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<h1 id="guarded-and-unguarded-elgot-algebras">Guarded and unguarded
Elgot algebras</h1>
<h2 id="guarded-elgot-algebras">Guarded Elgot algebras</h2>
<p>Guarded Elgot algebras are algebras on an endofunctor together with
an iteration operator that satisfies some axioms.</p>
<pre class="Agda"> <a id="629" class="Keyword">record</a> <a id="Guarded-Elgot-Algebra-on"></a><a id="636" href="Algebra.Elgot.html#636" class="Record">Guarded-Elgot-Algebra-on</a> <a id="661" class="Symbol">{</a><a id="662" href="Algebra.Elgot.html#662" class="Bound">F</a> <a id="664" class="Symbol">:</a> <a id="666" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="678" href="Algebra.Elgot.html#312" class="Bound">C</a><a id="679" class="Symbol">}</a> <a id="681" class="Symbol">(</a><a id="682" href="Algebra.Elgot.html#682" class="Bound">FA</a> <a id="685" class="Symbol">:</a> <a id="687" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="697" href="Algebra.Elgot.html#662" class="Bound">F</a><a id="698" class="Symbol">)</a> <a id="700" class="Symbol">:</a> <a id="702" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="706" class="Symbol">(</a><a id="707" href="Algebra.Elgot.html#304" class="Bound">o</a> <a id="709" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="711" href="Algebra.Elgot.html#306" class="Bound">ℓ</a> <a id="713" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="715" href="Algebra.Elgot.html#308" class="Bound">e</a><a id="716" class="Symbol">)</a> <a id="718" class="Keyword">where</a>
<a id="728" class="Keyword">open</a> <a id="733" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="741" href="Algebra.Elgot.html#662" class="Bound">F</a> <a id="743" class="Keyword">public</a>
<a id="754" class="Keyword">open</a> <a id="759" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="769" href="Algebra.Elgot.html#682" class="Bound">FA</a> <a id="772" class="Keyword">public</a>
<a id="783" class="Comment">-- iteration operator</a>
<a id="809" class="Keyword">field</a>
<a id="Guarded-Elgot-Algebra-on._#"></a><a id="821" href="Algebra.Elgot.html#821" class="Field Operator">_#</a> <a id="824" class="Symbol">:</a> <a id="826" class="Symbol">∀</a> <a id="828" class="Symbol">{</a><a id="829" href="Algebra.Elgot.html#829" class="Bound">X</a><a id="830" class="Symbol">}</a> <a id="832" class="Symbol">→</a> <a id="834" class="Symbol">(</a><a id="835" href="Algebra.Elgot.html#829" class="Bound">X</a> <a id="837" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="839" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="841" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="843" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="846" href="Algebra.Elgot.html#829" class="Bound">X</a><a id="847" class="Symbol">)</a> <a id="849" class="Symbol">→</a> <a id="851" class="Symbol">(</a><a id="852" href="Algebra.Elgot.html#829" class="Bound">X</a> <a id="854" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="856" href="Categories.Functor.Algebra.html#573" class="Function">A</a><a id="857" class="Symbol">)</a>
<a id="864" class="Comment">-- _# properties</a>
<a id="885" class="Keyword">field</a>
<a id="Guarded-Elgot-Algebra-on.#-Fixpoint"></a><a id="898" href="Algebra.Elgot.html#898" class="Field">#-Fixpoint</a> <a id="909" class="Symbol">:</a> <a id="911" class="Symbol">∀</a> <a id="913" class="Symbol">{</a><a id="914" href="Algebra.Elgot.html#914" class="Bound">X</a><a id="915" class="Symbol">}</a> <a id="917" class="Symbol">{</a><a id="918" href="Algebra.Elgot.html#918" class="Bound">f</a> <a id="920" class="Symbol">:</a> <a id="922" href="Algebra.Elgot.html#914" class="Bound">X</a> <a id="924" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="926" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="928" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="930" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="933" href="Algebra.Elgot.html#914" class="Bound">X</a> <a id="935" class="Symbol">}</a>
<a id="945" class="Symbol">→</a> <a id="947" href="Algebra.Elgot.html#918" class="Bound">f</a> <a id="949" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="951" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="953" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="955" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="958" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="960" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="962" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="964" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="967" class="Symbol">(</a><a id="968" href="Algebra.Elgot.html#918" class="Bound">f</a> <a id="970" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="971" class="Symbol">)</a> <a id="973" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="975" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="977" href="Algebra.Elgot.html#918" class="Bound">f</a>
<a id="Guarded-Elgot-Algebra-on.#-Uniformity"></a><a id="985" href="Algebra.Elgot.html#985" class="Field">#-Uniformity</a> <a id="998" class="Symbol">:</a> <a id="1000" class="Symbol">∀</a> <a id="1002" class="Symbol">{</a><a id="1003" href="Algebra.Elgot.html#1003" class="Bound">X</a> <a id="1005" href="Algebra.Elgot.html#1005" class="Bound">Y</a><a id="1006" class="Symbol">}</a> <a id="1008" class="Symbol">{</a><a id="1009" href="Algebra.Elgot.html#1009" class="Bound">f</a> <a id="1011" class="Symbol">:</a> <a id="1013" href="Algebra.Elgot.html#1003" class="Bound">X</a> <a id="1015" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1017" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1019" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1021" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1024" href="Algebra.Elgot.html#1003" class="Bound">X</a><a id="1025" class="Symbol">}</a> <a id="1027" class="Symbol">{</a><a id="1028" href="Algebra.Elgot.html#1028" class="Bound">g</a> <a id="1030" class="Symbol">:</a> <a id="1032" href="Algebra.Elgot.html#1005" class="Bound">Y</a> <a id="1034" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1036" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1038" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1040" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1043" href="Algebra.Elgot.html#1005" class="Bound">Y</a><a id="1044" class="Symbol">}</a> <a id="1046" class="Symbol">{</a><a id="1047" href="Algebra.Elgot.html#1047" class="Bound">h</a> <a id="1049" class="Symbol">:</a> <a id="1051" href="Algebra.Elgot.html#1003" class="Bound">X</a> <a id="1053" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1055" href="Algebra.Elgot.html#1005" class="Bound">Y</a><a id="1056" class="Symbol">}</a>
<a id="1067" class="Symbol">→</a> <a id="1069" class="Symbol">(</a><a id="1070" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1073" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1076" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1079" href="Algebra.Elgot.html#1047" class="Bound">h</a><a id="1080" class="Symbol">)</a> <a id="1082" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1084" href="Algebra.Elgot.html#1009" class="Bound">f</a> <a id="1086" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1088" href="Algebra.Elgot.html#1028" class="Bound">g</a> <a id="1090" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1092" href="Algebra.Elgot.html#1047" class="Bound">h</a>
<a id="1102" class="Symbol">→</a> <a id="1104" href="Algebra.Elgot.html#1009" class="Bound">f</a> <a id="1106" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="1108" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1110" href="Algebra.Elgot.html#1028" class="Bound">g</a> <a id="1112" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="1114" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1116" href="Algebra.Elgot.html#1047" class="Bound">h</a>
<a id="Guarded-Elgot-Algebra-on.#-Compositionality"></a><a id="1124" href="Algebra.Elgot.html#1124" class="Field">#-Compositionality</a> <a id="1143" class="Symbol">:</a> <a id="1145" class="Symbol">∀</a> <a id="1147" class="Symbol">{</a><a id="1148" href="Algebra.Elgot.html#1148" class="Bound">X</a> <a id="1150" href="Algebra.Elgot.html#1150" class="Bound">Y</a><a id="1151" class="Symbol">}</a> <a id="1153" class="Symbol">{</a><a id="1154" href="Algebra.Elgot.html#1154" class="Bound">f</a> <a id="1156" class="Symbol">:</a> <a id="1158" href="Algebra.Elgot.html#1148" class="Bound">X</a> <a id="1160" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1162" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1164" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1166" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1169" href="Algebra.Elgot.html#1148" class="Bound">X</a><a id="1170" class="Symbol">}</a> <a id="1172" class="Symbol">{</a><a id="1173" href="Algebra.Elgot.html#1173" class="Bound">h</a> <a id="1175" class="Symbol">:</a> <a id="1177" href="Algebra.Elgot.html#1150" class="Bound">Y</a> <a id="1179" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1181" href="Algebra.Elgot.html#1148" class="Bound">X</a> <a id="1183" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1185" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1188" href="Algebra.Elgot.html#1150" class="Bound">Y</a><a id="1189" class="Symbol">}</a>
<a id="1199" class="Symbol">→</a> <a id="1201" class="Symbol">(((</a><a id="1204" href="Algebra.Elgot.html#1154" class="Bound">f</a> <a id="1206" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="1207" class="Symbol">)</a> <a id="1209" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1212" href="Categories.Category.Core.html#630" class="Function">id</a><a id="1214" class="Symbol">)</a> <a id="1216" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1218" href="Algebra.Elgot.html#1173" class="Bound">h</a><a id="1219" class="Symbol">)</a><a id="1220" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="1222" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1224" class="Symbol">(</a><a id="1225" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1227" class="Symbol">(</a><a id="1228" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1231" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="1234" class="Symbol">(</a><a id="1235" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1238" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="1240" class="Symbol">))</a> <a id="1243" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1245" href="Algebra.Elgot.html#1154" class="Bound">f</a> <a id="1247" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1249" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="1252" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1254" class="Symbol">(</a><a id="1255" href="Categories.Functor.Core.html#455" class="Function">F₁</a> <a id="1258" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="1260" class="Symbol">)</a> <a id="1262" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="1264" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1266" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1268" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="1271" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1273" href="Algebra.Elgot.html#1173" class="Bound">h</a> <a id="1275" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="1276" class="Symbol">)</a><a id="1277" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="1279" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1281" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
<a id="Guarded-Elgot-Algebra-on.#-resp-≈"></a><a id="1290" href="Algebra.Elgot.html#1290" class="Field">#-resp-≈</a> <a id="1299" class="Symbol">:</a> <a id="1301" class="Symbol">∀</a> <a id="1303" class="Symbol">{</a><a id="1304" href="Algebra.Elgot.html#1304" class="Bound">X</a><a id="1305" class="Symbol">}</a> <a id="1307" class="Symbol">{</a><a id="1308" href="Algebra.Elgot.html#1308" class="Bound">f</a> <a id="1310" href="Algebra.Elgot.html#1310" class="Bound">g</a> <a id="1312" class="Symbol">:</a> <a id="1314" href="Algebra.Elgot.html#1304" class="Bound">X</a> <a id="1316" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1318" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="1320" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1322" href="Categories.Functor.Core.html#432" class="Function">F₀</a> <a id="1325" href="Algebra.Elgot.html#1304" class="Bound">X</a><a id="1326" class="Symbol">}</a>
<a id="1337" class="Symbol">→</a> <a id="1339" href="Algebra.Elgot.html#1308" class="Bound">f</a> <a id="1341" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1343" href="Algebra.Elgot.html#1310" class="Bound">g</a>
<a id="1354" class="Symbol">→</a> <a id="1356" class="Symbol">(</a><a id="1357" href="Algebra.Elgot.html#1308" class="Bound">f</a> <a id="1359" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="1360" class="Symbol">)</a> <a id="1362" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1364" class="Symbol">(</a><a id="1365" href="Algebra.Elgot.html#1310" class="Bound">g</a> <a id="1367" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="1368" class="Symbol">)</a>
<a id="1373" class="Keyword">record</a> <a id="Guarded-Elgot-Algebra"></a><a id="1380" href="Algebra.Elgot.html#1380" class="Record">Guarded-Elgot-Algebra</a> <a id="1402" class="Symbol">(</a><a id="1403" href="Algebra.Elgot.html#1403" class="Bound">F</a> <a id="1405" class="Symbol">:</a> <a id="1407" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="1419" href="Algebra.Elgot.html#312" class="Bound">C</a><a id="1420" class="Symbol">)</a> <a id="1422" class="Symbol">:</a> <a id="1424" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1428" class="Symbol">(</a><a id="1429" href="Algebra.Elgot.html#304" class="Bound">o</a> <a id="1431" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1433" href="Algebra.Elgot.html#306" class="Bound">ℓ</a> <a id="1435" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1437" href="Algebra.Elgot.html#308" class="Bound">e</a><a id="1438" class="Symbol">)</a> <a id="1440" class="Keyword">where</a>
<a id="1450" class="Keyword">field</a>
<a id="Guarded-Elgot-Algebra.algebra"></a><a id="1462" href="Algebra.Elgot.html#1462" class="Field">algebra</a> <a id="1470" class="Symbol">:</a> <a id="1472" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="1482" href="Algebra.Elgot.html#1403" class="Bound">F</a>
<a id="Guarded-Elgot-Algebra.guarded-elgot-algebra-on"></a><a id="1490" href="Algebra.Elgot.html#1490" class="Field">guarded-elgot-algebra-on</a> <a id="1515" class="Symbol">:</a> <a id="1517" href="Algebra.Elgot.html#636" class="Record">Guarded-Elgot-Algebra-on</a> <a id="1542" href="Algebra.Elgot.html#1462" class="Field">algebra</a>
<a id="1554" class="Keyword">open</a> <a id="1559" href="Algebra.Elgot.html#636" class="Module">Guarded-Elgot-Algebra-on</a> <a id="1584" href="Algebra.Elgot.html#1490" class="Field">guarded-elgot-algebra-on</a> <a id="1609" class="Keyword">public</a>
</pre>
<h2 id="unguarded-elgot-algebras">Unguarded Elgot algebras</h2>
<p>Unguarded elgot algebras are <code>Id</code>-guarded elgot algebras
where the functor algebra is also trivial. Here we give a different
(easier) Characterization and show that it is equal.</p>
<pre class="Agda"> <a id="1838" class="Keyword">record</a> <a id="Elgot-Algebra-on"></a><a id="1845" href="Algebra.Elgot.html#1845" class="Record">Elgot-Algebra-on</a> <a id="1862" class="Symbol">(</a><a id="1863" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="1865" class="Symbol">:</a> <a id="1867" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="1870" class="Symbol">)</a> <a id="1872" class="Symbol">:</a> <a id="1874" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1878" class="Symbol">(</a><a id="1879" href="Algebra.Elgot.html#304" class="Bound">o</a> <a id="1881" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1883" href="Algebra.Elgot.html#306" class="Bound">ℓ</a> <a id="1885" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="1887" href="Algebra.Elgot.html#308" class="Bound">e</a><a id="1888" class="Symbol">)</a> <a id="1890" class="Keyword">where</a>
<a id="1900" class="Comment">-- iteration operator</a>
<a id="1926" class="Keyword">field</a>
<a id="Elgot-Algebra-on._#"></a><a id="1938" href="Algebra.Elgot.html#1938" class="Field Operator">_#</a> <a id="1941" class="Symbol">:</a> <a id="1943" class="Symbol">∀</a> <a id="1945" class="Symbol">{</a><a id="1946" href="Algebra.Elgot.html#1946" class="Bound">X</a><a id="1947" class="Symbol">}</a> <a id="1949" class="Symbol">→</a> <a id="1951" class="Symbol">(</a><a id="1952" href="Algebra.Elgot.html#1946" class="Bound">X</a> <a id="1954" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1956" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="1958" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1960" href="Algebra.Elgot.html#1946" class="Bound">X</a><a id="1961" class="Symbol">)</a> <a id="1963" class="Symbol">→</a> <a id="1965" class="Symbol">(</a><a id="1966" href="Algebra.Elgot.html#1946" class="Bound">X</a> <a id="1968" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1970" href="Algebra.Elgot.html#1863" class="Bound">A</a><a id="1971" class="Symbol">)</a>
<a id="1978" class="Comment">-- _# properties</a>
<a id="1999" class="Keyword">field</a>
<a id="Elgot-Algebra-on.#-Fixpoint"></a><a id="2011" href="Algebra.Elgot.html#2011" class="Field">#-Fixpoint</a> <a id="2022" class="Symbol">:</a> <a id="2024" class="Symbol">∀</a> <a id="2026" class="Symbol">{</a><a id="2027" href="Algebra.Elgot.html#2027" class="Bound">X</a><a id="2028" class="Symbol">}</a> <a id="2030" class="Symbol">{</a><a id="2031" href="Algebra.Elgot.html#2031" class="Bound">f</a> <a id="2033" class="Symbol">:</a> <a id="2035" href="Algebra.Elgot.html#2027" class="Bound">X</a> <a id="2037" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2039" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2041" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2043" href="Algebra.Elgot.html#2027" class="Bound">X</a> <a id="2045" class="Symbol">}</a>
<a id="2055" class="Symbol">→</a> <a id="2057" href="Algebra.Elgot.html#2031" class="Bound">f</a> <a id="2059" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2061" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2063" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2065" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2068" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2070" href="Algebra.Elgot.html#2031" class="Bound">f</a> <a id="2072" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2074" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2076" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2078" href="Algebra.Elgot.html#2031" class="Bound">f</a>
<a id="Elgot-Algebra-on.#-Uniformity"></a><a id="2086" href="Algebra.Elgot.html#2086" class="Field">#-Uniformity</a> <a id="2099" class="Symbol">:</a> <a id="2101" class="Symbol">∀</a> <a id="2103" class="Symbol">{</a><a id="2104" href="Algebra.Elgot.html#2104" class="Bound">X</a> <a id="2106" href="Algebra.Elgot.html#2106" class="Bound">Y</a><a id="2107" class="Symbol">}</a> <a id="2109" class="Symbol">{</a><a id="2110" href="Algebra.Elgot.html#2110" class="Bound">f</a> <a id="2112" class="Symbol">:</a> <a id="2114" href="Algebra.Elgot.html#2104" class="Bound">X</a> <a id="2116" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2118" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2120" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2122" href="Algebra.Elgot.html#2104" class="Bound">X</a><a id="2123" class="Symbol">}</a> <a id="2125" class="Symbol">{</a><a id="2126" href="Algebra.Elgot.html#2126" class="Bound">g</a> <a id="2128" class="Symbol">:</a> <a id="2130" href="Algebra.Elgot.html#2106" class="Bound">Y</a> <a id="2132" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2134" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2136" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2138" href="Algebra.Elgot.html#2106" class="Bound">Y</a><a id="2139" class="Symbol">}</a> <a id="2141" class="Symbol">{</a><a id="2142" href="Algebra.Elgot.html#2142" class="Bound">h</a> <a id="2144" class="Symbol">:</a> <a id="2146" href="Algebra.Elgot.html#2104" class="Bound">X</a> <a id="2148" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2150" href="Algebra.Elgot.html#2106" class="Bound">Y</a><a id="2151" class="Symbol">}</a>
<a id="2162" class="Symbol">→</a> <a id="2164" class="Symbol">(</a><a id="2165" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2168" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2171" href="Algebra.Elgot.html#2142" class="Bound">h</a><a id="2172" class="Symbol">)</a> <a id="2174" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2176" href="Algebra.Elgot.html#2110" class="Bound">f</a> <a id="2178" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2180" href="Algebra.Elgot.html#2126" class="Bound">g</a> <a id="2182" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2184" href="Algebra.Elgot.html#2142" class="Bound">h</a>
<a id="2194" class="Symbol">→</a> <a id="2196" href="Algebra.Elgot.html#2110" class="Bound">f</a> <a id="2198" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2200" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2202" href="Algebra.Elgot.html#2126" class="Bound">g</a> <a id="2204" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2206" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2208" href="Algebra.Elgot.html#2142" class="Bound">h</a>
<a id="Elgot-Algebra-on.#-Folding"></a><a id="2216" href="Algebra.Elgot.html#2216" class="Field">#-Folding</a> <a id="2226" class="Symbol">:</a> <a id="2228" class="Symbol">∀</a> <a id="2230" class="Symbol">{</a><a id="2231" href="Algebra.Elgot.html#2231" class="Bound">X</a> <a id="2233" href="Algebra.Elgot.html#2233" class="Bound">Y</a><a id="2234" class="Symbol">}</a> <a id="2236" class="Symbol">{</a><a id="2237" href="Algebra.Elgot.html#2237" class="Bound">f</a> <a id="2239" class="Symbol">:</a> <a id="2241" href="Algebra.Elgot.html#2231" class="Bound">X</a> <a id="2243" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2245" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2247" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2249" href="Algebra.Elgot.html#2231" class="Bound">X</a><a id="2250" class="Symbol">}</a> <a id="2252" class="Symbol">{</a><a id="2253" href="Algebra.Elgot.html#2253" class="Bound">h</a> <a id="2255" class="Symbol">:</a> <a id="2257" href="Algebra.Elgot.html#2233" class="Bound">Y</a> <a id="2259" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2261" href="Algebra.Elgot.html#2231" class="Bound">X</a> <a id="2263" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2265" href="Algebra.Elgot.html#2233" class="Bound">Y</a><a id="2266" class="Symbol">}</a>
<a id="2277" class="Symbol">→</a> <a id="2279" class="Symbol">((</a><a id="2281" href="Algebra.Elgot.html#2237" class="Bound">f</a> <a id="2283" href="Algebra.Elgot.html#1938" class="Field Operator">#</a><a id="2284" class="Symbol">)</a> <a id="2286" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2289" href="Algebra.Elgot.html#2253" class="Bound">h</a><a id="2290" class="Symbol">)</a><a id="2291" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2293" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2295" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2297" class="Symbol">(</a><a id="2298" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2301" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2304" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="2306" class="Symbol">)</a> <a id="2308" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2310" href="Algebra.Elgot.html#2237" class="Bound">f</a> <a id="2312" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2314" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2317" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2319" href="Algebra.Elgot.html#2253" class="Bound">h</a> <a id="2321" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2323" href="Algebra.Elgot.html#1938" class="Field Operator">#</a>
<a id="Elgot-Algebra-on.#-resp-≈"></a><a id="2331" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a> <a id="2340" class="Symbol">:</a> <a id="2342" class="Symbol">∀</a> <a id="2344" class="Symbol">{</a><a id="2345" href="Algebra.Elgot.html#2345" class="Bound">X</a><a id="2346" class="Symbol">}</a> <a id="2348" class="Symbol">{</a><a id="2349" href="Algebra.Elgot.html#2349" class="Bound">f</a> <a id="2351" href="Algebra.Elgot.html#2351" class="Bound">g</a> <a id="2353" class="Symbol">:</a> <a id="2355" href="Algebra.Elgot.html#2345" class="Bound">X</a> <a id="2357" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2359" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2361" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2363" href="Algebra.Elgot.html#2345" class="Bound">X</a><a id="2364" class="Symbol">}</a> <a id="2366" class="Symbol">→</a> <a id="2368" href="Algebra.Elgot.html#2349" class="Bound">f</a> <a id="2370" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2372" href="Algebra.Elgot.html#2351" class="Bound">g</a> <a id="2374" class="Symbol">→</a> <a id="2376" class="Symbol">(</a><a id="2377" href="Algebra.Elgot.html#2349" class="Bound">f</a> <a id="2379" href="Algebra.Elgot.html#1938" class="Field Operator">#</a><a id="2380" class="Symbol">)</a> <a id="2382" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2384" class="Symbol">(</a><a id="2385" href="Algebra.Elgot.html#2351" class="Bound">g</a> <a id="2387" href="Algebra.Elgot.html#1938" class="Field Operator">#</a><a id="2388" class="Symbol">)</a>
<a id="2395" class="Keyword">open</a> <a id="2400" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="2417" class="Keyword">open</a> <a id="2422" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="2433" class="Comment">-- Compositionality is derivable</a>
<a id="Elgot-Algebra-on.#-Compositionality"></a><a id="2470" href="Algebra.Elgot.html#2470" class="Function">#-Compositionality</a> <a id="2489" class="Symbol">:</a> <a id="2491" class="Symbol">∀</a> <a id="2493" class="Symbol">{</a><a id="2494" href="Algebra.Elgot.html#2494" class="Bound">X</a> <a id="2496" href="Algebra.Elgot.html#2496" class="Bound">Y</a><a id="2497" class="Symbol">}</a> <a id="2499" class="Symbol">{</a><a id="2500" href="Algebra.Elgot.html#2500" class="Bound">f</a> <a id="2502" class="Symbol">:</a> <a id="2504" href="Algebra.Elgot.html#2494" class="Bound">X</a> <a id="2506" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2508" href="Algebra.Elgot.html#1863" class="Bound">A</a> <a id="2510" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2512" href="Algebra.Elgot.html#2494" class="Bound">X</a><a id="2513" class="Symbol">}</a> <a id="2515" class="Symbol">{</a><a id="2516" href="Algebra.Elgot.html#2516" class="Bound">h</a> <a id="2518" class="Symbol">:</a> <a id="2520" href="Algebra.Elgot.html#2496" class="Bound">Y</a> <a id="2522" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2524" href="Algebra.Elgot.html#2494" class="Bound">X</a> <a id="2526" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2528" href="Algebra.Elgot.html#2496" class="Bound">Y</a><a id="2529" class="Symbol">}</a>
<a id="2537" class="Symbol">→</a> <a id="2539" class="Symbol">(((</a><a id="2542" href="Algebra.Elgot.html#2500" class="Bound">f</a> <a id="2544" href="Algebra.Elgot.html#1938" class="Field Operator">#</a><a id="2545" class="Symbol">)</a> <a id="2547" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2550" href="Categories.Category.Core.html#630" class="Function">id</a><a id="2552" class="Symbol">)</a> <a id="2554" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2556" href="Algebra.Elgot.html#2516" class="Bound">h</a><a id="2557" class="Symbol">)</a><a id="2558" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2560" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2562" class="Symbol">(</a><a id="2563" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2565" class="Symbol">(</a><a id="2566" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="2569" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2572" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="2574" class="Symbol">)</a> <a id="2576" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2578" href="Algebra.Elgot.html#2500" class="Bound">f</a> <a id="2580" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2582" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2585" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2587" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="2590" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="2592" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2594" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="2596" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="2599" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="2601" href="Algebra.Elgot.html#2516" class="Bound">h</a> <a id="2603" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a><a id="2604" class="Symbol">)</a><a id="2605" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="2607" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2609" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a>
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<a id="15013" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15015" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15017" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15019" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15021" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15087" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="15090" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="15094" class="Symbol">(</a><a id="15095" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="15099" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="15101" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15110" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a> <a id="15120" href="Categories.Category.Core.html#1145" class="Function">identityʳ</a><a id="15129" class="Symbol">)</a> <a id="15131" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="15144" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15146" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15148" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15150" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15153" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15155" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15158" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15216" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="15226" href="Algebra.Elgot.html#15226" class="Function">by-fix</a> <a id="15233" class="Symbol">:</a> <a id="15235" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15237" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="15240" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15242" class="Symbol">(</a><a id="15243" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15246" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="15249" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="15251" class="Symbol">)</a> <a id="15253" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15255" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15257" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15259" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15261" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="15263" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15265" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15268" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15270" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15272" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15274" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="15284" href="Algebra.Elgot.html#15226" class="Function">by-fix</a> <a id="15291" class="Symbol">=</a> <a id="15293" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="15297" class="Symbol">(</a><a id="15298" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="15315" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15317" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15320" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15322" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15324" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15326" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15378" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="15381" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15390" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="15395" href="Algebra.Elgot.html#2011" class="Field">#-Fixpoint</a> <a id="15406" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="15419" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15421" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15424" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15426" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15428" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15431" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15433" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15435" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15437" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15439" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15441" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15443" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15481" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="15484" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15493" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="15498" class="Symbol">((</a><a id="15500" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15509" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="15514" class="Symbol">(</a><a id="15515" href="Algebra.Elgot.html#2086" class="Field">#-Uniformity</a> <a id="15528" class="Symbol">(</a><a id="15529" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="15533" href="Categories.Object.Coproduct.html#602" class="Function">inject₂</a><a id="15540" class="Symbol">)))</a> <a id="15544" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a><a id="15551" class="Symbol">)</a> <a id="15553" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="15566" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15568" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15571" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15573" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15575" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15578" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15580" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15582" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="15585" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15587" class="Symbol">(</a><a id="15588" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15591" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="15594" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="15596" class="Symbol">)</a> <a id="15598" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15600" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15602" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15604" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15606" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15608" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="15611" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15613" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15615" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15617" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15627" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="15631" href="Categories.Category.Cocartesian.html#2736" class="Function">∘[]</a> <a id="15635" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="15637" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15646" href="Categories.Object.Coproduct.html#569" class="Function">inject₁</a> <a id="15654" class="Symbol">(</a><a id="15655" href="Categories.Morphism.Reasoning.Core.html#2048" class="Function">pullˡ</a> <a id="15661" class="Symbol">(</a><a id="15662" href="Categories.Category.Cocartesian.html#2662" class="Function">[]∘+₁</a> <a id="15668" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="15670" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="15679" href="Categories.Category.Core.html#1339" class="Function">identity²</a> <a id="15689" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="15693" class="Symbol">))</a> <a id="15696" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="15709" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15711" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15714" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15716" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15718" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="15721" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15723" class="Symbol">(</a><a id="15724" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15727" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="15730" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="15732" class="Symbol">)</a> <a id="15734" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15736" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15738" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15740" href="Algebra.Elgot.html#1938" class="Field Operator">#</a> <a id="15742" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15744" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15746" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15748" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="15751" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15753" class="Symbol">(</a><a id="15754" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15757" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="15760" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="15762" class="Symbol">)</a> <a id="15764" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15766" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15768" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15770" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="15774" href="Algebra.Elgot.html#2011" class="Field">#-Fixpoint</a> <a id="15785" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="15798" class="Symbol">(</a><a id="15799" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="15801" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="15804" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="15806" class="Symbol">(</a><a id="15807" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="15810" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="15813" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="15815" class="Symbol">)</a> <a id="15817" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="15819" href="Algebra.Elgot.html#12802" class="Function">g</a> <a id="15821" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="15823" href="Algebra.Elgot.html#1938" class="Field Operator">#</a><a id="15824" class="Symbol">)</a> <a id="15861" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a><a id="15862" class="Symbol">)</a>
<a id="15869" class="Comment">-- every elgot-algebra comes with a divergence constant</a>
<a id="Elgot-Algebra-on.!ₑ"></a><a id="15929" href="Algebra.Elgot.html#15929" class="Function">!ₑ</a> <a id="15932" class="Symbol">:</a> <a id="15934" href="Categories.Object.Initial.html#804" class="Function">⊥</a> <a id="15936" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="15938" href="Algebra.Elgot.html#1863" class="Bound">A</a>
<a id="15944" href="Algebra.Elgot.html#15929" class="Function">!ₑ</a> <a id="15947" class="Symbol">=</a> <a id="15949" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="15952" href="Algebra.Elgot.html#1938" class="Field Operator">#</a>
<a id="15957" class="Keyword">record</a> <a id="Elgot-Algebra"></a><a id="15964" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a> <a id="15978" class="Symbol">:</a> <a id="15980" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="15984" class="Symbol">(</a><a id="15985" href="Algebra.Elgot.html#304" class="Bound">o</a> <a id="15987" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="15989" href="Algebra.Elgot.html#306" class="Bound">ℓ</a> <a id="15991" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="15993" href="Algebra.Elgot.html#308" class="Bound">e</a><a id="15994" class="Symbol">)</a> <a id="15996" class="Keyword">where</a>
<a id="16006" class="Keyword">field</a>
<a id="Elgot-Algebra.A"></a><a id="16018" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="16020" class="Symbol">:</a> <a id="16022" href="Categories.Category.Core.html#559" class="Function">Obj</a>
<a id="Elgot-Algebra.algebra"></a><a id="16032" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="16040" class="Symbol">:</a> <a id="16042" href="Algebra.Elgot.html#1845" class="Record">Elgot-Algebra-on</a> <a id="16059" href="Algebra.Elgot.html#16018" class="Field">A</a>
<a id="16065" class="Keyword">open</a> <a id="16070" href="Algebra.Elgot.html#1845" class="Module">Elgot-Algebra-on</a> <a id="16087" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="16095" class="Keyword">public</a>
</pre>
<p>Now we show that unguarded and Id-guarded Elgot algebras are the
same.</p>
<p>First we show how to get an Id-guarded algebra from a unguarded one
and vice versa:</p>
<pre class="Agda"> <a id="16273" class="Keyword">private</a>
<a id="16285" class="Comment">-- identity algebra</a>
<a id="Id-Algebra"></a><a id="16309" href="Algebra.Elgot.html#16309" class="Function">Id-Algebra</a> <a id="16320" class="Symbol">:</a> <a id="16322" href="Categories.Category.Core.html#559" class="Function">Obj</a> <a id="16326" class="Symbol">→</a> <a id="16328" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="16338" class="Symbol">(</a><a id="16339" href="Algebra.Elgot.html#79" class="Function">idF</a> <a id="16343" class="Symbol">{</a><a id="16344" class="Argument">C</a> <a id="16346" class="Symbol">=</a> <a id="16348" href="Algebra.Elgot.html#312" class="Bound">C</a><a id="16349" class="Symbol">})</a>
<a id="16356" href="Algebra.Elgot.html#16309" class="Function">Id-Algebra</a> <a id="16367" href="Algebra.Elgot.html#16367" class="Bound">A</a> <a id="16369" class="Symbol">=</a> <a id="16371" class="Keyword">record</a>
<a id="16384" class="Symbol">{</a> <a id="16386" href="Categories.Functor.Algebra.html#573" class="Field">A</a> <a id="16388" class="Symbol">=</a> <a id="16390" href="Algebra.Elgot.html#16367" class="Bound">A</a>
<a id="16398" class="Symbol">;</a> <a id="16400" href="Categories.Functor.Algebra.html#587" class="Field">α</a> <a id="16402" class="Symbol">=</a> <a id="16404" href="Categories.Category.Core.html#630" class="Function">id</a>
<a id="16413" class="Symbol">}</a>
<a id="16420" class="Keyword">where</a> <a id="16426" class="Keyword">open</a> <a id="16431" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="16439" class="Symbol">(</a><a id="16440" href="Algebra.Elgot.html#79" class="Function">idF</a> <a id="16444" class="Symbol">{</a><a id="16445" class="Argument">C</a> <a id="16447" class="Symbol">=</a> <a id="16449" href="Algebra.Elgot.html#312" class="Bound">C</a><a id="16450" class="Symbol">})</a>
<a id="16456" class="Comment">-- constructing an Id-Guarded Elgot-Algebra from an unguarded one</a>
<a id="Unguarded⇒Id-Guarded"></a><a id="16524" href="Algebra.Elgot.html#16524" class="Function">Unguarded⇒Id-Guarded</a> <a id="16545" class="Symbol">:</a> <a id="16547" class="Symbol">(</a><a id="16548" href="Algebra.Elgot.html#16548" class="Bound">EA</a> <a id="16551" class="Symbol">:</a> <a id="16553" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a><a id="16566" class="Symbol">)</a> <a id="16568" class="Symbol">→</a> <a id="16570" href="Algebra.Elgot.html#636" class="Record">Guarded-Elgot-Algebra-on</a> <a id="16595" class="Symbol">(</a><a id="16596" href="Algebra.Elgot.html#16309" class="Function">Id-Algebra</a> <a id="16607" class="Symbol">(</a><a id="16608" href="Algebra.Elgot.html#16018" class="Field">Elgot-Algebra.A</a> <a id="16624" href="Algebra.Elgot.html#16548" class="Bound">EA</a><a id="16626" class="Symbol">))</a>
<a id="16631" href="Algebra.Elgot.html#16524" class="Function">Unguarded⇒Id-Guarded</a> <a id="16652" href="Algebra.Elgot.html#16652" class="Bound">ea</a> <a id="16655" class="Symbol">=</a> <a id="16657" class="Keyword">record</a>
<a id="16670" class="Symbol">{</a> <a id="16672" href="Algebra.Elgot.html#821" class="Field Operator">_#</a> <a id="16675" class="Symbol">=</a> <a id="16677" href="Algebra.Elgot.html#1938" class="Function Operator">_#</a>
<a id="16686" class="Symbol">;</a> <a id="16688" href="Algebra.Elgot.html#898" class="Field">#-Fixpoint</a> <a id="16699" class="Symbol">=</a> <a id="16701" class="Symbol">λ</a> <a id="16703" class="Symbol">{</a><a id="16704" href="Algebra.Elgot.html#16704" class="Bound">X</a><a id="16705" class="Symbol">}</a> <a id="16707" class="Symbol">{</a><a id="16708" href="Algebra.Elgot.html#16708" class="Bound">f</a><a id="16709" class="Symbol">}</a> <a id="16711" class="Symbol">→</a> <a id="16713" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="16719" href="Algebra.Elgot.html#2011" class="Function">#-Fixpoint</a> <a id="16730" class="Symbol">(</a><a id="16731" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="16735" class="Symbol">(</a><a id="16736" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="16746" class="Symbol">(</a><a id="16747" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="16756" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="16765" class="Symbol">)))</a>
<a id="16775" class="Symbol">;</a> <a id="16777" href="Algebra.Elgot.html#985" class="Field">#-Uniformity</a> <a id="16790" class="Symbol">=</a> <a id="16792" href="Algebra.Elgot.html#2086" class="Function">#-Uniformity</a>
<a id="16811" class="Symbol">;</a> <a id="16813" href="Algebra.Elgot.html#1124" class="Field">#-Compositionality</a> <a id="16832" class="Symbol">=</a> <a id="16834" href="Algebra.Elgot.html#2470" class="Function">#-Compositionality</a>
<a id="16859" class="Symbol">;</a> <a id="16861" href="Algebra.Elgot.html#1290" class="Field">#-resp-≈</a> <a id="16870" class="Symbol">=</a> <a id="16872" href="Algebra.Elgot.html#2331" class="Function">#-resp-≈</a>
<a id="16887" class="Symbol">}</a>
<a id="16893" class="Keyword">where</a>
<a id="16904" class="Keyword">open</a> <a id="16909" href="Algebra.Elgot.html#15964" class="Module">Elgot-Algebra</a> <a id="16923" href="Algebra.Elgot.html#16652" class="Bound">ea</a>
<a id="16930" class="Keyword">open</a> <a id="16935" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="16952" class="Keyword">open</a> <a id="16957" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="16966" class="Comment">-- constructing an unguarded Elgot-Algebra from an Id-Guarded one </a>
<a id="Id-Guarded⇒Unguarded"></a><a id="17035" href="Algebra.Elgot.html#17035" class="Function">Id-Guarded⇒Unguarded</a> <a id="17056" class="Symbol">:</a> <a id="17058" class="Symbol">∀</a> <a id="17060" class="Symbol">{</a><a id="17061" href="Algebra.Elgot.html#17061" class="Bound">A</a><a id="17062" class="Symbol">}</a> <a id="17064" class="Symbol">→</a> <a id="17066" href="Algebra.Elgot.html#636" class="Record">Guarded-Elgot-Algebra-on</a> <a id="17091" class="Symbol">(</a><a id="17092" href="Algebra.Elgot.html#16309" class="Function">Id-Algebra</a> <a id="17103" href="Algebra.Elgot.html#17061" class="Bound">A</a><a id="17104" class="Symbol">)</a> <a id="17106" class="Symbol">→</a> <a id="17108" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a>
<a id="17124" href="Algebra.Elgot.html#17035" class="Function">Id-Guarded⇒Unguarded</a> <a id="17145" href="Algebra.Elgot.html#17145" class="Bound">gea</a> <a id="17149" class="Symbol">=</a> <a id="17151" class="Keyword">record</a>
<a id="17163" class="Symbol">{</a> <a id="17165" href="Algebra.Elgot.html#16018" class="Field">A</a> <a id="17167" class="Symbol">=</a> <a id="17169" href="Categories.Functor.Algebra.html#573" class="Function">A</a>
<a id="17176" class="Symbol">;</a> <a id="17178" href="Algebra.Elgot.html#16032" class="Field">algebra</a> <a id="17186" class="Symbol">=</a> <a id="17188" class="Keyword">record</a>
<a id="17201" class="Symbol">{</a> <a id="17203" href="Algebra.Elgot.html#1938" class="Field Operator">_#</a> <a id="17206" class="Symbol">=</a> <a id="17208" href="Algebra.Elgot.html#821" class="Field Operator">_#</a>
<a id="17217" class="Symbol">;</a> <a id="17219" href="Algebra.Elgot.html#2011" class="Field">#-Fixpoint</a> <a id="17230" class="Symbol">=</a> <a id="17232" class="Symbol">λ</a> <a id="17234" class="Symbol">{</a><a id="17235" href="Algebra.Elgot.html#17235" class="Bound">X</a><a id="17236" class="Symbol">}</a> <a id="17238" class="Symbol">{</a><a id="17239" href="Algebra.Elgot.html#17239" class="Bound">f</a><a id="17240" class="Symbol">}</a> <a id="17242" class="Symbol">→</a> <a id="17244" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="17250" href="Algebra.Elgot.html#898" class="Field">#-Fixpoint</a> <a id="17261" class="Symbol">(</a><a id="17262" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="17272" class="Symbol">(</a><a id="17273" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="17282" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a><a id="17291" class="Symbol">))</a>
<a id="17300" class="Symbol">;</a> <a id="17302" href="Algebra.Elgot.html#2086" class="Field">#-Uniformity</a> <a id="17315" class="Symbol">=</a> <a id="17317" href="Algebra.Elgot.html#985" class="Field">#-Uniformity</a>
<a id="17336" class="Symbol">;</a> <a id="17338" href="Algebra.Elgot.html#2216" class="Field">#-Folding</a> <a id="17348" class="Symbol">=</a> <a id="17350" class="Symbol">λ</a> <a id="17352" class="Symbol">{</a><a id="17353" href="Algebra.Elgot.html#17353" class="Bound">X</a><a id="17354" class="Symbol">}</a> <a id="17356" class="Symbol">{</a><a id="17357" href="Algebra.Elgot.html#17357" class="Bound">Y</a><a id="17358" class="Symbol">}</a> <a id="17360" class="Symbol">{</a><a id="17361" href="Algebra.Elgot.html#17361" class="Bound">f</a><a id="17362" class="Symbol">}</a> <a id="17364" class="Symbol">{</a><a id="17365" href="Algebra.Elgot.html#17365" class="Bound">h</a><a id="17366" class="Symbol">}</a> <a id="17368" class="Symbol">→</a> <a id="17370" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="17383" class="Symbol">((</a><a id="17385" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17387" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="17388" class="Symbol">)</a> <a id="17390" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17393" href="Algebra.Elgot.html#17365" class="Bound">h</a><a id="17394" class="Symbol">)</a> <a id="17396" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17461" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="17465" href="Categories.Category.Cocartesian.html#1741" class="Function">+-g-η</a> <a id="17471" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="17480" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17482" class="Symbol">(</a><a id="17483" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17485" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17487" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17490" href="Algebra.Elgot.html#17365" class="Bound">h</a><a id="17491" class="Symbol">)</a><a id="17492" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17494" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17496" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="17499" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17501" class="Symbol">(</a><a id="17502" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17504" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17506" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17509" href="Algebra.Elgot.html#17365" class="Bound">h</a><a id="17510" class="Symbol">)</a><a id="17511" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17513" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17515" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17518" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17558" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="17561" href="Categories.Object.Coproduct.html#816" class="Function">[]-cong₂</a> <a id="17570" href="Algebra.Elgot.html#17895" class="Function">left</a> <a id="17575" href="Algebra.Elgot.html#20728" class="Function">right</a> <a id="17581" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="17590" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17592" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17594" class="Symbol">(</a><a id="17595" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="17598" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17601" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="17603" class="Symbol">)</a> <a id="17605" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17607" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17609" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17611" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17614" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17616" href="Algebra.Elgot.html#17365" class="Bound">h</a> <a id="17618" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17620" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17622" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17624" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="17627" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17629" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17631" class="Symbol">(</a><a id="17632" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="17635" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17638" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="17640" class="Symbol">)</a> <a id="17642" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17644" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17646" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17648" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17651" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17653" href="Algebra.Elgot.html#17365" class="Bound">h</a> <a id="17655" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17657" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17659" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17661" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17664" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17666" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="17669" href="Categories.Category.Cocartesian.html#1741" class="Function">+-g-η</a> <a id="17675" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="17684" class="Symbol">(</a><a id="17685" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17687" class="Symbol">(</a><a id="17688" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="17691" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17694" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="17696" class="Symbol">)</a> <a id="17698" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17700" href="Algebra.Elgot.html#17361" class="Bound">f</a> <a id="17702" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17704" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17707" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17709" href="Algebra.Elgot.html#17365" class="Bound">h</a> <a id="17711" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17713" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="17714" class="Symbol">)</a> <a id="17761" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="17769" class="Symbol">;</a> <a id="17771" href="Algebra.Elgot.html#2331" class="Field">#-resp-≈</a> <a id="17780" class="Symbol">=</a> <a id="17782" href="Algebra.Elgot.html#1290" class="Field">#-resp-≈</a>
<a id="17797" class="Symbol">}</a>
<a id="17803" class="Symbol">}</a>
<a id="17809" class="Keyword">where</a>
<a id="17820" class="Keyword">open</a> <a id="17825" href="Algebra.Elgot.html#636" class="Module">Guarded-Elgot-Algebra-on</a> <a id="17850" href="Algebra.Elgot.html#17145" class="Bound">gea</a>
<a id="17858" class="Keyword">open</a> <a id="17863" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="17880" class="Keyword">open</a> <a id="17885" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="17895" href="Algebra.Elgot.html#17895" class="Function">left</a> <a id="17900" class="Symbol">:</a> <a id="17902" class="Symbol">∀</a> <a id="17904" class="Symbol">{</a><a id="17905" href="Algebra.Elgot.html#17905" class="Bound">X</a> <a id="17907" href="Algebra.Elgot.html#17907" class="Bound">Y</a><a id="17908" class="Symbol">}</a> <a id="17910" class="Symbol">{</a><a id="17911" href="Algebra.Elgot.html#17911" class="Bound">f</a> <a id="17913" class="Symbol">:</a> <a id="17915" href="Algebra.Elgot.html#17905" class="Bound">X</a> <a id="17917" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="17919" href="Categories.Functor.Algebra.html#573" class="Function">A</a> <a id="17921" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="17923" href="Algebra.Elgot.html#17905" class="Bound">X</a><a id="17924" class="Symbol">}</a> <a id="17926" class="Symbol">{</a><a id="17927" href="Algebra.Elgot.html#17927" class="Bound">h</a> <a id="17929" class="Symbol">:</a> <a id="17931" href="Algebra.Elgot.html#17907" class="Bound">Y</a> <a id="17933" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="17935" href="Algebra.Elgot.html#17905" class="Bound">X</a> <a id="17937" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="17939" href="Algebra.Elgot.html#17907" class="Bound">Y</a><a id="17940" class="Symbol">}</a>
<a id="17949" class="Symbol">→</a> <a id="17951" class="Symbol">(</a><a id="17952" href="Algebra.Elgot.html#17911" class="Bound">f</a> <a id="17954" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17956" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17959" href="Algebra.Elgot.html#17927" class="Bound">h</a><a id="17960" class="Symbol">)</a><a id="17961" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="17963" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17965" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="17968" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="17970" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="17972" class="Symbol">(</a><a id="17973" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="17976" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="17979" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="17981" class="Symbol">)</a> <a id="17983" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17985" href="Algebra.Elgot.html#17911" class="Bound">f</a> <a id="17987" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="17989" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="17992" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="17994" href="Algebra.Elgot.html#17927" class="Bound">h</a> <a id="17996" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="17998" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="18000" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="18002" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a>
<a id="18009" href="Algebra.Elgot.html#17895" class="Function">left</a> <a id="18014" class="Symbol">{</a><a id="18015" href="Algebra.Elgot.html#18015" class="Bound">X</a><a id="18016" class="Symbol">}</a> <a id="18018" class="Symbol">{</a><a id="18019" href="Algebra.Elgot.html#18019" class="Bound">Y</a><a id="18020" class="Symbol">}</a> <a id="18022" class="Symbol">{</a><a id="18023" href="Algebra.Elgot.html#18023" class="Bound">f</a><a id="18024" class="Symbol">}</a> <a id="18026" class="Symbol">{</a><a id="18027" href="Algebra.Elgot.html#18027" class="Bound">h</a><a id="18028" class="Symbol">}</a> <a id="18030" class="Symbol">=</a> <a id="18032" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="18046" class="Symbol">(</a><a id="18047" href="Algebra.Elgot.html#18023" class="Bound">f</a> <a id="18049" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="18051" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="18054" href="Algebra.Elgot.html#18027" class="Bound">h</a><a id="18055" class="Symbol">)</a><a id="18056" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="18058" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="18060" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="18101" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="18104" href="Algebra.Elgot.html#898" class="Field">#-Fixpoint</a> <a id="18115" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="18123" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="18132" class="Symbol">(</a><a id="18133" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="18135" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="18138" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="18140" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="18143" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="18145" class="Symbol">(((</a><a id="18148" href="Algebra.Elgot.html#18023" class="Bound">f</a> <a id="18150" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="18151" class="Symbol">)</a> <a id="18153" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="18156" href="Algebra.Elgot.html#18027" class="Bound">h</a><a id="18157" class="Symbol">)</a> <a id="18159" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="18160" class="Symbol">)</a> <a id="18162" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="18164" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="18166" class="Symbol">((</a><a id="18168" href="Algebra.Elgot.html#18023" class="Bound">f</a> <a id="18170" href="Algebra.Elgot.html#821" class="Field Operator">#</a><a id="18171" class="Symbol">)</a> <a id="18173" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="18176" href="Algebra.Elgot.html#18027" class="Bound">h</a><a id="18177" class="Symbol">))</a> <a id="18180" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="18182" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a> <a id="18185" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="18188" href="Categories.Morphism.Reasoning.Core.html#1914" class="Function">pullʳ</a> <a id="18194" href="Categories.Category.Cocartesian.html#2548" class="Function">+₁∘i₁</a> <a id="18200" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="22420" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22422" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22425" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22427" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22429" class="Symbol">(</a><a id="22430" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22433" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="22436" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="22438" class="Symbol">)</a> <a id="22440" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22442" href="Algebra.Elgot.html#20858" class="Bound">f</a> <a id="22444" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22446" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22449" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22451" href="Algebra.Elgot.html#20862" class="Bound">h</a> <a id="22453" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="22455" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="22457" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="22466" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22468" class="Symbol">(</a><a id="22469" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22471" class="Symbol">(</a><a id="22472" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22475" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="22478" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="22480" class="Symbol">)</a> <a id="22482" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22484" href="Algebra.Elgot.html#20858" class="Bound">f</a> <a id="22486" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22488" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22491" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22493" href="Algebra.Elgot.html#20862" class="Bound">h</a> <a id="22495" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="22497" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22499" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="22501" class="Symbol">)</a> <a id="22520" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="22524" href="Categories.Category.Cocartesian.html#2311" class="Function">[]-congˡ</a> <a id="22533" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="22543" href="Categories.Category.Core.html#2837" class="Function Operator">⟩∘⟨refl</a> <a id="22551" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="22559" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22561" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22564" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22566" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22569" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22571" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22573" class="Symbol">(</a><a id="22574" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22577" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="22580" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="22582" class="Symbol">)</a> <a id="22584" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22586" href="Algebra.Elgot.html#20858" class="Bound">f</a> <a id="22588" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22590" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22593" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22595" href="Algebra.Elgot.html#20862" class="Bound">h</a> <a id="22597" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="22599" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="22601" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="22610" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22612" class="Symbol">(</a><a id="22613" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22615" class="Symbol">(</a><a id="22616" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22619" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="22622" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="22624" class="Symbol">)</a> <a id="22626" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22628" href="Algebra.Elgot.html#20858" class="Bound">f</a> <a id="22630" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22632" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22635" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22637" href="Algebra.Elgot.html#20862" class="Bound">h</a> <a id="22639" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="22641" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22643" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a><a id="22645" class="Symbol">)</a> <a id="22664" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="22668" href="Categories.Morphism.Reasoning.Core.html#2485" class="Function">pushˡ</a> <a id="22674" href="Algebra.Elgot.html#898" class="Field">#-Fixpoint</a> <a id="22685" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="22694" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="22696" class="Symbol">(</a><a id="22697" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="22700" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="22703" href="Categories.Object.Coproduct.html#492" class="Function">i₁</a><a id="22705" class="Symbol">)</a> <a id="22707" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22709" href="Algebra.Elgot.html#20858" class="Bound">f</a> <a id="22711" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="22713" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22716" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22718" href="Algebra.Elgot.html#20862" class="Bound">h</a> <a id="22720" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a> <a id="22722" href="Algebra.Elgot.html#821" class="Field Operator">#</a> <a id="22724" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="22726" href="Categories.Object.Coproduct.html#512" class="Function">i₂</a> <a id="22748" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
</pre>
<p>and now we show that this transformation is isomorphic (this is just
a formality, it is of course obvious, since the record fields are the
<strong>same</strong>)</p>
<pre class="Agda"> <a id="Unguarded⇒Id-Guarded⇒Unguarded"></a><a id="22915" href="Algebra.Elgot.html#22915" class="Function">Unguarded⇒Id-Guarded⇒Unguarded</a> <a id="22946" class="Symbol">:</a> <a id="22948" class="Symbol">∀</a> <a id="22950" class="Symbol">(</a><a id="22951" href="Algebra.Elgot.html#22951" class="Bound">EA</a> <a id="22954" class="Symbol">:</a> <a id="22956" href="Algebra.Elgot.html#15964" class="Record">Elgot-Algebra</a><a id="22969" class="Symbol">)</a> <a id="22971" class="Symbol">{</a><a id="22972" href="Algebra.Elgot.html#22972" class="Bound">X</a> <a id="22974" class="Symbol">:</a> <a id="22976" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="22979" class="Symbol">}</a> <a id="22981" class="Symbol">(</a><a id="22982" href="Algebra.Elgot.html#22982" class="Bound">f</a> <a id="22984" class="Symbol">:</a> <a id="22986" href="Algebra.Elgot.html#22972" class="Bound">X</a> <a id="22988" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="22990" href="Algebra.Elgot.html#16018" class="Field">Elgot-Algebra.A</a> <a id="23006" href="Algebra.Elgot.html#22951" class="Bound">EA</a> <a id="23009" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="23011" href="Algebra.Elgot.html#22972" class="Bound">X</a><a id="23012" class="Symbol">)</a> <a id="23014" class="Symbol">→</a> <a id="23016" href="Algebra.Elgot.html#1938" class="Function Operator">Elgot-Algebra._#</a> <a id="23033" href="Algebra.Elgot.html#22951" class="Bound">EA</a> <a id="23036" href="Algebra.Elgot.html#22982" class="Bound">f</a> <a id="23038" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="23040" href="Algebra.Elgot.html#1938" class="Function Operator">Elgot-Algebra._#</a> <a id="23057" class="Symbol">(</a><a id="23058" href="Algebra.Elgot.html#17035" class="Function">Id-Guarded⇒Unguarded</a> <a id="23079" class="Symbol">(</a><a id="23080" href="Algebra.Elgot.html#16524" class="Function">Unguarded⇒Id-Guarded</a> <a id="23101" href="Algebra.Elgot.html#22951" class="Bound">EA</a><a id="23103" class="Symbol">))</a> <a id="23106" href="Algebra.Elgot.html#22982" class="Bound">f</a>
<a id="23110" href="Algebra.Elgot.html#22915" class="Function">Unguarded⇒Id-Guarded⇒Unguarded</a> <a id="23141" href="Algebra.Elgot.html#23141" class="Bound">EA</a> <a id="23144" class="Symbol">{</a><a id="23145" href="Algebra.Elgot.html#23145" class="Bound">X</a><a id="23146" class="Symbol">}</a> <a id="23148" href="Algebra.Elgot.html#23148" class="Bound">f</a> <a id="23150" class="Symbol">=</a> <a id="23152" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a>
<a id="Id-Guarded⇒Unguarded⇒Id-Guarded"></a><a id="23166" href="Algebra.Elgot.html#23166" class="Function">Id-Guarded⇒Unguarded⇒Id-Guarded</a> <a id="23198" class="Symbol">:</a> <a id="23200" class="Symbol">∀</a> <a id="23202" class="Symbol">(</a><a id="23203" href="Algebra.Elgot.html#23203" class="Bound">A</a> <a id="23205" class="Symbol">:</a> <a id="23207" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="23210" class="Symbol">)</a> <a id="23212" class="Symbol">(</a><a id="23213" href="Algebra.Elgot.html#23213" class="Bound">EA</a> <a id="23216" class="Symbol">:</a> <a id="23218" href="Algebra.Elgot.html#636" class="Record">Guarded-Elgot-Algebra-on</a> <a id="23243" class="Symbol">(</a><a id="23244" href="Algebra.Elgot.html#16309" class="Function">Id-Algebra</a> <a id="23255" href="Algebra.Elgot.html#23203" class="Bound">A</a><a id="23256" class="Symbol">))</a> <a id="23259" class="Symbol">{</a><a id="23260" href="Algebra.Elgot.html#23260" class="Bound">X</a> <a id="23262" class="Symbol">:</a> <a id="23264" href="Categories.Category.Core.html#559" class="Function">Obj</a><a id="23267" class="Symbol">}</a> <a id="23269" class="Symbol">(</a><a id="23270" href="Algebra.Elgot.html#23270" class="Bound">f</a> <a id="23272" class="Symbol">:</a> <a id="23274" href="Algebra.Elgot.html#23260" class="Bound">X</a> <a id="23276" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="23278" href="Algebra.Elgot.html#23203" class="Bound">A</a> <a id="23280" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="23282" href="Algebra.Elgot.html#23260" class="Bound">X</a><a id="23283" class="Symbol">)</a> <a id="23285" class="Symbol">→</a> <a id="23287" href="Algebra.Elgot.html#821" class="Field Operator">Guarded-Elgot-Algebra-on._#</a> <a id="23315" href="Algebra.Elgot.html#23213" class="Bound">EA</a> <a id="23318" href="Algebra.Elgot.html#23270" class="Bound">f</a> <a id="23320" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="23322" href="Algebra.Elgot.html#821" class="Field Operator">Guarded-Elgot-Algebra-on._#</a> <a id="23350" class="Symbol">(</a><a id="23351" href="Algebra.Elgot.html#16524" class="Function">Unguarded⇒Id-Guarded</a> <a id="23372" class="Symbol">(</a><a id="23373" href="Algebra.Elgot.html#17035" class="Function">Id-Guarded⇒Unguarded</a> <a id="23394" href="Algebra.Elgot.html#23213" class="Bound">EA</a><a id="23396" class="Symbol">))</a> <a id="23399" href="Algebra.Elgot.html#23270" class="Bound">f</a>
<a id="23403" href="Algebra.Elgot.html#23166" class="Function">Id-Guarded⇒Unguarded⇒Id-Guarded</a> <a id="23435" href="Algebra.Elgot.html#23435" class="Bound">A</a> <a id="23437" href="Algebra.Elgot.html#23437" class="Bound">EA</a> <a id="23440" class="Symbol">{</a><a id="23441" href="Algebra.Elgot.html#23441" class="Bound">X</a><a id="23442" class="Symbol">}</a> <a id="23444" href="Algebra.Elgot.html#23444" class="Bound">f</a> <a id="23446" class="Symbol">=</a> <a id="23448" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a>
</pre>
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