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16 KiB
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<html><head><meta charset="utf-8"><title>Categories.Category.Groupoid.Properties</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="36" class="Keyword">open</a> <a id="41" class="Keyword">import</a> <a id="48" href="Categories.Category.Groupoid.html" class="Module">Categories.Category.Groupoid</a>
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<a id="78" class="Keyword">module</a> <a id="85" href="Categories.Category.Groupoid.Properties.html" class="Module">Categories.Category.Groupoid.Properties</a> <a id="125" class="Symbol">{</a><a id="126" href="Categories.Category.Groupoid.Properties.html#126" class="Bound">o</a> <a id="128" href="Categories.Category.Groupoid.Properties.html#128" class="Bound">ℓ</a> <a id="130" href="Categories.Category.Groupoid.Properties.html#130" class="Bound">e</a><a id="131" class="Symbol">}</a> <a id="133" class="Symbol">(</a><a id="134" href="Categories.Category.Groupoid.Properties.html#134" class="Bound">G</a> <a id="136" class="Symbol">:</a> <a id="138" href="Categories.Category.Groupoid.html#807" class="Record">Groupoid</a> <a id="147" href="Categories.Category.Groupoid.Properties.html#126" class="Bound">o</a> <a id="149" href="Categories.Category.Groupoid.Properties.html#128" class="Bound">ℓ</a> <a id="151" href="Categories.Category.Groupoid.Properties.html#130" class="Bound">e</a><a id="152" class="Symbol">)</a> <a id="154" class="Keyword">where</a>
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<a id="161" class="Keyword">import</a> <a id="168" href="Categories.Morphism.html" class="Module">Categories.Morphism</a> <a id="188" class="Symbol">as</a> <a id="191" class="Module">Morphism</a>
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<a id="200" class="Keyword">import</a> <a id="207" href="Categories.Morphism.Properties.html" class="Module">Categories.Morphism.Properties</a> <a id="238" class="Symbol">as</a> <a id="241" class="Module">MorphismProps</a>
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<a id="255" class="Keyword">import</a> <a id="262" href="Categories.Morphism.Reasoning.html" class="Module">Categories.Morphism.Reasoning</a> <a id="292" class="Symbol">as</a> <a id="295" class="Module">MR</a>
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<a id="299" class="Keyword">open</a> <a id="304" href="Categories.Category.Groupoid.html#807" class="Module">Groupoid</a> <a id="313" href="Categories.Category.Groupoid.Properties.html#134" class="Bound">G</a>
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<a id="315" class="Keyword">open</a> <a id="320" href="Categories.Morphism.html" class="Module">Morphism</a> <a id="329" href="Categories.Category.Groupoid.html#874" class="Field">category</a>
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<a id="338" class="Keyword">open</a> <a id="343" href="Categories.Morphism.Properties.html" class="Module">MorphismProps</a> <a id="357" href="Categories.Category.Groupoid.html#874" class="Field">category</a>
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<a id="366" class="Keyword">open</a> <a id="371" href="Categories.Category.Core.html#2462" class="Module">HomReasoning</a>
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<a id="384" class="Keyword">open</a> <a id="389" href="Categories.Morphism.Reasoning.html" class="Module">MR</a> <a id="392" href="Categories.Category.Groupoid.html#874" class="Field">category</a>
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<a id="402" class="Keyword">private</a>
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<a id="412" class="Keyword">variable</a>
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<a id="425" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a> <a id="427" href="Categories.Category.Groupoid.Properties.html#427" class="Generalizable">B</a> <a id="429" href="Categories.Category.Groupoid.Properties.html#429" class="Generalizable">C</a> <a id="431" class="Symbol">:</a> <a id="433" href="Categories.Category.Core.html#559" class="Function">Obj</a>
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<a id="mono"></a><a id="438" href="Categories.Category.Groupoid.Properties.html#438" class="Function">mono</a> <a id="443" class="Symbol">:</a> <a id="445" class="Symbol">{</a><a id="446" href="Categories.Category.Groupoid.Properties.html#446" class="Bound">f</a> <a id="448" class="Symbol">:</a> <a id="450" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a> <a id="452" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="454" href="Categories.Category.Groupoid.Properties.html#427" class="Generalizable">B</a><a id="455" class="Symbol">}</a> <a id="457" class="Symbol">→</a> <a id="459" href="Categories.Morphism.html#488" class="Function">Mono</a> <a id="464" href="Categories.Category.Groupoid.Properties.html#446" class="Bound">f</a>
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<a id="466" href="Categories.Category.Groupoid.Properties.html#438" class="Function">mono</a> <a id="471" class="Symbol">=</a> <a id="473" href="Categories.Morphism.Properties.html#818" class="Function">Iso⇒Mono</a> <a id="482" href="Categories.Category.Groupoid.html#399" class="Function">iso</a>
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<a id="epi"></a><a id="487" href="Categories.Category.Groupoid.Properties.html#487" class="Function">epi</a> <a id="491" class="Symbol">:</a> <a id="493" class="Symbol">{</a><a id="494" href="Categories.Category.Groupoid.Properties.html#494" class="Bound">f</a> <a id="496" class="Symbol">:</a> <a id="498" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a> <a id="500" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="502" href="Categories.Category.Groupoid.Properties.html#427" class="Generalizable">B</a><a id="503" class="Symbol">}</a> <a id="505" class="Symbol">→</a> <a id="507" href="Categories.Morphism.html#875" class="Function">Epi</a> <a id="511" href="Categories.Category.Groupoid.Properties.html#494" class="Bound">f</a>
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<a id="513" href="Categories.Category.Groupoid.Properties.html#487" class="Function">epi</a> <a id="517" class="Symbol">=</a> <a id="519" href="Categories.Morphism.Properties.html#980" class="Function">Iso⇒Epi</a> <a id="527" href="Categories.Category.Groupoid.html#399" class="Function">iso</a>
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<a id="id-inverse"></a><a id="532" href="Categories.Category.Groupoid.Properties.html#532" class="Function">id-inverse</a> <a id="543" class="Symbol">:</a> <a id="545" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="548" class="Symbol">{</a><a id="549" class="Argument">A</a> <a id="551" class="Symbol">=</a> <a id="553" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a><a id="554" class="Symbol">}</a> <a id="556" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="559" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="561" href="Categories.Category.Core.html#630" class="Function">id</a>
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<a id="564" href="Categories.Category.Groupoid.Properties.html#532" class="Function">id-inverse</a> <a id="575" class="Symbol">=</a> <a id="577" href="Categories.Category.Core.html#3005" class="Function">⟺</a> <a id="579" href="Categories.Category.Core.html#1096" class="Function">identityˡ</a> <a id="589" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="591" href="Categories.Morphism.html#1612" class="Function">iso.isoʳ</a>
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<a id="⁻¹-involutive"></a><a id="601" href="Categories.Category.Groupoid.Properties.html#601" class="Function">⁻¹-involutive</a> <a id="615" class="Symbol">:</a> <a id="617" class="Symbol">{</a><a id="618" href="Categories.Category.Groupoid.Properties.html#618" class="Bound">f</a> <a id="620" class="Symbol">:</a> <a id="622" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a> <a id="624" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="626" href="Categories.Category.Groupoid.Properties.html#427" class="Generalizable">B</a><a id="627" class="Symbol">}</a> <a id="629" class="Symbol">→</a> <a id="631" href="Categories.Category.Groupoid.Properties.html#618" class="Bound">f</a> <a id="633" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="636" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="639" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="641" href="Categories.Category.Groupoid.Properties.html#618" class="Bound">f</a>
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<a id="643" href="Categories.Category.Groupoid.Properties.html#601" class="Function">⁻¹-involutive</a> <a id="657" class="Symbol">{</a><a id="658" class="Argument">f</a> <a id="660" class="Symbol">=</a> <a id="662" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a><a id="663" class="Symbol">}</a> <a id="665" class="Symbol">=</a> <a id="667" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="675" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a> <a id="677" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="680" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="694" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="697" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="704" href="Categories.Morphism.html#1586" class="Function">iso.isoˡ</a> <a id="713" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="717" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a> <a id="719" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="722" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="725" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="727" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a> <a id="729" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="732" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="734" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a> <a id="736" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="739" href="Categories.Category.Core.html#1004" class="Function">sym-assoc</a> <a id="749" href="Categories.Category.Core.html#3061" class="Function Operator">○</a> <a id="751" href="Categories.Morphism.Reasoning.Core.html#2948" class="Function">elimˡ</a> <a id="757" href="Categories.Morphism.html#1586" class="Function">iso.isoˡ</a> <a id="766" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="787" href="Categories.Category.Groupoid.Properties.html#662" class="Bound">f</a> <a id="789" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="⁻¹-commute"></a><a id="792" href="Categories.Category.Groupoid.Properties.html#792" class="Function">⁻¹-commute</a> <a id="803" class="Symbol">:</a> <a id="805" class="Symbol">{</a><a id="806" href="Categories.Category.Groupoid.Properties.html#806" class="Bound">f</a> <a id="808" class="Symbol">:</a> <a id="810" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a> <a id="812" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="814" href="Categories.Category.Groupoid.Properties.html#427" class="Generalizable">B</a><a id="815" class="Symbol">}</a> <a id="817" class="Symbol">{</a><a id="818" href="Categories.Category.Groupoid.Properties.html#818" class="Bound">g</a> <a id="820" class="Symbol">:</a> <a id="822" href="Categories.Category.Groupoid.Properties.html#429" class="Generalizable">C</a> <a id="824" href="Categories.Category.Core.html#575" class="Function Operator">⇒</a> <a id="826" href="Categories.Category.Groupoid.Properties.html#425" class="Generalizable">A</a><a id="827" class="Symbol">}</a> <a id="829" class="Symbol">→</a> <a id="831" class="Symbol">(</a><a id="832" href="Categories.Category.Groupoid.Properties.html#806" class="Bound">f</a> <a id="834" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="836" href="Categories.Category.Groupoid.Properties.html#818" class="Bound">g</a><a id="837" class="Symbol">)</a> <a id="839" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="842" href="Categories.Category.Core.html#595" class="Function Operator">≈</a> <a id="844" href="Categories.Category.Groupoid.Properties.html#818" class="Bound">g</a> <a id="846" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="849" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="851" href="Categories.Category.Groupoid.Properties.html#806" class="Bound">f</a> <a id="853" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a>
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<a id="856" href="Categories.Category.Groupoid.Properties.html#792" class="Function">⁻¹-commute</a> <a id="867" class="Symbol">{</a><a id="868" class="Argument">f</a> <a id="870" class="Symbol">=</a> <a id="872" href="Categories.Category.Groupoid.Properties.html#872" class="Bound">f</a><a id="873" class="Symbol">}</a> <a id="875" class="Symbol">{</a><a id="876" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a><a id="877" class="Symbol">}</a> <a id="879" class="Symbol">=</a> <a id="881" href="Categories.Category.Groupoid.Properties.html#487" class="Function">epi</a> <a id="885" class="Symbol">_</a> <a id="887" class="Symbol">_</a> <a id="889" class="Symbol">(</a> <a id="891" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
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<a id="899" class="Symbol">(</a><a id="900" href="Categories.Category.Groupoid.Properties.html#872" class="Bound">f</a> <a id="902" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="904" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a><a id="905" class="Symbol">)</a> <a id="907" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="910" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="912" href="Categories.Category.Groupoid.Properties.html#872" class="Bound">f</a> <a id="914" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="916" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a> <a id="921" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="924" href="Categories.Morphism.html#1586" class="Function">iso.isoˡ</a> <a id="933" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="937" href="Categories.Category.Core.html#630" class="Function">id</a> <a id="959" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="963" href="Categories.Morphism.html#1586" class="Function">iso.isoˡ</a> <a id="972" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="976" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a> <a id="978" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="981" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="983" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a> <a id="998" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="1002" href="Categories.Morphism.Reasoning.Core.html#7037" class="Function">cancelInner</a> <a id="1014" href="Categories.Morphism.html#1586" class="Function">iso.isoˡ</a> <a id="1023" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
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<a id="1027" class="Symbol">(</a><a id="1028" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a> <a id="1030" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a> <a id="1033" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1035" href="Categories.Category.Groupoid.Properties.html#872" class="Bound">f</a> <a id="1037" href="Categories.Category.Groupoid.html#365" class="Function Operator">⁻¹</a><a id="1039" class="Symbol">)</a> <a id="1041" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1043" href="Categories.Category.Groupoid.Properties.html#872" class="Bound">f</a> <a id="1045" href="Categories.Category.Core.html#656" class="Function Operator">∘</a> <a id="1047" href="Categories.Category.Groupoid.Properties.html#876" class="Bound">g</a> <a id="1049" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="1051" class="Symbol">)</a>
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