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<h1 id="the-maybe-monad">The maybe monad</h1>
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<a id="equationalLifting"></a><a id="1290" href="Monad.Instance.Maybe.EquationalLifting.html#1290" class="Function">equationalLifting</a> <a id="1308" class="Symbol">:</a> <a id="1310" class="Symbol">∀</a> <a id="1312" class="Symbol">{</a><a id="1313" href="Monad.Instance.Maybe.EquationalLifting.html#1313" class="Bound">X</a><a id="1314" class="Symbol">}</a> <a id="1316" class="Symbol">→</a> <a id="1318" class="Symbol">((</a><a id="1320" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1323" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1326" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1327" class="Symbol">)</a> <a id="1329" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1331" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="1344" class="Symbol">)</a> <a id="1346" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1348" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a> <a id="1350" class="Symbol">{</a><a id="1351" href="Monad.Instance.Maybe.EquationalLifting.html#1313" class="Bound">X</a> <a id="1353" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1355" href="Categories.Object.Terminal.html#905" class="Function">⊤</a><a id="1356" class="Symbol">}</a> <a id="1358" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1360" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1362" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1365" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1367" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1370" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="1372" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1375" href="Categories.Category.Core.html#630" class="Function">id</a>
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<a id="1429" class="Keyword">where</a>
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<a id="1813" class="Symbol">(</a><a id="1814" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1817" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1820" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1821" class="Symbol">)</a> <a id="1823" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1825" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1828" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1830" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1832" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1835" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1837" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1840" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="1867" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1870" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="1876" class="Symbol">(</a><a id="1877" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="1885" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1887" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="1896" class="Symbol">)</a> <a id="1898" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1907" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1910" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1912" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1914" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1917" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1919" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1922" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="1961" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="1967" href="Monad.Instance.Maybe.EquationalLifting.html#1967" class="Function">inj₂</a> <a id="1972" class="Symbol">:</a> <a id="1974" class="Symbol">(((</a><a id="1977" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1980" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1983" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="1984" class="Symbol">)</a> <a id="1986" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1988" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2001" class="Symbol">)</a> <a id="2003" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2005" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</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.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2013" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2015" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2018" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2020" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="2027" href="Monad.Instance.Maybe.EquationalLifting.html#1967" class="Function">inj₂</a> <a id="2032" class="Symbol">=</a> <a id="2034" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2047" class="Symbol">(((</a><a id="2050" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2053" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2056" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2057" class="Symbol">)</a> <a id="2059" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2061" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2074" class="Symbol">)</a> <a id="2076" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2078" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a><a id="2079" class="Symbol">)</a> <a id="2081" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2083" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2101" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2104" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2110" href="Categories.Category.BinaryProducts.html#6982" class="Function">Δ∘</a> <a id="2113" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2122" class="Symbol">((</a><a id="2124" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2127" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2130" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2131" class="Symbol">)</a> <a id="2133" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2135" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2148" class="Symbol">)</a> <a id="2150" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2152" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2154" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2157" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2159" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2162" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2176" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="2180" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2188" class="Symbol">(</a><a id="2189" href="Categories.Category.BinaryProducts.html#2599" class="Function">⁂∘⟨⟩</a> <a id="2194" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="2196" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="2205" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="2215" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="2224" class="Symbol">)</a> <a id="2226" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="2235" class="Symbol">((</a><a id="2237" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2240" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2243" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2244" class="Symbol">)</a> <a id="2246" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2248" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a><a id="2261" class="Symbol">)</a> <a id="2263" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2265" class="Symbol">(</a><a id="2266" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2269" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="2271" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="2273" class="Symbol">)</a> <a id="2275" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2277" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2279" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2282" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2284" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2287" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2289" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2292" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2298" class="Symbol">(</a><a id="2299" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2305" href="Categories.Category.Distributive.Properties.html#977" class="Function">distributeˡ⁻¹-i₂</a><a id="2321" class="Symbol">)</a> <a id="2323" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2332" class="Symbol">(</a><a id="2333" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2336" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2339" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2340" class="Symbol">)</a> <a id="2342" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2344" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2347" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2349" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2351" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2354" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2356" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2359" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2386" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2389" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2395" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a> <a id="2403" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2412" class="Symbol">(</a><a id="2413" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2416" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2418" href="Categories.Object.Terminal.html#577" class="Function">!</a><a id="2419" class="Symbol">)</a> <a id="2421" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2423" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2425" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2428" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2430" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2433" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2466" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2469" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="2475" class="Symbol">(</a><a id="2476" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="2480" class="Symbol">(</a><a id="2481" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a> <a id="2490" class="Symbol">(</a><a id="2491" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="2493" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2495" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2497" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2500" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2502" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2505" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a><a id="2506" class="Symbol">)))</a> <a id="2510" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2519" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2522" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2524" href="Categories.Object.Terminal.html#577" class="Function">!</a> <a id="2573" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2576" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="2584" class="Symbol">(</a><a id="2585" href="Categories.Object.Terminal.html#605" class="Function">!-unique</a> <a id="2594" href="Categories.Category.Core.html#630" class="Function">id</a><a id="2596" class="Symbol">)</a> <a id="2598" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2607" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2610" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2612" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2661" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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