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add some agda stuff
2022-06-23 22:12:24 +02:00
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<html><head><meta charset="utf-8"><title>Relation.Binary.Consequences</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">-- Some properties imply others</a>
<a id="138" class="Comment">------------------------------------------------------------------------</a>
<a id="212" class="Symbol">{-#</a> <a id="216" class="Keyword">OPTIONS</a> <a id="224" class="Pragma">--without-K</a> <a id="236" class="Pragma">--safe</a> <a id="243" class="Symbol">#-}</a>
<a id="248" class="Keyword">module</a> <a id="255" href="Relation.Binary.Consequences.html" class="Module">Relation.Binary.Consequences</a> <a id="284" class="Keyword">where</a>
<a id="291" class="Keyword">open</a> <a id="296" class="Keyword">import</a> <a id="303" href="Data.Maybe.Base.html" class="Module">Data.Maybe.Base</a> <a id="319" class="Keyword">using</a> <a id="325" class="Symbol">(</a><a id="326" href="Agda.Builtin.Maybe.html#174" class="InductiveConstructor">just</a><a id="330" class="Symbol">;</a> <a id="332" href="Agda.Builtin.Maybe.html#195" class="InductiveConstructor">nothing</a><a id="339" class="Symbol">;</a> <a id="341" href="Data.Maybe.Base.html#1212" class="Function">decToMaybe</a><a id="351" class="Symbol">)</a>
<a id="353" class="Keyword">open</a> <a id="358" class="Keyword">import</a> <a id="365" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="379" class="Symbol">as</a> <a id="382" class="Module">Sum</a> <a id="386" class="Keyword">using</a> <a id="392" class="Symbol">(</a><a id="393" href="Data.Sum.Base.html#784" class="InductiveConstructor">inj₁</a><a id="397" class="Symbol">;</a> <a id="399" href="Data.Sum.Base.html#809" class="InductiveConstructor">inj₂</a><a id="403" class="Symbol">)</a>
<a id="405" class="Keyword">open</a> <a id="410" class="Keyword">import</a> <a id="417" href="Data.Product.html" class="Module">Data.Product</a> <a id="430" class="Keyword">using</a> <a id="436" class="Symbol">(</a><a id="437" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">_,_</a><a id="440" class="Symbol">)</a>
<a id="442" class="Keyword">open</a> <a id="447" class="Keyword">import</a> <a id="454" href="Data.Empty.Irrelevant.html" class="Module">Data.Empty.Irrelevant</a> <a id="476" class="Keyword">using</a> <a id="482" class="Symbol">(</a><a id="483" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a><a id="489" class="Symbol">)</a>
<a id="491" class="Keyword">open</a> <a id="496" class="Keyword">import</a> <a id="503" href="Function.Base.html" class="Module">Function.Base</a> <a id="517" class="Keyword">using</a> <a id="523" class="Symbol">(</a><a id="524" href="Function.Base.html#1031" class="Function Operator">_∘_</a><a id="527" class="Symbol">;</a> <a id="529" href="Function.Base.html#1919" class="Function Operator">_$_</a><a id="532" class="Symbol">;</a> <a id="534" href="Function.Base.html#1554" class="Function">flip</a><a id="538" class="Symbol">)</a>
<a id="540" class="Keyword">open</a> <a id="545" class="Keyword">import</a> <a id="552" href="Level.html" class="Module">Level</a> <a id="558" class="Keyword">using</a> <a id="564" class="Symbol">(</a><a id="565" href="Agda.Primitive.html#597" class="Postulate">Level</a><a id="570" class="Symbol">)</a>
<a id="572" class="Keyword">open</a> <a id="577" class="Keyword">import</a> <a id="584" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a>
<a id="605" class="Keyword">open</a> <a id="610" class="Keyword">import</a> <a id="617" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
<a id="645" class="Keyword">open</a> <a id="650" class="Keyword">import</a> <a id="657" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="674" class="Keyword">using</a> <a id="680" class="Symbol">(</a><a id="681" href="Relation.Nullary.html#1648" class="InductiveConstructor">yes</a><a id="684" class="Symbol">;</a> <a id="686" href="Relation.Nullary.html#1685" class="InductiveConstructor">no</a><a id="688" class="Symbol">;</a> <a id="690" href="Relation.Nullary.html#1838" class="Function">recompute</a><a id="699" class="Symbol">)</a>
<a id="701" class="Keyword">open</a> <a id="706" class="Keyword">import</a> <a id="713" href="Relation.Nullary.Decidable.Core.html" class="Module">Relation.Nullary.Decidable.Core</a> <a id="745" class="Keyword">using</a> <a id="751" class="Symbol">(</a><a id="752" href="Relation.Nullary.Decidable.Core.html#3636" class="Function">map′</a><a id="756" class="Symbol">)</a>
<a id="758" class="Keyword">open</a> <a id="763" class="Keyword">import</a> <a id="770" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="785" class="Keyword">using</a> <a id="791" class="Symbol">(</a><a id="792" href="Relation.Unary.html#4278" class="Function">∁</a><a id="793" class="Symbol">;</a> <a id="795" href="Relation.Unary.html#1101" class="Function">Pred</a><a id="799" class="Symbol">)</a>
<a id="802" class="Keyword">private</a>
<a id="812" class="Keyword">variable</a>
<a id="825" href="Relation.Binary.Consequences.html#825" class="Generalizable">a</a> <a id="827" href="Relation.Binary.Consequences.html#827" class="Generalizable">ℓ</a> <a id="829" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a> <a id="832" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a> <a id="835" href="Relation.Binary.Consequences.html#835" class="Generalizable">ℓ₃</a> <a id="838" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₄</a> <a id="841" href="Relation.Binary.Consequences.html#841" class="Generalizable">p</a> <a id="843" class="Symbol">:</a> <a id="845" href="Agda.Primitive.html#597" class="Postulate">Level</a>
<a id="855" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="857" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="859" class="Symbol">:</a> <a id="861" href="Agda.Primitive.html#326" class="Primitive">Set</a> <a id="865" href="Relation.Binary.Consequences.html#825" class="Generalizable">a</a>
<a id="868" class="Comment">------------------------------------------------------------------------</a>
<a id="941" class="Comment">-- Substitutive properties</a>
<a id="969" class="Keyword">module</a> <a id="976" href="Relation.Binary.Consequences.html#976" class="Module">_</a> <a id="978" class="Symbol">{</a><a id="979" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a> <a id="983" class="Symbol">:</a> <a id="985" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="989" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="991" href="Relation.Binary.Consequences.html#827" class="Generalizable">ℓ</a><a id="992" class="Symbol">}</a> <a id="994" class="Symbol">(</a><a id="995" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="997" class="Symbol">:</a> <a id="999" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="1003" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="1005" href="Relation.Binary.Consequences.html#841" class="Generalizable">p</a><a id="1006" class="Symbol">)</a> <a id="1008" class="Keyword">where</a>
<a id="1017" href="Relation.Binary.Consequences.html#1017" class="Function">subst⇒respˡ</a> <a id="1029" class="Symbol">:</a> <a id="1031" href="Relation.Binary.Definitions.html#4369" class="Function">Substitutive</a> <a id="1044" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a> <a id="1048" href="Relation.Binary.Consequences.html#1005" class="Bound">p</a> <a id="1050" class="Symbol">→</a> <a id="1052" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="1054" href="Relation.Binary.Definitions.html#3914" class="Function Operator">Respectsˡ</a> <a id="1064" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a>
<a id="1070" href="Relation.Binary.Consequences.html#1017" class="Function">subst⇒respˡ</a> <a id="1082" href="Relation.Binary.Consequences.html#1082" class="Bound">subst</a> <a id="1088" class="Symbol">{</a><a id="1089" href="Relation.Binary.Consequences.html#1089" class="Bound">y</a><a id="1090" class="Symbol">}</a> <a id="1092" href="Relation.Binary.Consequences.html#1092" class="Bound">x′∼x</a> <a id="1097" href="Relation.Binary.Consequences.html#1097" class="Bound">Px′y</a> <a id="1102" class="Symbol">=</a> <a id="1104" href="Relation.Binary.Consequences.html#1082" class="Bound">subst</a> <a id="1110" class="Symbol">(</a><a id="1111" href="Function.Base.html#1554" class="Function">flip</a> <a id="1116" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="1118" href="Relation.Binary.Consequences.html#1089" class="Bound">y</a><a id="1119" class="Symbol">)</a> <a id="1121" href="Relation.Binary.Consequences.html#1092" class="Bound">x′∼x</a> <a id="1126" href="Relation.Binary.Consequences.html#1097" class="Bound">Px′y</a>
<a id="1134" href="Relation.Binary.Consequences.html#1134" class="Function">subst⇒respʳ</a> <a id="1146" class="Symbol">:</a> <a id="1148" href="Relation.Binary.Definitions.html#4369" class="Function">Substitutive</a> <a id="1161" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a> <a id="1165" href="Relation.Binary.Consequences.html#1005" class="Bound">p</a> <a id="1167" class="Symbol">→</a> <a id="1169" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="1171" href="Relation.Binary.Definitions.html#3749" class="Function Operator">Respectsʳ</a> <a id="1181" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a>
<a id="1187" href="Relation.Binary.Consequences.html#1134" class="Function">subst⇒respʳ</a> <a id="1199" href="Relation.Binary.Consequences.html#1199" class="Bound">subst</a> <a id="1205" class="Symbol">{</a><a id="1206" href="Relation.Binary.Consequences.html#1206" class="Bound">x</a><a id="1207" class="Symbol">}</a> <a id="1209" href="Relation.Binary.Consequences.html#1209" class="Bound">y′∼y</a> <a id="1214" href="Relation.Binary.Consequences.html#1214" class="Bound">Pxy′</a> <a id="1219" class="Symbol">=</a> <a id="1221" href="Relation.Binary.Consequences.html#1199" class="Bound">subst</a> <a id="1227" class="Symbol">(</a><a id="1228" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="1230" href="Relation.Binary.Consequences.html#1206" class="Bound">x</a><a id="1231" class="Symbol">)</a> <a id="1233" href="Relation.Binary.Consequences.html#1209" class="Bound">y′∼y</a> <a id="1238" href="Relation.Binary.Consequences.html#1214" class="Bound">Pxy′</a>
<a id="1246" href="Relation.Binary.Consequences.html#1246" class="Function">subst⇒resp₂</a> <a id="1258" class="Symbol">:</a> <a id="1260" href="Relation.Binary.Definitions.html#4369" class="Function">Substitutive</a> <a id="1273" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a> <a id="1277" href="Relation.Binary.Consequences.html#1005" class="Bound">p</a> <a id="1279" class="Symbol">→</a> <a id="1281" href="Relation.Binary.Consequences.html#995" class="Bound">R</a> <a id="1283" href="Relation.Binary.Definitions.html#4077" class="Function Operator">Respects₂</a> <a id="1293" href="Relation.Binary.Consequences.html#979" class="Bound Operator">_∼_</a>
<a id="1299" href="Relation.Binary.Consequences.html#1246" class="Function">subst⇒resp₂</a> <a id="1311" href="Relation.Binary.Consequences.html#1311" class="Bound">subst</a> <a id="1317" class="Symbol">=</a> <a id="1319" href="Relation.Binary.Consequences.html#1134" class="Function">subst⇒respʳ</a> <a id="1331" href="Relation.Binary.Consequences.html#1311" class="Bound">subst</a> <a id="1337" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1339" href="Relation.Binary.Consequences.html#1017" class="Function">subst⇒respˡ</a> <a id="1351" href="Relation.Binary.Consequences.html#1311" class="Bound">subst</a>
<a id="1358" class="Keyword">module</a> <a id="1365" href="Relation.Binary.Consequences.html#1365" class="Module">_</a> <a id="1367" class="Symbol">{</a><a id="1368" href="Relation.Binary.Consequences.html#1368" class="Bound Operator">_∼_</a> <a id="1372" class="Symbol">:</a> <a id="1374" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="1378" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="1380" href="Relation.Binary.Consequences.html#827" class="Generalizable">ℓ</a><a id="1381" class="Symbol">}</a> <a id="1383" class="Symbol">{</a><a id="1384" href="Relation.Binary.Consequences.html#1384" class="Bound">P</a> <a id="1386" class="Symbol">:</a> <a id="1388" href="Relation.Unary.html#1101" class="Function">Pred</a> <a id="1393" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="1395" href="Relation.Binary.Consequences.html#841" class="Generalizable">p</a><a id="1396" class="Symbol">}</a> <a id="1398" class="Keyword">where</a>
<a id="1407" href="Relation.Binary.Consequences.html#1407" class="Function">resp⇒¬-resp</a> <a id="1419" class="Symbol">:</a> <a id="1421" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="1431" href="Relation.Binary.Consequences.html#1368" class="Bound Operator">_∼_</a> <a id="1435" class="Symbol">→</a> <a id="1437" href="Relation.Binary.Consequences.html#1384" class="Bound">P</a> <a id="1439" href="Relation.Binary.Definitions.html#3593" class="Function Operator">Respects</a> <a id="1448" href="Relation.Binary.Consequences.html#1368" class="Bound Operator">_∼_</a> <a id="1452" class="Symbol">→</a> <a id="1454" class="Symbol">(</a><a id="1455" href="Relation.Unary.html#4278" class="Function">∁</a> <a id="1457" href="Relation.Binary.Consequences.html#1384" class="Bound">P</a><a id="1458" class="Symbol">)</a> <a id="1460" href="Relation.Binary.Definitions.html#3593" class="Function Operator">Respects</a> <a id="1469" href="Relation.Binary.Consequences.html#1368" class="Bound Operator">_∼_</a>
<a id="1475" href="Relation.Binary.Consequences.html#1407" class="Function">resp⇒¬-resp</a> <a id="1487" href="Relation.Binary.Consequences.html#1487" class="Bound">sym</a> <a id="1491" href="Relation.Binary.Consequences.html#1491" class="Bound">resp</a> <a id="1496" href="Relation.Binary.Consequences.html#1496" class="Bound">x∼y</a> <a id="1500" href="Relation.Binary.Consequences.html#1500" class="Bound">¬Px</a> <a id="1504" href="Relation.Binary.Consequences.html#1504" class="Bound">Py</a> <a id="1507" class="Symbol">=</a> <a id="1509" href="Relation.Binary.Consequences.html#1500" class="Bound">¬Px</a> <a id="1513" class="Symbol">(</a><a id="1514" href="Relation.Binary.Consequences.html#1491" class="Bound">resp</a> <a id="1519" class="Symbol">(</a><a id="1520" href="Relation.Binary.Consequences.html#1487" class="Bound">sym</a> <a id="1524" href="Relation.Binary.Consequences.html#1496" class="Bound">x∼y</a><a id="1527" class="Symbol">)</a> <a id="1529" href="Relation.Binary.Consequences.html#1504" class="Bound">Py</a><a id="1531" class="Symbol">)</a>
<a id="1534" class="Comment">------------------------------------------------------------------------</a>
<a id="1607" class="Comment">-- Proofs for non-strict orders</a>
<a id="1640" class="Keyword">module</a> <a id="1647" href="Relation.Binary.Consequences.html#1647" class="Module">_</a> <a id="1649" class="Symbol">{</a><a id="1650" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1654" class="Symbol">:</a> <a id="1656" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="1660" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="1662" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="1664" class="Symbol">}</a> <a id="1666" class="Symbol">{</a><a id="1667" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="1671" class="Symbol">:</a> <a id="1673" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="1677" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="1679" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="1681" class="Symbol">}</a> <a id="1683" class="Keyword">where</a>
<a id="1692" href="Relation.Binary.Consequences.html#1692" class="Function">total⇒refl</a> <a id="1703" class="Symbol">:</a> <a id="1705" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="1709" href="Relation.Binary.Definitions.html#4077" class="Function Operator">Respects₂</a> <a id="1719" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1723" class="Symbol">→</a> <a id="1725" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="1735" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1739" class="Symbol">→</a>
<a id="1756" href="Relation.Binary.Definitions.html#2584" class="Function">Total</a> <a id="1762" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="1766" class="Symbol">→</a> <a id="1768" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1772" href="Relation.Binary.Core.html#1254" class="Function Operator">⇒</a> <a id="1774" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a>
<a id="1780" href="Relation.Binary.Consequences.html#1692" class="Function">total⇒refl</a> <a id="1791" class="Symbol">(</a><a id="1792" href="Relation.Binary.Consequences.html#1792" class="Bound">respʳ</a> <a id="1798" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1800" href="Relation.Binary.Consequences.html#1800" class="Bound">respˡ</a><a id="1805" class="Symbol">)</a> <a id="1807" href="Relation.Binary.Consequences.html#1807" class="Bound">sym</a> <a id="1811" href="Relation.Binary.Consequences.html#1811" class="Bound">total</a> <a id="1817" class="Symbol">{</a><a id="1818" href="Relation.Binary.Consequences.html#1818" class="Bound">x</a><a id="1819" class="Symbol">}</a> <a id="1821" class="Symbol">{</a><a id="1822" href="Relation.Binary.Consequences.html#1822" class="Bound">y</a><a id="1823" class="Symbol">}</a> <a id="1825" href="Relation.Binary.Consequences.html#1825" class="Bound">x≈y</a> <a id="1829" class="Keyword">with</a> <a id="1834" href="Relation.Binary.Consequences.html#1811" class="Bound">total</a> <a id="1840" href="Relation.Binary.Consequences.html#1818" class="Bound">x</a> <a id="1842" href="Relation.Binary.Consequences.html#1822" class="Bound">y</a>
<a id="1846" class="Symbol">...</a> <a id="1850" class="Symbol">|</a> <a id="1852" href="Data.Sum.Base.html#784" class="InductiveConstructor">inj₁</a> <a id="1857" href="Relation.Binary.Consequences.html#1857" class="Bound">x∼y</a> <a id="1861" class="Symbol">=</a> <a id="1863" href="Relation.Binary.Consequences.html#1857" class="Bound">x∼y</a>
<a id="1869" class="Symbol">...</a> <a id="1873" class="Symbol">|</a> <a id="1875" href="Data.Sum.Base.html#809" class="InductiveConstructor">inj₂</a> <a id="1880" href="Relation.Binary.Consequences.html#1880" class="Bound">y∼x</a> <a id="1884" class="Symbol">=</a> <a id="1886" class="Bound">respʳ</a> <a id="1892" class="Bound">x≈y</a> <a id="1896" class="Symbol">(</a><a id="1897" class="Bound">respˡ</a> <a id="1903" class="Symbol">(</a><a id="1904" class="Bound">sym</a> <a id="1908" class="Bound">x≈y</a><a id="1911" class="Symbol">)</a> <a id="1913" href="Relation.Binary.Consequences.html#1880" class="Bound">y∼x</a><a id="1916" class="Symbol">)</a>
<a id="1921" href="Relation.Binary.Consequences.html#1921" class="Function">total∧dec⇒dec</a> <a id="1935" class="Symbol">:</a> <a id="1937" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1941" href="Relation.Binary.Core.html#1254" class="Function Operator">⇒</a> <a id="1943" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="1947" class="Symbol">→</a> <a id="1949" href="Relation.Binary.Definitions.html#2082" class="Function">Antisymmetric</a> <a id="1963" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="1967" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="1971" class="Symbol">→</a>
<a id="1991" href="Relation.Binary.Definitions.html#2584" class="Function">Total</a> <a id="1997" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a> <a id="2001" class="Symbol">→</a> <a id="2003" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="2013" href="Relation.Binary.Consequences.html#1650" class="Bound Operator">_≈_</a> <a id="2017" class="Symbol">→</a> <a id="2019" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="2029" href="Relation.Binary.Consequences.html#1667" class="Bound Operator">_≤_</a>
<a id="2035" href="Relation.Binary.Consequences.html#1921" class="Function">total∧dec⇒dec</a> <a id="2049" href="Relation.Binary.Consequences.html#2049" class="Bound">refl</a> <a id="2054" href="Relation.Binary.Consequences.html#2054" class="Bound">antisym</a> <a id="2062" href="Relation.Binary.Consequences.html#2062" class="Bound">total</a> <a id="2068" href="Relation.Binary.Consequences.html#2068" class="Bound Operator">_≟_</a> <a id="2072" href="Relation.Binary.Consequences.html#2072" class="Bound">x</a> <a id="2074" href="Relation.Binary.Consequences.html#2074" class="Bound">y</a> <a id="2076" class="Keyword">with</a> <a id="2081" href="Relation.Binary.Consequences.html#2062" class="Bound">total</a> <a id="2087" href="Relation.Binary.Consequences.html#2072" class="Bound">x</a> <a id="2089" href="Relation.Binary.Consequences.html#2074" class="Bound">y</a>
<a id="2093" class="Symbol">...</a> <a id="2097" class="Symbol">|</a> <a id="2099" href="Data.Sum.Base.html#784" class="InductiveConstructor">inj₁</a> <a id="2104" href="Relation.Binary.Consequences.html#2104" class="Bound">x≤y</a> <a id="2108" class="Symbol">=</a> <a id="2110" href="Relation.Nullary.html#1648" class="InductiveConstructor">yes</a> <a id="2114" href="Relation.Binary.Consequences.html#2104" class="Bound">x≤y</a>
<a id="2120" class="Symbol">...</a> <a id="2124" class="Symbol">|</a> <a id="2126" href="Data.Sum.Base.html#809" class="InductiveConstructor">inj₂</a> <a id="2131" href="Relation.Binary.Consequences.html#2131" class="Bound">y≤x</a> <a id="2135" class="Symbol">=</a> <a id="2137" href="Relation.Nullary.Decidable.Core.html#3636" class="Function">map′</a> <a id="2142" class="Bound">refl</a> <a id="2147" class="Symbol">(</a><a id="2148" href="Function.Base.html#1554" class="Function">flip</a> <a id="2153" class="Bound">antisym</a> <a id="2161" href="Relation.Binary.Consequences.html#2131" class="Bound">y≤x</a><a id="2164" class="Symbol">)</a> <a id="2166" class="Symbol">(</a><a id="2167" class="Bound">x</a> <a id="2169" class="Bound Operator">≟</a> <a id="2171" class="Bound">y</a><a id="2172" class="Symbol">)</a>
<a id="2175" class="Keyword">module</a> <a id="2182" href="Relation.Binary.Consequences.html#2182" class="Module">_</a> <a id="2184" class="Symbol">(</a><a id="2185" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2188" class="Symbol">:</a> <a id="2190" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2194" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="2196" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="2198" class="Symbol">)</a> <a id="2200" class="Symbol">(</a><a id="2201" href="Relation.Binary.Consequences.html#2201" class="Bound">≈₂</a> <a id="2204" class="Symbol">:</a> <a id="2206" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2210" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="2212" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="2214" class="Symbol">)</a> <a id="2216" class="Symbol">{</a><a id="2217" href="Relation.Binary.Consequences.html#2217" class="Bound">≤₁</a> <a id="2220" class="Symbol">:</a> <a id="2222" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2226" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="2228" href="Relation.Binary.Consequences.html#835" class="Generalizable">ℓ₃</a><a id="2230" class="Symbol">}</a> <a id="2232" class="Symbol">{</a><a id="2233" href="Relation.Binary.Consequences.html#2233" class="Bound">≤₂</a> <a id="2236" class="Symbol">:</a> <a id="2238" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2242" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="2244" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₄</a><a id="2246" class="Symbol">}</a> <a id="2248" class="Keyword">where</a>
<a id="2257" href="Relation.Binary.Consequences.html#2257" class="Function">mono⇒cong</a> <a id="2267" class="Symbol">:</a> <a id="2269" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="2279" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2282" class="Symbol">→</a> <a id="2284" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2287" href="Relation.Binary.Core.html#1254" class="Function Operator">⇒</a> <a id="2289" href="Relation.Binary.Consequences.html#2217" class="Bound">≤₁</a> <a id="2292" class="Symbol">→</a> <a id="2294" href="Relation.Binary.Definitions.html#2082" class="Function">Antisymmetric</a> <a id="2308" href="Relation.Binary.Consequences.html#2201" class="Bound">≈₂</a> <a id="2311" href="Relation.Binary.Consequences.html#2233" class="Bound">≤₂</a> <a id="2314" class="Symbol">→</a>
<a id="2330" class="Symbol">∀</a> <a id="2332" class="Symbol">{</a><a id="2333" href="Relation.Binary.Consequences.html#2333" class="Bound">f</a><a id="2334" class="Symbol">}</a> <a id="2336" class="Symbol">→</a> <a id="2338" href="Relation.Binary.Consequences.html#2333" class="Bound">f</a> <a id="2340" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="2350" href="Relation.Binary.Consequences.html#2217" class="Bound">≤₁</a> <a id="2353" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="2355" href="Relation.Binary.Consequences.html#2233" class="Bound">≤₂</a> <a id="2358" class="Symbol">→</a> <a id="2360" href="Relation.Binary.Consequences.html#2333" class="Bound">f</a> <a id="2362" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="2372" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2375" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="2377" href="Relation.Binary.Consequences.html#2201" class="Bound">≈₂</a>
<a id="2382" href="Relation.Binary.Consequences.html#2257" class="Function">mono⇒cong</a> <a id="2392" href="Relation.Binary.Consequences.html#2392" class="Bound">sym</a> <a id="2396" href="Relation.Binary.Consequences.html#2396" class="Bound">reflexive</a> <a id="2406" href="Relation.Binary.Consequences.html#2406" class="Bound">antisym</a> <a id="2414" href="Relation.Binary.Consequences.html#2414" class="Bound">mono</a> <a id="2419" href="Relation.Binary.Consequences.html#2419" class="Bound">x≈y</a> <a id="2423" class="Symbol">=</a> <a id="2425" href="Relation.Binary.Consequences.html#2406" class="Bound">antisym</a>
<a id="2437" class="Symbol">(</a><a id="2438" href="Relation.Binary.Consequences.html#2414" class="Bound">mono</a> <a id="2443" class="Symbol">(</a><a id="2444" href="Relation.Binary.Consequences.html#2396" class="Bound">reflexive</a> <a id="2454" href="Relation.Binary.Consequences.html#2419" class="Bound">x≈y</a><a id="2457" class="Symbol">))</a>
<a id="2464" class="Symbol">(</a><a id="2465" href="Relation.Binary.Consequences.html#2414" class="Bound">mono</a> <a id="2470" class="Symbol">(</a><a id="2471" href="Relation.Binary.Consequences.html#2396" class="Bound">reflexive</a> <a id="2481" class="Symbol">(</a><a id="2482" href="Relation.Binary.Consequences.html#2392" class="Bound">sym</a> <a id="2486" href="Relation.Binary.Consequences.html#2419" class="Bound">x≈y</a><a id="2489" class="Symbol">)))</a>
<a id="2496" href="Relation.Binary.Consequences.html#2496" class="Function">antimono⇒cong</a> <a id="2510" class="Symbol">:</a> <a id="2512" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="2522" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2525" class="Symbol">→</a> <a id="2527" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2530" href="Relation.Binary.Core.html#1254" class="Function Operator">⇒</a> <a id="2532" href="Relation.Binary.Consequences.html#2217" class="Bound">≤₁</a> <a id="2535" class="Symbol">→</a> <a id="2537" href="Relation.Binary.Definitions.html#2082" class="Function">Antisymmetric</a> <a id="2551" href="Relation.Binary.Consequences.html#2201" class="Bound">≈₂</a> <a id="2554" href="Relation.Binary.Consequences.html#2233" class="Bound">≤₂</a> <a id="2557" class="Symbol">→</a>
<a id="2577" class="Symbol">∀</a> <a id="2579" class="Symbol">{</a><a id="2580" href="Relation.Binary.Consequences.html#2580" class="Bound">f</a><a id="2581" class="Symbol">}</a> <a id="2583" class="Symbol">→</a> <a id="2585" href="Relation.Binary.Consequences.html#2580" class="Bound">f</a> <a id="2587" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="2597" href="Relation.Binary.Consequences.html#2217" class="Bound">≤₁</a> <a id="2600" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="2602" class="Symbol">(</a><a id="2603" href="Function.Base.html#1554" class="Function">flip</a> <a id="2608" href="Relation.Binary.Consequences.html#2233" class="Bound">≤₂</a><a id="2610" class="Symbol">)</a> <a id="2612" class="Symbol">→</a> <a id="2614" href="Relation.Binary.Consequences.html#2580" class="Bound">f</a> <a id="2616" href="Relation.Binary.Core.html#1563" class="Function Operator">Preserves</a> <a id="2626" href="Relation.Binary.Consequences.html#2185" class="Bound">≈₁</a> <a id="2629" href="Relation.Binary.Core.html#1563" class="Function Operator">⟶</a> <a id="2631" href="Relation.Binary.Consequences.html#2201" class="Bound">≈₂</a>
<a id="2636" href="Relation.Binary.Consequences.html#2496" class="Function">antimono⇒cong</a> <a id="2650" href="Relation.Binary.Consequences.html#2650" class="Bound">sym</a> <a id="2654" href="Relation.Binary.Consequences.html#2654" class="Bound">reflexive</a> <a id="2664" href="Relation.Binary.Consequences.html#2664" class="Bound">antisym</a> <a id="2672" href="Relation.Binary.Consequences.html#2672" class="Bound">antimono</a> <a id="2681" href="Relation.Binary.Consequences.html#2681" class="Bound">p≈q</a> <a id="2685" class="Symbol">=</a> <a id="2687" href="Relation.Binary.Consequences.html#2664" class="Bound">antisym</a>
<a id="2699" class="Symbol">(</a><a id="2700" href="Relation.Binary.Consequences.html#2672" class="Bound">antimono</a> <a id="2709" class="Symbol">(</a><a id="2710" href="Relation.Binary.Consequences.html#2654" class="Bound">reflexive</a> <a id="2720" class="Symbol">(</a><a id="2721" href="Relation.Binary.Consequences.html#2650" class="Bound">sym</a> <a id="2725" href="Relation.Binary.Consequences.html#2681" class="Bound">p≈q</a><a id="2728" class="Symbol">)))</a>
<a id="2736" class="Symbol">(</a><a id="2737" href="Relation.Binary.Consequences.html#2672" class="Bound">antimono</a> <a id="2746" class="Symbol">(</a><a id="2747" href="Relation.Binary.Consequences.html#2654" class="Bound">reflexive</a> <a id="2757" href="Relation.Binary.Consequences.html#2681" class="Bound">p≈q</a><a id="2760" class="Symbol">))</a>
<a id="2764" class="Comment">------------------------------------------------------------------------</a>
<a id="2837" class="Comment">-- Proofs for strict orders</a>
<a id="2866" class="Keyword">module</a> <a id="2873" href="Relation.Binary.Consequences.html#2873" class="Module">_</a> <a id="2875" class="Symbol">{</a><a id="2876" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="2880" class="Symbol">:</a> <a id="2882" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2886" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="2888" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="2890" class="Symbol">}</a> <a id="2892" class="Symbol">{</a><a id="2893" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="2897" class="Symbol">:</a> <a id="2899" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="2903" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="2905" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="2907" class="Symbol">}</a> <a id="2909" class="Keyword">where</a>
<a id="2918" href="Relation.Binary.Consequences.html#2918" class="Function">trans∧irr⇒asym</a> <a id="2933" class="Symbol">:</a> <a id="2935" href="Relation.Binary.Definitions.html#1339" class="Function">Reflexive</a> <a id="2945" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="2949" class="Symbol">→</a> <a id="2951" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="2962" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="2966" class="Symbol">→</a>
<a id="2987" href="Relation.Binary.Definitions.html#2241" class="Function">Irreflexive</a> <a id="2999" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3003" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3007" class="Symbol">→</a> <a id="3009" href="Relation.Binary.Definitions.html#2353" class="Function">Asymmetric</a> <a id="3020" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3026" href="Relation.Binary.Consequences.html#2918" class="Function">trans∧irr⇒asym</a> <a id="3041" href="Relation.Binary.Consequences.html#3041" class="Bound">refl</a> <a id="3046" href="Relation.Binary.Consequences.html#3046" class="Bound">trans</a> <a id="3052" href="Relation.Binary.Consequences.html#3052" class="Bound">irrefl</a> <a id="3059" href="Relation.Binary.Consequences.html#3059" class="Bound">x&lt;y</a> <a id="3063" href="Relation.Binary.Consequences.html#3063" class="Bound">y&lt;x</a> <a id="3067" class="Symbol">=</a>
<a id="3073" href="Relation.Binary.Consequences.html#3052" class="Bound">irrefl</a> <a id="3080" href="Relation.Binary.Consequences.html#3041" class="Bound">refl</a> <a id="3085" class="Symbol">(</a><a id="3086" href="Relation.Binary.Consequences.html#3046" class="Bound">trans</a> <a id="3092" href="Relation.Binary.Consequences.html#3059" class="Bound">x&lt;y</a> <a id="3096" href="Relation.Binary.Consequences.html#3063" class="Bound">y&lt;x</a><a id="3099" class="Symbol">)</a>
<a id="3104" href="Relation.Binary.Consequences.html#3104" class="Function">irr∧antisym⇒asym</a> <a id="3121" class="Symbol">:</a> <a id="3123" href="Relation.Binary.Definitions.html#2241" class="Function">Irreflexive</a> <a id="3135" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3139" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3143" class="Symbol">→</a> <a id="3145" href="Relation.Binary.Definitions.html#2082" class="Function">Antisymmetric</a> <a id="3159" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3163" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3167" class="Symbol">→</a>
<a id="3190" href="Relation.Binary.Definitions.html#2353" class="Function">Asymmetric</a> <a id="3201" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3207" href="Relation.Binary.Consequences.html#3104" class="Function">irr∧antisym⇒asym</a> <a id="3224" href="Relation.Binary.Consequences.html#3224" class="Bound">irrefl</a> <a id="3231" href="Relation.Binary.Consequences.html#3231" class="Bound">antisym</a> <a id="3239" href="Relation.Binary.Consequences.html#3239" class="Bound">x&lt;y</a> <a id="3243" href="Relation.Binary.Consequences.html#3243" class="Bound">y&lt;x</a> <a id="3247" class="Symbol">=</a>
<a id="3253" href="Relation.Binary.Consequences.html#3224" class="Bound">irrefl</a> <a id="3260" class="Symbol">(</a><a id="3261" href="Relation.Binary.Consequences.html#3231" class="Bound">antisym</a> <a id="3269" href="Relation.Binary.Consequences.html#3239" class="Bound">x&lt;y</a> <a id="3273" href="Relation.Binary.Consequences.html#3243" class="Bound">y&lt;x</a><a id="3276" class="Symbol">)</a> <a id="3278" href="Relation.Binary.Consequences.html#3239" class="Bound">x&lt;y</a>
<a id="3285" href="Relation.Binary.Consequences.html#3285" class="Function">asym⇒antisym</a> <a id="3298" class="Symbol">:</a> <a id="3300" href="Relation.Binary.Definitions.html#2353" class="Function">Asymmetric</a> <a id="3311" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3315" class="Symbol">→</a> <a id="3317" href="Relation.Binary.Definitions.html#2082" class="Function">Antisymmetric</a> <a id="3331" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3335" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3341" href="Relation.Binary.Consequences.html#3285" class="Function">asym⇒antisym</a> <a id="3354" href="Relation.Binary.Consequences.html#3354" class="Bound">asym</a> <a id="3359" href="Relation.Binary.Consequences.html#3359" class="Bound">x&lt;y</a> <a id="3363" href="Relation.Binary.Consequences.html#3363" class="Bound">y&lt;x</a> <a id="3367" class="Symbol">=</a> <a id="3369" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a> <a id="3376" class="Symbol">(</a><a id="3377" href="Relation.Binary.Consequences.html#3354" class="Bound">asym</a> <a id="3382" href="Relation.Binary.Consequences.html#3359" class="Bound">x&lt;y</a> <a id="3386" href="Relation.Binary.Consequences.html#3363" class="Bound">y&lt;x</a><a id="3389" class="Symbol">)</a>
<a id="3394" href="Relation.Binary.Consequences.html#3394" class="Function">asym⇒irr</a> <a id="3403" class="Symbol">:</a> <a id="3405" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3409" href="Relation.Binary.Definitions.html#4077" class="Function Operator">Respects₂</a> <a id="3419" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3423" class="Symbol">→</a> <a id="3425" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="3435" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3439" class="Symbol">→</a>
<a id="3454" href="Relation.Binary.Definitions.html#2353" class="Function">Asymmetric</a> <a id="3465" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3469" class="Symbol">→</a> <a id="3471" href="Relation.Binary.Definitions.html#2241" class="Function">Irreflexive</a> <a id="3483" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3487" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3493" href="Relation.Binary.Consequences.html#3394" class="Function">asym⇒irr</a> <a id="3502" class="Symbol">(</a><a id="3503" href="Relation.Binary.Consequences.html#3503" class="Bound">respʳ</a> <a id="3509" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3511" href="Relation.Binary.Consequences.html#3511" class="Bound">respˡ</a><a id="3516" class="Symbol">)</a> <a id="3518" href="Relation.Binary.Consequences.html#3518" class="Bound">sym</a> <a id="3522" href="Relation.Binary.Consequences.html#3522" class="Bound">asym</a> <a id="3527" class="Symbol">{</a><a id="3528" href="Relation.Binary.Consequences.html#3528" class="Bound">x</a><a id="3529" class="Symbol">}</a> <a id="3531" class="Symbol">{</a><a id="3532" href="Relation.Binary.Consequences.html#3532" class="Bound">y</a><a id="3533" class="Symbol">}</a> <a id="3535" href="Relation.Binary.Consequences.html#3535" class="Bound">x≈y</a> <a id="3539" href="Relation.Binary.Consequences.html#3539" class="Bound">x&lt;y</a> <a id="3543" class="Symbol">=</a>
<a id="3549" href="Relation.Binary.Consequences.html#3522" class="Bound">asym</a> <a id="3554" href="Relation.Binary.Consequences.html#3539" class="Bound">x&lt;y</a> <a id="3558" class="Symbol">(</a><a id="3559" href="Relation.Binary.Consequences.html#3503" class="Bound">respʳ</a> <a id="3565" class="Symbol">(</a><a id="3566" href="Relation.Binary.Consequences.html#3518" class="Bound">sym</a> <a id="3570" href="Relation.Binary.Consequences.html#3535" class="Bound">x≈y</a><a id="3573" class="Symbol">)</a> <a id="3575" class="Symbol">(</a><a id="3576" href="Relation.Binary.Consequences.html#3511" class="Bound">respˡ</a> <a id="3582" href="Relation.Binary.Consequences.html#3535" class="Bound">x≈y</a> <a id="3586" href="Relation.Binary.Consequences.html#3539" class="Bound">x&lt;y</a><a id="3589" class="Symbol">))</a>
<a id="3595" href="Relation.Binary.Consequences.html#3595" class="Function">tri⇒asym</a> <a id="3604" class="Symbol">:</a> <a id="3606" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="3619" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3623" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3627" class="Symbol">→</a> <a id="3629" href="Relation.Binary.Definitions.html#2353" class="Function">Asymmetric</a> <a id="3640" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3646" href="Relation.Binary.Consequences.html#3595" class="Function">tri⇒asym</a> <a id="3655" href="Relation.Binary.Consequences.html#3655" class="Bound">tri</a> <a id="3659" class="Symbol">{</a><a id="3660" href="Relation.Binary.Consequences.html#3660" class="Bound">x</a><a id="3661" class="Symbol">}</a> <a id="3663" class="Symbol">{</a><a id="3664" href="Relation.Binary.Consequences.html#3664" class="Bound">y</a><a id="3665" class="Symbol">}</a> <a id="3667" href="Relation.Binary.Consequences.html#3667" class="Bound">x&lt;y</a> <a id="3671" href="Relation.Binary.Consequences.html#3671" class="Bound">x&gt;y</a> <a id="3675" class="Keyword">with</a> <a id="3680" href="Relation.Binary.Consequences.html#3655" class="Bound">tri</a> <a id="3684" href="Relation.Binary.Consequences.html#3660" class="Bound">x</a> <a id="3686" href="Relation.Binary.Consequences.html#3664" class="Bound">y</a>
<a id="3690" class="Symbol">...</a> <a id="3694" class="Symbol">|</a> <a id="3696" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="3701" class="Symbol">_</a> <a id="3705" class="Symbol">_</a> <a id="3707" href="Relation.Binary.Consequences.html#3707" class="Bound">x≯y</a> <a id="3711" class="Symbol">=</a> <a id="3713" href="Relation.Binary.Consequences.html#3707" class="Bound">x≯y</a> <a id="3717" class="Bound">x&gt;y</a>
<a id="3723" class="Symbol">...</a> <a id="3727" class="Symbol">|</a> <a id="3729" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="3734" class="Symbol">_</a> <a id="3738" class="Symbol">_</a> <a id="3740" href="Relation.Binary.Consequences.html#3740" class="Bound">x≯y</a> <a id="3744" class="Symbol">=</a> <a id="3746" href="Relation.Binary.Consequences.html#3740" class="Bound">x≯y</a> <a id="3750" class="Bound">x&gt;y</a>
<a id="3756" class="Symbol">...</a> <a id="3760" class="Symbol">|</a> <a id="3762" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="3767" href="Relation.Binary.Consequences.html#3767" class="Bound">x≮y</a> <a id="3771" class="Symbol">_</a> <a id="3773" class="Symbol">_</a> <a id="3777" class="Symbol">=</a> <a id="3779" href="Relation.Binary.Consequences.html#3767" class="Bound">x≮y</a> <a id="3783" class="Bound">x&lt;y</a>
<a id="3790" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="3798" class="Symbol">:</a> <a id="3800" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="3813" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3817" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="3821" class="Symbol">→</a> <a id="3823" href="Relation.Binary.Definitions.html#2241" class="Function">Irreflexive</a> <a id="3835" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="3839" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="3845" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="3853" href="Relation.Binary.Consequences.html#3853" class="Bound">compare</a> <a id="3861" class="Symbol">{</a><a id="3862" href="Relation.Binary.Consequences.html#3862" class="Bound">x</a><a id="3863" class="Symbol">}</a> <a id="3865" class="Symbol">{</a><a id="3866" href="Relation.Binary.Consequences.html#3866" class="Bound">y</a><a id="3867" class="Symbol">}</a> <a id="3869" href="Relation.Binary.Consequences.html#3869" class="Bound">x≈y</a> <a id="3873" href="Relation.Binary.Consequences.html#3873" class="Bound">x&lt;y</a> <a id="3877" class="Keyword">with</a> <a id="3882" href="Relation.Binary.Consequences.html#3853" class="Bound">compare</a> <a id="3890" href="Relation.Binary.Consequences.html#3862" class="Bound">x</a> <a id="3892" href="Relation.Binary.Consequences.html#3866" class="Bound">y</a>
<a id="3896" class="Symbol">...</a> <a id="3900" class="Symbol">|</a> <a id="3902" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="3907" class="Symbol">_</a> <a id="3911" href="Relation.Binary.Consequences.html#3911" class="Bound">x≉y</a> <a id="3915" href="Relation.Binary.Consequences.html#3915" class="Bound">y≮x</a> <a id="3919" class="Symbol">=</a> <a id="3921" href="Relation.Binary.Consequences.html#3911" class="Bound">x≉y</a> <a id="3925" class="Bound">x≈y</a>
<a id="3931" class="Symbol">...</a> <a id="3935" class="Symbol">|</a> <a id="3937" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="3942" href="Relation.Binary.Consequences.html#3942" class="Bound">x≮y</a> <a id="3946" href="Relation.Binary.Consequences.html#3946" class="Bound">x≉y</a> <a id="3950" href="Relation.Binary.Consequences.html#3950" class="Bound">y&lt;x</a> <a id="3954" class="Symbol">=</a> <a id="3956" href="Relation.Binary.Consequences.html#3946" class="Bound">x≉y</a> <a id="3960" class="Bound">x≈y</a>
<a id="3966" class="Symbol">...</a> <a id="3970" class="Symbol">|</a> <a id="3972" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="3977" href="Relation.Binary.Consequences.html#3977" class="Bound">x≮y</a> <a id="3981" class="Symbol">_</a> <a id="3985" href="Relation.Binary.Consequences.html#3985" class="Bound">y≮x</a> <a id="3989" class="Symbol">=</a> <a id="3991" href="Relation.Binary.Consequences.html#3977" class="Bound">x≮y</a> <a id="3995" class="Bound">x&lt;y</a>
<a id="4002" href="Relation.Binary.Consequences.html#4002" class="Function">tri⇒dec≈</a> <a id="4011" class="Symbol">:</a> <a id="4013" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="4026" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4030" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4034" class="Symbol">→</a> <a id="4036" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="4046" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a>
<a id="4052" href="Relation.Binary.Consequences.html#4002" class="Function">tri⇒dec≈</a> <a id="4061" href="Relation.Binary.Consequences.html#4061" class="Bound">compare</a> <a id="4069" href="Relation.Binary.Consequences.html#4069" class="Bound">x</a> <a id="4071" href="Relation.Binary.Consequences.html#4071" class="Bound">y</a> <a id="4073" class="Keyword">with</a> <a id="4078" href="Relation.Binary.Consequences.html#4061" class="Bound">compare</a> <a id="4086" href="Relation.Binary.Consequences.html#4069" class="Bound">x</a> <a id="4088" href="Relation.Binary.Consequences.html#4071" class="Bound">y</a>
<a id="4092" class="Symbol">...</a> <a id="4096" class="Symbol">|</a> <a id="4098" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="4103" class="Symbol">_</a> <a id="4105" href="Relation.Binary.Consequences.html#4105" class="Bound">x≉y</a> <a id="4109" class="Symbol">_</a> <a id="4111" class="Symbol">=</a> <a id="4113" href="Relation.Nullary.html#1685" class="InductiveConstructor">no</a> <a id="4117" href="Relation.Binary.Consequences.html#4105" class="Bound">x≉y</a>
<a id="4123" class="Symbol">...</a> <a id="4127" class="Symbol">|</a> <a id="4129" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="4134" class="Symbol">_</a> <a id="4136" href="Relation.Binary.Consequences.html#4136" class="Bound">x≈y</a> <a id="4140" class="Symbol">_</a> <a id="4142" class="Symbol">=</a> <a id="4144" href="Relation.Nullary.html#1648" class="InductiveConstructor">yes</a> <a id="4148" href="Relation.Binary.Consequences.html#4136" class="Bound">x≈y</a>
<a id="4154" class="Symbol">...</a> <a id="4158" class="Symbol">|</a> <a id="4160" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="4165" class="Symbol">_</a> <a id="4167" href="Relation.Binary.Consequences.html#4167" class="Bound">x≉y</a> <a id="4171" class="Symbol">_</a> <a id="4173" class="Symbol">=</a> <a id="4175" href="Relation.Nullary.html#1685" class="InductiveConstructor">no</a> <a id="4179" href="Relation.Binary.Consequences.html#4167" class="Bound">x≉y</a>
<a id="4186" href="Relation.Binary.Consequences.html#4186" class="Function">tri⇒dec&lt;</a> <a id="4195" class="Symbol">:</a> <a id="4197" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="4210" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4214" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4218" class="Symbol">→</a> <a id="4220" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="4230" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a>
<a id="4236" href="Relation.Binary.Consequences.html#4186" class="Function">tri⇒dec&lt;</a> <a id="4245" href="Relation.Binary.Consequences.html#4245" class="Bound">compare</a> <a id="4253" href="Relation.Binary.Consequences.html#4253" class="Bound">x</a> <a id="4255" href="Relation.Binary.Consequences.html#4255" class="Bound">y</a> <a id="4257" class="Keyword">with</a> <a id="4262" href="Relation.Binary.Consequences.html#4245" class="Bound">compare</a> <a id="4270" href="Relation.Binary.Consequences.html#4253" class="Bound">x</a> <a id="4272" href="Relation.Binary.Consequences.html#4255" class="Bound">y</a>
<a id="4276" class="Symbol">...</a> <a id="4280" class="Symbol">|</a> <a id="4282" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="4287" href="Relation.Binary.Consequences.html#4287" class="Bound">x&lt;y</a> <a id="4291" class="Symbol">_</a> <a id="4293" class="Symbol">_</a> <a id="4295" class="Symbol">=</a> <a id="4297" href="Relation.Nullary.html#1648" class="InductiveConstructor">yes</a> <a id="4301" href="Relation.Binary.Consequences.html#4287" class="Bound">x&lt;y</a>
<a id="4307" class="Symbol">...</a> <a id="4311" class="Symbol">|</a> <a id="4313" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="4318" href="Relation.Binary.Consequences.html#4318" class="Bound">x≮y</a> <a id="4322" class="Symbol">_</a> <a id="4324" class="Symbol">_</a> <a id="4326" class="Symbol">=</a> <a id="4328" href="Relation.Nullary.html#1685" class="InductiveConstructor">no</a> <a id="4332" href="Relation.Binary.Consequences.html#4318" class="Bound">x≮y</a>
<a id="4338" class="Symbol">...</a> <a id="4342" class="Symbol">|</a> <a id="4344" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="4349" href="Relation.Binary.Consequences.html#4349" class="Bound">x≮y</a> <a id="4353" class="Symbol">_</a> <a id="4355" class="Symbol">_</a> <a id="4357" class="Symbol">=</a> <a id="4359" href="Relation.Nullary.html#1685" class="InductiveConstructor">no</a> <a id="4363" href="Relation.Binary.Consequences.html#4349" class="Bound">x≮y</a>
<a id="4370" href="Relation.Binary.Consequences.html#4370" class="Function">trans∧tri⇒respʳ</a> <a id="4386" class="Symbol">:</a> <a id="4388" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="4398" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4402" class="Symbol">→</a> <a id="4404" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="4415" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4419" class="Symbol">→</a>
<a id="4441" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="4452" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4456" class="Symbol">→</a> <a id="4458" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="4471" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4475" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4479" class="Symbol">→</a>
<a id="4501" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4505" href="Relation.Binary.Definitions.html#3749" class="Function Operator">Respectsʳ</a> <a id="4515" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a>
<a id="4521" href="Relation.Binary.Consequences.html#4370" class="Function">trans∧tri⇒respʳ</a> <a id="4537" href="Relation.Binary.Consequences.html#4537" class="Bound">sym</a> <a id="4541" href="Relation.Binary.Consequences.html#4541" class="Bound">≈-tr</a> <a id="4546" href="Relation.Binary.Consequences.html#4546" class="Bound">&lt;-tr</a> <a id="4551" href="Relation.Binary.Consequences.html#4551" class="Bound">tri</a> <a id="4555" class="Symbol">{</a><a id="4556" href="Relation.Binary.Consequences.html#4556" class="Bound">x</a><a id="4557" class="Symbol">}</a> <a id="4559" class="Symbol">{</a><a id="4560" href="Relation.Binary.Consequences.html#4560" class="Bound">y</a><a id="4561" class="Symbol">}</a> <a id="4563" class="Symbol">{</a><a id="4564" href="Relation.Binary.Consequences.html#4564" class="Bound">z</a><a id="4565" class="Symbol">}</a> <a id="4567" href="Relation.Binary.Consequences.html#4567" class="Bound">y≈z</a> <a id="4571" href="Relation.Binary.Consequences.html#4571" class="Bound">x&lt;y</a> <a id="4575" class="Keyword">with</a> <a id="4580" href="Relation.Binary.Consequences.html#4551" class="Bound">tri</a> <a id="4584" href="Relation.Binary.Consequences.html#4556" class="Bound">x</a> <a id="4586" href="Relation.Binary.Consequences.html#4564" class="Bound">z</a>
<a id="4590" class="Symbol">...</a> <a id="4594" class="Symbol">|</a> <a id="4596" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="4601" href="Relation.Binary.Consequences.html#4601" class="Bound">x&lt;z</a> <a id="4605" class="Symbol">_</a> <a id="4607" class="Symbol">_</a> <a id="4609" class="Symbol">=</a> <a id="4611" href="Relation.Binary.Consequences.html#4601" class="Bound">x&lt;z</a>
<a id="4617" class="Symbol">...</a> <a id="4621" class="Symbol">|</a> <a id="4623" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="4628" class="Symbol">_</a> <a id="4630" href="Relation.Binary.Consequences.html#4630" class="Bound">x≈z</a> <a id="4634" class="Symbol">_</a> <a id="4636" class="Symbol">=</a> <a id="4638" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a> <a id="4645" class="Symbol">(</a><a id="4646" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="4654" class="Bound">tri</a> <a id="4658" class="Symbol">(</a><a id="4659" class="Bound">≈-tr</a> <a id="4664" href="Relation.Binary.Consequences.html#4630" class="Bound">x≈z</a> <a id="4668" class="Symbol">(</a><a id="4669" class="Bound">sym</a> <a id="4673" class="Bound">y≈z</a><a id="4676" class="Symbol">))</a> <a id="4679" class="Bound">x&lt;y</a><a id="4682" class="Symbol">)</a>
<a id="4686" class="Symbol">...</a> <a id="4690" class="Symbol">|</a> <a id="4692" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="4697" class="Symbol">_</a> <a id="4699" class="Symbol">_</a> <a id="4701" href="Relation.Binary.Consequences.html#4701" class="Bound">z&lt;x</a> <a id="4705" class="Symbol">=</a> <a id="4707" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a> <a id="4714" class="Symbol">(</a><a id="4715" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="4723" class="Bound">tri</a> <a id="4727" class="Symbol">(</a><a id="4728" class="Bound">sym</a> <a id="4732" class="Bound">y≈z</a><a id="4735" class="Symbol">)</a> <a id="4737" class="Symbol">(</a><a id="4738" class="Bound">&lt;-tr</a> <a id="4743" href="Relation.Binary.Consequences.html#4701" class="Bound">z&lt;x</a> <a id="4747" class="Bound">x&lt;y</a><a id="4750" class="Symbol">))</a>
<a id="4756" href="Relation.Binary.Consequences.html#4756" class="Function">trans∧tri⇒respˡ</a> <a id="4772" class="Symbol">:</a> <a id="4774" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="4785" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4789" class="Symbol">→</a>
<a id="4811" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="4822" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4826" class="Symbol">→</a> <a id="4828" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="4841" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="4845" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4849" class="Symbol">→</a>
<a id="4871" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="4875" href="Relation.Binary.Definitions.html#3914" class="Function Operator">Respectsˡ</a> <a id="4885" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a>
<a id="4891" href="Relation.Binary.Consequences.html#4756" class="Function">trans∧tri⇒respˡ</a> <a id="4907" href="Relation.Binary.Consequences.html#4907" class="Bound">≈-tr</a> <a id="4912" href="Relation.Binary.Consequences.html#4912" class="Bound">&lt;-tr</a> <a id="4917" href="Relation.Binary.Consequences.html#4917" class="Bound">tri</a> <a id="4921" class="Symbol">{</a><a id="4922" href="Relation.Binary.Consequences.html#4922" class="Bound">z</a><a id="4923" class="Symbol">}</a> <a id="4925" class="Symbol">{_}</a> <a id="4929" class="Symbol">{</a><a id="4930" href="Relation.Binary.Consequences.html#4930" class="Bound">y</a><a id="4931" class="Symbol">}</a> <a id="4933" href="Relation.Binary.Consequences.html#4933" class="Bound">x≈y</a> <a id="4937" href="Relation.Binary.Consequences.html#4937" class="Bound">x&lt;z</a> <a id="4941" class="Keyword">with</a> <a id="4946" href="Relation.Binary.Consequences.html#4917" class="Bound">tri</a> <a id="4950" href="Relation.Binary.Consequences.html#4930" class="Bound">y</a> <a id="4952" href="Relation.Binary.Consequences.html#4922" class="Bound">z</a>
<a id="4956" class="Symbol">...</a> <a id="4960" class="Symbol">|</a> <a id="4962" href="Relation.Binary.Definitions.html#2778" class="InductiveConstructor">tri&lt;</a> <a id="4967" href="Relation.Binary.Consequences.html#4967" class="Bound">y&lt;z</a> <a id="4971" class="Symbol">_</a> <a id="4973" class="Symbol">_</a> <a id="4975" class="Symbol">=</a> <a id="4977" href="Relation.Binary.Consequences.html#4967" class="Bound">y&lt;z</a>
<a id="4983" class="Symbol">...</a> <a id="4987" class="Symbol">|</a> <a id="4989" href="Relation.Binary.Definitions.html#2832" class="InductiveConstructor">tri≈</a> <a id="4994" class="Symbol">_</a> <a id="4996" href="Relation.Binary.Consequences.html#4996" class="Bound">y≈z</a> <a id="5000" class="Symbol">_</a> <a id="5002" class="Symbol">=</a> <a id="5004" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a> <a id="5011" class="Symbol">(</a><a id="5012" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="5020" class="Bound">tri</a> <a id="5024" class="Symbol">(</a><a id="5025" class="Bound">≈-tr</a> <a id="5030" class="Bound">x≈y</a> <a id="5034" href="Relation.Binary.Consequences.html#4996" class="Bound">y≈z</a><a id="5037" class="Symbol">)</a> <a id="5039" class="Bound">x&lt;z</a><a id="5042" class="Symbol">)</a>
<a id="5046" class="Symbol">...</a> <a id="5050" class="Symbol">|</a> <a id="5052" href="Relation.Binary.Definitions.html#2886" class="InductiveConstructor">tri&gt;</a> <a id="5057" class="Symbol">_</a> <a id="5059" class="Symbol">_</a> <a id="5061" href="Relation.Binary.Consequences.html#5061" class="Bound">z&lt;y</a> <a id="5065" class="Symbol">=</a> <a id="5067" href="Data.Empty.Irrelevant.html#327" class="Function">⊥-elim</a> <a id="5074" class="Symbol">(</a><a id="5075" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a> <a id="5083" class="Bound">tri</a> <a id="5087" class="Bound">x≈y</a> <a id="5091" class="Symbol">(</a><a id="5092" class="Bound">&lt;-tr</a> <a id="5097" class="Bound">x&lt;z</a> <a id="5101" href="Relation.Binary.Consequences.html#5061" class="Bound">z&lt;y</a><a id="5104" class="Symbol">))</a>
<a id="5110" href="Relation.Binary.Consequences.html#5110" class="Function">trans∧tri⇒resp</a> <a id="5125" class="Symbol">:</a> <a id="5127" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="5137" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="5141" class="Symbol">→</a> <a id="5143" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="5154" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="5158" class="Symbol">→</a>
<a id="5179" href="Relation.Binary.Definitions.html#1866" class="Function">Transitive</a> <a id="5190" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="5194" class="Symbol">→</a> <a id="5196" href="Relation.Binary.Definitions.html#2955" class="Function">Trichotomous</a> <a id="5209" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a> <a id="5213" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="5217" class="Symbol">→</a>
<a id="5238" href="Relation.Binary.Consequences.html#2893" class="Bound Operator">_&lt;_</a> <a id="5242" href="Relation.Binary.Definitions.html#4077" class="Function Operator">Respects₂</a> <a id="5252" href="Relation.Binary.Consequences.html#2876" class="Bound Operator">_≈_</a>
<a id="5258" href="Relation.Binary.Consequences.html#5110" class="Function">trans∧tri⇒resp</a> <a id="5273" href="Relation.Binary.Consequences.html#5273" class="Bound">sym</a> <a id="5277" href="Relation.Binary.Consequences.html#5277" class="Bound">≈-tr</a> <a id="5282" href="Relation.Binary.Consequences.html#5282" class="Bound">&lt;-tr</a> <a id="5287" href="Relation.Binary.Consequences.html#5287" class="Bound">tri</a> <a id="5291" class="Symbol">=</a>
<a id="5297" href="Relation.Binary.Consequences.html#4370" class="Function">trans∧tri⇒respʳ</a> <a id="5313" href="Relation.Binary.Consequences.html#5273" class="Bound">sym</a> <a id="5317" href="Relation.Binary.Consequences.html#5277" class="Bound">≈-tr</a> <a id="5322" href="Relation.Binary.Consequences.html#5282" class="Bound">&lt;-tr</a> <a id="5327" href="Relation.Binary.Consequences.html#5287" class="Bound">tri</a> <a id="5331" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a>
<a id="5337" href="Relation.Binary.Consequences.html#4756" class="Function">trans∧tri⇒respˡ</a> <a id="5353" href="Relation.Binary.Consequences.html#5277" class="Bound">≈-tr</a> <a id="5358" href="Relation.Binary.Consequences.html#5282" class="Bound">&lt;-tr</a> <a id="5363" href="Relation.Binary.Consequences.html#5287" class="Bound">tri</a>
<a id="5368" class="Comment">------------------------------------------------------------------------</a>
<a id="5441" class="Comment">-- Without Loss of Generality</a>
<a id="5472" class="Keyword">module</a> <a id="5479" href="Relation.Binary.Consequences.html#5479" class="Module">_</a> <a id="5482" class="Symbol">{</a><a id="5483" href="Relation.Binary.Consequences.html#5483" class="Bound Operator">_R_</a> <a id="5487" class="Symbol">:</a> <a id="5489" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="5493" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="5495" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="5497" class="Symbol">}</a> <a id="5499" class="Symbol">{</a><a id="5500" href="Relation.Binary.Consequences.html#5500" class="Bound">Q</a> <a id="5502" class="Symbol">:</a> <a id="5504" href="Relation.Binary.Core.html#882" class="Function">Rel</a> <a id="5508" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="5510" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="5512" class="Symbol">}</a> <a id="5514" class="Keyword">where</a>
<a id="5523" href="Relation.Binary.Consequences.html#5523" class="Function">wlog</a> <a id="5528" class="Symbol">:</a> <a id="5530" href="Relation.Binary.Definitions.html#2584" class="Function">Total</a> <a id="5536" href="Relation.Binary.Consequences.html#5483" class="Bound Operator">_R_</a> <a id="5540" class="Symbol">→</a> <a id="5542" href="Relation.Binary.Definitions.html#1498" class="Function">Symmetric</a> <a id="5552" href="Relation.Binary.Consequences.html#5500" class="Bound">Q</a> <a id="5554" class="Symbol">→</a>
<a id="5565" class="Symbol">(∀</a> <a id="5568" href="Relation.Binary.Consequences.html#5568" class="Bound">a</a> <a id="5570" href="Relation.Binary.Consequences.html#5570" class="Bound">b</a> <a id="5572" class="Symbol">→</a> <a id="5574" href="Relation.Binary.Consequences.html#5568" class="Bound">a</a> <a id="5576" href="Relation.Binary.Consequences.html#5483" class="Bound Operator">R</a> <a id="5578" href="Relation.Binary.Consequences.html#5570" class="Bound">b</a> <a id="5580" class="Symbol">→</a> <a id="5582" href="Relation.Binary.Consequences.html#5500" class="Bound">Q</a> <a id="5584" href="Relation.Binary.Consequences.html#5568" class="Bound">a</a> <a id="5586" href="Relation.Binary.Consequences.html#5570" class="Bound">b</a><a id="5587" class="Symbol">)</a> <a id="5589" class="Symbol">→</a>
<a id="5600" class="Symbol">∀</a> <a id="5602" href="Relation.Binary.Consequences.html#5602" class="Bound">a</a> <a id="5604" href="Relation.Binary.Consequences.html#5604" class="Bound">b</a> <a id="5606" class="Symbol">→</a> <a id="5608" href="Relation.Binary.Consequences.html#5500" class="Bound">Q</a> <a id="5610" href="Relation.Binary.Consequences.html#5602" class="Bound">a</a> <a id="5612" href="Relation.Binary.Consequences.html#5604" class="Bound">b</a>
<a id="5616" href="Relation.Binary.Consequences.html#5523" class="Function">wlog</a> <a id="5621" href="Relation.Binary.Consequences.html#5621" class="Bound">r-total</a> <a id="5629" href="Relation.Binary.Consequences.html#5629" class="Bound">q-sym</a> <a id="5635" href="Relation.Binary.Consequences.html#5635" class="Bound">prf</a> <a id="5639" href="Relation.Binary.Consequences.html#5639" class="Bound">a</a> <a id="5641" href="Relation.Binary.Consequences.html#5641" class="Bound">b</a> <a id="5643" class="Keyword">with</a> <a id="5648" href="Relation.Binary.Consequences.html#5621" class="Bound">r-total</a> <a id="5656" href="Relation.Binary.Consequences.html#5639" class="Bound">a</a> <a id="5658" href="Relation.Binary.Consequences.html#5641" class="Bound">b</a>
<a id="5662" class="Symbol">...</a> <a id="5666" class="Symbol">|</a> <a id="5668" href="Data.Sum.Base.html#784" class="InductiveConstructor">inj₁</a> <a id="5673" href="Relation.Binary.Consequences.html#5673" class="Bound">aRb</a> <a id="5677" class="Symbol">=</a> <a id="5679" class="Bound">prf</a> <a id="5683" class="Bound">a</a> <a id="5685" class="Bound">b</a> <a id="5687" href="Relation.Binary.Consequences.html#5673" class="Bound">aRb</a>
<a id="5693" class="Symbol">...</a> <a id="5697" class="Symbol">|</a> <a id="5699" href="Data.Sum.Base.html#809" class="InductiveConstructor">inj₂</a> <a id="5704" href="Relation.Binary.Consequences.html#5704" class="Bound">bRa</a> <a id="5708" class="Symbol">=</a> <a id="5710" class="Bound">q-sym</a> <a id="5716" class="Symbol">(</a><a id="5717" class="Bound">prf</a> <a id="5721" class="Bound">b</a> <a id="5723" class="Bound">a</a> <a id="5725" href="Relation.Binary.Consequences.html#5704" class="Bound">bRa</a><a id="5728" class="Symbol">)</a>
<a id="5731" class="Comment">------------------------------------------------------------------------</a>
<a id="5804" class="Comment">-- Other proofs</a>
<a id="5821" class="Keyword">module</a> <a id="5828" href="Relation.Binary.Consequences.html#5828" class="Module">_</a> <a id="5830" class="Symbol">{</a><a id="5831" href="Relation.Binary.Consequences.html#5831" class="Bound">R</a> <a id="5833" class="Symbol">:</a> <a id="5835" href="Relation.Binary.Core.html#766" class="Function">REL</a> <a id="5839" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="5841" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="5843" href="Relation.Binary.Consequences.html#841" class="Generalizable">p</a><a id="5844" class="Symbol">}</a> <a id="5846" class="Keyword">where</a>
<a id="5855" href="Relation.Binary.Consequences.html#5855" class="Function">dec⇒weaklyDec</a> <a id="5869" class="Symbol">:</a> <a id="5871" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="5881" href="Relation.Binary.Consequences.html#5831" class="Bound">R</a> <a id="5883" class="Symbol">→</a> <a id="5885" href="Relation.Binary.Definitions.html#4748" class="Function">WeaklyDecidable</a> <a id="5901" href="Relation.Binary.Consequences.html#5831" class="Bound">R</a>
<a id="5905" href="Relation.Binary.Consequences.html#5855" class="Function">dec⇒weaklyDec</a> <a id="5919" href="Relation.Binary.Consequences.html#5919" class="Bound">dec</a> <a id="5923" href="Relation.Binary.Consequences.html#5923" class="Bound">x</a> <a id="5925" href="Relation.Binary.Consequences.html#5925" class="Bound">y</a> <a id="5927" class="Symbol">=</a> <a id="5929" href="Data.Maybe.Base.html#1212" class="Function">decToMaybe</a> <a id="5940" class="Symbol">(</a><a id="5941" href="Relation.Binary.Consequences.html#5919" class="Bound">dec</a> <a id="5945" href="Relation.Binary.Consequences.html#5923" class="Bound">x</a> <a id="5947" href="Relation.Binary.Consequences.html#5925" class="Bound">y</a><a id="5948" class="Symbol">)</a>
<a id="5953" href="Relation.Binary.Consequences.html#5953" class="Function">dec⇒recomputable</a> <a id="5970" class="Symbol">:</a> <a id="5972" href="Relation.Binary.Definitions.html#4575" class="Function">Decidable</a> <a id="5982" href="Relation.Binary.Consequences.html#5831" class="Bound">R</a> <a id="5984" class="Symbol">→</a> <a id="5986" href="Relation.Binary.Definitions.html#5226" class="Function">Recomputable</a> <a id="5999" href="Relation.Binary.Consequences.html#5831" class="Bound">R</a>
<a id="6003" href="Relation.Binary.Consequences.html#5953" class="Function">dec⇒recomputable</a> <a id="6020" href="Relation.Binary.Consequences.html#6020" class="Bound">dec</a> <a id="6024" class="Symbol">{</a><a id="6025" href="Relation.Binary.Consequences.html#6025" class="Bound">a</a><a id="6026" class="Symbol">}</a> <a id="6028" class="Symbol">{</a><a id="6029" href="Relation.Binary.Consequences.html#6029" class="Bound">b</a><a id="6030" class="Symbol">}</a> <a id="6032" class="Symbol">=</a> <a id="6034" href="Relation.Nullary.html#1838" class="Function">recompute</a> <a id="6044" href="Function.Base.html#1919" class="Function Operator">$</a> <a id="6046" href="Relation.Binary.Consequences.html#6020" class="Bound">dec</a> <a id="6050" href="Relation.Binary.Consequences.html#6025" class="Bound">a</a> <a id="6052" href="Relation.Binary.Consequences.html#6029" class="Bound">b</a>
<a id="6055" class="Keyword">module</a> <a id="6062" href="Relation.Binary.Consequences.html#6062" class="Module">_</a> <a id="6064" class="Symbol">{</a><a id="6065" href="Relation.Binary.Consequences.html#6065" class="Bound">R</a> <a id="6067" class="Symbol">:</a> <a id="6069" href="Relation.Binary.Core.html#766" class="Function">REL</a> <a id="6073" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="6075" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="6077" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="6079" class="Symbol">}</a> <a id="6081" class="Symbol">{</a><a id="6082" href="Relation.Binary.Consequences.html#6082" class="Bound">S</a> <a id="6084" class="Symbol">:</a> <a id="6086" href="Relation.Binary.Core.html#766" class="Function">REL</a> <a id="6090" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="6092" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="6094" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="6096" class="Symbol">}</a> <a id="6098" class="Keyword">where</a>
<a id="6107" href="Relation.Binary.Consequences.html#6107" class="Function">map-NonEmpty</a> <a id="6120" class="Symbol">:</a> <a id="6122" href="Relation.Binary.Consequences.html#6065" class="Bound">R</a> <a id="6124" href="Relation.Binary.Core.html#1254" class="Function Operator">⇒</a> <a id="6126" href="Relation.Binary.Consequences.html#6082" class="Bound">S</a> <a id="6128" class="Symbol">→</a> <a id="6130" href="Relation.Binary.Definitions.html#5487" class="Record">NonEmpty</a> <a id="6139" href="Relation.Binary.Consequences.html#6065" class="Bound">R</a> <a id="6141" class="Symbol">→</a> <a id="6143" href="Relation.Binary.Definitions.html#5487" class="Record">NonEmpty</a> <a id="6152" href="Relation.Binary.Consequences.html#6082" class="Bound">S</a>
<a id="6156" href="Relation.Binary.Consequences.html#6107" class="Function">map-NonEmpty</a> <a id="6169" href="Relation.Binary.Consequences.html#6169" class="Bound">f</a> <a id="6171" href="Relation.Binary.Consequences.html#6171" class="Bound">x</a> <a id="6173" class="Symbol">=</a> <a id="6175" href="Relation.Binary.Definitions.html#5590" class="InductiveConstructor">nonEmpty</a> <a id="6184" class="Symbol">(</a><a id="6185" href="Relation.Binary.Consequences.html#6169" class="Bound">f</a> <a id="6187" class="Symbol">(</a><a id="6188" href="Relation.Binary.Definitions.html#5639" class="Field">NonEmpty.proof</a> <a id="6203" href="Relation.Binary.Consequences.html#6171" class="Bound">x</a><a id="6204" class="Symbol">))</a>
<a id="6208" class="Keyword">module</a> <a id="6215" href="Relation.Binary.Consequences.html#6215" class="Module">_</a> <a id="6217" class="Symbol">{</a><a id="6218" href="Relation.Binary.Consequences.html#6218" class="Bound">R</a> <a id="6220" class="Symbol">:</a> <a id="6222" href="Relation.Binary.Core.html#766" class="Function">REL</a> <a id="6226" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="6228" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="6230" href="Relation.Binary.Consequences.html#829" class="Generalizable">ℓ₁</a><a id="6232" class="Symbol">}</a> <a id="6234" class="Symbol">{</a><a id="6235" href="Relation.Binary.Consequences.html#6235" class="Bound">S</a> <a id="6237" class="Symbol">:</a> <a id="6239" href="Relation.Binary.Core.html#766" class="Function">REL</a> <a id="6243" href="Relation.Binary.Consequences.html#857" class="Generalizable">B</a> <a id="6245" href="Relation.Binary.Consequences.html#855" class="Generalizable">A</a> <a id="6247" href="Relation.Binary.Consequences.html#832" class="Generalizable">ℓ₂</a><a id="6249" class="Symbol">}</a> <a id="6251" class="Keyword">where</a>
<a id="6260" href="Relation.Binary.Consequences.html#6260" class="Function">flip-Connex</a> <a id="6272" class="Symbol">:</a> <a id="6274" href="Relation.Binary.Definitions.html#2493" class="Function">Connex</a> <a id="6281" href="Relation.Binary.Consequences.html#6218" class="Bound">R</a> <a id="6283" href="Relation.Binary.Consequences.html#6235" class="Bound">S</a> <a id="6285" class="Symbol">→</a> <a id="6287" href="Relation.Binary.Definitions.html#2493" class="Function">Connex</a> <a id="6294" href="Relation.Binary.Consequences.html#6235" class="Bound">S</a> <a id="6296" href="Relation.Binary.Consequences.html#6218" class="Bound">R</a>
<a id="6300" href="Relation.Binary.Consequences.html#6260" class="Function">flip-Connex</a> <a id="6312" href="Relation.Binary.Consequences.html#6312" class="Bound">f</a> <a id="6314" href="Relation.Binary.Consequences.html#6314" class="Bound">x</a> <a id="6316" href="Relation.Binary.Consequences.html#6316" class="Bound">y</a> <a id="6318" class="Symbol">=</a> <a id="6320" href="Data.Sum.Base.html#1294" class="Function">Sum.swap</a> <a id="6329" class="Symbol">(</a><a id="6330" href="Relation.Binary.Consequences.html#6312" class="Bound">f</a> <a id="6332" href="Relation.Binary.Consequences.html#6316" class="Bound">y</a> <a id="6334" href="Relation.Binary.Consequences.html#6314" class="Bound">x</a><a id="6335" class="Symbol">)</a>
<a id="6340" class="Comment">------------------------------------------------------------------------</a>
<a id="6413" class="Comment">-- DEPRECATED NAMES</a>
<a id="6433" class="Comment">------------------------------------------------------------------------</a>
<a id="6506" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="6577" class="Comment">-- not guaranteed.</a>
<a id="6597" class="Comment">-- Version 1.6</a>
<a id="subst⟶respˡ"></a><a id="6613" href="Relation.Binary.Consequences.html#6613" class="Function">subst⟶respˡ</a> <a id="6625" class="Symbol">=</a> <a id="6627" href="Relation.Binary.Consequences.html#1017" class="Function">subst⇒respˡ</a>
<a id="6639" class="Symbol">{-#</a> <a id="6643" class="Keyword">WARNING_ON_USAGE</a> <a id="6660" class="Pragma">subst⟶respˡ</a>
<a id="6672" class="String">&quot;Warning: subst⟶respˡ was deprecated in v1.6.
Please use subst⇒respˡ instead.&quot;</a>
<a id="6751" class="Symbol">#-}</a>
<a id="subst⟶respʳ"></a><a id="6755" href="Relation.Binary.Consequences.html#6755" class="Function">subst⟶respʳ</a> <a id="6767" class="Symbol">=</a> <a id="6769" href="Relation.Binary.Consequences.html#1134" class="Function">subst⇒respʳ</a>
<a id="6781" class="Symbol">{-#</a> <a id="6785" class="Keyword">WARNING_ON_USAGE</a> <a id="6802" class="Pragma">subst⟶respʳ</a>
<a id="6814" class="String">&quot;Warning: subst⟶respʳ was deprecated in v1.6.
Please use subst⇒respʳ instead.&quot;</a>
<a id="6893" class="Symbol">#-}</a>
<a id="subst⟶resp₂"></a><a id="6897" href="Relation.Binary.Consequences.html#6897" class="Function">subst⟶resp₂</a> <a id="6909" class="Symbol">=</a> <a id="6911" href="Relation.Binary.Consequences.html#1246" class="Function">subst⇒resp₂</a>
<a id="6923" class="Symbol">{-#</a> <a id="6927" class="Keyword">WARNING_ON_USAGE</a> <a id="6944" class="Pragma">subst⟶resp₂</a>
<a id="6956" class="String">&quot;Warning: subst⟶resp₂ was deprecated in v1.6.
Please use subst⇒resp₂ instead.&quot;</a>
<a id="7035" class="Symbol">#-}</a>
<a id="P-resp⟶¬P-resp"></a><a id="7039" href="Relation.Binary.Consequences.html#7039" class="Function">P-resp⟶¬P-resp</a> <a id="7054" class="Symbol">=</a> <a id="7056" href="Relation.Binary.Consequences.html#1407" class="Function">resp⇒¬-resp</a>
<a id="7068" class="Symbol">{-#</a> <a id="7072" class="Keyword">WARNING_ON_USAGE</a> <a id="7089" class="Pragma">P-resp⟶¬P-resp</a>
<a id="7104" class="String">&quot;Warning: P-resp⟶¬P-resp was deprecated in v1.6.
Please use resp⇒¬-resp instead.&quot;</a>
<a id="7186" class="Symbol">#-}</a>
<a id="total⟶refl"></a><a id="7190" href="Relation.Binary.Consequences.html#7190" class="Function">total⟶refl</a> <a id="7201" class="Symbol">=</a> <a id="7203" href="Relation.Binary.Consequences.html#1692" class="Function">total⇒refl</a>
<a id="7214" class="Symbol">{-#</a> <a id="7218" class="Keyword">WARNING_ON_USAGE</a> <a id="7235" class="Pragma">total⟶refl</a>
<a id="7246" class="String">&quot;Warning: total⟶refl was deprecated in v1.6.
Please use total⇒refl instead.&quot;</a>
<a id="7323" class="Symbol">#-}</a>
<a id="total+dec⟶dec"></a><a id="7327" href="Relation.Binary.Consequences.html#7327" class="Function">total+dec⟶dec</a> <a id="7341" class="Symbol">=</a> <a id="7343" href="Relation.Binary.Consequences.html#1921" class="Function">total∧dec⇒dec</a>
<a id="7357" class="Symbol">{-#</a> <a id="7361" class="Keyword">WARNING_ON_USAGE</a> <a id="7378" class="Pragma">total+dec⟶dec</a>
<a id="7392" class="String">&quot;Warning: total+dec⟶dec was deprecated in v1.6.
Please use total∧dec⇒dec instead.&quot;</a>
<a id="7475" class="Symbol">#-}</a>
<a id="trans∧irr⟶asym"></a><a id="7479" href="Relation.Binary.Consequences.html#7479" class="Function">trans∧irr⟶asym</a> <a id="7494" class="Symbol">=</a> <a id="7496" href="Relation.Binary.Consequences.html#2918" class="Function">trans∧irr⇒asym</a>
<a id="7511" class="Symbol">{-#</a> <a id="7515" class="Keyword">WARNING_ON_USAGE</a> <a id="7532" class="Pragma">trans∧irr⟶asym</a>
<a id="7547" class="String">&quot;Warning: trans∧irr⟶asym was deprecated in v1.6.
Please use trans∧irr⇒asym instead.&quot;</a>
<a id="7632" class="Symbol">#-}</a>
<a id="irr∧antisym⟶asym"></a><a id="7636" href="Relation.Binary.Consequences.html#7636" class="Function">irr∧antisym⟶asym</a> <a id="7653" class="Symbol">=</a> <a id="7655" href="Relation.Binary.Consequences.html#3104" class="Function">irr∧antisym⇒asym</a>
<a id="7672" class="Symbol">{-#</a> <a id="7676" class="Keyword">WARNING_ON_USAGE</a> <a id="7693" class="Pragma">irr∧antisym⟶asym</a>
<a id="7710" class="String">&quot;Warning: irr∧antisym⟶asym was deprecated in v1.6.
Please use irr∧antisym⇒asym instead.&quot;</a>
<a id="7799" class="Symbol">#-}</a>
<a id="asym⟶antisym"></a><a id="7803" href="Relation.Binary.Consequences.html#7803" class="Function">asym⟶antisym</a> <a id="7816" class="Symbol">=</a> <a id="7818" href="Relation.Binary.Consequences.html#3285" class="Function">asym⇒antisym</a>
<a id="7831" class="Symbol">{-#</a> <a id="7835" class="Keyword">WARNING_ON_USAGE</a> <a id="7852" class="Pragma">asym⟶antisym</a>
<a id="7865" class="String">&quot;Warning: asym⟶antisym was deprecated in v1.6.
Please use asym⇒antisym instead.&quot;</a>
<a id="7946" class="Symbol">#-}</a>
<a id="asym⟶irr"></a><a id="7950" href="Relation.Binary.Consequences.html#7950" class="Function">asym⟶irr</a> <a id="7959" class="Symbol">=</a> <a id="7961" href="Relation.Binary.Consequences.html#3394" class="Function">asym⇒irr</a>
<a id="7970" class="Symbol">{-#</a> <a id="7974" class="Keyword">WARNING_ON_USAGE</a> <a id="7991" class="Pragma">asym⟶irr</a>
<a id="8000" class="String">&quot;Warning: asym⟶irr was deprecated in v1.6.
Please use asym⇒irr instead.&quot;</a>
<a id="8073" class="Symbol">#-}</a>
<a id="tri⟶asym"></a><a id="8077" href="Relation.Binary.Consequences.html#8077" class="Function">tri⟶asym</a> <a id="8086" class="Symbol">=</a> <a id="8088" href="Relation.Binary.Consequences.html#3595" class="Function">tri⇒asym</a>
<a id="8097" class="Symbol">{-#</a> <a id="8101" class="Keyword">WARNING_ON_USAGE</a> <a id="8118" class="Pragma">tri⟶asym</a>
<a id="8127" class="String">&quot;Warning: tri⟶asym was deprecated in v1.6.
Please use tri⇒asym instead.&quot;</a>
<a id="8200" class="Symbol">#-}</a>
<a id="tri⟶irr"></a><a id="8204" href="Relation.Binary.Consequences.html#8204" class="Function">tri⟶irr</a> <a id="8212" class="Symbol">=</a> <a id="8214" href="Relation.Binary.Consequences.html#3790" class="Function">tri⇒irr</a>
<a id="8222" class="Symbol">{-#</a> <a id="8226" class="Keyword">WARNING_ON_USAGE</a> <a id="8243" class="Pragma">tri⟶irr</a>
<a id="8251" class="String">&quot;Warning: tri⟶irr was deprecated in v1.6.
Please use tri⇒irr instead.&quot;</a>
<a id="8322" class="Symbol">#-}</a>
<a id="tri⟶dec≈"></a><a id="8326" href="Relation.Binary.Consequences.html#8326" class="Function">tri⟶dec≈</a> <a id="8335" class="Symbol">=</a> <a id="8337" href="Relation.Binary.Consequences.html#4002" class="Function">tri⇒dec≈</a>
<a id="8346" class="Symbol">{-#</a> <a id="8350" class="Keyword">WARNING_ON_USAGE</a> <a id="8367" class="Pragma">tri⟶dec≈</a>
<a id="8376" class="String">&quot;Warning: tri⟶dec≈ was deprecated in v1.6.
Please use tri⇒dec≈ instead.&quot;</a>
<a id="8449" class="Symbol">#-}</a>
<a id="tri⟶dec&lt;"></a><a id="8453" href="Relation.Binary.Consequences.html#8453" class="Function">tri⟶dec&lt;</a> <a id="8462" class="Symbol">=</a> <a id="8464" href="Relation.Binary.Consequences.html#4186" class="Function">tri⇒dec&lt;</a>
<a id="8473" class="Symbol">{-#</a> <a id="8477" class="Keyword">WARNING_ON_USAGE</a> <a id="8494" class="Pragma">tri⟶dec&lt;</a>
<a id="8503" class="String">&quot;Warning: tri⟶dec&lt; was deprecated in v1.6.
Please use tri⇒dec&lt; instead.&quot;</a>
<a id="8576" class="Symbol">#-}</a>
<a id="trans∧tri⟶respʳ≈"></a><a id="8580" href="Relation.Binary.Consequences.html#8580" class="Function">trans∧tri⟶respʳ≈</a> <a id="8597" class="Symbol">=</a> <a id="8599" href="Relation.Binary.Consequences.html#4370" class="Function">trans∧tri⇒respʳ</a>
<a id="8615" class="Symbol">{-#</a> <a id="8619" class="Keyword">WARNING_ON_USAGE</a> <a id="8636" class="Pragma">trans∧tri⟶respʳ≈</a>
<a id="8653" class="String">&quot;Warning: trans∧tri⟶respʳ≈ was deprecated in v1.6.
Please use trans∧tri⇒respʳ instead.&quot;</a>
<a id="8741" class="Symbol">#-}</a>
<a id="trans∧tri⟶respˡ≈"></a><a id="8745" href="Relation.Binary.Consequences.html#8745" class="Function">trans∧tri⟶respˡ≈</a> <a id="8762" class="Symbol">=</a> <a id="8764" href="Relation.Binary.Consequences.html#4756" class="Function">trans∧tri⇒respˡ</a>
<a id="8780" class="Symbol">{-#</a> <a id="8784" class="Keyword">WARNING_ON_USAGE</a> <a id="8801" class="Pragma">trans∧tri⟶respˡ≈</a>
<a id="8818" class="String">&quot;Warning: trans∧tri⟶respˡ≈ was deprecated in v1.6.
Please use trans∧tri⇒respˡ instead.&quot;</a>
<a id="8906" class="Symbol">#-}</a>
<a id="trans∧tri⟶resp≈"></a><a id="8910" href="Relation.Binary.Consequences.html#8910" class="Function">trans∧tri⟶resp≈</a> <a id="8926" class="Symbol">=</a> <a id="8928" href="Relation.Binary.Consequences.html#5110" class="Function">trans∧tri⇒resp</a>
<a id="8943" class="Symbol">{-#</a> <a id="8947" class="Keyword">WARNING_ON_USAGE</a> <a id="8964" class="Pragma">trans∧tri⟶resp≈</a>
<a id="8980" class="String">&quot;Warning: trans∧tri⟶resp≈ was deprecated in v1.6.
Please use trans∧tri⇒resp instead.&quot;</a>
<a id="9066" class="Symbol">#-}</a>
<a id="dec⟶weaklyDec"></a><a id="9070" href="Relation.Binary.Consequences.html#9070" class="Function">dec⟶weaklyDec</a> <a id="9084" class="Symbol">=</a> <a id="9086" href="Relation.Binary.Consequences.html#5855" class="Function">dec⇒weaklyDec</a>
<a id="9100" class="Symbol">{-#</a> <a id="9104" class="Keyword">WARNING_ON_USAGE</a> <a id="9121" class="Pragma">dec⟶weaklyDec</a>
<a id="9135" class="String">&quot;Warning: dec⟶weaklyDec was deprecated in v1.6.
Please use dec⇒weaklyDec instead.&quot;</a>
<a id="9218" class="Symbol">#-}</a>
<a id="dec⟶recomputable"></a><a id="9222" href="Relation.Binary.Consequences.html#9222" class="Function">dec⟶recomputable</a> <a id="9239" class="Symbol">=</a> <a id="9241" href="Relation.Binary.Consequences.html#5953" class="Function">dec⇒recomputable</a>
<a id="9258" class="Symbol">{-#</a> <a id="9262" class="Keyword">WARNING_ON_USAGE</a> <a id="9279" class="Pragma">dec⟶recomputable</a>
<a id="9296" class="String">&quot;Warning: dec⟶recomputable was deprecated in v1.6.
Please use dec⇒recomputable instead.&quot;</a>
<a id="9385" class="Symbol">#-}</a>
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