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18 KiB
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<html><head><meta charset="utf-8"><title>Categories.Diagram.Wedge</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
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<a id="37" class="Keyword">open</a> <a id="42" class="Keyword">import</a> <a id="49" href="Categories.Category.html" class="Module">Categories.Category</a>
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<a id="69" class="Keyword">open</a> <a id="74" class="Keyword">import</a> <a id="81" href="Categories.Functor.Bifunctor.html" class="Module">Categories.Functor.Bifunctor</a>
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<a id="111" class="Keyword">module</a> <a id="118" href="Categories.Diagram.Wedge.html" class="Module">Categories.Diagram.Wedge</a> <a id="143" class="Symbol">{</a><a id="144" href="Categories.Diagram.Wedge.html#144" class="Bound">o</a> <a id="146" href="Categories.Diagram.Wedge.html#146" class="Bound">ℓ</a> <a id="148" href="Categories.Diagram.Wedge.html#148" class="Bound">e</a> <a id="150" href="Categories.Diagram.Wedge.html#150" class="Bound">o′</a> <a id="153" href="Categories.Diagram.Wedge.html#153" class="Bound">ℓ′</a> <a id="156" href="Categories.Diagram.Wedge.html#156" class="Bound">e′</a><a id="158" class="Symbol">}</a> <a id="160" class="Symbol">{</a><a id="161" href="Categories.Diagram.Wedge.html#161" class="Bound">C</a> <a id="163" class="Symbol">:</a> <a id="165" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="174" href="Categories.Diagram.Wedge.html#144" class="Bound">o</a> <a id="176" href="Categories.Diagram.Wedge.html#146" class="Bound">ℓ</a> <a id="178" href="Categories.Diagram.Wedge.html#148" class="Bound">e</a><a id="179" class="Symbol">}</a> <a id="181" class="Symbol">{</a><a id="182" href="Categories.Diagram.Wedge.html#182" class="Bound">D</a> <a id="184" class="Symbol">:</a> <a id="186" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="195" href="Categories.Diagram.Wedge.html#150" class="Bound">o′</a> <a id="198" href="Categories.Diagram.Wedge.html#153" class="Bound">ℓ′</a> <a id="201" href="Categories.Diagram.Wedge.html#156" class="Bound">e′</a><a id="203" class="Symbol">}</a>
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<a id="207" class="Symbol">(</a><a id="208" href="Categories.Diagram.Wedge.html#208" class="Bound">F</a> <a id="210" class="Symbol">:</a> <a id="212" href="Categories.Functor.Bifunctor.html#441" class="Function">Bifunctor</a> <a id="222" class="Symbol">(</a><a id="223" href="Categories.Category.Core.html#3132" class="Function">Category.op</a> <a id="235" href="Categories.Diagram.Wedge.html#161" class="Bound">C</a><a id="236" class="Symbol">)</a> <a id="238" href="Categories.Diagram.Wedge.html#161" class="Bound">C</a> <a id="240" href="Categories.Diagram.Wedge.html#182" class="Bound">D</a><a id="241" class="Symbol">)</a> <a id="243" class="Keyword">where</a>
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<a id="250" class="Keyword">private</a>
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<a id="260" class="Keyword">module</a> <a id="C"></a><a id="267" href="Categories.Diagram.Wedge.html#267" class="Module">C</a> <a id="269" class="Symbol">=</a> <a id="271" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="280" href="Categories.Diagram.Wedge.html#161" class="Bound">C</a>
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<a id="284" class="Keyword">module</a> <a id="D"></a><a id="291" href="Categories.Diagram.Wedge.html#291" class="Module">D</a> <a id="293" class="Symbol">=</a> <a id="295" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="304" href="Categories.Diagram.Wedge.html#182" class="Bound">D</a>
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<a id="308" class="Keyword">open</a> <a id="313" href="Categories.Diagram.Wedge.html#291" class="Module">D</a>
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<a id="317" class="Keyword">open</a> <a id="322" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="337" class="Keyword">variable</a>
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<a id="350" href="Categories.Diagram.Wedge.html#350" class="Generalizable">A</a> <a id="352" class="Symbol">:</a> <a id="354" href="Categories.Category.Core.html#559" class="Function">Obj</a>
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<a id="359" class="Keyword">open</a> <a id="364" class="Keyword">import</a> <a id="371" href="Level.html" class="Module">Level</a>
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<a id="378" class="Keyword">open</a> <a id="383" class="Keyword">import</a> <a id="390" href="Categories.Functor.html" class="Module">Categories.Functor</a> <a id="409" class="Keyword">hiding</a> <a id="416" class="Symbol">(</a><a id="417" href="Categories.Functor.html#349" class="Function">id</a><a id="419" class="Symbol">)</a>
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<a id="421" class="Keyword">open</a> <a id="426" class="Keyword">import</a> <a id="433" href="Categories.Functor.Construction.Constant.html" class="Module">Categories.Functor.Construction.Constant</a>
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<a id="474" class="Keyword">open</a> <a id="479" class="Keyword">import</a> <a id="486" href="Categories.NaturalTransformation.Dinatural.html" class="Module">Categories.NaturalTransformation.Dinatural</a>
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<a id="530" class="Keyword">open</a> <a id="535" href="Categories.Functor.Core.html#248" class="Module">Functor</a> <a id="543" href="Categories.Diagram.Wedge.html#208" class="Bound">F</a>
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<a id="546" class="Keyword">record</a> <a id="Wedge"></a><a id="553" href="Categories.Diagram.Wedge.html#553" class="Record">Wedge</a> <a id="559" class="Symbol">:</a> <a id="561" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="565" class="Symbol">(</a><a id="566" href="Level.html#602" class="Function">levelOfTerm</a> <a id="578" href="Categories.Diagram.Wedge.html#208" class="Bound">F</a><a id="579" class="Symbol">)</a> <a id="581" class="Keyword">where</a>
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<a id="589" class="Keyword">field</a>
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<a id="Wedge.E"></a><a id="599" href="Categories.Diagram.Wedge.html#599" class="Field">E</a> <a id="609" class="Symbol">:</a> <a id="611" href="Categories.Category.Core.html#559" class="Function">Obj</a>
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<a id="Wedge.dinatural"></a><a id="619" href="Categories.Diagram.Wedge.html#619" class="Field">dinatural</a> <a id="629" class="Symbol">:</a> <a id="631" href="Categories.NaturalTransformation.Dinatural.html#578" class="Record">DinaturalTransformation</a> <a id="655" class="Symbol">(</a><a id="656" href="Categories.Functor.Construction.Constant.html#461" class="Function">const</a> <a id="662" href="Categories.Diagram.Wedge.html#599" class="Field">E</a><a id="663" class="Symbol">)</a> <a id="665" href="Categories.Diagram.Wedge.html#208" class="Bound">F</a>
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<a id="670" class="Keyword">module</a> <a id="Wedge.dinatural"></a><a id="677" href="Categories.Diagram.Wedge.html#677" class="Module">dinatural</a> <a id="687" class="Symbol">=</a> <a id="689" href="Categories.NaturalTransformation.Dinatural.html#578" class="Module">DinaturalTransformation</a> <a id="713" href="Categories.Diagram.Wedge.html#619" class="Field">dinatural</a>
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<a id="Wedge-∘"></a><a id="724" href="Categories.Diagram.Wedge.html#724" class="Function">Wedge-∘</a> <a id="732" class="Symbol">:</a> <a id="734" class="Symbol">(</a><a id="735" href="Categories.Diagram.Wedge.html#735" class="Bound">W</a> <a id="737" class="Symbol">:</a> <a id="739" href="Categories.Diagram.Wedge.html#553" class="Record">Wedge</a><a id="744" class="Symbol">)</a> <a id="746" class="Symbol">→</a> <a id="748" href="Categories.Diagram.Wedge.html#350" class="Generalizable">A</a> <a id="750" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="752" href="Categories.Diagram.Wedge.html#599" class="Field">Wedge.E</a> <a id="760" href="Categories.Diagram.Wedge.html#735" class="Bound">W</a> <a id="762" class="Symbol">→</a> <a id="764" href="Categories.Diagram.Wedge.html#553" class="Record">Wedge</a>
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<a id="770" href="Categories.Diagram.Wedge.html#724" class="Function">Wedge-∘</a> <a id="778" class="Symbol">{</a><a id="779" class="Argument">A</a> <a id="781" class="Symbol">=</a> <a id="783" href="Categories.Diagram.Wedge.html#783" class="Bound">A</a><a id="784" class="Symbol">}</a> <a id="786" href="Categories.Diagram.Wedge.html#786" class="Bound">W</a> <a id="788" href="Categories.Diagram.Wedge.html#788" class="Bound">f</a> <a id="790" class="Symbol">=</a> <a id="792" class="Keyword">record</a>
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<a id="801" class="Symbol">{</a> <a id="803" href="Categories.Diagram.Wedge.html#599" class="Field">E</a> <a id="813" class="Symbol">=</a> <a id="815" href="Categories.Diagram.Wedge.html#783" class="Bound">A</a>
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<a id="819" class="Symbol">;</a> <a id="821" href="Categories.Diagram.Wedge.html#619" class="Field">dinatural</a> <a id="831" class="Symbol">=</a> <a id="833" href="Categories.NaturalTransformation.Dinatural.html#7125" class="Function">extranaturalʳ</a> <a id="847" class="Symbol">(λ</a> <a id="850" href="Categories.Diagram.Wedge.html#850" class="Bound">X</a> <a id="852" class="Symbol">→</a> <a id="854" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">dinatural.α</a> <a id="866" href="Categories.Diagram.Wedge.html#850" class="Bound">X</a> <a id="868" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="870" href="Categories.Diagram.Wedge.html#788" class="Bound">f</a><a id="871" class="Symbol">)</a>
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<a id="903" class="Symbol">(</a><a id="904" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a> <a id="914" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="916" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="926" class="Symbol">(</a><a id="927" href="Categories.NaturalTransformation.Dinatural.html#7459" class="Function">extranatural-commʳ</a> <a id="946" href="Categories.Diagram.Wedge.html#619" class="Function">dinatural</a><a id="955" class="Symbol">)</a> <a id="957" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="959" href="Categories.Category.Core.html#715" class="Function">assoc</a><a id="964" class="Symbol">)</a>
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<a id="968" class="Symbol">}</a>
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<a id="972" class="Keyword">where</a> <a id="978" class="Keyword">open</a> <a id="983" href="Categories.Diagram.Wedge.html#553" class="Module">Wedge</a> <a id="989" href="Categories.Diagram.Wedge.html#786" class="Bound">W</a>
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<a id="992" class="Keyword">record</a> <a id="Wedge-Morphism"></a><a id="999" href="Categories.Diagram.Wedge.html#999" class="Record">Wedge-Morphism</a> <a id="1014" class="Symbol">(</a><a id="1015" href="Categories.Diagram.Wedge.html#1015" class="Bound">W₁</a> <a id="1018" href="Categories.Diagram.Wedge.html#1018" class="Bound">W₂</a> <a id="1021" class="Symbol">:</a> <a id="1023" href="Categories.Diagram.Wedge.html#553" class="Record">Wedge</a><a id="1028" class="Symbol">)</a> <a id="1030" class="Symbol">:</a> <a id="1032" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1036" class="Symbol">(</a><a id="1037" href="Level.html#602" class="Function">levelOfTerm</a> <a id="1049" href="Categories.Diagram.Wedge.html#208" class="Bound">F</a><a id="1050" class="Symbol">)</a> <a id="1052" class="Keyword">where</a>
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<a id="1060" class="Keyword">private</a>
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<a id="1072" class="Keyword">module</a> <a id="Wedge-Morphism.W₁"></a><a id="1079" href="Categories.Diagram.Wedge.html#1079" class="Module">W₁</a> <a id="1082" class="Symbol">=</a> <a id="1084" href="Categories.Diagram.Wedge.html#553" class="Module">Wedge</a> <a id="1090" href="Categories.Diagram.Wedge.html#1015" class="Bound">W₁</a>
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<a id="1097" class="Keyword">module</a> <a id="Wedge-Morphism.W₂"></a><a id="1104" href="Categories.Diagram.Wedge.html#1104" class="Module">W₂</a> <a id="1107" class="Symbol">=</a> <a id="1109" href="Categories.Diagram.Wedge.html#553" class="Module">Wedge</a> <a id="1115" href="Categories.Diagram.Wedge.html#1018" class="Bound">W₂</a>
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<a id="1122" class="Keyword">open</a> <a id="1127" href="Categories.NaturalTransformation.Dinatural.html#578" class="Module">DinaturalTransformation</a>
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<a id="1153" class="Keyword">field</a>
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<a id="Wedge-Morphism.u"></a><a id="1163" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1165" class="Symbol">:</a> <a id="1167" href="Categories.Diagram.Wedge.html#599" class="Function">W₁.E</a> <a id="1172" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="1174" href="Categories.Diagram.Wedge.html#599" class="Function">W₂.E</a>
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<a id="Wedge-Morphism.commute"></a><a id="1183" href="Categories.Diagram.Wedge.html#1183" class="Field">commute</a> <a id="1191" class="Symbol">:</a> <a id="1193" class="Symbol">∀</a> <a id="1195" class="Symbol">{</a><a id="1196" href="Categories.Diagram.Wedge.html#1196" class="Bound">C</a><a id="1197" class="Symbol">}</a> <a id="1199" class="Symbol">→</a> <a id="1201" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">W₂.dinatural.α</a> <a id="1216" href="Categories.Diagram.Wedge.html#1196" class="Bound">C</a> <a id="1218" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1220" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1222" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="1224" href="Categories.NaturalTransformation.Dinatural.html#851" class="Function">W₁.dinatural.α</a> <a id="1239" href="Categories.Diagram.Wedge.html#1196" class="Bound">C</a>
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<a id="Wedge-id"></a><a id="1242" href="Categories.Diagram.Wedge.html#1242" class="Function">Wedge-id</a> <a id="1251" class="Symbol">:</a> <a id="1253" class="Symbol">∀</a> <a id="1255" class="Symbol">{</a><a id="1256" href="Categories.Diagram.Wedge.html#1256" class="Bound">W</a><a id="1257" class="Symbol">}</a> <a id="1259" class="Symbol">→</a> <a id="1261" href="Categories.Diagram.Wedge.html#999" class="Record">Wedge-Morphism</a> <a id="1276" href="Categories.Diagram.Wedge.html#1256" class="Bound">W</a> <a id="1278" href="Categories.Diagram.Wedge.html#1256" class="Bound">W</a>
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<a id="1280" href="Categories.Diagram.Wedge.html#1242" class="Function">Wedge-id</a> <a id="1289" class="Symbol">{</a><a id="1290" href="Categories.Diagram.Wedge.html#1290" class="Bound">W</a><a id="1291" class="Symbol">}</a> <a id="1293" class="Symbol">=</a> <a id="1295" class="Keyword">record</a> <a id="1302" class="Symbol">{</a> <a id="1304" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1306" class="Symbol">=</a> <a id="1308" href="Categories.Category.Core.html#630" class="Function">D.id</a> <a id="1313" class="Symbol">;</a> <a id="1315" href="Categories.Diagram.Wedge.html#1183" class="Field">commute</a> <a id="1323" class="Symbol">=</a> <a id="1325" href="Categories.Category.Core.html#1145" class="Function">D.identityʳ</a> <a id="1337" class="Symbol">}</a>
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<a id="Wedge-Morphism-∘"></a><a id="1340" href="Categories.Diagram.Wedge.html#1340" class="Function">Wedge-Morphism-∘</a> <a id="1357" class="Symbol">:</a> <a id="1359" class="Symbol">{</a><a id="1360" href="Categories.Diagram.Wedge.html#1360" class="Bound">A</a> <a id="1362" href="Categories.Diagram.Wedge.html#1362" class="Bound">B</a> <a id="1364" href="Categories.Diagram.Wedge.html#1364" class="Bound">C</a> <a id="1366" class="Symbol">:</a> <a id="1368" href="Categories.Diagram.Wedge.html#553" class="Record">Wedge</a><a id="1373" class="Symbol">}</a> <a id="1375" class="Symbol">→</a> <a id="1377" href="Categories.Diagram.Wedge.html#999" class="Record">Wedge-Morphism</a> <a id="1392" href="Categories.Diagram.Wedge.html#1362" class="Bound">B</a> <a id="1394" href="Categories.Diagram.Wedge.html#1364" class="Bound">C</a> <a id="1396" class="Symbol">→</a> <a id="1398" href="Categories.Diagram.Wedge.html#999" class="Record">Wedge-Morphism</a> <a id="1413" href="Categories.Diagram.Wedge.html#1360" class="Bound">A</a> <a id="1415" href="Categories.Diagram.Wedge.html#1362" class="Bound">B</a> <a id="1417" class="Symbol">→</a> <a id="1419" href="Categories.Diagram.Wedge.html#999" class="Record">Wedge-Morphism</a> <a id="1434" href="Categories.Diagram.Wedge.html#1360" class="Bound">A</a> <a id="1436" href="Categories.Diagram.Wedge.html#1364" class="Bound">C</a>
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<a id="1438" href="Categories.Diagram.Wedge.html#1340" class="Function">Wedge-Morphism-∘</a> <a id="1455" href="Categories.Diagram.Wedge.html#1455" class="Bound">M</a> <a id="1457" href="Categories.Diagram.Wedge.html#1457" class="Bound">N</a> <a id="1459" class="Symbol">=</a> <a id="1461" class="Keyword">record</a> <a id="1468" class="Symbol">{</a> <a id="1470" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1472" class="Symbol">=</a> <a id="1474" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1476" href="Categories.Diagram.Wedge.html#1455" class="Bound">M</a> <a id="1478" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1480" href="Categories.Diagram.Wedge.html#1163" class="Field">u</a> <a id="1482" href="Categories.Diagram.Wedge.html#1457" class="Bound">N</a> <a id="1484" class="Symbol">;</a> <a id="1486" href="Categories.Diagram.Wedge.html#1183" class="Field">commute</a> <a id="1494" class="Symbol">=</a> <a id="1497" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a> <a id="1507" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1509" class="Symbol">(</a><a id="1510" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="1520" class="Symbol">(</a><a id="1521" href="Categories.Diagram.Wedge.html#1183" class="Field">commute</a> <a id="1529" href="Categories.Diagram.Wedge.html#1455" class="Bound">M</a><a id="1530" class="Symbol">)</a> <a id="1532" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="1534" href="Categories.Diagram.Wedge.html#1183" class="Field">commute</a> <a id="1542" href="Categories.Diagram.Wedge.html#1457" class="Bound">N</a><a id="1543" class="Symbol">)</a> <a id="1545" class="Symbol">}</a>
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<a id="1549" class="Keyword">where</a>
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<a id="1557" class="Keyword">open</a> <a id="1562" href="Categories.Diagram.Wedge.html#999" class="Module">Wedge-Morphism</a>
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<a id="1579" class="Keyword">open</a> <a id="1584" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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