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<html><head><meta charset="utf-8"><title>Data.Product.Relation.Binary.Pointwise.NonDependent</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Pointwise products of binary relations</a>
<a id="148" class="Comment">------------------------------------------------------------------------</a>
<a id="222" class="Symbol">{-#</a> <a id="226" class="Keyword">OPTIONS</a> <a id="234" class="Pragma">--cubical-compatible</a> <a id="255" class="Pragma">--safe</a> <a id="262" class="Symbol">#-}</a>
<a id="267" class="Keyword">module</a> <a id="274" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html" class="Module">Data.Product.Relation.Binary.Pointwise.NonDependent</a> <a id="326" class="Keyword">where</a>
<a id="333" class="Keyword">open</a> <a id="338" class="Keyword">import</a> <a id="345" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="363" class="Symbol">as</a> <a id="366" class="Module">Prod</a>
<a id="371" class="Keyword">open</a> <a id="376" class="Keyword">import</a> <a id="383" href="Data.Product.Properties.html" class="Module">Data.Product.Properties</a> <a id="407" class="Keyword">using</a> <a id="413" class="Symbol">(</a><a id="414" href="Data.Product.Properties.html#1321" class="Function">≡-dec</a><a id="419" class="Symbol">)</a>
<a id="421" class="Keyword">open</a> <a id="426" class="Keyword">import</a> <a id="433" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
<a id="447" class="Keyword">open</a> <a id="452" class="Keyword">import</a> <a id="459" href="Data.Unit.Base.html" class="Module">Data.Unit.Base</a> <a id="474" class="Keyword">using</a> <a id="480" class="Symbol">(</a><a id="481" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a><a id="482" class="Symbol">)</a>
<a id="484" class="Keyword">open</a> <a id="489" class="Keyword">import</a> <a id="496" href="Level.html" class="Module">Level</a> <a id="502" class="Keyword">using</a> <a id="508" class="Symbol">(</a><a id="509" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="514" class="Symbol">;</a> <a id="516" href="Agda.Primitive.html#961" class="Primitive Operator">_⊔_</a><a id="519" class="Symbol">;</a> <a id="521" href="Level.html#521" class="Function">0ℓ</a><a id="523" class="Symbol">)</a>
<a id="525" class="Keyword">open</a> <a id="530" class="Keyword">import</a> <a id="537" href="Function.html" class="Module">Function</a>
<a id="546" class="Keyword">open</a> <a id="551" class="Keyword">import</a> <a id="558" href="Relation.Nullary.Decidable.html" class="Module">Relation.Nullary.Decidable</a> <a id="585" class="Keyword">using</a> <a id="591" class="Symbol">(</a><a id="592" href="Relation.Nullary.Decidable.Core.html#2609" class="Function Operator">_×-dec_</a><a id="599" class="Symbol">)</a>
<a id="601" class="Keyword">open</a> <a id="606" class="Keyword">import</a> <a id="613" href="Relation.Binary.html" class="Module">Relation.Binary</a>
<a id="629" class="Keyword">open</a> <a id="634" class="Keyword">import</a> <a id="641" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a> <a id="684" class="Symbol">as</a> <a id="687" class="Module">P</a> <a id="689" class="Keyword">using</a> <a id="695" class="Symbol">(</a><a id="696" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="699" class="Symbol">)</a>
<a id="701" class="Keyword">import</a> <a id="708" href="Relation.Binary.PropositionalEquality.Properties.html" class="Module">Relation.Binary.PropositionalEquality.Properties</a> <a id="757" class="Symbol">as</a> <a id="760" class="Module">P</a>
<a id="763" class="Keyword">private</a>
<a id="773" class="Keyword">variable</a>
<a id="786" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="788" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="790" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="793" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="796" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#796" class="Generalizable">ℓ₃</a> <a id="799" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#799" class="Generalizable">ℓ₄</a> <a id="802" class="Symbol">:</a> <a id="804" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="814" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="816" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a> <a id="818" class="Symbol">:</a> <a id="820" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="824" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a>
<a id="830" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="832" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="834" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="837" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="840" class="Symbol">:</a> <a id="842" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="846" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="848" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a>
<a id="852" class="Comment">------------------------------------------------------------------------</a>
<a id="925" class="Comment">-- Definition</a>
<a id="Pointwise"></a><a id="940" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="950" class="Symbol">:</a> <a id="952" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="956" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="958" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="961" class="Symbol">→</a> <a id="963" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="967" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a> <a id="969" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="972" class="Symbol">→</a> <a id="974" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="978" class="Symbol">(</a><a id="979" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="981" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="983" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a><a id="984" class="Symbol">)</a> <a id="986" class="Symbol">(</a><a id="987" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="990" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="992" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a><a id="994" class="Symbol">)</a>
<a id="996" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1006" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1006" class="Bound">R</a> <a id="1008" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1008" class="Bound">S</a> <a id="1010" class="Symbol">=</a> <a id="1012" class="Symbol">(</a><a id="1013" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1006" class="Bound">R</a> <a id="1015" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1018" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="1023" class="Symbol">)</a> <a id="1025" href="Data.Product.Base.html#5099" class="Function Operator">-×-</a> <a id="1029" class="Symbol">(</a><a id="1030" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1008" class="Bound">S</a> <a id="1032" href="Function.Base.html#6209" class="Function Operator">on</a> <a id="1035" href="Data.Product.Base.html#650" class="Field">proj₂</a><a id="1040" class="Symbol">)</a>
<a id="1043" class="Comment">------------------------------------------------------------------------</a>
<a id="1116" class="Comment">-- Pointwise preserves many relational properties</a>
<a id="×-reflexive"></a><a id="1167" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1167" class="Function">×-reflexive</a> <a id="1179" class="Symbol">:</a> <a id="1181" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1184" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1186" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1188" class="Symbol">→</a> <a id="1190" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="1193" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1195" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1197" class="Symbol">→</a> <a id="1199" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1209" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1212" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="1215" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1217" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1227" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1229" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a>
<a id="1231" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1167" class="Function">×-reflexive</a> <a id="1243" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1243" class="Bound">refl₁</a> <a id="1249" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1249" class="Bound">refl₂</a> <a id="1255" class="Symbol">=</a> <a id="1257" href="Data.Product.Base.html#2173" class="Function">Prod.map</a> <a id="1266" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1243" class="Bound">refl₁</a> <a id="1272" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1249" class="Bound">refl₂</a>
<a id="×-refl"></a><a id="1279" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1279" class="Function">×-refl</a> <a id="1286" class="Symbol">:</a> <a id="1288" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1298" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1300" class="Symbol">→</a> <a id="1302" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1312" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1314" class="Symbol">→</a> <a id="1316" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="1326" class="Symbol">(</a><a id="1327" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1337" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1339" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="1340" class="Symbol">)</a>
<a id="1342" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1279" class="Function">×-refl</a> <a id="1349" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1349" class="Bound">refl₁</a> <a id="1355" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1355" class="Bound">refl₂</a> <a id="1361" class="Symbol">=</a> <a id="1363" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1349" class="Bound">refl₁</a> <a id="1369" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1371" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1355" class="Bound">refl₂</a>
<a id="×-irreflexive₁"></a><a id="1378" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1378" class="Function">×-irreflexive₁</a> <a id="1393" class="Symbol">:</a> <a id="1395" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1407" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1410" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1412" class="Symbol">→</a>
<a id="1431" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1443" class="Symbol">(</a><a id="1444" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1454" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1457" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="1459" class="Symbol">)</a> <a id="1461" class="Symbol">(</a><a id="1462" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1472" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1474" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="1475" class="Symbol">)</a>
<a id="1477" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1378" class="Function">×-irreflexive₁</a> <a id="1492" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1492" class="Bound">ir</a> <a id="1495" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1495" class="Bound">x≈y</a> <a id="1499" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1499" class="Bound">x&lt;y</a> <a id="1503" class="Symbol">=</a> <a id="1505" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1492" class="Bound">ir</a> <a id="1508" class="Symbol">(</a><a id="1509" href="Data.Product.Base.html#636" class="Field">proj₁</a> <a id="1515" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1495" class="Bound">x≈y</a><a id="1518" class="Symbol">)</a> <a id="1520" class="Symbol">(</a><a id="1521" href="Data.Product.Base.html#636" class="Field">proj₁</a> <a id="1527" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1499" class="Bound">x&lt;y</a><a id="1530" class="Symbol">)</a>
<a id="×-irreflexive₂"></a><a id="1533" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1533" class="Function">×-irreflexive₂</a> <a id="1548" class="Symbol">:</a> <a id="1550" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1562" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="1565" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1567" class="Symbol">→</a>
<a id="1586" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="1598" class="Symbol">(</a><a id="1599" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1609" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1612" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="1614" class="Symbol">)</a> <a id="1616" class="Symbol">(</a><a id="1617" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1627" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1629" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="1630" class="Symbol">)</a>
<a id="1632" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1533" class="Function">×-irreflexive₂</a> <a id="1647" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1647" class="Bound">ir</a> <a id="1650" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1650" class="Bound">x≈y</a> <a id="1654" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1654" class="Bound">x&lt;y</a> <a id="1658" class="Symbol">=</a> <a id="1660" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1647" class="Bound">ir</a> <a id="1663" class="Symbol">(</a><a id="1664" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="1670" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1650" class="Bound">x≈y</a><a id="1673" class="Symbol">)</a> <a id="1675" class="Symbol">(</a><a id="1676" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="1682" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1654" class="Bound">x&lt;y</a><a id="1685" class="Symbol">)</a>
<a id="×-symmetric"></a><a id="1688" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1688" class="Function">×-symmetric</a> <a id="1700" class="Symbol">:</a> <a id="1702" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1712" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1714" class="Symbol">→</a> <a id="1716" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1726" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1728" class="Symbol">→</a> <a id="1730" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1740" class="Symbol">(</a><a id="1741" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1751" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1753" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="1754" class="Symbol">)</a>
<a id="1756" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1688" class="Function">×-symmetric</a> <a id="1768" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1768" class="Bound">sym₁</a> <a id="1773" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1773" class="Bound">sym₂</a> <a id="1778" class="Symbol">=</a> <a id="1780" href="Data.Product.Base.html#2173" class="Function">Prod.map</a> <a id="1789" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1768" class="Bound">sym₁</a> <a id="1794" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1773" class="Bound">sym₂</a>
<a id="×-transitive"></a><a id="1800" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1800" class="Function">×-transitive</a> <a id="1813" class="Symbol">:</a> <a id="1815" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1826" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1828" class="Symbol">→</a> <a id="1830" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1841" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1843" class="Symbol">→</a> <a id="1845" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1856" class="Symbol">(</a><a id="1857" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="1867" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1869" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="1870" class="Symbol">)</a>
<a id="1872" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1800" class="Function">×-transitive</a> <a id="1885" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1885" class="Bound">trans₁</a> <a id="1892" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1892" class="Bound">trans₂</a> <a id="1899" class="Symbol">=</a> <a id="1901" href="Data.Product.Base.html#2747" class="Function">Prod.zip</a> <a id="1910" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1885" class="Bound">trans₁</a> <a id="1917" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1892" class="Bound">trans₂</a>
<a id="×-antisymmetric"></a><a id="1925" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1925" class="Function">×-antisymmetric</a> <a id="1941" class="Symbol">:</a> <a id="1943" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1957" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="1960" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="1962" class="Symbol">→</a> <a id="1964" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1978" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="1981" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="1983" class="Symbol">→</a>
<a id="2003" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2017" class="Symbol">(</a><a id="2018" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2028" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2031" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="2033" class="Symbol">)</a> <a id="2035" class="Symbol">(</a><a id="2036" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2046" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2048" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2049" class="Symbol">)</a>
<a id="2051" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1925" class="Function">×-antisymmetric</a> <a id="2067" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2067" class="Bound">antisym₁</a> <a id="2076" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2076" class="Bound">antisym₂</a> <a id="2085" class="Symbol">=</a> <a id="2087" href="Data.Product.Base.html#2747" class="Function">Prod.zip</a> <a id="2096" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2067" class="Bound">antisym₁</a> <a id="2105" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2076" class="Bound">antisym₂</a>
<a id="×-asymmetric₁"></a><a id="2115" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2115" class="Function">×-asymmetric₁</a> <a id="2129" class="Symbol">:</a> <a id="2131" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2142" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2144" class="Symbol">→</a> <a id="2146" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2157" class="Symbol">(</a><a id="2158" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2168" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2170" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2171" class="Symbol">)</a>
<a id="2173" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2115" class="Function">×-asymmetric₁</a> <a id="2187" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2187" class="Bound">asym₁</a> <a id="2193" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2193" class="Bound">x&lt;y</a> <a id="2197" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2197" class="Bound">y&lt;x</a> <a id="2201" class="Symbol">=</a> <a id="2203" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2187" class="Bound">asym₁</a> <a id="2209" class="Symbol">(</a><a id="2210" href="Data.Product.Base.html#636" class="Field">proj₁</a> <a id="2216" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2193" class="Bound">x&lt;y</a><a id="2219" class="Symbol">)</a> <a id="2221" class="Symbol">(</a><a id="2222" href="Data.Product.Base.html#636" class="Field">proj₁</a> <a id="2228" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2197" class="Bound">y&lt;x</a><a id="2231" class="Symbol">)</a>
<a id="×-asymmetric₂"></a><a id="2234" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2234" class="Function">×-asymmetric₂</a> <a id="2248" class="Symbol">:</a> <a id="2250" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2261" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="2263" class="Symbol">→</a> <a id="2265" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="2276" class="Symbol">(</a><a id="2277" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2287" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2289" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2290" class="Symbol">)</a>
<a id="2292" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2234" class="Function">×-asymmetric₂</a> <a id="2306" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2306" class="Bound">asym₂</a> <a id="2312" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2312" class="Bound">x&lt;y</a> <a id="2316" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2316" class="Bound">y&lt;x</a> <a id="2320" class="Symbol">=</a> <a id="2322" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2306" class="Bound">asym₂</a> <a id="2328" class="Symbol">(</a><a id="2329" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="2335" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2312" class="Bound">x&lt;y</a><a id="2338" class="Symbol">)</a> <a id="2340" class="Symbol">(</a><a id="2341" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="2347" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2316" class="Bound">y&lt;x</a><a id="2350" class="Symbol">)</a>
<a id="×-respectsʳ"></a><a id="2353" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2353" class="Function">×-respectsʳ</a> <a id="2365" class="Symbol">:</a> <a id="2367" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2369" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="2379" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2382" class="Symbol">→</a> <a id="2384" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="2386" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="2396" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="2399" class="Symbol">→</a>
<a id="2414" class="Symbol">(</a><a id="2415" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2425" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2427" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2428" class="Symbol">)</a> <a id="2430" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="2440" class="Symbol">(</a><a id="2441" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2451" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2454" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="2456" class="Symbol">)</a>
<a id="2458" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2353" class="Function">×-respectsʳ</a> <a id="2470" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2470" class="Bound">resp₁</a> <a id="2476" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2476" class="Bound">resp₂</a> <a id="2482" class="Symbol">=</a> <a id="2484" href="Data.Product.Base.html#2747" class="Function">Prod.zip</a> <a id="2493" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2470" class="Bound">resp₁</a> <a id="2499" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2476" class="Bound">resp₂</a>
<a id="×-respectsˡ"></a><a id="2506" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2506" class="Function">×-respectsˡ</a> <a id="2518" class="Symbol">:</a> <a id="2520" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2522" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="2532" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2535" class="Symbol">→</a> <a id="2537" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="2539" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="2549" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="2552" class="Symbol">→</a>
<a id="2567" class="Symbol">(</a><a id="2568" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2578" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2580" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2581" class="Symbol">)</a> <a id="2583" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="2593" class="Symbol">(</a><a id="2594" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2604" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2607" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="2609" class="Symbol">)</a>
<a id="2611" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2506" class="Function">×-respectsˡ</a> <a id="2623" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2623" class="Bound">resp₁</a> <a id="2629" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2629" class="Bound">resp₂</a> <a id="2635" class="Symbol">=</a> <a id="2637" href="Data.Product.Base.html#2747" class="Function">Prod.zip</a> <a id="2646" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2623" class="Bound">resp₁</a> <a id="2652" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2629" class="Bound">resp₂</a>
<a id="×-respects₂"></a><a id="2659" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2659" class="Function">×-respects₂</a> <a id="2671" class="Symbol">:</a> <a id="2673" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2675" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2685" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2688" class="Symbol">→</a> <a id="2690" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="2692" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2702" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="2705" class="Symbol">→</a>
<a id="2721" class="Symbol">(</a><a id="2722" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2732" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2734" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2735" class="Symbol">)</a> <a id="2737" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2747" class="Symbol">(</a><a id="2748" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2758" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="2761" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="2763" class="Symbol">)</a>
<a id="2765" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2659" class="Function">×-respects₂</a> <a id="2777" class="Symbol">=</a> <a id="2779" href="Data.Product.Base.html#2747" class="Function">Prod.zip</a> <a id="2788" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2353" class="Function">×-respectsʳ</a> <a id="2800" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2506" class="Function">×-respectsˡ</a>
<a id="×-total"></a><a id="2813" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2813" class="Function">×-total</a> <a id="2821" class="Symbol">:</a> <a id="2823" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2833" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2835" class="Symbol">→</a> <a id="2837" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2843" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2845" class="Symbol">→</a> <a id="2847" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2853" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="2855" class="Symbol">→</a> <a id="2857" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2863" class="Symbol">(</a><a id="2864" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="2874" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="2876" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="2877" class="Symbol">)</a>
<a id="2879" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2813" class="Function">×-total</a> <a id="2887" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2887" class="Bound">sym₁</a> <a id="2892" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2892" class="Bound">total₁</a> <a id="2899" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2899" class="Bound">total₂</a> <a id="2906" class="Symbol">(</a><a id="2907" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2907" class="Bound">x₁</a> <a id="2910" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2912" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2912" class="Bound">x₂</a><a id="2914" class="Symbol">)</a> <a id="2916" class="Symbol">(</a><a id="2917" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2917" class="Bound">y₁</a> <a id="2920" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2922" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2922" class="Bound">y₂</a><a id="2924" class="Symbol">)</a>
<a id="2928" class="Keyword">with</a> <a id="2933" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2892" class="Bound">total₁</a> <a id="2940" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2907" class="Bound">x₁</a> <a id="2943" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2917" class="Bound">y₁</a> <a id="2946" class="Symbol">|</a> <a id="2948" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2899" class="Bound">total₂</a> <a id="2955" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2912" class="Bound">x₂</a> <a id="2958" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2922" class="Bound">y₂</a>
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<a id="3019" class="Symbol">...</a> <a id="3023" class="Symbol">|</a> <a id="3025" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3030" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3030" class="Bound">x₁∼y₁</a> <a id="3036" class="Symbol">|</a> <a id="3038" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3043" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3043" class="Bound">y₂∼x₂</a> <a id="3049" class="Symbol">=</a> <a id="3051" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3056" class="Symbol">(</a><a id="3057" class="Bound">sym₁</a> <a id="3062" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3030" class="Bound">x₁∼y₁</a> <a id="3068" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3070" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3043" class="Bound">y₂∼x₂</a><a id="3075" class="Symbol">)</a>
<a id="3077" class="Symbol">...</a> <a id="3081" class="Symbol">|</a> <a id="3083" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3088" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3088" class="Bound">y₁∼x₁</a> <a id="3094" class="Symbol">|</a> <a id="3096" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3101" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3101" class="Bound">y₂∼x₂</a> <a id="3107" class="Symbol">=</a> <a id="3109" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3114" class="Symbol">(</a> <a id="3120" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3088" class="Bound">y₁∼x₁</a> <a id="3126" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3128" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3101" class="Bound">y₂∼x₂</a><a id="3133" class="Symbol">)</a>
<a id="3135" class="Symbol">...</a> <a id="3139" class="Symbol">|</a> <a id="3141" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3146" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3146" class="Bound">y₁∼x₁</a> <a id="3152" class="Symbol">|</a> <a id="3154" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3159" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3159" class="Bound">x₂∼y₂</a> <a id="3165" class="Symbol">=</a> <a id="3167" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3172" class="Symbol">(</a><a id="3173" class="Bound">sym₁</a> <a id="3178" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3146" class="Bound">y₁∼x₁</a> <a id="3184" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3186" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3159" class="Bound">x₂∼y₂</a><a id="3191" class="Symbol">)</a>
<a id="×-decidable"></a><a id="3194" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3194" class="Function">×-decidable</a> <a id="3206" class="Symbol">:</a> <a id="3208" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="3218" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3220" class="Symbol">→</a> <a id="3222" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="3232" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="3234" class="Symbol">→</a> <a id="3236" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="3246" class="Symbol">(</a><a id="3247" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="3257" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3259" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="3260" class="Symbol">)</a>
<a id="3262" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3194" class="Function">×-decidable</a> <a id="3274" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3274" class="Bound Operator">_≟₁_</a> <a id="3279" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3279" class="Bound Operator">_≟₂_</a> <a id="3284" class="Symbol">(</a><a id="3285" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3285" class="Bound">x₁</a> <a id="3288" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3290" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3290" class="Bound">x₂</a><a id="3292" class="Symbol">)</a> <a id="3294" class="Symbol">(</a><a id="3295" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3295" class="Bound">y₁</a> <a id="3298" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3300" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3300" class="Bound">y₂</a><a id="3302" class="Symbol">)</a> <a id="3304" class="Symbol">=</a> <a id="3306" class="Symbol">(</a><a id="3307" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3285" class="Bound">x₁</a> <a id="3310" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3274" class="Bound Operator">≟₁</a> <a id="3313" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3295" class="Bound">y₁</a><a id="3315" class="Symbol">)</a> <a id="3317" href="Relation.Nullary.Decidable.Core.html#2609" class="Function Operator">×-dec</a> <a id="3323" class="Symbol">(</a><a id="3324" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3290" class="Bound">x₂</a> <a id="3327" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3279" class="Bound Operator">≟₂</a> <a id="3330" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3300" class="Bound">y₂</a><a id="3332" class="Symbol">)</a>
<a id="3335" class="Comment">------------------------------------------------------------------------</a>
<a id="3408" class="Comment">-- Structures can also be combined.</a>
<a id="3445" class="Comment">-- Some collections of properties which are preserved by ×-Rel.</a>
<a id="×-isEquivalence"></a><a id="3510" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a> <a id="3526" class="Symbol">:</a> <a id="3528" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="3542" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3544" class="Symbol">→</a> <a id="3546" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="3560" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="3562" class="Symbol">→</a>
<a id="3582" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="3596" class="Symbol">(</a><a id="3597" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="3607" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3609" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="3610" class="Symbol">)</a>
<a id="3612" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a> <a id="3628" class="Symbol">{</a><a id="3629" class="Argument">R</a> <a id="3631" class="Symbol">=</a> <a id="3633" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3633" class="Bound">R</a><a id="3634" class="Symbol">}</a> <a id="3636" class="Symbol">{</a><a id="3637" class="Argument">S</a> <a id="3639" class="Symbol">=</a> <a id="3641" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3641" class="Bound">S</a><a id="3642" class="Symbol">}</a> <a id="3644" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3644" class="Bound">eq₁</a> <a id="3648" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3648" class="Bound">eq₂</a> <a id="3652" class="Symbol">=</a> <a id="3654" class="Keyword">record</a>
<a id="3663" class="Symbol">{</a> <a id="3665" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="3671" class="Symbol">=</a> <a id="3673" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1279" class="Function">×-refl</a> <a id="3680" class="Symbol">{</a><a id="3681" class="Argument">R</a> <a id="3683" class="Symbol">=</a> <a id="3685" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3633" class="Bound">R</a><a id="3686" class="Symbol">}</a> <a id="3688" class="Symbol">{</a><a id="3689" class="Argument">S</a> <a id="3691" class="Symbol">=</a> <a id="3693" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3641" class="Bound">S</a><a id="3694" class="Symbol">}</a> <a id="3696" class="Symbol">(</a><a id="3697" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="3702" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3644" class="Bound">eq₁</a><a id="3705" class="Symbol">)</a> <a id="3707" class="Symbol">(</a><a id="3708" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="3713" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3648" class="Bound">eq₂</a><a id="3716" class="Symbol">)</a>
<a id="3720" class="Symbol">;</a> <a id="3722" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="3728" class="Symbol">=</a> <a id="3730" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1688" class="Function">×-symmetric</a> <a id="3742" class="Symbol">{</a><a id="3743" class="Argument">R</a> <a id="3745" class="Symbol">=</a> <a id="3747" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3633" class="Bound">R</a><a id="3748" class="Symbol">}</a> <a id="3750" class="Symbol">{</a><a id="3751" class="Argument">S</a> <a id="3753" class="Symbol">=</a> <a id="3755" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3641" class="Bound">S</a><a id="3756" class="Symbol">}</a> <a id="3758" class="Symbol">(</a><a id="3759" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="3763" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3644" class="Bound">eq₁</a><a id="3766" class="Symbol">)</a> <a id="3768" class="Symbol">(</a><a id="3769" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="3773" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3648" class="Bound">eq₂</a><a id="3776" class="Symbol">)</a>
<a id="3780" class="Symbol">;</a> <a id="3782" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="3788" class="Symbol">=</a> <a id="3790" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1800" class="Function">×-transitive</a> <a id="3803" class="Symbol">{</a><a id="3804" class="Argument">R</a> <a id="3806" class="Symbol">=</a> <a id="3808" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3633" class="Bound">R</a><a id="3809" class="Symbol">}</a> <a id="3811" class="Symbol">{</a><a id="3812" class="Argument">S</a> <a id="3814" class="Symbol">=</a> <a id="3816" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3641" class="Bound">S</a><a id="3817" class="Symbol">}</a> <a id="3819" class="Symbol">(</a><a id="3820" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="3826" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3644" class="Bound">eq₁</a><a id="3829" class="Symbol">)</a> <a id="3831" class="Symbol">(</a><a id="3832" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="3838" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3648" class="Bound">eq₂</a><a id="3841" class="Symbol">)</a>
<a id="3845" class="Symbol">}</a> <a id="3847" class="Keyword">where</a> <a id="3853" class="Keyword">open</a> <a id="3858" href="Relation.Binary.Structures.html#1550" class="Module">IsEquivalence</a>
<a id="×-isDecEquivalence"></a><a id="3873" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3873" class="Function">×-isDecEquivalence</a> <a id="3892" class="Symbol">:</a> <a id="3894" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="3911" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3913" class="Symbol">→</a> <a id="3915" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="3932" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="3934" class="Symbol">→</a>
<a id="3957" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="3974" class="Symbol">(</a><a id="3975" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="3985" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="3987" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="3988" class="Symbol">)</a>
<a id="3990" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3873" class="Function">×-isDecEquivalence</a> <a id="4009" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4009" class="Bound">eq₁</a> <a id="4013" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4013" class="Bound">eq₂</a> <a id="4017" class="Symbol">=</a> <a id="4019" class="Keyword">record</a>
<a id="4028" class="Symbol">{</a> <a id="4030" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="4044" class="Symbol">=</a> <a id="4046" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a>
<a id="4084" class="Symbol">(</a><a id="4085" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="4099" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4009" class="Bound">eq₁</a><a id="4102" class="Symbol">)</a> <a id="4104" class="Symbol">(</a><a id="4105" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="4119" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4013" class="Bound">eq₂</a><a id="4122" class="Symbol">)</a>
<a id="4126" class="Symbol">;</a> <a id="4128" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="4142" class="Symbol">=</a> <a id="4144" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3194" class="Function">×-decidable</a> <a id="4156" class="Symbol">(</a><a id="4157" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="4161" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4009" class="Bound">eq₁</a><a id="4164" class="Symbol">)</a> <a id="4166" class="Symbol">(</a><a id="4167" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="4171" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4013" class="Bound">eq₂</a><a id="4174" class="Symbol">)</a>
<a id="4178" class="Symbol">}</a> <a id="4180" class="Keyword">where</a> <a id="4186" class="Keyword">open</a> <a id="4191" href="Relation.Binary.Structures.html#1852" class="Module">IsDecEquivalence</a>
<a id="×-isPreorder"></a><a id="4209" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4209" class="Function">×-isPreorder</a> <a id="4222" class="Symbol">:</a> <a id="4224" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="4235" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="4238" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="4240" class="Symbol">→</a> <a id="4242" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="4253" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="4256" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="4258" class="Symbol">→</a>
<a id="4275" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="4286" class="Symbol">(</a><a id="4287" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="4297" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="4300" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="4302" class="Symbol">)</a> <a id="4304" class="Symbol">(</a><a id="4305" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="4315" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="4317" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="4318" class="Symbol">)</a>
<a id="4320" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4209" class="Function">×-isPreorder</a> <a id="4333" class="Symbol">{</a><a id="4334" class="Argument">R</a> <a id="4336" class="Symbol">=</a> <a id="4338" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4338" class="Bound">R</a><a id="4339" class="Symbol">}</a> <a id="4341" class="Symbol">{</a><a id="4342" class="Argument">S</a> <a id="4344" class="Symbol">=</a> <a id="4346" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4346" class="Bound">S</a><a id="4347" class="Symbol">}</a> <a id="4349" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4349" class="Bound">pre₁</a> <a id="4354" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4354" class="Bound">pre₂</a> <a id="4359" class="Symbol">=</a> <a id="4361" class="Keyword">record</a>
<a id="4370" class="Symbol">{</a> <a id="4372" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="4386" class="Symbol">=</a> <a id="4388" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a>
<a id="4426" class="Symbol">(</a><a id="4427" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="4441" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4349" class="Bound">pre₁</a><a id="4445" class="Symbol">)</a> <a id="4447" class="Symbol">(</a><a id="4448" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="4462" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4354" class="Bound">pre₂</a><a id="4466" class="Symbol">)</a>
<a id="4470" class="Symbol">;</a> <a id="4472" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="4486" class="Symbol">=</a> <a id="4488" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1167" class="Function">×-reflexive</a> <a id="4500" class="Symbol">{</a><a id="4501" class="Argument">R</a> <a id="4503" class="Symbol">=</a> <a id="4505" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4338" class="Bound">R</a><a id="4506" class="Symbol">}</a> <a id="4508" class="Symbol">{</a><a id="4509" class="Argument">S</a> <a id="4511" class="Symbol">=</a> <a id="4513" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4346" class="Bound">S</a><a id="4514" class="Symbol">}</a>
<a id="4538" class="Symbol">(</a><a id="4539" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="4549" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4349" class="Bound">pre₁</a><a id="4553" class="Symbol">)</a> <a id="4555" class="Symbol">(</a><a id="4556" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="4566" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4354" class="Bound">pre₂</a><a id="4570" class="Symbol">)</a>
<a id="4574" class="Symbol">;</a> <a id="4576" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="4590" class="Symbol">=</a> <a id="4592" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1800" class="Function">×-transitive</a> <a id="4605" class="Symbol">{</a><a id="4606" class="Argument">R</a> <a id="4608" class="Symbol">=</a> <a id="4610" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4338" class="Bound">R</a><a id="4611" class="Symbol">}</a> <a id="4613" class="Symbol">{</a><a id="4614" class="Argument">S</a> <a id="4616" class="Symbol">=</a> <a id="4618" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4346" class="Bound">S</a><a id="4619" class="Symbol">}</a>
<a id="4643" class="Symbol">(</a><a id="4644" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="4650" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4349" class="Bound">pre₁</a><a id="4654" class="Symbol">)</a> <a id="4656" class="Symbol">(</a><a id="4657" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="4663" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4354" class="Bound">pre₂</a><a id="4667" class="Symbol">)</a>
<a id="4671" class="Symbol">}</a> <a id="4673" class="Keyword">where</a> <a id="4679" class="Keyword">open</a> <a id="4684" href="Relation.Binary.Structures.html#2191" class="Module">IsPreorder</a>
<a id="×-isPartialOrder"></a><a id="4696" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4696" class="Function">×-isPartialOrder</a> <a id="4713" class="Symbol">:</a> <a id="4715" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="4730" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="4733" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="4735" class="Symbol">→</a> <a id="4737" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="4752" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="4755" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="4757" class="Symbol">→</a>
<a id="4778" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="4793" class="Symbol">(</a><a id="4794" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="4804" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="4807" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="4809" class="Symbol">)</a> <a id="4811" class="Symbol">(</a><a id="4812" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="4822" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="4824" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="4825" class="Symbol">)</a>
<a id="4827" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4696" class="Function">×-isPartialOrder</a> <a id="4844" class="Symbol">{</a><a id="4845" class="Argument">R</a> <a id="4847" class="Symbol">=</a> <a id="4849" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4849" class="Bound">R</a><a id="4850" class="Symbol">}</a> <a id="4852" class="Symbol">{</a><a id="4853" class="Argument">S</a> <a id="4855" class="Symbol">=</a> <a id="4857" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4857" class="Bound">S</a><a id="4858" class="Symbol">}</a> <a id="4860" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4860" class="Bound">po₁</a> <a id="4864" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4864" class="Bound">po₂</a> <a id="4868" class="Symbol">=</a> <a id="4870" class="Keyword">record</a>
<a id="4879" class="Symbol">{</a> <a id="4881" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="4892" class="Symbol">=</a> <a id="4894" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4209" class="Function">×-isPreorder</a> <a id="4907" class="Symbol">(</a><a id="4908" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="4919" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4860" class="Bound">po₁</a><a id="4922" class="Symbol">)</a> <a id="4924" class="Symbol">(</a><a id="4925" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="4936" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4864" class="Bound">po₂</a><a id="4939" class="Symbol">)</a>
<a id="4943" class="Symbol">;</a> <a id="4945" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="4956" class="Symbol">=</a> <a id="4958" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1925" class="Function">×-antisymmetric</a> <a id="4974" class="Symbol">{</a><a id="4975" class="Argument">R</a> <a id="4977" class="Symbol">=</a> <a id="4979" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4849" class="Bound">R</a><a id="4980" class="Symbol">}</a> <a id="4982" class="Symbol">{</a><a id="4983" class="Argument">S</a> <a id="4985" class="Symbol">=</a> <a id="4987" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4857" class="Bound">S</a><a id="4988" class="Symbol">}</a>
<a id="5009" class="Symbol">(</a><a id="5010" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="5018" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4860" class="Bound">po₁</a><a id="5021" class="Symbol">)</a> <a id="5023" class="Symbol">(</a><a id="5024" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="5032" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4864" class="Bound">po₂</a><a id="5035" class="Symbol">)</a>
<a id="5039" class="Symbol">}</a> <a id="5041" class="Keyword">where</a> <a id="5047" class="Keyword">open</a> <a id="5052" href="Relation.Binary.Structures.html#3522" class="Module">IsPartialOrder</a>
<a id="×-isStrictPartialOrder"></a><a id="5068" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5068" class="Function">×-isStrictPartialOrder</a> <a id="5091" class="Symbol">:</a> <a id="5093" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="5114" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="5117" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="5119" class="Symbol">→</a>
<a id="5146" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="5167" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a> <a id="5170" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a> <a id="5172" class="Symbol">→</a>
<a id="5199" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="5220" class="Symbol">(</a><a id="5221" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="5231" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#834" class="Generalizable">≈₁</a> <a id="5234" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#837" class="Generalizable">≈₂</a><a id="5236" class="Symbol">)</a> <a id="5238" class="Symbol">(</a><a id="5239" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="5249" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#830" class="Generalizable">R</a> <a id="5251" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#832" class="Generalizable">S</a><a id="5252" class="Symbol">)</a>
<a id="5254" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5068" class="Function">×-isStrictPartialOrder</a> <a id="5277" class="Symbol">{</a><a id="5278" class="Argument">R</a> <a id="5280" class="Symbol">=</a> <a id="5282" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5282" class="Bound">R</a><a id="5283" class="Symbol">}</a> <a id="5285" class="Symbol">{</a><a id="5286" class="Argument">≈₂</a> <a id="5289" class="Symbol">=</a> <a id="5291" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5291" class="Bound">≈₂</a><a id="5293" class="Symbol">}</a> <a id="5295" class="Symbol">{</a><a id="5296" class="Argument">S</a> <a id="5298" class="Symbol">=</a> <a id="5300" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5300" class="Bound">S</a><a id="5301" class="Symbol">}</a> <a id="5303" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5303" class="Bound">spo₁</a> <a id="5308" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5308" class="Bound">spo₂</a> <a id="5313" class="Symbol">=</a> <a id="5315" class="Keyword">record</a>
<a id="5324" class="Symbol">{</a> <a id="5326" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="5340" class="Symbol">=</a> <a id="5342" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a>
<a id="5380" class="Symbol">(</a><a id="5381" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="5395" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5303" class="Bound">spo₁</a><a id="5399" class="Symbol">)</a> <a id="5401" class="Symbol">(</a><a id="5402" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="5416" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5308" class="Bound">spo₂</a><a id="5420" class="Symbol">)</a>
<a id="5424" class="Symbol">;</a> <a id="5426" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="5440" class="Symbol">=</a> <a id="5442" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1378" class="Function">×-irreflexive₁</a> <a id="5457" class="Symbol">{</a><a id="5458" class="Argument">R</a> <a id="5460" class="Symbol">=</a> <a id="5462" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5282" class="Bound">R</a><a id="5463" class="Symbol">}</a> <a id="5465" class="Symbol">{</a><a id="5466" class="Argument">≈₂</a> <a id="5469" class="Symbol">=</a> <a id="5471" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5291" class="Bound">≈₂</a><a id="5473" class="Symbol">}</a> <a id="5475" class="Symbol">{</a><a id="5476" class="Argument">S</a> <a id="5478" class="Symbol">=</a> <a id="5480" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5300" class="Bound">S</a><a id="5481" class="Symbol">}</a>
<a id="5505" class="Symbol">(</a><a id="5506" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="5513" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5303" class="Bound">spo₁</a><a id="5517" class="Symbol">)</a>
<a id="5521" class="Symbol">;</a> <a id="5523" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="5537" class="Symbol">=</a> <a id="5539" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#1800" class="Function">×-transitive</a> <a id="5552" class="Symbol">{</a><a id="5553" class="Argument">R</a> <a id="5555" class="Symbol">=</a> <a id="5557" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5282" class="Bound">R</a><a id="5558" class="Symbol">}</a> <a id="5560" class="Symbol">{</a><a id="5561" class="Argument">S</a> <a id="5563" class="Symbol">=</a> <a id="5565" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5300" class="Bound">S</a><a id="5566" class="Symbol">}</a>
<a id="5590" class="Symbol">(</a><a id="5591" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="5597" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5303" class="Bound">spo₁</a><a id="5601" class="Symbol">)</a> <a id="5603" class="Symbol">(</a><a id="5604" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="5610" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5308" class="Bound">spo₂</a><a id="5614" class="Symbol">)</a>
<a id="5618" class="Symbol">;</a> <a id="5620" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="5634" class="Symbol">=</a> <a id="5636" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#2659" class="Function">×-respects₂</a> <a id="5648" class="Symbol">(</a><a id="5649" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="5658" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5303" class="Bound">spo₁</a><a id="5662" class="Symbol">)</a> <a id="5664" class="Symbol">(</a><a id="5665" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="5674" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5308" class="Bound">spo₂</a><a id="5678" class="Symbol">)</a>
<a id="5682" class="Symbol">}</a> <a id="5684" class="Keyword">where</a> <a id="5690" class="Keyword">open</a> <a id="5695" href="Relation.Binary.Structures.html#4298" class="Module">IsStrictPartialOrder</a>
<a id="5717" class="Comment">------------------------------------------------------------------------</a>
<a id="5790" class="Comment">-- Bundles</a>
<a id="×-setoid"></a><a id="5802" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5802" class="Function">×-setoid</a> <a id="5811" class="Symbol">:</a> <a id="5813" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5820" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="5822" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="5825" class="Symbol">→</a> <a id="5827" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5834" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="5836" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="5839" class="Symbol">→</a> <a id="5841" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5848" class="Symbol">_</a> <a id="5850" class="Symbol">_</a>
<a id="5852" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5802" class="Function">×-setoid</a> <a id="5861" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5861" class="Bound">s₁</a> <a id="5864" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5864" class="Bound">s₂</a> <a id="5867" class="Symbol">=</a> <a id="5869" class="Keyword">record</a>
<a id="5878" class="Symbol">{</a> <a id="5880" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="5894" class="Symbol">=</a>
<a id="5902" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3510" class="Function">×-isEquivalence</a> <a id="5918" class="Symbol">(</a><a id="5919" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="5933" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5861" class="Bound">s₁</a><a id="5935" class="Symbol">)</a> <a id="5937" class="Symbol">(</a><a id="5938" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="5952" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5864" class="Bound">s₂</a><a id="5954" class="Symbol">)</a>
<a id="5958" class="Symbol">}</a> <a id="5960" class="Keyword">where</a> <a id="5966" class="Keyword">open</a> <a id="5971" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a>
<a id="×-decSetoid"></a><a id="5979" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5979" class="Function">×-decSetoid</a> <a id="5991" class="Symbol">:</a> <a id="5993" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6003" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="6005" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="6008" class="Symbol">→</a> <a id="6010" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6020" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="6022" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="6025" class="Symbol">→</a> <a id="6027" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="6037" class="Symbol">_</a> <a id="6039" class="Symbol">_</a>
<a id="6041" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5979" class="Function">×-decSetoid</a> <a id="6053" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6053" class="Bound">s₁</a> <a id="6056" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6056" class="Bound">s₂</a> <a id="6059" class="Symbol">=</a> <a id="6061" class="Keyword">record</a>
<a id="6070" class="Symbol">{</a> <a id="6072" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6089" class="Symbol">=</a>
<a id="6097" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#3873" class="Function">×-isDecEquivalence</a> <a id="6116" class="Symbol">(</a><a id="6117" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6134" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6053" class="Bound">s₁</a><a id="6136" class="Symbol">)</a> <a id="6138" class="Symbol">(</a><a id="6139" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="6156" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6056" class="Bound">s₂</a><a id="6158" class="Symbol">)</a>
<a id="6162" class="Symbol">}</a> <a id="6164" class="Keyword">where</a> <a id="6170" class="Keyword">open</a> <a id="6175" href="Relation.Binary.Bundles.html#1548" class="Module">DecSetoid</a>
<a id="×-preorder"></a><a id="6186" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6186" class="Function">×-preorder</a> <a id="6197" class="Symbol">:</a> <a id="6199" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6208" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="6210" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="6213" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="6216" class="Symbol">→</a> <a id="6218" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6227" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="6229" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#796" class="Generalizable">ℓ₃</a> <a id="6232" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#799" class="Generalizable">ℓ₄</a> <a id="6235" class="Symbol">→</a> <a id="6237" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6246" class="Symbol">_</a> <a id="6248" class="Symbol">_</a> <a id="6250" class="Symbol">_</a>
<a id="6252" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6186" class="Function">×-preorder</a> <a id="6263" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6263" class="Bound">p₁</a> <a id="6266" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6266" class="Bound">p₂</a> <a id="6269" class="Symbol">=</a> <a id="6271" class="Keyword">record</a>
<a id="6280" class="Symbol">{</a> <a id="6282" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="6293" class="Symbol">=</a> <a id="6295" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4209" class="Function">×-isPreorder</a> <a id="6308" class="Symbol">(</a><a id="6309" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="6320" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6263" class="Bound">p₁</a><a id="6322" class="Symbol">)</a> <a id="6324" class="Symbol">(</a><a id="6325" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="6336" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6266" class="Bound">p₂</a><a id="6338" class="Symbol">)</a>
<a id="6342" class="Symbol">}</a> <a id="6344" class="Keyword">where</a> <a id="6350" class="Keyword">open</a> <a id="6355" href="Relation.Binary.Bundles.html#2121" class="Module">Preorder</a>
<a id="×-poset"></a><a id="6365" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6365" class="Function">×-poset</a> <a id="6373" class="Symbol">:</a> <a id="6375" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6381" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="6383" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="6386" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="6389" class="Symbol">→</a> <a id="6391" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6397" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="6399" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#796" class="Generalizable">ℓ₃</a> <a id="6402" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#799" class="Generalizable">ℓ₄</a> <a id="6405" class="Symbol">→</a> <a id="6407" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6413" class="Symbol">_</a> <a id="6415" class="Symbol">_</a> <a id="6417" class="Symbol">_</a>
<a id="6419" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6365" class="Function">×-poset</a> <a id="6427" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6427" class="Bound">s₁</a> <a id="6430" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6430" class="Bound">s₂</a> <a id="6433" class="Symbol">=</a> <a id="6435" class="Keyword">record</a>
<a id="6444" class="Symbol">{</a> <a id="6446" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="6461" class="Symbol">=</a> <a id="6463" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#4696" class="Function">×-isPartialOrder</a> <a id="6480" class="Symbol">(</a><a id="6481" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="6496" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6427" class="Bound">s₁</a><a id="6498" class="Symbol">)</a>
<a id="6521" class="Symbol">(</a><a id="6522" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="6537" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6430" class="Bound">s₂</a><a id="6539" class="Symbol">)</a>
<a id="6543" class="Symbol">}</a> <a id="6545" class="Keyword">where</a> <a id="6551" class="Keyword">open</a> <a id="6556" href="Relation.Binary.Bundles.html#3530" class="Module">Poset</a>
<a id="×-strictPartialOrder"></a><a id="6563" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6563" class="Function">×-strictPartialOrder</a> <a id="6584" class="Symbol">:</a> <a id="6586" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6605" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="6607" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="6610" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="6613" class="Symbol">→</a>
<a id="6638" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6657" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="6659" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#796" class="Generalizable">ℓ₃</a> <a id="6662" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#799" class="Generalizable">ℓ₄</a> <a id="6665" class="Symbol">→</a>
<a id="6690" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="6709" class="Symbol">_</a> <a id="6711" class="Symbol">_</a> <a id="6713" class="Symbol">_</a>
<a id="6715" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6563" class="Function">×-strictPartialOrder</a> <a id="6736" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6736" class="Bound">s₁</a> <a id="6739" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6739" class="Bound">s₂</a> <a id="6742" class="Symbol">=</a> <a id="6744" class="Keyword">record</a>
<a id="6753" class="Symbol">{</a> <a id="6755" href="Relation.Binary.Bundles.html#5045" class="Field">isStrictPartialOrder</a> <a id="6776" class="Symbol">=</a> <a id="6778" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5068" class="Function">×-isStrictPartialOrder</a>
<a id="6830" class="Symbol">(</a><a id="6831" href="Relation.Binary.Bundles.html#5045" class="Field">isStrictPartialOrder</a> <a id="6852" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6736" class="Bound">s₁</a><a id="6854" class="Symbol">)</a>
<a id="6885" class="Symbol">(</a><a id="6886" href="Relation.Binary.Bundles.html#5045" class="Field">isStrictPartialOrder</a> <a id="6907" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#6739" class="Bound">s₂</a><a id="6909" class="Symbol">)</a>
<a id="6913" class="Symbol">}</a> <a id="6915" class="Keyword">where</a> <a id="6921" class="Keyword">open</a> <a id="6926" href="Relation.Binary.Bundles.html#4839" class="Module">StrictPartialOrder</a>
<a id="6946" class="Comment">------------------------------------------------------------------------</a>
<a id="7019" class="Comment">-- Additional notation</a>
<a id="7043" class="Comment">-- Infix combining setoids</a>
<a id="7070" class="Keyword">infix</a> <a id="7076" class="Number">4</a> <a id="7078" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7083" class="Function Operator">_×ₛ_</a>
<a id="_×ₛ_"></a><a id="7083" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7083" class="Function Operator">_×ₛ_</a> <a id="7088" class="Symbol">:</a> <a id="7090" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7097" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#786" class="Generalizable">a</a> <a id="7099" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#790" class="Generalizable">ℓ₁</a> <a id="7102" class="Symbol">→</a> <a id="7104" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7111" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#788" class="Generalizable">b</a> <a id="7113" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#793" class="Generalizable">ℓ₂</a> <a id="7116" class="Symbol">→</a> <a id="7118" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="7125" class="Symbol">_</a> <a id="7127" class="Symbol">_</a>
<a id="7129" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7083" class="Function Operator">_×ₛ_</a> <a id="7134" class="Symbol">=</a> <a id="7136" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#5802" class="Function">×-setoid</a>
<a id="7146" class="Comment">------------------------------------------------------------------------</a>
<a id="7219" class="Comment">-- The propositional equality setoid over products can be</a>
<a id="7277" class="Comment">-- decomposed using ×-Rel</a>
<a id="≡×≡⇒≡"></a><a id="7304" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7304" class="Function">≡×≡⇒≡</a> <a id="7310" class="Symbol">:</a> <a id="7312" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="7322" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7326" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7330" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="7332" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7336" class="Symbol">{</a><a id="7337" class="Argument">A</a> <a id="7339" class="Symbol">=</a> <a id="7341" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="7343" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="7345" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a><a id="7346" class="Symbol">}</a>
<a id="7348" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7304" class="Function">≡×≡⇒≡</a> <a id="7354" class="Symbol">(</a><a id="7355" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a> <a id="7362" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="7364" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a><a id="7370" class="Symbol">)</a> <a id="7372" class="Symbol">=</a> <a id="7374" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a>
<a id="≡⇒≡×≡"></a><a id="7382" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7382" class="Function">≡⇒≡×≡</a> <a id="7388" class="Symbol">:</a> <a id="7390" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7394" class="Symbol">{</a><a id="7395" class="Argument">A</a> <a id="7397" class="Symbol">=</a> <a id="7399" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="7401" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="7403" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a><a id="7404" class="Symbol">}</a> <a id="7406" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="7408" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#940" class="Function">Pointwise</a> <a id="7418" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="7422" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="7426" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7382" class="Function">≡⇒≡×≡</a> <a id="7432" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a> <a id="7439" class="Symbol">=</a> <a id="7441" class="Symbol">(</a><a id="7442" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a> <a id="7449" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="7451" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">P.refl</a><a id="7457" class="Symbol">)</a>
<a id="Pointwise-≡↔≡"></a><a id="7460" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7460" class="Function">Pointwise-≡↔≡</a> <a id="7474" class="Symbol">:</a> <a id="7476" href="Function.Bundles.html#7340" class="Record">Inverse</a> <a id="7484" class="Symbol">(</a><a id="7485" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="7494" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="7496" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7083" class="Function Operator">×ₛ</a> <a id="7499" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="7508" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a><a id="7509" class="Symbol">)</a> <a id="7511" class="Symbol">(</a><a id="7512" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">P.setoid</a> <a id="7521" class="Symbol">(</a><a id="7522" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#814" class="Generalizable">A</a> <a id="7524" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="7526" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#816" class="Generalizable">B</a><a id="7527" class="Symbol">))</a>
<a id="7530" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7460" class="Function">Pointwise-≡↔≡</a> <a id="7544" class="Symbol">=</a> <a id="7546" class="Keyword">record</a>
<a id="7555" class="Symbol">{</a> <a id="7557" href="Function.Bundles.html#7394" class="Field">to</a> <a id="7568" class="Symbol">=</a> <a id="7570" href="Function.Base.html#704" class="Function">id</a>
<a id="7575" class="Symbol">;</a> <a id="7577" href="Function.Bundles.html#7418" class="Field">from</a> <a id="7588" class="Symbol">=</a> <a id="7590" href="Function.Base.html#704" class="Function">id</a>
<a id="7595" class="Symbol">;</a> <a id="7597" href="Function.Bundles.html#7442" class="Field">to-cong</a> <a id="7608" class="Symbol">=</a> <a id="7610" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7304" class="Function">≡×≡⇒≡</a>
<a id="7618" class="Symbol">;</a> <a id="7620" href="Function.Bundles.html#7483" class="Field">from-cong</a> <a id="7631" class="Symbol">=</a> <a id="7633" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7382" class="Function">≡⇒≡×≡</a>
<a id="7641" class="Symbol">;</a> <a id="7643" href="Function.Bundles.html#7526" class="Field">inverse</a> <a id="7654" class="Symbol">=</a> <a id="7656" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7304" class="Function">≡×≡⇒≡</a> <a id="7662" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="7664" href="Data.Product.Relation.Binary.Pointwise.NonDependent.html#7382" class="Function">≡⇒≡×≡</a>
<a id="7672" class="Symbol">}</a>
</pre></body></html>
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