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

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# The maybe monad

<pre class="Agda"><a id="696" class="Keyword">module</a> <a id="703" href="Monad.Instance.Maybe.Strong.html" class="Module">Monad.Instance.Maybe.Strong</a> <a id="731" class="Symbol">{</a><a id="732" href="Monad.Instance.Maybe.Strong.html#732" class="Bound">o</a> <a id="734" href="Monad.Instance.Maybe.Strong.html#734" class="Bound">ℓ</a> <a id="736" href="Monad.Instance.Maybe.Strong.html#736" class="Bound">e</a><a id="737" class="Symbol">}</a> <a id="739" class="Symbol">{</a><a id="740" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a> <a id="742" class="Symbol">:</a> <a id="744" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="753" href="Monad.Instance.Maybe.Strong.html#732" class="Bound">o</a> <a id="755" href="Monad.Instance.Maybe.Strong.html#734" class="Bound">ℓ</a> <a id="757" href="Monad.Instance.Maybe.Strong.html#736" class="Bound">e</a><a id="758" class="Symbol">}</a> <a id="760" class="Symbol">(</a><a id="761" href="Monad.Instance.Maybe.Strong.html#761" class="Bound">distributive</a> <a id="774" class="Symbol">:</a> <a id="776" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a> <a id="789" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a><a id="790" class="Symbol">)</a> <a id="792" class="Keyword">where</a>
  <a id="800" class="Keyword">open</a> <a id="805" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="814" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a>
  <a id="818" class="Keyword">open</a> <a id="823" href="Categories.Morphism.html" class="Module">M</a> <a id="825" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a>
  <a id="829" class="Keyword">open</a> <a id="834" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="837" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a>
  <a id="841" class="Keyword">open</a> <a id="846" href="Categories.Morphism.Properties.html" class="Module">MP</a> <a id="849" href="Monad.Instance.Maybe.Strong.html#740" class="Bound">C</a>
  <a id="853" class="Keyword">open</a> <a id="858" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
  <a id="873" class="Keyword">open</a> <a id="878" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
  <a id="886" class="Keyword">open</a> <a id="891" href="Categories.Category.Distributive.html#712" class="Module">Distributive</a> <a id="904" href="Monad.Instance.Maybe.Strong.html#761" class="Bound">distributive</a>
  <a id="919" class="Keyword">open</a> <a id="924" class="Keyword">import</a> <a id="931" href="Categories.Category.Distributive.Properties.html" class="Module">Categories.Category.Distributive.Properties</a> <a id="975" href="Monad.Instance.Maybe.Strong.html#761" class="Bound">distributive</a>
  <a id="990" class="Keyword">open</a> <a id="995" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="1007" href="Categories.Category.Distributive.html#793" class="Field">cocartesian</a>
  <a id="1021" class="Keyword">open</a> <a id="1026" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="1036" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="1046" class="Keyword">using</a> <a id="1052" class="Symbol">(</a><a id="1053" href="Categories.Category.Cartesian.html#777" class="Field">terminal</a><a id="1061" class="Symbol">;</a> <a id="1063" href="Categories.Category.Cartesian.html#801" class="Field">products</a><a id="1071" class="Symbol">)</a>
  <a id="1075" class="Keyword">open</a> <a id="1080" href="Categories.Category.BinaryProducts.html#848" class="Module">BinaryProducts</a> <a id="1095" href="Categories.Category.Cartesian.html#801" class="Function">products</a> <a id="1104" class="Keyword">renaming</a> <a id="1113" class="Symbol">(</a><a id="1114" href="Categories.Object.Product.Core.html#671" class="Function">unique</a> <a id="1121" class="Symbol">to</a> <a id="1124" class="Function">⟨⟩-unique</a><a id="1133" class="Symbol">)</a>
  <a id="1137" class="Keyword">open</a> <a id="1142" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="1151" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>
  <a id="1162" class="Keyword">open</a> <a id="1167" href="Categories.Category.Cartesian.Monoidal.html#1139" class="Module">CartesianMonoidal</a> <a id="1185" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="1195" class="Keyword">using</a> <a id="1201" class="Symbol">(</a><a id="1202" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a><a id="1210" class="Symbol">)</a>
  <a id="1214" class="Keyword">open</a> <a id="1219" class="Keyword">import</a> <a id="1226" href="Monad.Instance.Maybe.html" class="Module">Monad.Instance.Maybe</a> <a id="1247" href="Monad.Instance.Maybe.Strong.html#761" class="Bound">distributive</a>

  <a id="1263" class="Keyword">open</a> <a id="1268" href="Categories.Monad.Strong.html#1832" class="Module">StrongMonad</a> <a id="1280" class="Keyword">using</a> <a id="1286" class="Symbol">(</a><a id="1287" href="Categories.Monad.Strong.html#1918" class="Field">M</a><a id="1288" class="Symbol">;</a> <a id="1290" href="Categories.Monad.Strong.html#1941" class="Field">strength</a><a id="1298" class="Symbol">)</a>
  <a id="1302" class="Keyword">open</a> <a id="1307" href="Categories.Monad.Strong.html#764" class="Module">Strength</a> <a id="1316" class="Keyword">renaming</a> <a id="1325" class="Symbol">(</a><a id="1326" href="Categories.Monad.Strong.html#1268" class="Field">identityˡ</a> <a id="1336" class="Symbol">to</a> <a id="1339" class="Field">s-identityˡ</a><a id="1350" class="Symbol">)</a>

  <a id="maybeStrong"></a><a id="1355" href="Monad.Instance.Maybe.Strong.html#1355" class="Function">maybeStrong</a> <a id="1367" class="Symbol">:</a> <a id="1369" href="Categories.Monad.Strong.html#1832" class="Record">StrongMonad</a> <a id="1381" href="Categories.Category.Cartesian.Monoidal.html#5400" class="Function">monoidal</a>
  <a id="1392" href="Monad.Instance.Maybe.Strong.html#1355" class="Function">maybeStrong</a> <a id="1404" class="Symbol">.</a><a id="1405" href="Categories.Monad.Strong.html#1918" class="Field">M</a> <a id="1407" class="Symbol">=</a> <a id="1409" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a>
  <a id="1422" href="Monad.Instance.Maybe.Strong.html#1355" class="Function">maybeStrong</a> <a id="1434" class="Symbol">.</a><a id="1435" href="Categories.Monad.Strong.html#1941" class="Field">strength</a> <a id="1444" class="Symbol">.</a><a id="1445" href="Categories.Monad.Strong.html#1060" class="Field">strengthen</a> <a id="1456" class="Symbol">=</a> <a id="1458" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1467" class="Symbol">(</a><a id="1468" class="Keyword">record</a> <a id="1475" class="Symbol">{</a> <a id="1477" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1479" class="Symbol">=</a> <a id="1481" class="Symbol">λ</a> <a id="1483" href="Monad.Instance.Maybe.Strong.html#1483" class="Bound">X</a> <a id="1485" class="Symbol">→</a> <a id="1487" class="Symbol">(</a><a id="1488" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1491" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1494" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1495" class="Symbol">)</a> <a id="1497" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1499" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1513" class="Symbol">;</a> <a id="1515" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1523" class="Symbol">=</a> <a id="1525" class="Symbol">λ</a> <a id="1527" class="Symbol">{</a><a id="1528" href="Monad.Instance.Maybe.Strong.html#1528" class="Bound">X</a><a id="1529" class="Symbol">}</a> <a id="1531" class="Symbol">{</a><a id="1532" href="Monad.Instance.Maybe.Strong.html#1532" class="Bound">Y</a><a id="1533" class="Symbol">}</a> <a id="1535" class="Symbol">(</a><a id="1536" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1538" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1540" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a><a id="1541" class="Symbol">)</a> <a id="1543" class="Symbol">→</a> <a id="1545" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a> 
    <a id="1556" class="Symbol">((</a><a id="1558" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1561" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1564" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1565" class="Symbol">)</a> <a id="1567" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1569" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="1582" class="Symbol">)</a> <a id="1584" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1586" class="Symbol">(</a><a id="1587" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1589" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1591" class="Symbol">(</a><a id="1592" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a> <a id="1594" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1597" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1599" class="Symbol">))</a> <a id="1602" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1605" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1611" class="Symbol">(</a><a id="1612" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1616" class="Symbol">(</a><a id="1617" href="Categories.Category.Distributive.Properties.html#3231" class="Function">distributeˡ⁻¹-natural</a> <a id="1639" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1641" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a> <a id="1643" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1645" class="Symbol">))</a> <a id="1648" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
    <a id="1655" class="Symbol">(</a><a id="1656" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1659" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1662" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1663" class="Symbol">)</a> <a id="1665" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1667" class="Symbol">(</a><a id="1668" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1670" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1672" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a> <a id="1674" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1677" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1679" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1681" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1683" class="Symbol">)</a> <a id="1685" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1687" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1701" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1704" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1710" class="Symbol">(</a><a id="1711" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="1717" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1719" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="1728" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="1738" class="Symbol">(</a><a id="1739" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1743" class="Symbol">(</a><a id="1744" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a> <a id="1753" class="Symbol">(</a><a id="1754" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="1756" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1758" class="Symbol">(</a><a id="1759" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1761" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1763" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1765" class="Symbol">)))))</a> <a id="1771" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
    <a id="1778" class="Symbol">(</a><a id="1779" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1781" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1783" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a> <a id="1785" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1788" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1789" class="Symbol">)</a> <a id="1791" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1793" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a>                  <a id="1824" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1828" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1834" class="Symbol">(</a><a id="1835" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="1841" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1843" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="1852" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="1862" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="1871" class="Symbol">)</a> <a id="1873" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a> 
    <a id="1880" class="Symbol">(</a><a id="1881" href="Monad.Instance.Maybe.Strong.html#1536" class="Bound">f</a> <a id="1883" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1885" href="Monad.Instance.Maybe.Strong.html#1540" class="Bound">g</a> <a id="1887" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1890" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1892" class="Symbol">)</a> <a id="1894" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1896" class="Symbol">(</a><a id="1897" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1900" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1903" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1904" class="Symbol">)</a> <a id="1906" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1908" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a>     <a id="1926" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="1928" class="Symbol">})</a>
  <a id="1933" href="Monad.Instance.Maybe.Strong.html#1355" class="Function">maybeStrong</a> <a id="1945" class="Symbol">.</a><a id="1946" href="Categories.Monad.Strong.html#1941" class="Field">strength</a> <a id="1955" class="Symbol">.</a><a id="1956" href="Monad.Instance.Maybe.Strong.html#1339" class="Field">s-identityˡ</a> <a id="1968" class="Symbol">=</a> <a id="1970" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a> 
    <a id="1981" class="Symbol">(</a><a id="1982" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a> <a id="1985" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1988" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1990" class="Symbol">)</a> <a id="1992" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1994" class="Symbol">(</a><a id="1995" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1998" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2001" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2002" class="Symbol">)</a> <a id="2004" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2006" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="2020" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2023" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2029" class="Symbol">(</a><a id="2030" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="2036" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2038" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="2047" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="2057" class="Symbol">(</a><a id="2058" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="2068" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2070" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a> <a id="2079" href="Categories.Object.Product.Core.html#546" class="Function">π₂</a><a id="2081" class="Symbol">))</a> <a id="2084" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
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</pre>