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<html><head><meta charset="utf-8"><title>Relation.Binary.PropositionalEquality.Core</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">-- Propositional equality</a>
<a id="132" class="Comment">--</a>
<a id="135" class="Comment">-- This file contains some core definitions which are re-exported by</a>
<a id="204" class="Comment">-- Relation.Binary.PropositionalEquality.</a>
<a id="246" class="Comment">------------------------------------------------------------------------</a>
<a id="320" class="Symbol">{-#</a> <a id="324" class="Keyword">OPTIONS</a> <a id="332" class="Pragma">--cubical-compatible</a> <a id="353" class="Pragma">--safe</a> <a id="360" class="Symbol">#-}</a>
<a id="365" class="Keyword">module</a> <a id="372" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a> <a id="415" class="Keyword">where</a>
<a id="422" class="Keyword">open</a> <a id="427" class="Keyword">import</a> <a id="434" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="452" class="Keyword">using</a> <a id="458" class="Symbol">(</a><a id="459" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="462" class="Symbol">)</a>
<a id="464" class="Keyword">open</a> <a id="469" class="Keyword">import</a> <a id="476" href="Function.Base.html" class="Module">Function.Base</a> <a id="490" class="Keyword">using</a> <a id="496" class="Symbol">(</a><a id="497" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="500" class="Symbol">)</a>
<a id="502" class="Keyword">open</a> <a id="507" class="Keyword">import</a> <a id="514" href="Level.html" class="Module">Level</a>
<a id="520" class="Keyword">open</a> <a id="525" class="Keyword">import</a> <a id="532" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a>
<a id="553" class="Keyword">open</a> <a id="558" class="Keyword">import</a> <a id="565" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
<a id="593" class="Keyword">open</a> <a id="598" class="Keyword">import</a> <a id="605" href="Relation.Nullary.Negation.Core.html" class="Module">Relation.Nullary.Negation.Core</a> <a id="636" class="Keyword">using</a> <a id="642" class="Symbol">(</a><a id="643" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬_</a><a id="645" class="Symbol">)</a>
<a id="648" class="Keyword">private</a>
<a id="658" class="Keyword">variable</a>
<a id="671" href="Relation.Binary.PropositionalEquality.Core.html#671" class="Generalizable">a</a> <a id="673" href="Relation.Binary.PropositionalEquality.Core.html#673" class="Generalizable">b</a> <a id="675" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a> <a id="677" class="Symbol">:</a> <a id="679" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="689" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="691" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a> <a id="693" href="Relation.Binary.PropositionalEquality.Core.html#693" class="Generalizable">C</a> <a id="695" class="Symbol">:</a> <a id="697" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="701" href="Relation.Binary.PropositionalEquality.Core.html#671" class="Generalizable">a</a>
<a id="704" class="Comment">------------------------------------------------------------------------</a>
<a id="777" class="Comment">-- Propositional equality</a>
<a id="804" class="Keyword">open</a> <a id="809" class="Keyword">import</a> <a id="816" href="Agda.Builtin.Equality.html" class="Module">Agda.Builtin.Equality</a> <a id="838" class="Keyword">public</a>
<a id="846" class="Keyword">infix</a> <a id="852" class="Number">4</a> <a id="854" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">_≢_</a>
<a id="_≢_"></a><a id="858" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">_≢_</a> <a id="862" class="Symbol">:</a> <a id="864" class="Symbol">{</a><a id="865" href="Relation.Binary.PropositionalEquality.Core.html#865" class="Bound">A</a> <a id="867" class="Symbol">:</a> <a id="869" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="873" href="Relation.Binary.PropositionalEquality.Core.html#671" class="Generalizable">a</a><a id="874" class="Symbol">}</a> <a id="876" class="Symbol">→</a> <a id="878" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="882" href="Relation.Binary.PropositionalEquality.Core.html#865" class="Bound">A</a> <a id="884" href="Relation.Binary.PropositionalEquality.Core.html#671" class="Generalizable">a</a>
<a id="886" href="Relation.Binary.PropositionalEquality.Core.html#886" class="Bound">x</a> <a id="888" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">≢</a> <a id="890" href="Relation.Binary.PropositionalEquality.Core.html#890" class="Bound">y</a> <a id="892" class="Symbol">=</a> <a id="894" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="896" href="Relation.Binary.PropositionalEquality.Core.html#886" class="Bound">x</a> <a id="898" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="900" href="Relation.Binary.PropositionalEquality.Core.html#890" class="Bound">y</a>
<a id="903" class="Comment">------------------------------------------------------------------------</a>
<a id="976" class="Comment">-- A variant of `refl` where the argument is explicit</a>
<a id="1031" class="Keyword">pattern</a> <a id="erefl"></a><a id="1039" href="Relation.Binary.PropositionalEquality.Core.html#1039" class="InductiveConstructor">erefl</a> <a id="1045" href="Relation.Binary.PropositionalEquality.Core.html#1059" class="Bound">x</a> <a id="1047" class="Symbol">=</a> <a id="1049" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1054" class="Symbol">{</a><a id="1055" class="Argument">x</a> <a id="1057" class="Symbol">=</a> <a id="1059" href="Relation.Binary.PropositionalEquality.Core.html#1059" class="Bound">x</a><a id="1060" class="Symbol">}</a>
<a id="1063" class="Comment">------------------------------------------------------------------------</a>
<a id="1136" class="Comment">-- Congruence lemmas</a>
<a id="cong"></a><a id="1158" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="1163" class="Symbol">:</a> <a id="1165" class="Symbol">∀</a> <a id="1167" class="Symbol">(</a><a id="1168" href="Relation.Binary.PropositionalEquality.Core.html#1168" class="Bound">f</a> <a id="1170" class="Symbol">:</a> <a id="1172" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1174" class="Symbol">→</a> <a id="1176" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a><a id="1177" class="Symbol">)</a> <a id="1179" class="Symbol">{</a><a id="1180" href="Relation.Binary.PropositionalEquality.Core.html#1180" class="Bound">x</a> <a id="1182" href="Relation.Binary.PropositionalEquality.Core.html#1182" class="Bound">y</a><a id="1183" class="Symbol">}</a> <a id="1185" class="Symbol">→</a> <a id="1187" href="Relation.Binary.PropositionalEquality.Core.html#1180" class="Bound">x</a> <a id="1189" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1191" href="Relation.Binary.PropositionalEquality.Core.html#1182" class="Bound">y</a> <a id="1193" class="Symbol">→</a> <a id="1195" href="Relation.Binary.PropositionalEquality.Core.html#1168" class="Bound">f</a> <a id="1197" href="Relation.Binary.PropositionalEquality.Core.html#1180" class="Bound">x</a> <a id="1199" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1201" href="Relation.Binary.PropositionalEquality.Core.html#1168" class="Bound">f</a> <a id="1203" href="Relation.Binary.PropositionalEquality.Core.html#1182" class="Bound">y</a>
<a id="1205" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="1210" href="Relation.Binary.PropositionalEquality.Core.html#1210" class="Bound">f</a> <a id="1212" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1217" class="Symbol">=</a> <a id="1219" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="cong′"></a><a id="1225" href="Relation.Binary.PropositionalEquality.Core.html#1225" class="Function">cong′</a> <a id="1231" class="Symbol">:</a> <a id="1233" class="Symbol">∀</a> <a id="1235" class="Symbol">{</a><a id="1236" href="Relation.Binary.PropositionalEquality.Core.html#1236" class="Bound">f</a> <a id="1238" class="Symbol">:</a> <a id="1240" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1242" class="Symbol">→</a> <a id="1244" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a><a id="1245" class="Symbol">}</a> <a id="1247" href="Relation.Binary.PropositionalEquality.Core.html#1247" class="Bound">x</a> <a id="1249" class="Symbol">→</a> <a id="1251" href="Relation.Binary.PropositionalEquality.Core.html#1236" class="Bound">f</a> <a id="1253" href="Relation.Binary.PropositionalEquality.Core.html#1247" class="Bound">x</a> <a id="1255" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1257" href="Relation.Binary.PropositionalEquality.Core.html#1236" class="Bound">f</a> <a id="1259" href="Relation.Binary.PropositionalEquality.Core.html#1247" class="Bound">x</a>
<a id="1261" href="Relation.Binary.PropositionalEquality.Core.html#1225" class="Function">cong′</a> <a id="1267" class="Symbol">_</a> <a id="1269" class="Symbol">=</a> <a id="1271" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="icong"></a><a id="1277" href="Relation.Binary.PropositionalEquality.Core.html#1277" class="Function">icong</a> <a id="1283" class="Symbol">:</a> <a id="1285" class="Symbol">∀</a> <a id="1287" class="Symbol">{</a><a id="1288" href="Relation.Binary.PropositionalEquality.Core.html#1288" class="Bound">f</a> <a id="1290" class="Symbol">:</a> <a id="1292" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1294" class="Symbol">→</a> <a id="1296" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a><a id="1297" class="Symbol">}</a> <a id="1299" class="Symbol">{</a><a id="1300" href="Relation.Binary.PropositionalEquality.Core.html#1300" class="Bound">x</a> <a id="1302" href="Relation.Binary.PropositionalEquality.Core.html#1302" class="Bound">y</a><a id="1303" class="Symbol">}</a> <a id="1305" class="Symbol">→</a> <a id="1307" href="Relation.Binary.PropositionalEquality.Core.html#1300" class="Bound">x</a> <a id="1309" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1311" href="Relation.Binary.PropositionalEquality.Core.html#1302" class="Bound">y</a> <a id="1313" class="Symbol">→</a> <a id="1315" href="Relation.Binary.PropositionalEquality.Core.html#1288" class="Bound">f</a> <a id="1317" href="Relation.Binary.PropositionalEquality.Core.html#1300" class="Bound">x</a> <a id="1319" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1321" href="Relation.Binary.PropositionalEquality.Core.html#1288" class="Bound">f</a> <a id="1323" href="Relation.Binary.PropositionalEquality.Core.html#1302" class="Bound">y</a>
<a id="1325" href="Relation.Binary.PropositionalEquality.Core.html#1277" class="Function">icong</a> <a id="1331" class="Symbol">=</a> <a id="1333" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="1338" class="Symbol">_</a>
<a id="icong′"></a><a id="1341" href="Relation.Binary.PropositionalEquality.Core.html#1341" class="Function">icong′</a> <a id="1348" class="Symbol">:</a> <a id="1350" class="Symbol">∀</a> <a id="1352" class="Symbol">{</a><a id="1353" href="Relation.Binary.PropositionalEquality.Core.html#1353" class="Bound">f</a> <a id="1355" class="Symbol">:</a> <a id="1357" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1359" class="Symbol">→</a> <a id="1361" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a><a id="1362" class="Symbol">}</a> <a id="1364" href="Relation.Binary.PropositionalEquality.Core.html#1364" class="Bound">x</a> <a id="1366" class="Symbol">→</a> <a id="1368" href="Relation.Binary.PropositionalEquality.Core.html#1353" class="Bound">f</a> <a id="1370" href="Relation.Binary.PropositionalEquality.Core.html#1364" class="Bound">x</a> <a id="1372" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1374" href="Relation.Binary.PropositionalEquality.Core.html#1353" class="Bound">f</a> <a id="1376" href="Relation.Binary.PropositionalEquality.Core.html#1364" class="Bound">x</a>
<a id="1378" href="Relation.Binary.PropositionalEquality.Core.html#1341" class="Function">icong′</a> <a id="1385" class="Symbol">_</a> <a id="1387" class="Symbol">=</a> <a id="1389" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="cong₂"></a><a id="1395" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="1401" class="Symbol">:</a> <a id="1403" class="Symbol">∀</a> <a id="1405" class="Symbol">(</a><a id="1406" href="Relation.Binary.PropositionalEquality.Core.html#1406" class="Bound">f</a> <a id="1408" class="Symbol">:</a> <a id="1410" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1412" class="Symbol">→</a> <a id="1414" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a> <a id="1416" class="Symbol">→</a> <a id="1418" href="Relation.Binary.PropositionalEquality.Core.html#693" class="Generalizable">C</a><a id="1419" class="Symbol">)</a> <a id="1421" class="Symbol">{</a><a id="1422" href="Relation.Binary.PropositionalEquality.Core.html#1422" class="Bound">x</a> <a id="1424" href="Relation.Binary.PropositionalEquality.Core.html#1424" class="Bound">y</a> <a id="1426" href="Relation.Binary.PropositionalEquality.Core.html#1426" class="Bound">u</a> <a id="1428" href="Relation.Binary.PropositionalEquality.Core.html#1428" class="Bound">v</a><a id="1429" class="Symbol">}</a> <a id="1431" class="Symbol">→</a> <a id="1433" href="Relation.Binary.PropositionalEquality.Core.html#1422" class="Bound">x</a> <a id="1435" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1437" href="Relation.Binary.PropositionalEquality.Core.html#1424" class="Bound">y</a> <a id="1439" class="Symbol">→</a> <a id="1441" href="Relation.Binary.PropositionalEquality.Core.html#1426" class="Bound">u</a> <a id="1443" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1445" href="Relation.Binary.PropositionalEquality.Core.html#1428" class="Bound">v</a> <a id="1447" class="Symbol">→</a> <a id="1449" href="Relation.Binary.PropositionalEquality.Core.html#1406" class="Bound">f</a> <a id="1451" href="Relation.Binary.PropositionalEquality.Core.html#1422" class="Bound">x</a> <a id="1453" href="Relation.Binary.PropositionalEquality.Core.html#1426" class="Bound">u</a> <a id="1455" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1457" href="Relation.Binary.PropositionalEquality.Core.html#1406" class="Bound">f</a> <a id="1459" href="Relation.Binary.PropositionalEquality.Core.html#1424" class="Bound">y</a> <a id="1461" href="Relation.Binary.PropositionalEquality.Core.html#1428" class="Bound">v</a>
<a id="1463" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="1469" href="Relation.Binary.PropositionalEquality.Core.html#1469" class="Bound">f</a> <a id="1471" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1476" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1481" class="Symbol">=</a> <a id="1483" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="cong-app"></a><a id="1489" href="Relation.Binary.PropositionalEquality.Core.html#1489" class="Function">cong-app</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.PropositionalEquality.Core.html#1503" class="Bound">A</a> <a id="1505" class="Symbol">:</a> <a id="1507" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1511" href="Relation.Binary.PropositionalEquality.Core.html#671" class="Generalizable">a</a><a id="1512" class="Symbol">}</a> <a id="1514" class="Symbol">{</a><a id="1515" href="Relation.Binary.PropositionalEquality.Core.html#1515" class="Bound">B</a> <a id="1517" class="Symbol">:</a> <a id="1519" href="Relation.Binary.PropositionalEquality.Core.html#1503" class="Bound">A</a> <a id="1521" class="Symbol">→</a> <a id="1523" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1527" href="Relation.Binary.PropositionalEquality.Core.html#673" class="Generalizable">b</a><a id="1528" class="Symbol">}</a> <a id="1530" class="Symbol">{</a><a id="1531" href="Relation.Binary.PropositionalEquality.Core.html#1531" class="Bound">f</a> <a id="1533" href="Relation.Binary.PropositionalEquality.Core.html#1533" class="Bound">g</a> <a id="1535" class="Symbol">:</a> <a id="1537" class="Symbol">(</a><a id="1538" href="Relation.Binary.PropositionalEquality.Core.html#1538" class="Bound">x</a> <a id="1540" class="Symbol">:</a> <a id="1542" href="Relation.Binary.PropositionalEquality.Core.html#1503" class="Bound">A</a><a id="1543" class="Symbol">)</a> <a id="1545" class="Symbol">→</a> <a id="1547" href="Relation.Binary.PropositionalEquality.Core.html#1515" class="Bound">B</a> <a id="1549" href="Relation.Binary.PropositionalEquality.Core.html#1538" class="Bound">x</a><a id="1550" class="Symbol">}</a> <a id="1552" class="Symbol">→</a>
<a id="1565" href="Relation.Binary.PropositionalEquality.Core.html#1531" class="Bound">f</a> <a id="1567" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1569" href="Relation.Binary.PropositionalEquality.Core.html#1533" class="Bound">g</a> <a id="1571" class="Symbol">→</a> <a id="1573" class="Symbol">(</a><a id="1574" href="Relation.Binary.PropositionalEquality.Core.html#1574" class="Bound">x</a> <a id="1576" class="Symbol">:</a> <a id="1578" href="Relation.Binary.PropositionalEquality.Core.html#1503" class="Bound">A</a><a id="1579" class="Symbol">)</a> <a id="1581" class="Symbol">→</a> <a id="1583" href="Relation.Binary.PropositionalEquality.Core.html#1531" class="Bound">f</a> <a id="1585" href="Relation.Binary.PropositionalEquality.Core.html#1574" class="Bound">x</a> <a id="1587" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1589" href="Relation.Binary.PropositionalEquality.Core.html#1533" class="Bound">g</a> <a id="1591" href="Relation.Binary.PropositionalEquality.Core.html#1574" class="Bound">x</a>
<a id="1593" href="Relation.Binary.PropositionalEquality.Core.html#1489" class="Function">cong-app</a> <a id="1602" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1607" href="Relation.Binary.PropositionalEquality.Core.html#1607" class="Bound">x</a> <a id="1609" class="Symbol">=</a> <a id="1611" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="1617" class="Comment">------------------------------------------------------------------------</a>
<a id="1690" class="Comment">-- Properties of _≡_</a>
<a id="sym"></a><a id="1712" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="1716" class="Symbol">:</a> <a id="1718" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1728" class="Symbol">{</a><a id="1729" class="Argument">A</a> <a id="1731" class="Symbol">=</a> <a id="1733" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a><a id="1734" class="Symbol">}</a> <a id="1736" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1740" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="1744" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1749" class="Symbol">=</a> <a id="1751" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="trans"></a><a id="1757" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="1763" class="Symbol">:</a> <a id="1765" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="1776" class="Symbol">{</a><a id="1777" class="Argument">A</a> <a id="1779" class="Symbol">=</a> <a id="1781" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a><a id="1782" class="Symbol">}</a> <a id="1784" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1788" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="1794" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1799" href="Relation.Binary.PropositionalEquality.Core.html#1799" class="Bound">eq</a> <a id="1802" class="Symbol">=</a> <a id="1804" href="Relation.Binary.PropositionalEquality.Core.html#1799" class="Bound">eq</a>
<a id="subst"></a><a id="1808" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="1814" class="Symbol">:</a> <a id="1816" href="Relation.Binary.Definitions.html#5955" class="Function">Substitutive</a> <a id="1829" class="Symbol">{</a><a id="1830" class="Argument">A</a> <a id="1832" class="Symbol">=</a> <a id="1834" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a><a id="1835" class="Symbol">}</a> <a id="1837" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1841" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a>
<a id="1843" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="1849" href="Relation.Binary.PropositionalEquality.Core.html#1849" class="Bound">P</a> <a id="1851" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1856" href="Relation.Binary.PropositionalEquality.Core.html#1856" class="Bound">p</a> <a id="1858" class="Symbol">=</a> <a id="1860" href="Relation.Binary.PropositionalEquality.Core.html#1856" class="Bound">p</a>
<a id="subst₂"></a><a id="1863" href="Relation.Binary.PropositionalEquality.Core.html#1863" class="Function">subst₂</a> <a id="1870" class="Symbol">:</a> <a id="1872" class="Symbol">∀</a> <a id="1874" class="Symbol">(</a><a id="1875" href="Relation.Binary.PropositionalEquality.Core.html#1875" class="Bound Operator">_∼_</a> <a id="1879" class="Symbol">:</a> <a id="1881" href="Relation.Binary.Core.html#780" class="Function">REL</a> <a id="1885" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1887" href="Relation.Binary.PropositionalEquality.Core.html#691" class="Generalizable">B</a> <a id="1889" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a><a id="1890" class="Symbol">)</a> <a id="1892" class="Symbol">{</a><a id="1893" href="Relation.Binary.PropositionalEquality.Core.html#1893" class="Bound">x</a> <a id="1895" href="Relation.Binary.PropositionalEquality.Core.html#1895" class="Bound">y</a> <a id="1897" href="Relation.Binary.PropositionalEquality.Core.html#1897" class="Bound">u</a> <a id="1899" href="Relation.Binary.PropositionalEquality.Core.html#1899" class="Bound">v</a><a id="1900" class="Symbol">}</a> <a id="1902" class="Symbol">→</a> <a id="1904" href="Relation.Binary.PropositionalEquality.Core.html#1893" class="Bound">x</a> <a id="1906" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1908" href="Relation.Binary.PropositionalEquality.Core.html#1895" class="Bound">y</a> <a id="1910" class="Symbol">→</a> <a id="1912" href="Relation.Binary.PropositionalEquality.Core.html#1897" class="Bound">u</a> <a id="1914" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1916" href="Relation.Binary.PropositionalEquality.Core.html#1899" class="Bound">v</a> <a id="1918" class="Symbol">→</a> <a id="1920" href="Relation.Binary.PropositionalEquality.Core.html#1893" class="Bound">x</a> <a id="1922" href="Relation.Binary.PropositionalEquality.Core.html#1875" class="Bound Operator">∼</a> <a id="1924" href="Relation.Binary.PropositionalEquality.Core.html#1897" class="Bound">u</a> <a id="1926" class="Symbol">→</a> <a id="1928" href="Relation.Binary.PropositionalEquality.Core.html#1895" class="Bound">y</a> <a id="1930" href="Relation.Binary.PropositionalEquality.Core.html#1875" class="Bound Operator">∼</a> <a id="1932" href="Relation.Binary.PropositionalEquality.Core.html#1899" class="Bound">v</a>
<a id="1934" href="Relation.Binary.PropositionalEquality.Core.html#1863" class="Function">subst₂</a> <a id="1941" class="Symbol">_</a> <a id="1943" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1948" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1953" href="Relation.Binary.PropositionalEquality.Core.html#1953" class="Bound">p</a> <a id="1955" class="Symbol">=</a> <a id="1957" href="Relation.Binary.PropositionalEquality.Core.html#1953" class="Bound">p</a>
<a id="resp"></a><a id="1960" href="Relation.Binary.PropositionalEquality.Core.html#1960" class="Function">resp</a> <a id="1965" class="Symbol">:</a> <a id="1967" class="Symbol">∀</a> <a id="1969" class="Symbol">(</a><a id="1970" href="Relation.Binary.PropositionalEquality.Core.html#1970" class="Bound">P</a> <a id="1972" class="Symbol">:</a> <a id="1974" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="1976" class="Symbol">→</a> <a id="1978" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1982" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a><a id="1983" class="Symbol">)</a> <a id="1985" class="Symbol">→</a> <a id="1987" href="Relation.Binary.PropositionalEquality.Core.html#1970" class="Bound">P</a> <a id="1989" href="Relation.Binary.Definitions.html#5179" class="Function Operator">Respects</a> <a id="1998" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="2002" href="Relation.Binary.PropositionalEquality.Core.html#1960" class="Function">resp</a> <a id="2007" href="Relation.Binary.PropositionalEquality.Core.html#2007" class="Bound">P</a> <a id="2009" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2014" href="Relation.Binary.PropositionalEquality.Core.html#2014" class="Bound">p</a> <a id="2016" class="Symbol">=</a> <a id="2018" href="Relation.Binary.PropositionalEquality.Core.html#2014" class="Bound">p</a>
<a id="respˡ"></a><a id="2021" href="Relation.Binary.PropositionalEquality.Core.html#2021" class="Function">respˡ</a> <a id="2027" class="Symbol">:</a> <a id="2029" class="Symbol">∀</a> <a id="2031" class="Symbol">(</a><a id="2032" href="Relation.Binary.PropositionalEquality.Core.html#2032" class="Bound">∼</a> <a id="2034" class="Symbol">:</a> <a id="2036" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2040" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="2042" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a><a id="2043" class="Symbol">)</a> <a id="2045" class="Symbol">→</a> <a id="2047" href="Relation.Binary.PropositionalEquality.Core.html#2032" class="Bound">∼</a> <a id="2049" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="2059" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="2063" href="Relation.Binary.PropositionalEquality.Core.html#2021" class="Function">respˡ</a> <a id="2069" href="Relation.Binary.PropositionalEquality.Core.html#2069" class="Bound Operator">_∼_</a> <a id="2073" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2078" href="Relation.Binary.PropositionalEquality.Core.html#2078" class="Bound">x∼y</a> <a id="2082" class="Symbol">=</a> <a id="2084" href="Relation.Binary.PropositionalEquality.Core.html#2078" class="Bound">x∼y</a>
<a id="respʳ"></a><a id="2089" href="Relation.Binary.PropositionalEquality.Core.html#2089" class="Function">respʳ</a> <a id="2095" class="Symbol">:</a> <a id="2097" class="Symbol">∀</a> <a id="2099" class="Symbol">(</a><a id="2100" href="Relation.Binary.PropositionalEquality.Core.html#2100" class="Bound">∼</a> <a id="2102" class="Symbol">:</a> <a id="2104" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2108" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="2110" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a><a id="2111" class="Symbol">)</a> <a id="2113" class="Symbol">→</a> <a id="2115" href="Relation.Binary.PropositionalEquality.Core.html#2100" class="Bound">∼</a> <a id="2117" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="2127" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="2131" href="Relation.Binary.PropositionalEquality.Core.html#2089" class="Function">respʳ</a> <a id="2137" href="Relation.Binary.PropositionalEquality.Core.html#2137" class="Bound Operator">_∼_</a> <a id="2141" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2146" href="Relation.Binary.PropositionalEquality.Core.html#2146" class="Bound">x∼y</a> <a id="2150" class="Symbol">=</a> <a id="2152" href="Relation.Binary.PropositionalEquality.Core.html#2146" class="Bound">x∼y</a>
<a id="resp₂"></a><a id="2157" href="Relation.Binary.PropositionalEquality.Core.html#2157" class="Function">resp₂</a> <a id="2163" class="Symbol">:</a> <a id="2165" class="Symbol">∀</a> <a id="2167" class="Symbol">(</a><a id="2168" href="Relation.Binary.PropositionalEquality.Core.html#2168" class="Bound">∼</a> <a id="2170" class="Symbol">:</a> <a id="2172" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2176" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a> <a id="2178" href="Relation.Binary.PropositionalEquality.Core.html#675" class="Generalizable">ℓ</a><a id="2179" class="Symbol">)</a> <a id="2181" class="Symbol">→</a> <a id="2183" href="Relation.Binary.PropositionalEquality.Core.html#2168" class="Bound">∼</a> <a id="2185" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="2195" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="2199" href="Relation.Binary.PropositionalEquality.Core.html#2157" class="Function">resp₂</a> <a id="2205" href="Relation.Binary.PropositionalEquality.Core.html#2205" class="Bound Operator">_∼_</a> <a id="2209" class="Symbol">=</a> <a id="2211" href="Relation.Binary.PropositionalEquality.Core.html#2089" class="Function">respʳ</a> <a id="2217" href="Relation.Binary.PropositionalEquality.Core.html#2205" class="Bound Operator">_∼_</a> <a id="2221" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2223" href="Relation.Binary.PropositionalEquality.Core.html#2021" class="Function">respˡ</a> <a id="2229" href="Relation.Binary.PropositionalEquality.Core.html#2205" class="Bound Operator">_∼_</a>
<a id="2234" class="Comment">------------------------------------------------------------------------</a>
<a id="2307" class="Comment">-- Properties of _≢_</a>
<a id="≢-sym"></a><a id="2329" href="Relation.Binary.PropositionalEquality.Core.html#2329" class="Function">≢-sym</a> <a id="2335" class="Symbol">:</a> <a id="2337" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2347" class="Symbol">{</a><a id="2348" class="Argument">A</a> <a id="2350" class="Symbol">=</a> <a id="2352" href="Relation.Binary.PropositionalEquality.Core.html#689" class="Generalizable">A</a><a id="2353" class="Symbol">}</a> <a id="2355" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">_≢_</a>
<a id="2359" href="Relation.Binary.PropositionalEquality.Core.html#2329" class="Function">≢-sym</a> <a id="2365" href="Relation.Binary.PropositionalEquality.Core.html#2365" class="Bound">x≢y</a> <a id="2369" class="Symbol">=</a> <a id="2372" href="Relation.Binary.PropositionalEquality.Core.html#2365" class="Bound">x≢y</a> <a id="2376" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2378" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a>
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