bsc-leon-vatthauer/agda/bsc-thesis/Monad.Instance.Maybe.Commutative.md

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<a id="817" class="Keyword">import</a> <a id="824" href="Categories.Morphism.Properties.html" class="Module">Categories.Morphism.Properties</a> <a id="855" class="Symbol">as</a> <a id="858" class="Module">MP</a>
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# The maybe monad

<pre class="Agda"><a id="897" class="Keyword">module</a> <a id="904" href="Monad.Instance.Maybe.Commutative.html" class="Module">Monad.Instance.Maybe.Commutative</a> <a id="937" class="Symbol">{</a><a id="938" href="Monad.Instance.Maybe.Commutative.html#938" class="Bound">o</a> <a id="940" href="Monad.Instance.Maybe.Commutative.html#940" class="Bound">ℓ</a> <a id="942" href="Monad.Instance.Maybe.Commutative.html#942" class="Bound">e</a><a id="943" class="Symbol">}</a> <a id="945" class="Symbol">{</a><a id="946" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a> <a id="948" class="Symbol">:</a> <a id="950" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="959" href="Monad.Instance.Maybe.Commutative.html#938" class="Bound">o</a> <a id="961" href="Monad.Instance.Maybe.Commutative.html#940" class="Bound">ℓ</a> <a id="963" href="Monad.Instance.Maybe.Commutative.html#942" class="Bound">e</a><a id="964" class="Symbol">}</a> <a id="966" class="Symbol">(</a><a id="967" href="Monad.Instance.Maybe.Commutative.html#967" class="Bound">distributive</a> <a id="980" class="Symbol">:</a> <a id="982" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a> <a id="995" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a><a id="996" class="Symbol">)</a> <a id="998" class="Keyword">where</a>
  <a id="1006" class="Keyword">open</a> <a id="1011" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="1020" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a>
  <a id="1024" class="Keyword">open</a> <a id="1029" href="Categories.Morphism.html" class="Module">M</a> <a id="1031" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a>
  <a id="1035" class="Keyword">open</a> <a id="1040" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1043" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a>
  <a id="1047" class="Keyword">open</a> <a id="1052" href="Categories.Morphism.Properties.html" class="Module">MP</a> <a id="1055" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a>
  <a id="1059" class="Keyword">open</a> <a id="1064" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
  <a id="1079" class="Keyword">open</a> <a id="1084" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
  <a id="1092" class="Keyword">open</a> <a id="1097" href="Categories.Category.Distributive.html#712" class="Module">Distributive</a> <a id="1110" href="Monad.Instance.Maybe.Commutative.html#967" class="Bound">distributive</a>
  <a id="1125" class="Keyword">open</a> <a id="1130" class="Keyword">import</a> <a id="1137" href="Categories.Category.Distributive.Properties.html" class="Module">Categories.Category.Distributive.Properties</a> <a id="1181" href="Monad.Instance.Maybe.Commutative.html#967" class="Bound">distributive</a>
  <a id="1196" class="Keyword">open</a> <a id="1201" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="1213" href="Categories.Category.Distributive.html#793" class="Field">cocartesian</a>
  <a id="1227" class="Keyword">open</a> <a id="1232" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="1242" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="1252" class="Keyword">using</a> <a id="1258" class="Symbol">(</a><a id="1259" href="Categories.Category.Cartesian.html#777" class="Field">terminal</a><a id="1267" class="Symbol">;</a> <a id="1269" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="1277" class="Symbol">)</a>
  <a id="1281" class="Keyword">open</a> <a id="1286" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="1301" href="Categories.Category.Cartesian.html#801" class="Function">products</a> <a id="1310" class="Keyword">renaming</a> <a id="1319" class="Symbol">(</a><a id="1320" href="Categories.Object.Product.Core.html#671" class="Function">unique</a> <a id="1327" class="Symbol">to</a> <a id="1330" class="Function">⟨⟩-unique</a><a id="1339" class="Symbol">)</a>
  <a id="1343" class="Keyword">open</a> <a id="1348" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1357" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>
  <a id="1368" class="Keyword">open</a> <a id="1373" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a> <a id="1391" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="1401" class="Keyword">using</a> <a id="1407" class="Symbol">(</a><a id="1408" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a><a id="1416" class="Symbol">)</a>

  <a id="1421" class="Keyword">open</a> <a id="1426" class="Keyword">import</a> <a id="1433" href="Monad.Instance.Maybe.html" class="Module">Monad.Instance.Maybe</a> <a id="1454" href="Monad.Instance.Maybe.Commutative.html#967" class="Bound">distributive</a>
  <a id="1469" class="Keyword">open</a> <a id="1474" class="Keyword">import</a> <a id="1481" href="Monad.Instance.Maybe.Strong.html" class="Module">Monad.Instance.Maybe.Strong</a> <a id="1509" href="Monad.Instance.Maybe.Commutative.html#967" class="Bound">distributive</a>

  <a id="1525" class="Keyword">open</a> <a id="1530" href="Categories.Category.Monoidal.Symmetric.html#997" class="Module">Symmetric</a> <a id="1540" class="Symbol">(</a><a id="1541" href="Categories.Category.Cartesian.SymmetricMonoidal.html#2255" class="Function">symmetric</a> <a id="1551" href="Monad.Instance.Maybe.Commutative.html#946" class="Bound">C</a> <a id="1553" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a><a id="1562" class="Symbol">)</a> <a id="1564" class="Keyword">using</a> <a id="1570" class="Symbol">(</a><a id="1571" href="Categories.Category.Monoidal.Symmetric.html#1047" class="Field">braided</a><a id="1578" class="Symbol">)</a>

  <a id="1583" class="Keyword">open</a> <a id="1588" href="Categories.Monad.Commutative.html#715" class="Module">Commutative</a>

  <a id="distribute₄"></a><a id="1603" href="Monad.Instance.Maybe.Commutative.html#1603" class="Function">distribute₄</a> <a id="1615" class="Symbol">:</a> <a id="1617" class="Symbol">∀</a> <a id="1619" class="Symbol">{</a><a id="1620" href="Monad.Instance.Maybe.Commutative.html#1620" class="Bound">A</a> <a id="1622" href="Monad.Instance.Maybe.Commutative.html#1622" class="Bound">B</a> <a id="1624" href="Monad.Instance.Maybe.Commutative.html#1624" class="Bound">C</a> <a id="1626" href="Monad.Instance.Maybe.Commutative.html#1626" class="Bound">D</a><a id="1627" class="Symbol">}</a> <a id="1629" class="Symbol">→</a> <a id="1631" class="Symbol">(</a><a id="1632" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="1646" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1649" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="1662" class="Symbol">)</a> <a id="1664" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1666" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1680" class="Symbol">{</a><a id="1681" href="Monad.Instance.Maybe.Commutative.html#1620" class="Bound">A</a> <a id="1683" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1685" href="Monad.Instance.Maybe.Commutative.html#1622" class="Bound">B</a><a id="1686" class="Symbol">}</a> <a id="1688" class="Symbol">{</a><a id="1689" href="Monad.Instance.Maybe.Commutative.html#1624" class="Bound">C</a><a id="1690" class="Symbol">}</a> <a id="1692" class="Symbol">{</a><a id="1693" href="Monad.Instance.Maybe.Commutative.html#1626" class="Bound">D</a><a id="1694" class="Symbol">}</a> <a id="1696" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1698" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1700" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1703" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1706" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1709" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1711" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1714" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1717" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1720" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1722" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1724" class="Symbol">(</a><a id="1725" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1739" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1742" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="1755" class="Symbol">)</a> <a id="1757" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1759" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a>
  <a id="1775" href="Monad.Instance.Maybe.Commutative.html#1603" class="Function">distribute₄</a> <a id="1787" class="Symbol">=</a> <a id="1789" href="Categories.Morphism.Properties.html#980" class="Function">Iso⇒Epi</a> <a id="1797" class="Symbol">(</a><a id="1798" href="Categories.Morphism.html#1895" class="Field">IsIso.iso</a> <a id="1808" href="Categories.Category.Distributive.html#1245" class="Function">isIsoʳ</a><a id="1814" class="Symbol">)</a> <a id="1816" class="Symbol">((</a><a id="1818" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="1832" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1835" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="1848" class="Symbol">)</a> <a id="1850" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1852" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="1865" class="Symbol">)</a> <a id="1867" class="Symbol">(</a><a id="1868" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1870" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1873" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1876" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1879" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1881" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1884" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1887" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1890" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1892" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1894" class="Symbol">(</a><a id="1895" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1909" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1912" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="1925" class="Symbol">)</a> <a id="1927" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1929" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="1942" class="Symbol">)</a> <a id="1944" class="Symbol">(</a><a id="1945" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a> 
    <a id="1956" class="Symbol">(((</a><a id="1959" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="1973" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1976" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="1989" class="Symbol">)</a> <a id="1991" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1993" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2006" class="Symbol">)</a> <a id="2008" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2010" href="Categories.Category.Distributive.html#1143" class="Function">distributeʳ</a><a id="2021" class="Symbol">)</a>                                                                                                <a id="2118" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2121" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="2125" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
    <a id="2132" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2134" class="Symbol">(((</a><a id="2137" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="2151" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2154" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="2167" class="Symbol">)</a> <a id="2169" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2171" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2184" class="Symbol">)</a> <a id="2186" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2188" class="Symbol">(</a><a id="2189" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="2192" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="2194" href="Categories.Category.Core.html#630" class="Function">id</a><a id="2196" class="Symbol">))</a> <a id="2199" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2201" class="Symbol">(((</a><a id="2204" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a> <a id="2218" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2221" href="Categories.Category.Distributive.html#3272" class="Function">distributeʳ⁻¹</a><a id="2234" class="Symbol">)</a> <a id="2236" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2238" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2251" class="Symbol">)</a> <a id="2253" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2255" class="Symbol">(</a><a id="2256" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2259" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="2261" href="Categories.Category.Core.html#630" class="Function">id</a><a id="2263" class="Symbol">))</a> <a id="2266" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>                           <a id="2294" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2297" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="2306" class="Symbol">(</a><a id="2307" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2313" class="Symbol">((</a><a id="2315" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2323" class="Symbol">(</a><a id="2324" href="Categories.Category.BinaryProducts.html#2515" class="Function">⁂-cong₂</a> <a id="2332" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="2337" class="Symbol">(</a><a id="2338" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="2342" class="Symbol">(</a><a id="2343" href="Categories.Category.Cocartesian.html#1714" class="Function">+-unique</a> <a id="2352" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a> <a id="2364" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a><a id="2375" class="Symbol">))))</a> <a id="2380" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2382" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="2386" class="Symbol">(</a><a id="2387" href="Categories.Category.Distributive.Properties.html#3231" class="Function">distributeˡ⁻¹-natural</a> <a id="2409" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="2412" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2415" href="Categories.Category.Core.html#630" class="Function">id</a><a id="2417" 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    <a id="8008" class="Symbol">(</a><a id="8009" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8011" class="Symbol">((</a><a id="8013" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8016" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8019" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="8020" class="Symbol">)</a> <a id="8022" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8024" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8037" class="Symbol">)</a> <a id="8039" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8041" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="8046" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8048" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8051" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8053" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="8055" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8057" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8059" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8072" class="Symbol">)</a> <a id="8074" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8076" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a>                                                                                                                           <a id="8203" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8207" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8213" class="Symbol">(</a><a id="8214" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8220" class="Symbol">(</a><a id="8221" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="8227" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="8229" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="8238" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="8248" class="Symbol">(</a><a id="8249" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="8257" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="8266" class="Symbol">)))</a> <a id="8270" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a> 
    <a id="8277" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8279" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8282" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8284" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8287" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8289" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8291" class="Symbol">(((((</a><a id="8296" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8299" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8302" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="8303" class="Symbol">)</a> <a id="8305" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8307" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8320" class="Symbol">)</a> <a id="8322" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8324" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a><a id="8328" class="Symbol">)</a> <a id="8330" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8333" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8336" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8338" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="8339" class="Symbol">)</a> <a id="8341" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8343" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8356" class="Symbol">)</a> <a id="8358" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8360" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a>                                                                                                            <a id="8472" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="8476" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="8484" class="Symbol">(</a><a id="8485" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8491" class="Symbol">(</a><a id="8492" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="8498" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="8500" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="8509" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="8514" href="Categories.Category.Core.html#1339" class="Function">identity²</a><a id="8523" class="Symbol">)</a> <a id="8525" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="8527" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8533" class="Symbol">(</a><a id="8534" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="8540" class="Symbol">(</a><a id="8541" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="8547" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="8549" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="8558" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="8568" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="8572" class="Symbol">)))</a> <a id="8576" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a> 
    <a id="8583" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="8585" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8588" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="8590" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="8593" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="8595" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8597" class="Symbol">((</a><a id="8599" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8602" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8605" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="8606" class="Symbol">)</a> <a id="8608" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8610" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="8624" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8627" href="Categories.Category.Core.html#630" class="Function">id</a><a id="8629" class="Symbol">)</a> <a id="8631" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8633" class="Symbol">(</a><a id="8634" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a> <a id="8639" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8642" href="Categories.Category.Core.html#630" class="Function">id</a><a id="8644" class="Symbol">)</a> <a id="8646" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8648" class="Symbol">((</a><a id="8650" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="8653" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="8656" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="8657" class="Symbol">)</a> <a id="8659" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8661" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="8674" class="Symbol">)</a> <a id="8676" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="8678" href="Categories.Category.BinaryProducts.html#2048" class="Function">swap</a>                                                                                                <a id="8778" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>


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