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<h2 id="summary">Summary</h2>
<p>This file introduces the delay monad <strong><em>D</em></strong></p>
<ul class="task-list">
<li><input type="checkbox" disabled="" checked="" /><em>Proposition
1</em> Characterization of the delay monad <strong><em>D</em></strong>
(here treated as definition)</li>
<li><input type="checkbox" disabled="" /><em>Proposition 2</em>
<strong><em>D</em></strong> is commutative</li>
</ul>
<h2 id="code">Code</h2>
<pre class="Agda"><a id="953" class="Keyword">module</a> <a id="960" href="Monad.Instance.Delay.html" class="Module">Monad.Instance.Delay</a> <a id="981" class="Symbol">{</a><a id="982" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="984" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="986" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="987" class="Symbol">}</a> <a id="989" class="Symbol">(</a><a id="990" href="Monad.Instance.Delay.html#990" class="Bound">ED</a> <a id="993" class="Symbol">:</a> <a id="995" href="Categories.Category.Extensive.Bundle.html#781" class="Record">ExtensiveDistributiveCategory</a> <a id="1025" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="1027" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="1029" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="1030" class="Symbol">)</a> <a id="1032" class="Keyword">where</a>
<a id="1040" class="Keyword">open</a> <a id="1045" href="Categories.Category.Extensive.Bundle.html#781" class="Module">ExtensiveDistributiveCategory</a> <a id="1075" href="Monad.Instance.Delay.html#990" class="Bound">ED</a> <a id="1078" class="Keyword">renaming</a> <a id="1087" class="Symbol">(</a><a id="1088" href="Categories.Category.Extensive.Bundle.html#859" class="Field">U</a> <a id="1090" class="Symbol">to</a> <a id="1093" class="Field">C</a><a id="1094" class="Symbol">;</a> <a id="1096" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="1099" class="Symbol">to</a> <a id="1102" class="Function">idC</a><a id="1105" class="Symbol">)</a>
<a id="1109" class="Keyword">open</a> <a id="1114" href="Categories.Category.Cocartesian.html#3385" class="Module">Cocartesian</a> <a id="1126" class="Symbol">(</a><a id="1127" href="Categories.Category.Extensive.html#734" class="Field">Extensive.cocartesian</a> <a id="1149" href="Categories.Category.Extensive.Bundle.html#913" class="Field">extensive</a><a id="1158" class="Symbol">)</a>
<a id="1162" class="Keyword">open</a> <a id="1167" href="Categories.Category.Cartesian.html#727" class="Module">Cartesian</a> <a id="1177" class="Symbol">(</a><a id="1178" href="Categories.Category.Extensive.Bundle.html#943" class="Field">ExtensiveDistributiveCategory.cartesian</a> <a id="1218" href="Monad.Instance.Delay.html#990" class="Bound">ED</a><a id="1220" class="Symbol">)</a>
<a id="1224" class="Keyword">open</a> <a id="1229" href="Categories.Category.BinaryProducts.html#812" class="Module">BinaryProducts</a> <a id="1244" href="Categories.Category.Cartesian.html#801" class="Function">products</a>
<a id="1256" class="Keyword">open</a> <a id="1261" href="Categories.Morphism.html" class="Module">M</a> <a id="1263" href="Monad.Instance.Delay.html#1093" class="Field">C</a>
<a id="1267" class="Keyword">open</a> <a id="1272" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="1275" href="Monad.Instance.Delay.html#1093" class="Field">C</a>
<a id="1279" class="Keyword">open</a> <a id="1284" href="Categories.Category.Core.html#1530" class="Module">Equiv</a>
<a id="1292" class="Keyword">open</a> <a id="1297" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
<a id="1312" class="Keyword">open</a> <a id="1317" href="Categories.Category.Construction.F-Coalgebras.html#2828" class="Module">CoLambek</a>
</pre>
<h3
id="proposition-1-characterization-of-the-delay-monad-d"><em>Proposition
1</em>: Characterization of the delay monad
<strong><em>D</em></strong></h3>
<p>First I postulate the Functor <em>D</em>, maybe I should derive it…
<strong>TODO</strong>: - how to define using final coalgebra(s)? - DX
can be defined as retract of infinite streams, how? - how to express
<strong>Theorem 8</strong> in agda?</p>
<pre class="Agda"> <a id="1620" class="Keyword">record</a> <a id="DelayFunctor"></a><a id="1627" href="Monad.Instance.Delay.html#1627" class="Record">DelayFunctor</a> <a id="1640" class="Symbol">:</a> <a id="1642" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1646" class="Symbol">(</a><a id="1647" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="1649" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1651" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="1653" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="1655" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="1656" class="Symbol">)</a> <a id="1658" class="Keyword">where</a>
<a id="1668" class="Keyword">field</a>
<a id="DelayFunctor.D"></a><a id="1680" href="Monad.Instance.Delay.html#1680" class="Field">D</a> <a id="1682" class="Symbol">:</a> <a id="1684" href="Categories.Functor.html#283" class="Function">Endofunctor</a> <a id="1696" href="Monad.Instance.Delay.html#1093" class="Field">C</a>
<a id="1703" class="Keyword">open</a> <a id="1708" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="1716" href="Monad.Instance.Delay.html#1680" class="Field">D</a> <a id="1718" class="Keyword">public</a> <a id="1725" class="Keyword">renaming</a> <a id="1734" class="Symbol">(</a><a id="1735" href="Categories.Functor.Core.html#432" class="Field">F₀</a> <a id="1738" class="Symbol">to</a> <a id="1741" class="Field">D₀</a><a id="1743" class="Symbol">;</a> <a id="1745" href="Categories.Functor.Core.html#455" class="Field">F₁</a> <a id="1748" class="Symbol">to</a> <a id="1751" class="Field">D₁</a><a id="1753" class="Symbol">)</a>
<a id="1760" class="Keyword">field</a>
<a id="DelayFunctor.now"></a><a id="1772" href="Monad.Instance.Delay.html#1772" class="Field">now</a> <a id="1776" class="Symbol">:</a> <a id="1778" class="Symbol">∀</a> <a id="1780" class="Symbol">{</a><a id="1781" href="Monad.Instance.Delay.html#1781" class="Bound">X</a><a id="1782" class="Symbol">}</a> <a id="1784" class="Symbol">→</a> <a id="1786" href="Monad.Instance.Delay.html#1781" class="Bound">X</a> <a id="1788" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1790" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="1793" href="Monad.Instance.Delay.html#1781" class="Bound">X</a>
<a id="DelayFunctor.later"></a><a id="1801" href="Monad.Instance.Delay.html#1801" class="Field">later</a> <a id="1807" class="Symbol">:</a> <a id="1809" class="Symbol">∀</a> <a id="1811" class="Symbol">{</a><a id="1812" href="Monad.Instance.Delay.html#1812" class="Bound">X</a><a id="1813" class="Symbol">}</a> <a id="1815" class="Symbol">→</a> <a id="1817" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="1820" href="Monad.Instance.Delay.html#1812" class="Bound">X</a> <a id="1822" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1824" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="1827" href="Monad.Instance.Delay.html#1812" class="Bound">X</a>
<a id="DelayFunctor.isIso"></a><a id="1835" href="Monad.Instance.Delay.html#1835" class="Field">isIso</a> <a id="1841" class="Symbol">:</a> <a id="1843" class="Symbol">∀</a> <a id="1845" class="Symbol">{</a><a id="1846" href="Monad.Instance.Delay.html#1846" class="Bound">X</a><a id="1847" class="Symbol">}</a> <a id="1849" class="Symbol">→</a> <a id="1851" href="Categories.Morphism.html#1826" class="Record">IsIso</a> <a id="1857" href="Categories.Object.Coproduct.html#532" class="Function Operator">[</a> <a id="1859" href="Monad.Instance.Delay.html#1772" class="Field">now</a> <a id="1863" class="Symbol">{</a><a id="1864" href="Monad.Instance.Delay.html#1846" class="Bound">X</a><a id="1865" class="Symbol">}</a> <a id="1867" href="Categories.Object.Coproduct.html#532" class="Function Operator">,</a> <a id="1869" href="Monad.Instance.Delay.html#1801" class="Field">later</a> <a id="1875" class="Symbol">{</a><a id="1876" href="Monad.Instance.Delay.html#1846" class="Bound">X</a><a id="1877" class="Symbol">}</a> <a id="1879" href="Categories.Object.Coproduct.html#532" class="Function Operator">]</a>
<a id="DelayFunctor.out"></a><a id="1886" href="Monad.Instance.Delay.html#1886" class="Function">out</a> <a id="1890" class="Symbol">:</a> <a id="1892" class="Symbol">∀</a> <a id="1894" class="Symbol">{</a><a id="1895" href="Monad.Instance.Delay.html#1895" class="Bound">X</a><a id="1896" class="Symbol">}</a> <a id="1898" class="Symbol">→</a> <a id="1900" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="1903" href="Monad.Instance.Delay.html#1895" class="Bound">X</a> <a id="1905" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1907" href="Monad.Instance.Delay.html#1895" class="Bound">X</a> <a id="1909" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1911" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="1914" href="Monad.Instance.Delay.html#1895" class="Bound">X</a>
<a id="1920" href="Monad.Instance.Delay.html#1886" class="Function">out</a> <a id="1924" class="Symbol">{</a><a id="1925" href="Monad.Instance.Delay.html#1925" class="Bound">X</a><a id="1926" class="Symbol">}</a> <a id="1928" class="Symbol">=</a> <a id="1930" href="Categories.Morphism.html#1879" class="Field">IsIso.inv</a> <a id="1940" class="Symbol">(</a><a id="1941" href="Monad.Instance.Delay.html#1835" class="Field">isIso</a> <a id="1947" class="Symbol">{</a><a id="1948" href="Monad.Instance.Delay.html#1925" class="Bound">X</a><a id="1949" class="Symbol">})</a>
<a id="1957" class="Keyword">field</a>
<a id="DelayFunctor.coit"></a><a id="1969" href="Monad.Instance.Delay.html#1969" class="Field">coit</a> <a id="1974" class="Symbol">:</a> <a id="1976" class="Symbol">∀</a> <a id="1978" class="Symbol">{</a><a id="1979" href="Monad.Instance.Delay.html#1979" class="Bound">X</a> <a id="1981" href="Monad.Instance.Delay.html#1981" class="Bound">Y</a><a id="1982" class="Symbol">}</a> <a id="1984" class="Symbol">→</a> <a id="1986" href="Monad.Instance.Delay.html#1981" class="Bound">Y</a> <a id="1988" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1990" href="Monad.Instance.Delay.html#1979" class="Bound">X</a> <a id="1992" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="1994" href="Monad.Instance.Delay.html#1981" class="Bound">Y</a> <a id="1996" class="Symbol">→</a> <a id="1998" href="Monad.Instance.Delay.html#1981" class="Bound">Y</a> <a id="2000" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2002" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="2005" href="Monad.Instance.Delay.html#1979" class="Bound">X</a>
<a id="DelayFunctor.coit-law"></a><a id="2013" href="Monad.Instance.Delay.html#2013" class="Field">coit-law</a> <a id="2022" class="Symbol">:</a> <a id="2024" class="Symbol">∀</a> <a id="2026" class="Symbol">{</a><a id="2027" href="Monad.Instance.Delay.html#2027" class="Bound">X</a> <a id="2029" href="Monad.Instance.Delay.html#2029" class="Bound">Y</a><a id="2030" class="Symbol">}</a> <a id="2032" class="Symbol">{</a><a id="2033" href="Monad.Instance.Delay.html#2033" class="Bound">f</a> <a id="2035" class="Symbol">:</a> <a id="2037" href="Monad.Instance.Delay.html#2029" class="Bound">Y</a> <a id="2039" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2041" href="Monad.Instance.Delay.html#2027" class="Bound">X</a> <a id="2043" href="Categories.Category.Cocartesian.html#1549" class="Function Operator">+</a> <a id="2045" href="Monad.Instance.Delay.html#2029" class="Bound">Y</a><a id="2046" class="Symbol">}</a> <a id="2048" class="Symbol">→</a> <a id="2050" href="Monad.Instance.Delay.html#1886" class="Function">out</a> <a id="2054" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2056" class="Symbol">(</a><a id="2057" href="Monad.Instance.Delay.html#1969" class="Field">coit</a> <a id="2062" href="Monad.Instance.Delay.html#2033" class="Bound">f</a><a id="2063" class="Symbol">)</a> <a id="2065" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="2067" class="Symbol">(</a><a id="2068" href="Monad.Instance.Delay.html#1102" class="Function">idC</a> <a id="2072" href="Categories.Category.Cocartesian.html#2133" class="Function Operator">+₁</a> <a id="2075" class="Symbol">(</a><a id="2076" href="Monad.Instance.Delay.html#1969" class="Field">coit</a> <a id="2081" href="Monad.Instance.Delay.html#2033" class="Bound">f</a><a id="2082" class="Symbol">))</a> <a id="2085" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="2087" href="Monad.Instance.Delay.html#2033" class="Bound">f</a>
</pre>
<p>Now let’s define the monad:</p>
<pre class="Agda"> <a id="2133" class="Keyword">record</a> <a id="DelayMonad"></a><a id="2140" href="Monad.Instance.Delay.html#2140" class="Record">DelayMonad</a> <a id="2151" class="Symbol">:</a> <a id="2153" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2157" class="Symbol">(</a><a id="2158" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="2160" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2162" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="2164" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="2166" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="2167" class="Symbol">)</a> <a id="2169" class="Keyword">where</a>
<a id="2179" class="Keyword">field</a>
<a id="DelayMonad.D"></a><a id="2191" href="Monad.Instance.Delay.html#2191" class="Field">D</a> <a id="2193" class="Symbol">:</a> <a id="2195" href="Monad.Instance.Delay.html#1627" class="Record">DelayFunctor</a>
<a id="2212" class="Keyword">open</a> <a id="2217" href="Monad.Instance.Delay.html#1627" class="Module">DelayFunctor</a> <a id="2230" href="Monad.Instance.Delay.html#2191" class="Field">D</a>
<a id="2237" class="Keyword">field</a>
<a id="DelayMonad._*"></a><a id="2249" href="Monad.Instance.Delay.html#2249" class="Field Operator">_*</a> <a id="2252" class="Symbol">:</a> <a id="2254" class="Symbol">∀</a> <a id="2256" class="Symbol">{</a><a id="2257" href="Monad.Instance.Delay.html#2257" class="Bound">X</a> <a id="2259" href="Monad.Instance.Delay.html#2259" class="Bound">Y</a><a id="2260" class="Symbol">}</a> <a id="2262" class="Symbol">→</a> <a id="2264" href="Monad.Instance.Delay.html#2257" class="Bound">X</a> <a id="2266" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2268" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="2271" href="Monad.Instance.Delay.html#2259" class="Bound">Y</a> <a id="2273" class="Symbol">→</a> <a id="2275" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="2278" href="Monad.Instance.Delay.html#2257" class="Bound">X</a> <a id="2280" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="2282" href="Monad.Instance.Delay.html#1741" class="Function">D₀</a> <a id="2285" href="Monad.Instance.Delay.html#2259" class="Bound">Y</a>
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<a id="3209" href="Monad.Instance.Delay.html#2825" class="Bound">k</a> <a id="3211" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
<a id="3219" class="Symbol">;</a> <a id="3221" href="Categories.Monad.Relative.html#1025" class="Field">identityˡ</a> <a id="3231" class="Symbol">=</a> <a id="3233" class="Symbol">λ</a> <a id="3235" class="Symbol">{</a><a id="3236" href="Monad.Instance.Delay.html#3236" class="Bound">X</a><a id="3237" class="Symbol">}</a> <a id="3239" class="Symbol">→</a> <a id="3241" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="3245" class="Symbol">(</a><a id="3246" href="Monad.Instance.Delay.html#2377" class="Field">*-unique</a> <a id="3255" href="Monad.Instance.Delay.html#1772" class="Function">now</a> <a id="3259" href="Monad.Instance.Delay.html#1102" class="Function">idC</a><a id="3262" class="Symbol">)</a>
<a id="3270" class="Symbol">;</a> <a id="3272" href="Categories.Monad.Relative.html#1078" class="Field">assoc</a> <a id="3278" class="Symbol">=</a> <a id="3280" class="Symbol">λ</a> <a id="3282" class="Symbol">{</a><a id="3283" href="Monad.Instance.Delay.html#3283" class="Bound">X</a><a id="3284" class="Symbol">}</a> <a id="3286" class="Symbol">{</a><a id="3287" href="Monad.Instance.Delay.html#3287" class="Bound">Y</a><a id="3288" class="Symbol">}</a> <a id="3290" class="Symbol">{</a><a id="3291" href="Monad.Instance.Delay.html#3291" class="Bound">Z</a><a id="3292" class="Symbol">}</a> <a id="3294" class="Symbol">{</a><a id="3295" href="Monad.Instance.Delay.html#3295" class="Bound">f</a><a id="3296" class="Symbol">}</a> <a id="3298" class="Symbol">{</a><a id="3299" href="Monad.Instance.Delay.html#3299" class="Bound">g</a><a id="3300" class="Symbol">}</a> <a id="3302" class="Symbol">→</a> <a id="3304" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="3308" class="Symbol">(</a><a id="3309" href="Monad.Instance.Delay.html#2377" class="Field">*-unique</a> <a id="3318" class="Symbol">((</a><a id="3320" href="Monad.Instance.Delay.html#3299" class="Bound">g</a> <a id="3322" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3323" class="Symbol">)</a> <a id="3325" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3327" href="Monad.Instance.Delay.html#3295" class="Bound">f</a><a id="3328" class="Symbol">)</a> <a id="3330" class="Symbol">((</a><a id="3332" href="Monad.Instance.Delay.html#3299" class="Bound">g</a> <a id="3334" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3335" class="Symbol">)</a> <a id="3337" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3339" class="Symbol">(</a><a id="3340" href="Monad.Instance.Delay.html#3295" class="Bound">f</a> <a id="3342" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3343" class="Symbol">)))</a>
<a id="3353" class="Symbol">;</a> <a id="3355" href="Categories.Monad.Relative.html#1190" class="Field">sym-assoc</a> <a id="3365" class="Symbol">=</a> <a id="3367" class="Symbol">λ</a> <a id="3369" class="Symbol">{</a><a id="3370" href="Monad.Instance.Delay.html#3370" class="Bound">X</a><a id="3371" class="Symbol">}</a> <a id="3373" class="Symbol">{</a><a id="3374" href="Monad.Instance.Delay.html#3374" class="Bound">Y</a><a id="3375" class="Symbol">}</a> <a id="3377" class="Symbol">{</a><a id="3378" href="Monad.Instance.Delay.html#3378" class="Bound">Z</a><a id="3379" class="Symbol">}</a> <a id="3381" class="Symbol">{</a><a id="3382" href="Monad.Instance.Delay.html#3382" class="Bound">f</a><a id="3383" class="Symbol">}</a> <a id="3385" class="Symbol">{</a><a id="3386" href="Monad.Instance.Delay.html#3386" class="Bound">g</a><a id="3387" class="Symbol">}</a> <a id="3389" class="Symbol">→</a> <a id="3391" href="Monad.Instance.Delay.html#2377" class="Field">*-unique</a> <a id="3400" class="Symbol">((</a><a id="3402" href="Monad.Instance.Delay.html#3386" class="Bound">g</a> <a id="3404" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3405" class="Symbol">)</a> <a id="3407" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3409" href="Monad.Instance.Delay.html#3382" class="Bound">f</a><a id="3410" class="Symbol">)</a> <a id="3412" class="Symbol">((</a><a id="3414" href="Monad.Instance.Delay.html#3386" class="Bound">g</a> <a id="3416" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3417" class="Symbol">)</a> <a id="3419" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3421" class="Symbol">(</a><a id="3422" href="Monad.Instance.Delay.html#3382" class="Bound">f</a> <a id="3424" href="Monad.Instance.Delay.html#2249" class="Field Operator">*</a><a id="3425" class="Symbol">))</a>
<a id="3434" class="Symbol">;</a> <a id="3436" href="Categories.Monad.Relative.html#1306" class="Field">extend-≈</a> <a id="3445" class="Symbol">=</a> <a id="3447" href="Monad.Instance.Delay.html#2445" class="Field">*-resp-≈</a>
<a id="3462" class="Symbol">}</a>
</pre>
<h3 id="definition-30-search-algebras">Definition 30:
Search-Algebras</h3>
<pre class="Agda"> <a id="3515" class="Keyword">record</a> <a id="SearchAlgebra"></a><a id="3522" href="Monad.Instance.Delay.html#3522" class="Record">SearchAlgebra</a> <a id="3536" class="Symbol">(</a><a id="3537" href="Monad.Instance.Delay.html#3537" class="Bound">DF</a> <a id="3540" class="Symbol">:</a> <a id="3542" href="Monad.Instance.Delay.html#1627" class="Record">DelayFunctor</a><a id="3554" class="Symbol">)</a> <a id="3556" class="Symbol">:</a> <a id="3558" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3562" class="Symbol">(</a><a id="3563" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="3565" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="3567" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="3569" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="3571" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="3572" class="Symbol">)</a> <a id="3574" class="Keyword">where</a>
<a id="3584" class="Keyword">open</a> <a id="3589" href="Monad.Instance.Delay.html#1627" class="Module">DelayFunctor</a> <a id="3602" href="Monad.Instance.Delay.html#3537" class="Bound">DF</a>
<a id="3614" class="Keyword">field</a>
<a id="SearchAlgebra.FA"></a><a id="3626" href="Monad.Instance.Delay.html#3626" class="Field">FA</a> <a id="3629" class="Symbol">:</a> <a id="3631" href="Categories.Functor.Algebra.html#487" class="Record">F-Algebra</a> <a id="3641" href="Monad.Instance.Delay.html#1680" class="Function">D</a>
<a id="3648" class="Keyword">open</a> <a id="3653" href="Categories.Functor.Algebra.html#487" class="Module">F-Algebra</a> <a id="3663" href="Monad.Instance.Delay.html#3626" class="Field">FA</a>
<a id="3671" class="Keyword">field</a>
<a id="SearchAlgebra.now-id"></a><a id="3683" href="Monad.Instance.Delay.html#3683" class="Field">now-id</a> <a id="3690" class="Symbol">:</a> <a id="3692" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="3694" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3696" href="Monad.Instance.Delay.html#1772" class="Function">now</a> <a id="3700" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3702" href="Monad.Instance.Delay.html#1102" class="Function">idC</a>
<a id="SearchAlgebra.later-same"></a><a id="3716" href="Monad.Instance.Delay.html#3716" class="Field">later-same</a> <a id="3727" class="Symbol">:</a> <a id="3729" href="Categories.Functor.Algebra.html#587" class="Function">α</a> <a id="3731" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="3733" href="Monad.Instance.Delay.html#1801" class="Function">later</a> <a id="3739" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="3741" href="Categories.Functor.Algebra.html#587" class="Function">α</a>
</pre>
<h3
id="proposition-31-the-category-of-uniform-iteration-algebras-coincides-with-the-category-of-search-algebras">Proposition
31 : the category of uniform-iteration algebras coincides with the
category of search-algebras</h3>
<p>TODOs:</p>
<ul class="task-list">
<li><input type="checkbox" disabled="" />Define SearchAlgebras (and
SearchAlgebra morphisms)</li>
<li><input type="checkbox" disabled="" />show StrongEquivalence</li>
<li><input type="checkbox" disabled="" />Show
‘ElgotAlgebra⇔Search+<strong><em>D</em></strong>’</li>
</ul>
<pre class="Agda"> <a id="4007" class="Keyword">record</a> <a id="SearchAlgebras"></a><a id="4014" href="Monad.Instance.Delay.html#4014" class="Record">SearchAlgebras</a> <a id="4029" class="Symbol">(</a><a id="4030" href="Monad.Instance.Delay.html#4030" class="Bound">DF</a> <a id="4033" class="Symbol">:</a> <a id="4035" href="Monad.Instance.Delay.html#1627" class="Record">DelayFunctor</a><a id="4047" class="Symbol">)</a> <a id="4049" class="Symbol">:</a> <a id="4051" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="4055" class="Symbol">(</a><a id="4056" href="Monad.Instance.Delay.html#982" class="Bound">o</a> <a id="4058" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4060" href="Monad.Instance.Delay.html#984" class="Bound">ℓ</a> <a id="4062" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="4064" href="Monad.Instance.Delay.html#986" class="Bound">e</a><a id="4065" class="Symbol">)</a> <a id="4067" class="Keyword">where</a>
<a id="4077" class="Keyword">open</a> <a id="4082" href="Monad.Instance.Delay.html#1627" class="Module">DelayFunctor</a> <a id="4095" href="Monad.Instance.Delay.html#4030" class="Bound">DF</a>
</pre>
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