binary.Structures.html#3623" class="Field">O.antisym</a>
|
||
<a id="3959" class="Symbol">}</a> <a id="3961" class="Keyword">where</a> <a id="3967" class="Keyword">module</a> <a id="3974" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3974" class="Module">O</a> <a id="3976" class="Symbol">=</a> <a id="3978" href="Relation.Binary.Structures.html#3522" class="Module">IsPartialOrder</a> <a id="3993" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3868" class="Bound">O</a>
|
||
|
||
<a id="isTotalOrder"></a><a id="3996" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3996" class="Function">isTotalOrder</a> <a id="4009" class="Symbol">:</a> <a id="4011" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="4024" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4026" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a> <a id="4028" class="Symbol">→</a> <a id="4030" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="4043" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4045" class="Symbol">(</a><a id="4046" href="Function.Base.html#1638" class="Function">flip</a> <a id="4051" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a><a id="4052" class="Symbol">)</a>
|
||
<a id="4054" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3996" class="Function">isTotalOrder</a> <a id="4067" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4067" class="Bound">O</a> <a id="4069" class="Symbol">=</a> <a id="4071" class="Keyword">record</a>
|
||
<a id="4080" class="Symbol">{</a> <a id="4082" href="Relation.Binary.Structures.html#5575" class="Field">isPartialOrder</a> <a id="4097" class="Symbol">=</a> <a id="4099" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3781" class="Function">isPartialOrder</a> <a id="4114" href="Relation.Binary.Structures.html#5575" class="Field">O.isPartialOrder</a>
|
||
<a id="4133" class="Symbol">;</a> <a id="4135" href="Relation.Binary.Structures.html#5615" class="Field">total</a> <a id="4150" class="Symbol">=</a> <a id="4152" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1665" class="Function">total</a> <a id="4158" class="Symbol">_</a> <a id="4160" href="Relation.Binary.Structures.html#5615" class="Field">O.total</a>
|
||
<a id="4170" class="Symbol">}</a> <a id="4172" class="Keyword">where</a> <a id="4178" class="Keyword">module</a> <a id="4185" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4185" class="Module">O</a> <a id="4187" class="Symbol">=</a> <a id="4189" href="Relation.Binary.Structures.html#5508" class="Module">IsTotalOrder</a> <a id="4202" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4067" class="Bound">O</a>
|
||
|
||
<a id="isDecTotalOrder"></a><a id="4205" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4205" class="Function">isDecTotalOrder</a> <a id="4221" class="Symbol">:</a> <a id="4223" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="4239" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4241" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a> <a id="4243" class="Symbol">→</a> <a id="4245" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="4261" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4263" class="Symbol">(</a><a id="4264" href="Function.Base.html#1638" class="Function">flip</a> <a id="4269" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1273" class="Generalizable">≤</a><a id="4270" class="Symbol">)</a>
|
||
<a id="4272" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4205" class="Function">isDecTotalOrder</a> <a id="4288" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4288" class="Bound">O</a> <a id="4290" class="Symbol">=</a> <a id="4292" class="Keyword">record</a>
|
||
<a id="4301" class="Symbol">{</a> <a id="4303" href="Relation.Binary.Structures.html#5914" class="Field">isTotalOrder</a> <a id="4316" class="Symbol">=</a> <a id="4318" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3996" class="Function">isTotalOrder</a> <a id="4331" href="Relation.Binary.Structures.html#5914" class="Field">O.isTotalOrder</a>
|
||
<a id="4348" class="Symbol">;</a> <a id="4350" href="Relation.Binary.Structures.html#5950" class="Field Operator">_≟_</a> <a id="4363" class="Symbol">=</a> <a id="4365" href="Relation.Binary.Structures.html#5950" class="Field Operator">O._≟_</a>
|
||
<a id="4373" class="Symbol">;</a> <a id="4375" href="Relation.Binary.Structures.html#5983" class="Field Operator">_≤?_</a> <a id="4388" class="Symbol">=</a> <a id="4390" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2726" class="Function">dec</a> <a id="4394" class="Symbol">_</a> <a id="4396" href="Relation.Binary.Structures.html#5983" class="Field Operator">O._≤?_</a>
|
||
<a id="4405" class="Symbol">}</a> <a id="4407" class="Keyword">where</a> <a id="4413" class="Keyword">module</a> <a id="4420" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4420" class="Module">O</a> <a id="4422" class="Symbol">=</a> <a id="4424" href="Relation.Binary.Structures.html#5825" class="Module">IsDecTotalOrder</a> <a id="4440" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4288" class="Bound">O</a>
|
||
|
||
<a id="isStrictPartialOrder"></a><a id="4443" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4443" class="Function">isStrictPartialOrder</a> <a id="4464" class="Symbol">:</a> <a id="4466" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="4487" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4489" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1275" class="Generalizable"><</a> <a id="4491" class="Symbol">→</a>
|
||
<a id="4516" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="4537" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4539" class="Symbol">(</a><a id="4540" href="Function.Base.html#1638" class="Function">flip</a> <a id="4545" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1275" class="Generalizable"><</a><a id="4546" class="Symbol">)</a>
|
||
<a id="4548" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4443" class="Function">isStrictPartialOrder</a> <a id="4569" class="Symbol">{</a><a id="4570" class="Argument"><</a> <a id="4572" class="Symbol">=</a> <a id="4574" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4574" class="Bound"><</a><a id="4575" class="Symbol">}</a> <a id="4577" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4577" class="Bound">O</a> <a id="4579" class="Symbol">=</a> <a id="4581" class="Keyword">record</a>
|
||
<a id="4590" class="Symbol">{</a> <a id="4592" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="4606" class="Symbol">=</a> <a id="4608" href="Relation.Binary.Structures.html#4373" class="Field">O.isEquivalence</a>
|
||
<a id="4626" class="Symbol">;</a> <a id="4628" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="4642" class="Symbol">=</a> <a id="4644" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2126" class="Function">irrefl</a> <a id="4651" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4574" class="Bound"><</a> <a id="4653" href="Relation.Binary.Structures.html#1622" class="Function">O.Eq.sym</a> <a id="4662" href="Relation.Binary.Structures.html#4407" class="Field">O.irrefl</a>
|
||
<a id="4673" class="Symbol">;</a> <a id="4675" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="4689" class="Symbol">=</a> <a id="4691" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1528" class="Function">trans</a> <a id="4697" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4574" class="Bound"><</a> <a id="4699" href="Relation.Binary.Structures.html#4447" class="Field">O.trans</a>
|
||
<a id="4709" class="Symbol">;</a> <a id="4711" href="Relation.Binary.Structures.html#4482" class="Field"><-resp-≈</a> <a id="4725" class="Symbol">=</a> <a id="4727" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2602" class="Function">resp₂</a> <a id="4733" class="Symbol">_</a> <a id="4735" class="Symbol">_</a> <a id="4737" href="Relation.Binary.Structures.html#4482" class="Field">O.<-resp-≈</a>
|
||
<a id="4750" class="Symbol">}</a> <a id="4752" class="Keyword">where</a> <a id="4758" class="Keyword">module</a> <a id="4765" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4765" class="Module">O</a> <a id="4767" class="Symbol">=</a> <a id="4769" href="Relation.Binary.Structures.html#4298" class="Module">IsStrictPartialOrder</a> <a id="4790" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4577" class="Bound">O</a>
|
||
|
||
<a id="isStrictTotalOrder"></a><a id="4793" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4793" class="Function">isStrictTotalOrder</a> <a id="4812" class="Symbol">:</a> <a id="4814" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="4833" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4835" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1275" class="Generalizable"><</a> <a id="4837" class="Symbol">→</a>
|
||
<a id="4860" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="4879" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1269" class="Generalizable">≈</a> <a id="4881" class="Symbol">(</a><a id="4882" href="Function.Base.html#1638" class="Function">flip</a> <a id="4887" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1275" class="Generalizable"><</a><a id="4888" class="Symbol">)</a>
|
||
<a id="4890" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4793" class="Function">isStrictTotalOrder</a> <a id="4909" class="Symbol">{</a><a id="4910" class="Argument"><</a> <a id="4912" class="Symbol">=</a> <a id="4914" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4914" class="Bound"><</a><a id="4915" class="Symbol">}</a> <a id="4917" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4917" class="Bound">O</a> <a id="4919" class="Symbol">=</a> <a id="4921" class="Keyword">record</a>
|
||
<a id="4930" class="Symbol">{</a> <a id="4932" href="Relation.Binary.Structures.html#6805" class="Field">isStrictPartialOrder</a> <a id="4953" class="Symbol">=</a> <a id="4955" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4443" class="Function">isStrictPartialOrder</a> <a id="4976" href="Relation.Binary.Structures.html#6805" class="Field">O.isStrictPartialOrder</a>
|
||
<a id="5001" class="Symbol">;</a> <a id="5003" href="Relation.Binary.Structures.html#6857" class="Field">compare</a> <a id="5024" class="Symbol">=</a> <a id="5026" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2335" class="Function">compare</a> <a id="5034" class="Symbol">_</a> <a id="5036" href="Relation.Binary.Structures.html#6857" class="Field">O.compare</a>
|
||
<a id="5048" class="Symbol">}</a> <a id="5050" class="Keyword">where</a> <a id="5056" class="Keyword">module</a> <a id="5063" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5063" class="Module">O</a> <a id="5065" class="Symbol">=</a> <a id="5067" href="Relation.Binary.Structures.html#6732" class="Module">IsStrictTotalOrder</a> <a id="5086" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4917" class="Bound">O</a>
|
||
|
||
<a id="5089" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="5162" class="Comment">-- Bundles</a>
|
||
|
||
<a id="setoid"></a><a id="5174" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5174" class="Function">setoid</a> <a id="5181" class="Symbol">:</a> <a id="5183" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5190" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5192" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a> <a id="5194" class="Symbol">→</a> <a id="5196" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5203" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5205" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a>
|
||
<a id="5207" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5174" class="Function">setoid</a> <a id="5214" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5214" class="Bound">S</a> <a id="5216" class="Symbol">=</a> <a id="5218" class="Keyword">record</a>
|
||
<a id="5227" class="Symbol">{</a> <a id="5229" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="5243" class="Symbol">=</a> <a id="5245" href="Relation.Binary.Construct.Flip.EqAndOrd.html#2875" class="Function">isEquivalence</a> <a id="5259" href="Relation.Binary.Bundles.html#1203" class="Field">S.isEquivalence</a>
|
||
<a id="5277" class="Symbol">}</a> <a id="5279" class="Keyword">where</a> <a id="5285" class="Keyword">module</a> <a id="5292" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5292" class="Module">S</a> <a id="5294" class="Symbol">=</a> <a id="5296" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a> <a id="5303" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5214" class="Bound">S</a>
|
||
|
||
<a id="decSetoid"></a><a id="5306" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5306" class="Function">decSetoid</a> <a id="5316" class="Symbol">:</a> <a id="5318" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="5328" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5330" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a> <a id="5332" class="Symbol">→</a> <a id="5334" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="5344" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5346" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1233" class="Generalizable">ℓ</a>
|
||
<a id="5348" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5306" class="Function">decSetoid</a> <a id="5358" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5358" class="Bound">S</a> <a id="5360" class="Symbol">=</a> <a id="5362" class="Keyword">record</a>
|
||
<a id="5371" class="Symbol">{</a> <a id="5373" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="5390" class="Symbol">=</a> <a id="5392" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3090" class="Function">isDecEquivalence</a> <a id="5409" href="Relation.Binary.Bundles.html#1680" class="Field">S.isDecEquivalence</a>
|
||
<a id="5430" class="Symbol">}</a> <a id="5432" class="Keyword">where</a> <a id="5438" class="Keyword">module</a> <a id="5445" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5445" class="Module">S</a> <a id="5447" class="Symbol">=</a> <a id="5449" href="Relation.Binary.Bundles.html#1548" class="Module">DecSetoid</a> <a id="5459" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5358" class="Bound">S</a>
|
||
|
||
<a id="preorder"></a><a id="5462" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5462" class="Function">preorder</a> <a id="5471" class="Symbol">:</a> <a id="5473" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="5482" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5484" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5487" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="5490" class="Symbol">→</a> <a id="5492" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="5501" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5503" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5506" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="5509" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5462" class="Function">preorder</a> <a id="5518" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5518" class="Bound">O</a> <a id="5520" class="Symbol">=</a> <a id="5522" class="Keyword">record</a>
|
||
<a id="5531" class="Symbol">{</a> <a id="5533" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="5544" class="Symbol">=</a> <a id="5546" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3323" class="Function">isPreorder</a> <a id="5557" href="Relation.Binary.Bundles.html#2334" class="Field">O.isPreorder</a>
|
||
<a id="5572" class="Symbol">}</a> <a id="5574" class="Keyword">where</a> <a id="5580" class="Keyword">module</a> <a id="5587" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5587" class="Module">O</a> <a id="5589" class="Symbol">=</a> <a id="5591" href="Relation.Binary.Bundles.html#2121" class="Module">Preorder</a> <a id="5600" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5518" class="Bound">O</a>
|
||
|
||
<a id="totalPreorder"></a><a id="5603" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5603" class="Function">totalPreorder</a> <a id="5617" class="Symbol">:</a> <a id="5619" href="Relation.Binary.Bundles.html#2842" class="Record">TotalPreorder</a> <a id="5633" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5635" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5638" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="5641" class="Symbol">→</a> <a id="5643" href="Relation.Binary.Bundles.html#2842" class="Record">TotalPreorder</a> <a id="5657" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5659" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5662" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="5665" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5603" class="Function">totalPreorder</a> <a id="5679" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5679" class="Bound">O</a> <a id="5681" class="Symbol">=</a> <a id="5683" class="Keyword">record</a>
|
||
<a id="5692" class="Symbol">{</a> <a id="5694" href="Relation.Binary.Bundles.html#3075" class="Field">isTotalPreorder</a> <a id="5710" class="Symbol">=</a> <a id="5712" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3573" class="Function">isTotalPreorder</a> <a id="5728" href="Relation.Binary.Bundles.html#3075" class="Field">O.isTotalPreorder</a>
|
||
<a id="5748" class="Symbol">}</a> <a id="5750" class="Keyword">where</a> <a id="5756" class="Keyword">module</a> <a id="5763" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5763" class="Module">O</a> <a id="5765" class="Symbol">=</a> <a id="5767" href="Relation.Binary.Bundles.html#2842" class="Module">TotalPreorder</a> <a id="5781" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5679" class="Bound">O</a>
|
||
|
||
<a id="poset"></a><a id="5784" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5784" class="Function">poset</a> <a id="5790" class="Symbol">:</a> <a id="5792" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="5798" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5800" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5803" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="5806" class="Symbol">→</a> <a id="5808" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="5814" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5816" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5819" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="5822" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5784" class="Function">poset</a> <a id="5828" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5828" class="Bound">O</a> <a id="5830" class="Symbol">=</a> <a id="5832" class="Keyword">record</a>
|
||
<a id="5841" class="Symbol">{</a> <a id="5843" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="5858" class="Symbol">=</a> <a id="5860" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3781" class="Function">isPartialOrder</a> <a id="5875" href="Relation.Binary.Bundles.html#3705" class="Field">O.isPartialOrder</a>
|
||
<a id="5894" class="Symbol">}</a> <a id="5896" class="Keyword">where</a> <a id="5902" class="Keyword">module</a> <a id="5909" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5909" class="Module">O</a> <a id="5911" class="Symbol">=</a> <a id="5913" href="Relation.Binary.Bundles.html#3530" class="Module">Poset</a> <a id="5919" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5828" class="Bound">O</a>
|
||
|
||
<a id="totalOrder"></a><a id="5922" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5922" class="Function">totalOrder</a> <a id="5933" class="Symbol">:</a> <a id="5935" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a> <a id="5946" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5948" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5951" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="5954" class="Symbol">→</a> <a id="5956" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a> <a id="5967" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="5969" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="5972" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="5975" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5922" class="Function">totalOrder</a> <a id="5986" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5986" class="Bound">O</a> <a id="5988" class="Symbol">=</a> <a id="5990" class="Keyword">record</a>
|
||
<a id="5999" class="Symbol">{</a> <a id="6001" href="Relation.Binary.Bundles.html#6632" class="Field">isTotalOrder</a> <a id="6014" class="Symbol">=</a> <a id="6016" href="Relation.Binary.Construct.Flip.EqAndOrd.html#3996" class="Function">isTotalOrder</a> <a id="6029" href="Relation.Binary.Bundles.html#6632" class="Field">O.isTotalOrder</a>
|
||
<a id="6046" class="Symbol">}</a> <a id="6048" class="Keyword">where</a> <a id="6054" class="Keyword">module</a> <a id="6061" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6061" class="Module">O</a> <a id="6063" class="Symbol">=</a> <a id="6065" href="Relation.Binary.Bundles.html#6458" class="Module">TotalOrder</a> <a id="6076" href="Relation.Binary.Construct.Flip.EqAndOrd.html#5986" class="Bound">O</a>
|
||
|
||
<a id="decTotalOrder"></a><a id="6079" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6079" class="Function">decTotalOrder</a> <a id="6093" class="Symbol">:</a> <a id="6095" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a> <a id="6109" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6111" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6114" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="6117" class="Symbol">→</a> <a id="6119" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a> <a id="6133" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6135" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6138" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="6141" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6079" class="Function">decTotalOrder</a> <a id="6155" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6155" class="Bound">O</a> <a id="6157" class="Symbol">=</a> <a id="6159" class="Keyword">record</a>
|
||
<a id="6168" class="Symbol">{</a> <a id="6170" href="Relation.Binary.Bundles.html#7227" class="Field">isDecTotalOrder</a> <a id="6186" class="Symbol">=</a> <a id="6188" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4205" class="Function">isDecTotalOrder</a> <a id="6204" href="Relation.Binary.Bundles.html#7227" class="Field">O.isDecTotalOrder</a>
|
||
<a id="6224" class="Symbol">}</a> <a id="6226" class="Keyword">where</a> <a id="6232" class="Keyword">module</a> <a id="6239" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6239" class="Module">O</a> <a id="6241" class="Symbol">=</a> <a id="6243" href="Relation.Binary.Bundles.html#7041" class="Module">DecTotalOrder</a> <a id="6257" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6155" class="Bound">O</a>
|
||
|
||
<a id="strictPartialOrder"></a><a id="6260" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6260" class="Function">strictPartialOrder</a> <a id="6279" class="Symbol">:</a> <a id="6281" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6300" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6302" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6305" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="6308" class="Symbol">→</a>
|
||
<a id="6331" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6350" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6352" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6355" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="6358" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6260" class="Function">strictPartialOrder</a> <a id="6377" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6377" class="Bound">O</a> <a id="6379" class="Symbol">=</a> <a id="6381" class="Keyword">record</a>
|
||
<a id="6390" class="Symbol">{</a> <a id="6392" href="Relation.Binary.Bundles.html#5045" class="Field">isStrictPartialOrder</a> <a id="6413" class="Symbol">=</a> <a id="6415" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4443" class="Function">isStrictPartialOrder</a> <a id="6436" href="Relation.Binary.Bundles.html#5045" class="Field">O.isStrictPartialOrder</a>
|
||
<a id="6461" class="Symbol">}</a> <a id="6463" class="Keyword">where</a> <a id="6469" class="Keyword">module</a> <a id="6476" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6476" class="Module">O</a> <a id="6478" class="Symbol">=</a> <a id="6480" href="Relation.Binary.Bundles.html#4839" class="Module">StrictPartialOrder</a> <a id="6499" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6377" class="Bound">O</a>
|
||
|
||
<a id="strictTotalOrder"></a><a id="6502" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6502" class="Function">strictTotalOrder</a> <a id="6519" class="Symbol">:</a> <a id="6521" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a> <a id="6538" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6540" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6543" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a> <a id="6546" class="Symbol">→</a>
|
||
<a id="6567" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a> <a id="6584" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1227" class="Generalizable">a</a> <a id="6586" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1235" class="Generalizable">ℓ₁</a> <a id="6589" href="Relation.Binary.Construct.Flip.EqAndOrd.html#1238" class="Generalizable">ℓ₂</a>
|
||
<a id="6592" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6502" class="Function">strictTotalOrder</a> <a id="6609" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6609" class="Bound">O</a> <a id="6611" class="Symbol">=</a> <a id="6613" class="Keyword">record</a>
|
||
<a id="6622" class="Symbol">{</a> <a id="6624" href="Relation.Binary.Bundles.html#8229" class="Field">isStrictTotalOrder</a> <a id="6643" class="Symbol">=</a> <a id="6645" href="Relation.Binary.Construct.Flip.EqAndOrd.html#4793" class="Function">isStrictTotalOrder</a> <a id="6664" href="Relation.Binary.Bundles.html#8229" class="Field">O.isStrictTotalOrder</a>
|
||
<a id="6687" class="Symbol">}</a> <a id="6689" class="Keyword">where</a> <a id="6695" class="Keyword">module</a> <a id="6702" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6702" class="Module">O</a> <a id="6704" class="Symbol">=</a> <a id="6706" href="Relation.Binary.Bundles.html#8031" class="Module">StrictTotalOrder</a> <a id="6723" href="Relation.Binary.Construct.Flip.EqAndOrd.html#6609" class="Bound">O</a>
|
||
</pre></body></html> |