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<html><head><meta charset="utf-8"><title>Relation.Binary.Indexed.Heterogeneous.Construct.Trivial</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">-- Creates trivially indexed records from their non-indexed counterpart.</a>
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<a id="179" class="Comment">------------------------------------------------------------------------</a>
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<a id="253" class="Symbol">{-#</a> <a id="257" class="Keyword">OPTIONS</a> <a id="265" class="Pragma">--cubical-compatible</a> <a id="286" class="Pragma">--safe</a> <a id="293" class="Symbol">#-}</a>
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<a id="298" class="Keyword">module</a> <a id="305" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html" class="Module">Relation.Binary.Indexed.Heterogeneous.Construct.Trivial</a>
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<a id="363" class="Symbol">{</a><a id="364" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#364" class="Bound">i</a><a id="365" class="Symbol">}</a> <a id="367" class="Symbol">{</a><a id="368" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="370" class="Symbol">:</a> <a id="372" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="376" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#364" class="Bound">i</a><a id="377" class="Symbol">}</a> <a id="379" class="Keyword">where</a>
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<a id="386" class="Keyword">open</a> <a id="391" class="Keyword">import</a> <a id="398" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="419" class="Keyword">using</a> <a id="425" class="Symbol">(</a><a id="426" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="429" 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="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a> <a id="467" class="Keyword">using</a> <a id="473" class="Symbol">(</a><a id="474" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="480" class="Symbol">;</a> <a id="482" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a><a id="490" class="Symbol">)</a>
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<a id="492" class="Keyword">open</a> <a id="497" class="Keyword">import</a> <a id="504" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a> <a id="531" class="Keyword">using</a> <a id="537" class="Symbol">(</a><a id="538" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="551" class="Symbol">;</a> <a id="553" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="563" class="Symbol">)</a>
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<a id="565" class="Keyword">open</a> <a id="570" class="Keyword">import</a> <a id="577" href="Relation.Binary.Indexed.Heterogeneous.html" class="Module">Relation.Binary.Indexed.Heterogeneous</a>
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<a id="616" class="Comment">------------------------------------------------------------------------</a>
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<a id="689" class="Comment">-- Structures</a>
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<a id="704" class="Keyword">module</a> <a id="711" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#711" class="Module">_</a> <a id="713" class="Symbol">{</a><a id="714" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#714" class="Bound">a</a><a id="715" class="Symbol">}</a> <a id="717" class="Symbol">{</a><a id="718" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#718" class="Bound">A</a> <a id="720" class="Symbol">:</a> <a id="722" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="726" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#714" class="Bound">a</a><a id="727" class="Symbol">}</a> <a id="729" class="Keyword">where</a>
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<a id="738" class="Keyword">private</a>
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<a id="750" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#750" class="Function">Aᵢ</a> <a id="753" class="Symbol">:</a> <a id="755" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="757" class="Symbol">→</a> <a id="759" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="763" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#714" class="Bound">a</a>
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<a id="769" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#750" class="Function">Aᵢ</a> <a id="772" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#772" class="Bound">i</a> <a id="774" class="Symbol">=</a> <a id="776" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#718" class="Bound">A</a>
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<a id="781" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#781" class="Function">isIndexedEquivalence</a> <a id="802" class="Symbol">:</a> <a id="804" class="Symbol">∀</a> <a id="806" class="Symbol">{</a><a id="807" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#807" class="Bound">ℓ</a><a id="808" class="Symbol">}</a> <a id="810" class="Symbol">{</a><a id="811" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#811" class="Bound Operator">_≈_</a> <a id="815" class="Symbol">:</a> <a id="817" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="821" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#718" class="Bound">A</a> <a id="823" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#807" class="Bound">ℓ</a><a id="824" class="Symbol">}</a> <a id="826" class="Symbol">→</a> <a id="828" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="842" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#811" class="Bound Operator">_≈_</a> <a id="846" class="Symbol">→</a>
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<a id="873" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#866" class="Record">IsIndexedEquivalence</a> <a id="894" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#750" class="Function">Aᵢ</a> <a id="897" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#811" class="Bound Operator">_≈_</a>
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<a id="903" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#781" class="Function">isIndexedEquivalence</a> <a id="924" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#924" class="Bound">isEq</a> <a id="929" class="Symbol">=</a> <a id="931" class="Keyword">record</a>
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<a id="942" class="Symbol">{</a> <a id="944" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#923" class="Field">refl</a> <a id="950" class="Symbol">=</a> <a id="952" href="Relation.Binary.Structures.html#1596" class="Field">refl</a>
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<a id="961" class="Symbol">;</a> <a id="963" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#952" class="Field">sym</a> <a id="969" class="Symbol">=</a> <a id="971" href="Relation.Binary.Structures.html#1622" class="Field">sym</a>
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<a id="979" class="Symbol">;</a> <a id="981" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#981" class="Field">trans</a> <a id="987" class="Symbol">=</a> <a id="989" href="Relation.Binary.Structures.html#1648" class="Field">trans</a>
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<a id="999" class="Symbol">}</a>
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<a id="1005" class="Keyword">where</a> <a id="1011" class="Keyword">open</a> <a id="1016" href="Relation.Binary.Structures.html#1550" class="Module">IsEquivalence</a> <a id="1030" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#924" class="Bound">isEq</a>
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<a id="1038" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1038" class="Function">isIndexedPreorder</a> <a id="1056" class="Symbol">:</a> <a id="1058" class="Symbol">∀</a> <a id="1060" class="Symbol">{</a><a id="1061" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1061" class="Bound">ℓ₁</a> <a id="1064" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1064" class="Bound">ℓ₂</a><a id="1066" class="Symbol">}</a> <a id="1068" class="Symbol">{</a><a id="1069" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1069" class="Bound Operator">_≈_</a> <a id="1073" class="Symbol">:</a> <a id="1075" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1079" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#718" class="Bound">A</a> <a id="1081" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1061" class="Bound">ℓ₁</a><a id="1083" class="Symbol">}</a> <a id="1085" class="Symbol">{</a><a id="1086" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1086" class="Bound Operator">_∼_</a> <a id="1090" class="Symbol">:</a> <a id="1092" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1096" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#718" class="Bound">A</a> <a id="1098" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1064" class="Bound">ℓ₂</a><a id="1100" class="Symbol">}</a> <a id="1102" class="Symbol">→</a>
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<a id="1126" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="1137" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1069" class="Bound Operator">_≈_</a> <a id="1141" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1086" class="Bound Operator">_∼_</a> <a id="1145" class="Symbol">→</a>
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<a id="1169" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1084" class="Record">IsIndexedPreorder</a> <a id="1187" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#750" class="Function">Aᵢ</a> <a id="1190" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1069" class="Bound Operator">_≈_</a> <a id="1194" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1086" class="Bound Operator">_∼_</a>
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<a id="1200" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1038" class="Function">isIndexedPreorder</a> <a id="1218" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1218" class="Bound">isPreorder</a> <a id="1229" class="Symbol">=</a> <a id="1231" class="Keyword">record</a>
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<a id="1242" class="Symbol">{</a> <a id="1244" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1166" class="Field">isEquivalence</a> <a id="1258" class="Symbol">=</a> <a id="1260" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#781" class="Function">isIndexedEquivalence</a> <a id="1281" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a>
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<a id="1299" class="Symbol">;</a> <a id="1301" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1207" class="Field">reflexive</a> <a id="1315" class="Symbol">=</a> <a id="1317" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a>
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<a id="1331" class="Symbol">;</a> <a id="1333" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1263" class="Field">trans</a> <a id="1347" class="Symbol">=</a> <a id="1349" href="Relation.Binary.Structures.html#2389" class="Field">trans</a>
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<a id="1359" class="Symbol">}</a>
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<a id="1365" class="Keyword">where</a> <a id="1371" class="Keyword">open</a> <a id="1376" href="Relation.Binary.Structures.html#2191" class="Module">IsPreorder</a> <a id="1387" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1218" class="Bound">isPreorder</a>
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<a id="1399" class="Comment">------------------------------------------------------------------------</a>
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<a id="1472" class="Comment">-- Bundles</a>
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<a id="indexedSetoid"></a><a id="1484" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1484" class="Function">indexedSetoid</a> <a id="1498" class="Symbol">:</a> <a id="1500" class="Symbol">∀</a> <a id="1502" class="Symbol">{</a><a id="1503" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1503" class="Bound">a</a> <a id="1505" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1505" class="Bound">ℓ</a><a id="1506" class="Symbol">}</a> <a id="1508" class="Symbol">→</a> <a id="1510" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="1517" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1503" class="Bound">a</a> <a id="1519" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1505" class="Bound">ℓ</a> <a id="1521" class="Symbol">→</a> <a id="1523" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#803" class="Record">IndexedSetoid</a> <a id="1537" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="1539" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1503" class="Bound">a</a> <a id="1541" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1505" class="Bound">ℓ</a>
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<a id="1543" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1484" class="Function">indexedSetoid</a> <a id="1557" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1557" class="Bound">S</a> <a id="1559" class="Symbol">=</a> <a id="1561" class="Keyword">record</a>
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<a id="1570" class="Symbol">{</a> <a id="1572" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#958" class="Field">isEquivalence</a> <a id="1586" class="Symbol">=</a> <a id="1588" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#781" class="Function">isIndexedEquivalence</a> <a id="1609" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a>
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<a id="1625" class="Symbol">}</a>
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<a id="1629" class="Keyword">where</a> <a id="1635" class="Keyword">open</a> <a id="1640" href="Relation.Binary.Bundles.html#1080" class="Module">Setoid</a> <a id="1647" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1557" class="Bound">S</a>
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<a id="indexedPreorder"></a><a id="1650" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1650" class="Function">indexedPreorder</a> <a id="1666" class="Symbol">:</a> <a id="1668" class="Symbol">∀</a> <a id="1670" class="Symbol">{</a><a id="1671" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1671" class="Bound">a</a> <a id="1673" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1673" class="Bound">ℓ₁</a> <a id="1676" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1676" class="Bound">ℓ₂</a><a id="1678" class="Symbol">}</a> <a id="1680" class="Symbol">→</a> <a id="1682" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="1691" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1671" class="Bound">a</a> <a id="1693" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1673" class="Bound">ℓ₁</a> <a id="1696" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1676" class="Bound">ℓ₂</a> <a id="1699" class="Symbol">→</a>
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<a id="1719" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#1066" class="Record">IndexedPreorder</a> <a id="1735" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="1737" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1671" class="Bound">a</a> <a id="1739" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1673" class="Bound">ℓ₁</a> <a id="1742" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1676" class="Bound">ℓ₂</a>
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<a id="1745" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1650" class="Function">indexedPreorder</a> <a id="1761" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1761" class="Bound">O</a> <a id="1763" class="Symbol">=</a> <a id="1765" class="Keyword">record</a>
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<a id="1774" class="Symbol">{</a> <a id="1776" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#1335" class="Field">isPreorder</a> <a id="1787" class="Symbol">=</a> <a id="1789" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1038" class="Function">isIndexedPreorder</a> <a id="1807" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a>
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<a id="1820" class="Symbol">}</a>
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<a id="1824" class="Keyword">where</a> <a id="1830" class="Keyword">open</a> <a id="1835" href="Relation.Binary.Bundles.html#2121" class="Module">Preorder</a> <a id="1844" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1761" class="Bound">O</a>
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</pre></body></html>
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