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<html><head><meta charset="utf-8"><title>Categories.Object.NaturalNumbers.Properties.F-Algebras</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="36" class="Keyword">module</a> <a id="43" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html" class="Module">Categories.Object.NaturalNumbers.Properties.F-Algebras</a> <a id="98" class="Keyword">where</a>
<a id="105" class="Keyword">open</a> <a id="110" class="Keyword">import</a> <a id="117" href="Level.html" class="Module">Level</a>
<a id="123" class="Keyword">open</a> <a id="128" class="Keyword">import</a> <a id="135" href="Function.html" class="Module">Function</a> <a id="144" class="Keyword">using</a> <a id="150" class="Symbol">(</a><a id="151" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="154" class="Symbol">)</a>
<a id="157" class="Keyword">open</a> <a id="162" class="Keyword">import</a> <a id="169" href="Categories.Category.Core.html" class="Module">Categories.Category.Core</a>
<a id="194" class="Keyword">open</a> <a id="199" class="Keyword">import</a> <a id="206" href="Categories.Category.Construction.F-Algebras.html" class="Module">Categories.Category.Construction.F-Algebras</a> <a id="250" class="Keyword">using</a> <a id="256" class="Symbol">(</a><a id="257" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a><a id="267" class="Symbol">)</a>
<a id="269" class="Keyword">open</a> <a id="274" class="Keyword">import</a> <a id="281" href="Categories.Category.Cocartesian.html" class="Module">Categories.Category.Cocartesian</a> <a id="313" class="Keyword">using</a> <a id="319" class="Symbol">(</a><a id="320" href="Categories.Category.Cocartesian.html#1360" class="Record">BinaryCoproducts</a><a id="336" class="Symbol">)</a>
<a id="338" class="Keyword">open</a> <a id="343" class="Keyword">import</a> <a id="350" href="Categories.Category.Cartesian.Bundle.html" class="Module">Categories.Category.Cartesian.Bundle</a> <a id="387" class="Keyword">using</a> <a id="393" class="Symbol">(</a><a id="394" href="Categories.Category.Cartesian.Bundle.html#422" class="Record">CartesianCategory</a><a id="411" class="Symbol">)</a>
<a id="413" class="Keyword">open</a> <a id="418" class="Keyword">import</a> <a id="425" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="460" class="Keyword">using</a> <a id="466" class="Symbol">(</a><a id="467" href="Categories.Category.BinaryProducts.html#848" class="Record">BinaryProducts</a><a id="481" class="Symbol">)</a>
<a id="483" class="Keyword">open</a> <a id="488" class="Keyword">import</a> <a id="495" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="514" class="Keyword">using</a> <a id="520" class="Symbol">(</a><a id="521" href="Categories.Functor.html#283" class="Function">Endofunctor</a><a id="532" class="Symbol">;</a> <a id="534" href="Categories.Functor.Core.html#248" class="Record">Functor</a><a id="541" class="Symbol">)</a>
<a id="543" class="Keyword">open</a> <a id="548" class="Keyword">import</a> <a id="555" href="Categories.Functor.Algebra.html" class="Module">Categories.Functor.Algebra</a> <a id="582" class="Keyword">using</a> <a id="588" class="Symbol">(</a><a id="589" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a><a id="598" class="Symbol">;</a> <a id="600" href="Categories.Functor.Algebra.html#1307" class="Record">F-Algebra-Morphism</a><a id="618" class="Symbol">)</a>
<a id="620" class="Keyword">open</a> <a id="625" class="Keyword">import</a> <a id="632" href="Categories.Object.Terminal.html" class="Module">Categories.Object.Terminal</a> <a id="659" class="Keyword">using</a> <a id="665" class="Symbol">(</a><a id="666" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a><a id="674" class="Symbol">)</a>
<a id="676" class="Keyword">open</a> <a id="681" class="Keyword">import</a> <a id="688" href="Categories.Object.Initial.html" class="Module">Categories.Object.Initial</a> <a id="714" class="Keyword">using</a> <a id="720" class="Symbol">(</a><a id="721" href="Categories.Object.Initial.html#760" class="Record">Initial</a><a id="728" class="Symbol">;</a> <a id="730" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a><a id="739" class="Symbol">)</a>
<a id="742" class="Keyword">import</a> <a id="749" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="779" class="Symbol">as</a> <a id="782" class="Module">MR</a>
<a id="785" class="Keyword">import</a> <a id="792" href="Categories.Object.NaturalNumbers.html" class="Module">Categories.Object.NaturalNumbers</a> <a id="825" class="Symbol">as</a> <a id="828" class="Module">NNOs</a>
<a id="833" class="Keyword">import</a> <a id="840" href="Categories.Object.NaturalNumbers.Parametrized.html" class="Module">Categories.Object.NaturalNumbers.Parametrized</a> <a id="886" class="Symbol">as</a> <a id="889" class="Module">PNNO</a>
<a id="895" class="Comment">-- A NNO is an inital algebra for the &#39;X ↦ ⊤ + X&#39; endofunctor.</a>
<a id="958" class="Keyword">module</a> <a id="965" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#965" class="Module">_</a> <a id="967" class="Symbol">{</a><a id="968" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#968" class="Bound">o</a> <a id="970" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#970" class="Bound">ℓ</a> <a id="972" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#972" class="Bound">e</a><a id="973" class="Symbol">}</a> <a id="975" class="Symbol">(</a><a id="976" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a> <a id="978" class="Symbol">:</a> <a id="980" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="989" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#968" class="Bound">o</a> <a id="991" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#970" class="Bound">ℓ</a> <a id="993" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#972" class="Bound">e</a><a id="994" class="Symbol">)</a> <a id="996" class="Symbol">(</a><a id="997" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#997" class="Bound">𝒞-Terminal</a> <a id="1008" class="Symbol">:</a> <a id="1010" href="Categories.Object.Terminal.html#860" class="Record">Terminal</a> <a id="1019" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a><a id="1020" class="Symbol">)</a> <a id="1022" class="Symbol">(</a><a id="1023" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1023" class="Bound">𝒞-Coproducts</a> <a id="1036" class="Symbol">:</a> <a id="1038" href="Categories.Category.Cocartesian.html#1360" class="Record">BinaryCoproducts</a> <a id="1055" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a><a id="1056" class="Symbol">)</a> <a id="1058" class="Keyword">where</a>
<a id="1067" class="Keyword">open</a> <a id="1072" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1081" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#997" class="Bound">𝒞-Terminal</a>
<a id="1094" class="Keyword">open</a> <a id="1099" href="Categories.Category.Cocartesian.html#1360" class="Module">BinaryCoproducts</a> <a id="1116" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1023" class="Bound">𝒞-Coproducts</a>
<a id="1131" class="Keyword">open</a> <a id="1136" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1145" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a>
<a id="1149" class="Keyword">open</a> <a id="1154" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1169" class="Keyword">open</a> <a id="1174" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="1182" class="Keyword">open</a> <a id="1187" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1190" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a>
<a id="1194" class="Keyword">open</a> <a id="1199" href="Categories.Object.NaturalNumbers.html" class="Module">NNOs</a> <a id="1204" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a> <a id="1206" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#997" class="Bound">𝒞-Terminal</a>
<a id="1220" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a> <a id="1226" class="Symbol">:</a> <a id="1228" href="Categories.Functor.Core.html#248" class="Record">Functor</a> <a id="1236" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a> <a id="1238" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#976" class="Bound">𝒞</a>
<a id="1242" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a> <a id="1248" class="Symbol">=</a> <a id="1250" class="Keyword">record</a>
<a id="1261" class="Symbol">{</a> <a id="1263" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="1266" class="Symbol">=</a> <a id="1268" class="Symbol">λ</a> <a id="1270" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1270" class="Bound">X</a> <a id="1272" class="Symbol">→</a> <a id="1274" href="Categories.Object.Terminal.html#905" class="Function">⊤</a> <a id="1276" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1278" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1270" class="Bound">X</a>
<a id="1284" class="Symbol">;</a> <a id="1286" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1289" class="Symbol">=</a> <a id="1291" class="Symbol">λ</a> <a id="1293" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1293" class="Bound">f</a> <a id="1295" class="Symbol">→</a> <a id="1297" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1299" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1302" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1304" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1307" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1309" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1293" class="Bound">f</a> <a id="1311" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="1317" class="Symbol">;</a> <a id="1319" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="1328" class="Symbol">=</a> <a id="1330" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="1339" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="1344" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="1354" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1356" href="Categories.Category.Cocartesian.html#1729" class="Function">+-η</a>
<a id="1365" class="Symbol">;</a> <a id="1367" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="1380" class="Symbol">=</a> <a id="1382" href="Categories.Category.Cocartesian.html#1714" class="Function">+-unique</a> <a id="1391" class="Symbol">(</a><a id="1392" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1398" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="1406" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1408" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="1415" class="Symbol">)</a> <a id="1417" class="Symbol">(</a><a id="1418" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1424" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="1432" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1434" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1440" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="1448" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1450" href="Categories.Category.Core.html#715" class="Function">assoc</a><a id="1455" class="Symbol">)</a>
<a id="1461" class="Symbol">;</a> <a id="1463" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="1472" class="Symbol">=</a> <a id="1474" class="Symbol">λ</a> <a id="1476" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1476" class="Bound">eq</a> <a id="1479" class="Symbol">→</a> <a id="1481" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="1490" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="1495" class="Symbol">(</a><a id="1496" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="1506" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1476" class="Bound">eq</a><a id="1508" class="Symbol">)</a>
<a id="1514" class="Symbol">}</a>
<a id="1519" class="Keyword">private</a>
<a id="1531" class="Keyword">module</a> <a id="1538" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1538" class="Module">Maybe</a> <a id="1544" class="Symbol">=</a> <a id="1546" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1554" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a>
<a id="1563" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1563" class="Function">Initial⇒NNO</a> <a id="1575" class="Symbol">:</a> <a id="1577" href="Categories.Object.Initial.html#760" class="Record">Initial</a> <a id="1585" class="Symbol">(</a><a id="1586" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a> <a id="1597" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a><a id="1602" class="Symbol">)</a> <a id="1604" class="Symbol">→</a> <a id="1606" href="Categories.Object.NaturalNumbers.html#1072" class="Record">NNO</a>
<a id="1612" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1563" class="Function">Initial⇒NNO</a> <a id="1624" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1624" class="Bound">initial</a> <a id="1632" class="Symbol">=</a> <a id="1634" class="Keyword">record</a>
<a id="1645" class="Symbol">{</a> <a id="1647" href="Categories.Object.NaturalNumbers.html#1112" class="Field">N</a> <a id="1649" class="Symbol">=</a> <a id="1651" href="Categories.Functor.Algebra.html#573" class="Function">⊥.A</a>
<a id="1659" class="Symbol">;</a> <a id="1661" href="Categories.Object.NaturalNumbers.html#1124" class="Field">isNNO</a> <a id="1667" class="Symbol">=</a> <a id="1669" class="Keyword">record</a>
<a id="1682" class="Symbol">{</a> <a id="1684" href="Categories.Object.NaturalNumbers.html#464" class="Field">z</a> <a id="1686" class="Symbol">=</a> <a id="1688" href="Categories.Functor.Algebra.html#587" class="Function">⊥.α</a> <a id="1692" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1694" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="1703" class="Symbol">;</a> <a id="1705" href="Categories.Object.NaturalNumbers.html#478" class="Field">s</a> <a id="1707" class="Symbol">=</a> <a id="1709" href="Categories.Functor.Algebra.html#587" class="Function">⊥.α</a> <a id="1713" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1715" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="1724" class="Symbol">;</a> <a id="1726" href="Categories.Object.NaturalNumbers.html#492" class="Field">universal</a> <a id="1736" class="Symbol">=</a> <a id="1738" class="Symbol">λ</a> <a id="1740" class="Symbol">{</a><a id="1741" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1741" class="Bound">A</a><a id="1742" class="Symbol">}</a> <a id="1744" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1744" class="Bound">q</a> <a id="1746" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1746" class="Bound">f</a> <a id="1748" class="Symbol">→</a>
<a id="1758" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="1779" class="Symbol">(</a><a id="1780" href="Categories.Object.Initial.html#478" class="Function">initial.!</a> <a id="1790" class="Symbol">{</a><a id="1791" class="Argument">A</a> <a id="1793" class="Symbol">=</a> <a id="1795" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3241" class="Function">alg</a> <a id="1799" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1744" class="Bound">q</a> <a id="1801" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1746" class="Bound">f</a><a id="1802" class="Symbol">})</a>
<a id="1811" class="Symbol">;</a> <a id="1813" href="Categories.Object.NaturalNumbers.html#538" class="Field">z-commute</a> <a id="1823" class="Symbol">=</a> <a id="1825" class="Symbol">λ</a> <a id="1827" class="Symbol">{</a><a id="1828" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1828" class="Bound">A</a><a id="1829" class="Symbol">}</a> <a id="1831" class="Symbol">{</a><a id="1832" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1832" class="Bound">q</a><a id="1833" class="Symbol">}</a> <a id="1835" class="Symbol">{</a><a id="1836" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1836" class="Bound">f</a><a id="1837" class="Symbol">}</a> <a id="1839" class="Symbol">→</a> <a id="1841" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1855" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1832" class="Bound">q</a> <a id="1921" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1924" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="1926" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="1934" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1944" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1946" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1832" class="Bound">q</a> <a id="1948" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1950" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1836" class="Bound">f</a> <a id="1952" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1954" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1956" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="2010" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2013" href="Categories.Morphism.Reasoning.Core.html#2347" class="Function">pushʳ</a> <a id="2019" class="Symbol">(</a><a id="2020" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="2022" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="2029" class="Symbol">)</a> <a id="2031" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="3428" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3379" class="Function">NNO⇒Initial</a> <a id="3440" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3440" class="Bound">nno</a> <a id="3444" class="Symbol">=</a> <a id="3446" class="Keyword">record</a>
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<a id="3580" class="Symbol">{</a> <a id="3582" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="3584" class="Symbol">=</a> <a id="3586" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="3596" class="Symbol">(</a><a id="3597" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="3609" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3558" class="Bound">alg</a> <a id="3613" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3615" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="3617" class="Symbol">)</a> <a id="3619" class="Symbol">(</a><a id="3620" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="3632" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3558" class="Bound">alg</a> <a id="3636" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3638" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="3640" class="Symbol">)</a>
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<a id="3679" href="Categories.Object.NaturalNumbers.html#492" class="Function">universal</a> <a id="3689" class="Symbol">(</a><a id="3690" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="3702" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3558" class="Bound">alg</a> <a id="3706" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3708" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="3710" class="Symbol">)</a> <a id="3712" class="Symbol">(</a><a id="3713" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="3725" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3558" class="Bound">alg</a> <a id="3729" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3731" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="3733" class="Symbol">)</a> <a id="3735" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3737" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3739" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a> <a id="3741" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3743" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="3745" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="3787" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3790" href="Categories.Object.Coproduct.html#980" class="Function">∘-distribˡ-[]</a> <a id="3804" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="4313" class="Symbol">}</a>
<a id="4321" class="Symbol">;</a> <a id="4323" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="4332" class="Symbol">=</a> <a id="4334" class="Symbol">λ</a> <a id="4336" class="Symbol">{</a><a id="4337" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4337" class="Bound">A</a><a id="4338" class="Symbol">}</a> <a id="4340" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="4342" class="Symbol">→</a>
<a id="4352" class="Keyword">let</a> <a id="4356" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4356" class="Bound">z-commutes</a> <a id="4367" class="Symbol">=</a> <a id="4369" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4389" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="4401" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4337" class="Bound">A</a> <a id="4403" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4405" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="4449" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4452" href="Categories.Morphism.Reasoning.Core.html#2347" class="Function">pushʳ</a> <a id="4458" class="Symbol">(</a><a id="4459" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="4461" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="4468" class="Symbol">)</a> <a id="4470" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="4604" class="Symbol">(</a><a id="4605" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="4626" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="4628" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4630" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="4632" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a> <a id="4634" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="4636" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="4638" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="4639" class="Symbol">)</a> <a id="4641" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4643" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="4664" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4667" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="4673" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="4681" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4697" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="4718" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="4720" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4722" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a> <a id="4757" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4771" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4771" class="Bound">s-commutes</a> <a id="4782" class="Symbol">=</a> <a id="4784" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4804" class="Symbol">(</a><a id="4805" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="4817" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4337" class="Bound">A</a> <a id="4819" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4821" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="4823" class="Symbol">)</a> <a id="4825" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4827" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="4848" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="4864" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4867" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="4873" class="Symbol">(</a><a id="4874" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="4876" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="4883" class="Symbol">)</a> <a id="4885" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="4887" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="4889" href="Categories.Category.Core.html#715" class="Function">assoc</a> <a id="4895" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4911" class="Symbol">(</a><a id="4912" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="4924" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4337" class="Bound">A</a> <a id="4926" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4928" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="4930" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="4933" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="4935" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="4938" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4940" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="4961" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="4963" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="4964" class="Symbol">)</a> <a id="4966" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4968" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="4971" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4975" href="Categories.Functor.Algebra.html#1388" class="Field">F-Algebra-Morphism.commutes</a> <a id="5003" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="5005" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5013" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5029" class="Symbol">(</a><a id="5030" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="5051" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="5053" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5055" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5057" href="Categories.Object.NaturalNumbers.html#464" class="Function">z</a> <a id="5059" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5061" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="5063" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="5064" class="Symbol">)</a> <a id="5066" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5068" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5089" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5092" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5098" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="5106" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5122" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="5143" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4340" class="Bound">f</a> <a id="5145" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5147" href="Categories.Object.NaturalNumbers.html#478" class="Function">s</a> <a id="5182" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5192" class="Keyword">in</a> <a id="5195" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="5197" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5199" href="Categories.Object.NaturalNumbers.html#694" class="Function">unique</a> <a id="5206" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4356" class="Bound">z-commutes</a> <a id="5217" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#4771" class="Bound">s-commutes</a>
<a id="5234" class="Symbol">}</a>
<a id="5240" class="Symbol">}</a>
<a id="5246" class="Keyword">where</a>
<a id="5258" class="Keyword">open</a> <a id="5263" href="Categories.Object.NaturalNumbers.html#1072" class="Module">NNO</a> <a id="5267" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3440" class="Bound">nno</a>
<a id="5272" class="Comment">-- A parametrized NNO corresponds to existence of a Maybe algebra and initiality of the PNNO algebra</a>
<a id="5373" class="Keyword">module</a> <a id="5380" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5380" class="Module">_</a> <a id="5382" class="Symbol">{</a><a id="5383" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5383" class="Bound">o</a> <a id="5385" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5385" class="Bound">ℓ</a> <a id="5387" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5387" class="Bound">e</a><a id="5388" class="Symbol">}</a> <a id="5390" class="Symbol">(</a><a id="5391" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5391" class="Bound">CC</a> <a id="5394" class="Symbol">:</a> <a id="5396" href="Categories.Category.Cartesian.Bundle.html#422" class="Record">CartesianCategory</a> <a id="5414" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5383" class="Bound">o</a> <a id="5416" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5385" class="Bound">ℓ</a> <a id="5418" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5387" class="Bound">e</a><a id="5419" class="Symbol">)</a> <a id="5421" class="Symbol">(</a><a id="5422" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5422" class="Bound">𝒞-Coproducts</a> <a id="5435" class="Symbol">:</a> <a id="5437" href="Categories.Category.Cocartesian.html#1360" class="Record">BinaryCoproducts</a> <a id="5454" class="Symbol">(</a><a id="5455" href="Categories.Category.Cartesian.Bundle.html#488" class="Field">CartesianCategory.U</a> <a id="5475" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5391" class="Bound">CC</a><a id="5477" class="Symbol">))</a> <a id="5480" class="Keyword">where</a>
<a id="5488" class="Keyword">open</a> <a id="5493" href="Categories.Category.Cartesian.Bundle.html#422" class="Module">CartesianCategory</a> <a id="5511" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5391" class="Bound">CC</a> <a id="5514" class="Keyword">renaming</a> <a id="5523" class="Symbol">(</a><a id="5524" href="Categories.Category.Cartesian.Bundle.html#488" class="Field">U</a> <a id="5526" class="Symbol">to</a> <a id="5529" class="Field">𝒞</a><a id="5530" class="Symbol">)</a>
<a id="5534" class="Keyword">open</a> <a id="5539" href="Categories.Category.Cocartesian.html#1360" class="Module">BinaryCoproducts</a> <a id="5556" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5422" class="Bound">𝒞-Coproducts</a>
<a id="5571" class="Keyword">open</a> <a id="5576" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="5591" href="Categories.Category.Cartesian.html#801" class="Function">products</a> <a id="5600" class="Keyword">hiding</a> <a id="5607" class="Symbol">(</a><a id="5608" href="Categories.Object.Product.Core.html#671" class="Function">unique</a><a id="5614" class="Symbol">)</a>
<a id="5618" class="Keyword">open</a> <a id="5623" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="5638" class="Keyword">open</a> <a id="5643" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="5651" class="Keyword">open</a> <a id="5656" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="5659" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a>
<a id="5663" class="Keyword">open</a> <a id="5668" href="Categories.Object.NaturalNumbers.Parametrized.html" class="Module">PNNO</a> <a id="5673" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5391" class="Bound">CC</a>
<a id="5678" class="Keyword">open</a> <a id="5683" href="Categories.Object.NaturalNumbers.html" class="Module">NNOs</a> <a id="5688" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a> <a id="5690" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>
<a id="5701" class="Keyword">open</a> <a id="5706" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="5715" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>
<a id="5727" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="5738" class="Symbol">:</a> <a id="5740" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="5744" class="Symbol">→</a> <a id="5746" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="5758" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a>
<a id="5762" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="5773" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5773" class="Bound">A</a> <a id="5775" class="Symbol">=</a> <a id="5777" class="Keyword">record</a>
<a id="5788" class="Symbol">{</a> <a id="5790" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="5793" class="Symbol">=</a> <a id="5795" class="Symbol">λ</a> <a id="5797" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5797" class="Bound">X</a> <a id="5799" class="Symbol">→</a> <a id="5801" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5773" class="Bound">A</a> <a id="5803" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="5805" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5797" class="Bound">X</a>
<a id="5811" class="Symbol">;</a> <a id="5813" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="5816" class="Symbol">=</a> <a id="5818" class="Symbol">λ</a> <a id="5820" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5820" class="Bound">f</a> <a id="5822" class="Symbol">→</a> <a id="5824" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5826" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5829" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5831" class="Symbol">(</a><a id="5832" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5835" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5837" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5820" class="Bound">f</a><a id="5838" class="Symbol">)</a> <a id="5840" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="5846" class="Symbol">;</a> <a id="5848" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="5857" class="Symbol">=</a> <a id="5859" class="Symbol">λ</a> <a id="5861" class="Symbol">{</a><a id="5862" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5862" class="Bound">A</a><a id="5863" class="Symbol">}</a> <a id="5865" class="Symbol">→</a> <a id="5867" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="5873" class="Symbol">(</a><a id="5874" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="5883" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="5892" class="Symbol">)</a>
<a id="5926" class="Symbol">(</a><a id="5927" href="Categories.Category.Cocartesian.html#1714" class="Function">coproduct.unique</a> <a id="5944" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="5954" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="5963" class="Symbol">)</a>
<a id="5970" class="Symbol">;</a> <a id="5972" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="5985" class="Symbol">=</a> <a id="5987" class="Symbol">λ</a> <a id="5989" class="Symbol">{</a><a id="5990" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5990" class="Bound">X</a><a id="5991" class="Symbol">}</a> <a id="5993" class="Symbol">{</a><a id="5994" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5994" class="Bound">Y</a><a id="5995" class="Symbol">}</a> <a id="5997" class="Symbol">{</a><a id="5998" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5998" class="Bound">Z</a><a id="5999" class="Symbol">}</a> <a id="6001" class="Symbol">{</a><a id="6002" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6002" class="Bound">f</a><a id="6003" class="Symbol">}</a> <a id="6005" class="Symbol">{</a><a id="6006" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6006" class="Bound">g</a><a id="6007" class="Symbol">}</a> <a id="6009" class="Symbol">→</a> <a id="6011" href="Categories.Category.Cocartesian.html#1714" class="Function">coproduct.unique</a>
<a id="6035" class="Symbol">(</a><a id="6036" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="6042" class="Symbol">(</a><a id="6043" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="6049" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="6056" class="Symbol">)</a> <a id="6058" class="Symbol">(</a><a id="6059" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="6066" class="Symbol">))</a>
<a id="6076" class="Symbol">(</a><a id="6077" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="6083" class="Symbol">(</a><a id="6084" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="6090" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="6097" class="Symbol">)</a> <a id="6099" class="Symbol">(</a><a id="6100" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="6106" class="Symbol">(</a><a id="6107" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6113" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="6120" class="Symbol">)</a> <a id="6122" href="Categories.Category.Core.html#715" class="Function">assoc</a><a id="6127" class="Symbol">))</a>
<a id="6134" class="Symbol">;</a> <a id="6136" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="6145" class="Symbol">=</a> <a id="6147" class="Symbol">λ</a> <a id="6149" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6149" class="Bound">fg</a> <a id="6152" class="Symbol">→</a> <a id="6154" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="6163" class="Symbol">(</a><a id="6164" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="6174" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6149" class="Bound">fg</a><a id="6176" class="Symbol">)</a>
<a id="6182" class="Symbol">}</a>
<a id="6187" class="Keyword">private</a>
<a id="6199" class="Keyword">module</a> <a id="6206" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6206" class="Module">coproductF</a> <a id="6217" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6217" class="Bound">A</a> <a id="6219" class="Symbol">=</a> <a id="6221" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6229" class="Symbol">(</a><a id="6230" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="6241" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6217" class="Bound">A</a><a id="6242" class="Symbol">)</a>
<a id="6247" class="Comment">-- the algebra that corresponds to a PNNO (if it is initial)</a>
<a id="6310" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6310" class="Function">PNNO-Algebra</a> <a id="6323" class="Symbol">:</a> <a id="6325" class="Symbol">∀</a> <a id="6327" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6327" class="Bound">A</a> <a id="6329" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6329" class="Bound">N</a> <a id="6331" class="Symbol">→</a> <a id="6333" href="Categories.Object.Terminal.html#905" class="Function">⊤</a> <a id="6335" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="6337" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6329" class="Bound">N</a> <a id="6339" class="Symbol">→</a> <a id="6341" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6329" class="Bound">N</a> <a id="6343" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="6345" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6329" class="Bound">N</a> <a id="6347" class="Symbol">→</a> <a id="6349" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="6359" class="Symbol">(</a><a id="6360" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="6371" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6327" class="Bound">A</a><a id="6372" class="Symbol">)</a>
<a id="6376" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6310" class="Function">PNNO-Algebra</a> <a id="6389" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6389" class="Bound">A</a> <a id="6391" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6391" class="Bound">N</a> <a id="6393" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6393" class="Bound">z</a> <a id="6395" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6395" class="Bound">s</a> <a id="6397" class="Symbol">=</a> <a id="6399" class="Keyword">record</a>
<a id="6410" class="Symbol">{</a> <a id="6412" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="6414" class="Symbol">=</a> <a id="6416" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6389" class="Bound">A</a> <a id="6418" href="Categories.Category.BinaryProducts.html#1053" class="Function Operator">×</a> <a id="6420" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6391" class="Bound">N</a>
<a id="6426" class="Symbol">;</a> <a id="6428" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="6430" class="Symbol">=</a> <a id="6432" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6434" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="6436" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6439" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="6441" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6393" class="Bound">z</a> <a id="6443" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6445" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="6447" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="6449" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6451" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6454" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6456" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6395" class="Bound">s</a> <a id="6458" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="6465" class="Symbol">}</a>
<a id="6470" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6470" class="Function">Initial⇒PNNO</a> <a id="6483" class="Symbol">:</a> <a id="6485" class="Symbol">(</a><a id="6486" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6486" class="Bound">algebra</a> <a id="6494" class="Symbol">:</a> <a id="6496" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="6506" class="Symbol">(</a><a id="6507" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a> <a id="6513" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a> <a id="6515" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="6524" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5422" class="Bound">𝒞-Coproducts</a><a id="6536" class="Symbol">))</a>
<a id="6544" class="Symbol">→</a> <a id="6546" class="Symbol">(∀</a> <a id="6549" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6549" class="Bound">A</a> <a id="6551" class="Symbol">→</a> <a id="6553" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="6563" class="Symbol">(</a><a id="6564" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a> <a id="6575" class="Symbol">(</a><a id="6576" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="6587" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6549" class="Bound">A</a><a id="6588" class="Symbol">))</a>
<a id="6615" class="Symbol">(</a><a id="6616" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6310" class="Function">PNNO-Algebra</a> <a id="6629" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6549" class="Bound">A</a> <a id="6631" class="Symbol">(</a><a id="6632" href="Categories.Functor.Algebra.html#573" class="Field">F-Algebra.A</a> <a id="6644" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6486" class="Bound">algebra</a><a id="6651" class="Symbol">)</a> <a id="6653" class="Symbol">(</a><a id="6654" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="6666" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6486" class="Bound">algebra</a> <a id="6674" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6676" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="6678" class="Symbol">)</a> <a id="6680" class="Symbol">(</a><a id="6681" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="6693" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6486" class="Bound">algebra</a> <a id="6701" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6703" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="6705" class="Symbol">)))</a>
<a id="6714" class="Symbol">→</a> <a id="6716" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a>
<a id="6734" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6470" class="Function">Initial⇒PNNO</a> <a id="6747" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6747" class="Bound">algebra</a> <a id="6755" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6755" class="Bound">isInitial</a> <a id="6765" class="Symbol">=</a> <a id="6767" class="Keyword">record</a>
<a id="6779" class="Symbol">{</a> <a id="6781" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Field">N</a> <a id="6783" class="Symbol">=</a> <a id="6785" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8911" class="Function">N</a>
<a id="6791" class="Symbol">;</a> <a id="6793" href="Categories.Object.NaturalNumbers.Parametrized.html#2940" class="Field">isParametrizedNNO</a> <a id="6811" class="Symbol">=</a> <a id="6813" class="Keyword">record</a>
<a id="6826" class="Symbol">{</a> <a id="6828" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Field">z</a> <a id="6830" class="Symbol">=</a> <a id="6832" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8920" class="Function">z</a>
<a id="6840" class="Symbol">;</a> <a id="6842" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Field">s</a> <a id="6844" class="Symbol">=</a> <a id="6846" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8937" class="Function">s</a>
<a id="6854" class="Symbol">;</a> <a id="6856" href="Categories.Object.NaturalNumbers.Parametrized.html#988" class="Field">universal</a> <a id="6866" class="Symbol">=</a> <a id="6868" class="Symbol">λ</a> <a id="6870" class="Symbol">{</a><a id="6871" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6871" class="Bound">A</a><a id="6872" class="Symbol">}</a> <a id="6874" class="Symbol">{</a><a id="6875" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6875" class="Bound">X</a><a id="6876" class="Symbol">}</a> <a id="6878" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6878" class="Bound">f</a> <a id="6880" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6880" class="Bound">g</a> <a id="6882" class="Symbol">→</a> <a id="6884" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="6905" class="Symbol">(</a><a id="6906" href="Categories.Object.Initial.html#478" class="Function">isInitial.!</a> <a id="6918" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6871" class="Bound">A</a> <a id="6920" class="Symbol">{</a><a id="6921" class="Argument">A</a> <a id="6923" class="Symbol">=</a> <a id="6925" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9007" class="Function">alg′</a> <a id="6930" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6878" class="Bound">f</a> <a id="6932" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6880" class="Bound">g</a><a id="6933" class="Symbol">})</a>
<a id="6942" class="Symbol">;</a> <a id="6944" href="Categories.Object.NaturalNumbers.Parametrized.html#1040" class="Field">commute₁</a> <a id="6953" class="Symbol">=</a> <a id="6955" class="Symbol">λ</a> <a id="6957" class="Symbol">{</a><a id="6958" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6958" class="Bound">A</a><a id="6959" class="Symbol">}</a> <a id="6961" class="Symbol">{</a><a id="6962" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6962" class="Bound">X</a><a id="6963" class="Symbol">}</a> <a id="6965" class="Symbol">{</a><a id="6966" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6966" class="Bound">f</a><a id="6967" class="Symbol">}</a> <a id="6969" class="Symbol">{</a><a id="6970" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6970" class="Bound">g</a><a id="6971" class="Symbol">}</a> <a id="6973" class="Symbol">→</a> <a id="6975" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6990" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6966" class="Bound">f</a> <a id="7065" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="7069" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="7077" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="7088" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7090" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6966" class="Bound">f</a> <a id="7092" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7094" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6970" class="Bound">g</a> <a id="7096" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7098" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7100" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7163" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7166" href="Categories.Morphism.Reasoning.Core.html#2347" class="Function">pushʳ</a> <a id="7172" class="Symbol">(</a><a id="7173" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="7175" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="7182" class="Symbol">)</a> <a id="7184" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7194" class="Symbol">((</a><a id="7196" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7198" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6966" class="Bound">f</a> <a id="7200" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7202" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6970" class="Bound">g</a> <a id="7204" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7206" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7208" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7210" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7213" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7215" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7218" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7220" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="7241" class="Symbol">(</a><a id="7242" href="Categories.Object.Initial.html#478" class="Function">isInitial.!</a> <a id="7254" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6958" class="Bound">A</a><a id="7255" class="Symbol">)</a> <a id="7257" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="7258" class="Symbol">)</a> <a id="7260" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7262" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="7264" class="Symbol">)</a> <a id="7269" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7272" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="7278" class="Symbol">(</a><a id="7279" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="7281" class="Symbol">(</a><a id="7282" href="Categories.Functor.Algebra.html#1388" class="Field">F-Algebra-Morphism.commutes</a> <a id="7310" class="Symbol">(</a><a id="7311" href="Categories.Object.Initial.html#478" class="Function">isInitial.!</a> <a id="7323" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6958" class="Bound">A</a> <a id="7325" class="Symbol">{</a><a id="7326" class="Argument">A</a> <a id="7328" class="Symbol">=</a> <a id="7330" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9007" class="Function">alg′</a> <a id="7335" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6966" class="Bound">f</a> <a id="7337" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6970" class="Bound">g</a><a id="7338" class="Symbol">})))</a> <a id="7343" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7353" class="Symbol">(</a><a id="7354" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="7375" class="Symbol">(</a><a id="7376" href="Categories.Object.Initial.html#478" class="Function">isInitial.!</a> <a id="7388" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6958" class="Bound">A</a><a id="7389" class="Symbol">)</a> <a id="7391" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7393" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7395" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="7397" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="7400" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="7402" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8920" class="Function">z</a> <a id="7404" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7406" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="7408" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="7410" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7412" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="7415" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="7417" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8937" class="Function">s</a> <a id="7419" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7421" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7424" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="7426" class="Symbol">)</a> <a id="7428" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7431" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="7439" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="7447" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7457" class="Symbol">(</a><a id="7458" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="7479" class="Symbol">(</a><a id="7480" href="Categories.Object.Initial.html#478" class="Field">IsInitial.!</a> <a id="7492" class="Symbol">(</a><a id="7493" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6755" class="Bound">isInitial</a> <a id="7503" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6958" class="Bound">A</a><a id="7504" class="Symbol">)))</a> <a id="7508" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7510" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="7512" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="7515" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="7517" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8920" class="Function">z</a> <a id="7519" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7521" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="7523" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="7532" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="7540" class="Symbol">;</a> <a id="7542" href="Categories.Object.NaturalNumbers.Parametrized.html#1124" class="Field">commute₂</a> <a id="7551" class="Symbol">=</a> <a id="7553" class="Symbol">λ</a> <a id="7555" class="Symbol">{</a><a id="7556" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7556" class="Bound">A</a><a id="7557" class="Symbol">}</a> <a id="7559" class="Symbol">{</a><a id="7560" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7560" class="Bound">X</a><a id="7561" class="Symbol">}</a> <a id="7563" class="Symbol">{</a><a id="7564" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7564" class="Bound">f</a><a id="7565" class="Symbol">}</a> <a id="7567" class="Symbol">{</a><a id="7568" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7568" class="Bound">g</a><a id="7569" class="Symbol">}</a> <a id="7571" class="Symbol">→</a> <a id="7573" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="7588" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7568" class="Bound">g</a> <a id="7590" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7592" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="7613" class="Symbol">(</a><a id="7614" href="Categories.Object.Initial.html#478" class="Field">IsInitial.!</a> <a id="7626" class="Symbol">(</a><a id="7627" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6755" class="Bound">isInitial</a> <a id="7637" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#7556" class="Bound">A</a><a id="7638" class="Symbol">))</a> <a id="7672" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7675" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="7681" class="Symbol">(</a><a id="7682" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="7684" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="7691" class="Symbol">)</a> <a id="7693" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="8773" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8775" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8216" class="Bound">f</a> <a id="8777" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8779" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8220" class="Bound">g</a> <a id="8781" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8783" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8785" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8787" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="8790" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8792" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8795" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8797" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8224" class="Bound">u</a> <a id="8799" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8818" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="8829" class="Symbol">})</a>
<a id="8838" class="Symbol">}</a>
<a id="8845" class="Symbol">}</a>
<a id="8851" class="Keyword">where</a>
<a id="8863" class="Keyword">open</a> <a id="8868" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="8878" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6747" class="Bound">algebra</a> <a id="8886" class="Keyword">using</a> <a id="8892" class="Symbol">(</a><a id="8893" href="Categories.Functor.Algebra.html#587" class="Field">α</a><a id="8894" class="Symbol">)</a> <a id="8896" class="Keyword">renaming</a> <a id="8905" class="Symbol">(</a><a id="8906" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="8908" class="Symbol">to</a> <a id="8911" class="Field">N</a><a id="8912" class="Symbol">)</a>
<a id="8920" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8920" class="Function">z</a> <a id="8922" class="Symbol">=</a> <a id="8924" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="8926" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8928" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="8937" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8937" class="Function">s</a> <a id="8939" class="Symbol">=</a> <a id="8941" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="8943" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8945" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="8955" class="Keyword">module</a> <a id="8962" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8962" class="Module">isInitial</a> <a id="8972" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8972" class="Bound">A</a> <a id="8974" class="Symbol">=</a> <a id="8976" href="Categories.Object.Initial.html#422" class="Module">IsInitial</a> <a id="8986" class="Symbol">(</a><a id="8987" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6755" class="Bound">isInitial</a> <a id="8997" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#8972" class="Bound">A</a><a id="8998" class="Symbol">)</a>
<a id="9007" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9007" class="Function">alg′</a> <a id="9013" class="Symbol">:</a> <a id="9015" class="Symbol">∀</a> <a id="9017" class="Symbol">{</a><a id="9018" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9018" class="Bound">A</a> <a id="9020" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9020" class="Bound">X</a><a id="9021" class="Symbol">}</a> <a id="9023" class="Symbol">→</a> <a id="9025" class="Symbol">(</a><a id="9026" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9026" class="Bound">f</a> <a id="9028" class="Symbol">:</a> <a id="9030" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9018" class="Bound">A</a> <a id="9032" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="9034" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9020" class="Bound">X</a><a id="9035" class="Symbol">)</a> <a id="9037" class="Symbol">→</a> <a id="9039" class="Symbol">(</a><a id="9040" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9040" class="Bound">g</a> <a id="9042" class="Symbol">:</a> <a id="9044" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9020" class="Bound">X</a> <a id="9046" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="9048" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9020" class="Bound">X</a><a id="9049" class="Symbol">)</a> <a id="9051" class="Symbol">→</a> <a id="9053" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="9063" class="Symbol">(</a><a id="9064" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="9075" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9018" class="Bound">A</a><a id="9076" class="Symbol">)</a>
<a id="9084" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9007" class="Function">alg′</a> <a id="9089" class="Symbol">{</a><a id="9090" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9090" class="Bound">A</a><a id="9091" class="Symbol">}</a> <a id="9093" class="Symbol">{</a><a id="9094" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9094" class="Bound">X</a><a id="9095" class="Symbol">}</a> <a id="9097" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9097" class="Bound">f</a> <a id="9099" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9099" class="Bound">g</a> <a id="9101" class="Symbol">=</a> <a id="9103" class="Keyword">record</a>
<a id="9119" class="Symbol">{</a> <a id="9121" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="9123" class="Symbol">=</a> <a id="9125" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9094" class="Bound">X</a>
<a id="9136" class="Symbol">;</a> <a id="9138" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="9140" class="Symbol">=</a> <a id="9142" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="9144" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9097" class="Bound">f</a> <a id="9146" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="9148" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9099" class="Bound">g</a> <a id="9150" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="9161" class="Symbol">}</a>
<a id="9166" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9166" class="Function">PNNO⇒Initial₁</a> <a id="9180" class="Symbol">:</a> <a id="9182" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a> <a id="9198" class="Symbol">→</a> <a id="9200" href="Categories.Object.Initial.html#760" class="Record">Initial</a> <a id="9208" class="Symbol">(</a><a id="9209" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a> <a id="9220" class="Symbol">(</a><a id="9221" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#1220" class="Function">Maybe</a> <a id="9227" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a> <a id="9229" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="9238" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5422" class="Bound">𝒞-Coproducts</a><a id="9250" class="Symbol">))</a>
<a id="9255" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9166" class="Function">PNNO⇒Initial₁</a> <a id="9269" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9269" class="Bound">pnno</a> <a id="9274" class="Symbol">=</a> <a id="9276" class="Symbol">(</a><a id="9277" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#3379" class="Function">NNO⇒Initial</a> <a id="9289" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5529" class="Function">𝒞</a> <a id="9291" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a> <a id="9300" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5422" class="Bound">𝒞-Coproducts</a><a id="9312" class="Symbol">)</a> <a id="9314" class="Symbol">(</a><a id="9315" href="Categories.Object.NaturalNumbers.Parametrized.html#3101" class="Function">PNNO⇒NNO</a> <a id="9324" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9269" class="Bound">pnno</a><a id="9328" class="Symbol">)</a>
<a id="9333" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9333" class="Function">PNNO⇒Initial₂</a> <a id="9347" class="Symbol">:</a> <a id="9349" class="Symbol">(</a><a id="9350" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9350" class="Bound">pnno</a> <a id="9355" class="Symbol">:</a> <a id="9357" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Record">ParametrizedNNO</a><a id="9372" class="Symbol">)</a>
<a id="9378" class="Symbol">→</a> <a id="9380" class="Symbol">(∀</a> <a id="9383" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9383" class="Bound">A</a> <a id="9385" class="Symbol">→</a> <a id="9387" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="9397" class="Symbol">(</a><a id="9398" href="Categories.Category.Construction.F-Algebras.html#462" class="Function">F-Algebras</a> <a id="9409" class="Symbol">(</a><a id="9410" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#5727" class="Function">coproductF</a> <a id="9421" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9383" class="Bound">A</a><a id="9422" class="Symbol">))</a>
<a id="9449" class="Symbol">(</a><a id="9450" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#6310" class="Function">PNNO-Algebra</a> <a id="9463" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9383" class="Bound">A</a> <a id="9465" class="Symbol">(</a><a id="9466" href="Categories.Object.NaturalNumbers.Parametrized.html#2928" class="Field">ParametrizedNNO.N</a> <a id="9484" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9350" class="Bound">pnno</a><a id="9488" class="Symbol">)</a> <a id="9490" class="Symbol">(</a><a id="9491" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Function">ParametrizedNNO.z</a> <a id="9509" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9350" class="Bound">pnno</a><a id="9513" class="Symbol">)</a> <a id="9515" class="Symbol">(</a><a id="9516" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Function">ParametrizedNNO.s</a> <a id="9534" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9350" class="Bound">pnno</a><a id="9538" class="Symbol">)))</a>
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<a id="9579" class="Symbol">{</a> <a id="9581" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="9583" class="Symbol">=</a> <a id="9585" class="Symbol">λ</a> <a id="9587" class="Symbol">{</a><a id="9588" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9588" class="Bound">alg</a><a id="9591" class="Symbol">}</a> <a id="9593" class="Symbol">→</a> <a id="9595" class="Keyword">record</a>
<a id="9609" class="Symbol">{</a> <a id="9611" href="Categories.Functor.Algebra.html#1366" class="Field">f</a> <a id="9613" class="Symbol">=</a> <a id="9615" href="Categories.Object.NaturalNumbers.Parametrized.html#988" class="Function">universal</a> <a id="9625" class="Symbol">(</a><a id="9626" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="9638" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9588" class="Bound">alg</a> <a id="9642" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9644" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="9646" class="Symbol">)</a> <a id="9648" class="Symbol">(</a><a id="9649" href="Categories.Functor.Algebra.html#587" class="Field">F-Algebra.α</a> <a id="9661" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9588" class="Bound">alg</a> <a id="9665" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="9667" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="9669" class="Symbol">)</a>
<a id="9678" class="Symbol">;</a> <a id="9680" href="Categories.Functor.Algebra.html#1388" class="Field">commutes</a> <a id="9689" class="Symbol">=</a> <a id="9691" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="11026" class="Symbol">((</a><a id="11028" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="11049" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#10340" class="Bound">f</a> <a id="11051" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="11053" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="11055" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="11057" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="11060" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="11062" href="Categories.Object.NaturalNumbers.Parametrized.html#960" class="Function">z</a> <a id="11064" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="11066" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="11068" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="11070" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="11072" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="11075" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="11077" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Function">s</a> <a id="11079" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="11080" class="Symbol">)</a> <a id="11082" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="11084" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="11086" class="Symbol">)</a> <a id="11088" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11091" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="11097" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="11105" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11119" href="Categories.Functor.Algebra.html#1366" class="Field">F-Algebra-Morphism.f</a> <a id="11140" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#10340" class="Bound">f</a> <a id="11142" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="11144" class="Symbol">(</a><a id="11145" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="11148" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="11150" href="Categories.Object.NaturalNumbers.Parametrized.html#974" class="Function">s</a><a id="11151" class="Symbol">)</a> <a id="11181" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="11189" class="Keyword">in</a> <a id="11192" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="11194" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="11196" href="Categories.Object.NaturalNumbers.Parametrized.html#1222" class="Function">unique</a> <a id="11203" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#10355" class="Bound">commute₁</a> <a id="11212" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#10768" class="Bound">commute₂</a>
<a id="11225" class="Symbol">}</a>
<a id="11231" class="Keyword">where</a>
<a id="11243" class="Keyword">open</a> <a id="11248" href="Categories.Object.NaturalNumbers.Parametrized.html#2876" class="Module">ParametrizedNNO</a> <a id="11264" href="Categories.Object.NaturalNumbers.Properties.F-Algebras.html#9558" class="Bound">pnno</a></pre></body></html>
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