bsc-leon-vatthauer/agda/bsc-thesis/Monad.Instance.K.EquationalLifting.md

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<pre class="Agda"><a id="343" class="Keyword">module</a> <a id="350" href="Monad.Instance.K.EquationalLifting.html" class="Module">Monad.Instance.K.EquationalLifting</a> <a id="385" class="Symbol">{</a><a id="386" href="Monad.Instance.K.EquationalLifting.html#386" class="Bound">o</a> <a id="388" href="Monad.Instance.K.EquationalLifting.html#388" class="Bound">ℓ</a> <a id="390" href="Monad.Instance.K.EquationalLifting.html#390" class="Bound">e</a><a id="391" class="Symbol">}</a> <a id="393" class="Symbol">(</a><a id="394" href="Monad.Instance.K.EquationalLifting.html#394" class="Bound">ambient</a> <a id="402" class="Symbol">:</a> <a id="404" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="412" href="Monad.Instance.K.EquationalLifting.html#386" class="Bound">o</a> <a id="414" href="Monad.Instance.K.EquationalLifting.html#388" class="Bound">ℓ</a> <a id="416" href="Monad.Instance.K.EquationalLifting.html#390" class="Bound">e</a><a id="417" class="Symbol">)</a> <a id="419" class="Symbol">(</a><a id="420" href="Monad.Instance.K.EquationalLifting.html#420" class="Bound">MK</a> <a id="423" class="Symbol">:</a> <a id="425" href="Monad.Instance.K.html#1123" class="Record">MIK.MonadK</a> <a id="436" href="Monad.Instance.K.EquationalLifting.html#394" class="Bound">ambient</a><a id="443" class="Symbol">)</a> <a id="445" class="Keyword">where</a>
<a id="451" class="Keyword">open</a> <a id="456" href="Category.Ambient.html#1681" class="Module">Ambient</a> <a id="464" href="Monad.Instance.K.EquationalLifting.html#394" class="Bound">ambient</a>
<a id="472" class="Keyword">open</a> <a id="477" href="Monad.Instance.K.html" class="Module">MIK</a> <a id="481" href="Monad.Instance.K.EquationalLifting.html#394" class="Bound">ambient</a>
<a id="489" class="Keyword">open</a> <a id="494" href="Monad.Instance.K.html#1123" class="Module">MonadK</a> <a id="501" href="Monad.Instance.K.EquationalLifting.html#420" class="Bound">MK</a>
<a id="504" class="Keyword">open</a> <a id="509" class="Keyword">import</a> <a id="516" href="Monad.Instance.K.Strong.html" class="Module">Monad.Instance.K.Strong</a> <a id="540" href="Monad.Instance.K.EquationalLifting.html#394" class="Bound">ambient</a> <a id="548" href="Monad.Instance.K.EquationalLifting.html#420" class="Bound">MK</a>
<a id="551" class="Keyword">open</a> <a id="556" class="Keyword">import</a> <a id="563" href="Category.Construction.ElgotAlgebras.html" class="Module">Category.Construction.ElgotAlgebras</a> <a id="599" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>
<a id="611" class="Keyword">open</a> <a id="616" class="Keyword">import</a> <a id="623" href="Algebra.Elgot.html" class="Module">Algebra.Elgot</a> <a id="637" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>
<a id="649" class="Keyword">open</a> <a id="654" class="Keyword">import</a> <a id="661" href="Algebra.Elgot.Stable.html" class="Module">Algebra.Elgot.Stable</a> <a id="682" href="Category.Ambient.html#2762" class="Function">distributive</a> <a id="695" class="Keyword">using</a> <a id="701" class="Symbol">(</a><a id="702" href="Algebra.Elgot.Stable.html#2437" class="Record">IsStableFreeElgotAlgebra</a><a id="726" class="Symbol">)</a>

<a id="729" class="Keyword">open</a> <a id="734" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="747" class="Keyword">open</a> <a id="752" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="758" class="Keyword">open</a> <a id="763" href="Category.Ambient.html#9069" class="Module">M</a> <a id="765" href="Category.Ambient.html#1757" class="Function">C</a>
<a id="767" class="Keyword">open</a> <a id="772" href="Category.Ambient.html#9087" class="Module">MR</a> <a id="775" href="Category.Ambient.html#1757" class="Function">C</a>
<a id="777" class="Keyword">open</a> <a id="782" href="Monad.Instance.K.html#1775" class="Module">kleisliK</a> <a id="791" class="Keyword">using</a> <a id="797" class="Symbol">(</a><a id="798" href="Categories.Monad.Relative.html#895" class="Function">extend</a><a id="804" class="Symbol">)</a>
<a id="806" class="Keyword">open</a> <a id="811" href="Monad.Instance.K.html#1672" class="Module">monadK</a> <a id="818" class="Keyword">using</a> <a id="824" class="Symbol">(</a><a id="825" href="Categories.Monad.html#583" class="Function">μ</a><a id="826" class="Symbol">)</a>
<a id="828" class="Keyword">open</a> <a id="833" href="Categories.FreeObjects.Free.html#485" class="Module">FreeObject</a> <a id="844" class="Keyword">using</a> <a id="850" class="Symbol">(</a><a id="851" href="Categories.FreeObjects.Free.html#1017" class="Field">*-uniq</a><a id="857" class="Symbol">)</a>
<a id="859" class="Keyword">open</a> <a id="864" href="Algebra.Elgot.html#15964" class="Module">Elgot-Algebra</a> <a id="878" class="Keyword">using</a> <a id="884" class="Symbol">(</a><a id="885" href="Algebra.Elgot.html#2086" class="Function">#-Uniformity</a><a id="897" class="Symbol">;</a> <a id="899" href="Algebra.Elgot.html#2011" class="Function">#-Fixpoint</a><a id="909" class="Symbol">;</a> <a id="911" href="Algebra.Elgot.html#2331" class="Function">#-resp-≈</a><a id="919" class="Symbol">)</a>
<a id="921" class="Keyword">open</a> <a id="926" href="Monad.Instance.K.Strong.html#22135" class="Module">strongK</a> <a id="934" class="Keyword">using</a> <a id="940" class="Symbol">(</a><a id="941" href="Categories.Monad.Strong.html#1060" class="Function">strengthen</a><a id="951" class="Symbol">)</a>

<a id="954" class="Comment">-- some helper definitions to make our life easier</a>
<a id="1005" class="Keyword">private</a>
  <a id="η"></a><a id="1015" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="1017" class="Symbol">=</a> <a id="1019" class="Symbol">λ</a> <a id="1021" href="Monad.Instance.K.EquationalLifting.html#1021" class="Bound">Z</a> <a id="1023" class="Symbol">→</a> <a id="1025" href="Categories.FreeObjects.Free.html#854" class="Field">FreeObject.η</a> <a id="1038" class="Symbol">(</a><a id="1039" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1052" href="Monad.Instance.K.EquationalLifting.html#1021" class="Bound">Z</a><a id="1053" class="Symbol">)</a>
  <a id="_♯"></a><a id="1057" href="Monad.Instance.K.EquationalLifting.html#1057" class="Function Operator">_♯</a> <a id="1060" class="Symbol">=</a> <a id="1062" class="Symbol">λ</a> <a id="1064" class="Symbol">{</a><a id="1065" href="Monad.Instance.K.EquationalLifting.html#1065" class="Bound">A</a> <a id="1067" href="Monad.Instance.K.EquationalLifting.html#1067" class="Bound">X</a> <a id="1069" href="Monad.Instance.K.EquationalLifting.html#1069" class="Bound">Y</a><a id="1070" class="Symbol">}</a> <a id="1072" href="Monad.Instance.K.EquationalLifting.html#1072" class="Bound">f</a> <a id="1074" class="Symbol">→</a> <a id="1076" href="Algebra.Elgot.Stable.html#2658" class="Field Operator">IsStableFreeElgotAlgebra.[_,_]♯</a> <a id="1108" class="Symbol">{</a><a id="1109" class="Argument">Y</a> <a id="1111" class="Symbol">=</a> <a id="1113" href="Monad.Instance.K.EquationalLifting.html#1067" class="Bound">X</a><a id="1114" class="Symbol">}</a> <a id="1116" class="Symbol">(</a><a id="1117" href="Monad.Instance.K.html#1228" class="Field">stable</a> <a id="1124" href="Monad.Instance.K.EquationalLifting.html#1067" class="Bound">X</a><a id="1125" class="Symbol">)</a> <a id="1127" class="Symbol">{</a><a id="1128" class="Argument">X</a> <a id="1130" class="Symbol">=</a> <a id="1132" href="Monad.Instance.K.EquationalLifting.html#1065" class="Bound">A</a><a id="1133" class="Symbol">}</a> <a id="1135" class="Symbol">(</a><a id="1136" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="1145" href="Monad.Instance.K.EquationalLifting.html#1069" class="Bound">Y</a><a id="1146" class="Symbol">)</a> <a id="1148" href="Monad.Instance.K.EquationalLifting.html#1072" class="Bound">f</a>
  <a id="_#"></a><a id="1152" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">_#</a> <a id="1155" class="Symbol">=</a> <a id="1157" class="Symbol">λ</a> <a id="1159" class="Symbol">{</a><a id="1160" href="Monad.Instance.K.EquationalLifting.html#1160" class="Bound">A</a><a id="1161" class="Symbol">}</a> <a id="1163" class="Symbol">{</a><a id="1164" href="Monad.Instance.K.EquationalLifting.html#1164" class="Bound">X</a><a id="1165" class="Symbol">}</a> <a id="1167" href="Monad.Instance.K.EquationalLifting.html#1167" class="Bound">f</a> <a id="1169" class="Symbol">→</a> <a id="1171" href="Algebra.Elgot.html#1938" class="Function Operator">Elgot-Algebra._#</a> <a id="1188" class="Symbol">(</a><a id="1189" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="1198" href="Monad.Instance.K.EquationalLifting.html#1160" class="Bound">A</a><a id="1199" class="Symbol">)</a> <a id="1201" class="Symbol">{</a><a id="1202" class="Argument">X</a> <a id="1204" class="Symbol">=</a> <a id="1206" href="Monad.Instance.K.EquationalLifting.html#1164" class="Bound">X</a><a id="1207" class="Symbol">}</a> <a id="1209" href="Monad.Instance.K.EquationalLifting.html#1167" class="Bound">f</a>
</pre>
# **K** is an equational lifting monad

<pre class="Agda"><a id="equationalLifting"></a><a id="1264" href="Monad.Instance.K.EquationalLifting.html#1264" class="Function">equationalLifting</a> <a id="1282" class="Symbol">:</a> <a id="1284" class="Symbol">∀</a> <a id="1286" class="Symbol">{</a><a id="1287" href="Monad.Instance.K.EquationalLifting.html#1287" class="Bound">X</a><a id="1288" class="Symbol">}</a> <a id="1290" class="Symbol">→</a> <a id="1292" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1294" class="Symbol">(</a><a id="1295" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1299" href="Monad.Instance.K.EquationalLifting.html#1287" class="Bound">X</a> <a id="1301" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1303" href="Monad.Instance.K.EquationalLifting.html#1287" class="Bound">X</a><a id="1304" class="Symbol">)</a> <a id="1306" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1308" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a> <a id="1310" class="Symbol">{</a><a id="1311" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1315" href="Monad.Instance.K.EquationalLifting.html#1287" class="Bound">X</a><a id="1316" class="Symbol">}</a> <a id="1318" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1320" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="1324" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1326" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="1328" href="Monad.Instance.K.EquationalLifting.html#1287" class="Bound">X</a> <a id="1330" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1332" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="1336" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a>
<a id="1338" href="Monad.Instance.K.EquationalLifting.html#1264" class="Function">equationalLifting</a> <a id="1356" class="Symbol">{</a><a id="1357" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1358" class="Symbol">}</a> <a id="1360" class="Symbol">=</a> <a id="1362" href="Categories.FreeObjects.Free.html#1017" class="Field">*-uniq</a> <a id="1369" class="Symbol">(</a><a id="1370" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="1383" class="Symbol">_)</a> <a id="1386" class="Symbol">(</a><a id="1387" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="1389" class="Symbol">_</a> <a id="1391" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1393" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1395" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="1397" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1399" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1401" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="1405" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a><a id="1406" class="Symbol">)</a> <a id="1408" class="Symbol">(</a><a id="1409" class="Keyword">record</a> <a id="1416" class="Symbol">{</a> <a id="1418" href="Category.Construction.ElgotAlgebras.html#794" class="Field">h</a> <a id="1420" class="Symbol">=</a> <a id="1422" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1424" class="Symbol">(</a><a id="1425" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1429" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1431" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1433" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1434" class="Symbol">)</a> <a id="1436" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1438" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a> <a id="1440" class="Symbol">;</a> <a id="1442" href="Category.Construction.ElgotAlgebras.html#812" class="Field">preserves</a> <a id="1452" class="Symbol">=</a> <a id="1454" href="Monad.Instance.K.EquationalLifting.html#1488" class="Function">preserves&#39;</a> <a id="1465" class="Symbol">})</a> <a id="1468" href="Monad.Instance.K.EquationalLifting.html#4090" class="Function">commute</a>
  <a id="1478" class="Keyword">where</a>
    <a id="1488" href="Monad.Instance.K.EquationalLifting.html#1488" class="Function">preserves&#39;</a> <a id="1499" class="Symbol">:</a> <a id="1501" class="Symbol">∀</a> <a id="1503" class="Symbol">{</a><a id="1504" href="Monad.Instance.K.EquationalLifting.html#1504" class="Bound">Z</a><a id="1505" class="Symbol">}</a> <a id="1507" class="Symbol">{</a><a id="1508" href="Monad.Instance.K.EquationalLifting.html#1508" class="Bound">f</a> <a id="1510" class="Symbol">:</a> <a id="1512" href="Monad.Instance.K.EquationalLifting.html#1504" class="Bound">Z</a> <a id="1514" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1516" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1520" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1522" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1524" href="Monad.Instance.K.EquationalLifting.html#1504" class="Bound">Z</a><a id="1525" class="Symbol">}</a> <a id="1527" class="Symbol">→</a> <a id="1529" class="Symbol">(</a><a id="1530" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1532" class="Symbol">(</a><a id="1533" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1537" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1539" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1541" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1542" class="Symbol">)</a> <a id="1544" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1546" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a><a id="1547" class="Symbol">)</a> <a id="1549" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1551" href="Monad.Instance.K.EquationalLifting.html#1508" class="Bound">f</a> <a id="1553" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1555" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1557" class="Symbol">((</a><a id="1559" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1561" class="Symbol">(</a><a id="1562" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1566" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1568" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1570" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1571" class="Symbol">)</a> <a id="1573" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1575" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a> <a id="1577" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1580" href="Category.Ambient.html#2115" class="Function">idC</a><a id="1583" class="Symbol">)</a> <a id="1585" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1587" href="Monad.Instance.K.EquationalLifting.html#1508" class="Bound">f</a><a id="1588" class="Symbol">)</a> <a id="1590" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a>
    <a id="1596" href="Monad.Instance.K.EquationalLifting.html#1488" class="Function">preserves&#39;</a> <a id="1607" class="Symbol">{</a><a id="1608" href="Monad.Instance.K.EquationalLifting.html#1608" class="Bound">Z</a><a id="1609" class="Symbol">}</a> <a id="1611" class="Symbol">{</a><a id="1612" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a><a id="1613" class="Symbol">}</a> <a id="1615" class="Symbol">=</a> <a id="1617" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a> 
      <a id="1630" class="Symbol">(</a><a id="1631" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1633" class="Symbol">(</a><a id="1634" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1638" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1640" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1642" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1643" class="Symbol">)</a> <a id="1645" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1647" href="Categories.Category.BinaryProducts.html#6931" class="Function">Δ</a><a id="1648" class="Symbol">)</a> <a id="1650" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1652" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="1654" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1656" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1659" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="1665" href="Categories.Category.BinaryProducts.html#6982" class="Function">Δ∘</a> <a id="1668" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
      <a id="1677" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1679" class="Symbol">(</a><a id="1680" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="1684" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="1686" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1688" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="1689" class="Symbol">)</a> <a id="1691" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1693" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1695" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="1697" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1699" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1701" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="1703" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1705" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="1707" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1710" href="Monad.Instance.K.EquationalLifting.html#1873" class="Function">helper₁</a> <a id="1718" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
      <a id="1726" class="Symbol">((</a><a id="1728" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="1730" class="Symbol">_</a> <a id="1732" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1735" href="Category.Ambient.html#2115" class="Function">idC</a><a id="1738" class="Symbol">)</a> <a id="1740" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1742" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="1756" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1758" class="Symbol">(</a><a id="1759" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="1763" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="1765" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a><a id="1766" class="Symbol">))</a><a id="1768" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1770" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1772" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="1774" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="1776" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="1778" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="1780" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="1784" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="1786" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1789" href="Monad.Instance.K.EquationalLifting.html#2267" class="Function">helper₂</a> <a id="1797" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> <a id="1799" class="Comment">-- lemma20</a>
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          <a id="2120" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="2122" class="Symbol">(</a><a id="2123" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2127" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="2129" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2131" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="2132" class="Symbol">)</a> <a id="2134" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2136" class="Symbol">(</a><a id="2137" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="2141" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="2143" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="2145" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a><a id="2146" class="Symbol">)</a> <a id="2148" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2150" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="2152" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a> <a id="2154" href="Monad.Instance.K.EquationalLifting.html#1152" class="Function Operator">#</a> <a id="2156" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="2158" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="2162" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="2164" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2167" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="2173" class="Symbol">(</a><a id="2174" href="Monad.Instance.K.Strong.html#5557" class="Function">τ-comm</a> <a id="2181" href="Monad.Instance.K.EquationalLifting.html#1612" class="Bound">f</a><a id="2182" class="Symbol">)</a> <a id="2184" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
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      <a id="4222" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="4224" class="Symbol">(</a><a id="4225" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="4229" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4231" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4233" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="4234" class="Symbol">)</a> <a id="4236" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4238" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="4240" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4242" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4244" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="4246" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4248" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4250" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="4252" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4255" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="4263" class="Symbol">(</a><a id="4264" href="Categories.Object.Product.Core.html#854" class="Function">⟨⟩-cong₂</a> <a id="4273" class="Symbol">(</a><a id="4274" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4278" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="4287" class="Symbol">)</a> <a id="4289" class="Symbol">(</a><a id="4290" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4294" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="4303" class="Symbol">)</a> <a id="4305" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="4307" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4311" href="Categories.Category.BinaryProducts.html#2599" class="Function">⁂∘⟨⟩</a><a id="4315" class="Symbol">)</a> <a id="4317" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
      <a id="4326" href="Monad.Instance.K.Strong.html#3259" class="Function">τ</a> <a id="4328" class="Symbol">(</a><a id="4329" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="4333" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4335" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4337" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="4338" class="Symbol">)</a> <a id="4340" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4342" class="Symbol">(</a><a id="4343" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="4347" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="4349" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4351" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a><a id="4352" class="Symbol">)</a> <a id="4354" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4356" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="4358" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4360" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4362" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="4364" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="4368" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="4370" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4373" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="4379" class="Symbol">(</a><a id="4380" href="Monad.Instance.K.Strong.html#3312" class="Function">τ-η</a> <a id="4384" class="Symbol">_)</a> <a id="4387" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
      <a id="4396" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4398" class="Symbol">_</a> <a id="4400" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4402" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟨</a> <a id="4404" href="Monad.Instance.K.EquationalLifting.html#1015" class="Function">η</a> <a id="4406" href="Monad.Instance.K.EquationalLifting.html#1357" class="Bound">X</a> <a id="4408" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">,</a> <a id="4410" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="4414" href="Categories.Category.BinaryProducts.html#1396" class="Function Operator">⟩</a> <a id="4416" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
</pre>