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287 lines
89 KiB
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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>
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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">-- Some properties imply others</a>
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<a id="138" class="Comment">------------------------------------------------------------------------</a>
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<a id="212" class="Symbol">{-#</a> <a id="216" class="Keyword">OPTIONS</a> <a id="224" class="Pragma">--cubical-compatible</a> <a id="245" class="Pragma">--safe</a> <a id="252" class="Symbol">#-}</a>
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<a id="257" class="Keyword">module</a> <a id="264" href="Relation.Binary.Consequences.html" class="Module">Relation.Binary.Consequences</a> <a id="293" class="Keyword">where</a>
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<a id="300" class="Keyword">open</a> <a id="305" class="Keyword">import</a> <a id="312" href="Data.Maybe.Base.html" class="Module">Data.Maybe.Base</a> <a id="328" class="Keyword">using</a> <a id="334" class="Symbol">(</a><a id="335" href="Agda.Builtin.Maybe.html#156" class="InductiveConstructor">just</a><a id="339" class="Symbol">;</a> <a id="341" href="Agda.Builtin.Maybe.html#177" class="InductiveConstructor">nothing</a><a id="348" class="Symbol">;</a> <a id="350" href="Data.Maybe.Base.html#1221" class="Function">decToMaybe</a><a id="360" class="Symbol">)</a>
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<a id="362" class="Keyword">open</a> <a id="367" class="Keyword">import</a> <a id="374" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="388" class="Symbol">as</a> <a id="391" class="Module">Sum</a> <a id="395" class="Keyword">using</a> <a id="401" class="Symbol">(</a><a id="402" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a><a id="406" class="Symbol">;</a> <a id="408" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a><a id="412" class="Symbol">)</a>
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<a id="414" class="Keyword">open</a> <a id="419" class="Keyword">import</a> <a id="426" href="Data.Product.html" class="Module">Data.Product</a> <a id="439" class="Keyword">using</a> <a id="445" class="Symbol">(</a><a id="446" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="449" class="Symbol">)</a>
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<a id="451" class="Keyword">open</a> <a id="456" class="Keyword">import</a> <a id="463" href="Data.Empty.Irrelevant.html" class="Module">Data.Empty.Irrelevant</a> <a id="485" class="Keyword">using</a> <a id="491" class="Symbol">(</a><a id="492" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a><a id="498" class="Symbol">)</a>
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<a id="500" class="Keyword">open</a> <a id="505" class="Keyword">import</a> <a id="512" href="Function.Base.html" class="Module">Function.Base</a> <a id="526" class="Keyword">using</a> <a id="532" class="Symbol">(</a><a id="533" href="Function.Base.html#1040" class="Function Operator">_∘_</a><a id="536" class="Symbol">;</a> <a id="538" href="Function.Base.html#1928" class="Function Operator">_$_</a><a id="541" class="Symbol">;</a> <a id="543" href="Function.Base.html#1563" class="Function">flip</a><a id="547" class="Symbol">)</a>
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<a id="549" class="Keyword">open</a> <a id="554" class="Keyword">import</a> <a id="561" href="Level.html" class="Module">Level</a> <a id="567" class="Keyword">using</a> <a id="573" class="Symbol">(</a><a id="574" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="579" class="Symbol">)</a>
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<a id="581" class="Keyword">open</a> <a id="586" class="Keyword">import</a> <a id="593" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a>
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<a id="614" class="Keyword">open</a> <a id="619" class="Keyword">import</a> <a id="626" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
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<a id="654" class="Keyword">open</a> <a id="659" class="Keyword">import</a> <a id="666" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="683" class="Keyword">using</a> <a id="689" class="Symbol">(</a><a id="690" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a><a id="693" class="Symbol">;</a> <a id="695" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a><a id="697" class="Symbol">;</a> <a id="699" href="Relation.Nullary.html#1847" class="Function">recompute</a><a id="708" class="Symbol">)</a>
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<a id="710" class="Keyword">open</a> <a id="715" class="Keyword">import</a> <a id="722" href="Relation.Nullary.Decidable.Core.html" class="Module">Relation.Nullary.Decidable.Core</a> <a id="754" class="Keyword">using</a> <a id="760" class="Symbol">(</a><a id="761" href="Relation.Nullary.Decidable.Core.html#3645" class="Function">map′</a><a id="765" class="Symbol">)</a>
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<a id="767" class="Keyword">open</a> <a id="772" class="Keyword">import</a> <a id="779" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="794" class="Keyword">using</a> <a id="800" class="Symbol">(</a><a id="801" href="Relation.Unary.html#4287" class="Function">∁</a><a id="802" class="Symbol">;</a> <a id="804" href="Relation.Unary.html#1110" class="Function">Pred</a><a id="808" class="Symbol">)</a>
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<a id="811" class="Keyword">private</a>
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<a id="821" class="Keyword">variable</a>
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<a id="834" href="Relation.Binary.Consequences.html#834" class="Generalizable">a</a> <a id="836" href="Relation.Binary.Consequences.html#836" class="Generalizable">ℓ</a> <a id="838" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a> <a id="841" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a> <a id="844" href="Relation.Binary.Consequences.html#844" class="Generalizable">ℓ₃</a> <a id="847" href="Relation.Binary.Consequences.html#847" class="Generalizable">ℓ₄</a> <a id="850" href="Relation.Binary.Consequences.html#850" class="Generalizable">p</a> <a id="852" class="Symbol">:</a> <a id="854" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="864" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="866" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="868" class="Symbol">:</a> <a id="870" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="874" href="Relation.Binary.Consequences.html#834" class="Generalizable">a</a>
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<a id="877" class="Comment">------------------------------------------------------------------------</a>
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<a id="950" class="Comment">-- Substitutive properties</a>
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<a id="978" class="Keyword">module</a> <a id="985" href="Relation.Binary.Consequences.html#985" class="Module">_</a> <a id="987" class="Symbol">{</a><a id="988" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a> <a id="992" class="Symbol">:</a> <a id="994" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="998" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1000" href="Relation.Binary.Consequences.html#836" class="Generalizable">ℓ</a><a id="1001" class="Symbol">}</a> <a id="1003" class="Symbol">(</a><a id="1004" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1006" class="Symbol">:</a> <a id="1008" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1012" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1014" href="Relation.Binary.Consequences.html#850" class="Generalizable">p</a><a id="1015" class="Symbol">)</a> <a id="1017" class="Keyword">where</a>
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<a id="1026" href="Relation.Binary.Consequences.html#1026" class="Function">subst⇒respˡ</a> <a id="1038" class="Symbol">:</a> <a id="1040" href="Relation.Binary.Definitions.html#4378" class="Function">Substitutive</a> <a id="1053" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a> <a id="1057" href="Relation.Binary.Consequences.html#1014" class="Bound">p</a> <a id="1059" class="Symbol">→</a> <a id="1061" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1063" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="1073" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a>
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<a id="1079" href="Relation.Binary.Consequences.html#1026" class="Function">subst⇒respˡ</a> <a id="1091" href="Relation.Binary.Consequences.html#1091" class="Bound">subst</a> <a id="1097" class="Symbol">{</a><a id="1098" href="Relation.Binary.Consequences.html#1098" class="Bound">y</a><a id="1099" class="Symbol">}</a> <a id="1101" href="Relation.Binary.Consequences.html#1101" class="Bound">x′∼x</a> <a id="1106" href="Relation.Binary.Consequences.html#1106" class="Bound">Px′y</a> <a id="1111" class="Symbol">=</a> <a id="1113" href="Relation.Binary.Consequences.html#1091" class="Bound">subst</a> <a id="1119" class="Symbol">(</a><a id="1120" href="Function.Base.html#1563" class="Function">flip</a> <a id="1125" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1127" href="Relation.Binary.Consequences.html#1098" class="Bound">y</a><a id="1128" class="Symbol">)</a> <a id="1130" href="Relation.Binary.Consequences.html#1101" class="Bound">x′∼x</a> <a id="1135" href="Relation.Binary.Consequences.html#1106" class="Bound">Px′y</a>
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<a id="1143" href="Relation.Binary.Consequences.html#1143" class="Function">subst⇒respʳ</a> <a id="1155" class="Symbol">:</a> <a id="1157" href="Relation.Binary.Definitions.html#4378" class="Function">Substitutive</a> <a id="1170" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a> <a id="1174" href="Relation.Binary.Consequences.html#1014" class="Bound">p</a> <a id="1176" class="Symbol">→</a> <a id="1178" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1180" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="1190" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a>
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<a id="1196" href="Relation.Binary.Consequences.html#1143" class="Function">subst⇒respʳ</a> <a id="1208" href="Relation.Binary.Consequences.html#1208" class="Bound">subst</a> <a id="1214" class="Symbol">{</a><a id="1215" href="Relation.Binary.Consequences.html#1215" class="Bound">x</a><a id="1216" class="Symbol">}</a> <a id="1218" href="Relation.Binary.Consequences.html#1218" class="Bound">y′∼y</a> <a id="1223" href="Relation.Binary.Consequences.html#1223" class="Bound">Pxy′</a> <a id="1228" class="Symbol">=</a> <a id="1230" href="Relation.Binary.Consequences.html#1208" class="Bound">subst</a> <a id="1236" class="Symbol">(</a><a id="1237" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1239" href="Relation.Binary.Consequences.html#1215" class="Bound">x</a><a id="1240" class="Symbol">)</a> <a id="1242" href="Relation.Binary.Consequences.html#1218" class="Bound">y′∼y</a> <a id="1247" href="Relation.Binary.Consequences.html#1223" class="Bound">Pxy′</a>
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<a id="1255" href="Relation.Binary.Consequences.html#1255" class="Function">subst⇒resp₂</a> <a id="1267" class="Symbol">:</a> <a id="1269" href="Relation.Binary.Definitions.html#4378" class="Function">Substitutive</a> <a id="1282" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a> <a id="1286" href="Relation.Binary.Consequences.html#1014" class="Bound">p</a> <a id="1288" class="Symbol">→</a> <a id="1290" href="Relation.Binary.Consequences.html#1004" class="Bound">R</a> <a id="1292" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="1302" href="Relation.Binary.Consequences.html#988" class="Bound Operator">_∼_</a>
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<a id="1308" href="Relation.Binary.Consequences.html#1255" class="Function">subst⇒resp₂</a> <a id="1320" href="Relation.Binary.Consequences.html#1320" class="Bound">subst</a> <a id="1326" class="Symbol">=</a> <a id="1328" href="Relation.Binary.Consequences.html#1143" class="Function">subst⇒respʳ</a> <a id="1340" href="Relation.Binary.Consequences.html#1320" class="Bound">subst</a> <a id="1346" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1348" href="Relation.Binary.Consequences.html#1026" class="Function">subst⇒respˡ</a> <a id="1360" href="Relation.Binary.Consequences.html#1320" class="Bound">subst</a>
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<a id="1367" class="Keyword">module</a> <a id="1374" href="Relation.Binary.Consequences.html#1374" class="Module">_</a> <a id="1376" class="Symbol">{</a><a id="1377" href="Relation.Binary.Consequences.html#1377" class="Bound Operator">_∼_</a> <a id="1381" class="Symbol">:</a> <a id="1383" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1387" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1389" href="Relation.Binary.Consequences.html#836" class="Generalizable">ℓ</a><a id="1390" class="Symbol">}</a> <a id="1392" class="Symbol">{</a><a id="1393" href="Relation.Binary.Consequences.html#1393" class="Bound">P</a> <a id="1395" class="Symbol">:</a> <a id="1397" href="Relation.Unary.html#1110" class="Function">Pred</a> <a id="1402" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1404" href="Relation.Binary.Consequences.html#850" class="Generalizable">p</a><a id="1405" class="Symbol">}</a> <a id="1407" class="Keyword">where</a>
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<a id="1416" href="Relation.Binary.Consequences.html#1416" class="Function">resp⇒¬-resp</a> <a id="1428" class="Symbol">:</a> <a id="1430" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1440" href="Relation.Binary.Consequences.html#1377" class="Bound Operator">_∼_</a> <a id="1444" class="Symbol">→</a> <a id="1446" href="Relation.Binary.Consequences.html#1393" class="Bound">P</a> <a id="1448" href="Relation.Binary.Definitions.html#3602" class="Function Operator">Respects</a> <a id="1457" href="Relation.Binary.Consequences.html#1377" class="Bound Operator">_∼_</a> <a id="1461" class="Symbol">→</a> <a id="1463" class="Symbol">(</a><a id="1464" href="Relation.Unary.html#4287" class="Function">∁</a> <a id="1466" href="Relation.Binary.Consequences.html#1393" class="Bound">P</a><a id="1467" class="Symbol">)</a> <a id="1469" href="Relation.Binary.Definitions.html#3602" class="Function Operator">Respects</a> <a id="1478" href="Relation.Binary.Consequences.html#1377" class="Bound Operator">_∼_</a>
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<a id="1484" href="Relation.Binary.Consequences.html#1416" class="Function">resp⇒¬-resp</a> <a id="1496" href="Relation.Binary.Consequences.html#1496" class="Bound">sym</a> <a id="1500" href="Relation.Binary.Consequences.html#1500" class="Bound">resp</a> <a id="1505" href="Relation.Binary.Consequences.html#1505" class="Bound">x∼y</a> <a id="1509" href="Relation.Binary.Consequences.html#1509" class="Bound">¬Px</a> <a id="1513" href="Relation.Binary.Consequences.html#1513" class="Bound">Py</a> <a id="1516" class="Symbol">=</a> <a id="1518" href="Relation.Binary.Consequences.html#1509" class="Bound">¬Px</a> <a id="1522" class="Symbol">(</a><a id="1523" href="Relation.Binary.Consequences.html#1500" class="Bound">resp</a> <a id="1528" class="Symbol">(</a><a id="1529" href="Relation.Binary.Consequences.html#1496" class="Bound">sym</a> <a id="1533" href="Relation.Binary.Consequences.html#1505" class="Bound">x∼y</a><a id="1536" class="Symbol">)</a> <a id="1538" href="Relation.Binary.Consequences.html#1513" class="Bound">Py</a><a id="1540" class="Symbol">)</a>
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<a id="1543" class="Comment">------------------------------------------------------------------------</a>
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<a id="1616" class="Comment">-- Proofs for non-strict orders</a>
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<a id="1649" class="Keyword">module</a> <a id="1656" href="Relation.Binary.Consequences.html#1656" class="Module">_</a> <a id="1658" class="Symbol">{</a><a id="1659" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1663" class="Symbol">:</a> <a id="1665" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1669" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1671" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="1673" class="Symbol">}</a> <a id="1675" class="Symbol">{</a><a id="1676" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="1680" class="Symbol">:</a> <a id="1682" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1686" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="1688" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="1690" class="Symbol">}</a> <a id="1692" class="Keyword">where</a>
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<a id="1701" href="Relation.Binary.Consequences.html#1701" class="Function">total⇒refl</a> <a id="1712" class="Symbol">:</a> <a id="1714" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="1718" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="1728" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1732" class="Symbol">→</a> <a id="1734" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1744" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1748" class="Symbol">→</a>
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<a id="1765" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="1771" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="1775" class="Symbol">→</a> <a id="1777" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1781" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1783" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a>
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<a id="1789" href="Relation.Binary.Consequences.html#1701" class="Function">total⇒refl</a> <a id="1800" class="Symbol">(</a><a id="1801" href="Relation.Binary.Consequences.html#1801" class="Bound">respʳ</a> <a id="1807" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1809" href="Relation.Binary.Consequences.html#1809" class="Bound">respˡ</a><a id="1814" class="Symbol">)</a> <a id="1816" href="Relation.Binary.Consequences.html#1816" class="Bound">sym</a> <a id="1820" href="Relation.Binary.Consequences.html#1820" class="Bound">total</a> <a id="1826" class="Symbol">{</a><a id="1827" href="Relation.Binary.Consequences.html#1827" class="Bound">x</a><a id="1828" class="Symbol">}</a> <a id="1830" class="Symbol">{</a><a id="1831" href="Relation.Binary.Consequences.html#1831" class="Bound">y</a><a id="1832" class="Symbol">}</a> <a id="1834" href="Relation.Binary.Consequences.html#1834" class="Bound">x≈y</a> <a id="1838" class="Keyword">with</a> <a id="1843" href="Relation.Binary.Consequences.html#1820" class="Bound">total</a> <a id="1849" href="Relation.Binary.Consequences.html#1827" class="Bound">x</a> <a id="1851" href="Relation.Binary.Consequences.html#1831" class="Bound">y</a>
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<a id="1855" class="Symbol">...</a> <a id="1859" class="Symbol">|</a> <a id="1861" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="1866" href="Relation.Binary.Consequences.html#1866" class="Bound">x∼y</a> <a id="1870" class="Symbol">=</a> <a id="1872" href="Relation.Binary.Consequences.html#1866" class="Bound">x∼y</a>
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<a id="1878" class="Symbol">...</a> <a id="1882" class="Symbol">|</a> <a id="1884" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="1889" href="Relation.Binary.Consequences.html#1889" class="Bound">y∼x</a> <a id="1893" class="Symbol">=</a> <a id="1895" class="Bound">respʳ</a> <a id="1901" class="Bound">x≈y</a> <a id="1905" class="Symbol">(</a><a id="1906" class="Bound">respˡ</a> <a id="1912" class="Symbol">(</a><a id="1913" class="Bound">sym</a> <a id="1917" class="Bound">x≈y</a><a id="1920" class="Symbol">)</a> <a id="1922" href="Relation.Binary.Consequences.html#1889" class="Bound">y∼x</a><a id="1925" class="Symbol">)</a>
|
|||
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<a id="1930" href="Relation.Binary.Consequences.html#1930" class="Function">total∧dec⇒dec</a> <a id="1944" class="Symbol">:</a> <a id="1946" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1950" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1952" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="1956" class="Symbol">→</a> <a id="1958" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1972" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="1976" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="1980" class="Symbol">→</a>
|
|||
|
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<a id="2000" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="2006" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a> <a id="2010" class="Symbol">→</a> <a id="2012" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="2022" href="Relation.Binary.Consequences.html#1659" class="Bound Operator">_≈_</a> <a id="2026" class="Symbol">→</a> <a id="2028" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="2038" href="Relation.Binary.Consequences.html#1676" class="Bound Operator">_≤_</a>
|
|||
|
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<a id="2044" href="Relation.Binary.Consequences.html#1930" class="Function">total∧dec⇒dec</a> <a id="2058" href="Relation.Binary.Consequences.html#2058" class="Bound">refl</a> <a id="2063" href="Relation.Binary.Consequences.html#2063" class="Bound">antisym</a> <a id="2071" href="Relation.Binary.Consequences.html#2071" class="Bound">total</a> <a id="2077" href="Relation.Binary.Consequences.html#2077" class="Bound Operator">_≟_</a> <a id="2081" href="Relation.Binary.Consequences.html#2081" class="Bound">x</a> <a id="2083" href="Relation.Binary.Consequences.html#2083" class="Bound">y</a> <a id="2085" class="Keyword">with</a> <a id="2090" href="Relation.Binary.Consequences.html#2071" class="Bound">total</a> <a id="2096" href="Relation.Binary.Consequences.html#2081" class="Bound">x</a> <a id="2098" href="Relation.Binary.Consequences.html#2083" class="Bound">y</a>
|
|||
|
|
<a id="2102" class="Symbol">...</a> <a id="2106" class="Symbol">|</a> <a id="2108" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="2113" href="Relation.Binary.Consequences.html#2113" class="Bound">x≤y</a> <a id="2117" class="Symbol">=</a> <a id="2119" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="2123" href="Relation.Binary.Consequences.html#2113" class="Bound">x≤y</a>
|
|||
|
|
<a id="2129" class="Symbol">...</a> <a id="2133" class="Symbol">|</a> <a id="2135" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="2140" href="Relation.Binary.Consequences.html#2140" class="Bound">y≤x</a> <a id="2144" class="Symbol">=</a> <a id="2146" href="Relation.Nullary.Decidable.Core.html#3645" class="Function">map′</a> <a id="2151" class="Bound">refl</a> <a id="2156" class="Symbol">(</a><a id="2157" href="Function.Base.html#1563" class="Function">flip</a> <a id="2162" class="Bound">antisym</a> <a id="2170" href="Relation.Binary.Consequences.html#2140" class="Bound">y≤x</a><a id="2173" class="Symbol">)</a> <a id="2175" class="Symbol">(</a><a id="2176" class="Bound">x</a> <a id="2178" class="Bound Operator">≟</a> <a id="2180" class="Bound">y</a><a id="2181" class="Symbol">)</a>
|
|||
|
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|
|
<a id="2184" class="Keyword">module</a> <a id="2191" href="Relation.Binary.Consequences.html#2191" class="Module">_</a> <a id="2193" class="Symbol">(</a><a id="2194" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2197" class="Symbol">:</a> <a id="2199" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2203" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="2205" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="2207" class="Symbol">)</a> <a id="2209" class="Symbol">(</a><a id="2210" href="Relation.Binary.Consequences.html#2210" class="Bound">≈₂</a> <a id="2213" class="Symbol">:</a> <a id="2215" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2219" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="2221" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="2223" class="Symbol">)</a> <a id="2225" class="Symbol">{</a><a id="2226" href="Relation.Binary.Consequences.html#2226" class="Bound">≤₁</a> <a id="2229" class="Symbol">:</a> <a id="2231" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2235" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="2237" href="Relation.Binary.Consequences.html#844" class="Generalizable">ℓ₃</a><a id="2239" class="Symbol">}</a> <a id="2241" class="Symbol">{</a><a id="2242" href="Relation.Binary.Consequences.html#2242" class="Bound">≤₂</a> <a id="2245" class="Symbol">:</a> <a id="2247" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2251" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="2253" href="Relation.Binary.Consequences.html#847" class="Generalizable">ℓ₄</a><a id="2255" class="Symbol">}</a> <a id="2257" class="Keyword">where</a>
|
|||
|
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|
|||
|
|
<a id="2266" href="Relation.Binary.Consequences.html#2266" class="Function">mono⇒cong</a> <a id="2276" class="Symbol">:</a> <a id="2278" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="2288" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2291" class="Symbol">→</a> <a id="2293" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2296" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="2298" href="Relation.Binary.Consequences.html#2226" class="Bound">≤₁</a> <a id="2301" class="Symbol">→</a> <a id="2303" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="2317" href="Relation.Binary.Consequences.html#2210" class="Bound">≈₂</a> <a id="2320" href="Relation.Binary.Consequences.html#2242" class="Bound">≤₂</a> <a id="2323" class="Symbol">→</a>
|
|||
|
|
<a id="2339" class="Symbol">∀</a> <a id="2341" class="Symbol">{</a><a id="2342" href="Relation.Binary.Consequences.html#2342" class="Bound">f</a><a id="2343" class="Symbol">}</a> <a id="2345" class="Symbol">→</a> <a id="2347" href="Relation.Binary.Consequences.html#2342" class="Bound">f</a> <a id="2349" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2359" href="Relation.Binary.Consequences.html#2226" class="Bound">≤₁</a> <a id="2362" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2364" href="Relation.Binary.Consequences.html#2242" class="Bound">≤₂</a> <a id="2367" class="Symbol">→</a> <a id="2369" href="Relation.Binary.Consequences.html#2342" class="Bound">f</a> <a id="2371" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2381" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2384" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2386" href="Relation.Binary.Consequences.html#2210" class="Bound">≈₂</a>
|
|||
|
|
<a id="2391" href="Relation.Binary.Consequences.html#2266" class="Function">mono⇒cong</a> <a id="2401" href="Relation.Binary.Consequences.html#2401" class="Bound">sym</a> <a id="2405" href="Relation.Binary.Consequences.html#2405" class="Bound">reflexive</a> <a id="2415" href="Relation.Binary.Consequences.html#2415" class="Bound">antisym</a> <a id="2423" href="Relation.Binary.Consequences.html#2423" class="Bound">mono</a> <a id="2428" href="Relation.Binary.Consequences.html#2428" class="Bound">x≈y</a> <a id="2432" class="Symbol">=</a> <a id="2434" href="Relation.Binary.Consequences.html#2415" class="Bound">antisym</a>
|
|||
|
|
<a id="2446" class="Symbol">(</a><a id="2447" href="Relation.Binary.Consequences.html#2423" class="Bound">mono</a> <a id="2452" class="Symbol">(</a><a id="2453" href="Relation.Binary.Consequences.html#2405" class="Bound">reflexive</a> <a id="2463" href="Relation.Binary.Consequences.html#2428" class="Bound">x≈y</a><a id="2466" class="Symbol">))</a>
|
|||
|
|
<a id="2473" class="Symbol">(</a><a id="2474" href="Relation.Binary.Consequences.html#2423" class="Bound">mono</a> <a id="2479" class="Symbol">(</a><a id="2480" href="Relation.Binary.Consequences.html#2405" class="Bound">reflexive</a> <a id="2490" class="Symbol">(</a><a id="2491" href="Relation.Binary.Consequences.html#2401" class="Bound">sym</a> <a id="2495" href="Relation.Binary.Consequences.html#2428" class="Bound">x≈y</a><a id="2498" class="Symbol">)))</a>
|
|||
|
|
|
|||
|
|
<a id="2505" href="Relation.Binary.Consequences.html#2505" class="Function">antimono⇒cong</a> <a id="2519" class="Symbol">:</a> <a id="2521" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="2531" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2534" class="Symbol">→</a> <a id="2536" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2539" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="2541" href="Relation.Binary.Consequences.html#2226" class="Bound">≤₁</a> <a id="2544" class="Symbol">→</a> <a id="2546" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="2560" href="Relation.Binary.Consequences.html#2210" class="Bound">≈₂</a> <a id="2563" href="Relation.Binary.Consequences.html#2242" class="Bound">≤₂</a> <a id="2566" class="Symbol">→</a>
|
|||
|
|
<a id="2586" class="Symbol">∀</a> <a id="2588" class="Symbol">{</a><a id="2589" href="Relation.Binary.Consequences.html#2589" class="Bound">f</a><a id="2590" class="Symbol">}</a> <a id="2592" class="Symbol">→</a> <a id="2594" href="Relation.Binary.Consequences.html#2589" class="Bound">f</a> <a id="2596" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2606" href="Relation.Binary.Consequences.html#2226" class="Bound">≤₁</a> <a id="2609" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2611" class="Symbol">(</a><a id="2612" href="Function.Base.html#1563" class="Function">flip</a> <a id="2617" href="Relation.Binary.Consequences.html#2242" class="Bound">≤₂</a><a id="2619" class="Symbol">)</a> <a id="2621" class="Symbol">→</a> <a id="2623" href="Relation.Binary.Consequences.html#2589" class="Bound">f</a> <a id="2625" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2635" href="Relation.Binary.Consequences.html#2194" class="Bound">≈₁</a> <a id="2638" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2640" href="Relation.Binary.Consequences.html#2210" class="Bound">≈₂</a>
|
|||
|
|
<a id="2645" href="Relation.Binary.Consequences.html#2505" class="Function">antimono⇒cong</a> <a id="2659" href="Relation.Binary.Consequences.html#2659" class="Bound">sym</a> <a id="2663" href="Relation.Binary.Consequences.html#2663" class="Bound">reflexive</a> <a id="2673" href="Relation.Binary.Consequences.html#2673" class="Bound">antisym</a> <a id="2681" href="Relation.Binary.Consequences.html#2681" class="Bound">antimono</a> <a id="2690" href="Relation.Binary.Consequences.html#2690" class="Bound">p≈q</a> <a id="2694" class="Symbol">=</a> <a id="2696" href="Relation.Binary.Consequences.html#2673" class="Bound">antisym</a>
|
|||
|
|
<a id="2708" class="Symbol">(</a><a id="2709" href="Relation.Binary.Consequences.html#2681" class="Bound">antimono</a> <a id="2718" class="Symbol">(</a><a id="2719" href="Relation.Binary.Consequences.html#2663" class="Bound">reflexive</a> <a id="2729" class="Symbol">(</a><a id="2730" href="Relation.Binary.Consequences.html#2659" class="Bound">sym</a> <a id="2734" href="Relation.Binary.Consequences.html#2690" class="Bound">p≈q</a><a id="2737" class="Symbol">)))</a>
|
|||
|
|
<a id="2745" class="Symbol">(</a><a id="2746" href="Relation.Binary.Consequences.html#2681" class="Bound">antimono</a> <a id="2755" class="Symbol">(</a><a id="2756" href="Relation.Binary.Consequences.html#2663" class="Bound">reflexive</a> <a id="2766" href="Relation.Binary.Consequences.html#2690" class="Bound">p≈q</a><a id="2769" class="Symbol">))</a>
|
|||
|
|
|
|||
|
|
<a id="2773" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="2846" class="Comment">-- Proofs for strict orders</a>
|
|||
|
|
|
|||
|
|
<a id="2875" class="Keyword">module</a> <a id="2882" href="Relation.Binary.Consequences.html#2882" class="Module">_</a> <a id="2884" class="Symbol">{</a><a id="2885" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="2889" class="Symbol">:</a> <a id="2891" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2895" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="2897" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="2899" class="Symbol">}</a> <a id="2901" class="Symbol">{</a><a id="2902" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="2906" class="Symbol">:</a> <a id="2908" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2912" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="2914" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="2916" class="Symbol">}</a> <a id="2918" class="Keyword">where</a>
|
|||
|
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|
|||
|
|
<a id="2927" href="Relation.Binary.Consequences.html#2927" class="Function">trans∧irr⇒asym</a> <a id="2942" class="Symbol">:</a> <a id="2944" href="Relation.Binary.Definitions.html#1348" class="Function">Reflexive</a> <a id="2954" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="2958" class="Symbol">→</a> <a id="2960" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2971" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="2975" class="Symbol">→</a>
|
|||
|
|
<a id="2996" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="3008" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3012" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3016" class="Symbol">→</a> <a id="3018" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3029" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
|
<a id="3035" href="Relation.Binary.Consequences.html#2927" class="Function">trans∧irr⇒asym</a> <a id="3050" href="Relation.Binary.Consequences.html#3050" class="Bound">refl</a> <a id="3055" href="Relation.Binary.Consequences.html#3055" class="Bound">trans</a> <a id="3061" href="Relation.Binary.Consequences.html#3061" class="Bound">irrefl</a> <a id="3068" href="Relation.Binary.Consequences.html#3068" class="Bound">x<y</a> <a id="3072" href="Relation.Binary.Consequences.html#3072" class="Bound">y<x</a> <a id="3076" class="Symbol">=</a>
|
|||
|
|
<a id="3082" href="Relation.Binary.Consequences.html#3061" class="Bound">irrefl</a> <a id="3089" href="Relation.Binary.Consequences.html#3050" class="Bound">refl</a> <a id="3094" class="Symbol">(</a><a id="3095" href="Relation.Binary.Consequences.html#3055" class="Bound">trans</a> <a id="3101" href="Relation.Binary.Consequences.html#3068" class="Bound">x<y</a> <a id="3105" href="Relation.Binary.Consequences.html#3072" class="Bound">y<x</a><a id="3108" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="3113" href="Relation.Binary.Consequences.html#3113" class="Function">irr∧antisym⇒asym</a> <a id="3130" class="Symbol">:</a> <a id="3132" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="3144" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3148" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3152" class="Symbol">→</a> <a id="3154" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="3168" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3172" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3176" class="Symbol">→</a>
|
|||
|
|
<a id="3199" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3210" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
|
<a id="3216" href="Relation.Binary.Consequences.html#3113" class="Function">irr∧antisym⇒asym</a> <a id="3233" href="Relation.Binary.Consequences.html#3233" class="Bound">irrefl</a> <a id="3240" href="Relation.Binary.Consequences.html#3240" class="Bound">antisym</a> <a id="3248" href="Relation.Binary.Consequences.html#3248" class="Bound">x<y</a> <a id="3252" href="Relation.Binary.Consequences.html#3252" class="Bound">y<x</a> <a id="3256" class="Symbol">=</a>
|
|||
|
|
<a id="3262" href="Relation.Binary.Consequences.html#3233" class="Bound">irrefl</a> <a id="3269" class="Symbol">(</a><a id="3270" href="Relation.Binary.Consequences.html#3240" class="Bound">antisym</a> <a id="3278" href="Relation.Binary.Consequences.html#3248" class="Bound">x<y</a> <a id="3282" href="Relation.Binary.Consequences.html#3252" class="Bound">y<x</a><a id="3285" class="Symbol">)</a> <a id="3287" href="Relation.Binary.Consequences.html#3248" class="Bound">x<y</a>
|
|||
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|
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|
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<a id="3294" href="Relation.Binary.Consequences.html#3294" class="Function">asym⇒antisym</a> <a id="3307" class="Symbol">:</a> <a id="3309" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3320" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3324" class="Symbol">→</a> <a id="3326" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="3340" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3344" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
|
<a id="3350" href="Relation.Binary.Consequences.html#3294" class="Function">asym⇒antisym</a> <a id="3363" href="Relation.Binary.Consequences.html#3363" class="Bound">asym</a> <a id="3368" href="Relation.Binary.Consequences.html#3368" class="Bound">x<y</a> <a id="3372" href="Relation.Binary.Consequences.html#3372" class="Bound">y<x</a> <a id="3376" class="Symbol">=</a> <a id="3378" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a> <a id="3385" class="Symbol">(</a><a id="3386" href="Relation.Binary.Consequences.html#3363" class="Bound">asym</a> <a id="3391" href="Relation.Binary.Consequences.html#3368" class="Bound">x<y</a> <a id="3395" href="Relation.Binary.Consequences.html#3372" class="Bound">y<x</a><a id="3398" class="Symbol">)</a>
|
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<a id="3403" href="Relation.Binary.Consequences.html#3403" class="Function">asym⇒irr</a> <a id="3412" class="Symbol">:</a> <a id="3414" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3418" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="3428" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3432" class="Symbol">→</a> <a id="3434" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="3444" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3448" class="Symbol">→</a>
|
|||
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<a id="3463" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3474" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3478" class="Symbol">→</a> <a id="3480" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="3492" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3496" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
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<a id="3502" href="Relation.Binary.Consequences.html#3403" class="Function">asym⇒irr</a> <a id="3511" class="Symbol">(</a><a id="3512" href="Relation.Binary.Consequences.html#3512" class="Bound">respʳ</a> <a id="3518" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3520" href="Relation.Binary.Consequences.html#3520" class="Bound">respˡ</a><a id="3525" class="Symbol">)</a> <a id="3527" href="Relation.Binary.Consequences.html#3527" class="Bound">sym</a> <a id="3531" href="Relation.Binary.Consequences.html#3531" class="Bound">asym</a> <a id="3536" class="Symbol">{</a><a id="3537" href="Relation.Binary.Consequences.html#3537" class="Bound">x</a><a id="3538" class="Symbol">}</a> <a id="3540" class="Symbol">{</a><a id="3541" href="Relation.Binary.Consequences.html#3541" class="Bound">y</a><a id="3542" class="Symbol">}</a> <a id="3544" href="Relation.Binary.Consequences.html#3544" class="Bound">x≈y</a> <a id="3548" href="Relation.Binary.Consequences.html#3548" class="Bound">x<y</a> <a id="3552" class="Symbol">=</a>
|
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|
<a id="3558" href="Relation.Binary.Consequences.html#3531" class="Bound">asym</a> <a id="3563" href="Relation.Binary.Consequences.html#3548" class="Bound">x<y</a> <a id="3567" class="Symbol">(</a><a id="3568" href="Relation.Binary.Consequences.html#3512" class="Bound">respʳ</a> <a id="3574" class="Symbol">(</a><a id="3575" href="Relation.Binary.Consequences.html#3527" class="Bound">sym</a> <a id="3579" href="Relation.Binary.Consequences.html#3544" class="Bound">x≈y</a><a id="3582" class="Symbol">)</a> <a id="3584" class="Symbol">(</a><a id="3585" href="Relation.Binary.Consequences.html#3520" class="Bound">respˡ</a> <a id="3591" href="Relation.Binary.Consequences.html#3544" class="Bound">x≈y</a> <a id="3595" href="Relation.Binary.Consequences.html#3548" class="Bound">x<y</a><a id="3598" class="Symbol">))</a>
|
|||
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<a id="3604" href="Relation.Binary.Consequences.html#3604" class="Function">tri⇒asym</a> <a id="3613" class="Symbol">:</a> <a id="3615" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="3628" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3632" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3636" class="Symbol">→</a> <a id="3638" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3649" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
|
<a id="3655" href="Relation.Binary.Consequences.html#3604" class="Function">tri⇒asym</a> <a id="3664" href="Relation.Binary.Consequences.html#3664" class="Bound">tri</a> <a id="3668" class="Symbol">{</a><a id="3669" href="Relation.Binary.Consequences.html#3669" class="Bound">x</a><a id="3670" class="Symbol">}</a> <a id="3672" class="Symbol">{</a><a id="3673" href="Relation.Binary.Consequences.html#3673" class="Bound">y</a><a id="3674" class="Symbol">}</a> <a id="3676" href="Relation.Binary.Consequences.html#3676" class="Bound">x<y</a> <a id="3680" href="Relation.Binary.Consequences.html#3680" class="Bound">x>y</a> <a id="3684" class="Keyword">with</a> <a id="3689" href="Relation.Binary.Consequences.html#3664" class="Bound">tri</a> <a id="3693" href="Relation.Binary.Consequences.html#3669" class="Bound">x</a> <a id="3695" href="Relation.Binary.Consequences.html#3673" class="Bound">y</a>
|
|||
|
|
<a id="3699" class="Symbol">...</a> <a id="3703" class="Symbol">|</a> <a id="3705" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="3710" class="Symbol">_</a> <a id="3714" class="Symbol">_</a> <a id="3716" href="Relation.Binary.Consequences.html#3716" class="Bound">x≯y</a> <a id="3720" class="Symbol">=</a> <a id="3722" href="Relation.Binary.Consequences.html#3716" class="Bound">x≯y</a> <a id="3726" class="Bound">x>y</a>
|
|||
|
|
<a id="3732" class="Symbol">...</a> <a id="3736" class="Symbol">|</a> <a id="3738" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="3743" class="Symbol">_</a> <a id="3747" class="Symbol">_</a> <a id="3749" href="Relation.Binary.Consequences.html#3749" class="Bound">x≯y</a> <a id="3753" class="Symbol">=</a> <a id="3755" href="Relation.Binary.Consequences.html#3749" class="Bound">x≯y</a> <a id="3759" class="Bound">x>y</a>
|
|||
|
|
<a id="3765" class="Symbol">...</a> <a id="3769" class="Symbol">|</a> <a id="3771" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="3776" href="Relation.Binary.Consequences.html#3776" class="Bound">x≮y</a> <a id="3780" class="Symbol">_</a> <a id="3782" class="Symbol">_</a> <a id="3786" class="Symbol">=</a> <a id="3788" href="Relation.Binary.Consequences.html#3776" class="Bound">x≮y</a> <a id="3792" class="Bound">x<y</a>
|
|||
|
|
|
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|
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<a id="3799" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="3807" class="Symbol">:</a> <a id="3809" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="3822" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3826" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="3830" class="Symbol">→</a> <a id="3832" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="3844" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="3848" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
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<a id="3854" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="3862" href="Relation.Binary.Consequences.html#3862" class="Bound">compare</a> <a id="3870" class="Symbol">{</a><a id="3871" href="Relation.Binary.Consequences.html#3871" class="Bound">x</a><a id="3872" class="Symbol">}</a> <a id="3874" class="Symbol">{</a><a id="3875" href="Relation.Binary.Consequences.html#3875" class="Bound">y</a><a id="3876" class="Symbol">}</a> <a id="3878" href="Relation.Binary.Consequences.html#3878" class="Bound">x≈y</a> <a id="3882" href="Relation.Binary.Consequences.html#3882" class="Bound">x<y</a> <a id="3886" class="Keyword">with</a> <a id="3891" href="Relation.Binary.Consequences.html#3862" class="Bound">compare</a> <a id="3899" href="Relation.Binary.Consequences.html#3871" class="Bound">x</a> <a id="3901" href="Relation.Binary.Consequences.html#3875" class="Bound">y</a>
|
|||
|
|
<a id="3905" class="Symbol">...</a> <a id="3909" class="Symbol">|</a> <a id="3911" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="3916" class="Symbol">_</a> <a id="3920" href="Relation.Binary.Consequences.html#3920" class="Bound">x≉y</a> <a id="3924" href="Relation.Binary.Consequences.html#3924" class="Bound">y≮x</a> <a id="3928" class="Symbol">=</a> <a id="3930" href="Relation.Binary.Consequences.html#3920" class="Bound">x≉y</a> <a id="3934" class="Bound">x≈y</a>
|
|||
|
|
<a id="3940" class="Symbol">...</a> <a id="3944" class="Symbol">|</a> <a id="3946" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="3951" href="Relation.Binary.Consequences.html#3951" class="Bound">x≮y</a> <a id="3955" href="Relation.Binary.Consequences.html#3955" class="Bound">x≉y</a> <a id="3959" href="Relation.Binary.Consequences.html#3959" class="Bound">y<x</a> <a id="3963" class="Symbol">=</a> <a id="3965" href="Relation.Binary.Consequences.html#3955" class="Bound">x≉y</a> <a id="3969" class="Bound">x≈y</a>
|
|||
|
|
<a id="3975" class="Symbol">...</a> <a id="3979" class="Symbol">|</a> <a id="3981" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="3986" href="Relation.Binary.Consequences.html#3986" class="Bound">x≮y</a> <a id="3990" class="Symbol">_</a> <a id="3994" href="Relation.Binary.Consequences.html#3994" class="Bound">y≮x</a> <a id="3998" class="Symbol">=</a> <a id="4000" href="Relation.Binary.Consequences.html#3986" class="Bound">x≮y</a> <a id="4004" class="Bound">x<y</a>
|
|||
|
|
|
|||
|
|
<a id="4011" href="Relation.Binary.Consequences.html#4011" class="Function">tri⇒dec≈</a> <a id="4020" class="Symbol">:</a> <a id="4022" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="4035" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4039" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4043" class="Symbol">→</a> <a id="4045" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="4055" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="4061" href="Relation.Binary.Consequences.html#4011" class="Function">tri⇒dec≈</a> <a id="4070" href="Relation.Binary.Consequences.html#4070" class="Bound">compare</a> <a id="4078" href="Relation.Binary.Consequences.html#4078" class="Bound">x</a> <a id="4080" href="Relation.Binary.Consequences.html#4080" class="Bound">y</a> <a id="4082" class="Keyword">with</a> <a id="4087" href="Relation.Binary.Consequences.html#4070" class="Bound">compare</a> <a id="4095" href="Relation.Binary.Consequences.html#4078" class="Bound">x</a> <a id="4097" href="Relation.Binary.Consequences.html#4080" class="Bound">y</a>
|
|||
|
|
<a id="4101" class="Symbol">...</a> <a id="4105" class="Symbol">|</a> <a id="4107" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="4112" class="Symbol">_</a> <a id="4114" href="Relation.Binary.Consequences.html#4114" class="Bound">x≉y</a> <a id="4118" class="Symbol">_</a> <a id="4120" class="Symbol">=</a> <a id="4122" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="4126" href="Relation.Binary.Consequences.html#4114" class="Bound">x≉y</a>
|
|||
|
|
<a id="4132" class="Symbol">...</a> <a id="4136" class="Symbol">|</a> <a id="4138" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="4143" class="Symbol">_</a> <a id="4145" href="Relation.Binary.Consequences.html#4145" class="Bound">x≈y</a> <a id="4149" class="Symbol">_</a> <a id="4151" class="Symbol">=</a> <a id="4153" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="4157" href="Relation.Binary.Consequences.html#4145" class="Bound">x≈y</a>
|
|||
|
|
<a id="4163" class="Symbol">...</a> <a id="4167" class="Symbol">|</a> <a id="4169" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="4174" class="Symbol">_</a> <a id="4176" href="Relation.Binary.Consequences.html#4176" class="Bound">x≉y</a> <a id="4180" class="Symbol">_</a> <a id="4182" class="Symbol">=</a> <a id="4184" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="4188" href="Relation.Binary.Consequences.html#4176" class="Bound">x≉y</a>
|
|||
|
|
|
|||
|
|
<a id="4195" href="Relation.Binary.Consequences.html#4195" class="Function">tri⇒dec<</a> <a id="4204" class="Symbol">:</a> <a id="4206" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="4219" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4223" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4227" class="Symbol">→</a> <a id="4229" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="4239" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a>
|
|||
|
|
<a id="4245" href="Relation.Binary.Consequences.html#4195" class="Function">tri⇒dec<</a> <a id="4254" href="Relation.Binary.Consequences.html#4254" class="Bound">compare</a> <a id="4262" href="Relation.Binary.Consequences.html#4262" class="Bound">x</a> <a id="4264" href="Relation.Binary.Consequences.html#4264" class="Bound">y</a> <a id="4266" class="Keyword">with</a> <a id="4271" href="Relation.Binary.Consequences.html#4254" class="Bound">compare</a> <a id="4279" href="Relation.Binary.Consequences.html#4262" class="Bound">x</a> <a id="4281" href="Relation.Binary.Consequences.html#4264" class="Bound">y</a>
|
|||
|
|
<a id="4285" class="Symbol">...</a> <a id="4289" class="Symbol">|</a> <a id="4291" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="4296" href="Relation.Binary.Consequences.html#4296" class="Bound">x<y</a> <a id="4300" class="Symbol">_</a> <a id="4302" class="Symbol">_</a> <a id="4304" class="Symbol">=</a> <a id="4306" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="4310" href="Relation.Binary.Consequences.html#4296" class="Bound">x<y</a>
|
|||
|
|
<a id="4316" class="Symbol">...</a> <a id="4320" class="Symbol">|</a> <a id="4322" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="4327" href="Relation.Binary.Consequences.html#4327" class="Bound">x≮y</a> <a id="4331" class="Symbol">_</a> <a id="4333" class="Symbol">_</a> <a id="4335" class="Symbol">=</a> <a id="4337" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="4341" href="Relation.Binary.Consequences.html#4327" class="Bound">x≮y</a>
|
|||
|
|
<a id="4347" class="Symbol">...</a> <a id="4351" class="Symbol">|</a> <a id="4353" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="4358" href="Relation.Binary.Consequences.html#4358" class="Bound">x≮y</a> <a id="4362" class="Symbol">_</a> <a id="4364" class="Symbol">_</a> <a id="4366" class="Symbol">=</a> <a id="4368" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="4372" href="Relation.Binary.Consequences.html#4358" class="Bound">x≮y</a>
|
|||
|
|
|
|||
|
|
<a id="4379" href="Relation.Binary.Consequences.html#4379" class="Function">trans∧tri⇒respʳ</a> <a id="4395" class="Symbol">:</a> <a id="4397" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="4407" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4411" class="Symbol">→</a> <a id="4413" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="4424" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4428" class="Symbol">→</a>
|
|||
|
|
<a id="4450" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="4461" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4465" class="Symbol">→</a> <a id="4467" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="4480" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4484" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4488" class="Symbol">→</a>
|
|||
|
|
<a id="4510" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4514" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="4524" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="4530" href="Relation.Binary.Consequences.html#4379" class="Function">trans∧tri⇒respʳ</a> <a id="4546" href="Relation.Binary.Consequences.html#4546" class="Bound">sym</a> <a id="4550" href="Relation.Binary.Consequences.html#4550" class="Bound">≈-tr</a> <a id="4555" href="Relation.Binary.Consequences.html#4555" class="Bound"><-tr</a> <a id="4560" href="Relation.Binary.Consequences.html#4560" class="Bound">tri</a> <a id="4564" class="Symbol">{</a><a id="4565" href="Relation.Binary.Consequences.html#4565" class="Bound">x</a><a id="4566" class="Symbol">}</a> <a id="4568" class="Symbol">{</a><a id="4569" href="Relation.Binary.Consequences.html#4569" class="Bound">y</a><a id="4570" class="Symbol">}</a> <a id="4572" class="Symbol">{</a><a id="4573" href="Relation.Binary.Consequences.html#4573" class="Bound">z</a><a id="4574" class="Symbol">}</a> <a id="4576" href="Relation.Binary.Consequences.html#4576" class="Bound">y≈z</a> <a id="4580" href="Relation.Binary.Consequences.html#4580" class="Bound">x<y</a> <a id="4584" class="Keyword">with</a> <a id="4589" href="Relation.Binary.Consequences.html#4560" class="Bound">tri</a> <a id="4593" href="Relation.Binary.Consequences.html#4565" class="Bound">x</a> <a id="4595" href="Relation.Binary.Consequences.html#4573" class="Bound">z</a>
|
|||
|
|
<a id="4599" class="Symbol">...</a> <a id="4603" class="Symbol">|</a> <a id="4605" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="4610" href="Relation.Binary.Consequences.html#4610" class="Bound">x<z</a> <a id="4614" class="Symbol">_</a> <a id="4616" class="Symbol">_</a> <a id="4618" class="Symbol">=</a> <a id="4620" href="Relation.Binary.Consequences.html#4610" class="Bound">x<z</a>
|
|||
|
|
<a id="4626" class="Symbol">...</a> <a id="4630" class="Symbol">|</a> <a id="4632" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="4637" class="Symbol">_</a> <a id="4639" href="Relation.Binary.Consequences.html#4639" class="Bound">x≈z</a> <a id="4643" class="Symbol">_</a> <a id="4645" class="Symbol">=</a> <a id="4647" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a> <a id="4654" class="Symbol">(</a><a id="4655" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="4663" class="Bound">tri</a> <a id="4667" class="Symbol">(</a><a id="4668" class="Bound">≈-tr</a> <a id="4673" href="Relation.Binary.Consequences.html#4639" class="Bound">x≈z</a> <a id="4677" class="Symbol">(</a><a id="4678" class="Bound">sym</a> <a id="4682" class="Bound">y≈z</a><a id="4685" class="Symbol">))</a> <a id="4688" class="Bound">x<y</a><a id="4691" class="Symbol">)</a>
|
|||
|
|
<a id="4695" class="Symbol">...</a> <a id="4699" class="Symbol">|</a> <a id="4701" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="4706" class="Symbol">_</a> <a id="4708" class="Symbol">_</a> <a id="4710" href="Relation.Binary.Consequences.html#4710" class="Bound">z<x</a> <a id="4714" class="Symbol">=</a> <a id="4716" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a> <a id="4723" class="Symbol">(</a><a id="4724" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="4732" class="Bound">tri</a> <a id="4736" class="Symbol">(</a><a id="4737" class="Bound">sym</a> <a id="4741" class="Bound">y≈z</a><a id="4744" class="Symbol">)</a> <a id="4746" class="Symbol">(</a><a id="4747" class="Bound"><-tr</a> <a id="4752" href="Relation.Binary.Consequences.html#4710" class="Bound">z<x</a> <a id="4756" class="Bound">x<y</a><a id="4759" class="Symbol">))</a>
|
|||
|
|
|
|||
|
|
<a id="4765" href="Relation.Binary.Consequences.html#4765" class="Function">trans∧tri⇒respˡ</a> <a id="4781" class="Symbol">:</a> <a id="4783" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="4794" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4798" class="Symbol">→</a>
|
|||
|
|
<a id="4820" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="4831" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4835" class="Symbol">→</a> <a id="4837" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="4850" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="4854" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4858" class="Symbol">→</a>
|
|||
|
|
<a id="4880" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="4884" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="4894" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="4900" href="Relation.Binary.Consequences.html#4765" class="Function">trans∧tri⇒respˡ</a> <a id="4916" href="Relation.Binary.Consequences.html#4916" class="Bound">≈-tr</a> <a id="4921" href="Relation.Binary.Consequences.html#4921" class="Bound"><-tr</a> <a id="4926" href="Relation.Binary.Consequences.html#4926" class="Bound">tri</a> <a id="4930" class="Symbol">{</a><a id="4931" href="Relation.Binary.Consequences.html#4931" class="Bound">z</a><a id="4932" class="Symbol">}</a> <a id="4934" class="Symbol">{_}</a> <a id="4938" class="Symbol">{</a><a id="4939" href="Relation.Binary.Consequences.html#4939" class="Bound">y</a><a id="4940" class="Symbol">}</a> <a id="4942" href="Relation.Binary.Consequences.html#4942" class="Bound">x≈y</a> <a id="4946" href="Relation.Binary.Consequences.html#4946" class="Bound">x<z</a> <a id="4950" class="Keyword">with</a> <a id="4955" href="Relation.Binary.Consequences.html#4926" class="Bound">tri</a> <a id="4959" href="Relation.Binary.Consequences.html#4939" class="Bound">y</a> <a id="4961" href="Relation.Binary.Consequences.html#4931" class="Bound">z</a>
|
|||
|
|
<a id="4965" class="Symbol">...</a> <a id="4969" class="Symbol">|</a> <a id="4971" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="4976" href="Relation.Binary.Consequences.html#4976" class="Bound">y<z</a> <a id="4980" class="Symbol">_</a> <a id="4982" class="Symbol">_</a> <a id="4984" class="Symbol">=</a> <a id="4986" href="Relation.Binary.Consequences.html#4976" class="Bound">y<z</a>
|
|||
|
|
<a id="4992" class="Symbol">...</a> <a id="4996" class="Symbol">|</a> <a id="4998" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="5003" class="Symbol">_</a> <a id="5005" href="Relation.Binary.Consequences.html#5005" class="Bound">y≈z</a> <a id="5009" class="Symbol">_</a> <a id="5011" class="Symbol">=</a> <a id="5013" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a> <a id="5020" class="Symbol">(</a><a id="5021" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="5029" class="Bound">tri</a> <a id="5033" class="Symbol">(</a><a id="5034" class="Bound">≈-tr</a> <a id="5039" class="Bound">x≈y</a> <a id="5043" href="Relation.Binary.Consequences.html#5005" class="Bound">y≈z</a><a id="5046" class="Symbol">)</a> <a id="5048" class="Bound">x<z</a><a id="5051" class="Symbol">)</a>
|
|||
|
|
<a id="5055" class="Symbol">...</a> <a id="5059" class="Symbol">|</a> <a id="5061" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="5066" class="Symbol">_</a> <a id="5068" class="Symbol">_</a> <a id="5070" href="Relation.Binary.Consequences.html#5070" class="Bound">z<y</a> <a id="5074" class="Symbol">=</a> <a id="5076" href="Data.Empty.Irrelevant.html#336" class="Function">⊥-elim</a> <a id="5083" class="Symbol">(</a><a id="5084" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a> <a id="5092" class="Bound">tri</a> <a id="5096" class="Bound">x≈y</a> <a id="5100" class="Symbol">(</a><a id="5101" class="Bound"><-tr</a> <a id="5106" class="Bound">x<z</a> <a id="5110" href="Relation.Binary.Consequences.html#5070" class="Bound">z<y</a><a id="5113" class="Symbol">))</a>
|
|||
|
|
|
|||
|
|
<a id="5119" href="Relation.Binary.Consequences.html#5119" class="Function">trans∧tri⇒resp</a> <a id="5134" class="Symbol">:</a> <a id="5136" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="5146" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="5150" class="Symbol">→</a> <a id="5152" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="5163" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="5167" class="Symbol">→</a>
|
|||
|
|
<a id="5188" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="5199" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="5203" class="Symbol">→</a> <a id="5205" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="5218" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a> <a id="5222" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="5226" class="Symbol">→</a>
|
|||
|
|
<a id="5247" href="Relation.Binary.Consequences.html#2902" class="Bound Operator">_<_</a> <a id="5251" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="5261" href="Relation.Binary.Consequences.html#2885" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="5267" href="Relation.Binary.Consequences.html#5119" class="Function">trans∧tri⇒resp</a> <a id="5282" href="Relation.Binary.Consequences.html#5282" class="Bound">sym</a> <a id="5286" href="Relation.Binary.Consequences.html#5286" class="Bound">≈-tr</a> <a id="5291" href="Relation.Binary.Consequences.html#5291" class="Bound"><-tr</a> <a id="5296" href="Relation.Binary.Consequences.html#5296" class="Bound">tri</a> <a id="5300" class="Symbol">=</a>
|
|||
|
|
<a id="5306" href="Relation.Binary.Consequences.html#4379" class="Function">trans∧tri⇒respʳ</a> <a id="5322" href="Relation.Binary.Consequences.html#5282" class="Bound">sym</a> <a id="5326" href="Relation.Binary.Consequences.html#5286" class="Bound">≈-tr</a> <a id="5331" href="Relation.Binary.Consequences.html#5291" class="Bound"><-tr</a> <a id="5336" href="Relation.Binary.Consequences.html#5296" class="Bound">tri</a> <a id="5340" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a>
|
|||
|
|
<a id="5346" href="Relation.Binary.Consequences.html#4765" class="Function">trans∧tri⇒respˡ</a> <a id="5362" href="Relation.Binary.Consequences.html#5286" class="Bound">≈-tr</a> <a id="5367" href="Relation.Binary.Consequences.html#5291" class="Bound"><-tr</a> <a id="5372" href="Relation.Binary.Consequences.html#5296" class="Bound">tri</a>
|
|||
|
|
|
|||
|
|
<a id="5377" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="5450" class="Comment">-- Without Loss of Generality</a>
|
|||
|
|
|
|||
|
|
<a id="5481" class="Keyword">module</a> <a id="5488" href="Relation.Binary.Consequences.html#5488" class="Module">_</a> <a id="5491" class="Symbol">{</a><a id="5492" href="Relation.Binary.Consequences.html#5492" class="Bound Operator">_R_</a> <a id="5496" class="Symbol">:</a> <a id="5498" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="5502" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="5504" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="5506" class="Symbol">}</a> <a id="5508" class="Symbol">{</a><a id="5509" href="Relation.Binary.Consequences.html#5509" class="Bound">Q</a> <a id="5511" class="Symbol">:</a> <a id="5513" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="5517" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="5519" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="5521" class="Symbol">}</a> <a id="5523" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="5532" href="Relation.Binary.Consequences.html#5532" class="Function">wlog</a> <a id="5537" class="Symbol">:</a> <a id="5539" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="5545" href="Relation.Binary.Consequences.html#5492" class="Bound Operator">_R_</a> <a id="5549" class="Symbol">→</a> <a id="5551" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="5561" href="Relation.Binary.Consequences.html#5509" class="Bound">Q</a> <a id="5563" class="Symbol">→</a>
|
|||
|
|
<a id="5574" class="Symbol">(∀</a> <a id="5577" href="Relation.Binary.Consequences.html#5577" class="Bound">a</a> <a id="5579" href="Relation.Binary.Consequences.html#5579" class="Bound">b</a> <a id="5581" class="Symbol">→</a> <a id="5583" href="Relation.Binary.Consequences.html#5577" class="Bound">a</a> <a id="5585" href="Relation.Binary.Consequences.html#5492" class="Bound Operator">R</a> <a id="5587" href="Relation.Binary.Consequences.html#5579" class="Bound">b</a> <a id="5589" class="Symbol">→</a> <a id="5591" href="Relation.Binary.Consequences.html#5509" class="Bound">Q</a> <a id="5593" href="Relation.Binary.Consequences.html#5577" class="Bound">a</a> <a id="5595" href="Relation.Binary.Consequences.html#5579" class="Bound">b</a><a id="5596" class="Symbol">)</a> <a id="5598" class="Symbol">→</a>
|
|||
|
|
<a id="5609" class="Symbol">∀</a> <a id="5611" href="Relation.Binary.Consequences.html#5611" class="Bound">a</a> <a id="5613" href="Relation.Binary.Consequences.html#5613" class="Bound">b</a> <a id="5615" class="Symbol">→</a> <a id="5617" href="Relation.Binary.Consequences.html#5509" class="Bound">Q</a> <a id="5619" href="Relation.Binary.Consequences.html#5611" class="Bound">a</a> <a id="5621" href="Relation.Binary.Consequences.html#5613" class="Bound">b</a>
|
|||
|
|
<a id="5625" href="Relation.Binary.Consequences.html#5532" class="Function">wlog</a> <a id="5630" href="Relation.Binary.Consequences.html#5630" class="Bound">r-total</a> <a id="5638" href="Relation.Binary.Consequences.html#5638" class="Bound">q-sym</a> <a id="5644" href="Relation.Binary.Consequences.html#5644" class="Bound">prf</a> <a id="5648" href="Relation.Binary.Consequences.html#5648" class="Bound">a</a> <a id="5650" href="Relation.Binary.Consequences.html#5650" class="Bound">b</a> <a id="5652" class="Keyword">with</a> <a id="5657" href="Relation.Binary.Consequences.html#5630" class="Bound">r-total</a> <a id="5665" href="Relation.Binary.Consequences.html#5648" class="Bound">a</a> <a id="5667" href="Relation.Binary.Consequences.html#5650" class="Bound">b</a>
|
|||
|
|
<a id="5671" class="Symbol">...</a> <a id="5675" class="Symbol">|</a> <a id="5677" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="5682" href="Relation.Binary.Consequences.html#5682" class="Bound">aRb</a> <a id="5686" class="Symbol">=</a> <a id="5688" class="Bound">prf</a> <a id="5692" class="Bound">a</a> <a id="5694" class="Bound">b</a> <a id="5696" href="Relation.Binary.Consequences.html#5682" class="Bound">aRb</a>
|
|||
|
|
<a id="5702" class="Symbol">...</a> <a id="5706" class="Symbol">|</a> <a id="5708" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="5713" href="Relation.Binary.Consequences.html#5713" class="Bound">bRa</a> <a id="5717" class="Symbol">=</a> <a id="5719" class="Bound">q-sym</a> <a id="5725" class="Symbol">(</a><a id="5726" class="Bound">prf</a> <a id="5730" class="Bound">b</a> <a id="5732" class="Bound">a</a> <a id="5734" href="Relation.Binary.Consequences.html#5713" class="Bound">bRa</a><a id="5737" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="5740" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="5813" class="Comment">-- Other proofs</a>
|
|||
|
|
|
|||
|
|
<a id="5830" class="Keyword">module</a> <a id="5837" href="Relation.Binary.Consequences.html#5837" class="Module">_</a> <a id="5839" class="Symbol">{</a><a id="5840" href="Relation.Binary.Consequences.html#5840" class="Bound">R</a> <a id="5842" class="Symbol">:</a> <a id="5844" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="5848" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="5850" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="5852" href="Relation.Binary.Consequences.html#850" class="Generalizable">p</a><a id="5853" class="Symbol">}</a> <a id="5855" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="5864" href="Relation.Binary.Consequences.html#5864" class="Function">dec⇒weaklyDec</a> <a id="5878" class="Symbol">:</a> <a id="5880" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="5890" href="Relation.Binary.Consequences.html#5840" class="Bound">R</a> <a id="5892" class="Symbol">→</a> <a id="5894" href="Relation.Binary.Definitions.html#4757" class="Function">WeaklyDecidable</a> <a id="5910" href="Relation.Binary.Consequences.html#5840" class="Bound">R</a>
|
|||
|
|
<a id="5914" href="Relation.Binary.Consequences.html#5864" class="Function">dec⇒weaklyDec</a> <a id="5928" href="Relation.Binary.Consequences.html#5928" class="Bound">dec</a> <a id="5932" href="Relation.Binary.Consequences.html#5932" class="Bound">x</a> <a id="5934" href="Relation.Binary.Consequences.html#5934" class="Bound">y</a> <a id="5936" class="Symbol">=</a> <a id="5938" href="Data.Maybe.Base.html#1221" class="Function">decToMaybe</a> <a id="5949" class="Symbol">(</a><a id="5950" href="Relation.Binary.Consequences.html#5928" class="Bound">dec</a> <a id="5954" href="Relation.Binary.Consequences.html#5932" class="Bound">x</a> <a id="5956" href="Relation.Binary.Consequences.html#5934" class="Bound">y</a><a id="5957" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="5962" href="Relation.Binary.Consequences.html#5962" class="Function">dec⇒recomputable</a> <a id="5979" class="Symbol">:</a> <a id="5981" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="5991" href="Relation.Binary.Consequences.html#5840" class="Bound">R</a> <a id="5993" class="Symbol">→</a> <a id="5995" href="Relation.Binary.Definitions.html#5235" class="Function">Recomputable</a> <a id="6008" href="Relation.Binary.Consequences.html#5840" class="Bound">R</a>
|
|||
|
|
<a id="6012" href="Relation.Binary.Consequences.html#5962" class="Function">dec⇒recomputable</a> <a id="6029" href="Relation.Binary.Consequences.html#6029" class="Bound">dec</a> <a id="6033" class="Symbol">{</a><a id="6034" href="Relation.Binary.Consequences.html#6034" class="Bound">a</a><a id="6035" class="Symbol">}</a> <a id="6037" class="Symbol">{</a><a id="6038" href="Relation.Binary.Consequences.html#6038" class="Bound">b</a><a id="6039" class="Symbol">}</a> <a id="6041" class="Symbol">=</a> <a id="6043" href="Relation.Nullary.html#1847" class="Function">recompute</a> <a id="6053" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6055" href="Relation.Binary.Consequences.html#6029" class="Bound">dec</a> <a id="6059" href="Relation.Binary.Consequences.html#6034" class="Bound">a</a> <a id="6061" href="Relation.Binary.Consequences.html#6038" class="Bound">b</a>
|
|||
|
|
|
|||
|
|
<a id="6064" class="Keyword">module</a> <a id="6071" href="Relation.Binary.Consequences.html#6071" class="Module">_</a> <a id="6073" class="Symbol">{</a><a id="6074" href="Relation.Binary.Consequences.html#6074" class="Bound">R</a> <a id="6076" class="Symbol">:</a> <a id="6078" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="6082" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="6084" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="6086" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="6088" class="Symbol">}</a> <a id="6090" class="Symbol">{</a><a id="6091" href="Relation.Binary.Consequences.html#6091" class="Bound">S</a> <a id="6093" class="Symbol">:</a> <a id="6095" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="6099" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="6101" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="6103" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="6105" class="Symbol">}</a> <a id="6107" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="6116" href="Relation.Binary.Consequences.html#6116" class="Function">map-NonEmpty</a> <a id="6129" class="Symbol">:</a> <a id="6131" href="Relation.Binary.Consequences.html#6074" class="Bound">R</a> <a id="6133" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="6135" href="Relation.Binary.Consequences.html#6091" class="Bound">S</a> <a id="6137" class="Symbol">→</a> <a id="6139" href="Relation.Binary.Definitions.html#5496" class="Record">NonEmpty</a> <a id="6148" href="Relation.Binary.Consequences.html#6074" class="Bound">R</a> <a id="6150" class="Symbol">→</a> <a id="6152" href="Relation.Binary.Definitions.html#5496" class="Record">NonEmpty</a> <a id="6161" href="Relation.Binary.Consequences.html#6091" class="Bound">S</a>
|
|||
|
|
<a id="6165" href="Relation.Binary.Consequences.html#6116" class="Function">map-NonEmpty</a> <a id="6178" href="Relation.Binary.Consequences.html#6178" class="Bound">f</a> <a id="6180" href="Relation.Binary.Consequences.html#6180" class="Bound">x</a> <a id="6182" class="Symbol">=</a> <a id="6184" href="Relation.Binary.Definitions.html#5599" class="InductiveConstructor">nonEmpty</a> <a id="6193" class="Symbol">(</a><a id="6194" href="Relation.Binary.Consequences.html#6178" class="Bound">f</a> <a id="6196" class="Symbol">(</a><a id="6197" href="Relation.Binary.Definitions.html#5648" class="Field">NonEmpty.proof</a> <a id="6212" href="Relation.Binary.Consequences.html#6180" class="Bound">x</a><a id="6213" class="Symbol">))</a>
|
|||
|
|
|
|||
|
|
<a id="6217" class="Keyword">module</a> <a id="6224" href="Relation.Binary.Consequences.html#6224" class="Module">_</a> <a id="6226" class="Symbol">{</a><a id="6227" href="Relation.Binary.Consequences.html#6227" class="Bound">R</a> <a id="6229" class="Symbol">:</a> <a id="6231" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="6235" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="6237" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="6239" href="Relation.Binary.Consequences.html#838" class="Generalizable">ℓ₁</a><a id="6241" class="Symbol">}</a> <a id="6243" class="Symbol">{</a><a id="6244" href="Relation.Binary.Consequences.html#6244" class="Bound">S</a> <a id="6246" class="Symbol">:</a> <a id="6248" href="Relation.Binary.Core.html#775" class="Function">REL</a> <a id="6252" href="Relation.Binary.Consequences.html#866" class="Generalizable">B</a> <a id="6254" href="Relation.Binary.Consequences.html#864" class="Generalizable">A</a> <a id="6256" href="Relation.Binary.Consequences.html#841" class="Generalizable">ℓ₂</a><a id="6258" class="Symbol">}</a> <a id="6260" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="6269" href="Relation.Binary.Consequences.html#6269" class="Function">flip-Connex</a> <a id="6281" class="Symbol">:</a> <a id="6283" href="Relation.Binary.Definitions.html#2502" class="Function">Connex</a> <a id="6290" href="Relation.Binary.Consequences.html#6227" class="Bound">R</a> <a id="6292" href="Relation.Binary.Consequences.html#6244" class="Bound">S</a> <a id="6294" class="Symbol">→</a> <a id="6296" href="Relation.Binary.Definitions.html#2502" class="Function">Connex</a> <a id="6303" href="Relation.Binary.Consequences.html#6244" class="Bound">S</a> <a id="6305" href="Relation.Binary.Consequences.html#6227" class="Bound">R</a>
|
|||
|
|
<a id="6309" href="Relation.Binary.Consequences.html#6269" class="Function">flip-Connex</a> <a id="6321" href="Relation.Binary.Consequences.html#6321" class="Bound">f</a> <a id="6323" href="Relation.Binary.Consequences.html#6323" class="Bound">x</a> <a id="6325" href="Relation.Binary.Consequences.html#6325" class="Bound">y</a> <a id="6327" class="Symbol">=</a> <a id="6329" href="Data.Sum.Base.html#1303" class="Function">Sum.swap</a> <a id="6338" class="Symbol">(</a><a id="6339" href="Relation.Binary.Consequences.html#6321" class="Bound">f</a> <a id="6341" href="Relation.Binary.Consequences.html#6325" class="Bound">y</a> <a id="6343" href="Relation.Binary.Consequences.html#6323" class="Bound">x</a><a id="6344" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<a id="6349" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="6422" class="Comment">-- DEPRECATED NAMES</a>
|
|||
|
|
<a id="6442" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="6515" class="Comment">-- Please use the new names as continuing support for the old names is</a>
|
|||
|
|
<a id="6586" class="Comment">-- not guaranteed.</a>
|
|||
|
|
|
|||
|
|
<a id="6606" class="Comment">-- Version 1.6</a>
|
|||
|
|
|
|||
|
|
<a id="subst⟶respˡ"></a><a id="6622" href="Relation.Binary.Consequences.html#6622" class="Function">subst⟶respˡ</a> <a id="6634" class="Symbol">=</a> <a id="6636" href="Relation.Binary.Consequences.html#1026" class="Function">subst⇒respˡ</a>
|
|||
|
|
<a id="6648" class="Symbol">{-#</a> <a id="6652" class="Keyword">WARNING_ON_USAGE</a> <a id="6669" class="Pragma">subst⟶respˡ</a>
|
|||
|
|
<a id="6681" class="String">"Warning: subst⟶respˡ was deprecated in v1.6.
|
|||
|
|
Please use subst⇒respˡ instead."</a>
|
|||
|
|
<a id="6760" class="Symbol">#-}</a>
|
|||
|
|
<a id="subst⟶respʳ"></a><a id="6764" href="Relation.Binary.Consequences.html#6764" class="Function">subst⟶respʳ</a> <a id="6776" class="Symbol">=</a> <a id="6778" href="Relation.Binary.Consequences.html#1143" class="Function">subst⇒respʳ</a>
|
|||
|
|
<a id="6790" class="Symbol">{-#</a> <a id="6794" class="Keyword">WARNING_ON_USAGE</a> <a id="6811" class="Pragma">subst⟶respʳ</a>
|
|||
|
|
<a id="6823" class="String">"Warning: subst⟶respʳ was deprecated in v1.6.
|
|||
|
|
Please use subst⇒respʳ instead."</a>
|
|||
|
|
<a id="6902" class="Symbol">#-}</a>
|
|||
|
|
<a id="subst⟶resp₂"></a><a id="6906" href="Relation.Binary.Consequences.html#6906" class="Function">subst⟶resp₂</a> <a id="6918" class="Symbol">=</a> <a id="6920" href="Relation.Binary.Consequences.html#1255" class="Function">subst⇒resp₂</a>
|
|||
|
|
<a id="6932" class="Symbol">{-#</a> <a id="6936" class="Keyword">WARNING_ON_USAGE</a> <a id="6953" class="Pragma">subst⟶resp₂</a>
|
|||
|
|
<a id="6965" class="String">"Warning: subst⟶resp₂ was deprecated in v1.6.
|
|||
|
|
Please use subst⇒resp₂ instead."</a>
|
|||
|
|
<a id="7044" class="Symbol">#-}</a>
|
|||
|
|
<a id="P-resp⟶¬P-resp"></a><a id="7048" href="Relation.Binary.Consequences.html#7048" class="Function">P-resp⟶¬P-resp</a> <a id="7063" class="Symbol">=</a> <a id="7065" href="Relation.Binary.Consequences.html#1416" class="Function">resp⇒¬-resp</a>
|
|||
|
|
<a id="7077" class="Symbol">{-#</a> <a id="7081" class="Keyword">WARNING_ON_USAGE</a> <a id="7098" class="Pragma">P-resp⟶¬P-resp</a>
|
|||
|
|
<a id="7113" class="String">"Warning: P-resp⟶¬P-resp was deprecated in v1.6.
|
|||
|
|
Please use resp⇒¬-resp instead."</a>
|
|||
|
|
<a id="7195" class="Symbol">#-}</a>
|
|||
|
|
<a id="total⟶refl"></a><a id="7199" href="Relation.Binary.Consequences.html#7199" class="Function">total⟶refl</a> <a id="7210" class="Symbol">=</a> <a id="7212" href="Relation.Binary.Consequences.html#1701" class="Function">total⇒refl</a>
|
|||
|
|
<a id="7223" class="Symbol">{-#</a> <a id="7227" class="Keyword">WARNING_ON_USAGE</a> <a id="7244" class="Pragma">total⟶refl</a>
|
|||
|
|
<a id="7255" class="String">"Warning: total⟶refl was deprecated in v1.6.
|
|||
|
|
Please use total⇒refl instead."</a>
|
|||
|
|
<a id="7332" class="Symbol">#-}</a>
|
|||
|
|
<a id="total+dec⟶dec"></a><a id="7336" href="Relation.Binary.Consequences.html#7336" class="Function">total+dec⟶dec</a> <a id="7350" class="Symbol">=</a> <a id="7352" href="Relation.Binary.Consequences.html#1930" class="Function">total∧dec⇒dec</a>
|
|||
|
|
<a id="7366" class="Symbol">{-#</a> <a id="7370" class="Keyword">WARNING_ON_USAGE</a> <a id="7387" class="Pragma">total+dec⟶dec</a>
|
|||
|
|
<a id="7401" class="String">"Warning: total+dec⟶dec was deprecated in v1.6.
|
|||
|
|
Please use total∧dec⇒dec instead."</a>
|
|||
|
|
<a id="7484" class="Symbol">#-}</a>
|
|||
|
|
<a id="trans∧irr⟶asym"></a><a id="7488" href="Relation.Binary.Consequences.html#7488" class="Function">trans∧irr⟶asym</a> <a id="7503" class="Symbol">=</a> <a id="7505" href="Relation.Binary.Consequences.html#2927" class="Function">trans∧irr⇒asym</a>
|
|||
|
|
<a id="7520" class="Symbol">{-#</a> <a id="7524" class="Keyword">WARNING_ON_USAGE</a> <a id="7541" class="Pragma">trans∧irr⟶asym</a>
|
|||
|
|
<a id="7556" class="String">"Warning: trans∧irr⟶asym was deprecated in v1.6.
|
|||
|
|
Please use trans∧irr⇒asym instead."</a>
|
|||
|
|
<a id="7641" class="Symbol">#-}</a>
|
|||
|
|
<a id="irr∧antisym⟶asym"></a><a id="7645" href="Relation.Binary.Consequences.html#7645" class="Function">irr∧antisym⟶asym</a> <a id="7662" class="Symbol">=</a> <a id="7664" href="Relation.Binary.Consequences.html#3113" class="Function">irr∧antisym⇒asym</a>
|
|||
|
|
<a id="7681" class="Symbol">{-#</a> <a id="7685" class="Keyword">WARNING_ON_USAGE</a> <a id="7702" class="Pragma">irr∧antisym⟶asym</a>
|
|||
|
|
<a id="7719" class="String">"Warning: irr∧antisym⟶asym was deprecated in v1.6.
|
|||
|
|
Please use irr∧antisym⇒asym instead."</a>
|
|||
|
|
<a id="7808" class="Symbol">#-}</a>
|
|||
|
|
<a id="asym⟶antisym"></a><a id="7812" href="Relation.Binary.Consequences.html#7812" class="Function">asym⟶antisym</a> <a id="7825" class="Symbol">=</a> <a id="7827" href="Relation.Binary.Consequences.html#3294" class="Function">asym⇒antisym</a>
|
|||
|
|
<a id="7840" class="Symbol">{-#</a> <a id="7844" class="Keyword">WARNING_ON_USAGE</a> <a id="7861" class="Pragma">asym⟶antisym</a>
|
|||
|
|
<a id="7874" class="String">"Warning: asym⟶antisym was deprecated in v1.6.
|
|||
|
|
Please use asym⇒antisym instead."</a>
|
|||
|
|
<a id="7955" class="Symbol">#-}</a>
|
|||
|
|
<a id="asym⟶irr"></a><a id="7959" href="Relation.Binary.Consequences.html#7959" class="Function">asym⟶irr</a> <a id="7968" class="Symbol">=</a> <a id="7970" href="Relation.Binary.Consequences.html#3403" class="Function">asym⇒irr</a>
|
|||
|
|
<a id="7979" class="Symbol">{-#</a> <a id="7983" class="Keyword">WARNING_ON_USAGE</a> <a id="8000" class="Pragma">asym⟶irr</a>
|
|||
|
|
<a id="8009" class="String">"Warning: asym⟶irr was deprecated in v1.6.
|
|||
|
|
Please use asym⇒irr instead."</a>
|
|||
|
|
<a id="8082" class="Symbol">#-}</a>
|
|||
|
|
<a id="tri⟶asym"></a><a id="8086" href="Relation.Binary.Consequences.html#8086" class="Function">tri⟶asym</a> <a id="8095" class="Symbol">=</a> <a id="8097" href="Relation.Binary.Consequences.html#3604" class="Function">tri⇒asym</a>
|
|||
|
|
<a id="8106" class="Symbol">{-#</a> <a id="8110" class="Keyword">WARNING_ON_USAGE</a> <a id="8127" class="Pragma">tri⟶asym</a>
|
|||
|
|
<a id="8136" class="String">"Warning: tri⟶asym was deprecated in v1.6.
|
|||
|
|
Please use tri⇒asym instead."</a>
|
|||
|
|
<a id="8209" class="Symbol">#-}</a>
|
|||
|
|
<a id="tri⟶irr"></a><a id="8213" href="Relation.Binary.Consequences.html#8213" class="Function">tri⟶irr</a> <a id="8221" class="Symbol">=</a> <a id="8223" href="Relation.Binary.Consequences.html#3799" class="Function">tri⇒irr</a>
|
|||
|
|
<a id="8231" class="Symbol">{-#</a> <a id="8235" class="Keyword">WARNING_ON_USAGE</a> <a id="8252" class="Pragma">tri⟶irr</a>
|
|||
|
|
<a id="8260" class="String">"Warning: tri⟶irr was deprecated in v1.6.
|
|||
|
|
Please use tri⇒irr instead."</a>
|
|||
|
|
<a id="8331" class="Symbol">#-}</a>
|
|||
|
|
<a id="tri⟶dec≈"></a><a id="8335" href="Relation.Binary.Consequences.html#8335" class="Function">tri⟶dec≈</a> <a id="8344" class="Symbol">=</a> <a id="8346" href="Relation.Binary.Consequences.html#4011" class="Function">tri⇒dec≈</a>
|
|||
|
|
<a id="8355" class="Symbol">{-#</a> <a id="8359" class="Keyword">WARNING_ON_USAGE</a> <a id="8376" class="Pragma">tri⟶dec≈</a>
|
|||
|
|
<a id="8385" class="String">"Warning: tri⟶dec≈ was deprecated in v1.6.
|
|||
|
|
Please use tri⇒dec≈ instead."</a>
|
|||
|
|
<a id="8458" class="Symbol">#-}</a>
|
|||
|
|
<a id="tri⟶dec<"></a><a id="8462" href="Relation.Binary.Consequences.html#8462" class="Function">tri⟶dec<</a> <a id="8471" class="Symbol">=</a> <a id="8473" href="Relation.Binary.Consequences.html#4195" class="Function">tri⇒dec<</a>
|
|||
|
|
<a id="8482" class="Symbol">{-#</a> <a id="8486" class="Keyword">WARNING_ON_USAGE</a> <a id="8503" class="Pragma">tri⟶dec<</a>
|
|||
|
|
<a id="8512" class="String">"Warning: tri⟶dec< was deprecated in v1.6.
|
|||
|
|
Please use tri⇒dec< instead."</a>
|
|||
|
|
<a id="8585" class="Symbol">#-}</a>
|
|||
|
|
<a id="trans∧tri⟶respʳ≈"></a><a id="8589" href="Relation.Binary.Consequences.html#8589" class="Function">trans∧tri⟶respʳ≈</a> <a id="8606" class="Symbol">=</a> <a id="8608" href="Relation.Binary.Consequences.html#4379" class="Function">trans∧tri⇒respʳ</a>
|
|||
|
|
<a id="8624" class="Symbol">{-#</a> <a id="8628" class="Keyword">WARNING_ON_USAGE</a> <a id="8645" class="Pragma">trans∧tri⟶respʳ≈</a>
|
|||
|
|
<a id="8662" class="String">"Warning: trans∧tri⟶respʳ≈ was deprecated in v1.6.
|
|||
|
|
Please use trans∧tri⇒respʳ instead."</a>
|
|||
|
|
<a id="8750" class="Symbol">#-}</a>
|
|||
|
|
<a id="trans∧tri⟶respˡ≈"></a><a id="8754" href="Relation.Binary.Consequences.html#8754" class="Function">trans∧tri⟶respˡ≈</a> <a id="8771" class="Symbol">=</a> <a id="8773" href="Relation.Binary.Consequences.html#4765" class="Function">trans∧tri⇒respˡ</a>
|
|||
|
|
<a id="8789" class="Symbol">{-#</a> <a id="8793" class="Keyword">WARNING_ON_USAGE</a> <a id="8810" class="Pragma">trans∧tri⟶respˡ≈</a>
|
|||
|
|
<a id="8827" class="String">"Warning: trans∧tri⟶respˡ≈ was deprecated in v1.6.
|
|||
|
|
Please use trans∧tri⇒respˡ instead."</a>
|
|||
|
|
<a id="8915" class="Symbol">#-}</a>
|
|||
|
|
<a id="trans∧tri⟶resp≈"></a><a id="8919" href="Relation.Binary.Consequences.html#8919" class="Function">trans∧tri⟶resp≈</a> <a id="8935" class="Symbol">=</a> <a id="8937" href="Relation.Binary.Consequences.html#5119" class="Function">trans∧tri⇒resp</a>
|
|||
|
|
<a id="8952" class="Symbol">{-#</a> <a id="8956" class="Keyword">WARNING_ON_USAGE</a> <a id="8973" class="Pragma">trans∧tri⟶resp≈</a>
|
|||
|
|
<a id="8989" class="String">"Warning: trans∧tri⟶resp≈ was deprecated in v1.6.
|
|||
|
|
Please use trans∧tri⇒resp instead."</a>
|
|||
|
|
<a id="9075" class="Symbol">#-}</a>
|
|||
|
|
<a id="dec⟶weaklyDec"></a><a id="9079" href="Relation.Binary.Consequences.html#9079" class="Function">dec⟶weaklyDec</a> <a id="9093" class="Symbol">=</a> <a id="9095" href="Relation.Binary.Consequences.html#5864" class="Function">dec⇒weaklyDec</a>
|
|||
|
|
<a id="9109" class="Symbol">{-#</a> <a id="9113" class="Keyword">WARNING_ON_USAGE</a> <a id="9130" class="Pragma">dec⟶weaklyDec</a>
|
|||
|
|
<a id="9144" class="String">"Warning: dec⟶weaklyDec was deprecated in v1.6.
|
|||
|
|
Please use dec⇒weaklyDec instead."</a>
|
|||
|
|
<a id="9227" class="Symbol">#-}</a>
|
|||
|
|
<a id="dec⟶recomputable"></a><a id="9231" href="Relation.Binary.Consequences.html#9231" class="Function">dec⟶recomputable</a> <a id="9248" class="Symbol">=</a> <a id="9250" href="Relation.Binary.Consequences.html#5962" class="Function">dec⇒recomputable</a>
|
|||
|
|
<a id="9267" class="Symbol">{-#</a> <a id="9271" class="Keyword">WARNING_ON_USAGE</a> <a id="9288" class="Pragma">dec⟶recomputable</a>
|
|||
|
|
<a id="9305" class="String">"Warning: dec⟶recomputable was deprecated in v1.6.
|
|||
|
|
Please use dec⇒recomputable instead."</a>
|
|||
|
|
<a id="9394" class="Symbol">#-}</a>
|
|||
|
|
</pre></body></html>
|