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< h2 id = "summary" > Summary< / h2 >
< p > This file introduces Elgot Monads.< / p >
< ul class = "task-list" >
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< li > < label > < input type = "checkbox" checked = "" / > < em > Definition 13< / em >
Pre-Elgot Monads< / label > < / li >
< li > < label > < input type = "checkbox" / > < em > Definition 13< / em > strong
pre-Elgot< / label > < / li >
< li > < label > < input type = "checkbox" checked = "" / > < em > Definition 14< / em >
Elgot Monads< / label > < / li >
< li > < label > < input type = "checkbox" / > < em > Definition 14< / em > strong
Elgot< / label > < / li >
< li > < label > < input type = "checkbox" / > < em > Proposition 15< / em > (Strong)
Elgot monads are (strong) pre-Elgot< / label > < / li >
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< / ul >
< h2 id = "code" > Code< / h2 >
< pre class = "Agda" > < a id = "752" class = "Keyword" > module< / a > < a id = "759" href = "Monad.ElgotMonad.html" class = "Module" > Monad.ElgotMonad< / a > < a id = "776" class = "Symbol" > {< / a > < a id = "777" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "779" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "781" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "782" class = "Symbol" > }< / a > < a id = "784" class = "Symbol" > (< / a > < a id = "785" href = "Monad.ElgotMonad.html#785" class = "Bound" > ED< / a > < a id = "788" class = "Symbol" > :< / a > < a id = "790" href = "Categories.Category.Extensive.Bundle.html#781" class = "Record" > ExtensiveDistributiveCategory< / a > < a id = "820" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "822" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "824" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "825" class = "Symbol" > )< / a > < a id = "827" class = "Keyword" > where< / a >
< a id = "835" class = "Keyword" > open< / a > < a id = "840" href = "Categories.Category.Extensive.Bundle.html#781" class = "Module" > ExtensiveDistributiveCategory< / a > < a id = "870" href = "Monad.ElgotMonad.html#785" class = "Bound" > ED< / a > < a id = "873" class = "Keyword" > renaming< / a > < a id = "882" class = "Symbol" > (< / a > < a id = "883" href = "Categories.Category.Extensive.Bundle.html#859" class = "Field" > U< / a > < a id = "885" class = "Symbol" > to< / a > < a id = "888" class = "Field" > C< / a > < a id = "889" class = "Symbol" > ;< / a > < a id = "891" href = "Categories.Category.Core.html#630" class = "Function" > id< / a > < a id = "894" class = "Symbol" > to< / a > < a id = "897" class = "Function" > idC< / a > < a id = "900" class = "Symbol" > )< / a >
< a id = "904" class = "Keyword" > open< / a > < a id = "909" href = "Categories.Category.Core.html#2462" class = "Module" > HomReasoning< / a >
< a id = "924" class = "Keyword" > open< / a > < a id = "929" href = "Categories.Category.Cocartesian.html#3385" class = "Module" > Cocartesian< / a > < a id = "941" class = "Symbol" > (< / a > < a id = "942" href = "Categories.Category.Extensive.html#734" class = "Field" > Extensive.cocartesian< / a > < a id = "964" href = "Categories.Category.Extensive.Bundle.html#913" class = "Field" > extensive< / a > < a id = "973" class = "Symbol" > )< / a >
< a id = "977" class = "Keyword" > open< / a > < a id = "982" href = "Categories.Category.Cartesian.html#727" class = "Module" > Cartesian< / a > < a id = "992" class = "Symbol" > (< / a > < a id = "993" href = "Categories.Category.Extensive.Bundle.html#943" class = "Field" > ExtensiveDistributiveCategory.cartesian< / a > < a id = "1033" href = "Monad.ElgotMonad.html#785" class = "Bound" > ED< / a > < a id = "1035" class = "Symbol" > )< / a >
< a id = "1039" class = "Keyword" > open< / a > < a id = "1044" href = "Categories.Category.BinaryProducts.html#812" class = "Module" > BinaryProducts< / a > < a id = "1059" href = "Categories.Category.Cartesian.html#801" class = "Function" > products< / a > < a id = "1068" class = "Keyword" > hiding< / a > < a id = "1075" class = "Symbol" > (< / a > < a id = "1076" href = "Categories.Object.Product.Core.html#797" class = "Function" > η< / a > < a id = "1077" class = "Symbol" > )< / a >
< a id = "1081" class = "Keyword" > open< / a > < a id = "1086" href = "Categories.Morphism.Reasoning.html" class = "Module" > MR< / a > < a id = "1089" href = "Monad.ElgotMonad.html#888" class = "Field" > C< / a >
< a id = "1093" class = "Keyword" > open< / a > < a id = "1098" href = "Categories.Category.Core.html#1530" class = "Module" > Equiv< / a >
< / pre >
< h3 id = "definition-13-pre-elgot-monads" > < em > Definition 13< / em > :
Pre-Elgot Monads< / h3 >
< pre class = "Agda" > < a id = "1158" class = "Keyword" > record< / a > < a id = "IsPreElgot" > < / a > < a id = "1165" href = "Monad.ElgotMonad.html#1165" class = "Record" > IsPreElgot< / a > < a id = "1176" class = "Symbol" > (< / a > < a id = "1177" href = "Monad.ElgotMonad.html#1177" class = "Bound" > T< / a > < a id = "1179" class = "Symbol" > :< / a > < a id = "1181" href = "Categories.Monad.html#454" class = "Record" > Monad< / a > < a id = "1187" href = "Monad.ElgotMonad.html#888" class = "Field" > C< / a > < a id = "1188" class = "Symbol" > )< / a > < a id = "1190" class = "Symbol" > :< / a > < a id = "1192" href = "Agda.Primitive.html#320" class = "Primitive" > Set< / a > < a id = "1196" class = "Symbol" > (< / a > < a id = "1197" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "1199" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1201" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "1203" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1205" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "1206" class = "Symbol" > )< / a > < a id = "1208" class = "Keyword" > where< / a >
< a id = "1218" class = "Keyword" > open< / a > < a id = "1223" href = "Categories.Monad.html#454" class = "Module" > Monad< / a > < a id = "1229" href = "Monad.ElgotMonad.html#1177" class = "Bound" > T< / a >
< a id = "1235" class = "Keyword" > open< / a > < a id = "1240" href = "Categories.Functor.Core.html#248" class = "Module" > Functor< / a > < a id = "1248" href = "Categories.Monad.html#525" class = "Function" > F< / a > < a id = "1250" class = "Keyword" > renaming< / a > < a id = "1259" class = "Symbol" > (< / a > < a id = "1260" href = "Categories.Functor.Core.html#432" class = "Field" > F₀< / a > < a id = "1263" class = "Symbol" > to< / a > < a id = "1266" class = "Field" > T₀< / a > < a id = "1268" class = "Symbol" > ;< / a > < a id = "1270" href = "Categories.Functor.Core.html#455" class = "Field" > F₁< / a > < a id = "1273" class = "Symbol" > to< / a > < a id = "1276" class = "Field" > T₁< / a > < a id = "1278" class = "Symbol" > )< / a >
< a id = "1285" class = "Comment" > -- every TX needs to be equipped with an elgot algebra structure< / a >
< a id = "1354" class = "Keyword" > field< / a >
< a id = "IsPreElgot.elgotalgebras" > < / a > < a id = "1366" href = "Monad.ElgotMonad.html#1366" class = "Field" > elgotalgebras< / a > < a id = "1380" class = "Symbol" > :< / a > < a id = "1382" class = "Symbol" > ∀< / a > < a id = "1384" class = "Symbol" > {< / a > < a id = "1385" href = "Monad.ElgotMonad.html#1385" class = "Bound" > X< / a > < a id = "1386" class = "Symbol" > }< / a > < a id = "1388" class = "Symbol" > →< / a > < a id = "1390" href = "ElgotAlgebra.html#2093" class = "Record" > Elgot-Algebra-on< / a > < a id = "1407" href = "Monad.ElgotMonad.html#785" class = "Bound" > ED< / a > < a id = "1410" class = "Symbol" > (< / a > < a id = "1411" href = "Monad.ElgotMonad.html#1266" class = "Function" > T₀< / a > < a id = "1414" href = "Monad.ElgotMonad.html#1385" class = "Bound" > X< / a > < a id = "1415" class = "Symbol" > )< / a >
< a id = "1422" class = "Keyword" > module< / a > < a id = "IsPreElgot.elgotalgebras" > < / a > < a id = "1429" href = "Monad.ElgotMonad.html#1429" class = "Module" > elgotalgebras< / a > < a id = "1443" class = "Symbol" > {< / a > < a id = "1444" href = "Monad.ElgotMonad.html#1444" class = "Bound" > X< / a > < a id = "1445" class = "Symbol" > }< / a > < a id = "1447" class = "Symbol" > =< / a > < a id = "1449" href = "ElgotAlgebra.html#2093" class = "Module" > Elgot-Algebra-on< / a > < a id = "1466" class = "Symbol" > (< / a > < a id = "1467" href = "Monad.ElgotMonad.html#1366" class = "Field" > elgotalgebras< / a > < a id = "1481" class = "Symbol" > {< / a > < a id = "1482" href = "Monad.ElgotMonad.html#1444" class = "Bound" > X< / a > < a id = "1483" class = "Symbol" > })< / a >
< a id = "1491" class = "Comment" > -- with the following associativity< / a >
< a id = "1531" class = "Keyword" > field< / a >
< a id = "IsPreElgot.assoc" > < / a > < a id = "1543" href = "Monad.ElgotMonad.html#1543" class = "Field" > assoc< / a > < a id = "1549" class = "Symbol" > :< / a > < a id = "1551" class = "Symbol" > ∀< / a > < a id = "1553" class = "Symbol" > {< / a > < a id = "1554" href = "Monad.ElgotMonad.html#1554" class = "Bound" > X< / a > < a id = "1556" href = "Monad.ElgotMonad.html#1556" class = "Bound" > Y< / a > < a id = "1558" href = "Monad.ElgotMonad.html#1558" class = "Bound" > Z< / a > < a id = "1559" class = "Symbol" > }< / a > < a id = "1561" class = "Symbol" > (< / a > < a id = "1562" href = "Monad.ElgotMonad.html#1562" class = "Bound" > f< / a > < a id = "1564" class = "Symbol" > :< / a > < a id = "1566" href = "Monad.ElgotMonad.html#1558" class = "Bound" > Z< / a > < a id = "1568" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "1570" href = "Monad.ElgotMonad.html#1266" class = "Function" > T₀< / a > < a id = "1573" href = "Monad.ElgotMonad.html#1554" class = "Bound" > X< / a > < a id = "1575" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "1577" href = "Monad.ElgotMonad.html#1558" class = "Bound" > Z< / a > < a id = "1578" class = "Symbol" > )< / a > < a id = "1580" class = "Symbol" > (< / a > < a id = "1581" href = "Monad.ElgotMonad.html#1581" class = "Bound" > h< / a > < a id = "1583" class = "Symbol" > :< / a > < a id = "1585" href = "Monad.ElgotMonad.html#1554" class = "Bound" > X< / a > < a id = "1587" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "1589" href = "Monad.ElgotMonad.html#1266" class = "Function" > T₀< / a > < a id = "1592" href = "Monad.ElgotMonad.html#1556" class = "Bound" > Y< / a > < a id = "1593" class = "Symbol" > )< / a >
< a id = "1604" class = "Symbol" > →< / a > < a id = "1606" href = "ElgotAlgebra.html#2186" class = "Function Operator" > elgotalgebras._#< / a > < a id = "1623" class = "Symbol" > (((< / a > < a id = "1626" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "1630" class = "Symbol" > _< / a > < a id = "1632" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "1634" href = "Monad.ElgotMonad.html#1276" class = "Function" > T₁< / a > < a id = "1637" href = "Monad.ElgotMonad.html#1581" class = "Bound" > h< / a > < a id = "1638" class = "Symbol" > )< / a > < a id = "1640" href = "Categories.Category.Cocartesian.html#2133" class = "Function Operator" > +₁< / a > < a id = "1643" href = "Monad.ElgotMonad.html#897" class = "Function" > idC< / a > < a id = "1646" class = "Symbol" > )< / a > < a id = "1648" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "1650" href = "Monad.ElgotMonad.html#1562" class = "Bound" > f< / a > < a id = "1651" class = "Symbol" > )< / a > < a id = "1653" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "1655" class = "Symbol" > (< / a > < a id = "1656" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "1660" class = "Symbol" > _< / a > < a id = "1662" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "1664" href = "Monad.ElgotMonad.html#1276" class = "Function" > T₁< / a > < a id = "1667" href = "Monad.ElgotMonad.html#1581" class = "Bound" > h< / a > < a id = "1668" class = "Symbol" > )< / a > < a id = "1670" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "1672" class = "Symbol" > (< / a > < a id = "1673" href = "ElgotAlgebra.html#2186" class = "Function Operator" > elgotalgebras._#< / a > < a id = "1690" class = "Symbol" > {< / a > < a id = "1691" href = "Monad.ElgotMonad.html#1554" class = "Bound" > X< / a > < a id = "1692" class = "Symbol" > })< / a > < a id = "1695" href = "Monad.ElgotMonad.html#1562" class = "Bound" > f< / a >
< a id = "1702" class = "Keyword" > record< / a > < a id = "PreElgotMonad" > < / a > < a id = "1709" href = "Monad.ElgotMonad.html#1709" class = "Record" > PreElgotMonad< / a > < a id = "1723" class = "Symbol" > :< / a > < a id = "1725" href = "Agda.Primitive.html#320" class = "Primitive" > Set< / a > < a id = "1729" class = "Symbol" > (< / a > < a id = "1730" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "1732" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1734" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "1736" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1738" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "1739" class = "Symbol" > )< / a > < a id = "1741" class = "Keyword" > where< / a >
< a id = "1751" class = "Keyword" > field< / a >
< a id = "PreElgotMonad.T" > < / a > < a id = "1763" href = "Monad.ElgotMonad.html#1763" class = "Field" > T< / a > < a id = "1765" class = "Symbol" > :< / a > < a id = "1767" href = "Categories.Monad.html#454" class = "Record" > Monad< / a > < a id = "1773" href = "Monad.ElgotMonad.html#888" class = "Field" > C< / a >
< a id = "PreElgotMonad.isPreElgot" > < / a > < a id = "1781" href = "Monad.ElgotMonad.html#1781" class = "Field" > isPreElgot< / a > < a id = "1792" class = "Symbol" > :< / a > < a id = "1794" href = "Monad.ElgotMonad.html#1165" class = "Record" > IsPreElgot< / a > < a id = "1805" href = "Monad.ElgotMonad.html#1763" class = "Field" > T< / a >
< a id = "1812" class = "Keyword" > open< / a > < a id = "1817" href = "Monad.ElgotMonad.html#1165" class = "Module" > IsPreElgot< / a > < a id = "1828" href = "Monad.ElgotMonad.html#1781" class = "Field" > isPreElgot< / a > < a id = "1839" class = "Keyword" > public< / a >
< / pre >
< h3 id = "definition-14-elgot-monads" > < em > Definition 14< / em > : Elgot
Monads< / h3 >
< pre class = "Agda" > < a id = "1895" class = "Keyword" > record< / a > < a id = "IsElgot" > < / a > < a id = "1902" href = "Monad.ElgotMonad.html#1902" class = "Record" > IsElgot< / a > < a id = "1910" class = "Symbol" > (< / a > < a id = "1911" href = "Monad.ElgotMonad.html#1911" class = "Bound" > T< / a > < a id = "1913" class = "Symbol" > :< / a > < a id = "1915" href = "Categories.Monad.html#454" class = "Record" > Monad< / a > < a id = "1921" href = "Monad.ElgotMonad.html#888" class = "Field" > C< / a > < a id = "1922" class = "Symbol" > )< / a > < a id = "1924" class = "Symbol" > :< / a > < a id = "1926" href = "Agda.Primitive.html#320" class = "Primitive" > Set< / a > < a id = "1930" class = "Symbol" > (< / a > < a id = "1931" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "1933" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1935" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "1937" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "1939" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "1940" class = "Symbol" > )< / a > < a id = "1942" class = "Keyword" > where< / a >
< a id = "1952" class = "Keyword" > open< / a > < a id = "1957" href = "Categories.Monad.html#454" class = "Module" > Monad< / a > < a id = "1963" href = "Monad.ElgotMonad.html#1911" class = "Bound" > T< / a >
< a id = "1969" class = "Keyword" > open< / a > < a id = "1974" href = "Categories.Functor.Core.html#248" class = "Module" > Functor< / a > < a id = "1982" href = "Categories.Monad.html#525" class = "Function" > F< / a > < a id = "1984" class = "Keyword" > renaming< / a > < a id = "1993" class = "Symbol" > (< / a > < a id = "1994" href = "Categories.Functor.Core.html#432" class = "Field" > F₀< / a > < a id = "1997" class = "Symbol" > to< / a > < a id = "2000" class = "Field" > T₀< / a > < a id = "2002" class = "Symbol" > ;< / a > < a id = "2004" href = "Categories.Functor.Core.html#455" class = "Field" > F₁< / a > < a id = "2007" class = "Symbol" > to< / a > < a id = "2010" class = "Field" > T₁< / a > < a id = "2012" class = "Symbol" > )< / a >
< a id = "2019" class = "Comment" > -- iteration operator< / a >
< a id = "2045" class = "Keyword" > field< / a >
< a id = "IsElgot._†" > < / a > < a id = "2057" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > _†< / a > < a id = "2060" class = "Symbol" > :< / a > < a id = "2062" class = "Symbol" > ∀< / a > < a id = "2064" class = "Symbol" > {< / a > < a id = "2065" href = "Monad.ElgotMonad.html#2065" class = "Bound" > X< / a > < a id = "2067" href = "Monad.ElgotMonad.html#2067" class = "Bound" > Y< / a > < a id = "2068" class = "Symbol" > }< / a > < a id = "2070" class = "Symbol" > →< / a > < a id = "2072" href = "Monad.ElgotMonad.html#2065" class = "Bound" > X< / a > < a id = "2074" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2076" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2079" class = "Symbol" > (< / a > < a id = "2080" href = "Monad.ElgotMonad.html#2067" class = "Bound" > Y< / a > < a id = "2082" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2084" href = "Monad.ElgotMonad.html#2065" class = "Bound" > X< / a > < a id = "2085" class = "Symbol" > )< / a > < a id = "2087" class = "Symbol" > →< / a > < a id = "2089" href = "Monad.ElgotMonad.html#2065" class = "Bound" > X< / a > < a id = "2091" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2093" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2096" href = "Monad.ElgotMonad.html#2067" class = "Bound" > Y< / a >
< a id = "IsElgot.†-resp-≈" > < / a > < a id = "2104" href = "Monad.ElgotMonad.html#2104" class = "Field" > †-resp-≈< / a > < a id = "2113" class = "Symbol" > :< / a > < a id = "2115" class = "Symbol" > ∀< / a > < a id = "2117" class = "Symbol" > {< / a > < a id = "2118" href = "Monad.ElgotMonad.html#2118" class = "Bound" > X< / a > < a id = "2120" href = "Monad.ElgotMonad.html#2120" class = "Bound" > Y< / a > < a id = "2121" class = "Symbol" > }< / a > < a id = "2123" class = "Symbol" > {< / a > < a id = "2124" href = "Monad.ElgotMonad.html#2124" class = "Bound" > f< / a > < a id = "2126" href = "Monad.ElgotMonad.html#2126" class = "Bound" > g< / a > < a id = "2128" class = "Symbol" > :< / a > < a id = "2130" href = "Monad.ElgotMonad.html#2118" class = "Bound" > X< / a > < a id = "2132" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2134" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2137" class = "Symbol" > (< / a > < a id = "2138" href = "Monad.ElgotMonad.html#2120" class = "Bound" > Y< / a > < a id = "2140" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2142" href = "Monad.ElgotMonad.html#2118" class = "Bound" > X< / a > < a id = "2143" class = "Symbol" > )}< / a > < a id = "2146" class = "Symbol" > →< / a > < a id = "2148" href = "Monad.ElgotMonad.html#2124" class = "Bound" > f< / a > < a id = "2150" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2152" href = "Monad.ElgotMonad.html#2126" class = "Bound" > g< / a > < a id = "2154" class = "Symbol" > →< / a > < a id = "2156" href = "Monad.ElgotMonad.html#2124" class = "Bound" > f< / a > < a id = "2158" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2160" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2162" href = "Monad.ElgotMonad.html#2126" class = "Bound" > g< / a > < a id = "2164" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a >
< a id = "2171" class = "Comment" > -- laws< / a >
< a id = "2183" class = "Keyword" > field< / a >
< a id = "IsElgot.Fixpoint" > < / a > < a id = "2195" href = "Monad.ElgotMonad.html#2195" class = "Field" > Fixpoint< / a > < a id = "2204" class = "Symbol" > :< / a > < a id = "2206" class = "Symbol" > ∀< / a > < a id = "2208" class = "Symbol" > {< / a > < a id = "2209" href = "Monad.ElgotMonad.html#2209" class = "Bound" > X< / a > < a id = "2211" href = "Monad.ElgotMonad.html#2211" class = "Bound" > Y< / a > < a id = "2212" class = "Symbol" > }< / a > < a id = "2214" class = "Symbol" > {< / a > < a id = "2215" href = "Monad.ElgotMonad.html#2215" class = "Bound" > f< / a > < a id = "2217" class = "Symbol" > :< / a > < a id = "2219" href = "Monad.ElgotMonad.html#2209" class = "Bound" > X< / a > < a id = "2221" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2223" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2226" class = "Symbol" > (< / a > < a id = "2227" href = "Monad.ElgotMonad.html#2211" class = "Bound" > Y< / a > < a id = "2229" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2231" href = "Monad.ElgotMonad.html#2209" class = "Bound" > X< / a > < a id = "2232" class = "Symbol" > )}< / a >
< a id = "2244" class = "Symbol" > →< / a > < a id = "2246" href = "Monad.ElgotMonad.html#2215" class = "Bound" > f< / a > < a id = "2248" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2250" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2252" class = "Symbol" > (< / a > < a id = "2253" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "2257" class = "Symbol" > _< / a > < a id = "2259" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2261" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2264" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "2266" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "2270" class = "Symbol" > _< / a > < a id = "2272" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "2274" href = "Monad.ElgotMonad.html#2215" class = "Bound" > f< / a > < a id = "2276" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2278" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "2279" class = "Symbol" > )< / a > < a id = "2281" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2283" href = "Monad.ElgotMonad.html#2215" class = "Bound" > f< / a >
< a id = "IsElgot.Naturality" > < / a > < a id = "2291" href = "Monad.ElgotMonad.html#2291" class = "Field" > Naturality< / a > < a id = "2302" class = "Symbol" > :< / a > < a id = "2304" class = "Symbol" > ∀< / a > < a id = "2306" class = "Symbol" > {< / a > < a id = "2307" href = "Monad.ElgotMonad.html#2307" class = "Bound" > X< / a > < a id = "2309" href = "Monad.ElgotMonad.html#2309" class = "Bound" > Y< / a > < a id = "2311" href = "Monad.ElgotMonad.html#2311" class = "Bound" > Z< / a > < a id = "2312" class = "Symbol" > }< / a > < a id = "2314" class = "Symbol" > {< / a > < a id = "2315" href = "Monad.ElgotMonad.html#2315" class = "Bound" > f< / a > < a id = "2317" class = "Symbol" > :< / a > < a id = "2319" href = "Monad.ElgotMonad.html#2307" class = "Bound" > X< / a > < a id = "2321" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2323" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2326" class = "Symbol" > (< / a > < a id = "2327" href = "Monad.ElgotMonad.html#2309" class = "Bound" > Y< / a > < a id = "2329" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2331" href = "Monad.ElgotMonad.html#2307" class = "Bound" > X< / a > < a id = "2332" class = "Symbol" > )}< / a > < a id = "2335" class = "Symbol" > {< / a > < a id = "2336" href = "Monad.ElgotMonad.html#2336" class = "Bound" > g< / a > < a id = "2338" class = "Symbol" > :< / a > < a id = "2340" href = "Monad.ElgotMonad.html#2309" class = "Bound" > Y< / a > < a id = "2342" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2344" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2347" href = "Monad.ElgotMonad.html#2311" class = "Bound" > Z< / a > < a id = "2348" class = "Symbol" > }< / a >
< a id = "2359" class = "Symbol" > →< / a > < a id = "2361" class = "Symbol" > (< / a > < a id = "2362" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "2366" class = "Symbol" > _< / a > < a id = "2368" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2370" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2373" href = "Monad.ElgotMonad.html#2336" class = "Bound" > g< / a > < a id = "2374" class = "Symbol" > )< / a > < a id = "2376" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2378" href = "Monad.ElgotMonad.html#2315" class = "Bound" > f< / a > < a id = "2380" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2382" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2384" class = "Symbol" > ((< / a > < a id = "2386" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "2390" class = "Symbol" > _< / a > < a id = "2392" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2394" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2397" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "2399" class = "Symbol" > (< / a > < a id = "2400" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2403" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "2405" class = "Symbol" > )< / a > < a id = "2407" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2409" href = "Monad.ElgotMonad.html#2336" class = "Bound" > g< / a > < a id = "2411" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "2413" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "2417" class = "Symbol" > _< / a > < a id = "2419" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2421" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "2424" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "2425" class = "Symbol" > )< / a > < a id = "2427" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2429" href = "Monad.ElgotMonad.html#2315" class = "Bound" > f< / a > < a id = "2430" class = "Symbol" > )< / a > < a id = "2431" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a >
< a id = "IsElgot.Codiagonal" > < / a > < a id = "2439" href = "Monad.ElgotMonad.html#2439" class = "Field" > Codiagonal< / a > < a id = "2450" class = "Symbol" > :< / a > < a id = "2452" class = "Symbol" > ∀< / a > < a id = "2454" class = "Symbol" > {< / a > < a id = "2455" href = "Monad.ElgotMonad.html#2455" class = "Bound" > X< / a > < a id = "2457" href = "Monad.ElgotMonad.html#2457" class = "Bound" > Y< / a > < a id = "2458" class = "Symbol" > }< / a > < a id = "2460" class = "Symbol" > {< / a > < a id = "2461" href = "Monad.ElgotMonad.html#2461" class = "Bound" > f< / a > < a id = "2463" class = "Symbol" > :< / a > < a id = "2465" href = "Monad.ElgotMonad.html#2455" class = "Bound" > X< / a > < a id = "2467" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2469" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2472" class = "Symbol" > ((< / a > < a id = "2474" href = "Monad.ElgotMonad.html#2457" class = "Bound" > Y< / a > < a id = "2476" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2478" href = "Monad.ElgotMonad.html#2455" class = "Bound" > X< / a > < a id = "2479" class = "Symbol" > )< / a > < a id = "2481" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2483" href = "Monad.ElgotMonad.html#2455" class = "Bound" > X< / a > < a id = "2484" class = "Symbol" > )}< / a >
< a id = "2496" class = "Symbol" > →< / a > < a id = "2498" class = "Symbol" > (< / a > < a id = "2499" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2502" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "2504" href = "Monad.ElgotMonad.html#897" class = "Function" > idC< / a > < a id = "2508" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "2510" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "2513" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "2515" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2517" href = "Monad.ElgotMonad.html#2461" class = "Bound" > f< / a > < a id = "2519" class = "Symbol" > )< / a > < a id = "2520" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2522" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2524" href = "Monad.ElgotMonad.html#2461" class = "Bound" > f< / a > < a id = "2526" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2528" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a >
< a id = "IsElgot.Uniformity" > < / a > < a id = "2536" href = "Monad.ElgotMonad.html#2536" class = "Field" > Uniformity< / a > < a id = "2547" class = "Symbol" > :< / a > < a id = "2549" class = "Symbol" > ∀< / a > < a id = "2551" class = "Symbol" > {< / a > < a id = "2552" href = "Monad.ElgotMonad.html#2552" class = "Bound" > X< / a > < a id = "2554" href = "Monad.ElgotMonad.html#2554" class = "Bound" > Y< / a > < a id = "2556" href = "Monad.ElgotMonad.html#2556" class = "Bound" > Z< / a > < a id = "2557" class = "Symbol" > }< / a > < a id = "2559" class = "Symbol" > {< / a > < a id = "2560" href = "Monad.ElgotMonad.html#2560" class = "Bound" > f< / a > < a id = "2562" class = "Symbol" > :< / a > < a id = "2564" href = "Monad.ElgotMonad.html#2552" class = "Bound" > X< / a > < a id = "2566" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2568" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2571" class = "Symbol" > (< / a > < a id = "2572" href = "Monad.ElgotMonad.html#2554" class = "Bound" > Y< / a > < a id = "2574" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2576" href = "Monad.ElgotMonad.html#2552" class = "Bound" > X< / a > < a id = "2577" class = "Symbol" > )}< / a > < a id = "2580" class = "Symbol" > {< / a > < a id = "2581" href = "Monad.ElgotMonad.html#2581" class = "Bound" > g< / a > < a id = "2583" class = "Symbol" > :< / a > < a id = "2585" href = "Monad.ElgotMonad.html#2556" class = "Bound" > Z< / a > < a id = "2587" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2589" href = "Monad.ElgotMonad.html#2000" class = "Function" > T₀< / a > < a id = "2592" class = "Symbol" > (< / a > < a id = "2593" href = "Monad.ElgotMonad.html#2554" class = "Bound" > Y< / a > < a id = "2595" href = "Categories.Category.Cocartesian.html#1549" class = "Function Operator" > +< / a > < a id = "2597" href = "Monad.ElgotMonad.html#2556" class = "Bound" > Z< / a > < a id = "2598" class = "Symbol" > )}< / a > < a id = "2601" class = "Symbol" > {< / a > < a id = "2602" href = "Monad.ElgotMonad.html#2602" class = "Bound" > h< / a > < a id = "2604" class = "Symbol" > :< / a > < a id = "2606" href = "Monad.ElgotMonad.html#2556" class = "Bound" > Z< / a > < a id = "2608" href = "Categories.Category.Core.html#575" class = "Function Operator" > ⇒< / a > < a id = "2610" href = "Monad.ElgotMonad.html#2552" class = "Bound" > X< / a > < a id = "2611" class = "Symbol" > }< / a >
< a id = "2622" class = "Symbol" > →< / a > < a id = "2624" href = "Monad.ElgotMonad.html#2560" class = "Bound" > f< / a > < a id = "2626" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2628" href = "Monad.ElgotMonad.html#2602" class = "Bound" > h< / a > < a id = "2630" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2632" class = "Symbol" > (< / a > < a id = "2633" href = "Monad.ElgotMonad.html#2010" class = "Function" > T₁< / a > < a id = "2636" class = "Symbol" > (< / a > < a id = "2637" href = "Monad.ElgotMonad.html#897" class = "Function" > idC< / a > < a id = "2641" href = "Categories.Category.Cocartesian.html#2133" class = "Function Operator" > +₁< / a > < a id = "2644" href = "Monad.ElgotMonad.html#2602" class = "Bound" > h< / a > < a id = "2645" class = "Symbol" > ))< / a > < a id = "2648" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2650" href = "Monad.ElgotMonad.html#2581" class = "Bound" > g< / a > < a id = "2652" class = "Symbol" > →< / a > < a id = "2654" href = "Monad.ElgotMonad.html#2560" class = "Bound" > f< / a > < a id = "2656" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a > < a id = "2658" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "2660" href = "Monad.ElgotMonad.html#2602" class = "Bound" > h< / a > < a id = "2662" href = "Categories.Category.Core.html#595" class = "Function Operator" > ≈< / a > < a id = "2664" href = "Monad.ElgotMonad.html#2581" class = "Bound" > g< / a > < a id = "2666" href = "Monad.ElgotMonad.html#2057" class = "Field Operator" > †< / a >
< a id = "2671" class = "Keyword" > record< / a > < a id = "ElgotMonad" > < / a > < a id = "2678" href = "Monad.ElgotMonad.html#2678" class = "Record" > ElgotMonad< / a > < a id = "2689" class = "Symbol" > :< / a > < a id = "2691" href = "Agda.Primitive.html#320" class = "Primitive" > Set< / a > < a id = "2695" class = "Symbol" > (< / a > < a id = "2696" href = "Monad.ElgotMonad.html#777" class = "Bound" > o< / a > < a id = "2698" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "2700" href = "Monad.ElgotMonad.html#779" class = "Bound" > ℓ < / a > < a id = "2702" href = "Agda.Primitive.html#804" class = "Primitive Operator" > ⊔< / a > < a id = "2704" href = "Monad.ElgotMonad.html#781" class = "Bound" > e< / a > < a id = "2705" class = "Symbol" > )< / a > < a id = "2707" class = "Keyword" > where< / a >
< a id = "2717" class = "Keyword" > field< / a >
< a id = "ElgotMonad.T" > < / a > < a id = "2729" href = "Monad.ElgotMonad.html#2729" class = "Field" > T< / a > < a id = "2731" class = "Symbol" > :< / a > < a id = "2733" href = "Categories.Monad.html#454" class = "Record" > Monad< / a > < a id = "2739" href = "Monad.ElgotMonad.html#888" class = "Field" > C< / a >
< a id = "ElgotMonad.isElgot" > < / a > < a id = "2747" href = "Monad.ElgotMonad.html#2747" class = "Field" > isElgot< / a > < a id = "2755" class = "Symbol" > :< / a > < a id = "2757" href = "Monad.ElgotMonad.html#1902" class = "Record" > IsElgot< / a > < a id = "2765" href = "Monad.ElgotMonad.html#2729" class = "Field" > T< / a >
< a id = "2772" class = "Keyword" > open< / a > < a id = "2777" href = "Monad.ElgotMonad.html#1902" class = "Module" > IsElgot< / a > < a id = "2785" href = "Monad.ElgotMonad.html#2747" class = "Field" > isElgot< / a > < a id = "2793" class = "Keyword" > public< / a >
< / pre >
< h3
id="proposition-15-strong-elgot-monads-are-strong-pre-elgot">< em > Proposition
15< / em > : (Strong) Elgot monads are (strong) pre-Elgot< / h3 >
< pre class = "Agda" > < a id = "2883" class = "Comment" > -- elgot monads are pre-elgot< / a >
< a id = "Elgot⇒PreElgot" > < / a > < a id = "2915" href = "Monad.ElgotMonad.html#2915" class = "Function" > Elgot⇒PreElgot< / a > < a id = "2930" class = "Symbol" > :< / a > < a id = "2932" href = "Monad.ElgotMonad.html#2678" class = "Record" > ElgotMonad< / a > < a id = "2943" class = "Symbol" > →< / a > < a id = "2945" href = "Monad.ElgotMonad.html#1709" class = "Record" > PreElgotMonad< / a >
< a id = "2961" href = "Monad.ElgotMonad.html#2915" class = "Function" > Elgot⇒PreElgot< / a > < a id = "2976" href = "Monad.ElgotMonad.html#2976" class = "Bound" > EM< / a > < a id = "2979" class = "Symbol" > =< / a > < a id = "2981" class = "Keyword" > record< / a >
< a id = "2993" class = "Symbol" > {< / a > < a id = "2995" href = "Monad.ElgotMonad.html#1763" class = "Field" > T< / a > < a id = "2997" class = "Symbol" > =< / a > < a id = "2999" href = "Monad.ElgotMonad.html#2729" class = "Field" > T< / a >
< a id = "3006" class = "Symbol" > ;< / a > < a id = "3008" href = "Monad.ElgotMonad.html#1781" class = "Field" > isPreElgot< / a > < a id = "3019" class = "Symbol" > =< / a > < a id = "3021" class = "Keyword" > record< / a >
< a id = "3035" class = "Symbol" > {< / a > < a id = "3037" href = "Monad.ElgotMonad.html#1366" class = "Field" > elgotalgebras< / a > < a id = "3051" class = "Symbol" > =< / a > < a id = "3053" class = "Symbol" > λ< / a > < a id = "3055" class = "Symbol" > {< / a > < a id = "3056" href = "Monad.ElgotMonad.html#3056" class = "Bound" > X< / a > < a id = "3057" class = "Symbol" > }< / a > < a id = "3059" class = "Symbol" > →< / a > < a id = "3061" class = "Keyword" > record< / a >
< a id = "3076" class = "Symbol" > {< / a > < a id = "3078" href = "ElgotAlgebra.html#2186" class = "Field Operator" > _#< / a > < a id = "3081" class = "Symbol" > =< / a > < a id = "3083" class = "Symbol" > λ< / a > < a id = "3085" href = "Monad.ElgotMonad.html#3085" class = "Bound" > f< / a > < a id = "3087" class = "Symbol" > →< / a > < a id = "3089" class = "Symbol" > (< / a > < a id = "3090" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3092" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3095" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "3098" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3100" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3104" class = "Symbol" > _< / a > < a id = "3106" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3108" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "3111" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3113" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3115" href = "Monad.ElgotMonad.html#3085" class = "Bound" > f< / a > < a id = "3116" class = "Symbol" > )< / a > < a id = "3118" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a >
< a id = "3128" class = "Symbol" > ;< / a > < a id = "3130" href = "ElgotAlgebra.html#2259" class = "Field" > #-Fixpoint< / a > < a id = "3141" class = "Symbol" > =< / a > < a id = "3143" class = "Symbol" > λ< / a > < a id = "3145" class = "Symbol" > {< / a > < a id = "3146" href = "Monad.ElgotMonad.html#3146" class = "Bound" > Y< / a > < a id = "3147" class = "Symbol" > }< / a > < a id = "3149" class = "Symbol" > {< / a > < a id = "3150" href = "Monad.ElgotMonad.html#3150" class = "Bound" > f< / a > < a id = "3151" class = "Symbol" > }< / a > < a id = "3153" class = "Symbol" > →< / a > < a id = "3155" href = "Relation.Binary.Reasoning.Base.Single.html#1925" class = "Function Operator" > begin< / a >
< a id = "3172" class = "Symbol" > (< / a > < a id = "3173" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3175" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3178" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "3181" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3183" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3187" class = "Symbol" > _< / a > < a id = "3189" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3191" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "3194" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3196" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3198" href = "Monad.ElgotMonad.html#3150" class = "Bound" > f< / a > < a id = "3199" class = "Symbol" > )< / a > < a id = "3201" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "3259" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ≈⟨< / a > < a id = "3262" href = "Monad.ElgotMonad.html#2195" class = "Function" > Fixpoint< / a > < a id = "3271" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ⟩< / a >
< a id = "3284" class = "Symbol" > (< / a > < a id = "3285" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "3289" class = "Symbol" > _< / a > < a id = "3291" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3293" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3296" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3298" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3302" class = "Symbol" > _< / a > < a id = "3304" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3306" class = "Symbol" > (< / a > < a id = "3307" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3309" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3312" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "3315" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3317" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3321" class = "Symbol" > _< / a > < a id = "3323" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3325" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "3328" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3330" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3332" href = "Monad.ElgotMonad.html#3150" class = "Bound" > f< / a > < a id = "3333" class = "Symbol" > )< / a > < a id = "3335" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "3337" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3338" class = "Symbol" > )< / a > < a id = "3340" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3342" class = "Symbol" > (< / a > < a id = "3343" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3345" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3348" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "3351" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3353" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3357" class = "Symbol" > _< / a > < a id = "3359" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3361" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "3364" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3366" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3368" href = "Monad.ElgotMonad.html#3150" class = "Bound" > f< / a > < a id = "3369" class = "Symbol" > )< / a > < a id = "3371" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ≈⟨< / a > < a id = "3374" href = "Categories.Morphism.Reasoning.Core.html#2048" class = "Function" > pullˡ< / a > < a id = "3380" href = "Categories.Category.Cocartesian.html#2736" class = "Function" > ∘[]< / a > < a id = "3384" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ⟩< / a >
< a id = "3396" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3398" class = "Symbol" > (< / a > < a id = "3399" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "3403" class = "Symbol" > _< / a > < a id = "3405" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3407" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3410" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3412" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3416" class = "Symbol" > _< / a > < a id = "3418" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3420" class = "Symbol" > (< / a > < a id = "3421" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3423" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3426" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "3429" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3431" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3435" class = "Symbol" > _< / a > < a id = "3437" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3439" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "3442" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3444" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3446" href = "Monad.ElgotMonad.html#3150" class = "Bound" > f< / a > < a id = "3447" class = "Symbol" > )< / a > < a id = "3449" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "3451" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "3452" class = "Symbol" > )< / a > < a id = "3454" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3456" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3459" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a >
< a id = "3473" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "3475" class = "Symbol" > (< / a > < a id = "3476" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "3480" class = "Symbol" > _< / a > < a id = "3482" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "3484" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "3487" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "3489" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "3493" class = "Symbol" > _< / a >
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< a id = "3707" class = "Symbol" > (< / a > < a id = "3708" href = "Categories.Morphism.Reasoning.Core.html#2347" class = "Function" > pushʳ< / a > < a id = "3714" class = "Symbol" > (< / a > < a id = "3715" href = "Categories.Functor.Core.html#565" class = "Function" > homomorphism< / a > < a id = "3727" class = "Symbol" > ))< / a >
< a id = "3831" class = "Symbol" > (< / a > < a id = "3832" href = "Categories.Morphism.Reasoning.Core.html#2485" class = "Function" > pushˡ< / a > < a id = "3838" class = "Symbol" > (< / a > < a id = "3839" href = "Categories.Morphism.Reasoning.Core.html#2347" class = "Function" > pushʳ< / a > < a id = "3845" class = "Symbol" > (< / a > < a id = "3846" href = "Categories.NaturalTransformation.Core.html#827" class = "Function" > η.commute< / a > < a id = "3856" class = "Symbol" > _)))< / a >
< a id = "3959" href = "Categories.Category.Core.html#2837" class = "Function Operator" > ⟩∘⟨refl< / a > < a id = "3967" href = "Relation.Binary.Reasoning.Setoid.html#1162" class = "Function" > ⟩< / a >
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< a id = "5881" class = "Symbol" > ;< / a > < a id = "5883" href = "ElgotAlgebra.html#2582" class = "Field" > #-resp-≈< / a > < a id = "5892" class = "Symbol" > =< / a > < a id = "5894" class = "Symbol" > λ< / a > < a id = "5896" href = "Monad.ElgotMonad.html#5896" class = "Bound" > fg< / a > < a id = "5899" class = "Symbol" > →< / a > < a id = "5901" href = "Monad.ElgotMonad.html#2104" class = "Function" > †-resp-≈< / a > < a id = "5910" class = "Symbol" > (< / a > < a id = "5911" href = "Categories.Category.Core.html#1706" class = "Function" > ∘-resp-≈ʳ< / a > < a id = "5921" href = "Monad.ElgotMonad.html#5896" class = "Bound" > fg< / a > < a id = "5923" class = "Symbol" > )< / a >
< a id = "5933" class = "Symbol" > }< / a >
< a id = "5941" class = "Symbol" > ;< / a > < a id = "5943" href = "Monad.ElgotMonad.html#1543" class = "Field" > assoc< / a > < a id = "5949" class = "Symbol" > =< / a > < a id = "5951" class = "Symbol" > λ< / a > < a id = "5953" class = "Symbol" > {< / a > < a id = "5954" href = "Monad.ElgotMonad.html#5954" class = "Bound" > X< / a > < a id = "5955" class = "Symbol" > }< / a > < a id = "5957" class = "Symbol" > {< / a > < a id = "5958" href = "Monad.ElgotMonad.html#5958" class = "Bound" > Y< / a > < a id = "5959" class = "Symbol" > }< / a > < a id = "5961" class = "Symbol" > {< / a > < a id = "5962" href = "Monad.ElgotMonad.html#5962" class = "Bound" > Z< / a > < a id = "5963" class = "Symbol" > }< / a > < a id = "5965" href = "Monad.ElgotMonad.html#5965" class = "Bound" > f< / a > < a id = "5967" href = "Monad.ElgotMonad.html#5967" class = "Bound" > h< / a > < a id = "5969" class = "Symbol" > →< / a > < a id = "5971" href = "Relation.Binary.Reasoning.Base.Single.html#1925" class = "Function Operator" > begin< / a >
< a id = "5986" class = "Comment" > -- TODO tidy up by moving doing sym outside, apply Naturality and then do `†-resp-≈ pullˡ` once.< / a >
< a id = "6091" class = "Symbol" > (< / a > < a id = "6092" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "6094" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6097" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "6100" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "6102" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "6106" class = "Symbol" > _< / a > < a id = "6108" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6110" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "6113" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "6115" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6117" class = "Symbol" > (< / a > < a id = "6118" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "6122" href = "Monad.ElgotMonad.html#5958" class = "Bound" > Y< / a > < a id = "6124" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6126" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6129" href = "Monad.ElgotMonad.html#5967" class = "Bound" > h< / a > < a id = "6131" href = "Categories.Category.Cocartesian.html#2133" class = "Function Operator" > +₁< / a > < a id = "6134" href = "Monad.ElgotMonad.html#897" class = "Function" > idC< / a > < a id = "6137" class = "Symbol" > )< / a > < a id = "6139" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6141" href = "Monad.ElgotMonad.html#5965" class = "Bound" > f< / a > < a id = "6142" class = "Symbol" > )< / a > < a id = "6143" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "6205" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ≈⟨< / a > < a id = "6208" href = "Monad.ElgotMonad.html#2104" class = "Function" > †-resp-≈< / a > < a id = "6217" class = "Symbol" > (< / a > < a id = "6218" href = "Categories.Morphism.Reasoning.Core.html#2048" class = "Function" > pullˡ< / a > < a id = "6224" href = "Categories.Category.Cocartesian.html#2662" class = "Function" > []∘+₁< / a > < a id = "6229" class = "Symbol" > )< / a > < a id = "6231" href = "Relation.Binary.Reasoning.Setoid.html#1061" class = "Function" > ⟩< / a >
< a id = "6242" class = "Symbol" > ((< / a > < a id = "6244" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "6246" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6249" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "6252" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6254" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "6258" class = "Symbol" > _< / a > < a id = "6260" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6262" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6265" href = "Monad.ElgotMonad.html#5967" class = "Bound" > h< / a > < a id = "6267" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "6269" class = "Symbol" > (< / a > < a id = "6270" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "6274" class = "Symbol" > _< / a > < a id = "6276" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6278" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "6280" class = "Symbol" > )< / a > < a id = "6282" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6284" href = "Monad.ElgotMonad.html#897" class = "Function" > idC< / a > < a id = "6288" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "6290" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6292" href = "Monad.ElgotMonad.html#5965" class = "Bound" > f< / a > < a id = "6293" class = "Symbol" > )< / a > < a id = "6294" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "6295" class = "Symbol" > )< / a > < a id = "6356" href = "Relation.Binary.Reasoning.Setoid.html#1162" class = "Function" > ≈˘⟨< / a > < a id = "6360" href = "Monad.ElgotMonad.html#2104" class = "Function" > †-resp-≈< / a > < a id = "6369" class = "Symbol" > (< / a > < a id = "6370" href = "Categories.Category.Core.html#1597" class = "Function" > ∘-resp-≈ˡ< / a > < a id = "6380" class = "Symbol" > (< / a > < a id = "6381" href = "Categories.Object.Coproduct.html#816" class = "Function" > []-cong₂< / a > < a id = "6390" href = "Categories.Category.Core.html#715" class = "Function" > assoc< / a > < a id = "6396" class = "Symbol" > (< / a > < a id = "6397" href = "Relation.Binary.Structures.html#1603" class = "Function" > sym< / a > < a id = "6401" href = "Categories.Category.Core.html#1145" class = "Function" > identityʳ< / a > < a id = "6410" class = "Symbol" > )))< / a > < a id = "6414" href = "Relation.Binary.Reasoning.Setoid.html#1162" class = "Function" > ⟩< / a >
< a id = "6424" class = "Symbol" > (< / a > < a id = "6425" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > [< / a > < a id = "6427" class = "Symbol" > (< / a > < a id = "6428" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6431" href = "Categories.Object.Coproduct.html#492" class = "Function" > i₁< / a > < a id = "6434" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6436" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > μ.η< / a > < a id = "6440" class = "Symbol" > _)< / a > < a id = "6443" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6445" href = "Monad.ElgotMonad.html#7809" class = "Function" > T₁< / a > < a id = "6448" href = "Monad.ElgotMonad.html#5967" class = "Bound" > h< / a > < a id = "6450" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ,< / a > < a id = "6452" href = "Categories.NaturalTransformation.Core.html#783" class = "Function" > η.η< / a > < a id = "6456" class = "Symbol" > _< / a > < a id = "6458" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6460" href = "Categories.Object.Coproduct.html#512" class = "Function" > i₂< / a > < a id = "6463" href = "Categories.Object.Coproduct.html#532" class = "Function Operator" > ]< / a > < a id = "6465" href = "Categories.Category.Core.html#656" class = "Function Operator" > ∘< / a > < a id = "6467" href = "Monad.ElgotMonad.html#5965" class = "Bound" > f< / a > < a id = "6468" class = "Symbol" > )< / a > < a id = "6469" href = "Monad.ElgotMonad.html#2057" class = "Function Operator" > †< / a > < a id = "6538" href = "Relation.Binary.Reasoning.Setoid.html#1162" class = "Function" > ≈˘⟨< / a > < a id = "6542" href = "Monad.ElgotMonad.html#2104" class = "Function" > †-resp-≈< / a > < a id = "6551" class = "Symbol" > (< / a > < a id = "6552" href = "Categories.Category.Core.html#1597" class = "Function" > ∘-resp-≈ˡ< / a > < a id = "6562" class = "Symbol" > (< / a > < a id = "6563" href = "Categories.Category.Cocartesian.html#2271" class = "Function" > []-congʳ< / a > < a id = "6572" class = "Symbol" > (< / a > < a id = "6573" href = "Categories.Morphism.Reasoning.Core.html#2048" class = "Function" > pullˡ< / a > < a id = "6579" class = "Symbol" > (< / a > < a id = "6580" href = "Categories.NaturalTransformation.Core.html#827" class = "Function" > μ.commute< / a > < a id = "6590" class = "Symbol" > _))))< / a > < a id = "6596" href = "Relation.Binary.Reasoning.Setoid.html#1162" class = "Function" > ⟩< / a >
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