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<html><head><meta charset="utf-8"><title>Categories.Category.Instance.EmptySet</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="Level.html" class="Module">Level</a>
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<a id="56" class="Comment">-- This is really a degenerate version of Categories.Category.Instance.Zero</a>
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<a id="132" class="Comment">-- Here EmptySet is not given an explicit name, it is an alias for Lift o ⊥</a>
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<a id="208" class="Keyword">module</a> <a id="215" href="Categories.Category.Instance.EmptySet.html" class="Module">Categories.Category.Instance.EmptySet</a> <a id="253" class="Keyword">where</a>
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<a id="260" class="Keyword">open</a> <a id="265" class="Keyword">import</a> <a id="272" href="Data.Unit.html" class="Module">Data.Unit</a>
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<a id="282" class="Keyword">open</a> <a id="287" class="Keyword">import</a> <a id="294" href="Data.Empty.html" class="Module">Data.Empty</a> <a id="305" class="Keyword">using</a> <a id="311" class="Symbol">(</a><a id="312" href="Data.Empty.html#895" class="Function">⊥</a><a id="313" class="Symbol">;</a> <a id="315" href="Data.Empty.html#1050" class="Function">⊥-elim</a><a id="321" class="Symbol">)</a>
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<a id="324" class="Keyword">open</a> <a id="329" class="Keyword">import</a> <a id="336" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="352" class="Keyword">using</a> <a id="358" class="Symbol">(</a><a id="359" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="365" class="Symbol">)</a>
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<a id="367" class="Keyword">open</a> <a id="372" class="Keyword">import</a> <a id="379" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a> <a id="417" class="Keyword">using</a> <a id="423" class="Symbol">(</a><a id="424" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="428" class="Symbol">)</a>
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<a id="431" class="Keyword">open</a> <a id="436" class="Keyword">import</a> <a id="443" href="Categories.Category.Instance.Sets.html" class="Module">Categories.Category.Instance.Sets</a>
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<a id="477" class="Keyword">open</a> <a id="482" class="Keyword">import</a> <a id="489" href="Categories.Category.Instance.Setoids.html" class="Module">Categories.Category.Instance.Setoids</a>
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<a id="526" class="Keyword">import</a> <a id="533" href="Categories.Object.Initial.html" class="Module">Categories.Object.Initial</a> <a id="559" class="Symbol">as</a> <a id="562" class="Module">Init</a>
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<a id="568" class="Keyword">module</a> <a id="575" href="Categories.Category.Instance.EmptySet.html#575" class="Module">_</a> <a id="577" class="Symbol">{</a><a id="578" href="Categories.Category.Instance.EmptySet.html#578" class="Bound">o</a> <a id="580" class="Symbol">:</a> <a id="582" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="587" class="Symbol">}</a> <a id="589" class="Keyword">where</a>
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<a id="597" class="Keyword">open</a> <a id="602" href="Categories.Object.Initial.html" class="Module">Init</a> <a id="607" class="Symbol">(</a><a id="608" href="Categories.Category.Instance.Sets.html#416" class="Function">Sets</a> <a id="613" href="Categories.Category.Instance.EmptySet.html#578" class="Bound">o</a><a id="614" class="Symbol">)</a>
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<a id="619" href="Categories.Category.Instance.EmptySet.html#619" class="Function">EmptySet-⊥</a> <a id="630" class="Symbol">:</a> <a id="632" href="Categories.Object.Initial.html#760" class="Record">Initial</a>
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<a id="642" href="Categories.Category.Instance.EmptySet.html#619" class="Function">EmptySet-⊥</a> <a id="653" class="Symbol">=</a> <a id="655" class="Keyword">record</a> <a id="662" class="Symbol">{</a> <a id="664" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="666" class="Symbol">=</a> <a id="668" href="Level.html#409" class="Record">Lift</a> <a id="673" href="Categories.Category.Instance.EmptySet.html#578" class="Bound">o</a> <a id="675" href="Data.Empty.html#895" class="Function">⊥</a> <a id="677" class="Symbol">;</a> <a id="679" href="Categories.Object.Initial.html#816" class="Field">⊥-is-initial</a> <a id="692" class="Symbol">=</a> <a id="694" class="Keyword">record</a> <a id="701" class="Symbol">{</a> <a id="703" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="705" class="Symbol">=</a> <a id="707" class="Symbol">λ</a> <a id="709" class="Symbol">{</a> <a id="711" class="Symbol">{</a><a id="712" href="Categories.Category.Instance.EmptySet.html#712" class="Bound">A</a><a id="713" class="Symbol">}</a> <a id="715" class="Symbol">(</a><a id="716" href="Level.html#466" class="InductiveConstructor">lift</a> <a id="721" href="Categories.Category.Instance.EmptySet.html#721" class="Bound">x</a><a id="722" class="Symbol">)</a> <a id="724" class="Symbol">→</a> <a id="726" href="Data.Empty.html#1050" class="Function">⊥-elim</a> <a id="733" href="Categories.Category.Instance.EmptySet.html#721" class="Bound">x</a> <a id="735" class="Symbol">}</a> <a id="737" class="Symbol">;</a> <a id="739" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="748" class="Symbol">=</a> <a id="750" class="Symbol">λ</a> <a id="752" class="Symbol">{</a> <a id="754" href="Categories.Category.Instance.EmptySet.html#754" class="Bound">f</a> <a id="756" class="Symbol">{()}</a> <a id="761" class="Symbol">}</a> <a id="763" class="Symbol">}</a> <a id="765" class="Symbol">}</a>
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<a id="768" class="Keyword">module</a> <a id="775" href="Categories.Category.Instance.EmptySet.html#775" class="Module">_</a> <a id="777" class="Symbol">{</a><a id="778" href="Categories.Category.Instance.EmptySet.html#778" class="Bound">c</a> <a id="780" href="Categories.Category.Instance.EmptySet.html#780" class="Bound">ℓ</a> <a id="782" class="Symbol">:</a> <a id="784" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="789" class="Symbol">}</a> <a id="791" class="Keyword">where</a>
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<a id="799" class="Keyword">open</a> <a id="804" href="Categories.Object.Initial.html" class="Module">Init</a> <a id="809" class="Symbol">(</a><a id="810" href="Categories.Category.Instance.Setoids.html#555" class="Function">Setoids</a> <a id="818" href="Categories.Category.Instance.EmptySet.html#778" class="Bound">c</a> <a id="820" href="Categories.Category.Instance.EmptySet.html#780" class="Bound">ℓ</a><a id="821" class="Symbol">)</a>
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<a id="826" href="Categories.Category.Instance.EmptySet.html#826" class="Function">EmptySetoid</a> <a id="838" class="Symbol">:</a> <a id="840" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="847" href="Categories.Category.Instance.EmptySet.html#778" class="Bound">c</a> <a id="849" href="Categories.Category.Instance.EmptySet.html#780" class="Bound">ℓ</a>
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<a id="853" href="Categories.Category.Instance.EmptySet.html#826" class="Function">EmptySetoid</a> <a id="865" class="Symbol">=</a> <a id="867" class="Keyword">record</a>
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<a id="878" class="Symbol">{</a> <a id="880" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="888" class="Symbol">=</a> <a id="890" href="Level.html#409" class="Record">Lift</a> <a id="895" href="Categories.Category.Instance.EmptySet.html#778" class="Bound">c</a> <a id="897" href="Data.Empty.html#895" class="Function">⊥</a>
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<a id="903" class="Symbol">;</a> <a id="905" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="913" class="Symbol">=</a> <a id="915" class="Symbol">λ</a> <a id="917" href="Categories.Category.Instance.EmptySet.html#917" class="Bound">_</a> <a id="919" href="Categories.Category.Instance.EmptySet.html#919" class="Bound">_</a> <a id="921" class="Symbol">→</a> <a id="923" href="Level.html#409" class="Record">Lift</a> <a id="928" href="Categories.Category.Instance.EmptySet.html#780" class="Bound">ℓ</a> <a id="930" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a>
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<a id="936" class="Symbol">}</a>
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<a id="941" href="Categories.Category.Instance.EmptySet.html#941" class="Function">EmptySetoid-⊥</a> <a id="955" class="Symbol">:</a> <a id="957" href="Categories.Object.Initial.html#760" class="Record">Initial</a>
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<a id="967" href="Categories.Category.Instance.EmptySet.html#941" class="Function">EmptySetoid-⊥</a> <a id="981" class="Symbol">=</a> <a id="983" class="Keyword">record</a>
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<a id="994" class="Symbol">{</a> <a id="996" href="Categories.Object.Initial.html#804" class="Field">⊥</a> <a id="1009" class="Symbol">=</a> <a id="1011" href="Categories.Category.Instance.EmptySet.html#826" class="Function">EmptySetoid</a>
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<a id="1027" class="Symbol">;</a> <a id="1029" href="Categories.Object.Initial.html#816" class="Field">⊥-is-initial</a> <a id="1042" class="Symbol">=</a> <a id="1044" class="Keyword">record</a>
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<a id="1057" class="Symbol">{</a> <a id="1059" href="Categories.Object.Initial.html#478" class="Field">!</a> <a id="1068" class="Symbol">=</a> <a id="1070" class="Keyword">record</a>
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<a id="1085" class="Symbol">{</a> <a id="1087" href="Function.Bundles.html#2094" class="Field">to</a> <a id="1090" class="Symbol">=</a> <a id="1092" class="Symbol">λ</a> <a id="1094" class="Symbol">{</a> <a id="1096" class="Symbol">()</a> <a id="1099" class="Symbol">}</a>
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<a id="1109" class="Symbol">;</a> <a id="1111" href="Function.Bundles.html#2113" class="Field">cong</a> <a id="1117" class="Symbol">=</a> <a id="1119" class="Symbol">λ</a> <a id="1121" class="Symbol">{</a> <a id="1123" class="Symbol">{()}</a> <a id="1128" class="Symbol">}</a>
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<a id="1138" class="Symbol">}</a>
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<a id="1146" class="Symbol">;</a> <a id="1148" href="Categories.Object.Initial.html#506" class="Field">!-unique</a> <a id="1157" class="Symbol">=</a> <a id="1159" class="Symbol">λ</a> <a id="1161" class="Symbol">{</a> <a id="1163" class="Symbol">_</a> <a id="1165" class="Symbol">{()}</a> <a id="1170" class="Symbol">}</a>
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<a id="1178" class="Symbol">}</a>
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<a id="1184" class="Symbol">}</a>
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</pre></body></html>
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