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

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# The maybe monad

<pre class="Agda"><a id="449" class="Keyword">module</a> <a id="456" href="Monad.Instance.Maybe.html" class="Module">Monad.Instance.Maybe</a> <a id="477" class="Symbol">{</a><a id="478" href="Monad.Instance.Maybe.html#478" class="Bound">o</a> <a id="480" href="Monad.Instance.Maybe.html#480" class="Bound">ℓ</a> <a id="482" href="Monad.Instance.Maybe.html#482" class="Bound">e</a><a id="483" class="Symbol">}</a> <a id="485" class="Symbol">{</a><a id="486" href="Monad.Instance.Maybe.html#486" class="Bound">C</a> <a id="488" class="Symbol">:</a> <a id="490" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="499" href="Monad.Instance.Maybe.html#478" class="Bound">o</a> <a id="501" href="Monad.Instance.Maybe.html#480" class="Bound">ℓ</a> <a id="503" href="Monad.Instance.Maybe.html#482" class="Bound">e</a><a id="504" class="Symbol">}</a> <a id="506" class="Symbol">(</a><a id="507" href="Monad.Instance.Maybe.html#507" class="Bound">distributive</a> <a id="520" class="Symbol">:</a> <a id="522" href="Categories.Category.Distributive.html#712" class="Record">Distributive</a> <a id="535" href="Monad.Instance.Maybe.html#486" class="Bound">C</a><a id="536" class="Symbol">)</a> <a id="538" class="Keyword">where</a>
  <a id="546" class="Keyword">open</a> <a id="551" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="560" href="Monad.Instance.Maybe.html#486" class="Bound">C</a>
  <a id="564" class="Keyword">open</a> <a id="569" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="572" href="Monad.Instance.Maybe.html#486" class="Bound">C</a>
  <a id="576" class="Keyword">open</a> <a id="581" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
  <a id="596" class="Keyword">open</a> <a id="601" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
  <a id="609" class="Keyword">open</a> <a id="614" href="Categories.Category.Distributive.html#712" class="Module">Distributive</a> <a id="627" href="Monad.Instance.Maybe.html#507" class="Bound">distributive</a>
  <a id="642" class="Keyword">open</a> <a id="647" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="659" href="Categories.Category.Distributive.html#793" class="Field">cocartesian</a>
  <a id="673" class="Keyword">open</a> <a id="678" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="688" href="Categories.Category.Distributive.html#765" class="Field">cartesian</a> <a id="698" class="Keyword">using</a> <a id="704" class="Symbol">(</a><a id="705" href="Categories.Category.Cartesian.html#777" class="Field">terminal</a><a id="713" class="Symbol">)</a>
  <a id="717" class="Keyword">open</a> <a id="722" href="Categories.Object.Terminal.html#860" class="Module">Terminal</a> <a id="731" href="Categories.Category.Cartesian.html#777" class="Function">terminal</a>

  <a id="maybeFunctor"></a><a id="743" href="Monad.Instance.Maybe.html#743" class="Function">maybeFunctor</a> <a id="756" class="Symbol">:</a> <a id="758" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="770" href="Monad.Instance.Maybe.html#486" class="Bound">C</a>
  <a id="774" href="Monad.Instance.Maybe.html#743" class="Function">maybeFunctor</a> <a id="787" class="Symbol">=</a> <a id="789" class="Keyword">record</a>
    <a id="800" class="Symbol">{</a> <a id="802" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="805" class="Symbol">=</a> <a id="807" class="Symbol">λ</a> <a id="809" href="Monad.Instance.Maybe.html#809" class="Bound">X</a> <a id="811" class="Symbol">→</a> <a id="813" href="Monad.Instance.Maybe.html#809" class="Bound">X</a> <a id="815" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="817" href="Categories.Object.Terminal.html#905" class="Function">⊤</a>
    <a id="823" class="Symbol">;</a> <a id="825" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="828" class="Symbol">=</a> <a id="830" class="Symbol">λ</a> <a id="832" href="Monad.Instance.Maybe.html#832" class="Bound">f</a> <a id="834" class="Symbol">→</a> <a id="836" href="Monad.Instance.Maybe.html#832" class="Bound">f</a> <a id="838" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="841" href="Categories.Category.Core.html#630" class="Function">id</a>
    <a id="848" class="Symbol">;</a> <a id="850" href="Categories.Functor.Core.html#511" class="Field">identity</a> <a id="859" class="Symbol">=</a> <a id="861" href="Categories.Category.Cocartesian.html#1714" class="Function">+-unique</a> <a id="870" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a> <a id="882" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a>
    <a id="898" class="Symbol">;</a> <a id="900" href="Categories.Functor.Core.html#565" class="Field">homomorphism</a> <a id="913" class="Symbol">=</a> <a id="915" class="Symbol">λ</a> <a id="917" class="Symbol">{</a><a id="918" href="Monad.Instance.Maybe.html#918" class="Bound">X</a><a id="919" class="Symbol">}</a> <a id="921" class="Symbol">{</a><a id="922" href="Monad.Instance.Maybe.html#922" class="Bound">Y</a><a id="923" class="Symbol">}</a> <a id="925" class="Symbol">{</a><a id="926" href="Monad.Instance.Maybe.html#926" class="Bound">Z</a><a id="927" class="Symbol">}</a> <a id="929" class="Symbol">{</a><a id="930" href="Monad.Instance.Maybe.html#930" class="Bound">f</a><a id="931" class="Symbol">}</a> <a id="933" class="Symbol">{</a><a id="934" href="Monad.Instance.Maybe.html#934" class="Bound">g</a><a id="935" class="Symbol">}</a> <a id="937" class="Symbol">→</a> <a id="939" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="943" class="Symbol">(</a><a id="944" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="950" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="952" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="961" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="966" href="Categories.Category.Core.html#1339" class="Function">identity²</a><a id="975" class="Symbol">)</a>
    <a id="981" class="Symbol">;</a> <a id="983" href="Categories.Functor.Core.html#696" class="Field">F-resp-≈</a> <a id="992" class="Symbol">=</a> <a id="994" class="Symbol">λ</a> <a id="996" href="Monad.Instance.Maybe.html#996" class="Bound">eq</a> <a id="999" class="Symbol">→</a> <a id="1001" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="1010" href="Monad.Instance.Maybe.html#996" class="Bound">eq</a> <a id="1013" href="Relation.Binary.Structures.html#1596" class="Function">refl</a>
    <a id="1022" class="Symbol">}</a>

  <a id="1027" class="Keyword">open</a> <a id="1032" href="Categories.Monad.html#454" class="Module">Monad</a> <a id="1038" class="Keyword">renaming</a> <a id="1047" class="Symbol">(</a><a id="1048" href="Categories.Monad.html#899" class="Field">identityˡ</a> <a id="1058" class="Symbol">to</a> <a id="1061" class="Field">m-identityˡ</a><a id="1072" class="Symbol">;</a> <a id="1074" href="Categories.Monad.html#953" class="Field">identityʳ</a> <a id="1084" class="Symbol">to</a> <a id="1087" class="Field">m-identityʳ</a><a id="1098" class="Symbol">;</a> <a id="1100" href="Categories.Monad.html#759" class="Field">assoc</a> <a id="1106" class="Symbol">to</a> <a id="1109" class="Field">m-assoc</a><a id="1116" class="Symbol">;</a> <a id="1118" href="Categories.Monad.html#829" class="Field">sym-assoc</a> <a id="1128" class="Symbol">to</a> <a id="1131" class="Field">m-sym-assoc</a><a id="1142" class="Symbol">)</a>

  <a id="maybeMonad"></a><a id="1147" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1158" class="Symbol">:</a> <a id="1160" href="Categories.Monad.html#454" class="Record">Monad</a> <a id="1166" href="Monad.Instance.Maybe.html#486" class="Bound">C</a>
  <a id="1170" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1181" class="Symbol">.</a><a id="1182" href="Categories.Monad.html#525" class="Field">F</a> <a id="1184" class="Symbol">=</a> <a id="1186" href="Monad.Instance.Maybe.html#743" class="Function">maybeFunctor</a>
  <a id="1201" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1212" class="Symbol">.</a><a id="1213" href="Categories.Monad.html#547" class="Field">η</a> <a id="1215" class="Symbol">=</a> <a id="1217" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1226" class="Symbol">(</a><a id="1227" class="Keyword">record</a> <a id="1234" class="Symbol">{</a> <a id="1236" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1238" class="Symbol">=</a> <a id="1240" class="Symbol">λ</a> <a id="1242" href="Monad.Instance.Maybe.html#1242" class="Bound">X</a> <a id="1244" class="Symbol">→</a> <a id="1246" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1249" class="Symbol">;</a> <a id="1251" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1259" class="Symbol">=</a> <a id="1261" class="Symbol">λ</a> <a id="1263" href="Monad.Instance.Maybe.html#1263" class="Bound">f</a> <a id="1265" class="Symbol">→</a> <a id="1267" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1271" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="1279" class="Symbol">})</a>
  <a id="1284" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1295" class="Symbol">.</a><a id="1296" href="Categories.Monad.html#583" class="Field">μ</a> <a id="1298" class="Symbol">=</a> <a id="1300" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1309" class="Symbol">(</a><a id="1310" class="Keyword">record</a> <a id="1317" class="Symbol">{</a> <a id="1319" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1321" class="Symbol">=</a> <a id="1323" class="Symbol">λ</a> <a id="1325" href="Monad.Instance.Maybe.html#1325" class="Bound">X</a> <a id="1327" class="Symbol">→</a> <a id="1329" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1331" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1334" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1336" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1339" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1341" class="Symbol">;</a> <a id="1343" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1351" class="Symbol">=</a> <a id="1353" class="Symbol">λ</a> <a id="1355" href="Monad.Instance.Maybe.html#1355" class="Bound">f</a> <a id="1357" class="Symbol">→</a> <a id="1359" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="1365" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1367" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="1376" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a> <a id="1388" class="Symbol">(</a><a id="1389" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1393" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="1400" class="Symbol">)</a> <a id="1402" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1404" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1408" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a><a id="1411" class="Symbol">})</a>
  <a id="1416" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1427" class="Symbol">.</a><a id="1428" href="Monad.Instance.Maybe.html#1109" class="Field">m-assoc</a> <a id="1436" class="Symbol">=</a> <a id="1438" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a> 
    <a id="1449" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1451" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1454" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1456" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1459" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1461" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1463" class="Symbol">(</a><a id="1464" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1466" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1469" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1471" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1474" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1476" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1479" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1481" class="Symbol">)</a> <a id="1483" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1486" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="1492" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
    <a id="1499" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1501" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1504" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1506" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1508" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1511" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1513" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1516" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1518" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1520" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1523" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1525" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1528" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1530" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1534" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="1543" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a> <a id="1551" class="Symbol">(</a><a id="1552" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="1560" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1562" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="1569" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="1573" class="Symbol">)</a> <a id="1575" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
    <a id="1581" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1583" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1585" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1588" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1590" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1593" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1595" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1597" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1600" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1602" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1604" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1607" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1609" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1612" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1614" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1616" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1619" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1621" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1625" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="1629" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
    <a id="1635" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1637" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1640" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1642" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1645" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1647" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1649" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1651" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1654" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1656" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1659" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1661" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
  <a id="1665" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1676" class="Symbol">.</a><a id="1677" href="Monad.Instance.Maybe.html#1131" class="Field">m-sym-assoc</a> <a id="1689" class="Symbol">=</a> <a id="1691" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1695" class="Symbol">(</a><a id="1696" href="Monad.Instance.Maybe.html#1109" class="Field">m-assoc</a> <a id="1704" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a><a id="1714" class="Symbol">)</a>
  <a id="1718" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1729" class="Symbol">.</a><a id="1730" href="Monad.Instance.Maybe.html#1061" class="Field">m-identityˡ</a> <a id="1742" class="Symbol">=</a> <a id="1744" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="1750" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1752" href="Categories.Category.Cocartesian.html#1714" class="Function">+-unique</a> <a id="1761" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="1766" href="Categories.Morphism.Reasoning.Core.html#1309" class="Function">id-comm-sym</a>
  <a id="1780" href="Monad.Instance.Maybe.html#1147" class="Function">maybeMonad</a> <a id="1791" class="Symbol">.</a><a id="1792" href="Monad.Instance.Maybe.html#1087" class="Field">m-identityʳ</a> <a id="1804" class="Symbol">=</a> <a id="1806" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a>
</pre>