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

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<pre class="Agda"><a id="558" class="Keyword">module</a> <a id="565" href="Monad.Instance.K.StrongPreElgot.html" class="Module">Monad.Instance.K.StrongPreElgot</a> <a id="597" class="Symbol">{</a><a id="598" href="Monad.Instance.K.StrongPreElgot.html#598" class="Bound">o</a> <a id="600" href="Monad.Instance.K.StrongPreElgot.html#600" class="Bound">ℓ</a> <a id="602" href="Monad.Instance.K.StrongPreElgot.html#602" class="Bound">e</a><a id="603" class="Symbol">}</a> <a id="605" class="Symbol">(</a><a id="606" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a> <a id="614" class="Symbol">:</a> <a id="616" href="Category.Ambient.html#1681" class="Record">Ambient</a> <a id="624" href="Monad.Instance.K.StrongPreElgot.html#598" class="Bound">o</a> <a id="626" href="Monad.Instance.K.StrongPreElgot.html#600" class="Bound">ℓ</a> <a id="628" href="Monad.Instance.K.StrongPreElgot.html#602" class="Bound">e</a><a id="629" class="Symbol">)</a> <a id="631" class="Symbol">(</a><a id="632" href="Monad.Instance.K.StrongPreElgot.html#632" class="Bound">MK</a> <a id="635" class="Symbol">:</a> <a id="637" href="Monad.Instance.K.html#1123" class="Record">MIK.MonadK</a> <a id="648" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a><a id="655" class="Symbol">)</a> <a id="657" class="Keyword">where</a>
<a id="663" class="Keyword">open</a> <a id="668" href="Category.Ambient.html#1681" class="Module">Ambient</a> <a id="676" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a>
<a id="684" class="Keyword">open</a> <a id="689" href="Monad.Instance.K.html" class="Module">MIK</a> <a id="693" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a>
<a id="701" class="Keyword">open</a> <a id="706" href="Monad.Instance.K.html#1123" class="Module">MonadK</a> <a id="713" href="Monad.Instance.K.StrongPreElgot.html#632" class="Bound">MK</a>
<a id="716" class="Keyword">open</a> <a id="721" class="Keyword">import</a> <a id="728" href="Algebra.Elgot.html" class="Module">Algebra.Elgot</a> <a id="742" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>
<a id="754" class="Keyword">open</a> <a id="759" class="Keyword">import</a> <a id="766" href="Algebra.Elgot.Free.html" class="Module">Algebra.Elgot.Free</a> <a id="785" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>
<a id="797" class="Keyword">open</a> <a id="802" class="Keyword">import</a> <a id="809" href="Algebra.Elgot.Stable.html" class="Module">Algebra.Elgot.Stable</a> <a id="830" href="Category.Ambient.html#2762" class="Function">distributive</a>
<a id="843" class="Keyword">open</a> <a id="848" class="Keyword">import</a> <a id="855" href="Monad.PreElgot.html" class="Module">Monad.PreElgot</a> <a id="870" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a>
<a id="878" class="Keyword">open</a> <a id="883" class="Keyword">import</a> <a id="890" href="Monad.Instance.K.html" class="Module">Monad.Instance.K</a> <a id="907" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a>
<a id="915" class="Keyword">open</a> <a id="920" class="Keyword">import</a> <a id="927" href="Monad.Instance.K.Commutative.html" class="Module">Monad.Instance.K.Commutative</a> <a id="956" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a> <a id="964" href="Monad.Instance.K.StrongPreElgot.html#632" class="Bound">MK</a>
<a id="967" class="Keyword">open</a> <a id="972" class="Keyword">import</a> <a id="979" href="Monad.Instance.K.Strong.html" class="Module">Monad.Instance.K.Strong</a> <a id="1003" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a> <a id="1011" href="Monad.Instance.K.StrongPreElgot.html#632" class="Bound">MK</a>
<a id="1014" class="Keyword">open</a> <a id="1019" class="Keyword">import</a> <a id="1026" href="Monad.Instance.K.PreElgot.html" class="Module">Monad.Instance.K.PreElgot</a> <a id="1052" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a> <a id="1060" href="Monad.Instance.K.StrongPreElgot.html#632" class="Bound">MK</a>
<a id="1063" class="Keyword">open</a> <a id="1068" class="Keyword">import</a> <a id="1075" href="Category.Construction.StrongPreElgotMonads.html" class="Module">Category.Construction.StrongPreElgotMonads</a> <a id="1118" href="Monad.Instance.K.StrongPreElgot.html#606" class="Bound">ambient</a>
<a id="1126" class="Keyword">open</a> <a id="1131" class="Keyword">import</a> <a id="1138" href="Category.Construction.ElgotAlgebras.html" class="Module">Category.Construction.ElgotAlgebras</a> <a id="1174" href="Categories.Category.Extensive.html#734" class="Function">cocartesian</a>

<a id="1187" class="Keyword">open</a> <a id="1192" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="1198" class="Keyword">open</a> <a id="1203" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1216" class="Keyword">open</a> <a id="1221" href="Category.Ambient.html#9087" class="Module">MR</a> <a id="1224" href="Category.Ambient.html#1757" class="Function">C</a>
<a id="1226" class="Keyword">open</a> <a id="1231" href="Category.Ambient.html#9069" class="Module">M</a> <a id="1233" href="Category.Ambient.html#1757" class="Function">C</a>
</pre>
# K is the initial strong pre-Elgot monad

<pre class="Agda"><a id="isStrongPreElgot"></a><a id="1291" href="Monad.Instance.K.StrongPreElgot.html#1291" class="Function">isStrongPreElgot</a> <a id="1308" class="Symbol">:</a> <a id="1310" href="Monad.PreElgot.html#1307" class="Record">IsStrongPreElgot</a> <a id="1327" href="Monad.Instance.K.Strong.html#21664" class="Function">KStrong</a>
<a id="1335" href="Monad.Instance.K.StrongPreElgot.html#1291" class="Function">isStrongPreElgot</a> <a id="1352" class="Symbol">=</a> <a id="1354" class="Keyword">record</a> 
  <a id="1364" class="Symbol">{</a> <a id="1366" href="Monad.PreElgot.html#1488" class="Field">preElgot</a> <a id="1375" class="Symbol">=</a> <a id="1377" href="Monad.Instance.K.PreElgot.html#1025" class="Function">isPreElgot</a> 
  <a id="1391" class="Symbol">;</a> <a id="1393" href="Monad.PreElgot.html#1614" class="Field">strengthen-preserves</a> <a id="1414" class="Symbol">=</a> <a id="1416" href="Monad.Instance.K.Strong.html#5557" class="Function">τ-comm</a> 
  <a id="1426" class="Symbol">}</a>

<a id="strongPreElgot"></a><a id="1429" href="Monad.Instance.K.StrongPreElgot.html#1429" class="Function">strongPreElgot</a> <a id="1444" class="Symbol">:</a> <a id="1446" href="Monad.PreElgot.html#1819" class="Record">StrongPreElgotMonad</a>
<a id="1466" href="Monad.Instance.K.StrongPreElgot.html#1429" class="Function">strongPreElgot</a> <a id="1481" class="Symbol">=</a> <a id="1483" class="Keyword">record</a> 
  <a id="1493" class="Symbol">{</a> <a id="1495" href="Monad.PreElgot.html#1879" class="Field">SM</a> <a id="1498" class="Symbol">=</a> <a id="1500" href="Monad.Instance.K.Strong.html#21664" class="Function">KStrong</a> 
  <a id="1511" class="Symbol">;</a> <a id="1513" href="Monad.PreElgot.html#1911" class="Field">isStrongPreElgot</a> <a id="1530" class="Symbol">=</a> <a id="1532" href="Monad.Instance.K.StrongPreElgot.html#1291" class="Function">isStrongPreElgot</a> 
  <a id="1552" class="Symbol">}</a>

<a id="isInitialStrongPreElgot"></a><a id="1555" href="Monad.Instance.K.StrongPreElgot.html#1555" class="Function">isInitialStrongPreElgot</a> <a id="1579" class="Symbol">:</a> <a id="1581" href="Categories.Object.Initial.html#422" class="Record">IsInitial</a> <a id="1591" href="Category.Construction.StrongPreElgotMonads.html#1914" class="Function">StrongPreElgotMonads</a> <a id="1612" href="Monad.Instance.K.StrongPreElgot.html#1429" class="Function">strongPreElgot</a>
<a id="1627" href="Monad.Instance.K.StrongPreElgot.html#1555" class="Function">isInitialStrongPreElgot</a> <a id="1651" class="Symbol">=</a> <a id="1653" class="Keyword">record</a> <a id="1660" class="Symbol">{</a> <a id="1662" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="1664" class="Symbol">=</a> <a id="1666" href="Monad.Instance.K.StrongPreElgot.html#1706" class="Function">!′</a> <a id="1669" class="Symbol">;</a> <a id="1671" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1680" class="Symbol">=</a> <a id="1682" href="Monad.Instance.K.StrongPreElgot.html#14727" class="Function">!-unique′</a> <a id="1692" class="Symbol">}</a>
  <a id="1696" class="Keyword">where</a>
    <a id="1706" href="Monad.Instance.K.StrongPreElgot.html#1706" class="Function">!′</a> <a id="1709" class="Symbol">:</a> <a id="1711" class="Symbol">∀</a> <a id="1713" class="Symbol">{</a><a id="1714" href="Monad.Instance.K.StrongPreElgot.html#1714" class="Bound">A</a> <a id="1716" class="Symbol">:</a> <a id="1718" href="Monad.PreElgot.html#1819" class="Record">StrongPreElgotMonad</a><a id="1737" class="Symbol">}</a> <a id="1739" class="Symbol">→</a> <a id="1741" href="Category.Construction.StrongPreElgotMonads.html#1408" class="Record">StrongPreElgotMonad-Morphism</a> <a id="1770" href="Monad.Instance.K.StrongPreElgot.html#1429" class="Function">strongPreElgot</a> <a id="1785" href="Monad.Instance.K.StrongPreElgot.html#1714" class="Bound">A</a>
    <a id="1791" href="Monad.Instance.K.StrongPreElgot.html#1706" class="Function">!′</a> <a id="1794" class="Symbol">{</a><a id="1795" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a><a id="1796" class="Symbol">}</a> <a id="1798" class="Symbol">=</a> <a id="1800" class="Keyword">record</a>
      <a id="1813" class="Symbol">{</a> <a id="1815" href="Category.Construction.StrongPreElgotMonads.html#1477" class="Field">α</a> <a id="1817" class="Symbol">=</a> <a id="1819" href="Categories.NaturalTransformation.Core.html#1750" class="Function">ntHelper</a> <a id="1828" class="Symbol">(</a><a id="1829" class="Keyword">record</a> <a id="1836" class="Symbol">{</a> <a id="1838" href="Categories.NaturalTransformation.Core.html#1637" class="Field">η</a> <a id="1840" class="Symbol">=</a> <a id="1842" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="1845" class="Symbol">;</a> <a id="1847" href="Categories.NaturalTransformation.Core.html#1681" class="Field">commute</a> <a id="1855" class="Symbol">=</a> <a id="1857" href="Monad.Instance.K.StrongPreElgot.html#3900" class="Function">commute</a> <a id="1865" class="Symbol">})</a>
      <a id="1874" class="Symbol">;</a> <a id="1876" href="Category.Construction.StrongPreElgotMonads.html#1568" class="Field">α-η</a> <a id="1880" class="Symbol">=</a> <a id="1882" href="Monad.Instance.K.StrongPreElgot.html#7035" class="Function">α-η</a>
      <a id="1892" class="Symbol">;</a> <a id="1894" href="Category.Construction.StrongPreElgotMonads.html#1624" class="Field">α-μ</a> <a id="1898" class="Symbol">=</a> <a id="1900" href="Monad.Instance.K.StrongPreElgot.html#7151" class="Function">α-μ</a> 
      <a id="1911" class="Symbol">;</a> <a id="1913" href="Category.Construction.StrongPreElgotMonads.html#1714" class="Field">α-strength</a> <a id="1924" class="Symbol">=</a> <a id="1926" href="Monad.Instance.K.StrongPreElgot.html#11162" class="Function">α-strength</a> 
      <a id="1944" class="Symbol">;</a> <a id="1946" href="Category.Construction.StrongPreElgotMonads.html#1827" class="Field">α-preserves</a> <a id="1958" class="Symbol">=</a> <a id="1960" class="Symbol">λ</a> <a id="1962" class="Symbol">{</a><a id="1963" href="Monad.Instance.K.StrongPreElgot.html#1963" class="Bound">X</a><a id="1964" class="Symbol">}</a> <a id="1966" class="Symbol">{</a><a id="1967" href="Monad.Instance.K.StrongPreElgot.html#1967" class="Bound">B</a><a id="1968" class="Symbol">}</a> <a id="1970" href="Monad.Instance.K.StrongPreElgot.html#1970" class="Bound">f</a> <a id="1972" class="Symbol">→</a> <a id="1974" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="2007" class="Symbol">(((</a><a id="2010" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="2023" href="Monad.Instance.K.StrongPreElgot.html#1967" class="Bound">B</a><a id="2024" class="Symbol">)</a> <a id="2026" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="2038" class="Symbol">)</a> <a id="2040" class="Symbol">{</a><a id="2041" class="Argument">A</a> <a id="2043" class="Symbol">=</a> <a id="2045" class="Keyword">record</a> <a id="2052" class="Symbol">{</a> <a id="2054" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="2056" class="Symbol">=</a> <a id="2058" href="Categories.Functor.Core.html#432" class="Function">T.F.F₀</a> <a id="2065" href="Monad.Instance.K.StrongPreElgot.html#1967" class="Bound">B</a> <a id="2067" class="Symbol">;</a> <a id="2069" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="2077" class="Symbol">=</a> <a id="2079" href="Monad.PreElgot.html#815" class="Function">StrongPreElgotMonad.elgotalgebras</a> <a id="2113" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a> <a id="2115" class="Symbol">}}</a> <a id="2118" class="Symbol">(</a><a id="2119" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="2125" href="Monad.Instance.K.StrongPreElgot.html#1967" class="Bound">B</a><a id="2126" class="Symbol">))</a> 
      <a id="2136" class="Symbol">}</a>
      <a id="2144" class="Keyword">where</a>
        <a id="2158" class="Keyword">open</a> <a id="2163" href="Monad.PreElgot.html#1819" class="Module">StrongPreElgotMonad</a> <a id="2183" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a> <a id="2185" class="Keyword">using</a> <a id="2191" class="Symbol">(</a><a id="2192" href="Monad.PreElgot.html#1879" class="Field">SM</a><a id="2194" class="Symbol">)</a>
        <a id="2204" class="Keyword">module</a> <a id="2211" href="Monad.Instance.K.StrongPreElgot.html#2211" class="Module">SM</a> <a id="2214" class="Symbol">=</a> <a id="2216" href="Categories.Monad.Strong.html#1832" class="Module">StrongMonad</a> <a id="2228" href="Monad.PreElgot.html#1879" class="Field">SM</a>
        <a id="2239" class="Keyword">open</a> <a id="2244" href="Monad.Instance.K.StrongPreElgot.html#2211" class="Module">SM</a> <a id="2247" class="Keyword">using</a> <a id="2253" class="Symbol">(</a><a id="2254" href="Categories.Monad.Strong.html#1060" class="Function">strengthen</a><a id="2264" class="Symbol">)</a> <a id="2266" class="Keyword">renaming</a> <a id="2275" class="Symbol">(</a><a id="2276" href="Categories.Monad.Strong.html#1918" class="Function">M</a> <a id="2278" class="Symbol">to</a> <a id="2281" class="Function">T</a><a id="2282" class="Symbol">)</a>
        <a id="2292" class="Keyword">open</a> <a id="2297" href="Monad.Instance.K.StrongPreElgot.html#379" class="Module">RMonad</a> <a id="2304" class="Symbol">(</a><a id="2305" href="Categories.Monad.Construction.Kleisli.html#2851" class="Function">Monad⇒Kleisli</a> <a id="2319" href="Category.Ambient.html#1757" class="Function">C</a> <a id="2321" href="Monad.Instance.K.StrongPreElgot.html#2281" class="Function">T</a><a id="2322" class="Symbol">)</a> <a id="2324" class="Keyword">using</a> <a id="2330" class="Symbol">(</a><a id="2331" href="Categories.Monad.Relative.html#895" class="Field">extend</a><a id="2337" class="Symbol">)</a>
        <a id="2347" class="Keyword">open</a> <a id="2352" href="Monad.Instance.K.html#1672" class="Module">monadK</a> <a id="2359" class="Keyword">using</a> <a id="2365" class="Symbol">()</a> <a id="2368" class="Keyword">renaming</a> <a id="2377" class="Symbol">(</a><a id="2378" href="Categories.Monad.html#547" class="Function">η</a> <a id="2380" class="Symbol">to</a> <a id="2383" class="Function">ηK</a><a id="2385" class="Symbol">;</a> <a id="2387" href="Categories.Monad.html#583" class="Function">μ</a> <a id="2389" class="Symbol">to</a> <a id="2392" class="Function">μK</a><a id="2394" class="Symbol">)</a>
        <a id="2404" class="Keyword">open</a> <a id="2409" href="Monad.Instance.K.Strong.html#22135" class="Module">strongK</a> <a id="2417" class="Keyword">using</a> <a id="2423" class="Symbol">()</a> <a id="2426" class="Keyword">renaming</a> <a id="2435" class="Symbol">(</a><a id="2436" href="Categories.Monad.Strong.html#1060" class="Function">strengthen</a> <a id="2447" class="Symbol">to</a> <a id="2450" class="Function">strengthenK</a><a id="2461" class="Symbol">)</a>
        <a id="2471" class="Keyword">open</a> <a id="2476" href="Algebra.Elgot.html#1845" class="Module">Elgot-Algebra-on</a> <a id="2493" class="Keyword">using</a> <a id="2499" class="Symbol">(</a><a id="2500" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a><a id="2508" class="Symbol">)</a>
        <a id="2518" href="Monad.Instance.K.StrongPreElgot.html#2518" class="Function">T-Alg</a> <a id="2524" class="Symbol">:</a> <a id="2526" class="Symbol">∀</a> <a id="2528" class="Symbol">(</a><a id="2529" href="Monad.Instance.K.StrongPreElgot.html#2529" class="Bound">X</a> <a id="2531" class="Symbol">:</a> <a id="2533" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2536" class="Symbol">)</a> <a id="2538" class="Symbol">→</a> <a id="2540" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
        <a id="2562" href="Monad.Instance.K.StrongPreElgot.html#2518" class="Function">T-Alg</a> <a id="2568" href="Monad.Instance.K.StrongPreElgot.html#2568" class="Bound">X</a> <a id="2570" class="Symbol">=</a> <a id="2572" class="Keyword">record</a> <a id="2579" class="Symbol">{</a> <a id="2581" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="2583" class="Symbol">=</a> <a id="2585" href="Categories.Functor.Core.html#797" class="Function">T.F.₀</a> <a id="2591" href="Monad.Instance.K.StrongPreElgot.html#2568" class="Bound">X</a> <a id="2593" class="Symbol">;</a> <a id="2595" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="2603" class="Symbol">=</a> <a id="2605" href="Monad.PreElgot.html#815" class="Function">StrongPreElgotMonad.elgotalgebras</a> <a id="2639" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a> <a id="2641" class="Symbol">}</a>
        <a id="2651" href="Monad.Instance.K.StrongPreElgot.html#2651" class="Function">K-Alg</a> <a id="2657" class="Symbol">:</a> <a id="2659" class="Symbol">∀</a> <a id="2661" class="Symbol">(</a><a id="2662" href="Monad.Instance.K.StrongPreElgot.html#2662" class="Bound">X</a> <a id="2664" class="Symbol">:</a> <a id="2666" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2669" class="Symbol">)</a> <a id="2671" class="Symbol">→</a> <a id="2673" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
        <a id="2695" href="Monad.Instance.K.StrongPreElgot.html#2651" class="Function">K-Alg</a> <a id="2701" href="Monad.Instance.K.StrongPreElgot.html#2701" class="Bound">X</a> <a id="2703" class="Symbol">=</a> <a id="2705" class="Keyword">record</a> <a id="2712" class="Symbol">{</a> <a id="2714" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="2716" class="Symbol">=</a> <a id="2718" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2722" href="Monad.Instance.K.StrongPreElgot.html#2701" class="Bound">X</a> <a id="2724" class="Symbol">;</a> <a id="2726" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="2734" class="Symbol">=</a> <a id="2736" href="Algebra.Elgot.html#16032" class="Field">Elgot-Algebra.algebra</a> <a id="2758" class="Symbol">(</a><a id="2759" href="Monad.Instance.K.html#1333" class="Function">algebras</a> <a id="2768" href="Monad.Instance.K.StrongPreElgot.html#2701" class="Bound">X</a><a id="2769" class="Symbol">)</a> <a id="2771" class="Symbol">}</a>
        <a id="2781" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="2784" class="Symbol">:</a> <a id="2786" class="Symbol">∀</a> <a id="2788" class="Symbol">(</a><a id="2789" href="Monad.Instance.K.StrongPreElgot.html#2789" class="Bound">X</a> <a id="2791" class="Symbol">:</a> <a id="2793" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2796" class="Symbol">)</a> <a id="2798" class="Symbol">→</a> <a id="2800" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="2804" href="Monad.Instance.K.StrongPreElgot.html#2789" class="Bound">X</a> <a id="2806" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2808" href="Categories.Functor.Core.html#797" class="Function">T.F.₀</a> <a id="2814" href="Monad.Instance.K.StrongPreElgot.html#2789" class="Bound">X</a>
        <a id="2824" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="2827" href="Monad.Instance.K.StrongPreElgot.html#2827" class="Bound">X</a> <a id="2829" class="Symbol">=</a> <a id="2831" href="Category.Construction.ElgotAlgebras.html#794" class="Field">Elgot-Algebra-Morphism.h</a> <a id="2856" class="Symbol">(</a><a id="2857" href="Categories.FreeObjects.Free.html#880" class="Function Operator">_*</a> <a id="2860" class="Symbol">{</a><a id="2861" class="Argument">A</a> <a id="2863" class="Symbol">=</a> <a id="2865" href="Monad.Instance.K.StrongPreElgot.html#2518" class="Function">T-Alg</a> <a id="2871" href="Monad.Instance.K.StrongPreElgot.html#2827" class="Bound">X</a><a id="2872" class="Symbol">}</a> <a id="2874" class="Symbol">(</a><a id="2875" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="2881" href="Monad.Instance.K.StrongPreElgot.html#2827" class="Bound">X</a><a id="2882" class="Symbol">))</a>
          <a id="2895" class="Keyword">where</a> <a id="2901" class="Keyword">open</a> <a id="2906" href="Categories.FreeObjects.Free.html#485" class="Module">FreeObject</a> <a id="2917" class="Symbol">(</a><a id="2918" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="2931" href="Monad.Instance.K.StrongPreElgot.html#2827" class="Bound">X</a><a id="2932" class="Symbol">)</a>
        <a id="2942" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">_#K</a> <a id="2946" class="Symbol">=</a> <a id="2948" class="Symbol">λ</a> <a id="2950" class="Symbol">{</a><a id="2951" href="Monad.Instance.K.StrongPreElgot.html#2951" class="Bound">B</a><a id="2952" class="Symbol">}</a> <a id="2954" class="Symbol">{</a><a id="2955" href="Monad.Instance.K.StrongPreElgot.html#2955" class="Bound">C</a><a id="2956" class="Symbol">}</a> <a id="2958" href="Monad.Instance.K.StrongPreElgot.html#2958" class="Bound">f</a> <a id="2960" class="Symbol">→</a> <a id="2962" href="Algebra.Elgot.html#1938" class="Function Operator">Elgot-Algebra._#</a> <a id="2979" class="Symbol">(</a><a id="2980" href="Categories.FreeObjects.Free.html#838" class="Field">FreeObject.FX</a> <a id="2994" class="Symbol">(</a><a id="2995" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3008" href="Monad.Instance.K.StrongPreElgot.html#2955" class="Bound">C</a><a id="3009" class="Symbol">))</a> <a id="3012" class="Symbol">{</a><a id="3013" href="Monad.Instance.K.StrongPreElgot.html#2951" class="Bound">B</a><a id="3014" class="Symbol">}</a> <a id="3016" href="Monad.Instance.K.StrongPreElgot.html#2958" class="Bound">f</a>
        <a id="3026" href="Monad.Instance.K.StrongPreElgot.html#3026" class="Function Operator">_#T</a> <a id="3030" class="Symbol">=</a> <a id="3032" class="Symbol">λ</a> <a id="3034" class="Symbol">{</a><a id="3035" href="Monad.Instance.K.StrongPreElgot.html#3035" class="Bound">B</a><a id="3036" class="Symbol">}</a> <a id="3038" class="Symbol">{</a><a id="3039" href="Monad.Instance.K.StrongPreElgot.html#3039" class="Bound">C</a><a id="3040" class="Symbol">}</a> <a id="3042" href="Monad.Instance.K.StrongPreElgot.html#3042" class="Bound">f</a> <a id="3044" class="Symbol">→</a> <a id="3046" href="Algebra.Elgot.html#1938" class="Function Operator">StrongPreElgotMonad.elgotalgebras._#</a> <a id="3083" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a> <a id="3085" class="Symbol">{</a><a id="3086" href="Monad.Instance.K.StrongPreElgot.html#3035" class="Bound">B</a><a id="3087" class="Symbol">}</a> <a id="3089" class="Symbol">{</a><a id="3090" href="Monad.Instance.K.StrongPreElgot.html#3039" class="Bound">C</a><a id="3091" class="Symbol">}</a> <a id="3093" href="Monad.Instance.K.StrongPreElgot.html#3042" class="Bound">f</a>
        <a id="3103" class="Comment">-- some preservation facts that follow immediately, since these things are elgot-algebra-morphisms.</a>
        <a id="3211" href="Monad.Instance.K.StrongPreElgot.html#3211" class="Function">K₁-preserves</a> <a id="3224" class="Symbol">:</a> <a id="3226" class="Symbol">∀</a> <a id="3228" class="Symbol">{</a><a id="3229" href="Monad.Instance.K.StrongPreElgot.html#3229" class="Bound">X</a> <a id="3231" href="Monad.Instance.K.StrongPreElgot.html#3231" class="Bound">Y</a> <a id="3233" href="Monad.Instance.K.StrongPreElgot.html#3233" class="Bound">Z</a> <a id="3235" class="Symbol">:</a> <a id="3237" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3240" class="Symbol">}</a> <a id="3242" class="Symbol">(</a><a id="3243" href="Monad.Instance.K.StrongPreElgot.html#3243" class="Bound">f</a> <a id="3245" class="Symbol">:</a> <a id="3247" href="Monad.Instance.K.StrongPreElgot.html#3229" class="Bound">X</a> <a id="3249" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3251" href="Monad.Instance.K.StrongPreElgot.html#3231" class="Bound">Y</a><a id="3252" class="Symbol">)</a> <a id="3254" class="Symbol">(</a><a id="3255" href="Monad.Instance.K.StrongPreElgot.html#3255" class="Bound">g</a> <a id="3257" class="Symbol">:</a> <a id="3259" href="Monad.Instance.K.StrongPreElgot.html#3233" class="Bound">Z</a> <a id="3261" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3263" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3267" href="Monad.Instance.K.StrongPreElgot.html#3229" class="Bound">X</a> <a id="3269" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3271" href="Monad.Instance.K.StrongPreElgot.html#3233" class="Bound">Z</a><a id="3272" class="Symbol">)</a> <a id="3274" class="Symbol">→</a> <a id="3276" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="3280" href="Monad.Instance.K.StrongPreElgot.html#3243" class="Bound">f</a> <a id="3282" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3284" class="Symbol">(</a><a id="3285" href="Monad.Instance.K.StrongPreElgot.html#3255" class="Bound">g</a> <a id="3287" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">#K</a><a id="3289" class="Symbol">)</a> <a id="3291" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3293" class="Symbol">((</a><a id="3295" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="3299" href="Monad.Instance.K.StrongPreElgot.html#3243" class="Bound">f</a> <a id="3301" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3304" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3307" class="Symbol">)</a> <a id="3309" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3311" href="Monad.Instance.K.StrongPreElgot.html#3255" class="Bound">g</a><a id="3312" class="Symbol">)</a> <a id="3314" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">#K</a>
        <a id="3325" href="Monad.Instance.K.StrongPreElgot.html#3211" class="Function">K₁-preserves</a> <a id="3338" class="Symbol">{</a><a id="3339" href="Monad.Instance.K.StrongPreElgot.html#3339" class="Bound">X</a><a id="3340" class="Symbol">}</a> <a id="3342" class="Symbol">{</a><a id="3343" href="Monad.Instance.K.StrongPreElgot.html#3343" class="Bound">Y</a><a id="3344" class="Symbol">}</a> <a id="3346" class="Symbol">{</a><a id="3347" href="Monad.Instance.K.StrongPreElgot.html#3347" class="Bound">Z</a><a id="3348" class="Symbol">}</a> <a id="3350" href="Monad.Instance.K.StrongPreElgot.html#3350" class="Bound">f</a> <a id="3352" href="Monad.Instance.K.StrongPreElgot.html#3352" class="Bound">g</a> <a id="3354" class="Symbol">=</a> <a id="3356" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="3389" class="Symbol">(((</a><a id="3392" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3405" href="Monad.Instance.K.StrongPreElgot.html#3339" class="Bound">X</a><a id="3406" class="Symbol">)</a> <a id="3408" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="3420" class="Symbol">)</a> <a id="3422" class="Symbol">{</a><a id="3423" class="Argument">A</a> <a id="3425" class="Symbol">=</a> <a id="3427" href="Monad.Instance.K.StrongPreElgot.html#2651" class="Function">K-Alg</a> <a id="3433" href="Monad.Instance.K.StrongPreElgot.html#3343" class="Bound">Y</a><a id="3434" class="Symbol">}</a> <a id="3436" class="Symbol">(</a><a id="3437" href="Categories.NaturalTransformation.Core.html#783" class="Function">ηK.η</a> <a id="3442" class="Symbol">_</a> <a id="3444" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3446" href="Monad.Instance.K.StrongPreElgot.html#3350" class="Bound">f</a><a id="3447" class="Symbol">))</a>
        <a id="3458" href="Monad.Instance.K.StrongPreElgot.html#3458" class="Function">μK-preserves</a> <a id="3471" class="Symbol">:</a> <a id="3473" class="Symbol">∀</a> <a id="3475" class="Symbol">{</a><a id="3476" href="Monad.Instance.K.StrongPreElgot.html#3476" class="Bound">X</a> <a id="3478" href="Monad.Instance.K.StrongPreElgot.html#3478" class="Bound">Y</a> <a id="3480" class="Symbol">:</a> <a id="3482" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3485" class="Symbol">}</a> <a id="3487" class="Symbol">(</a><a id="3488" href="Monad.Instance.K.StrongPreElgot.html#3488" class="Bound">g</a> <a id="3490" class="Symbol">:</a> <a id="3492" href="Monad.Instance.K.StrongPreElgot.html#3478" class="Bound">Y</a> <a id="3494" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3496" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3500" class="Symbol">(</a><a id="3501" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3505" href="Monad.Instance.K.StrongPreElgot.html#3476" class="Bound">X</a><a id="3506" class="Symbol">)</a> <a id="3508" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3510" href="Monad.Instance.K.StrongPreElgot.html#3478" class="Bound">Y</a><a id="3511" class="Symbol">)</a> <a id="3513" class="Symbol">→</a> <a id="3515" href="Categories.NaturalTransformation.Core.html#783" class="Function">μK.η</a> <a id="3520" href="Monad.Instance.K.StrongPreElgot.html#3476" class="Bound">X</a> <a id="3522" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3524" href="Monad.Instance.K.StrongPreElgot.html#3488" class="Bound">g</a> <a id="3526" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">#K</a> <a id="3529" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3531" class="Symbol">((</a><a id="3533" href="Categories.NaturalTransformation.Core.html#783" class="Function">μK.η</a> <a id="3538" href="Monad.Instance.K.StrongPreElgot.html#3476" class="Bound">X</a> <a id="3540" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3543" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3546" class="Symbol">)</a> <a id="3548" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3550" href="Monad.Instance.K.StrongPreElgot.html#3488" class="Bound">g</a><a id="3551" class="Symbol">)</a> <a id="3553" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">#K</a>
        <a id="3564" href="Monad.Instance.K.StrongPreElgot.html#3458" class="Function">μK-preserves</a> <a id="3577" class="Symbol">{</a><a id="3578" href="Monad.Instance.K.StrongPreElgot.html#3578" class="Bound">X</a><a id="3579" class="Symbol">}</a> <a id="3581" href="Monad.Instance.K.StrongPreElgot.html#3581" class="Bound">g</a> <a id="3583" class="Symbol">=</a> <a id="3585" href="Category.Construction.ElgotAlgebras.html#812" class="Field">Elgot-Algebra-Morphism.preserves</a> <a id="3618" class="Symbol">(((</a><a id="3621" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="3634" class="Symbol">(</a><a id="3635" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3639" href="Monad.Instance.K.StrongPreElgot.html#3578" class="Bound">X</a><a id="3640" class="Symbol">))</a> <a id="3643" href="Categories.FreeObjects.Free.html#880" class="Field Operator">FreeObject.*</a><a id="3655" class="Symbol">)</a> <a id="3657" class="Symbol">{</a><a id="3658" class="Argument">A</a> <a id="3660" class="Symbol">=</a> <a id="3662" href="Monad.Instance.K.StrongPreElgot.html#2651" class="Function">K-Alg</a> <a id="3668" href="Monad.Instance.K.StrongPreElgot.html#3578" class="Bound">X</a><a id="3669" class="Symbol">}</a> <a id="3671" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3674" class="Symbol">)</a>
        <a id="3684" href="Monad.Instance.K.StrongPreElgot.html#3684" class="Function">η&#39;-preserves</a> <a id="3697" class="Symbol">:</a> <a id="3699" class="Symbol">∀</a> <a id="3701" class="Symbol">{</a><a id="3702" href="Monad.Instance.K.StrongPreElgot.html#3702" class="Bound">X</a> <a id="3704" href="Monad.Instance.K.StrongPreElgot.html#3704" class="Bound">Y</a> <a id="3706" class="Symbol">:</a> <a id="3708" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="3711" class="Symbol">}</a> <a id="3713" class="Symbol">(</a><a id="3714" href="Monad.Instance.K.StrongPreElgot.html#3714" class="Bound">g</a> <a id="3716" class="Symbol">:</a> <a id="3718" href="Monad.Instance.K.StrongPreElgot.html#3704" class="Bound">Y</a> <a id="3720" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="3722" href="Categories.Functor.Core.html#797" class="Function">K.₀</a> <a id="3726" href="Monad.Instance.K.StrongPreElgot.html#3702" class="Bound">X</a> <a id="3728" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="3730" href="Monad.Instance.K.StrongPreElgot.html#3704" class="Bound">Y</a><a id="3731" class="Symbol">)</a> <a id="3733" class="Symbol">→</a> <a id="3735" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="3738" href="Monad.Instance.K.StrongPreElgot.html#3702" class="Bound">X</a> <a id="3740" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3742" href="Monad.Instance.K.StrongPreElgot.html#3714" class="Bound">g</a> <a id="3744" href="Monad.Instance.K.StrongPreElgot.html#2942" class="Function Operator">#K</a> <a id="3747" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3749" class="Symbol">((</a><a id="3751" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="3754" href="Monad.Instance.K.StrongPreElgot.html#3702" class="Bound">X</a> <a id="3756" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="3759" href="Category.Ambient.html#2115" class="Function">idC</a><a id="3762" class="Symbol">)</a> <a id="3764" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3766" href="Monad.Instance.K.StrongPreElgot.html#3714" class="Bound">g</a><a id="3767" class="Symbol">)</a> <a id="3769" href="Monad.Instance.K.StrongPreElgot.html#3026" class="Function Operator">#T</a>
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                                                                                <a id="4177" class="Symbol">(</a><a id="4178" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="4191" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="4192" class="Symbol">)</a>
                                                                                <a id="4274" class="Symbol">{</a><a id="4275" class="Argument">A</a> <a id="4277" class="Symbol">=</a> <a id="4279" href="Monad.Instance.K.StrongPreElgot.html#2518" class="Function">T-Alg</a> <a id="4285" href="Monad.Instance.K.StrongPreElgot.html#3989" class="Bound">Y</a><a id="4286" class="Symbol">}</a> 
                                                                                <a id="4369" class="Symbol">(</a><a id="4370" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="4376" href="Monad.Instance.K.StrongPreElgot.html#3992" class="Bound">f</a> <a id="4378" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4380" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="4386" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="4387" class="Symbol">)</a> 
                                                                                <a id="4470" class="Symbol">(</a><a id="4471" class="Keyword">record</a> <a id="4478" class="Symbol">{</a> <a id="4480" href="Category.Construction.ElgotAlgebras.html#794" class="Field">h</a> <a id="4482" class="Symbol">=</a> <a id="4484" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="4487" href="Monad.Instance.K.StrongPreElgot.html#3989" class="Bound">Y</a> <a id="4489" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="4491" href="Categories.Functor.Core.html#806" class="Function">K.₁</a> <a id="4495" href="Monad.Instance.K.StrongPreElgot.html#3992" class="Bound">f</a> <a id="4497" class="Symbol">;</a> <a id="4499" href="Category.Construction.ElgotAlgebras.html#812" class="Field">preserves</a> <a id="4509" class="Symbol">=</a> <a id="4511" href="Monad.Instance.K.StrongPreElgot.html#5416" class="Function">pres₁</a> <a id="4517" class="Symbol">})</a> 
                                                                                <a id="4601" href="Monad.Instance.K.StrongPreElgot.html#6702" class="Function">comm₁</a> <a id="4607" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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                                                                                  <a id="4820" class="Symbol">(</a><a id="4821" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="4834" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="4835" class="Symbol">)</a>
                                                                                  <a id="4919" class="Symbol">{</a><a id="4920" class="Argument">A</a> <a id="4922" class="Symbol">=</a> <a id="4924" href="Monad.Instance.K.StrongPreElgot.html#2518" class="Function">T-Alg</a> <a id="4930" href="Monad.Instance.K.StrongPreElgot.html#3989" class="Bound">Y</a><a id="4931" class="Symbol">}</a> 
                                                                                  <a id="5016" class="Symbol">(</a><a id="5017" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="5023" href="Monad.Instance.K.StrongPreElgot.html#3992" class="Bound">f</a> <a id="5025" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5027" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="5033" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="5034" class="Symbol">)</a> 
                                                                                  <a id="5119" class="Symbol">(</a><a id="5120" class="Keyword">record</a> <a id="5127" class="Symbol">{</a> <a id="5129" href="Category.Construction.ElgotAlgebras.html#794" class="Field">h</a> <a id="5131" class="Symbol">=</a> <a id="5133" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="5139" href="Monad.Instance.K.StrongPreElgot.html#3992" class="Bound">f</a> <a id="5141" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5143" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="5146" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a> <a id="5148" class="Symbol">;</a> <a id="5150" href="Category.Construction.ElgotAlgebras.html#812" class="Field">preserves</a> <a id="5160" class="Symbol">=</a> <a id="5162" href="Monad.Instance.K.StrongPreElgot.html#5942" class="Function">pres₂</a> <a id="5168" class="Symbol">})</a> 
                                                                                  <a id="5254" class="Symbol">(</a><a id="5255" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="5261" class="Symbol">(</a><a id="5262" href="Categories.FreeObjects.Free.html#937" class="Field">FreeObject.*-lift</a> <a id="5280" class="Symbol">(</a><a id="5281" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="5294" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="5295" class="Symbol">)</a> <a id="5297" class="Symbol">(</a><a id="5298" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="5304" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a><a id="5305" class="Symbol">))))</a> <a id="5310" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
          <a id="5322" href="Categories.Functor.Core.html#806" class="Function">T.F.₁</a> <a id="5328" href="Monad.Instance.K.StrongPreElgot.html#3992" class="Bound">f</a> <a id="5330" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5332" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="5335" href="Monad.Instance.K.StrongPreElgot.html#3985" class="Bound">X</a>                                                  <a id="5386" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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              <a id="14149" class="Symbol">(((</a><a id="14152" href="Categories.NaturalTransformation.Core.html#783" class="Function">strengthen.η</a> <a id="14165" class="Symbol">(</a><a id="14166" href="Monad.Instance.K.StrongPreElgot.html#11284" class="Bound">X</a> <a id="14168" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="14170" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a><a id="14171" class="Symbol">)</a> <a id="14173" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="14176" href="Category.Ambient.html#2115" class="Function">idC</a><a id="14179" class="Symbol">)</a> <a id="14181" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14183" class="Symbol">(</a><a id="14184" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="14198" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14200" class="Symbol">(</a><a id="14201" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14205" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14207" class="Symbol">(</a><a id="14208" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="14211" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a> <a id="14213" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="14216" href="Category.Ambient.html#2115" class="Function">idC</a><a id="14219" class="Symbol">)))</a> <a id="14223" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14225" class="Symbol">(</a><a id="14226" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14230" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14232" href="Monad.Instance.K.StrongPreElgot.html#13577" class="Bound">h</a><a id="14233" class="Symbol">))</a> <a id="14236" href="Monad.Instance.K.StrongPreElgot.html#3026" class="Function Operator">#T</a><a id="14238" class="Symbol">)</a> <a id="14240" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="14243" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="14247" class="Symbol">(</a><a id="14248" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a> <a id="14257" class="Symbol">(</a><a id="14258" href="Monad.PreElgot.html#815" class="Function">StrongPreElgotMonad.elgotalgebras</a> <a id="14292" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a><a id="14293" class="Symbol">)</a> <a id="14295" class="Symbol">(</a><a id="14296" href="Categories.Category.Core.html#2734" class="Function Operator">refl⟩∘⟨</a> <a id="14304" class="Symbol">(</a><a id="14305" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="14311" class="Symbol">((</a><a id="14313" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="14322" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="14327" class="Symbol">(</a><a id="14328" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="14332" class="Symbol">(</a><a id="14333" href="Category.Ambient.html#2592" class="Function">⟨⟩-unique</a> <a id="14343" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a> <a id="14351" href="Categories.Morphism.Reasoning.Core.html#1202" class="Function">id-comm</a><a id="14358" class="Symbol">)))</a> <a id="14362" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="14370" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="14372" href="Categories.Category.Distributive.Properties.html#3231" class="Function">distributeˡ⁻¹-natural</a> <a id="14394" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14398" class="Symbol">(</a><a id="14399" href="Monad.Instanc
              <a id="14430" class="Symbol">((</a><a id="14432" href="Categories.NaturalTransformation.Core.html#783" class="Function">strengthen.η</a> <a id="14445" class="Symbol">(</a><a id="14446" href="Monad.Instance.K.StrongPreElgot.html#11284" class="Bound">X</a> <a id="14448" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="14450" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a><a id="14451" class="Symbol">)</a> <a id="14453" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="14456" href="Category.Ambient.html#2115" class="Function">idC</a><a id="14459" class="Symbol">)</a> <a id="14461" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14463" class="Symbol">((</a><a id="14465" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14469" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14471" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="14474" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a><a id="14475" class="Symbol">)</a> <a id="14477" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="14480" href="Category.Ambient.html#2115" class="Function">idC</a><a id="14483" class="Symbol">)</a> <a id="14485" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14487" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="14501" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14503" class="Symbol">(</a><a id="14504" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14508" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14510" href="Monad.Instance.K.StrongPreElgot.html#13577" class="Bound">h</a><a id="14511" class="Symbol">))</a> <a id="14514" href="Monad.Instance.K.StrongPreElgot.html#3026" class="Function Operator">#T</a>     <a id="14521" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="14524" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a> <a id="14533" class="Symbol">(</a><a id="14534" href="Monad.PreElgot.html#815" class="Function">StrongPreElgotMonad.elgotalgebras</a> <a id="14568" href="Monad.Instance.K.StrongPreElgot.html#1795" class="Bound">A</a><a id="14569" class="Symbol">)</a> <a id="14571" class="Symbol">(</a><a id="14572" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="14578" class="Symbol">(</a><a id="14579" href="Categories.Category.Cocartesian.html#2699" class="Function">+₁∘+₁</a> <a id="14585" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="14587" href="Categories.Category.Cocartesian.html#2622" class="Function">+₁-cong₂</a> <a id="14596" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="14601" href="Categories.Category.Core.html#1339" class="Function">identity²</a><a id="14610" class="Symbol">))</a> <a id="14613" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a> 
              <a id="14630" class="Symbol">((</a><a id="14632" href="Categories.NaturalTransformation.Core.html#783" class="Function">strengthen.η</a> <a id="14645" class="Symbol">(</a><a id="14646" href="Monad.Instance.K.StrongPreElgot.html#11284" class="Bound">X</a> <a id="14648" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="14650" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a><a id="14651" class="Symbol">)</a> <a id="14653" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14655" class="Symbol">(</a><a id="14656" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14660" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14662" href="Monad.Instance.K.StrongPreElgot.html#2781" class="Function">η&#39;</a> <a id="14665" href="Monad.Instance.K.StrongPreElgot.html#11288" class="Bound">Y</a><a id="14666" class="Symbol">)</a> <a id="14668" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="14671" href="Category.Ambient.html#2115" class="Function">idC</a><a id="14674" class="Symbol">)</a> <a id="14676" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14678" href="Categories.Category.Distributive.html#3168" class="Function">distributeˡ⁻¹</a> <a id="14692" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="14694" class="Symbol">(</a><a id="14695" href="Category.Ambient.html#2115" class="Function">idC</a> <a id="14699" href="Categories.Category.BinaryProducts.html#1465" class="Function Operator">⁂</a> <a id="14701" href="Monad.Instance.K.StrongPreElgot.html#13577" class="Bound">h</a><a id="14702" class="Symbol">))</a> <a id="14705" href="Monad.Instance.K.StrongPreElgot.html#3026" class="Function Operator">#T</a>              <a id="14721" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
    <a id="14727" href="Monad.Instance.K.StrongPreElgot.html#14727" class="Function">!-unique′</a> <a id="14737" class="Symbol">:</a> <a id="14739" class="Symbol">∀</a> <a id="14741" class="Symbol">{</a><a id="14742" href="Monad.Instance.K.StrongPreElgot.html#14742" class="Bound">A</a> <a id="14744" class="Symbol">:</a> <a id="14746" href="Monad.PreElgot.html#1819" class="Record">StrongPreElgotMonad</a><a id="14765" class="Symbol">}</a> <a id="14767" class="Symbol">(</a><a id="14768" href="Monad.Instance.K.StrongPreElgot.html#14768" class="Bound">f</a> <a id="14770" class="Symbol">:</a> <a id="14772" href="Category.Construction.StrongPreElgotMonads.html#1408" class="Record">StrongPreElgotMonad-Morphism</a> <a id="14801" href="Monad.Instance.K.StrongPreElgot.html#1429" class="Function">strongPreElgot</a> <a id="14816" href="Monad.Instance.K.StrongPreElgot.html#14742" class="Bound">A</a><a id="14817" class="Symbol">)</a> <a id="14819" class="Symbol">→</a> <a id="14821" href="Category.Construction.StrongPreElgotMonads.html#1477" class="Field">StrongPreElgotMonad-Morphism.α</a> <a id="14852" class="Symbol">(</a><a id="14853" href="Monad.Instance.K.StrongPreElgot.html#1706" class="Function">!′</a> <a id="14856" class="Symbol">{</a><a id="14857" class="Argument">A</a> <a id="14859" class="Symbol">=</a> <a id="14861" href="Monad.Instance.K.StrongPreElgot.html#14742" class="Bound">A</a><a id="14862" class="Symbol">})</a> <a id="14865" href="Categories.NaturalTransformation.Equivalence.html#630" class="Function Operator">≃</a> <a id="14867" href="Category.Construction.StrongPreElgotMonads.html#1477" class="Field">StrongPreElgotMonad-Morphism.α</a> <a id="14898" href="Monad.Instance.K.StrongPreElgot.html#14768" class="Bound">f</a>
    <a id="14904" href="Monad.Instance.K.StrongPreElgot.html#14727" class="Function">!-unique′</a> <a id="14914" class="Symbol">{</a><a id="14915" href="Monad.Instance.K.StrongPreElgot.html#14915" class="Bound">A</a><a id="14916" class="Symbol">}</a> <a id="14918" href="Monad.Instance.K.StrongPreElgot.html#14918" class="Bound">f</a> <a id="14920" class="Symbol">{</a><a id="14921" href="Monad.Instance.K.StrongPreElgot.html#14921" class="Bound">X</a><a id="14922" class="Symbol">}</a> <a id="14924" class="Symbol">=</a> <a id="14926" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="14930" class="Symbol">(</a><a id="14931" href="Categories.FreeObjects.Free.html#1017" class="Field">FreeObject.*-uniq</a>
                                <a id="14981" class="Symbol">(</a><a id="14982" href="Monad.Instance.K.html#1182" class="Field">freealgebras</a> <a id="14995" href="Monad.Instance.K.StrongPreElgot.html#14921" class="Bound">X</a><a id="14996" class="Symbol">)</a> 
                                <a id="15031" class="Symbol">{</a><a id="15032" class="Argument">A</a> <a id="15034" class="Symbol">=</a> <a id="15036" class="Keyword">record</a> <a id="15043" class="Symbol">{</a> <a id="15045" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="15047" class="Symbol">=</a> <a id="15049" href="Categories.Functor.Core.html#432" class="Function">T.F.F₀</a> <a id="15056" href="Monad.Instance.K.StrongPreElgot.html#14921" class="Bound">X</a> <a id="15058" class="Symbol">;</a> <a id="15060" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="15068" class="Symbol">=</a> <a id="15070" href="Monad.PreElgot.html#815" class="Function">StrongPreElgotMonad.elgotalgebras</a> <a id="15104" href="Monad.Instance.K.StrongPreElgot.html#14915" class="Bound">A</a> <a id="15106" class="Symbol">}}</a> 
                                <a id="15142" class="Symbol">(</a><a id="15143" href="Categories.NaturalTransformation.Core.html#783" class="Function">T.η.η</a> <a id="15149" href="Monad.Instance.K.StrongPreElgot.html#14921" class="Bound">X</a><a id="15150" class="Symbol">)</a> 
                                <a id="15185" class="Symbol">(</a><a id="15186" class="Keyword">record</a> <a id="15193" class="Symbol">{</a> <a id="15195" href="Category.Construction.ElgotAlgebras.html#794" class="Field">h</a> <a id="15197" class="Symbol">=</a> <a id="15199" href="Categories.NaturalTransformation.Core.html#783" class="Function">α.η</a> <a id="15203" href="Monad.Instance.K.StrongPreElgot.html#14921" class="Bound">X</a> <a id="15205" class="Symbol">;</a> <a id="15207" href="Category.Construction.ElgotAlgebras.html#812" class="Field">preserves</a> <a id="15217" class="Symbol">=</a> <a id="15219" href="Category.Construction.StrongPreElgotMonads.html#1827" class="Function">α-preserves</a> <a id="15231" class="Symbol">_</a> <a id="15233" class="Symbol">})</a> 
                                <a id="15269" href="Category.Construction.StrongPreElgotMonads.html#1568" class="Function">α-η</a><a id="15272" class="Symbol">)</a>
      <a id="15280" class="Keyword">where</a>
        <a id="15294" class="Keyword">open</a> <a id="15299" href="Category.Construction.StrongPreElgotMonads.html#1408" class="Module">StrongPreElgotMonad-Morphism</a> <a id="15328" href="Monad.Instance.K.StrongPreElgot.html#14918" class="Bound">f</a> <a id="15330" class="Keyword">using</a> <a id="15336" class="Symbol">(</a><a id="15337" href="Category.Construction.StrongPreElgotMonads.html#1477" class="Field">α</a><a id="15338" class="Symbol">;</a> <a id="15340" href="Category.Construction.StrongPreElgotMonads.html#1568" class="Field">α-η</a><a id="15343" class="Symbol">;</a> <a id="15345" href="Category.Construction.StrongPreElgotMonads.html#1827" class="Field">α-preserves</a><a id="15356" class="Symbol">)</a>
        <a id="15366" class="Keyword">open</a> <a id="15371" href="Monad.PreElgot.html#1819" class="Module">StrongPreElgotMonad</a> <a id="15391" href="Monad.Instance.K.StrongPreElgot.html#14915" class="Bound">A</a> <a id="15393" class="Keyword">using</a> <a id="15399" class="Symbol">(</a><a id="15400" href="Monad.PreElgot.html#1879" class="Field">SM</a><a id="15402" class="Symbol">)</a>
        <a id="15412" class="Keyword">open</a> <a id="15417" href="Categories.Monad.Strong.html#1832" class="Module">StrongMonad</a> <a id="15429" href="Monad.PreElgot.html#1879" class="Function">SM</a> <a id="15432" class="Keyword">using</a> <a id="15438" class="Symbol">()</a> <a id="15441" class="Keyword">renaming</a> <a id="15450" class="Symbol">(</a><a id="15451" href="Categories.Monad.Strong.html#1918" class="Field">M</a> <a id="15453" class="Symbol">to</a> <a id="15456" class="Field">T</a><a id="15457" class="Symbol">)</a>
</pre>