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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Object.NaturalNumbers.Properties.Parametrized.html
Leon Vatthauer 2bf4655eae
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<html><head><meta charset="utf-8"><title>Categories.Object.NaturalNumbers.Properties.Parametrized</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a>
<a id="74" class="Keyword">open</a> <a id="79" class="Keyword">import</a> <a id="86" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a> <a id="113" class="Keyword">using</a> <a id="119" class="Symbol">(</a><a id="120" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a><a id="128" class="Symbol">)</a>
<a id="130" class="Keyword">open</a> <a id="135" class="Keyword">import</a> <a id="142" href="Categories.Category.CartesianClosed.Bundle.html" class="Module">Categories.Category.CartesianClosed.Bundle</a> <a id="185" class="Keyword">using</a> <a id="191" class="Symbol">(</a><a id="192" href="Categories.Category.CartesianClosed.Bundle.html#346" class="Record">CartesianClosedCategory</a><a id="215" class="Symbol">)</a>
<a id="217" class="Keyword">open</a> <a id="222" class="Keyword">import</a> <a id="229" href="Categories.Category.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="259" class="Keyword">using</a> <a id="265" class="Symbol">(</a><a id="266" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="275" class="Symbol">)</a>
<a id="277" class="Keyword">open</a> <a id="282" class="Keyword">import</a> <a id="289" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="324" class="Keyword">using</a> <a id="330" class="Symbol">(</a><a id="331" href="Categories.Category.BinaryProducts.html#848" class="Record">BinaryProducts</a><a id="345" class="Symbol">)</a>
<a id="347" class="Keyword">open</a> <a id="352" class="Keyword">import</a> <a id="359" href="Categories.Category.Cocartesian.html" class="Module">Categories.Category.Cocartesian</a> <a id="391" class="Keyword">using</a> <a id="397" class="Symbol">(</a><a id="398" href="Categories.Category.Cocartesian.html#1360" class="Record">BinaryCoproducts</a><a id="414" class="Symbol">)</a>
<a id="416" class="Keyword">open</a> <a id="421" class="Keyword">import</a> <a id="428" href="Categories.Category.CartesianClosed.html" class="Module">Categories.Category.CartesianClosed</a> <a id="464" class="Keyword">using</a> <a id="470" class="Symbol">(</a><a id="471" href="Categories.Category.CartesianClosed.html#1233" class="Record">CartesianClosed</a><a id="486" class="Symbol">)</a>
<a id="489" class="Comment">-- In CCCs NNOs and PNNOs coincide</a>
<a id="525" class="Keyword">module</a> <a id="532" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html" class="Module">Categories.Object.NaturalNumbers.Properties.Parametrized</a> <a id="589" class="Symbol">{</a><a id="590" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#590" class="Bound">o</a> <a id="592" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#592" class="Bound">ℓ</a> <a id="594" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#594" class="Bound">e</a><a id="595" class="Symbol">}</a> <a id="597" class="Symbol">(</a><a id="598" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#598" class="Bound">CCC</a> <a id="602" class="Symbol">:</a> <a id="604" href="Categories.Category.CartesianClosed.Bundle.html#346" class="Record">CartesianClosedCategory</a> <a id="628" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#590" class="Bound">o</a> <a id="630" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#592" class="Bound">ℓ</a> <a id="632" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#594" class="Bound">e</a><a id="633" class="Symbol">)</a> <a id="635" class="Symbol">(</a><a id="636" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#636" class="Bound">𝒞-Coproducts</a> <a id="649" class="Symbol">:</a> <a id="651" href="Categories.Category.Cocartesian.html#1360" class="Record">BinaryCoproducts</a> <a id="668" class="Symbol">(</a><a id="669" href="Categories.Category.CartesianClosed.Bundle.html#418" class="Field">CartesianClosedCategory.U</a> <a id="695" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#598" class="Bound">CCC</a><a id="698" class="Symbol">))</a> <a id="701" class="Keyword">where</a>
<a id="708" class="Keyword">open</a> <a id="713" class="Keyword">import</a> <a id="720" href="Level.html" class="Module">Level</a>
<a id="727" class="Keyword">open</a> <a id="732" href="Categories.Category.CartesianClosed.Bundle.html#346" class="Module">CartesianClosedCategory</a> <a id="756" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#598" class="Bound">CCC</a> <a id="760" class="Keyword">renaming</a> <a id="769" class="Symbol">(</a><a id="770" href="Categories.Category.CartesianClosed.Bundle.html#418" class="Field">U</a> <a id="772" class="Symbol">to</a> <a id="775" class="Field">𝒞</a><a id="776" class="Symbol">)</a>
<a id="778" class="Keyword">open</a> <a id="783" href="Categories.Category.CartesianClosed.html#1233" class="Module">CartesianClosed</a> <a id="799" href="Categories.Category.CartesianClosed.Bundle.html#470" class="Function">cartesianClosed</a> <a id="815" class="Keyword">using</a> <a id="821" class="Symbol">(</a><a id="822" href="Categories.Category.CartesianClosed.html#1408" class="Field">cartesian</a><a id="831" class="Symbol">;</a> <a id="833" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a><a id="835" class="Symbol">;</a> <a id="837" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="842" class="Symbol">;</a> <a id="844" href="Categories.Category.CartesianClosed.html#4217" class="Function">β′</a><a id="846" class="Symbol">;</a> <a id="848" href="Categories.Category.CartesianClosed.html#2254" class="Function">λ-inj</a><a id="853" class="Symbol">;</a> <a id="855" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a><a id="861" class="Symbol">;</a> <a id="863" href="Categories.Category.CartesianClosed.html#4406" class="Function">η-exp′</a><a id="869" class="Symbol">;</a> <a id="871" href="Categories.Category.CartesianClosed.html#3957" class="Function">λ-unique′</a><a id="880" class="Symbol">;</a> <a id="882" href="Categories.Category.CartesianClosed.html#2454" class="Function">subst</a><a id="887" class="Symbol">)</a>
<a id="889" class="Keyword">open</a> <a id="894" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="904" href="Categories.Category.CartesianClosed.html#1408" class="Function">cartesian</a> <a id="914" class="Keyword">using</a> <a id="920" class="Symbol">(</a><a id="921" href="Categories.Category.Cartesian.html#777" class="Field">terminal</a><a id="929" class="Symbol">;</a> <a id="931" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="939" class="Symbol">)</a>
<a id="941" class="Keyword">open</a> <a id="946" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="961" href="Categories.Category.Cartesian.html#801" class="Function">products</a> <a id="970" class="Keyword">renaming</a> <a id="979" class="Symbol">(</a><a id="980" href="Categories.Object.Product.Core.html#1170" class="Function">unique′</a> <a id="988" class="Symbol">to</a> <a id="991" class="Function">bp-unique′</a><a id="1001" class="Symbol">)</a>
<a id="1004" class="Keyword">open</a> <a id="1009" class="Keyword">import</a> <a id="1016" href="Categories.Object.NaturalNumbers.html" class="Module">Categories.Object.NaturalNumbers</a> <a id="1049" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#775" class="Function">𝒞</a> <a id="1051" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="1060" class="Keyword">using</a> <a id="1066" class="Symbol">(</a><a id="1067" href="Categories.Object.NaturalNumbers.html#1072" class="Record">NNO</a><a id="1070" class="Symbol">)</a>
<a id="1072" class="Keyword">open</a> <a id="1077" class="Keyword">import</a> <a id="1084" href="Categories.Object.NaturalNumbers.Parametrized.html" class="Module">Categories.Object.NaturalNumbers.Parametrized</a> <a id="1130" href="Categories.Category.CartesianClosed.Bundle.html#535" class="Function">cartesianCategory</a> <a id="1148" class="Keyword">using</a> <a id="1154" class="Symbol">(</a><a id="1155" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a><a id="1170" class="Symbol">)</a>
<a id="1172" class="Keyword">open</a> <a id="1177" class="Keyword">import</a> <a id="1184" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="1204" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#775" class="Function">𝒞</a>
<a id="1206" class="Keyword">open</a> <a id="1211" class="Keyword">import</a> <a id="1218" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="1248" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#775" class="Function">𝒞</a>
<a id="1251" class="Keyword">open</a> <a id="1256" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1269" class="Keyword">open</a> <a id="1274" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="1281" class="Keyword">open</a> <a id="1286" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1295" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>
<a id="NNO×CCC⇒PNNO"></a><a id="1305" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1305" class="Function">NNO×CCC⇒PNNO</a> <a id="1318" class="Symbol">:</a> <a id="1320" href="Categories.Object.NaturalNumbers.html#1072" class="Record">NNO</a> <a id="1324" class="Symbol">→</a> <a id="1326" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a>
<a id="1342" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1305" class="Function">NNO×CCC⇒PNNO</a> <a id="1355" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1355" class="Bound">nno</a> <a id="1359" class="Symbol">=</a> <a id="1361" class="Keyword">record</a>
<a id="1371" class="Symbol">{</a> <a id="1373" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Field">N</a> <a id="1375" class="Symbol">=</a> <a id="1377" href="Categories.Object.NaturalNumbers.html#1112" class="Field">N</a>
<a id="1382" class="Symbol">;</a> <a id="1384" href="Categories.Object.NaturalNumbers.Parametrized.html#2940" class="Field">isParametrizedNNO</a> <a id="1402" class="Symbol">=</a> <a id="1404" class="Keyword">record</a>
<a id="1415" class="Symbol">{</a> <a id="1417" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Field">z</a> <a id="1419" class="Symbol">=</a> <a id="1421" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a>
<a id="1427" class="Symbol">;</a> <a id="1429" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Field">s</a> <a id="1431" class="Symbol">=</a> <a id="1433" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a>
<a id="1439" class="Symbol">;</a> <a id="1441" href="Categories.Object.NaturalNumbers.Parametrized.html#988" class="Field">universal</a> <a id="1451" class="Symbol">=</a> <a id="1453" class="Symbol">λ</a> <a id="1455" class="Symbol">{</a><a id="1456" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1456" class="Bound">A</a><a id="1457" class="Symbol">}</a> <a id="1459" class="Symbol">{</a><a id="1460" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1460" class="Bound">X</a><a id="1461" class="Symbol">}</a> <a id="1463" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1463" class="Bound">f</a> <a id="1465" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1465" class="Bound">g</a> <a id="1467" class="Symbol">→</a> <a id="1469" class="Symbol">(</a><a id="1470" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="1476" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1478" class="Symbol">(</a><a id="1479" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="1489" class="Symbol">(</a><a id="1490" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="1493" class="Symbol">(</a><a id="1494" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1463" class="Bound">f</a> <a id="1496" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1498" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="1500" class="Symbol">))</a> <a id="1503" class="Symbol">(</a><a id="1504" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="1507" class="Symbol">(</a><a id="1508" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1465" class="Bound">g</a> <a id="1510" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1512" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="1517" class="Symbol">))</a> <a id="1520" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1522" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1524" class="Symbol">))</a> <a id="1527" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1529" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a>
<a id="1538" class="Symbol">;</a> <a id="1540" href="Categories.Object.NaturalNumbers.Parametrized.html#1040" class="Field">commute₁</a> <a id="1549" class="Symbol">=</a> <a id="1551" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1684" class="Function">commute₁&#39;</a>
<a id="1565" class="Symbol">;</a> <a id="1567" href="Categories.Object.NaturalNumbers.Parametrized.html#1124" class="Field">commute₂</a> <a id="1576" class="Symbol">=</a> <a id="1578" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#2667" class="Function">commute₂&#39;</a>
<a id="1592" class="Symbol">;</a> <a id="1594" href="Categories.Object.NaturalNumbers.Parametrized.html#1222" class="Field">unique</a> <a id="1601" class="Symbol">=</a> <a id="1603" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3782" class="Function">unique&#39;</a>
<a id="1615" class="Symbol">}</a>
<a id="1620" class="Symbol">}</a>
<a id="1624" class="Keyword">where</a>
<a id="1634" class="Keyword">open</a> <a id="1639" href="Categories.Object.NaturalNumbers.html#1072" class="Module">NNO</a> <a id="1643" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1355" class="Bound">nno</a> <a id="1647" class="Keyword">renaming</a> <a id="1656" class="Symbol">(</a><a id="1657" href="Categories.Object.NaturalNumbers.html#694" class="Function">unique</a> <a id="1664" class="Symbol">to</a> <a id="1667" class="Function">nno-unique</a><a id="1677" class="Symbol">)</a>
<a id="1684" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1684" class="Function">commute₁&#39;</a> <a id="1694" class="Symbol">:</a> <a id="1696" class="Symbol">∀</a> <a id="1698" class="Symbol">{</a><a id="1699" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1699" class="Bound">A</a> <a id="1701" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1701" class="Bound">X</a><a id="1702" class="Symbol">}</a> <a id="1704" class="Symbol">{</a><a id="1705" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1705" class="Bound">f</a> <a id="1707" class="Symbol">:</a> <a id="1709" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1699" class="Bound">A</a> <a id="1711" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1713" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1701" class="Bound">X</a><a id="1714" class="Symbol">}</a> <a id="1716" class="Symbol">{</a><a id="1717" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1717" class="Bound">g</a> <a id="1719" class="Symbol">:</a> <a id="1721" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1701" class="Bound">X</a> <a id="1723" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1725" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1701" class="Bound">X</a><a id="1726" class="Symbol">}</a> <a id="1728" class="Symbol">→</a> <a id="1730" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1705" class="Bound">f</a> <a id="1732" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1734" class="Symbol">((</a><a id="1736" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="1742" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1744" class="Symbol">(</a><a id="1745" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="1755" class="Symbol">(</a><a id="1756" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="1759" class="Symbol">(</a><a id="1760" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1705" class="Bound">f</a> <a id="1762" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1764" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="1766" class="Symbol">))</a> <a id="1769" class="Symbol">(</a><a id="1770" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="1773" class="Symbol">(</a><a id="1774" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1717" class="Bound">g</a> <a id="1776" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1778" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="1783" class="Symbol">))</a> <a id="1786" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1788" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1790" class="Symbol">))</a> <a id="1793" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1795" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="1799" class="Symbol">)</a> <a id="1801" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1803" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1805" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1808" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1810" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a> <a id="1812" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1814" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1816" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a>
<a id="1822" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1684" class="Function">commute₁&#39;</a> <a id="1832" class="Symbol">{</a><a id="1833" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1833" class="Bound">A</a><a id="1834" class="Symbol">}</a> <a id="1836" class="Symbol">{</a><a id="1837" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1837" class="Bound">X</a><a id="1838" class="Symbol">}</a> <a id="1840" class="Symbol">{</a><a id="1841" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1841" class="Bound">f</a><a id="1842" class="Symbol">}</a> <a id="1844" class="Symbol">{</a><a id="1845" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1845" class="Bound">g</a><a id="1846" class="Symbol">}</a> <a id="1848" class="Symbol">=</a> <a id="1850" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1863" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1841" class="Bound">f</a> <a id="1947" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1950" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="1957" href="Categories.Object.Product.Core.html#637" class="Function">project₂</a> <a id="1966" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1974" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#1841" class="Bound">f</a> <a id="1976" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1978" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1981" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1983" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1985" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1987" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1989" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1992" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2058" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2061" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2067" class="Symbol">(</a><a id="2068" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2070" href="Categories.Category.CartesianClosed.html#4217" class="Function">β′</a><a id="2072" class="Symbol">)</a> <a id="2074" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="3160" class="Symbol">(</a><a id="3161" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="3167" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3169" class="Symbol">(</a><a id="3170" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="3173" class="Symbol">(</a><a id="3174" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#2898" class="Bound">g</a> <a id="3176" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3178" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="3183" class="Symbol">)</a> <a id="3185" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3188" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="3198" class="Symbol">(</a><a id="3199" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="3202" class="Symbol">(</a><a id="3203" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#2894" class="Bound">f</a> <a id="3205" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3207" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="3209" class="Symbol">))</a> <a id="3212" class="Symbol">(</a><a id="3213" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="3216" class="Symbol">(</a><a id="3217" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#2898" class="Bound">g</a> <a id="3219" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3221" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="3226" class="Symbol">))</a> <a id="3229" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3231" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3234" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3236" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3238" class="Symbol">))</a> <a id="3241" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3243" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="3251" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3254" class="Symbol">(</a><a id="3255" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3263" class="Symbol">(</a><a id="3264" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="3272" href="Categories.Object.NaturalNumbers.html#608" class="Function">s-commute</a> <a id="3282" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="3286" class="Symbol">))</a> <a id="3289" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3297" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="5180" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5183" class="Symbol">(</a><a id="5184" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3991" class="Bound">g</a> <a id="5186" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5188" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="5193" class="Symbol">)</a> <a id="5195" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5197" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5200" class="Symbol">(</a><a id="5201" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3995" class="Bound">u</a> <a id="5203" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5205" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5209" class="Symbol">)</a> <a id="5220" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5223" href="Categories.Category.CartesianClosed.html#2454" class="Function">subst</a> <a id="5229" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5242" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5245" class="Symbol">((</a><a id="5247" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3991" class="Bound">g</a> <a id="5249" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5251" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="5256" class="Symbol">)</a> <a id="5258" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5260" class="Symbol">(</a><a id="5261" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5264" class="Symbol">(</a><a id="5265" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3995" class="Bound">u</a> <a id="5267" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5269" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5273" class="Symbol">)</a> <a id="5275" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5277" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5279" class="Symbol">))</a> <a id="5282" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5285" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="5292" class="Symbol">(</a><a id="5293" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5299" href="Categories.Category.CartesianClosed.html#4217" class="Function">β′</a> <a id="5302" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5304" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5310" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#4002" class="Bound">eqˢ</a><a id="5313" class="Symbol">)</a> <a id="5315" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5327" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5330" class="Symbol">((</a><a id="5332" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3995" class="Bound">u</a> <a id="5334" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5336" class="Symbol">(</a><a id="5337" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5340" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5342" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a><a id="5343" class="Symbol">))</a> <a id="5346" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5348" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5352" class="Symbol">)</a> <a id="5367" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5370" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="5377" class="Symbol">(</a><a id="5378" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5384" class="Symbol">(</a><a id="5385" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="5387" href="Categories.Category.BinaryProducts.html#4512" class="Function">swap∘⁂</a><a id="5393" class="Symbol">)</a> <a id="5395" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="5397" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a><a id="5406" class="Symbol">)</a> <a id="5408" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5420" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5423" class="Symbol">((</a><a id="5425" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3995" class="Bound">u</a> <a id="5427" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5429" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5433" class="Symbol">)</a> <a id="5435" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5437" class="Symbol">(</a><a id="5438" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="5440" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5442" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5444" class="Symbol">))</a> <a id="5460" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5464" href="Categories.Category.CartesianClosed.html#2454" class="Function">subst</a> <a id="5470" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5482" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5485" class="Symbol">(</a><a id="5486" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3995" class="Bound">u</a> <a id="5488" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5490" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5494" class="Symbol">)</a> <a id="5496" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5498" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="5522" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="5523" class="Symbol">)</a> <a id="5525" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5534" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="5544" class="Symbol">(</a><a id="5545" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5548" class="Symbol">(</a><a id="5549" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3987" class="Bound">f</a> <a id="5551" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5553" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="5555" class="Symbol">))</a> <a id="5558" class="Symbol">(</a><a id="5559" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5562" class="Symbol">(</a><a id="5563" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3991" class="Bound">g</a> <a id="5565" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5567" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="5572" class="Symbol">))</a> <a id="5613" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5617" href="Categories.Category.CartesianClosed.html#4406" class="Function">η-exp′</a> <a id="5624" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5632" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5635" class="Symbol">(</a><a id="5636" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="5642" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5644" class="Symbol">(</a><a id="5645" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="5655" class="Symbol">(</a><a id="5656" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5659" class="Symbol">(</a><a id="5660" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3987" class="Bound">f</a> <a id="5662" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5664" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="5666" class="Symbol">))</a> <a id="5669" class="Symbol">(</a><a id="5670" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5673" class="Symbol">(</a><a id="5674" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3991" class="Bound">g</a> <a id="5676" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5678" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="5683" class="Symbol">))</a> <a id="5686" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5688" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5690" class="Symbol">))</a> <a id="5711" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5715" href="Categories.Category.CartesianClosed.html#2186" class="Function">λ-cong</a> <a id="5722" class="Symbol">(</a><a id="5723" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="5731" href="Categories.Category.BinaryProducts.html#4688" class="Function">swap∘swap</a><a id="5740" class="Symbol">)</a> <a id="5742" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5750" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5753" class="Symbol">(((</a><a id="5756" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a> <a id="5762" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5764" class="Symbol">(</a><a id="5765" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="5775" class="Symbol">(</a><a id="5776" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5779" class="Symbol">(</a><a id="5780" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3987" class="Bound">f</a> <a id="5782" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5784" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="5786" class="Symbol">))</a> <a id="5789" class="Symbol">(</a><a id="5790" href="Categories.Category.CartesianClosed.html#2130" class="Function">λg</a> <a id="5793" class="Symbol">(</a><a id="5794" href="Categories.Object.NaturalNumbers.Properties.Parametrized.html#3991" class="Bound">g</a> <a id="5796" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5798" href="Categories.Category.CartesianClosed.html#3883" class="Function">eval′</a><a id="5803" class="Symbol">))</a> <a id="5806" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5808" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5810" class="Symbol">))</a> <a id="5813" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5815" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5819" class="Symbol">)</a> <a id="5821" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5823" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5827" class="Symbol">)</a> <a id="5829" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="5830" class="Symbol">))</a>
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