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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>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Creates trivially indexed records from their non-indexed counterpart.</a>
<a id="179" class="Comment">------------------------------------------------------------------------</a>
<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>
<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>
<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#320" 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>
<a id="386" class="Keyword">open</a> <a id="391" class="Keyword">import</a> <a id="398" href="Relation.Binary.html" class="Module">Relation.Binary</a>
<a id="414" class="Keyword">open</a> <a id="419" class="Keyword">import</a> <a id="426" href="Relation.Binary.Indexed.Heterogeneous.html" class="Module">Relation.Binary.Indexed.Heterogeneous</a> <a id="464" class="Keyword">hiding</a> <a id="471" class="Symbol">(</a><a id="472" href="Relation.Binary.Indexed.Heterogeneous.html#1076" class="Function">Rel</a><a id="475" class="Symbol">)</a>
<a id="479" class="Keyword">hiding</a> <a id="486" class="Symbol">(</a><a id="487" href="Relation.Binary.Indexed.Heterogeneous.html#1313" class="Function">IsEquivalence</a><a id="500" class="Symbol">;</a> <a id="502" href="Relation.Binary.Indexed.Heterogeneous.html#1181" class="Function">Setoid</a><a id="508" class="Symbol">)</a>
<a id="511" class="Comment">------------------------------------------------------------------------</a>
<a id="584" class="Comment">-- Structures</a>
<a id="599" class="Keyword">module</a> <a id="606" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#606" class="Module">_</a> <a id="608" class="Symbol">{</a><a id="609" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#609" class="Bound">a</a><a id="610" class="Symbol">}</a> <a id="612" class="Symbol">{</a><a id="613" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#613" class="Bound">A</a> <a id="615" class="Symbol">:</a> <a id="617" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="621" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#609" class="Bound">a</a><a id="622" class="Symbol">}</a> <a id="624" class="Keyword">where</a>
<a id="633" class="Keyword">private</a>
<a id="645" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#645" class="Function">Aᵢ</a> <a id="648" class="Symbol">:</a> <a id="650" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="652" class="Symbol">→</a> <a id="654" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="658" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#609" class="Bound">a</a>
<a id="664" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#645" class="Function">Aᵢ</a> <a id="667" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#667" class="Bound">i</a> <a id="669" class="Symbol">=</a> <a id="671" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#613" class="Bound">A</a>
<a id="676" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#676" class="Function">isIndexedEquivalence</a> <a id="697" class="Symbol">:</a> <a id="699" class="Symbol">∀</a> <a id="701" class="Symbol">{</a><a id="702" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#702" class="Bound">ℓ</a><a id="703" class="Symbol">}</a> <a id="705" class="Symbol">{</a><a id="706" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#706" class="Bound Operator">_≈_</a> <a id="710" class="Symbol">:</a> <a id="712" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="716" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#613" class="Bound">A</a> <a id="718" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#702" class="Bound">ℓ</a><a id="719" class="Symbol">}</a> <a id="721" class="Symbol">→</a> <a id="723" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a> <a id="737" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#706" class="Bound Operator">_≈_</a> <a id="741" class="Symbol">→</a>
<a id="768" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#861" class="Record">IsIndexedEquivalence</a> <a id="789" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#645" class="Function">Aᵢ</a> <a id="792" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#706" class="Bound Operator">_≈_</a>
<a id="798" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#676" class="Function">isIndexedEquivalence</a> <a id="819" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#819" class="Bound">isEq</a> <a id="824" class="Symbol">=</a> <a id="826" class="Keyword">record</a>
<a id="837" class="Symbol">{</a> <a id="839" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#918" class="Field">refl</a> <a id="845" class="Symbol">=</a> <a id="847" href="Relation.Binary.Structures.html#1577" class="Field">refl</a>
<a id="856" class="Symbol">;</a> <a id="858" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#947" class="Field">sym</a> <a id="864" class="Symbol">=</a> <a id="866" href="Relation.Binary.Structures.html#1603" class="Field">sym</a>
<a id="874" class="Symbol">;</a> <a id="876" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#976" class="Field">trans</a> <a id="882" class="Symbol">=</a> <a id="884" href="Relation.Binary.Structures.html#1629" class="Field">trans</a>
<a id="894" class="Symbol">}</a>
<a id="900" class="Keyword">where</a> <a id="906" class="Keyword">open</a> <a id="911" href="Relation.Binary.Structures.html#1531" class="Module">IsEquivalence</a> <a id="925" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#819" class="Bound">isEq</a>
<a id="933" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#933" class="Function">isIndexedPreorder</a> <a id="951" class="Symbol">:</a> <a id="953" class="Symbol">∀</a> <a id="955" class="Symbol">{</a><a id="956" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#956" class="Bound">ℓ₁</a> <a id="959" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#959" class="Bound">ℓ₂</a><a id="961" class="Symbol">}</a> <a id="963" class="Symbol">{</a><a id="964" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#964" class="Bound Operator">_≈_</a> <a id="968" class="Symbol">:</a> <a id="970" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="974" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#613" class="Bound">A</a> <a id="976" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#956" class="Bound">ℓ₁</a><a id="978" class="Symbol">}</a> <a id="980" class="Symbol">{</a><a id="981" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#981" class="Bound Operator">_∼_</a> <a id="985" class="Symbol">:</a> <a id="987" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="991" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#613" class="Bound">A</a> <a id="993" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#959" class="Bound">ℓ₂</a><a id="995" class="Symbol">}</a> <a id="997" class="Symbol">→</a>
<a id="1021" href="Relation.Binary.Structures.html#2172" class="Record">IsPreorder</a> <a id="1032" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#964" class="Bound Operator">_≈_</a> <a id="1036" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#981" class="Bound Operator">_∼_</a> <a id="1040" class="Symbol">→</a>
<a id="1064" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1079" class="Record">IsIndexedPreorder</a> <a id="1082" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#645" class="Function">Aᵢ</a> <a id="1085" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#964" class="Bound Operator">_≈_</a> <a id="1089" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#981" class="Bound Operator">_∼_</a>
<a id="1095" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#933" class="Function">isIndexedPreorder</a> <a id="1113" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1113" class="Bound">isPreorder</a> <a id="1124" class="Symbol">=</a> <a id="1126" class="Keyword">record</a>
<a id="1137" class="Symbol">{</a> <a id="1139" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1161" class="Field">isEquivalence</a> <a id="1153" class="Symbol">=</a> <a id="1155" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#676" class="Function">isIndexedEquivalence</a> <a id="1176" href="Relation.Binary.Structures.html#2237" class="Field">isEquivalence</a>
<a id="1194" class="Symbol">;</a> <a id="1196" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1202" class="Field">reflexive</a> <a id="1210" class="Symbol">=</a> <a id="1212" href="Relation.Binary.Structures.html#2340" class="Field">reflexive</a>
<a id="1226" class="Symbol">;</a> <a id="1228" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#1258" class="Field">trans</a> <a id="1242" class="Symbol">=</a> <a id="1244" href="Relation.Binary.Structures.html#2370" class="Field">trans</a>
<a id="1254" class="Symbol">}</a>
<a id="1260" class="Keyword">where</a> <a id="1266" class="Keyword">open</a> <a id="1271" href="Relation.Binary.Structures.html#2172" class="Module">IsPreorder</a> <a id="1282" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1113" class="Bound">isPreorder</a>
<a id="1294" class="Comment">------------------------------------------------------------------------</a>
<a id="1367" class="Comment">-- Bundles</a>
<a id="indexedSetoid"></a><a id="1379" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1379" class="Function">indexedSetoid</a> <a id="1393" class="Symbol">:</a> <a id="1395" class="Symbol">∀</a> <a id="1397" class="Symbol">{</a><a id="1398" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1398" class="Bound">a</a> <a id="1400" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1400" class="Bound">ℓ</a><a id="1401" class="Symbol">}</a> <a id="1403" class="Symbol">→</a> <a id="1405" href="Relation.Binary.Bundles.html#1018" class="Record">Setoid</a> <a id="1412" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1398" class="Bound">a</a> <a id="1414" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1400" class="Bound">ℓ</a> <a id="1416" class="Symbol">→</a> <a id="1418" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#798" class="Record">IndexedSetoid</a> <a id="1432" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="1434" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1398" class="Bound">a</a> <a id="1436" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1400" class="Bound">ℓ</a>
<a id="1438" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1379" class="Function">indexedSetoid</a> <a id="1452" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1452" class="Bound">S</a> <a id="1454" class="Symbol">=</a> <a id="1456" class="Keyword">record</a>
<a id="1465" class="Symbol">{</a> <a id="1467" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#953" class="Field">isEquivalence</a> <a id="1481" class="Symbol">=</a> <a id="1483" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#676" class="Function">isIndexedEquivalence</a> <a id="1504" href="Relation.Binary.Bundles.html#1141" class="Field">isEquivalence</a>
<a id="1520" class="Symbol">}</a>
<a id="1524" class="Keyword">where</a> <a id="1530" class="Keyword">open</a> <a id="1535" href="Relation.Binary.Bundles.html#1018" class="Module">Setoid</a> <a id="1542" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1452" class="Bound">S</a>
<a id="indexedPreorder"></a><a id="1545" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1545" class="Function">indexedPreorder</a> <a id="1561" class="Symbol">:</a> <a id="1563" class="Symbol">∀</a> <a id="1565" class="Symbol">{</a><a id="1566" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1566" class="Bound">a</a> <a id="1568" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1568" class="Bound">ℓ₁</a> <a id="1571" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1571" class="Bound">ℓ₂</a><a id="1573" class="Symbol">}</a> <a id="1575" class="Symbol">→</a> <a id="1577" href="Relation.Binary.Bundles.html#1929" class="Record">Preorder</a> <a id="1586" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1566" class="Bound">a</a> <a id="1588" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1568" class="Bound">ℓ₁</a> <a id="1591" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1571" class="Bound">ℓ₂</a> <a id="1594" class="Symbol">→</a>
<a id="1614" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#1061" class="Record">IndexedPreorder</a> <a id="1630" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#368" class="Bound">I</a> <a id="1632" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1566" class="Bound">a</a> <a id="1634" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1568" class="Bound">ℓ₁</a> <a id="1637" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1571" class="Bound">ℓ₂</a>
<a id="1640" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1545" class="Function">indexedPreorder</a> <a id="1656" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1656" class="Bound">O</a> <a id="1658" class="Symbol">=</a> <a id="1660" class="Keyword">record</a>
<a id="1669" class="Symbol">{</a> <a id="1671" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#1330" class="Field">isPreorder</a> <a id="1682" class="Symbol">=</a> <a id="1684" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#933" class="Function">isIndexedPreorder</a> <a id="1702" href="Relation.Binary.Bundles.html#2142" class="Field">isPreorder</a>
<a id="1715" class="Symbol">}</a>
<a id="1719" class="Keyword">where</a> <a id="1725" class="Keyword">open</a> <a id="1730" href="Relation.Binary.Bundles.html#1929" class="Module">Preorder</a> <a id="1739" href="Relation.Binary.Indexed.Heterogeneous.Construct.Trivial.html#1656" class="Bound">O</a>
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