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bsc-leon-vatthauer/agda/bsc-thesis/Categories.Category.Distributive.Properties.html
Leon Vatthauer 2bf4655eae
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2024-02-09 17:53:52 +01:00

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<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Categories.Category.Distributive.Properties</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">using</a> <a id="80" class="Symbol">(</a><a id="81" href="Categories.Category.Core.html#442" class="Record">Category</a><a id="89" class="Symbol">)</a>
<a id="91" class="Keyword">open</a> <a id="96" class="Keyword">import</a> <a id="103" href="Categories.Category.Cartesian.html" class="Module">Categories.Category.Cartesian</a> <a id="133" class="Keyword">using</a> <a id="139" class="Symbol">(</a><a id="140" href="Categories.Category.Cartesian.html#727" class="Record">Cartesian</a><a id="149" class="Symbol">)</a>
<a id="151" class="Keyword">open</a> <a id="156" class="Keyword">import</a> <a id="163" href="Categories.Category.BinaryProducts.html" class="Module">Categories.Category.BinaryProducts</a> <a id="198" class="Keyword">using</a> <a id="204" class="Symbol">(</a><a id="205" href="Categories.Category.BinaryProducts.html#848" class="Record">BinaryProducts</a><a id="219" class="Symbol">)</a>
<a id="221" class="Keyword">open</a> <a id="226" class="Keyword">import</a> <a id="233" href="Categories.Category.Cocartesian.html" class="Module">Categories.Category.Cocartesian</a> <a id="265" class="Keyword">using</a> <a id="271" class="Symbol">(</a><a id="272" href="Categories.Category.Cocartesian.html#3385" class="Record">Cocartesian</a><a id="283" class="Symbol">)</a>
<a id="285" class="Keyword">open</a> <a id="290" class="Keyword">import</a> <a id="297" href="Categories.Category.Distributive.html" class="Module">Categories.Category.Distributive</a> <a id="330" class="Keyword">using</a> <a id="336" class="Symbol">(</a><a id="337" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a><a id="349" class="Symbol">)</a>
<a id="352" class="Keyword">import</a> <a id="359" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="379" class="Symbol">as</a> <a id="382" class="Module">M</a>
<a id="384" class="Keyword">import</a> <a id="391" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="421" class="Symbol">as</a> <a id="424" class="Module">MR</a>
<a id="427" class="Keyword">import</a> <a id="434" href="Categories.Morphism.Properties.html" class="Module">Categories.Morphism.Properties</a> <a id="465" class="Symbol">as</a> <a id="468" class="Module">MP</a>
<a id="472" class="Keyword">module</a> <a id="479" href="Categories.Category.Distributive.Properties.html" class="Module">Categories.Category.Distributive.Properties</a> <a id="523" class="Symbol">{</a><a id="524" href="Categories.Category.Distributive.Properties.html#524" class="Bound">o</a> <a id="526" href="Categories.Category.Distributive.Properties.html#526" class="Bound">ℓ</a> <a id="528" href="Categories.Category.Distributive.Properties.html#528" class="Bound">e</a><a id="529" class="Symbol">}</a> <a id="531" class="Symbol">{</a><a id="532" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a> <a id="534" class="Symbol">:</a> <a id="536" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="545" href="Categories.Category.Distributive.Properties.html#524" class="Bound">o</a> <a id="547" href="Categories.Category.Distributive.Properties.html#526" class="Bound">ℓ</a> <a id="549" href="Categories.Category.Distributive.Properties.html#528" class="Bound">e</a><a id="550" class="Symbol">}</a> <a id="552" class="Symbol">(</a><a id="553" href="Categories.Category.Distributive.Properties.html#553" class="Bound">distributive</a> <a id="566" class="Symbol">:</a> <a id="568" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a> <a id="581" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a><a id="582" class="Symbol">)</a> <a id="584" class="Keyword">where</a>
<a id="590" class="Keyword">open</a> <a id="595" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="604" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a>
<a id="606" class="Keyword">open</a> <a id="611" href="Categories.Morphism.html" class="Module">M</a> <a id="613" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a>
<a id="615" class="Keyword">open</a> <a id="620" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="623" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a>
<a id="625" class="Keyword">open</a> <a id="630" href="Categories.Morphism.Properties.html" class="Module">MP</a> <a id="633" href="Categories.Category.Distributive.Properties.html#532" class="Bound">𝒞</a>
<a id="635" class="Keyword">open</a> <a id="640" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="653" class="Keyword">open</a> <a id="658" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="665" class="Keyword">open</a> <a id="670" href="Categories.Category.Distributive.html#712" class="Module">Distributive</a> <a id="683" href="Categories.Category.Distributive.Properties.html#553" class="Bound">distributive</a>
<a id="696" class="Keyword">open</a> <a id="701" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="711" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="721" class="Keyword">using</a> <a id="727" class="Symbol">(</a><a id="728" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="736" class="Symbol">)</a>
<a id="738" class="Keyword">open</a> <a id="743" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="758" href="Categories.Category.Cartesian.html#801" class="Function">products</a>
<a id="767" class="Keyword">open</a> <a id="772" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="784" href="Categories.Category.Distributive.html#793" class="Field">cocartesian</a>
<a id="797" class="Comment">-- distribution and injection</a>
<a id="distributeˡ⁻¹-i₁"></a><a id="827" href="Categories.Category.Distributive.Properties.html#827" class="Function">distributeˡ⁻¹-i₁</a> <a id="844" class="Symbol">:</a> <a id="846" class="Symbol">∀</a> <a id="848" class="Symbol">{</a><a id="849" href="Categories.Category.Distributive.Properties.html#849" class="Bound">A</a> <a id="851" href="Categories.Category.Distributive.Properties.html#851" class="Bound">B</a> <a id="853" href="Categories.Category.Distributive.Properties.html#853" class="Bound">C</a><a id="854" class="Symbol">}</a> <a id="856" class="Symbol">→</a> <a id="858" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="872" class="Symbol">{</a><a id="873" href="Categories.Category.Distributive.Properties.html#849" class="Bound">A</a><a id="874" class="Symbol">}</a> <a id="876" class="Symbol">{</a><a id="877" href="Categories.Category.Distributive.Properties.html#851" class="Bound">B</a><a id="878" class="Symbol">}</a> <a id="880" class="Symbol">{</a><a id="881" href="Categories.Category.Distributive.Properties.html#853" class="Bound">C</a><a id="882" class="Symbol">}</a> <a id="884" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="886" class="Symbol">(</a><a id="887" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="890" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="892" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="894" class="Symbol">)</a> <a id="896" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="898" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="901" href="Categories.Category.Distributive.Properties.html#827" class="Function">distributeˡ⁻¹-i₁</a> <a id="918" class="Symbol">=</a> <a id="920" class="Symbol">(</a><a id="921" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="929" class="Symbol">(</a><a id="930" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="934" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="941" class="Symbol">))</a> <a id="944" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="946" class="Symbol">(</a><a id="947" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="955" class="Symbol">(</a><a id="956" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="967" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="973" class="Symbol">))</a>
<a id="distributeˡ⁻¹-i₂"></a><a id="977" href="Categories.Category.Distributive.Properties.html#977" class="Function">distributeˡ⁻¹-i₂</a> <a id="994" class="Symbol">:</a> <a id="996" class="Symbol">∀</a> <a id="998" class="Symbol">{</a><a id="999" href="Categories.Category.Distributive.Properties.html#999" class="Bound">A</a> <a id="1001" href="Categories.Category.Distributive.Properties.html#1001" class="Bound">B</a> <a id="1003" href="Categories.Category.Distributive.Properties.html#1003" class="Bound">C</a><a id="1004" class="Symbol">}</a> <a id="1006" class="Symbol">→</a> <a id="1008" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1022" class="Symbol">{</a><a id="1023" href="Categories.Category.Distributive.Properties.html#999" class="Bound">A</a><a id="1024" class="Symbol">}</a> <a id="1026" class="Symbol">{</a><a id="1027" href="Categories.Category.Distributive.Properties.html#1001" class="Bound">B</a><a id="1028" class="Symbol">}</a> <a id="1030" class="Symbol">{</a><a id="1031" href="Categories.Category.Distributive.Properties.html#1003" class="Bound">C</a><a id="1032" class="Symbol">}</a> <a id="1034" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1036" class="Symbol">(</a><a id="1037" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1040" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1042" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="1044" class="Symbol">)</a> <a id="1046" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1048" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="1051" href="Categories.Category.Distributive.Properties.html#977" class="Function">distributeˡ⁻¹-i₂</a> <a id="1068" class="Symbol">=</a> <a id="1070" class="Symbol">(</a><a id="1071" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1079" class="Symbol">(</a><a id="1080" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1084" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="1091" class="Symbol">))</a> <a id="1094" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1096" class="Symbol">(</a><a id="1097" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="1105" class="Symbol">(</a><a id="1106" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="1117" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="1123" class="Symbol">))</a>
<a id="distributeʳ⁻¹-i₁"></a><a id="1127" href="Categories.Category.Distributive.Properties.html#1127" class="Function">distributeʳ⁻¹-i₁</a> <a id="1144" class="Symbol">:</a> <a id="1146" class="Symbol">∀</a> <a id="1148" class="Symbol">{</a><a id="1149" href="Categories.Category.Distributive.Properties.html#1149" class="Bound">A</a> <a id="1151" href="Categories.Category.Distributive.Properties.html#1151" class="Bound">B</a> <a id="1153" href="Categories.Category.Distributive.Properties.html#1153" class="Bound">C</a><a id="1154" class="Symbol">}</a> <a id="1156" class="Symbol">→</a> <a id="1158" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="1172" class="Symbol">{</a><a id="1173" href="Categories.Category.Distributive.Properties.html#1149" class="Bound">A</a><a id="1174" class="Symbol">}</a> <a id="1176" class="Symbol">{</a><a id="1177" href="Categories.Category.Distributive.Properties.html#1151" class="Bound">B</a><a id="1178" class="Symbol">}</a> <a id="1180" class="Symbol">{</a><a id="1181" href="Categories.Category.Distributive.Properties.html#1153" class="Bound">C</a><a id="1182" class="Symbol">}</a> <a id="1184" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1186" class="Symbol">(</a><a id="1187" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1190" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1192" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1194" class="Symbol">)</a> <a id="1196" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1198" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="1201" href="Categories.Category.Distributive.Properties.html#1127" class="Function">distributeʳ⁻¹-i₁</a> <a id="1218" class="Symbol">=</a> <a id="1220" class="Symbol">(</a><a id="1221" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1229" class="Symbol">(</a><a id="1230" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1234" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a><a id="1241" class="Symbol">))</a> <a id="1244" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1246" class="Symbol">(</a><a id="1247" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="1255" class="Symbol">(</a><a id="1256" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="1267" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="1273" class="Symbol">))</a>
<a id="distributeʳ⁻¹-i₂"></a><a id="1277" href="Categories.Category.Distributive.Properties.html#1277" class="Function">distributeʳ⁻¹-i₂</a> <a id="1294" class="Symbol">:</a> <a id="1296" class="Symbol">∀</a> <a id="1298" class="Symbol">{</a><a id="1299" href="Categories.Category.Distributive.Properties.html#1299" class="Bound">A</a> <a id="1301" href="Categories.Category.Distributive.Properties.html#1301" class="Bound">B</a> <a id="1303" href="Categories.Category.Distributive.Properties.html#1303" class="Bound">C</a><a id="1304" class="Symbol">}</a> <a id="1306" class="Symbol">→</a> <a id="1308" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="1322" class="Symbol">{</a><a id="1323" href="Categories.Category.Distributive.Properties.html#1299" class="Bound">A</a><a id="1324" class="Symbol">}</a> <a id="1326" class="Symbol">{</a><a id="1327" href="Categories.Category.Distributive.Properties.html#1301" class="Bound">B</a><a id="1328" class="Symbol">}</a> <a id="1330" class="Symbol">{</a><a id="1331" href="Categories.Category.Distributive.Properties.html#1303" class="Bound">C</a><a id="1332" class="Symbol">}</a> <a id="1334" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1336" class="Symbol">(</a><a id="1337" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1340" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1342" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1344" class="Symbol">)</a> <a id="1346" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1348" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="1351" href="Categories.Category.Distributive.Properties.html#1277" class="Function">distributeʳ⁻¹-i₂</a> <a id="1368" class="Symbol">=</a> <a id="1370" class="Symbol">(</a><a id="1371" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="1379" class="Symbol">(</a><a id="1380" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1384" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="1391" class="Symbol">))</a> <a id="1394" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1396" class="Symbol">(</a><a id="1397" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="1405" class="Symbol">(</a><a id="1406" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="1417" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="1423" class="Symbol">))</a>
<a id="1427" class="Comment">-- distribution and projection</a>
<a id="distributeˡ⁻¹-π₁"></a><a id="1458" href="Categories.Category.Distributive.Properties.html#1458" class="Function">distributeˡ⁻¹-π₁</a> <a id="1475" class="Symbol">:</a> <a id="1477" class="Symbol">∀</a> <a id="1479" class="Symbol">{</a><a id="1480" href="Categories.Category.Distributive.Properties.html#1480" class="Bound">A</a> <a id="1482" href="Categories.Category.Distributive.Properties.html#1482" class="Bound">B</a> <a id="1484" href="Categories.Category.Distributive.Properties.html#1484" class="Bound">C</a><a id="1485" class="Symbol">}</a> <a id="1487" class="Symbol">→</a> <a id="1489" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1491" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1494" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1496" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1499" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1501" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1503" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1517" class="Symbol">{</a><a id="1518" href="Categories.Category.Distributive.Properties.html#1480" class="Bound">A</a><a id="1519" class="Symbol">}</a> <a id="1521" class="Symbol">{</a><a id="1522" href="Categories.Category.Distributive.Properties.html#1482" class="Bound">B</a><a id="1523" class="Symbol">}</a> <a id="1525" class="Symbol">{</a><a id="1526" href="Categories.Category.Distributive.Properties.html#1484" class="Bound">C</a><a id="1527" class="Symbol">}</a> <a id="1529" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1531" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a>
<a id="1534" href="Categories.Category.Distributive.Properties.html#1458" class="Function">distributeˡ⁻¹-π₁</a> <a id="1551" class="Symbol">=</a> <a id="1553" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1557" class="Symbol">(</a><a id="1558" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1567" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1623" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1626" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="1633" class="Symbol">(</a><a id="1634" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="1645" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="1651" class="Symbol">)</a> <a id="1653" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1658" href="Categories.Object.Product.Core.html#526" class="Function">π₁</a> <a id="1661" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1663" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a> <a id="1675" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1677" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1714" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1717" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1723" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="1727" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="2855" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2911" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2914" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="2921" class="Symbol">(</a><a id="2922" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="2933" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="2939" class="Symbol">)</a> <a id="2941" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2946" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="2949" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2951" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a> <a id="2963" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2965" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="3002" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3005" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3011" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="3015" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3020" class="Symbol">(</a><a id="3021" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3023" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="3026" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3028" class="Symbol">((</a><a id="3030" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="3033" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3035" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3037" class="Symbol">))</a> <a id="3040" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3042" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="3045" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3047" class="Symbol">(</a><a id="3048" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="3051" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3053" href="Categories.Category.Core.html#630" class="Function">id</a><a id="3055" class="Symbol">)</a> <a id="3057" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="3059" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3061" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="3074" class="Symbol">)</a> <a id="3076" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3079" class="Symbol">((</a><a id="3081" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="3090" class="Symbol">(</a><a id="3091" href="Categories.Category.BinaryProducts.html#2453" class="Function">π₂∘⁂</a> <a id="3096" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3098" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="3107" class="Symbol">)</a> <a id="3109" class="Symbol">(</a><a id="3110" href="Categories.Category.BinaryProducts.html#2453" class="Function">π₂∘⁂</a> <a id="3115" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="3117" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="3126" class="Symbol">))</a> <a id="3129" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="3136" class="Symbol">)</a> <a id="3138" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3143" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3145" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="3148" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3150" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="3153" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="3155" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3157" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="3199" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="3200" class="Symbol">)</a>
<a id="3203" class="Comment">-- distribute over products</a>
<a id="distributeˡ⁻¹-natural"></a><a id="3231" href="Categories.Category.Distributive.Properties.html#3231" class="Function">distributeˡ⁻¹-natural</a> <a id="3253" class="Symbol">:</a> <a id="3255" class="Symbol">∀</a> <a id="3257" class="Symbol">{</a><a id="3258" href="Categories.Category.Distributive.Properties.html#3258" class="Bound">X</a> <a id="3260" href="Categories.Category.Distributive.Properties.html#3260" class="Bound">Y</a> <a id="3262" href="Categories.Category.Distributive.Properties.html#3262" class="Bound">Z</a> <a id="3264" href="Categories.Category.Distributive.Properties.html#3264" class="Bound">U</a> <a id="3266" href="Categories.Category.Distributive.Properties.html#3266" class="Bound">V</a> <a id="3268" href="Categories.Category.Distributive.Properties.html#3268" class="Bound">W</a><a id="3269" class="Symbol">}</a> <a id="3271" class="Symbol">(</a><a id="3272" href="Categories.Category.Distributive.Properties.html#3272" class="Bound">f</a> <a id="3274" class="Symbol">:</a> <a id="3276" href="Categories.Category.Distributive.Properties.html#3258" class="Bound">X</a> <a id="3278" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3280" href="Categories.Category.Distributive.Properties.html#3264" class="Bound">U</a><a id="3281" class="Symbol">)</a> <a id="3283" class="Symbol">(</a><a id="3284" href="Categories.Category.Distributive.Properties.html#3284" class="Bound">g</a> <a id="3286" class="Symbol">:</a> <a id="3288" href="Categories.Category.Distributive.Properties.html#3260" class="Bound">Y</a> <a id="3290" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3292" href="Categories.Category.Distributive.Properties.html#3266" class="Bound">V</a><a id="3293" class="Symbol">)</a> <a id="3295" class="Symbol">(</a><a id="3296" href="Categories.Category.Distributive.Properties.html#3296" class="Bound">h</a> <a id="3298" class="Symbol">:</a> <a id="3300" href="Categories.Category.Distributive.Properties.html#3262" class="Bound">Z</a> <a id="3302" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3304" href="Categories.Category.Distributive.Properties.html#3268" class="Bound">W</a><a id="3305" class="Symbol">)</a> <a id="3307" class="Symbol">→</a> <a id="3309" class="Symbol">((</a><a id="3311" href="Categories.Category.Distributive.Properties.html#3272" class="Bound">f</a> <a id="3313" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3315" href="Categories.Category.Distributive.Properties.html#3284" class="Bound">g</a><a id="3316" class="Symbol">)</a> <a id="3318" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3321" class="Symbol">(</a><a id="3322" href="Categories.Category.Distributive.Properties.html#3272" class="Bound">f</a> <a id="3324" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3326" href="Categories.Category.Distributive.Properties.html#3296" class="Bound">h</a><a id="3327" class="Symbol">))</a> <a id="3330" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3332" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3346" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3348" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3362" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3364" class="Symbol">(</a><a id="3365" href="Categories.Category.Distributive.Properties.html#3272" class="Bound">f</a> <a id="3367" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3369" class="Symbol">(</a><a id="3370" href="Categories.Category.Distributive.Properties.html#3284" class="Bound">g</a> <a id="3372" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3375" href="Categories.Category.Distributive.Properties.html#3296" class="Bound">h</a><a id="3376" class="Symbol">))</a>
<a id="3379" href="Categories.Category.Distributive.Properties.html#3231" class="Function">distributeˡ⁻¹-natural</a> <a id="3401" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3403" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a> <a id="3405" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a> <a id="3407" class="Symbol">=</a> <a id="3409" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="3418" class="Symbol">((</a><a id="3420" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3422" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3424" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a><a id="3425" class="Symbol">)</a> <a id="3427" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3430" class="Symbol">(</a><a id="3431" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3433" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3435" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a><a id="3436" class="Symbol">))</a> <a id="3439" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3441" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3516" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3519" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="3526" class="Symbol">(</a><a id="3527" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="3538" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="3544" class="Symbol">)</a> <a id="3546" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3551" class="Symbol">(</a><a id="3552" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3566" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3568" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a><a id="3579" class="Symbol">)</a> <a id="3581" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3583" class="Symbol">((</a><a id="3585" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3587" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3589" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a><a id="3590" class="Symbol">)</a> <a id="3592" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3595" class="Symbol">(</a><a id="3596" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3598" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3600" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a><a id="3601" class="Symbol">))</a> <a id="3604" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3606" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3649" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3652" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="3658" class="Symbol">(</a><a id="3659" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="3665" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a><a id="3670" class="Symbol">)</a> <a id="3672" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3676" class="Symbol">(</a><a id="3677" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3691" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3693" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a><a id="3694" class="Symbol">(</a><a id="3695" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3698" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3700" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="3702" class="Symbol">)</a> <a id="3704" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3706" class="Symbol">(</a><a id="3707" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3709" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3711" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a><a id="3712" class="Symbol">)</a> <a id="3714" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3716" class="Symbol">(</a><a id="3717" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3720" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3722" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="3724" class="Symbol">)</a> <a id="3726" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3728" class="Symbol">(</a><a id="3729" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3731" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3733" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a><a id="3734" class="Symbol">)</a><a id="3735" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="3736" class="Symbol">)</a> <a id="3738" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3740" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3774" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3777" class="Symbol">(</a><a id="3778" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3786" class="Symbol">(</a><a id="3787" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="3796" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a> <a id="3800" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a><a id="3803" class="Symbol">))</a> <a id="3806" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3814" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3818" class="Symbol">(</a><a id="3819" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3833" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3835" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="3837" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3840" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3842" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3844" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3846" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="3849" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3851" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a> <a id="3853" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="3855" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="3858" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3860" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="3862" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="3864" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="3867" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3869" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a> <a id="3871" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="3872" class="Symbol">)</a> <a id="3874" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3876" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="3916" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="3920" class="Symbol">(</a><a id="3921" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="3929" class="Symbol">(</a><a id="3930" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="3939" class="Symbol">(</a><a id="3940" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="3948" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a> <a id="3956" href="Categories.Category.Cocartesian.html#2548" class="Function">+₁∘i₁</a><a id="3961" class="Symbol">)</a> <a id="3963" class="Symbol">(</a><a id="3964" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="3972" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a> <a id="3980" href="Categories.Category.Cocartesian.html#2585" class="Function">+₁∘i₂</a><a id="3985" class="Symbol">)))</a> <a id="3989" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="3997" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="4001" class="Symbol">(</a><a id="4002" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="4016" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4018" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="4020" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="4022" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4024" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4027" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4029" class="Symbol">(</a><a id="4030" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a> <a id="4032" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4035" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a><a id="4036" class="Symbol">)</a> <a id="4038" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4040" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="4043" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="4045" href="Categories.Category.Distributive.Properties.html#3401" class="Bound">f</a> <a id="4047" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4049" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="4052" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4054" class="Symbol">(</a><a id="4055" href="Categories.Category.Distributive.Properties.html#3403" class="Bound">g</a> <a id="4057" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4060" href="Categories.Category.Distributive.Properties.html#3405" class="Bound">h</a><a id="4061" class="Symbol">)</a> <a id="4063" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4065" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="4068" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="4069" class="Symbol">)</a> <a id="4071" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4073" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="4099" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4103" class="Symbol">(</a><a id="4104" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4112" class="Symbol">(</a><a id="4113" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="4122" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a> <a id="4126" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a><a id="4129" class="Symbol">))</a> <a id="4132" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="4140" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="4688" class="Symbol">((</a><a id="4690" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="4692" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4694" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="4695" class="Symbol">)</a> <a id="4697" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4700" class="Symbol">(</a><a id="4701" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a> <a id="4703" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4705" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="4706" class="Symbol">))</a> <a id="4709" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4711" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4782" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4785" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="4792" class="Symbol">(</a><a id="4793" href="Categories.Morphism.html#1586" class="Function">IsIso.isoˡ</a> <a id="4804" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="4810" class="Symbol">)</a> <a id="4812" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4816" class="Symbol">(</a><a id="4817" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4831" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4833" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a><a id="4844" class="Symbol">)</a> <a id="4846" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4848" class="Symbol">(</a><a id="4849" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="4851" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4853" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a> <a id="4855" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="4858" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a> <a id="4860" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4862" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="4863" class="Symbol">)</a> <a id="4865" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4867" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4910" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4913" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4919" class="Symbol">(</a><a id="4920" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="4926" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a><a id="4931" class="Symbol">)</a> <a id="4933" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4937" class="Symbol">(</a><a id="4938" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="4952" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4954" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="4956" class="Symbol">(</a><a id="4957" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="4960" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4962" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4964" class="Symbol">)</a> <a id="4966" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4968" class="Symbol">(</a><a id="4969" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="4971" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4973" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="4974" class="Symbol">)</a> <a id="4976" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="4978" class="Symbol">(</a><a id="4979" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="4982" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4984" href="Categories.Category.Core.html#630" class="Function">id</a><a id="4986" class="Symbol">)</a> <a id="4988" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4990" class="Symbol">(</a><a id="4991" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a> <a id="4993" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4995" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="4996" class="Symbol">)</a> <a id="4998" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="4999" class="Symbol">)</a> <a id="5001" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5003" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5031" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5034" class="Symbol">(</a><a id="5035" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5043" class="Symbol">(</a><a id="5044" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="5053" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a> <a id="5057" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a><a id="5060" class="Symbol">))</a> <a id="5063" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5071" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5075" class="Symbol">(</a><a id="5076" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5090" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5092" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5094" class="Symbol">(</a><a id="5095" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5098" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5100" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5102" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5104" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5107" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5109" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5110" class="Symbol">)</a> <a id="5112" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5114" class="Symbol">(</a><a id="5115" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5118" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5120" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a> <a id="5122" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5124" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="5127" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5129" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5130" class="Symbol">)</a> <a id="5132" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="5133" class="Symbol">)</a> <a id="5135" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5137" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5169" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5173" class="Symbol">(</a><a id="5174" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5182" class="Symbol">(</a><a id="5183" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="5192" class="Symbol">(</a><a id="5193" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="5201" href="Categories.Category.Cocartesian.html#2548" class="Function">+₁∘i₁</a> <a id="5207" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a><a id="5214" class="Symbol">)</a> <a id="5216" class="Symbol">(</a><a id="5217" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="5225" href="Categories.Category.Cocartesian.html#2585" class="Function">+₁∘i₂</a> <a id="5231" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a><a id="5238" class="Symbol">)))</a> <a id="5242" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5250" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5254" class="Symbol">(</a><a id="5255" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5269" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5271" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5273" class="Symbol">((</a><a id="5275" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5277" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5280" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5281" class="Symbol">)</a> <a id="5283" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5285" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5288" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5290" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a> <a id="5292" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5294" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5296" class="Symbol">)</a> <a id="5298" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5300" class="Symbol">((</a><a id="5302" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5304" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5307" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5308" class="Symbol">)</a> <a id="5310" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5312" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5315" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5317" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a> <a id="5319" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5321" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5323" class="Symbol">)</a> <a id="5325" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="5326" class="Symbol">)</a> <a id="5328" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5330" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5348" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5352" class="Symbol">(</a><a id="5353" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="5361" class="Symbol">(</a><a id="5362" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="5371" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a> <a id="5375" href="Categories.Category.BinaryProducts.html#2881" class="Function">⁂∘⁂</a><a id="5378" class="Symbol">))</a> <a id="5381" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="5389" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5393" class="Symbol">(</a><a id="5394" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5408" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5410" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5412" class="Symbol">((</a><a id="5414" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5416" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5419" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5420" class="Symbol">)</a> <a id="5422" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5424" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5425" class="Symbol">)</a> <a id="5427" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5429" class="Symbol">(</a><a id="5430" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="5433" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5435" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5437" class="Symbol">)</a> <a id="5439" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5441" class="Symbol">((</a><a id="5443" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5445" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5448" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5449" class="Symbol">)</a> <a id="5451" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5453" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5454" class="Symbol">)</a> <a id="5456" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5458" class="Symbol">(</a><a id="5459" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5462" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5464" href="Categories.Category.Core.html#630" class="Function">id</a><a id="5466" class="Symbol">)</a> <a id="5468" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="5469" class="Symbol">)</a> <a id="5471" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5473" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5487" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5491" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="5497" class="Symbol">(</a><a id="5498" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5504" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a><a id="5507" class="Symbol">)</a> <a id="5509" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5513" class="Symbol">(</a><a id="5514" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5528" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5530" class="Symbol">((</a><a id="5532" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5534" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5537" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5538" class="Symbol">)</a> <a id="5540" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5542" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5543" class="Symbol">))</a> <a id="5546" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5548" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a> <a id="5560" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5562" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5607" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="5611" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="5618" class="Symbol">(</a><a id="5619" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="5630" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="5636" class="Symbol">)</a> <a id="5638" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="5642" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5656" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5658" class="Symbol">((</a><a id="5660" href="Categories.Category.Distributive.Properties.html#4674" class="Bound">g</a> <a id="5662" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5665" href="Categories.Category.Distributive.Properties.html#4676" class="Bound">h</a><a id="5666" class="Symbol">)</a> <a id="5668" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="5670" href="Categories.Category.Distributive.Properties.html#4672" class="Bound">f</a><a id="5671" class="Symbol">)</a> <a id="5736" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5739" class="Comment">-- distribute and swap</a>
<a id="distributeˡ⁻¹∘swap"></a><a id="5762" href="Categories.Category.Distributive.Properties.html#5762" class="Function">distributeˡ⁻¹∘swap</a> <a id="5781" class="Symbol">:</a> <a id="5783" class="Symbol">∀</a> <a id="5785" class="Symbol">{</a><a id="5786" href="Categories.Category.Distributive.Properties.html#5786" class="Bound">A</a> <a id="5788" href="Categories.Category.Distributive.Properties.html#5788" class="Bound">B</a> <a id="5790" href="Categories.Category.Distributive.Properties.html#5790" class="Bound">C</a> <a id="5792" class="Symbol">:</a> <a id="5794" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="5797" class="Symbol">}</a> <a id="5799" class="Symbol">→</a> <a id="5801" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="5815" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5817" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="5822" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="5824" class="Symbol">(</a><a id="5825" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="5830" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5833" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5837" class="Symbol">)</a> <a id="5839" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5841" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="5855" class="Symbol">{</a><a id="5856" href="Categories.Category.Distributive.Properties.html#5786" class="Bound">A</a><a id="5857" class="Symbol">}</a> <a id="5859" class="Symbol">{</a><a id="5860" href="Categories.Category.Distributive.Properties.html#5788" class="Bound">B</a><a id="5861" class="Symbol">}</a> <a id="5863" class="Symbol">{</a><a id="5864" href="Categories.Category.Distributive.Properties.html#5790" class="Bound">C</a><a id="5865" class="Symbol">}</a>
<a id="5867" href="Categories.Category.Distributive.Properties.html#5762" class="Function">distributeˡ⁻¹∘swap</a> <a id="5886" class="Symbol">=</a> <a id="5888" href="Categories.Morphism.Properties.html#818" class="Function">Iso⇒Mono</a> <a id="5897" class="Symbol">(</a><a id="5898" href="Categories.Morphism.html#1895" class="Field">IsIso.iso</a> <a id="5908" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="5914" class="Symbol">)</a> <a id="5916" class="Symbol">(</a><a id="5917" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="5931" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5933" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5937" class="Symbol">)</a> <a id="5939" class="Symbol">((</a><a id="5941" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="5946" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5949" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="5953" class="Symbol">)</a> <a id="5955" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5957" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="5970" class="Symbol">)</a> <a id="5972" class="Symbol">(</a><a id="5973" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5982" class="Symbol">(</a><a id="5983" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a> <a id="5995" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5997" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="6011" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6013" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6017" class="Symbol">)</a> <a id="6038" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6041" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6049" class="Symbol">(</a><a id="6050" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="6061" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="6067" class="Symbol">)</a> <a id="6069" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6073" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6129" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6133" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="6141" class="Symbol">(</a><a id="6142" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="6153" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="6159" class="Symbol">)</a> <a id="6161" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6166" class="Symbol">((</a><a id="6168" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6173" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6175" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a><a id="6186" class="Symbol">)</a> <a id="6188" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6190" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="6203" class="Symbol">)</a> <a id="6222" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6225" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="6229" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="6237" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6242" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6244" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6249" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6251" class="Symbol">(</a><a id="6252" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6255" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6257" href="Categories.Category.Core.html#630" class="Function">id</a><a id="6259" class="Symbol">)</a> <a id="6261" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6263" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6268" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6270" class="Symbol">(</a><a id="6271" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6274" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6276" href="Categories.Category.Core.html#630" class="Function">id</a><a id="6278" class="Symbol">)</a> <a id="6280" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6282" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6284" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6298" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6302" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="6311" class="Symbol">(</a><a id="6312" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="6316" href="Categories.Category.BinaryProducts.html#4512" class="Function">swap∘⁂</a><a id="6322" class="Symbol">)</a> <a id="6324" class="Symbol">(</a><a id="6325" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="6329" href="Categories.Category.BinaryProducts.html#4512" class="Function">swap∘⁂</a><a id="6335" class="Symbol">)</a> <a id="6337" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="6345" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6350" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6352" class="Symbol">(</a><a id="6353" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6356" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6358" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="6360" class="Symbol">)</a> <a id="6362" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6364" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6369" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6371" class="Symbol">(</a><a id="6372" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6375" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6377" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="6379" class="Symbol">)</a> <a id="6381" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6383" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6388" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6390" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6392" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6406" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6410" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="6416" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="6422" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6427" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a> <a id="6439" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6441" class="Symbol">(</a><a id="6442" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6447" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="6450" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6454" class="Symbol">)</a> <a id="6456" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6458" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6483" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="6484" class="Symbol">)</a>
<a id="distributeʳ⁻¹∘swap"></a><a id="6487" href="Categories.Category.Distributive.Properties.html#6487" class="Function">distributeʳ⁻¹∘swap</a> <a id="6506" class="Symbol">:</a> <a id="6508" class="Symbol">∀</a> <a id="6510" class="Symbol">{</a><a id="6511" href="Categories.Category.Distributive.Properties.html#6511" class="Bound">A</a> <a id="6513" href="Categories.Category.Distributive.Properties.html#6513" class="Bound">B</a> <a id="6515" href="Categories.Category.Distributive.Properties.html#6515" class="Bound">C</a> <a id="6517" class="Symbol">:</a> <a id="6519" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="6522" class="Symbol">}</a> <a id="6524" class="Symbol">→</a> <a id="6526" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6540" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6542" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6547" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="6549" class="Symbol">(</a><a id="6550" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6555" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="6558" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6562" class="Symbol">)</a> <a id="6564" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6566" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="6580" class="Symbol">{</a><a id="6581" href="Categories.Category.Distributive.Properties.html#6511" class="Bound">A</a><a id="6582" class="Symbol">}</a> <a id="6584" class="Symbol">{</a><a id="6585" href="Categories.Category.Distributive.Properties.html#6513" class="Bound">B</a><a id="6586" class="Symbol">}</a> <a id="6588" class="Symbol">{</a><a id="6589" href="Categories.Category.Distributive.Properties.html#6515" class="Bound">C</a><a id="6590" class="Symbol">}</a>
<a id="6592" href="Categories.Category.Distributive.Properties.html#6487" class="Function">distributeʳ⁻¹∘swap</a> <a id="6611" class="Symbol">=</a> <a id="6613" href="Categories.Morphism.Properties.html#818" class="Function">Iso⇒Mono</a> <a id="6622" class="Symbol">(</a><a id="6623" href="Categories.Morphism.html#1895" class="Field">IsIso.iso</a> <a id="6633" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="6639" class="Symbol">)</a> <a id="6641" class="Symbol">(</a><a id="6642" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6656" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6658" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6662" class="Symbol">)</a> <a id="6664" class="Symbol">((</a><a id="6666" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6671" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="6674" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6678" class="Symbol">)</a> <a id="6680" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6682" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="6695" class="Symbol">)</a> <a id="6697" class="Symbol">(</a><a id="6698" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6707" class="Symbol">(</a><a id="6708" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a> <a id="6720" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6722" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="6736" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6738" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="6742" class="Symbol">)</a> <a id="6763" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6766" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="6774" class="Symbol">(</a><a id="6775" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="6786" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="6792" class="Symbol">)</a> <a id="6794" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6799" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6855" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="6859" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="6867" class="Symbol">(</a><a id="6868" href="Categories.Morphism.html#1612" class="Function">IsIso.isoʳ</a> <a id="6879" href="Categories.Category.Distributive.html#1042" class="Field">isIsoˡ</a><a id="6885" class="Symbol">)</a> <a id="6887" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="6892" class="Symbol">((</a><a id="6894" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6899" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6901" href="Categories.Category.Distributive.html#931" class="Function">distributeˡ</a><a id="6912" class="Symbol">)</a> <a id="6914" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6916" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="6929" class="Symbol">)</a> <a id="6948" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6951" class="Symbol">(</a><a id="6952" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="6956" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="6963" class="Symbol">)</a> <a id="6965" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6970" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6972" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6977" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6979" class="Symbol">(</a><a id="6980" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="6983" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="6985" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="6987" class="Symbol">)</a> <a id="6989" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6991" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="6996" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6998" class="Symbol">(</a><a id="6999" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="7002" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="7004" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="7006" class="Symbol">)</a> <a id="7008" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7010" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7012" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="7026" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="7030" class="Symbol">(</a><a id="7031" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="7040" class="Symbol">(</a><a id="7041" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="7045" href="Categories.Category.BinaryProducts.html#4512" class="Function">swap∘⁂</a><a id="7051" class="Symbol">)</a> <a id="7053" class="Symbol">(</a><a id="7054" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="7058" href="Categories.Category.BinaryProducts.html#4512" class="Function">swap∘⁂</a><a id="7064" class="Symbol">))</a> <a id="7067" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="7075" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="7080" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7082" class="Symbol">(</a><a id="7083" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7086" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="7088" href="Categories.Category.Core.html#630" class="Function">id</a><a id="7090" class="Symbol">)</a> <a id="7092" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7094" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="7099" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7101" class="Symbol">(</a><a id="7102" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7105" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="7107" href="Categories.Category.Core.html#630" class="Function">id</a><a id="7109" class="Symbol">)</a> <a id="7111" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7113" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="7118" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7120" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7122" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="7136" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="7140" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="7146" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="7152" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="7157" class="Symbol">(</a><a id="7158" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a> <a id="7170" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7172" class="Symbol">(</a><a id="7173" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="7178" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="7181" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="7185" class="Symbol">)</a> <a id="7187" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7189" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="7202" class="Symbol">)</a> <a id="7213" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="7214" class="Symbol">)</a>
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