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<html><head><meta charset="utf-8"><title>Categories.Morphism.Reasoning</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="70" class="Comment">-- Reasoning facilities about morphism equivalences (not necessarily 'squares')</a>
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<a id="151" class="Keyword">module</a> <a id="158" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="188" class="Symbol">{</a><a id="189" href="Categories.Morphism.Reasoning.html#189" class="Bound">o</a> <a id="191" href="Categories.Morphism.Reasoning.html#191" class="Bound">ℓ</a> <a id="193" href="Categories.Morphism.Reasoning.html#193" class="Bound">e</a><a id="194" class="Symbol">}</a> <a id="196" class="Symbol">(</a><a id="197" href="Categories.Morphism.Reasoning.html#197" class="Bound">C</a> <a id="199" class="Symbol">:</a> <a id="201" href="Categories.Category.Core.html#442" class="Record">Category</a> <a id="210" href="Categories.Morphism.Reasoning.html#189" class="Bound">o</a> <a id="212" href="Categories.Morphism.Reasoning.html#191" class="Bound">ℓ</a> <a id="214" href="Categories.Morphism.Reasoning.html#193" class="Bound">e</a><a id="215" class="Symbol">)</a> <a id="217" class="Keyword">where</a>
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<a id="224" class="Comment">-- some items are defined in sub-modules</a>
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<a id="265" class="Keyword">open</a> <a id="270" class="Keyword">import</a> <a id="277" href="Categories.Morphism.Reasoning.Core.html" class="Module">Categories.Morphism.Reasoning.Core</a> <a id="312" href="Categories.Morphism.Reasoning.html#197" class="Bound">C</a> <a id="314" class="Keyword">public</a>
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<a id="321" class="Keyword">open</a> <a id="326" class="Keyword">import</a> <a id="333" href="Categories.Morphism.Reasoning.Iso.html" class="Module">Categories.Morphism.Reasoning.Iso</a> <a id="367" href="Categories.Morphism.Reasoning.html#197" class="Bound">C</a> <a id="369" class="Keyword">public</a>
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<a id="377" class="Keyword">open</a> <a id="382" href="Categories.Category.Core.html#442" class="Module">Category</a> <a id="391" href="Categories.Morphism.Reasoning.html#197" class="Bound">C</a>
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<a id="393" class="Keyword">open</a> <a id="398" href="Categories.Category.html#824" class="Module">Definitions</a> <a id="410" href="Categories.Morphism.Reasoning.html#197" class="Bound">C</a>
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<a id="412" class="Keyword">open</a> <a id="417" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="431" class="Comment">-- create a commutative square from an equivalence</a>
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<a id="toSquare"></a><a id="482" href="Categories.Morphism.Reasoning.html#482" class="Function">toSquare</a> <a id="491" class="Symbol">:</a> <a id="493" class="Symbol">∀</a> <a id="495" class="Symbol">{</a><a id="496" href="Categories.Morphism.Reasoning.html#496" class="Bound">A</a> <a id="498" href="Categories.Morphism.Reasoning.html#498" class="Bound">B</a><a id="499" class="Symbol">}</a> <a id="501" class="Symbol">{</a><a id="502" href="Categories.Morphism.Reasoning.html#502" class="Bound">f</a> <a id="504" href="Categories.Morphism.Reasoning.html#504" class="Bound">g</a> <a id="506" class="Symbol">:</a> <a id="508" href="Categories.Morphism.Reasoning.html#496" class="Bound">A</a> <a id="510" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="512" href="Categories.Morphism.Reasoning.html#498" class="Bound">B</a><a id="513" class="Symbol">}</a> <a id="515" class="Symbol">→</a> <a id="517" href="Categories.Morphism.Reasoning.html#502" class="Bound">f</a> <a id="519" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="521" href="Categories.Morphism.Reasoning.html#504" class="Bound">g</a> <a id="523" class="Symbol">→</a> <a id="525" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="543" href="Categories.Morphism.Reasoning.html#502" class="Bound">f</a> <a id="545" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="548" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="551" href="Categories.Morphism.Reasoning.html#504" class="Bound">g</a>
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<a id="553" href="Categories.Morphism.Reasoning.html#482" class="Function">toSquare</a> <a id="562" class="Symbol">{_}</a> <a id="566" class="Symbol">{_}</a> <a id="570" class="Symbol">{</a><a id="571" href="Categories.Morphism.Reasoning.html#571" class="Bound">f</a><a id="572" class="Symbol">}</a> <a id="574" class="Symbol">{</a><a id="575" href="Categories.Morphism.Reasoning.html#575" class="Bound">g</a><a id="576" class="Symbol">}</a> <a id="578" href="Categories.Morphism.Reasoning.html#578" class="Bound">f≈g</a> <a id="582" class="Symbol">=</a> <a id="584" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="596" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="599" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="601" href="Categories.Morphism.Reasoning.html#571" class="Bound">f</a> <a id="605" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="608" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="618" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="626" href="Categories.Morphism.Reasoning.html#571" class="Bound">f</a> <a id="635" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="638" href="Categories.Morphism.Reasoning.html#578" class="Bound">f≈g</a> <a id="642" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="650" href="Categories.Morphism.Reasoning.html#575" class="Bound">g</a> <a id="659" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="663" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="673" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="681" href="Categories.Morphism.Reasoning.html#575" class="Bound">g</a> <a id="683" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="685" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="690" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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