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<h2 id="summary">Summary</h2>
<p>This file introduces (guarded) elgot algebras</p>
<ul class="task-list">
<li><label><input type="checkbox" checked="" /><em>Definition 7</em>
Guarded Elgot Algebras</label></li>
<li><label><input type="checkbox" /><em>Theorem 8</em> Existence of
final coalgebras is equivalent to existence of free H-guarded Elgot
algebras</label></li>
<li><label><input type="checkbox" checked="" /><em>Proposition 10</em>
Characterization of unguarded elgot algebras</label></li>
</ul>
<h2 id="code">Code</h2>
<pre class="Agda"><a id="727" class="Keyword">module</a> <a id="734" href="ElgotAlgebra.html" class="Module">ElgotAlgebra</a> <a id="747" class="Keyword">where</a>
<a id="754" class="Keyword">private</a>
<a id="764" class="Keyword">variable</a>
<a id="777" href="ElgotAlgebra.html#777" class="Generalizable">o</a> <a id="779" href="ElgotAlgebra.html#779" class="Generalizable">ℓ</a> <a id="781" href="ElgotAlgebra.html#781" class="Generalizable">e</a> <a id="783" class="Symbol">:</a> <a id="785" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="793" class="Keyword">module</a> <a id="800" href="ElgotAlgebra.html#800" class="Module">_</a> <a id="802" class="Symbol">(</a><a id="803" href="ElgotAlgebra.html#803" class="Bound">D</a> <a id="805" class="Symbol">:</a> <a id="807" href="Categories.Category.Extensive.Bundle.html#781" class="Record">ExtensiveDistributiveCategory</a> <a id="837" href="ElgotAlgebra.html#777" class="Generalizable">o</a> <a id="839" href="ElgotAlgebra.html#779" class="Generalizable">ℓ</a> <a id="841" href="ElgotAlgebra.html#781" class="Generalizable">e</a><a id="842" class="Symbol">)</a> <a id="844" class="Keyword">where</a>
<a id="852" class="Keyword">open</a> <a id="857" href="Categories.Category.Extensive.Bundle.html#781" class="Module">ExtensiveDistributiveCategory</a> <a id="887" href="ElgotAlgebra.html#803" class="Bound">D</a> <a id="889" class="Keyword">renaming</a> <a id="898" class="Symbol">(</a><a id="899" href="Categories.Category.Extensive.Bundle.html#859" class="Field">U</a> <a id="901" class="Symbol">to</a> <a id="904" class="Field">C</a><a id="905" class="Symbol">;</a> <a id="907" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="910" class="Symbol">to</a> <a id="913" class="Function">idC</a><a id="916" class="Symbol">)</a>
<a id="920" class="Keyword">open</a> <a id="925" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="937" class="Symbol">(</a><a id="938" href="Categories.Category.Extensive.html#734" class="Field">Extensive.cocartesian</a> <a id="960" href="Categories.Category.Extensive.Bundle.html#913" class="Field">extensive</a><a id="969" class="Symbol">)</a>
<a id="973" class="Keyword">open</a> <a id="978" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="988" class="Symbol">(</a><a id="989" href="Categories.Category.Extensive.Bundle.html#943" class="Field">ExtensiveDistributiveCategory.cartesian</a> <a id="1029" href="ElgotAlgebra.html#803" class="Bound">D</a><a id="1030" class="Symbol">)</a>
<a id="1034" class="Keyword">open</a> <a id="1039" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1042" href="ElgotAlgebra.html#904" class="Field">C</a>
</pre>
<h3 id="definition-7-guarded-elgot-algebras"><em>Definition 7</em>
Guarded Elgot Algebras</h3>
<pre class="Agda"> <a id="1101" class="Keyword">module</a> <a id="1108" href="ElgotAlgebra.html#1108" class="Module">_</a> <a id="1110" class="Symbol">{</a><a id="1111" href="ElgotAlgebra.html#1111" class="Bound">F</a> <a id="1113" class="Symbol">:</a> <a id="1115" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="1127" href="ElgotAlgebra.html#904" class="Field">C</a><a id="1128" class="Symbol">}</a> <a id="1130" class="Symbol">(</a><a id="1131" href="ElgotAlgebra.html#1131" class="Bound">FA</a> <a id="1134" class="Symbol">:</a> <a id="1136" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="1146" href="ElgotAlgebra.html#1111" class="Bound">F</a><a id="1147" class="Symbol">)</a> <a id="1149" class="Keyword">where</a>
<a id="1159" class="Keyword">record</a> <a id="1166" href="ElgotAlgebra.html#1166" class="Record">Guarded-Elgot-Algebra</a> <a id="1188" class="Symbol">:</a> <a id="1190" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1194" class="Symbol">(</a><a id="1195" href="ElgotAlgebra.html#837" class="Bound">o</a> <a id="1197" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1199" href="ElgotAlgebra.html#839" class="Bound">ℓ</a> <a id="1201" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1203" href="ElgotAlgebra.html#841" class="Bound">e</a><a id="1204" class="Symbol">)</a> <a id="1206" class="Keyword">where</a>
<a id="1218" class="Keyword">open</a> <a id="1223" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1231" href="ElgotAlgebra.html#1111" class="Bound">F</a> <a id="1233" class="Keyword">public</a>
<a id="1246" class="Keyword">open</a> <a id="1251" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="1261" href="ElgotAlgebra.html#1131" class="Bound">FA</a> <a id="1264" class="Keyword">public</a>
<a id="1277" class="Comment">-- iteration operator</a>
<a id="1305" class="Keyword">field</a>
<a id="1319" href="ElgotAlgebra.html#1319" class="Field Operator">_#</a> <a id="1322" class="Symbol">:</a> <a id="1324" class="Symbol">∀</a> <a id="1326" class="Symbol">{</a><a id="1327" href="ElgotAlgebra.html#1327" class="Bound">X</a><a id="1328" class="Symbol">}</a> <a id="1330" class="Symbol">→</a> <a id="1332" class="Symbol">(</a><a id="1333" href="ElgotAlgebra.html#1327" class="Bound">X</a> <a id="1335" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1337" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1339" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1341" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1344" href="ElgotAlgebra.html#1327" class="Bound">X</a><a id="1345" class="Symbol">)</a> <a id="1347" class="Symbol">→</a> <a id="1349" class="Symbol">(</a><a id="1350" href="ElgotAlgebra.html#1327" class="Bound">X</a> <a id="1352" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1354" href="Categories.Functor.Algebra.html#573" class="Function">A</a><a id="1355" class="Symbol">)</a>
<a id="1364" class="Comment">-- _# properties</a>
<a id="1387" class="Keyword">field</a>
<a id="1402" href="ElgotAlgebra.html#1402" class="Field">#-Fixpoint</a> <a id="1413" class="Symbol">:</a> <a id="1415" class="Symbol">∀</a> <a id="1417" class="Symbol">{</a><a id="1418" href="ElgotAlgebra.html#1418" class="Bound">X</a><a id="1419" class="Symbol">}</a> <a id="1421" class="Symbol">{</a><a id="1422" href="ElgotAlgebra.html#1422" class="Bound">f</a> <a id="1424" class="Symbol">:</a> <a id="1426" href="ElgotAlgebra.html#1418" class="Bound">X</a> <a id="1428" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1430" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1432" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1434" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1437" href="ElgotAlgebra.html#1418" class="Bound">X</a> <a id="1439" class="Symbol">}</a>
<a id="1451" class="Symbol">→</a> <a id="1453" href="ElgotAlgebra.html#1422" class="Bound">f</a> <a id="1455" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="1457" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1459" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1461" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="1465" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1467" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="1469" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1471" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1474" class="Symbol">(</a><a id="1475" href="ElgotAlgebra.html#1422" class="Bound">f</a> <a id="1477" href="ElgotAlgebra.html#1319" class="Field Operator">#</a><a id="1478" class="Symbol">)</a> <a id="1480" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1482" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1484" href="ElgotAlgebra.html#1422" class="Bound">f</a>
<a id="1494" href="ElgotAlgebra.html#1494" class="Field">#-Uniformity</a> <a id="1507" class="Symbol">:</a> <a id="1509" class="Symbol">∀</a> <a id="1511" class="Symbol">{</a><a id="1512" href="ElgotAlgebra.html#1512" class="Bound">X</a> <a id="1514" href="ElgotAlgebra.html#1514" class="Bound">Y</a><a id="1515" class="Symbol">}</a> <a id="1517" class="Symbol">{</a><a id="1518" href="ElgotAlgebra.html#1518" class="Bound">f</a> <a id="1520" class="Symbol">:</a> <a id="1522" href="ElgotAlgebra.html#1512" class="Bound">X</a> <a id="1524" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1526" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1528" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1530" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1533" href="ElgotAlgebra.html#1512" class="Bound">X</a><a id="1534" class="Symbol">}</a> <a id="1536" class="Symbol">{</a><a id="1537" href="ElgotAlgebra.html#1537" class="Bound">g</a> <a id="1539" class="Symbol">:</a> <a id="1541" href="ElgotAlgebra.html#1514" class="Bound">Y</a> <a id="1543" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1545" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1547" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1549" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1552" href="ElgotAlgebra.html#1514" class="Bound">Y</a><a id="1553" class="Symbol">}</a> <a id="1555" class="Symbol">{</a><a id="1556" href="ElgotAlgebra.html#1556" class="Bound">h</a> <a id="1558" class="Symbol">:</a> <a id="1560" href="ElgotAlgebra.html#1512" class="Bound">X</a> <a id="1562" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1564" href="ElgotAlgebra.html#1514" class="Bound">Y</a><a id="1565" class="Symbol">}</a>
<a id="1578" class="Symbol">→</a> <a id="1580" class="Symbol">(</a><a id="1581" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="1585" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1588" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1591" href="ElgotAlgebra.html#1556" class="Bound">h</a><a id="1592" class="Symbol">)</a> <a id="1594" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1596" href="ElgotAlgebra.html#1518" class="Bound">f</a> <a id="1598" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1600" href="ElgotAlgebra.html#1537" class="Bound">g</a> <a id="1602" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1604" href="ElgotAlgebra.html#1556" class="Bound">h</a>
<a id="1616" class="Symbol">→</a> <a id="1618" href="ElgotAlgebra.html#1518" class="Bound">f</a> <a id="1620" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="1622" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1624" href="ElgotAlgebra.html#1537" class="Bound">g</a> <a id="1626" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="1628" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1630" href="ElgotAlgebra.html#1556" class="Bound">h</a>
<a id="1640" href="ElgotAlgebra.html#1640" class="Field">#-Compositionality</a> <a id="1659" class="Symbol">:</a> <a id="1661" class="Symbol">∀</a> <a id="1663" class="Symbol">{</a><a id="1664" href="ElgotAlgebra.html#1664" class="Bound">X</a> <a id="1666" href="ElgotAlgebra.html#1666" class="Bound">Y</a><a id="1667" class="Symbol">}</a> <a id="1669" class="Symbol">{</a><a id="1670" href="ElgotAlgebra.html#1670" class="Bound">f</a> <a id="1672" class="Symbol">:</a> <a id="1674" href="ElgotAlgebra.html#1664" class="Bound">X</a> <a id="1676" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1678" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1680" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1682" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1685" href="ElgotAlgebra.html#1664" class="Bound">X</a><a id="1686" class="Symbol">}</a> <a id="1688" class="Symbol">{</a><a id="1689" href="ElgotAlgebra.html#1689" class="Bound">h</a> <a id="1691" class="Symbol">:</a> <a id="1693" href="ElgotAlgebra.html#1666" class="Bound">Y</a> <a id="1695" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1697" href="ElgotAlgebra.html#1664" class="Bound">X</a> <a id="1699" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1701" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1704" href="ElgotAlgebra.html#1666" class="Bound">Y</a><a id="1705" class="Symbol">}</a>
<a id="1717" class="Symbol">→</a> <a id="1719" class="Symbol">(((</a><a id="1722" href="ElgotAlgebra.html#1670" class="Bound">f</a> <a id="1724" href="ElgotAlgebra.html#1319" class="Field Operator">#</a><a id="1725" class="Symbol">)</a> <a id="1727" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1730" href="ElgotAlgebra.html#913" class="Function">idC</a><a id="1733" class="Symbol">)</a> <a id="1735" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1737" href="ElgotAlgebra.html#1689" class="Bound">h</a><a id="1738" class="Symbol">)</a><a id="1739" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="1741" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1743" class="Symbol">(</a><a id="1744" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1746" class="Symbol">(</a><a id="1747" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="1751" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1754" class="Symbol">(</a><a id="1755" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1758" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="1760" class="Symbol">))</a> <a id="1763" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1765" href="ElgotAlgebra.html#1670" class="Bound">f</a> <a id="1767" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1769" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1772" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1774" class="Symbol">(</a><a id="1775" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1778" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="1780" class="Symbol">)</a> <a id="1782" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1784" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1786" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1788" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1791" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1793" href="ElgotAlgebra.html#1689" class="Bound">h</a> <a id="1795" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="1796" class="Symbol">)</a><a id="1797" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="1799" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1801" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="1812" href="ElgotAlgebra.html#1812" class="Field">#-resp-≈</a> <a id="1821" class="Symbol">:</a> <a id="1823" class="Symbol">∀</a> <a id="1825" class="Symbol">{</a><a id="1826" href="ElgotAlgebra.html#1826" class="Bound">X</a><a id="1827" class="Symbol">}</a> <a id="1829" class="Symbol">{</a><a id="1830" href="ElgotAlgebra.html#1830" class="Bound">f</a> <a id="1832" href="ElgotAlgebra.html#1832" class="Bound">g</a> <a id="1834" class="Symbol">:</a> <a id="1836" href="ElgotAlgebra.html#1826" class="Bound">X</a> <a id="1838" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1840" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1842" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1844" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1847" href="ElgotAlgebra.html#1826" class="Bound">X</a><a id="1848" class="Symbol">}</a>
<a id="1861" class="Symbol">→</a> <a id="1863" href="ElgotAlgebra.html#1830" class="Bound">f</a> <a id="1865" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1867" href="ElgotAlgebra.html#1832" class="Bound">g</a>
<a id="1880" class="Symbol">→</a> <a id="1882" class="Symbol">(</a><a id="1883" href="ElgotAlgebra.html#1830" class="Bound">f</a> <a id="1885" href="ElgotAlgebra.html#1319" class="Field Operator">#</a><a id="1886" class="Symbol">)</a> <a id="1888" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1890" class="Symbol">(</a><a id="1891" href="ElgotAlgebra.html#1832" class="Bound">g</a> <a id="1893" href="ElgotAlgebra.html#1319" class="Field Operator">#</a><a id="1894" class="Symbol">)</a>
</pre>
<h3 id="proposition-10-unguarded-elgot-algebras"><em>Proposition 10</em>
Unguarded Elgot Algebras</h3>
<p>Unguarded elgot algebras are <code>Id</code>-guarded elgot algebras.
Here we give a different Characterization and show that it is equal.</p>
<pre class="Agda"> <a id="2086" class="Keyword">record</a> <a id="2093" href="ElgotAlgebra.html#2093" class="Record">Elgot-Algebra-on</a> <a id="2110" class="Symbol">(</a><a id="2111" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2113" class="Symbol">:</a> <a id="2115" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="2118" class="Symbol">)</a> <a id="2120" class="Symbol">:</a> <a id="2122" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2126" class="Symbol">(</a><a id="2127" href="ElgotAlgebra.html#837" class="Bound">o</a> <a id="2129" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2131" href="ElgotAlgebra.html#839" class="Bound">ℓ</a> <a id="2133" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2135" href="ElgotAlgebra.html#841" class="Bound">e</a><a id="2136" class="Symbol">)</a> <a id="2138" class="Keyword">where</a>
<a id="2148" class="Comment">-- iteration operator</a>
<a id="2174" class="Keyword">field</a>
<a id="2186" href="ElgotAlgebra.html#2186" class="Field Operator">_#</a> <a id="2189" class="Symbol">:</a> <a id="2191" class="Symbol">∀</a> <a id="2193" class="Symbol">{</a><a id="2194" href="ElgotAlgebra.html#2194" class="Bound">X</a><a id="2195" class="Symbol">}</a> <a id="2197" class="Symbol">→</a> <a id="2199" class="Symbol">(</a><a id="2200" href="ElgotAlgebra.html#2194" class="Bound">X</a> <a id="2202" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2204" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2206" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2208" href="ElgotAlgebra.html#2194" class="Bound">X</a><a id="2209" class="Symbol">)</a> <a id="2211" class="Symbol">→</a> <a id="2213" class="Symbol">(</a><a id="2214" href="ElgotAlgebra.html#2194" class="Bound">X</a> <a id="2216" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2218" href="ElgotAlgebra.html#2111" class="Bound">A</a><a id="2219" class="Symbol">)</a>
<a id="2226" class="Comment">-- _# properties</a>
<a id="2247" class="Keyword">field</a>
<a id="2259" href="ElgotAlgebra.html#2259" class="Field">#-Fixpoint</a> <a id="2270" class="Symbol">:</a> <a id="2272" class="Symbol">∀</a> <a id="2274" class="Symbol">{</a><a id="2275" href="ElgotAlgebra.html#2275" class="Bound">X</a><a id="2276" class="Symbol">}</a> <a id="2278" class="Symbol">{</a><a id="2279" href="ElgotAlgebra.html#2279" class="Bound">f</a> <a id="2281" class="Symbol">:</a> <a id="2283" href="ElgotAlgebra.html#2275" class="Bound">X</a> <a id="2285" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2287" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2289" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2291" href="ElgotAlgebra.html#2275" class="Bound">X</a> <a id="2293" class="Symbol">}</a>
<a id="2303" class="Symbol">→</a> <a id="2305" href="ElgotAlgebra.html#2279" class="Bound">f</a> <a id="2307" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2309" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2311" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2313" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="2317" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2319" href="ElgotAlgebra.html#2279" class="Bound">f</a> <a id="2321" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2323" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2325" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2327" href="ElgotAlgebra.html#2279" class="Bound">f</a>
<a id="2335" href="ElgotAlgebra.html#2335" class="Field">#-Uniformity</a> <a id="2348" class="Symbol">:</a> <a id="2350" class="Symbol">∀</a> <a id="2352" class="Symbol">{</a><a id="2353" href="ElgotAlgebra.html#2353" class="Bound">X</a> <a id="2355" href="ElgotAlgebra.html#2355" class="Bound">Y</a><a id="2356" class="Symbol">}</a> <a id="2358" class="Symbol">{</a><a id="2359" href="ElgotAlgebra.html#2359" class="Bound">f</a> <a id="2361" class="Symbol">:</a> <a id="2363" href="ElgotAlgebra.html#2353" class="Bound">X</a> <a id="2365" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2367" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2369" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2371" href="ElgotAlgebra.html#2353" class="Bound">X</a><a id="2372" class="Symbol">}</a> <a id="2374" class="Symbol">{</a><a id="2375" href="ElgotAlgebra.html#2375" class="Bound">g</a> <a id="2377" class="Symbol">:</a> <a id="2379" href="ElgotAlgebra.html#2355" class="Bound">Y</a> <a id="2381" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2383" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2385" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2387" href="ElgotAlgebra.html#2355" class="Bound">Y</a><a id="2388" class="Symbol">}</a> <a id="2390" class="Symbol">{</a><a id="2391" href="ElgotAlgebra.html#2391" class="Bound">h</a> <a id="2393" class="Symbol">:</a> <a id="2395" href="ElgotAlgebra.html#2353" class="Bound">X</a> <a id="2397" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2399" href="ElgotAlgebra.html#2355" class="Bound">Y</a><a id="2400" class="Symbol">}</a>
<a id="2411" class="Symbol">→</a> <a id="2413" class="Symbol">(</a><a id="2414" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="2418" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2421" href="ElgotAlgebra.html#2391" class="Bound">h</a><a id="2422" class="Symbol">)</a> <a id="2424" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2426" href="ElgotAlgebra.html#2359" class="Bound">f</a> <a id="2428" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2430" href="ElgotAlgebra.html#2375" class="Bound">g</a> <a id="2432" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2434" href="ElgotAlgebra.html#2391" class="Bound">h</a>
<a id="2444" class="Symbol">→</a> <a id="2446" href="ElgotAlgebra.html#2359" class="Bound">f</a> <a id="2448" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2450" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2452" href="ElgotAlgebra.html#2375" class="Bound">g</a> <a id="2454" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2456" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2458" href="ElgotAlgebra.html#2391" class="Bound">h</a>
<a id="2466" href="ElgotAlgebra.html#2466" class="Field">#-Folding</a> <a id="2476" class="Symbol">:</a> <a id="2478" class="Symbol">∀</a> <a id="2480" class="Symbol">{</a><a id="2481" href="ElgotAlgebra.html#2481" class="Bound">X</a> <a id="2483" href="ElgotAlgebra.html#2483" class="Bound">Y</a><a id="2484" class="Symbol">}</a> <a id="2486" class="Symbol">{</a><a id="2487" href="ElgotAlgebra.html#2487" class="Bound">f</a> <a id="2489" class="Symbol">:</a> <a id="2491" href="ElgotAlgebra.html#2481" class="Bound">X</a> <a id="2493" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2495" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2497" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2499" href="ElgotAlgebra.html#2481" class="Bound">X</a><a id="2500" class="Symbol">}</a> <a id="2502" class="Symbol">{</a><a id="2503" href="ElgotAlgebra.html#2503" class="Bound">h</a> <a id="2505" class="Symbol">:</a> <a id="2507" href="ElgotAlgebra.html#2483" class="Bound">Y</a> <a id="2509" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2511" href="ElgotAlgebra.html#2481" class="Bound">X</a> <a id="2513" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2515" href="ElgotAlgebra.html#2483" class="Bound">Y</a><a id="2516" class="Symbol">}</a>
<a id="2527" class="Symbol">→</a> <a id="2529" class="Symbol">((</a><a id="2531" href="ElgotAlgebra.html#2487" class="Bound">f</a> <a id="2533" href="ElgotAlgebra.html#2186" class="Field Operator">#</a><a id="2534" class="Symbol">)</a> <a id="2536" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2539" href="ElgotAlgebra.html#2503" class="Bound">h</a><a id="2540" class="Symbol">)</a><a id="2541" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2543" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2545" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2547" class="Symbol">(</a><a id="2548" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="2552" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2555" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="2557" class="Symbol">)</a> <a id="2559" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2561" href="ElgotAlgebra.html#2487" class="Bound">f</a> <a id="2563" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2565" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2568" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2570" href="ElgotAlgebra.html#2503" class="Bound">h</a> <a id="2572" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2574" href="ElgotAlgebra.html#2186" class="Field Operator">#</a>
<a id="2582" href="ElgotAlgebra.html#2582" class="Field">#-resp-≈</a> <a id="2591" class="Symbol">:</a> <a id="2593" class="Symbol">∀</a> <a id="2595" class="Symbol">{</a><a id="2596" href="ElgotAlgebra.html#2596" class="Bound">X</a><a id="2597" class="Symbol">}</a> <a id="2599" class="Symbol">{</a><a id="2600" href="ElgotAlgebra.html#2600" class="Bound">f</a> <a id="2602" href="ElgotAlgebra.html#2602" class="Bound">g</a> <a id="2604" class="Symbol">:</a> <a id="2606" href="ElgotAlgebra.html#2596" class="Bound">X</a> <a id="2608" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2610" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2612" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2614" href="ElgotAlgebra.html#2596" class="Bound">X</a><a id="2615" class="Symbol">}</a> <a id="2617" class="Symbol">→</a> <a id="2619" href="ElgotAlgebra.html#2600" class="Bound">f</a> <a id="2621" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2623" href="ElgotAlgebra.html#2602" class="Bound">g</a> <a id="2625" class="Symbol">→</a> <a id="2627" class="Symbol">(</a><a id="2628" href="ElgotAlgebra.html#2600" class="Bound">f</a> <a id="2630" href="ElgotAlgebra.html#2186" class="Field Operator">#</a><a id="2631" class="Symbol">)</a> <a id="2633" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2635" class="Symbol">(</a><a id="2636" href="ElgotAlgebra.html#2602" class="Bound">g</a> <a id="2638" href="ElgotAlgebra.html#2186" class="Field Operator">#</a><a id="2639" class="Symbol">)</a>
<a id="2646" class="Keyword">open</a> <a id="2651" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="2668" class="Keyword">open</a> <a id="2673" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="2683" class="Comment">-- Compositionality is derivable</a>
<a id="2720" href="ElgotAlgebra.html#2720" class="Function">#-Compositionality</a> <a id="2739" class="Symbol">:</a> <a id="2741" class="Symbol">∀</a> <a id="2743" class="Symbol">{</a><a id="2744" href="ElgotAlgebra.html#2744" class="Bound">X</a> <a id="2746" href="ElgotAlgebra.html#2746" class="Bound">Y</a><a id="2747" class="Symbol">}</a> <a id="2749" class="Symbol">{</a><a id="2750" href="ElgotAlgebra.html#2750" class="Bound">f</a> <a id="2752" class="Symbol">:</a> <a id="2754" href="ElgotAlgebra.html#2744" class="Bound">X</a> <a id="2756" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2758" href="ElgotAlgebra.html#2111" class="Bound">A</a> <a id="2760" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2762" href="ElgotAlgebra.html#2744" class="Bound">X</a><a id="2763" class="Symbol">}</a> <a id="2765" class="Symbol">{</a><a id="2766" href="ElgotAlgebra.html#2766" class="Bound">h</a> <a id="2768" class="Symbol">:</a> <a id="2770" href="ElgotAlgebra.html#2746" class="Bound">Y</a> <a id="2772" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2774" href="ElgotAlgebra.html#2744" class="Bound">X</a> <a id="2776" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2778" href="ElgotAlgebra.html#2746" class="Bound">Y</a><a id="2779" class="Symbol">}</a>
<a id="2787" class="Symbol">→</a> <a id="2789" class="Symbol">(((</a><a id="2792" href="ElgotAlgebra.html#2750" class="Bound">f</a> <a id="2794" href="ElgotAlgebra.html#2186" class="Field Operator">#</a><a id="2795" class="Symbol">)</a> <a id="2797" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2800" href="ElgotAlgebra.html#913" class="Function">idC</a><a id="2803" class="Symbol">)</a> <a id="2805" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2807" href="ElgotAlgebra.html#2766" class="Bound">h</a><a id="2808" class="Symbol">)</a><a id="2809" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2811" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2813" class="Symbol">(</a><a id="2814" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2816" class="Symbol">(</a><a id="2817" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="2821" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2824" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="2826" class="Symbol">)</a> <a id="2828" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2830" href="ElgotAlgebra.html#2750" class="Bound">f</a> <a id="2832" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2834" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2837" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2839" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2842" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2844" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2846" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2848" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="2851" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2853" href="ElgotAlgebra.html#2766" class="Bound">h</a> <a id="2855" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="2856" class="Symbol">)</a><a id="2857" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="2859" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2861" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
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<a id="5845" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5847" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5849" class="Symbol">(</a><a id="5850" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="5854" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5857" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="5859" class="Symbol">)</a> <a id="5861" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5863" href="ElgotAlgebra.html#2896" class="Bound">f</a> <a id="5865" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5867" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5870" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5872" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="5874" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5876" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5878" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="5890" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5892" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5894" class="Symbol">(</a><a id="5895" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="5899" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5902" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="5904" class="Symbol">)</a> <a id="5906" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5908" href="ElgotAlgebra.html#2896" class="Bound">f</a> <a id="5910" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5912" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5915" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5917" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="5919" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5921" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5923" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="5925" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5952" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5956" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="5960" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="5970" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="5972" class="Symbol">(</a><a id="5973" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="5977" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="5980" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="5982" class="Symbol">)</a> <a id="5984" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5986" href="ElgotAlgebra.html#2896" class="Bound">f</a> <a id="5988" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="5990" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="5993" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="5995" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="5997" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="5999" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6001" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6003" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6006" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6008" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="6010" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6032" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a><a id="6033" class="Symbol">))</a>
<a id="6042" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="6050" class="Symbol">(</a><a id="6051" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6053" class="Symbol">(</a><a id="6054" href="ElgotAlgebra.html#913" class="Function">idC</a> <a id="6058" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="6061" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="6063" class="Symbol">)</a> <a id="6065" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6067" href="ElgotAlgebra.html#2896" class="Bound">f</a> <a id="6069" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6071" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6074" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6076" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6079" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="6081" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6083" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="6085" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="6088" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="6090" href="ElgotAlgebra.html#2900" class="Bound">h</a> <a id="6092" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="6093" class="Symbol">)</a><a id="6094" href="ElgotAlgebra.html#2186" class="Field Operator">#</a> <a id="6096" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="6098" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6102" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="6109" class="Comment">-- every elgot-algebra comes with a divergence constant</a>
<a id="6169" href="ElgotAlgebra.html#6169" class="Function">!ₑ</a> <a id="6172" class="Symbol">:</a> <a id="6174" href="Categories.Object.Initial.html#804" class="Function">⊥</a> <a id="6176" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="6178" href="ElgotAlgebra.html#2111" class="Bound">A</a>
<a id="6184" href="ElgotAlgebra.html#6169" class="Function">!ₑ</a> <a id="6187" class="Symbol">=</a> <a id="6189" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="6192" href="ElgotAlgebra.html#2186" class="Field Operator">#</a>
<a id="6196" class="Keyword">record</a> <a id="6203" href="ElgotAlgebra.html#6203" class="Record">Elgot-Algebra</a> <a id="6217" class="Symbol">:</a> <a id="6219" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="6223" class="Symbol">(</a><a id="6224" href="ElgotAlgebra.html#837" class="Bound">o</a> <a id="6226" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="6228" href="ElgotAlgebra.html#839" class="Bound">ℓ</a> <a id="6230" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="6232" href="ElgotAlgebra.html#841" class="Bound">e</a><a id="6233" class="Symbol">)</a> <a id="6235" class="Keyword">where</a>
<a id="6245" class="Keyword">field</a>
<a id="6257" href="ElgotAlgebra.html#6257" class="Field">A</a> <a id="6259" class="Symbol">:</a> <a id="6261" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="6271" href="ElgotAlgebra.html#6271" class="Field">algebra</a> <a id="6279" class="Symbol">:</a> <a id="6281" href="ElgotAlgebra.html#2093" class="Record">Elgot-Algebra-on</a> <a id="6298" href="ElgotAlgebra.html#6257" class="Field">A</a>
<a id="6304" class="Keyword">open</a> <a id="6309" href="ElgotAlgebra.html#2093" class="Module">Elgot-Algebra-on</a> <a id="6326" href="ElgotAlgebra.html#6271" class="Field">algebra</a> <a id="6334" class="Keyword">public</a>
<a id="6344" class="Comment">--*</a>
<a id="6350" class="Comment">-- Here follows the proof of equivalence for unguarded and Id-guarded Elgot-Algebras</a>
<a id="6437" class="Comment">--*</a>
<a id="6444" class="Keyword">private</a>
<a id="6456" class="Comment">-- identity algebra</a>
<a id="6480" href="ElgotAlgebra.html#6480" class="Function">Id-Algebra</a> <a id="6491" class="Symbol">:</a> <a id="6493" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="6497" class="Symbol">→</a> <a id="6499" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="6509" class="Symbol">(</a><a id="6510" href="ElgotAlgebra.html#79" class="Function">idF</a> <a id="6514" class="Symbol">{</a><a id="6515" class="Argument">C</a> <a id="6517" class="Symbol">=</a> <a id="6519" href="ElgotAlgebra.html#904" class="Field">C</a><a id="6520" class="Symbol">})</a>
<a id="6527" href="ElgotAlgebra.html#6480" class="Function">Id-Algebra</a> <a id="6538" href="ElgotAlgebra.html#6538" class="Bound">A</a> <a id="6540" class="Symbol">=</a> <a id="6542" class="Keyword">record</a>
<a id="6555" class="Symbol">{</a> <a id="6557" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="6559" class="Symbol">=</a> <a id="6561" href="ElgotAlgebra.html#6538" class="Bound">A</a>
<a id="6569" class="Symbol">;</a> <a id="6571" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="6573" class="Symbol">=</a> <a id="6575" href="ElgotAlgebra.html#913" class="Function">idC</a>
<a id="6585" class="Symbol">}</a>
<a id="6592" class="Keyword">where</a> <a id="6598" class="Keyword">open</a> <a id="6603" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="6611" class="Symbol">(</a><a id="6612" href="ElgotAlgebra.html#79" class="Function">idF</a> <a id="6616" class="Symbol">{</a><a id="6617" class="Argument">C</a> <a id="6619" class="Symbol">=</a> <a id="6621" href="ElgotAlgebra.html#904" class="Field">C</a><a id="6622" class="Symbol">})</a>
<a id="6628" class="Comment">-- constructing an Id-Guarded Elgot-Algebra from an unguarded one</a>
<a id="6696" href="ElgotAlgebra.html#6696" class="Function">Unguarded→Id-Guarded</a> <a id="6717" class="Symbol">:</a> <a id="6719" class="Symbol">(</a><a id="6720" href="ElgotAlgebra.html#6720" class="Bound">EA</a> <a id="6723" class="Symbol">:</a> <a id="6725" href="ElgotAlgebra.html#6203" class="Record">Elgot-Algebra</a><a id="6738" class="Symbol">)</a> <a id="6740" class="Symbol">→</a> <a id="6742" href="ElgotAlgebra.html#1166" class="Record">Guarded-Elgot-Algebra</a> <a id="6764" class="Symbol">(</a><a id="6765" href="ElgotAlgebra.html#6480" class="Function">Id-Algebra</a> <a id="6776" class="Symbol">(</a><a id="6777" href="ElgotAlgebra.html#6257" class="Field">Elgot-Algebra.A</a> <a id="6793" href="ElgotAlgebra.html#6720" class="Bound">EA</a><a id="6795" class="Symbol">))</a>
<a id="6800" href="ElgotAlgebra.html#6696" class="Function">Unguarded→Id-Guarded</a> <a id="6821" href="ElgotAlgebra.html#6821" class="Bound">ea</a> <a id="6824" class="Symbol">=</a> <a id="6826" class="Keyword">record</a>
<a id="6837" class="Symbol">{</a> <a id="6839" href="ElgotAlgebra.html#1319" class="Field Operator">_#</a> <a id="6842" class="Symbol">=</a> <a id="6844" href="ElgotAlgebra.html#2186" class="Function Operator">_#</a>
<a id="6851" class="Symbol">;</a> <a id="6853" href="ElgotAlgebra.html#1402" class="Field">#-Fixpoint</a> <a id="6864" class="Symbol">=</a> <a id="6866" class="Symbol">λ</a> <a id="6868" class="Symbol">{</a><a id="6869" href="ElgotAlgebra.html#6869" class="Bound">X</a><a id="6870" class="Symbol">}</a> <a id="6872" class="Symbol">{</a><a id="6873" href="ElgotAlgebra.html#6873" class="Bound">f</a><a id="6874" class="Symbol">}</a> <a id="6876" class="Symbol">→</a> <a id="6878" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="6884" href="ElgotAlgebra.html#2259" class="Function">#-Fixpoint</a> <a id="6895" class="Symbol">(</a><a id="6896" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="6900" class="Symbol">(</a><a id="6901" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="6911" class="Symbol">(</a><a id="6912" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="6921" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="6930" class="Symbol">)))</a>
<a id="6938" class="Symbol">;</a> <a id="6940" href="ElgotAlgebra.html#1494" class="Field">#-Uniformity</a> <a id="6953" class="Symbol">=</a> <a id="6955" href="ElgotAlgebra.html#2335" class="Function">#-Uniformity</a>
<a id="6972" class="Symbol">;</a> <a id="6974" href="ElgotAlgebra.html#1640" class="Field">#-Compositionality</a> <a id="6993" class="Symbol">=</a> <a id="6995" href="ElgotAlgebra.html#2720" class="Function">#-Compositionality</a>
<a id="7018" class="Symbol">;</a> <a id="7020" href="ElgotAlgebra.html#1812" class="Field">#-resp-≈</a> <a id="7029" class="Symbol">=</a> <a id="7031" href="ElgotAlgebra.html#2582" class="Function">#-resp-≈</a>
<a id="7044" class="Symbol">}</a>
<a id="7050" class="Keyword">where</a>
<a id="7061" class="Keyword">open</a> <a id="7066" href="ElgotAlgebra.html#6203" class="Module">Elgot-Algebra</a> <a id="7080" href="ElgotAlgebra.html#6821" class="Bound">ea</a>
<a id="7087" class="Keyword">open</a> <a id="7092" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="7109" class="Keyword">open</a> <a id="7114" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="7123" class="Comment">-- constructing an unguarded Elgot-Algebra from an Id-Guarded one </a>
<a id="7192" href="ElgotAlgebra.html#7192" class="Function">Id-Guarded→Unguarded</a> <a id="7213" class="Symbol">:</a> <a id="7215" class="Symbol">∀</a> <a id="7217" class="Symbol">{</a><a id="7218" href="ElgotAlgebra.html#7218" class="Bound">A</a> <a id="7220" class="Symbol">:</a> <a id="7222" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="7225" class="Symbol">}</a> <a id="7227" class="Symbol">→</a> <a id="7229" href="ElgotAlgebra.html#1166" class="Record">Guarded-Elgot-Algebra</a> <a id="7251" class="Symbol">(</a><a id="7252" href="ElgotAlgebra.html#6480" class="Function">Id-Algebra</a> <a id="7263" href="ElgotAlgebra.html#7218" class="Bound">A</a><a id="7264" class="Symbol">)</a> <a id="7266" class="Symbol">→</a> <a id="7268" href="ElgotAlgebra.html#6203" class="Record">Elgot-Algebra</a>
<a id="7284" href="ElgotAlgebra.html#7192" class="Function">Id-Guarded→Unguarded</a> <a id="7305" href="ElgotAlgebra.html#7305" class="Bound">gea</a> <a id="7309" class="Symbol">=</a> <a id="7311" class="Keyword">record</a>
<a id="7323" class="Symbol">{</a> <a id="7325" href="ElgotAlgebra.html#6257" class="Field">A</a> <a id="7327" class="Symbol">=</a> <a id="7329" href="Categories.Functor.Algebra.html#573" class="Function">A</a>
<a id="7336" class="Symbol">;</a> <a id="7338" href="ElgotAlgebra.html#6271" class="Field">algebra</a> <a id="7346" class="Symbol">=</a> <a id="7348" class="Keyword">record</a>
<a id="7361" class="Symbol">{</a> <a id="7363" href="ElgotAlgebra.html#2186" class="Field Operator">_#</a> <a id="7366" class="Symbol">=</a> <a id="7368" href="ElgotAlgebra.html#1319" class="Field Operator">_#</a>
<a id="7377" class="Symbol">;</a> <a id="7379" href="ElgotAlgebra.html#2259" class="Field">#-Fixpoint</a> <a id="7390" class="Symbol">=</a> <a id="7392" class="Symbol">λ</a> <a id="7394" class="Symbol">{</a><a id="7395" href="ElgotAlgebra.html#7395" class="Bound">X</a><a id="7396" class="Symbol">}</a> <a id="7398" class="Symbol">{</a><a id="7399" href="ElgotAlgebra.html#7399" class="Bound">f</a><a id="7400" class="Symbol">}</a> <a id="7402" class="Symbol">→</a> <a id="7404" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="7410" href="ElgotAlgebra.html#1402" class="Field">#-Fixpoint</a> <a id="7421" class="Symbol">(</a><a id="7422" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="7432" class="Symbol">(</a><a id="7433" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="7442" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="7451" class="Symbol">))</a>
<a id="7460" class="Symbol">;</a> <a id="7462" href="ElgotAlgebra.html#2335" class="Field">#-Uniformity</a> <a id="7475" class="Symbol">=</a> <a id="7477" href="ElgotAlgebra.html#1494" class="Field">#-Uniformity</a>
<a id="7496" class="Symbol">;</a> <a id="7498" href="ElgotAlgebra.html#2466" class="Field">#-Folding</a> <a id="7508" class="Symbol">=</a> <a id="7510" class="Symbol">λ</a> <a id="7512" class="Symbol">{</a><a id="7513" href="ElgotAlgebra.html#7513" class="Bound">X</a><a id="7514" class="Symbol">}</a> <a id="7516" class="Symbol">{</a><a id="7517" href="ElgotAlgebra.html#7517" class="Bound">Y</a><a id="7518" class="Symbol">}</a> <a id="7520" class="Symbol">{</a><a id="7521" href="ElgotAlgebra.html#7521" class="Bound">f</a><a id="7522" class="Symbol">}</a> <a id="7524" class="Symbol">{</a><a id="7525" href="ElgotAlgebra.html#7525" class="Bound">h</a><a id="7526" class="Symbol">}</a> <a id="7528" class="Symbol">→</a> <a id="7530" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
<a id="7543" class="Symbol">((</a><a id="7545" href="ElgotAlgebra.html#7521" class="Bound">f</a> <a id="7547" href="ElgotAlgebra.html#1319" class="Field Operator">#</a><a id="7548" class="Symbol">)</a> <a id="7550" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="7553" href="ElgotAlgebra.html#7525" class="Bound">h</a><a id="7554" class="Symbol">)</a> <a id="7556" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="7621" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="7625" href="Categories.Category.Cocartesian.html#1741" class="Function">+-g-η</a> <a id="7631" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
<a id="7640" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="7642" class="Symbol">(</a><a id="7643" href="ElgotAlgebra.html#7521" class="Bound">f</a> <a id="7645" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="7647" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="7650" href="ElgotAlgebra.html#7525" class="Bound">h</a><a id="7651" class="Symbol">)</a><a id="7652" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="7654" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7656" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="7659" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="7661" class="Symbol">(</a><a id="7662" href="ElgotAlgebra.html#7521" class="Bound">f</a> <a id="7664" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="7666" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="7669" href="ElgotAlgebra.html#7525" class="Bound">h</a><a id="7670" class="Symbol">)</a><a id="7671" href="ElgotAlgebra.html#1319" class="Field Operator">#</a> <a id="7673" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="7675" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="7678" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="7718" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7721" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="7730" href="ElgotAlgebra.html#8055" class="Function">left</a> <a id="7735" href="ElgotAlgebra.html#10930" class="Function">right</a> <a id="7741" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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</pre>
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