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<html><head><meta charset="utf-8"><title>Relation.Binary.Indexed.Heterogeneous.Definitions</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
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<a id="74" class="Comment">-- The Agda standard library</a>
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<a id="103" class="Comment">--</a>
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<a id="106" class="Comment">-- Indexed binary relations</a>
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<a id="134" class="Comment">------------------------------------------------------------------------</a>
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<a id="208" class="Comment">-- The contents of this module should be accessed via</a>
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<a id="262" class="Comment">-- `Relation.Binary.Indexed.Heterogeneous`.</a>
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<a id="307" class="Symbol">{-#</a> <a id="311" class="Keyword">OPTIONS</a> <a id="319" class="Pragma">--cubical-compatible</a> <a id="340" class="Pragma">--safe</a> <a id="347" class="Symbol">#-}</a>
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<a id="352" class="Keyword">module</a> <a id="359" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html" class="Module">Relation.Binary.Indexed.Heterogeneous.Definitions</a> <a id="409" class="Keyword">where</a>
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<a id="416" class="Keyword">open</a> <a id="421" class="Keyword">import</a> <a id="428" href="Level.html" class="Module">Level</a>
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<a id="434" class="Keyword">import</a> <a id="441" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="462" class="Symbol">as</a> <a id="465" class="Module">B</a>
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<a id="467" class="Keyword">import</a> <a id="474" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a> <a id="502" class="Symbol">as</a> <a id="505" class="Module">B</a>
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<a id="507" class="Keyword">import</a> <a id="514" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a> <a id="557" class="Symbol">as</a> <a id="560" class="Module">P</a>
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<a id="562" class="Keyword">open</a> <a id="567" class="Keyword">import</a> <a id="574" href="Relation.Binary.Indexed.Heterogeneous.Core.html" class="Module">Relation.Binary.Indexed.Heterogeneous.Core</a>
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<a id="618" class="Keyword">private</a>
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<a id="628" class="Keyword">variable</a>
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<a id="641" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#641" class="Generalizable">i</a> <a id="643" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#643" class="Generalizable">a</a> <a id="645" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#645" class="Generalizable">ℓ</a> <a id="647" class="Symbol">:</a> <a id="649" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="659" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#659" class="Generalizable">I</a> <a id="661" class="Symbol">:</a> <a id="663" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="667" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#641" class="Generalizable">i</a>
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<a id="670" class="Comment">------------------------------------------------------------------------</a>
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<a id="743" class="Comment">-- Simple properties of indexed binary relations</a>
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<a id="Reflexive"></a><a id="793" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#793" class="Function">Reflexive</a> <a id="803" class="Symbol">:</a> <a id="805" class="Symbol">(</a><a id="806" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#806" class="Bound">A</a> <a id="808" class="Symbol">:</a> <a id="810" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#659" class="Generalizable">I</a> <a id="812" class="Symbol">→</a> <a id="814" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="818" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#643" class="Generalizable">a</a><a id="819" class="Symbol">)</a> <a id="821" class="Symbol">→</a> <a id="823" href="Relation.Binary.Indexed.Heterogeneous.Core.html#865" class="Function">IRel</a> <a id="828" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#806" class="Bound">A</a> <a id="830" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#645" class="Generalizable">ℓ</a> <a id="832" class="Symbol">→</a> <a id="834" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="838" class="Symbol">_</a>
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<a id="840" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#793" class="Function">Reflexive</a> <a id="850" class="Symbol">_</a> <a id="852" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#852" class="Bound Operator">_∼_</a> <a id="856" class="Symbol">=</a> <a id="858" class="Symbol">∀</a> <a id="860" class="Symbol">{</a><a id="861" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#861" class="Bound">i</a><a id="862" class="Symbol">}</a> <a id="864" class="Symbol">→</a> <a id="866" href="Relation.Binary.Definitions.html#1348" class="Function">B.Reflexive</a> <a id="878" class="Symbol">(</a><a id="879" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#852" class="Bound Operator">_∼_</a> <a id="883" class="Symbol">{</a><a id="884" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#861" class="Bound">i</a><a id="885" class="Symbol">})</a>
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<a id="Symmetric"></a><a id="889" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#889" class="Function">Symmetric</a> <a id="899" class="Symbol">:</a> <a id="901" class="Symbol">(</a><a id="902" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#902" class="Bound">A</a> <a id="904" class="Symbol">:</a> <a id="906" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#659" class="Generalizable">I</a> <a id="908" class="Symbol">→</a> <a id="910" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="914" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#643" class="Generalizable">a</a><a id="915" class="Symbol">)</a> <a id="917" class="Symbol">→</a> <a id="919" href="Relation.Binary.Indexed.Heterogeneous.Core.html#865" class="Function">IRel</a> <a id="924" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#902" class="Bound">A</a> <a id="926" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#645" class="Generalizable">ℓ</a> <a id="928" class="Symbol">→</a> <a id="930" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="934" class="Symbol">_</a>
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<a id="936" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#889" class="Function">Symmetric</a> <a id="946" class="Symbol">_</a> <a id="948" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#948" class="Bound Operator">_∼_</a> <a id="952" class="Symbol">=</a> <a id="954" class="Symbol">∀</a> <a id="956" class="Symbol">{</a><a id="957" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#957" class="Bound">i</a> <a id="959" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#959" class="Bound">j</a><a id="960" class="Symbol">}</a> <a id="962" class="Symbol">→</a> <a id="964" href="Relation.Binary.Definitions.html#1433" class="Function">B.Sym</a> <a id="970" class="Symbol">(</a><a id="971" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#948" class="Bound Operator">_∼_</a> <a id="975" class="Symbol">{</a><a id="976" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#957" class="Bound">i</a><a id="977" class="Symbol">}</a> <a id="979" class="Symbol">{</a><a id="980" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#959" class="Bound">j</a><a id="981" class="Symbol">})</a> <a id="984" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#948" class="Bound Operator">_∼_</a>
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<a id="Transitive"></a><a id="989" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#989" class="Function">Transitive</a> <a id="1000" class="Symbol">:</a> <a id="1002" class="Symbol">(</a><a id="1003" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1003" class="Bound">A</a> <a id="1005" class="Symbol">:</a> <a id="1007" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#659" class="Generalizable">I</a> <a id="1009" class="Symbol">→</a> <a id="1011" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1015" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#643" class="Generalizable">a</a><a id="1016" class="Symbol">)</a> <a id="1018" class="Symbol">→</a> <a id="1020" href="Relation.Binary.Indexed.Heterogeneous.Core.html#865" class="Function">IRel</a> <a id="1025" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1003" class="Bound">A</a> <a id="1027" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#645" class="Generalizable">ℓ</a> <a id="1029" class="Symbol">→</a> <a id="1031" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1035" class="Symbol">_</a>
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<a id="1037" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#989" class="Function">Transitive</a> <a id="1048" class="Symbol">_</a> <a id="1050" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1050" class="Bound Operator">_∼_</a> <a id="1054" class="Symbol">=</a> <a id="1056" class="Symbol">∀</a> <a id="1058" class="Symbol">{</a><a id="1059" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1059" class="Bound">i</a> <a id="1061" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1061" class="Bound">j</a> <a id="1063" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1063" class="Bound">k</a><a id="1064" class="Symbol">}</a> <a id="1066" class="Symbol">→</a> <a id="1068" href="Relation.Binary.Definitions.html#1594" class="Function">B.Trans</a> <a id="1076" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1050" class="Bound Operator">_∼_</a> <a id="1080" class="Symbol">(</a><a id="1081" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1050" class="Bound Operator">_∼_</a> <a id="1085" class="Symbol">{</a><a id="1086" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1061" class="Bound">j</a><a id="1087" class="Symbol">})</a> <a id="1090" class="Symbol">(</a><a id="1091" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1050" class="Bound Operator">_∼_</a> <a id="1095" class="Symbol">{</a><a id="1096" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1059" class="Bound">i</a><a id="1097" class="Symbol">}</a> <a id="1099" class="Symbol">{</a><a id="1100" href="Relation.Binary.Indexed.Heterogeneous.Definitions.html#1063" class="Bound">k</a><a id="1101" class="Symbol">})</a>
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