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Cubical.Data.Sigma.Properties.html
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Cubical.Data.Sigma.Properties.html
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<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Cubical.Data.Sigma.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">{-
Basic properties about Σ-types
- Action of Σ on functions ([map-fst], [map-snd])
- Characterization of equality in Σ-types using dependent paths ([ΣPath{Iso,≃,≡}PathΣ], [Σ≡Prop])
- Proof that discrete types are closed under Σ ([discreteΣ])
- Commutativity and associativity ([Σ-swap-*, Σ-assoc-*])
- Distributivity of Π over Σ ([Σ-Π-*])
- Action of Σ on isomorphisms, equivalences, and paths ([Σ-cong-fst], [Σ-cong-snd], ...)
- Characterization of equality in Σ-types using transport ([ΣPathTransport{≃,≡}PathΣ])
- Σ with a contractible base is its fiber ([Σ-contractFst, ΣUnit])
-}</a>
<a id="590" class="Symbol">{-#</a> <a id="594" class="Keyword">OPTIONS</a> <a id="602" class="Pragma">--safe</a> <a id="609" class="Symbol">#-}</a>
<a id="613" class="Keyword">module</a> <a id="620" href="Cubical.Data.Sigma.Properties.html" class="Module">Cubical.Data.Sigma.Properties</a> <a id="650" class="Keyword">where</a>
<a id="657" class="Keyword">open</a> <a id="662" class="Keyword">import</a> <a id="669" href="Cubical.Data.Sigma.Base.html" class="Module">Cubical.Data.Sigma.Base</a>
<a id="694" class="Keyword">open</a> <a id="699" class="Keyword">import</a> <a id="706" href="Cubical.Core.Everything.html" class="Module">Cubical.Core.Everything</a>
<a id="731" class="Keyword">open</a> <a id="736" class="Keyword">import</a> <a id="743" href="Cubical.Foundations.Prelude.html" class="Module">Cubical.Foundations.Prelude</a>
<a id="771" class="Keyword">open</a> <a id="776" class="Keyword">import</a> <a id="783" href="Cubical.Foundations.Function.html" class="Module">Cubical.Foundations.Function</a>
<a id="812" class="Keyword">open</a> <a id="817" class="Keyword">import</a> <a id="824" href="Cubical.Foundations.Isomorphism.html" class="Module">Cubical.Foundations.Isomorphism</a>
<a id="856" class="Keyword">open</a> <a id="861" class="Keyword">import</a> <a id="868" href="Cubical.Foundations.Equiv.html" class="Module">Cubical.Foundations.Equiv</a>
<a id="894" class="Keyword">open</a> <a id="899" class="Keyword">import</a> <a id="906" href="Cubical.Foundations.Equiv.HalfAdjoint.html" class="Module">Cubical.Foundations.Equiv.HalfAdjoint</a>
<a id="944" class="Keyword">open</a> <a id="949" class="Keyword">import</a> <a id="956" href="Cubical.Foundations.GroupoidLaws.html" class="Module">Cubical.Foundations.GroupoidLaws</a>
<a id="989" class="Keyword">open</a> <a id="994" class="Keyword">import</a> <a id="1001" href="Cubical.Foundations.Path.html" class="Module">Cubical.Foundations.Path</a>
<a id="1026" class="Keyword">open</a> <a id="1031" class="Keyword">import</a> <a id="1038" href="Cubical.Foundations.Transport.html" class="Module">Cubical.Foundations.Transport</a>
<a id="1068" class="Keyword">open</a> <a id="1073" class="Keyword">import</a> <a id="1080" href="Cubical.Foundations.Univalence.html" class="Module">Cubical.Foundations.Univalence</a>
<a id="1111" class="Keyword">open</a> <a id="1116" class="Keyword">import</a> <a id="1123" href="Cubical.Relation.Nullary.html" class="Module">Cubical.Relation.Nullary</a>
<a id="1148" class="Keyword">open</a> <a id="1153" class="Keyword">import</a> <a id="1160" href="Cubical.Data.Unit.Base.html" class="Module">Cubical.Data.Unit.Base</a>
<a id="1183" class="Keyword">open</a> <a id="1188" class="Keyword">import</a> <a id="1195" href="Cubical.Data.Empty.Base.html" class="Module">Cubical.Data.Empty.Base</a>
<a id="1220" class="Keyword">open</a> <a id="1225" class="Keyword">import</a> <a id="1232" href="Cubical.Reflection.StrictEquiv.html" class="Module">Cubical.Reflection.StrictEquiv</a>
<a id="1264" class="Keyword">open</a> <a id="1269" href="Cubical.Foundations.Isomorphism.html#773" class="Module">Iso</a>
<a id="1274" class="Keyword">private</a>
<a id="1284" class="Keyword">variable</a>
<a id="1297" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a> <a id="1299" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a> <a id="1302" href="Cubical.Data.Sigma.Properties.html#1302" class="Generalizable">ℓ''</a> <a id="1306" class="Symbol">:</a> <a id="1308" href="Agda.Primitive.html#597" class="Postulate">Level</a>
<a id="1318" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1320" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="1323" class="Symbol">:</a> <a id="1325" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1330" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a>
<a id="1336" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="1338" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="1341" class="Symbol">:</a> <a id="1343" class="Symbol">(</a><a id="1344" href="Cubical.Data.Sigma.Properties.html#1344" class="Bound">a</a> <a id="1346" class="Symbol">:</a> <a id="1348" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="1349" class="Symbol">)</a> <a id="1351" class="Symbol">→</a> <a id="1353" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1358" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a>
<a id="1364" href="Cubical.Data.Sigma.Properties.html#1364" class="Generalizable">C</a> <a id="1366" class="Symbol">:</a> <a id="1368" class="Symbol">(</a><a id="1369" href="Cubical.Data.Sigma.Properties.html#1369" class="Bound">a</a> <a id="1371" class="Symbol">:</a> <a id="1373" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="1374" class="Symbol">)</a> <a id="1376" class="Symbol">(</a><a id="1377" href="Cubical.Data.Sigma.Properties.html#1377" class="Bound">b</a> <a id="1379" class="Symbol">:</a> <a id="1381" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="1383" href="Cubical.Data.Sigma.Properties.html#1369" class="Bound">a</a><a id="1384" class="Symbol">)</a> <a id="1386" class="Symbol">→</a> <a id="1388" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1393" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a>
<a id="map-fst"></a><a id="1396" href="Cubical.Data.Sigma.Properties.html#1396" class="Function">map-fst</a> <a id="1404" class="Symbol">:</a> <a id="1406" class="Symbol">{</a><a id="1407" href="Cubical.Data.Sigma.Properties.html#1407" class="Bound">B</a> <a id="1409" class="Symbol">:</a> <a id="1411" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1416" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="1417" class="Symbol">}</a> <a id="1419" class="Symbol">→</a> <a id="1421" class="Symbol">(</a><a id="1422" href="Cubical.Data.Sigma.Properties.html#1422" class="Bound">f</a> <a id="1424" class="Symbol">:</a> <a id="1426" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1428" class="Symbol">→</a> <a id="1430" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="1432" class="Symbol">)</a> <a id="1434" class="Symbol">→</a> <a id="1436" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1438" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="1440" href="Cubical.Data.Sigma.Properties.html#1407" class="Bound">B</a> <a id="1442" class="Symbol">→</a> <a id="1444" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="1447" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="1449" href="Cubical.Data.Sigma.Properties.html#1407" class="Bound">B</a>
<a id="1451" href="Cubical.Data.Sigma.Properties.html#1396" class="Function">map-fst</a> <a id="1459" href="Cubical.Data.Sigma.Properties.html#1459" class="Bound">f</a> <a id="1461" class="Symbol">(</a><a id="1462" href="Cubical.Data.Sigma.Properties.html#1462" class="Bound">a</a> <a id="1464" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1466" href="Cubical.Data.Sigma.Properties.html#1466" class="Bound">b</a><a id="1467" class="Symbol">)</a> <a id="1469" class="Symbol">=</a> <a id="1471" class="Symbol">(</a><a id="1472" href="Cubical.Data.Sigma.Properties.html#1459" class="Bound">f</a> <a id="1474" href="Cubical.Data.Sigma.Properties.html#1462" class="Bound">a</a> <a id="1476" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1478" href="Cubical.Data.Sigma.Properties.html#1466" class="Bound">b</a><a id="1479" class="Symbol">)</a>
<a id="map-snd"></a><a id="1482" href="Cubical.Data.Sigma.Properties.html#1482" class="Function">map-snd</a> <a id="1490" class="Symbol">:</a> <a id="1492" class="Symbol">(∀</a> <a id="1495" class="Symbol">{</a><a id="1496" href="Cubical.Data.Sigma.Properties.html#1496" class="Bound">a</a><a id="1497" class="Symbol">}</a> <a id="1499" class="Symbol">→</a> <a id="1501" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="1503" href="Cubical.Data.Sigma.Properties.html#1496" class="Bound">a</a> <a id="1505" class="Symbol">→</a> <a id="1507" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="1510" href="Cubical.Data.Sigma.Properties.html#1496" class="Bound">a</a><a id="1511" class="Symbol">)</a> <a id="1513" class="Symbol">→</a> <a id="1515" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="1517" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1519" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="1521" class="Symbol">→</a> <a id="1523" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="1525" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1527" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="1530" href="Cubical.Data.Sigma.Properties.html#1482" class="Function">map-snd</a> <a id="1538" href="Cubical.Data.Sigma.Properties.html#1538" class="Bound">f</a> <a id="1540" class="Symbol">(</a><a id="1541" href="Cubical.Data.Sigma.Properties.html#1541" class="Bound">a</a> <a id="1543" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1545" href="Cubical.Data.Sigma.Properties.html#1545" class="Bound">b</a><a id="1546" class="Symbol">)</a> <a id="1548" class="Symbol">=</a> <a id="1550" class="Symbol">(</a><a id="1551" href="Cubical.Data.Sigma.Properties.html#1541" class="Bound">a</a> <a id="1553" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1555" href="Cubical.Data.Sigma.Properties.html#1538" class="Bound">f</a> <a id="1557" href="Cubical.Data.Sigma.Properties.html#1545" class="Bound">b</a><a id="1558" class="Symbol">)</a>
<a id="map-×"></a><a id="1561" href="Cubical.Data.Sigma.Properties.html#1561" class="Function">map-×</a> <a id="1567" class="Symbol">:</a> <a id="1569" class="Symbol">{</a><a id="1570" href="Cubical.Data.Sigma.Properties.html#1570" class="Bound">B</a> <a id="1572" class="Symbol">:</a> <a id="1574" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1579" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="1580" class="Symbol">}</a> <a id="1582" class="Symbol">{</a><a id="1583" href="Cubical.Data.Sigma.Properties.html#1583" class="Bound">B'</a> <a id="1586" class="Symbol">:</a> <a id="1588" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1593" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="1595" class="Symbol">}</a> <a id="1597" class="Symbol">→</a> <a id="1599" class="Symbol">(</a><a id="1600" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1602" class="Symbol">→</a> <a id="1604" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="1606" class="Symbol">)</a> <a id="1608" class="Symbol">→</a> <a id="1610" class="Symbol">(</a><a id="1611" href="Cubical.Data.Sigma.Properties.html#1570" class="Bound">B</a> <a id="1613" class="Symbol">→</a> <a id="1615" href="Cubical.Data.Sigma.Properties.html#1583" class="Bound">B'</a><a id="1617" class="Symbol">)</a> <a id="1619" class="Symbol">→</a> <a id="1621" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="1623" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="1625" href="Cubical.Data.Sigma.Properties.html#1570" class="Bound">B</a> <a id="1627" class="Symbol">→</a> <a id="1629" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="1632" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="1634" href="Cubical.Data.Sigma.Properties.html#1583" class="Bound">B'</a>
<a id="1637" href="Cubical.Data.Sigma.Properties.html#1561" class="Function">map-×</a> <a id="1643" href="Cubical.Data.Sigma.Properties.html#1643" class="Bound">f</a> <a id="1645" href="Cubical.Data.Sigma.Properties.html#1645" class="Bound">g</a> <a id="1647" class="Symbol">(</a><a id="1648" href="Cubical.Data.Sigma.Properties.html#1648" class="Bound">a</a> <a id="1650" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1652" href="Cubical.Data.Sigma.Properties.html#1652" class="Bound">b</a><a id="1653" class="Symbol">)</a> <a id="1655" class="Symbol">=</a> <a id="1657" class="Symbol">(</a><a id="1658" href="Cubical.Data.Sigma.Properties.html#1643" class="Bound">f</a> <a id="1660" href="Cubical.Data.Sigma.Properties.html#1648" class="Bound">a</a> <a id="1662" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1664" href="Cubical.Data.Sigma.Properties.html#1645" class="Bound">g</a> <a id="1666" href="Cubical.Data.Sigma.Properties.html#1652" class="Bound">b</a><a id="1667" class="Symbol">)</a>
<a id="≡-×"></a><a id="1670" href="Cubical.Data.Sigma.Properties.html#1670" class="Function">≡-×</a> <a id="1674" class="Symbol">:</a> <a id="1676" class="Symbol">{</a><a id="1677" href="Cubical.Data.Sigma.Properties.html#1677" class="Bound">A</a> <a id="1679" class="Symbol">:</a> <a id="1681" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1686" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="1687" class="Symbol">}</a> <a id="1689" class="Symbol">{</a><a id="1690" href="Cubical.Data.Sigma.Properties.html#1690" class="Bound">B</a> <a id="1692" class="Symbol">:</a> <a id="1694" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1699" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="1701" class="Symbol">}</a> <a id="1703" class="Symbol">{</a><a id="1704" href="Cubical.Data.Sigma.Properties.html#1704" class="Bound">x</a> <a id="1706" href="Cubical.Data.Sigma.Properties.html#1706" class="Bound">y</a> <a id="1708" class="Symbol">:</a> <a id="1710" href="Cubical.Data.Sigma.Properties.html#1677" class="Bound">A</a> <a id="1712" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="1714" href="Cubical.Data.Sigma.Properties.html#1690" class="Bound">B</a><a id="1715" class="Symbol">}</a> <a id="1717" class="Symbol">→</a> <a id="1719" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="1723" href="Cubical.Data.Sigma.Properties.html#1704" class="Bound">x</a> <a id="1725" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="1727" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="1731" href="Cubical.Data.Sigma.Properties.html#1706" class="Bound">y</a> <a id="1733" class="Symbol">→</a> <a id="1735" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="1739" href="Cubical.Data.Sigma.Properties.html#1704" class="Bound">x</a> <a id="1741" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="1743" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="1747" href="Cubical.Data.Sigma.Properties.html#1706" class="Bound">y</a> <a id="1749" class="Symbol">→</a> <a id="1751" href="Cubical.Data.Sigma.Properties.html#1704" class="Bound">x</a> <a id="1753" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="1755" href="Cubical.Data.Sigma.Properties.html#1706" class="Bound">y</a>
<a id="1757" href="Cubical.Data.Sigma.Properties.html#1670" class="Function">≡-×</a> <a id="1761" href="Cubical.Data.Sigma.Properties.html#1761" class="Bound">p</a> <a id="1763" href="Cubical.Data.Sigma.Properties.html#1763" class="Bound">q</a> <a id="1765" href="Cubical.Data.Sigma.Properties.html#1765" class="Bound">i</a> <a id="1767" class="Symbol">=</a> <a id="1769" class="Symbol">(</a><a id="1770" href="Cubical.Data.Sigma.Properties.html#1761" class="Bound">p</a> <a id="1772" href="Cubical.Data.Sigma.Properties.html#1765" class="Bound">i</a><a id="1773" class="Symbol">)</a> <a id="1775" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="1777" class="Symbol">(</a><a id="1778" href="Cubical.Data.Sigma.Properties.html#1763" class="Bound">q</a> <a id="1780" href="Cubical.Data.Sigma.Properties.html#1765" class="Bound">i</a><a id="1781" class="Symbol">)</a>
<a id="1785" class="Comment">-- Characterization of paths in Σ using dependent paths</a>
<a id="1842" class="Keyword">module</a> <a id="1849" href="Cubical.Data.Sigma.Properties.html#1849" class="Module">_</a> <a id="1851" class="Symbol">{</a><a id="1852" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="1854" class="Symbol">:</a> <a id="1856" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a> <a id="1858" class="Symbol">→</a> <a id="1860" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1865" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="1866" class="Symbol">}</a> <a id="1868" class="Symbol">{</a><a id="1869" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="1871" class="Symbol">:</a> <a id="1873" class="Symbol">(</a><a id="1874" href="Cubical.Data.Sigma.Properties.html#1874" class="Bound">i</a> <a id="1876" class="Symbol">:</a> <a id="1878" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a><a id="1879" class="Symbol">)</a> <a id="1881" class="Symbol">→</a> <a id="1883" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="1885" href="Cubical.Data.Sigma.Properties.html#1874" class="Bound">i</a> <a id="1887" class="Symbol">→</a> <a id="1889" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="1894" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="1896" class="Symbol">}</a>
<a id="1900" class="Symbol">{</a><a id="1901" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="1903" class="Symbol">:</a> <a id="1905" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="1907" class="Symbol">(</a><a id="1908" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="1910" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="1912" class="Symbol">)</a> <a id="1914" class="Symbol">(</a><a id="1915" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="1917" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="1919" class="Symbol">)}</a> <a id="1922" class="Symbol">{</a><a id="1923" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a> <a id="1925" class="Symbol">:</a> <a id="1927" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="1929" class="Symbol">(</a><a id="1930" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="1932" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="1934" class="Symbol">)</a> <a id="1936" class="Symbol">(</a><a id="1937" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="1939" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="1941" class="Symbol">)}</a>
<a id="1946" class="Keyword">where</a>
<a id="1955" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="1962" class="Symbol">:</a> <a id="1964" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="1967" href="Cubical.Data.Sigma.Properties.html#1967" class="Bound">p</a> <a id="1969" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="1971" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="1977" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="1979" class="Symbol">(</a><a id="1980" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="1984" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="1985" class="Symbol">)</a> <a id="1987" class="Symbol">(</a><a id="1988" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="1992" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="1993" class="Symbol">)</a> <a id="1995" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="1997" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2003" class="Symbol">(λ</a> <a id="2006" href="Cubical.Data.Sigma.Properties.html#2006" class="Bound">i</a> <a id="2008" class="Symbol">→</a> <a id="2010" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2012" href="Cubical.Data.Sigma.Properties.html#2006" class="Bound">i</a> <a id="2014" class="Symbol">(</a><a id="2015" href="Cubical.Data.Sigma.Properties.html#1967" class="Bound">p</a> <a id="2017" href="Cubical.Data.Sigma.Properties.html#2006" class="Bound">i</a><a id="2018" class="Symbol">))</a> <a id="2021" class="Symbol">(</a><a id="2022" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2026" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2027" class="Symbol">)</a> <a id="2029" class="Symbol">(</a><a id="2030" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2034" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2035" class="Symbol">)</a>
<a id="2046" class="Symbol">→</a> <a id="2048" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2054" class="Symbol">(λ</a> <a id="2057" href="Cubical.Data.Sigma.Properties.html#2057" class="Bound">i</a> <a id="2059" class="Symbol">→</a> <a id="2061" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="2063" class="Symbol">(</a><a id="2064" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2066" href="Cubical.Data.Sigma.Properties.html#2057" class="Bound">i</a><a id="2067" class="Symbol">)</a> <a id="2069" class="Symbol">(</a><a id="2070" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2072" href="Cubical.Data.Sigma.Properties.html#2057" class="Bound">i</a><a id="2073" class="Symbol">))</a> <a id="2076" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="2078" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a>
<a id="2082" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="2089" href="Cubical.Data.Sigma.Properties.html#2089" class="Bound">eq</a> <a id="2092" href="Cubical.Data.Sigma.Properties.html#2092" class="Bound">i</a> <a id="2094" class="Symbol">=</a> <a id="2096" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2100" href="Cubical.Data.Sigma.Properties.html#2089" class="Bound">eq</a> <a id="2103" href="Cubical.Data.Sigma.Properties.html#2092" class="Bound">i</a> <a id="2105" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="2107" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2111" href="Cubical.Data.Sigma.Properties.html#2089" class="Bound">eq</a> <a id="2114" href="Cubical.Data.Sigma.Properties.html#2092" class="Bound">i</a>
<a id="2119" href="Cubical.Data.Sigma.Properties.html#2119" class="Function">PathPΣ</a> <a id="2126" class="Symbol">:</a> <a id="2128" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2134" class="Symbol">(λ</a> <a id="2137" href="Cubical.Data.Sigma.Properties.html#2137" class="Bound">i</a> <a id="2139" class="Symbol">→</a> <a id="2141" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="2143" class="Symbol">(</a><a id="2144" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2146" href="Cubical.Data.Sigma.Properties.html#2137" class="Bound">i</a><a id="2147" class="Symbol">)</a> <a id="2149" class="Symbol">(</a><a id="2150" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2152" href="Cubical.Data.Sigma.Properties.html#2137" class="Bound">i</a><a id="2153" class="Symbol">))</a> <a id="2156" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="2158" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a>
<a id="2169" class="Symbol">→</a> <a id="2171" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="2174" href="Cubical.Data.Sigma.Properties.html#2174" class="Bound">p</a> <a id="2176" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="2178" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2184" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2186" class="Symbol">(</a><a id="2187" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2191" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2192" class="Symbol">)</a> <a id="2194" class="Symbol">(</a><a id="2195" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2199" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2200" class="Symbol">)</a> <a id="2202" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="2204" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2210" class="Symbol">(λ</a> <a id="2213" href="Cubical.Data.Sigma.Properties.html#2213" class="Bound">i</a> <a id="2215" class="Symbol">→</a> <a id="2217" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2219" href="Cubical.Data.Sigma.Properties.html#2213" class="Bound">i</a> <a id="2221" class="Symbol">(</a><a id="2222" href="Cubical.Data.Sigma.Properties.html#2174" class="Bound">p</a> <a id="2224" href="Cubical.Data.Sigma.Properties.html#2213" class="Bound">i</a><a id="2225" class="Symbol">))</a> <a id="2228" class="Symbol">(</a><a id="2229" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2233" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2234" class="Symbol">)</a> <a id="2236" class="Symbol">(</a><a id="2237" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2241" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2242" class="Symbol">)</a>
<a id="2246" href="Cubical.Data.Sigma.Properties.html#2119" class="Function">PathPΣ</a> <a id="2253" href="Cubical.Data.Sigma.Properties.html#2253" class="Bound">eq</a> <a id="2256" class="Symbol">=</a> <a id="2258" class="Symbol">(λ</a> <a id="2261" href="Cubical.Data.Sigma.Properties.html#2261" class="Bound">i</a> <a id="2263" class="Symbol">→</a> <a id="2265" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2269" class="Symbol">(</a><a id="2270" href="Cubical.Data.Sigma.Properties.html#2253" class="Bound">eq</a> <a id="2273" href="Cubical.Data.Sigma.Properties.html#2261" class="Bound">i</a><a id="2274" class="Symbol">))</a> <a id="2277" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="2279" class="Symbol">(λ</a> <a id="2282" href="Cubical.Data.Sigma.Properties.html#2282" class="Bound">i</a> <a id="2284" class="Symbol">→</a> <a id="2286" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2290" class="Symbol">(</a><a id="2291" href="Cubical.Data.Sigma.Properties.html#2253" class="Bound">eq</a> <a id="2294" href="Cubical.Data.Sigma.Properties.html#2282" class="Bound">i</a><a id="2295" class="Symbol">))</a>
<a id="2301" class="Comment">-- allows one to write</a>
<a id="2326" class="Comment">-- open PathPΣ somePathInΣAB renaming (fst ... )</a>
<a id="2377" class="Keyword">module</a> <a id="2384" href="Cubical.Data.Sigma.Properties.html#2384" class="Module">PathPΣ</a> <a id="2391" class="Symbol">(</a><a id="2392" href="Cubical.Data.Sigma.Properties.html#2392" class="Bound">p</a> <a id="2394" class="Symbol">:</a> <a id="2396" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2402" class="Symbol">(λ</a> <a id="2405" href="Cubical.Data.Sigma.Properties.html#2405" class="Bound">i</a> <a id="2407" class="Symbol">→</a> <a id="2409" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="2411" class="Symbol">(</a><a id="2412" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2414" href="Cubical.Data.Sigma.Properties.html#2405" class="Bound">i</a><a id="2415" class="Symbol">)</a> <a id="2417" class="Symbol">(</a><a id="2418" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2420" href="Cubical.Data.Sigma.Properties.html#2405" class="Bound">i</a><a id="2421" class="Symbol">))</a> <a id="2424" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="2426" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2427" class="Symbol">)</a> <a id="2429" class="Keyword">where</a>
<a id="2439" class="Keyword">open</a> <a id="2444" href="Agda.Builtin.Sigma.html#166" class="Module">Σ</a> <a id="2446" class="Symbol">(</a><a id="2447" href="Cubical.Data.Sigma.Properties.html#2119" class="Function">PathPΣ</a> <a id="2454" href="Cubical.Data.Sigma.Properties.html#2392" class="Bound">p</a><a id="2455" class="Symbol">)</a> <a id="2457" class="Keyword">public</a>
<a id="2467" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a> <a id="2481" class="Symbol">:</a> <a id="2483" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="2487" class="Symbol">(</a><a id="2488" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="2491" href="Cubical.Data.Sigma.Properties.html#2491" class="Bound">p</a> <a id="2493" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="2495" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2501" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2503" class="Symbol">(</a><a id="2504" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2508" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2509" class="Symbol">)</a> <a id="2511" class="Symbol">(</a><a id="2512" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2516" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2517" class="Symbol">)</a> <a id="2519" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="2521" class="Symbol">(</a><a id="2522" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2528" class="Symbol">(λ</a> <a id="2531" href="Cubical.Data.Sigma.Properties.html#2531" class="Bound">i</a> <a id="2533" class="Symbol">→</a> <a id="2535" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2537" href="Cubical.Data.Sigma.Properties.html#2531" class="Bound">i</a> <a id="2539" class="Symbol">(</a><a id="2540" href="Cubical.Data.Sigma.Properties.html#2491" class="Bound">p</a> <a id="2542" href="Cubical.Data.Sigma.Properties.html#2531" class="Bound">i</a><a id="2543" class="Symbol">))</a> <a id="2546" class="Symbol">(</a><a id="2547" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2551" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2552" class="Symbol">)</a> <a id="2554" class="Symbol">(</a><a id="2555" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2559" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2560" class="Symbol">)))</a>
<a id="2586" class="Symbol">(</a><a id="2587" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2593" class="Symbol">(λ</a> <a id="2596" href="Cubical.Data.Sigma.Properties.html#2596" class="Bound">i</a> <a id="2598" class="Symbol">→</a> <a id="2600" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="2602" class="Symbol">(</a><a id="2603" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2605" href="Cubical.Data.Sigma.Properties.html#2596" class="Bound">i</a><a id="2606" class="Symbol">)</a> <a id="2608" class="Symbol">(</a><a id="2609" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2611" href="Cubical.Data.Sigma.Properties.html#2596" class="Bound">i</a><a id="2612" class="Symbol">))</a> <a id="2615" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="2617" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2618" class="Symbol">)</a>
<a id="2622" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="2626" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a> <a id="2647" class="Symbol">=</a> <a id="2649" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a>
<a id="2658" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="2662" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a> <a id="2683" class="Symbol">=</a> <a id="2685" href="Cubical.Data.Sigma.Properties.html#2119" class="Function">PathPΣ</a>
<a id="2694" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="2703" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a> <a id="2717" class="Symbol">_</a> <a id="2719" class="Symbol">=</a> <a id="2721" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="2728" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="2736" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a> <a id="2750" class="Symbol">_</a> <a id="2753" class="Symbol">=</a> <a id="2755" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="2763" class="Keyword">unquoteDecl</a> <a id="2775" href="Cubical.Data.Sigma.Properties.html#2775" class="Function">ΣPath≃PathΣ</a> <a id="2787" class="Symbol">=</a> <a id="2789" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="2810" href="Cubical.Data.Sigma.Properties.html#2775" class="Function">ΣPath≃PathΣ</a> <a id="2822" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a>
<a id="2839" href="Cubical.Data.Sigma.Properties.html#2839" class="Function">ΣPath≡PathΣ</a> <a id="2851" class="Symbol">:</a> <a id="2853" class="Symbol">(</a><a id="2854" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="2857" href="Cubical.Data.Sigma.Properties.html#2857" class="Bound">p</a> <a id="2859" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="2861" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2867" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2869" class="Symbol">(</a><a id="2870" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2874" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2875" class="Symbol">)</a> <a id="2877" class="Symbol">(</a><a id="2878" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="2882" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2883" class="Symbol">)</a> <a id="2885" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="2887" class="Symbol">(</a><a id="2888" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2894" class="Symbol">(λ</a> <a id="2897" href="Cubical.Data.Sigma.Properties.html#2897" class="Bound">i</a> <a id="2899" class="Symbol">→</a> <a id="2901" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2903" href="Cubical.Data.Sigma.Properties.html#2897" class="Bound">i</a> <a id="2905" class="Symbol">(</a><a id="2906" href="Cubical.Data.Sigma.Properties.html#2857" class="Bound">p</a> <a id="2908" href="Cubical.Data.Sigma.Properties.html#2897" class="Bound">i</a><a id="2909" class="Symbol">))</a> <a id="2912" class="Symbol">(</a><a id="2913" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2917" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a><a id="2918" class="Symbol">)</a> <a id="2920" class="Symbol">(</a><a id="2921" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="2925" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2926" class="Symbol">)))</a>
<a id="2944" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="2946" class="Symbol">(</a><a id="2947" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="2953" class="Symbol">(λ</a> <a id="2956" href="Cubical.Data.Sigma.Properties.html#2956" class="Bound">i</a> <a id="2958" class="Symbol">→</a> <a id="2960" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="2962" class="Symbol">(</a><a id="2963" href="Cubical.Data.Sigma.Properties.html#1852" class="Bound">A</a> <a id="2965" href="Cubical.Data.Sigma.Properties.html#2956" class="Bound">i</a><a id="2966" class="Symbol">)</a> <a id="2968" class="Symbol">(</a><a id="2969" href="Cubical.Data.Sigma.Properties.html#1869" class="Bound">B</a> <a id="2971" href="Cubical.Data.Sigma.Properties.html#2956" class="Bound">i</a><a id="2972" class="Symbol">))</a> <a id="2975" href="Cubical.Data.Sigma.Properties.html#1901" class="Bound">x</a> <a id="2977" href="Cubical.Data.Sigma.Properties.html#1923" class="Bound">y</a><a id="2978" class="Symbol">)</a>
<a id="2982" href="Cubical.Data.Sigma.Properties.html#2839" class="Function">ΣPath≡PathΣ</a> <a id="2994" class="Symbol">=</a> <a id="2996" href="Cubical.Foundations.Univalence.html#928" class="Function">ua</a> <a id="2999" href="Cubical.Data.Sigma.Properties.html#2775" class="Function">ΣPath≃PathΣ</a>
<a id="×≡Prop"></a><a id="3012" href="Cubical.Data.Sigma.Properties.html#3012" class="Function">×≡Prop</a> <a id="3019" class="Symbol">:</a> <a id="3021" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="3028" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="3031" class="Symbol">→</a> <a id="3033" class="Symbol">{</a><a id="3034" href="Cubical.Data.Sigma.Properties.html#3034" class="Bound">u</a> <a id="3036" href="Cubical.Data.Sigma.Properties.html#3036" class="Bound">v</a> <a id="3038" class="Symbol">:</a> <a id="3040" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="3042" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="3044" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="3046" class="Symbol">}</a> <a id="3048" class="Symbol">→</a> <a id="3050" href="Cubical.Data.Sigma.Properties.html#3034" class="Bound">u</a> <a id="3052" class="Symbol">.</a><a id="3053" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="3057" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="3059" href="Cubical.Data.Sigma.Properties.html#3036" class="Bound">v</a> <a id="3061" class="Symbol">.</a><a id="3062" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="3066" class="Symbol">→</a> <a id="3068" href="Cubical.Data.Sigma.Properties.html#3034" class="Bound">u</a> <a id="3070" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="3072" href="Cubical.Data.Sigma.Properties.html#3036" class="Bound">v</a>
<a id="3074" href="Cubical.Data.Sigma.Properties.html#3012" class="Function">×≡Prop</a> <a id="3081" href="Cubical.Data.Sigma.Properties.html#3081" class="Bound">pB</a> <a id="3084" class="Symbol">{</a><a id="3085" href="Cubical.Data.Sigma.Properties.html#3085" class="Bound">u</a><a id="3086" class="Symbol">}</a> <a id="3088" class="Symbol">{</a><a id="3089" href="Cubical.Data.Sigma.Properties.html#3089" class="Bound">v</a><a id="3090" class="Symbol">}</a> <a id="3092" href="Cubical.Data.Sigma.Properties.html#3092" class="Bound">p</a> <a id="3094" href="Cubical.Data.Sigma.Properties.html#3094" class="Bound">i</a> <a id="3096" class="Symbol">=</a> <a id="3098" class="Symbol">(</a><a id="3099" href="Cubical.Data.Sigma.Properties.html#3092" class="Bound">p</a> <a id="3101" href="Cubical.Data.Sigma.Properties.html#3094" class="Bound">i</a><a id="3102" class="Symbol">)</a> <a id="3104" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3106" class="Symbol">(</a><a id="3107" href="Cubical.Data.Sigma.Properties.html#3081" class="Bound">pB</a> <a id="3110" class="Symbol">(</a><a id="3111" href="Cubical.Data.Sigma.Properties.html#3085" class="Bound">u</a> <a id="3113" class="Symbol">.</a><a id="3114" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="3117" class="Symbol">)</a> <a id="3119" class="Symbol">(</a><a id="3120" href="Cubical.Data.Sigma.Properties.html#3089" class="Bound">v</a> <a id="3122" class="Symbol">.</a><a id="3123" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="3126" class="Symbol">)</a> <a id="3128" href="Cubical.Data.Sigma.Properties.html#3094" class="Bound">i</a><a id="3129" class="Symbol">)</a>
<a id="3132" class="Comment">-- Useful lemma to prove unique existence</a>
<a id="uniqueExists"></a><a id="3174" href="Cubical.Data.Sigma.Properties.html#3174" class="Function">uniqueExists</a> <a id="3187" class="Symbol">:</a> <a id="3189" class="Symbol">(</a><a id="3190" href="Cubical.Data.Sigma.Properties.html#3190" class="Bound">a</a> <a id="3192" class="Symbol">:</a> <a id="3194" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="3195" class="Symbol">)</a> <a id="3197" class="Symbol">(</a><a id="3198" href="Cubical.Data.Sigma.Properties.html#3198" class="Bound">b</a> <a id="3200" class="Symbol">:</a> <a id="3202" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="3204" href="Cubical.Data.Sigma.Properties.html#3190" class="Bound">a</a><a id="3205" class="Symbol">)</a> <a id="3207" class="Symbol">(</a><a id="3208" href="Cubical.Data.Sigma.Properties.html#3208" class="Bound">h</a> <a id="3210" class="Symbol">:</a> <a id="3212" class="Symbol">(</a><a id="3213" href="Cubical.Data.Sigma.Properties.html#3213" class="Bound">a'</a> <a id="3216" class="Symbol">:</a> <a id="3218" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="3219" class="Symbol">)</a> <a id="3221" class="Symbol">→</a> <a id="3223" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="3230" class="Symbol">(</a><a id="3231" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="3233" href="Cubical.Data.Sigma.Properties.html#3213" class="Bound">a'</a><a id="3235" class="Symbol">))</a> <a id="3238" class="Symbol">(</a><a id="3239" href="Cubical.Data.Sigma.Properties.html#3239" class="Bound">H</a> <a id="3241" class="Symbol">:</a> <a id="3243" class="Symbol">(</a><a id="3244" href="Cubical.Data.Sigma.Properties.html#3244" class="Bound">a'</a> <a id="3247" class="Symbol">:</a> <a id="3249" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="3250" class="Symbol">)</a> <a id="3252" class="Symbol">→</a> <a id="3254" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="3256" href="Cubical.Data.Sigma.Properties.html#3244" class="Bound">a'</a> <a id="3259" class="Symbol">→</a> <a id="3261" href="Cubical.Data.Sigma.Properties.html#3190" class="Bound">a</a> <a id="3263" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="3265" href="Cubical.Data.Sigma.Properties.html#3244" class="Bound">a'</a><a id="3267" class="Symbol">)</a> <a id="3269" class="Symbol">→</a> <a id="3271" href="Cubical.Data.Sigma.Base.html#943" class="Function">∃![</a> <a id="3275" href="Cubical.Data.Sigma.Properties.html#3275" class="Bound">a</a> <a id="3277" href="Cubical.Data.Sigma.Base.html#943" class="Function">∈</a> <a id="3279" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="3281" href="Cubical.Data.Sigma.Base.html#943" class="Function">]</a> <a id="3283" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="3285" href="Cubical.Data.Sigma.Properties.html#3275" class="Bound">a</a>
<a id="3287" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="3291" class="Symbol">(</a><a id="3292" href="Cubical.Data.Sigma.Properties.html#3174" class="Function">uniqueExists</a> <a id="3305" href="Cubical.Data.Sigma.Properties.html#3305" class="Bound">a</a> <a id="3307" href="Cubical.Data.Sigma.Properties.html#3307" class="Bound">b</a> <a id="3309" href="Cubical.Data.Sigma.Properties.html#3309" class="Bound">h</a> <a id="3311" href="Cubical.Data.Sigma.Properties.html#3311" class="Bound">H</a><a id="3312" class="Symbol">)</a> <a id="3314" class="Symbol">=</a> <a id="3316" class="Symbol">(</a><a id="3317" href="Cubical.Data.Sigma.Properties.html#3305" class="Bound">a</a> <a id="3319" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3321" href="Cubical.Data.Sigma.Properties.html#3307" class="Bound">b</a><a id="3322" class="Symbol">)</a>
<a id="3324" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="3328" class="Symbol">(</a><a id="3329" href="Cubical.Data.Sigma.Properties.html#3174" class="Function">uniqueExists</a> <a id="3342" href="Cubical.Data.Sigma.Properties.html#3342" class="Bound">a</a> <a id="3344" href="Cubical.Data.Sigma.Properties.html#3344" class="Bound">b</a> <a id="3346" href="Cubical.Data.Sigma.Properties.html#3346" class="Bound">h</a> <a id="3348" href="Cubical.Data.Sigma.Properties.html#3348" class="Bound">H</a><a id="3349" class="Symbol">)</a> <a id="3351" class="Symbol">(</a><a id="3352" href="Cubical.Data.Sigma.Properties.html#3352" class="Bound">a'</a> <a id="3355" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3357" href="Cubical.Data.Sigma.Properties.html#3357" class="Bound">b'</a><a id="3359" class="Symbol">)</a> <a id="3361" class="Symbol">=</a> <a id="3363" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="3370" class="Symbol">(</a><a id="3371" href="Cubical.Data.Sigma.Properties.html#3348" class="Bound">H</a> <a id="3373" href="Cubical.Data.Sigma.Properties.html#3352" class="Bound">a'</a> <a id="3376" href="Cubical.Data.Sigma.Properties.html#3357" class="Bound">b'</a> <a id="3379" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3381" href="Cubical.Foundations.Prelude.html#17730" class="Function">isProp→PathP</a> <a id="3394" class="Symbol">(λ</a> <a id="3397" href="Cubical.Data.Sigma.Properties.html#3397" class="Bound">i</a> <a id="3399" class="Symbol">→</a> <a id="3401" href="Cubical.Data.Sigma.Properties.html#3346" class="Bound">h</a> <a id="3403" class="Symbol">(</a><a id="3404" href="Cubical.Data.Sigma.Properties.html#3348" class="Bound">H</a> <a id="3406" href="Cubical.Data.Sigma.Properties.html#3352" class="Bound">a'</a> <a id="3409" href="Cubical.Data.Sigma.Properties.html#3357" class="Bound">b'</a> <a id="3412" href="Cubical.Data.Sigma.Properties.html#3397" class="Bound">i</a><a id="3413" class="Symbol">))</a> <a id="3416" href="Cubical.Data.Sigma.Properties.html#3344" class="Bound">b</a> <a id="3418" href="Cubical.Data.Sigma.Properties.html#3357" class="Bound">b'</a><a id="3420" class="Symbol">)</a>
<a id="3424" class="Comment">-- Characterization of dependent paths in Σ</a>
<a id="3469" class="Keyword">module</a> <a id="3476" href="Cubical.Data.Sigma.Properties.html#3476" class="Module">_</a> <a id="3478" class="Symbol">{</a><a id="3479" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3481" class="Symbol">:</a> <a id="3483" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a> <a id="3485" class="Symbol">→</a> <a id="3487" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="3492" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="3493" class="Symbol">}</a> <a id="3495" class="Symbol">{</a><a id="3496" href="Cubical.Data.Sigma.Properties.html#3496" class="Bound">B</a> <a id="3498" class="Symbol">:</a> <a id="3500" class="Symbol">(</a><a id="3501" href="Cubical.Data.Sigma.Properties.html#3501" class="Bound">i</a> <a id="3503" class="Symbol">:</a> <a id="3505" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a><a id="3506" class="Symbol">)</a> <a id="3508" class="Symbol">→</a> <a id="3510" class="Symbol">(</a><a id="3511" href="Cubical.Data.Sigma.Properties.html#3511" class="Bound">a</a> <a id="3513" class="Symbol">:</a> <a id="3515" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3517" href="Cubical.Data.Sigma.Properties.html#3501" class="Bound">i</a><a id="3518" class="Symbol">)</a> <a id="3520" class="Symbol">→</a> <a id="3522" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="3527" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="3529" class="Symbol">}</a>
<a id="3533" class="Symbol">{</a><a id="3534" href="Cubical.Data.Sigma.Properties.html#3534" class="Bound">x</a> <a id="3536" class="Symbol">:</a> <a id="3538" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="3540" class="Symbol">(</a><a id="3541" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3543" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="3545" class="Symbol">)</a> <a id="3547" class="Symbol">(</a><a id="3548" href="Cubical.Data.Sigma.Properties.html#3496" class="Bound">B</a> <a id="3550" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="3552" class="Symbol">)}</a> <a id="3555" class="Symbol">{</a><a id="3556" href="Cubical.Data.Sigma.Properties.html#3556" class="Bound">y</a> <a id="3558" class="Symbol">:</a> <a id="3560" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="3562" class="Symbol">(</a><a id="3563" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3565" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="3567" class="Symbol">)</a> <a id="3569" class="Symbol">(</a><a id="3570" href="Cubical.Data.Sigma.Properties.html#3496" class="Bound">B</a> <a id="3572" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="3574" class="Symbol">)}</a>
<a id="3579" class="Keyword">where</a>
<a id="3588" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3604" class="Symbol">:</a>
<a id="3610" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="3614" class="Symbol">(</a><a id="3615" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="3618" href="Cubical.Data.Sigma.Properties.html#3618" class="Bound">p</a> <a id="3620" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="3622" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="3628" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3630" class="Symbol">(</a><a id="3631" href="Cubical.Data.Sigma.Properties.html#3534" class="Bound">x</a> <a id="3633" class="Symbol">.</a><a id="3634" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="3637" class="Symbol">)</a> <a id="3639" class="Symbol">(</a><a id="3640" href="Cubical.Data.Sigma.Properties.html#3556" class="Bound">y</a> <a id="3642" class="Symbol">.</a><a id="3643" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="3646" class="Symbol">)</a> <a id="3648" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="3650" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="3656" class="Symbol">(λ</a> <a id="3659" href="Cubical.Data.Sigma.Properties.html#3659" class="Bound">i</a> <a id="3661" class="Symbol">→</a> <a id="3663" href="Cubical.Data.Sigma.Properties.html#3496" class="Bound">B</a> <a id="3665" href="Cubical.Data.Sigma.Properties.html#3659" class="Bound">i</a> <a id="3667" class="Symbol">(</a><a id="3668" href="Cubical.Data.Sigma.Properties.html#3618" class="Bound">p</a> <a id="3670" href="Cubical.Data.Sigma.Properties.html#3659" class="Bound">i</a><a id="3671" class="Symbol">))</a> <a id="3674" class="Symbol">(</a><a id="3675" href="Cubical.Data.Sigma.Properties.html#3534" class="Bound">x</a> <a id="3677" class="Symbol">.</a><a id="3678" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="3681" class="Symbol">)</a> <a id="3683" class="Symbol">(</a><a id="3684" href="Cubical.Data.Sigma.Properties.html#3556" class="Bound">y</a> <a id="3686" class="Symbol">.</a><a id="3687" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="3690" class="Symbol">))</a>
<a id="3701" class="Symbol">(</a><a id="3702" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="3708" class="Symbol">(λ</a> <a id="3711" href="Cubical.Data.Sigma.Properties.html#3711" class="Bound">i</a> <a id="3713" class="Symbol">→</a> <a id="3715" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="3717" class="Symbol">(</a><a id="3718" href="Cubical.Data.Sigma.Properties.html#3479" class="Bound">A</a> <a id="3720" href="Cubical.Data.Sigma.Properties.html#3711" class="Bound">i</a><a id="3721" class="Symbol">)</a> <a id="3723" class="Symbol">(</a><a id="3724" href="Cubical.Data.Sigma.Properties.html#3496" class="Bound">B</a> <a id="3726" href="Cubical.Data.Sigma.Properties.html#3711" class="Bound">i</a><a id="3727" class="Symbol">))</a> <a id="3730" href="Cubical.Data.Sigma.Properties.html#3534" class="Bound">x</a> <a id="3732" href="Cubical.Data.Sigma.Properties.html#3556" class="Bound">y</a><a id="3733" class="Symbol">)</a>
<a id="3737" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3753" class="Symbol">.</a><a id="3754" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="3758" class="Symbol">(</a><a id="3759" href="Cubical.Data.Sigma.Properties.html#3759" class="Bound">p</a> <a id="3761" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3763" href="Cubical.Data.Sigma.Properties.html#3763" class="Bound">q</a><a id="3764" class="Symbol">)</a> <a id="3766" href="Cubical.Data.Sigma.Properties.html#3766" class="Bound">i</a> <a id="3768" class="Symbol">=</a> <a id="3770" href="Cubical.Data.Sigma.Properties.html#3759" class="Bound">p</a> <a id="3772" href="Cubical.Data.Sigma.Properties.html#3766" class="Bound">i</a> <a id="3774" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="3776" href="Cubical.Data.Sigma.Properties.html#3763" class="Bound">q</a> <a id="3778" href="Cubical.Data.Sigma.Properties.html#3766" class="Bound">i</a>
<a id="3782" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3798" class="Symbol">.</a><a id="3799" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="3803" href="Cubical.Data.Sigma.Properties.html#3803" class="Bound">pq</a> <a id="3806" class="Symbol">.</a><a id="3807" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="3811" href="Cubical.Data.Sigma.Properties.html#3811" class="Bound">i</a> <a id="3813" class="Symbol">=</a> <a id="3815" href="Cubical.Data.Sigma.Properties.html#3803" class="Bound">pq</a> <a id="3818" href="Cubical.Data.Sigma.Properties.html#3811" class="Bound">i</a> <a id="3820" class="Symbol">.</a><a id="3821" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a>
<a id="3827" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3843" class="Symbol">.</a><a id="3844" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="3848" href="Cubical.Data.Sigma.Properties.html#3848" class="Bound">pq</a> <a id="3851" class="Symbol">.</a><a id="3852" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="3856" href="Cubical.Data.Sigma.Properties.html#3856" class="Bound">i</a> <a id="3858" class="Symbol">=</a> <a id="3860" href="Cubical.Data.Sigma.Properties.html#3848" class="Bound">pq</a> <a id="3863" href="Cubical.Data.Sigma.Properties.html#3856" class="Bound">i</a> <a id="3865" class="Symbol">.</a><a id="3866" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a>
<a id="3872" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3888" class="Symbol">.</a><a id="3889" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="3898" class="Symbol">_</a> <a id="3900" class="Symbol">=</a> <a id="3902" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="3909" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a> <a id="3925" class="Symbol">.</a><a id="3926" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="3934" class="Symbol">_</a> <a id="3936" class="Symbol">=</a> <a id="3938" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="3946" class="Keyword">unquoteDecl</a> <a id="3958" href="Cubical.Data.Sigma.Properties.html#3958" class="Function">ΣPathP≃PathPΣ</a> <a id="3972" class="Symbol">=</a> <a id="3974" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="3995" href="Cubical.Data.Sigma.Properties.html#3958" class="Function">ΣPathP≃PathPΣ</a> <a id="4009" href="Cubical.Data.Sigma.Properties.html#3588" class="Function">ΣPathPIsoPathPΣ</a>
<a id="4028" href="Cubical.Data.Sigma.Properties.html#4028" class="Function">ΣPathP≡PathPΣ</a> <a id="4042" class="Symbol">=</a> <a id="4044" href="Cubical.Foundations.Univalence.html#928" class="Function">ua</a> <a id="4047" href="Cubical.Data.Sigma.Properties.html#3958" class="Function">ΣPathP≃PathPΣ</a>
<a id="4062" class="Comment">-- Σ of discrete types</a>
<a id="discreteΣ"></a><a id="4086" href="Cubical.Data.Sigma.Properties.html#4086" class="Function">discreteΣ</a> <a id="4096" class="Symbol">:</a> <a id="4098" href="Cubical.Relation.Nullary.Base.html#1384" class="Function">Discrete</a> <a id="4107" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="4109" class="Symbol">→</a> <a id="4111" class="Symbol">((</a><a id="4113" href="Cubical.Data.Sigma.Properties.html#4113" class="Bound">a</a> <a id="4115" class="Symbol">:</a> <a id="4117" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="4118" class="Symbol">)</a> <a id="4120" class="Symbol">→</a> <a id="4122" href="Cubical.Relation.Nullary.Base.html#1384" class="Function">Discrete</a> <a id="4131" class="Symbol">(</a><a id="4132" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="4134" href="Cubical.Data.Sigma.Properties.html#4113" class="Bound">a</a><a id="4135" class="Symbol">))</a> <a id="4138" class="Symbol">→</a> <a id="4140" href="Cubical.Relation.Nullary.Base.html#1384" class="Function">Discrete</a> <a id="4149" class="Symbol">(</a><a id="4150" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="4152" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="4154" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="4155" class="Symbol">)</a>
<a id="4157" href="Cubical.Data.Sigma.Properties.html#4086" class="Function">discreteΣ</a> <a id="4167" class="Symbol">{</a><a id="4168" class="Argument">B</a> <a id="4170" class="Symbol">=</a> <a id="4172" href="Cubical.Data.Sigma.Properties.html#4172" class="Bound">B</a><a id="4173" class="Symbol">}</a> <a id="4175" href="Cubical.Data.Sigma.Properties.html#4175" class="Bound">Adis</a> <a id="4180" href="Cubical.Data.Sigma.Properties.html#4180" class="Bound">Bdis</a> <a id="4185" class="Symbol">(</a><a id="4186" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4189" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4191" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a><a id="4193" class="Symbol">)</a> <a id="4195" class="Symbol">(</a><a id="4196" href="Cubical.Data.Sigma.Properties.html#4196" class="Bound">a1</a> <a id="4199" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4201" href="Cubical.Data.Sigma.Properties.html#4201" class="Bound">b1</a><a id="4203" class="Symbol">)</a> <a id="4205" class="Symbol">=</a> <a id="4207" href="Cubical.Data.Sigma.Properties.html#4243" class="Function">discreteΣ'</a> <a id="4218" class="Symbol">(</a><a id="4219" href="Cubical.Data.Sigma.Properties.html#4175" class="Bound">Adis</a> <a id="4224" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4227" href="Cubical.Data.Sigma.Properties.html#4196" class="Bound">a1</a><a id="4229" class="Symbol">)</a>
<a id="4233" class="Keyword">where</a>
<a id="4243" href="Cubical.Data.Sigma.Properties.html#4243" class="Function">discreteΣ'</a> <a id="4254" class="Symbol">:</a> <a id="4256" href="Cubical.Relation.Nullary.Base.html#444" class="Datatype">Dec</a> <a id="4260" class="Symbol">(</a><a id="4261" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4264" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="4266" href="Cubical.Data.Sigma.Properties.html#4196" class="Bound">a1</a><a id="4268" class="Symbol">)</a> <a id="4270" class="Symbol">→</a> <a id="4272" href="Cubical.Relation.Nullary.Base.html#444" class="Datatype">Dec</a> <a id="4276" class="Symbol">((</a><a id="4278" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4281" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4283" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a><a id="4285" class="Symbol">)</a> <a id="4287" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="4289" class="Symbol">(</a><a id="4290" href="Cubical.Data.Sigma.Properties.html#4196" class="Bound">a1</a> <a id="4293" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4295" href="Cubical.Data.Sigma.Properties.html#4201" class="Bound">b1</a><a id="4297" class="Symbol">))</a>
<a id="4304" href="Cubical.Data.Sigma.Properties.html#4243" class="Function">discreteΣ'</a> <a id="4315" class="Symbol">(</a><a id="4316" href="Cubical.Relation.Nullary.Base.html#478" class="InductiveConstructor">yes</a> <a id="4320" href="Cubical.Data.Sigma.Properties.html#4320" class="Bound">p</a><a id="4321" class="Symbol">)</a> <a id="4323" class="Symbol">=</a> <a id="4325" href="Cubical.Foundations.Prelude.html#11155" class="Function">J</a> <a id="4327" class="Symbol">(λ</a> <a id="4330" href="Cubical.Data.Sigma.Properties.html#4330" class="Bound">a1</a> <a id="4333" href="Cubical.Data.Sigma.Properties.html#4333" class="Bound">p</a> <a id="4335" class="Symbol">→</a> <a id="4337" class="Symbol">∀</a> <a id="4339" href="Cubical.Data.Sigma.Properties.html#4339" class="Bound">b1</a> <a id="4342" class="Symbol">→</a> <a id="4344" href="Cubical.Relation.Nullary.Base.html#444" class="Datatype">Dec</a> <a id="4348" class="Symbol">((</a><a id="4350" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4353" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4355" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a><a id="4357" class="Symbol">)</a> <a id="4359" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="4361" class="Symbol">(</a><a id="4362" href="Cubical.Data.Sigma.Properties.html#4330" class="Bound">a1</a> <a id="4365" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4367" href="Cubical.Data.Sigma.Properties.html#4339" class="Bound">b1</a><a id="4369" class="Symbol">)))</a> <a id="4373" class="Symbol">(</a><a id="4374" href="Cubical.Data.Sigma.Properties.html#4412" class="Function">discreteΣ''</a><a id="4385" class="Symbol">)</a> <a id="4387" href="Cubical.Data.Sigma.Properties.html#4320" class="Bound">p</a> <a id="4389" href="Cubical.Data.Sigma.Properties.html#4201" class="Bound">b1</a>
<a id="4398" class="Keyword">where</a>
<a id="4412" href="Cubical.Data.Sigma.Properties.html#4412" class="Function">discreteΣ''</a> <a id="4424" class="Symbol">:</a> <a id="4426" class="Symbol">(</a><a id="4427" href="Cubical.Data.Sigma.Properties.html#4427" class="Bound">b1</a> <a id="4430" class="Symbol">:</a> <a id="4432" href="Cubical.Data.Sigma.Properties.html#4172" class="Bound">B</a> <a id="4434" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a><a id="4436" class="Symbol">)</a> <a id="4438" class="Symbol">→</a> <a id="4440" href="Cubical.Relation.Nullary.Base.html#444" class="Datatype">Dec</a> <a id="4444" class="Symbol">((</a><a id="4446" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4449" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4451" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a><a id="4453" class="Symbol">)</a> <a id="4455" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="4457" class="Symbol">(</a><a id="4458" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4461" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4463" href="Cubical.Data.Sigma.Properties.html#4427" class="Bound">b1</a><a id="4465" class="Symbol">))</a>
<a id="4476" href="Cubical.Data.Sigma.Properties.html#4412" class="Function">discreteΣ''</a> <a id="4488" href="Cubical.Data.Sigma.Properties.html#4488" class="Bound">b1</a> <a id="4491" class="Keyword">with</a> <a id="4496" href="Cubical.Data.Sigma.Properties.html#4180" class="Bound">Bdis</a> <a id="4501" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4504" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a> <a id="4507" href="Cubical.Data.Sigma.Properties.html#4488" class="Bound">b1</a>
<a id="4518" class="Symbol">...</a> <a id="4522" class="Symbol">|</a> <a id="4524" class="Symbol">(</a><a id="4525" href="Cubical.Relation.Nullary.Base.html#478" class="InductiveConstructor">yes</a> <a id="4529" href="Cubical.Data.Sigma.Properties.html#4529" class="Bound">q</a><a id="4530" class="Symbol">)</a> <a id="4532" class="Symbol">=</a> <a id="4534" href="Cubical.Relation.Nullary.Base.html#478" class="InductiveConstructor">yes</a> <a id="4538" class="Symbol">(</a><a id="4539" href="Cubical.Foundations.Prelude.html#8597" class="Function">transport</a> <a id="4549" href="Cubical.Data.Sigma.Properties.html#2839" class="Function">ΣPath≡PathΣ</a> <a id="4561" class="Symbol">(</a><a id="4562" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a> <a id="4567" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="4569" href="Cubical.Data.Sigma.Properties.html#4529" class="Bound">q</a><a id="4570" class="Symbol">))</a>
<a id="4581" class="Symbol">...</a> <a id="4585" class="Symbol">|</a> <a id="4587" class="Symbol">(</a><a id="4588" href="Cubical.Relation.Nullary.Base.html#505" class="InductiveConstructor">no</a> <a id="4591" href="Cubical.Data.Sigma.Properties.html#4591" class="Bound">¬q</a><a id="4593" class="Symbol">)</a> <a id="4595" class="Symbol">=</a> <a id="4597" href="Cubical.Relation.Nullary.Base.html#505" class="InductiveConstructor">no</a> <a id="4600" class="Symbol">(λ</a> <a id="4603" href="Cubical.Data.Sigma.Properties.html#4603" class="Bound">r</a> <a id="4605" class="Symbol">→</a> <a id="4607" href="Cubical.Data.Sigma.Properties.html#4591" class="Bound">¬q</a> <a id="4610" class="Symbol">(</a><a id="4611" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="4617" class="Symbol">(λ</a> <a id="4620" href="Cubical.Data.Sigma.Properties.html#4620" class="Bound">X</a> <a id="4622" class="Symbol">→</a> <a id="4624" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="4630" class="Symbol">(λ</a> <a id="4633" href="Cubical.Data.Sigma.Properties.html#4633" class="Bound">i</a> <a id="4635" class="Symbol">→</a> <a id="4637" href="Cubical.Data.Sigma.Properties.html#4172" class="Bound">B</a> <a id="4639" class="Symbol">(</a><a id="4640" href="Cubical.Data.Sigma.Properties.html#4620" class="Bound">X</a> <a id="4642" href="Cubical.Data.Sigma.Properties.html#4633" class="Bound">i</a><a id="4643" class="Symbol">))</a> <a id="4646" href="Cubical.Data.Sigma.Properties.html#4191" class="Bound">b0</a> <a id="4649" class="Bound">b1</a><a id="4651" class="Symbol">)</a> <a id="4653" class="Symbol">(</a><a id="4654" href="Cubical.Relation.Nullary.Properties.html#6131" class="Function">Discrete→isSet</a> <a id="4669" href="Cubical.Data.Sigma.Properties.html#4175" class="Bound">Adis</a> <a id="4674" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4677" href="Cubical.Data.Sigma.Properties.html#4186" class="Bound">a0</a> <a id="4680" class="Symbol">(</a><a id="4681" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="4686" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="4690" href="Cubical.Data.Sigma.Properties.html#4603" class="Bound">r</a><a id="4691" class="Symbol">)</a> <a id="4693" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a><a id="4697" class="Symbol">)</a> <a id="4699" class="Symbol">(</a><a id="4700" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="4705" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="4709" href="Cubical.Data.Sigma.Properties.html#4603" class="Bound">r</a><a id="4710" class="Symbol">)))</a>
<a id="4718" href="Cubical.Data.Sigma.Properties.html#4243" class="Function">discreteΣ'</a> <a id="4729" class="Symbol">(</a><a id="4730" href="Cubical.Relation.Nullary.Base.html#505" class="InductiveConstructor">no</a> <a id="4733" href="Cubical.Data.Sigma.Properties.html#4733" class="Bound">¬p</a><a id="4735" class="Symbol">)</a> <a id="4737" class="Symbol">=</a> <a id="4739" href="Cubical.Relation.Nullary.Base.html#505" class="InductiveConstructor">no</a> <a id="4742" class="Symbol">(λ</a> <a id="4745" href="Cubical.Data.Sigma.Properties.html#4745" class="Bound">r</a> <a id="4747" class="Symbol">→</a> <a id="4749" href="Cubical.Data.Sigma.Properties.html#4733" class="Bound">¬p</a> <a id="4752" class="Symbol">(</a><a id="4753" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="4758" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="4762" href="Cubical.Data.Sigma.Properties.html#4745" class="Bound">r</a><a id="4763" class="Symbol">))</a>
<a id="lUnit×Iso"></a><a id="4767" href="Cubical.Data.Sigma.Properties.html#4767" class="Function">lUnit×Iso</a> <a id="4777" class="Symbol">:</a> <a id="4779" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="4783" class="Symbol">(</a><a id="4784" href="Cubical.Data.Unit.Base.html#171" class="Record">Unit</a> <a id="4789" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="4791" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="4792" class="Symbol">)</a> <a id="4794" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a>
<a id="4796" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="4800" href="Cubical.Data.Sigma.Properties.html#4767" class="Function">lUnit×Iso</a> <a id="4810" class="Symbol">=</a> <a id="4812" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a>
<a id="4816" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="4820" href="Cubical.Data.Sigma.Properties.html#4767" class="Function">lUnit×Iso</a> <a id="4830" class="Symbol">=</a> <a id="4832" href="Agda.Builtin.Unit.html#201" class="InductiveConstructor">tt</a> <a id="4835" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,_</a>
<a id="4838" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="4847" href="Cubical.Data.Sigma.Properties.html#4767" class="Function">lUnit×Iso</a> <a id="4857" class="Symbol">_</a> <a id="4859" class="Symbol">=</a> <a id="4861" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="4866" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="4874" href="Cubical.Data.Sigma.Properties.html#4767" class="Function">lUnit×Iso</a> <a id="4884" class="Symbol">_</a> <a id="4886" class="Symbol">=</a> <a id="4888" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="rUnit×Iso"></a><a id="4894" href="Cubical.Data.Sigma.Properties.html#4894" class="Function">rUnit×Iso</a> <a id="4904" class="Symbol">:</a> <a id="4906" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="4910" class="Symbol">(</a><a id="4911" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="4913" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="4915" href="Cubical.Data.Unit.Base.html#171" class="Record">Unit</a><a id="4919" class="Symbol">)</a> <a id="4921" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a>
<a id="4923" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="4927" href="Cubical.Data.Sigma.Properties.html#4894" class="Function">rUnit×Iso</a> <a id="4937" class="Symbol">=</a> <a id="4939" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a>
<a id="4943" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="4947" href="Cubical.Data.Sigma.Properties.html#4894" class="Function">rUnit×Iso</a> <a id="4957" class="Symbol">=</a> <a id="4959" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">_,</a> <a id="4962" href="Agda.Builtin.Unit.html#201" class="InductiveConstructor">tt</a>
<a id="4965" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="4974" href="Cubical.Data.Sigma.Properties.html#4894" class="Function">rUnit×Iso</a> <a id="4984" class="Symbol">_</a> <a id="4986" class="Symbol">=</a> <a id="4988" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="4993" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="5001" href="Cubical.Data.Sigma.Properties.html#4894" class="Function">rUnit×Iso</a> <a id="5011" class="Symbol">_</a> <a id="5013" class="Symbol">=</a> <a id="5015" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5021" class="Keyword">module</a> <a id="5028" href="Cubical.Data.Sigma.Properties.html#5028" class="Module">_</a> <a id="5030" class="Symbol">{</a><a id="5031" href="Cubical.Data.Sigma.Properties.html#5031" class="Bound">A</a> <a id="5033" class="Symbol">:</a> <a id="5035" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="5040" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="5041" class="Symbol">}</a> <a id="5043" class="Symbol">{</a><a id="5044" href="Cubical.Data.Sigma.Properties.html#5044" class="Bound">A'</a> <a id="5047" class="Symbol">:</a> <a id="5049" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="5054" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="5056" class="Symbol">}</a> <a id="5058" class="Keyword">where</a>
<a id="5066" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a> <a id="5077" class="Symbol">:</a> <a id="5079" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="5083" class="Symbol">(</a><a id="5084" href="Cubical.Data.Sigma.Properties.html#5031" class="Bound">A</a> <a id="5086" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="5088" href="Cubical.Data.Sigma.Properties.html#5044" class="Bound">A'</a><a id="5090" class="Symbol">)</a> <a id="5092" class="Symbol">(</a><a id="5093" href="Cubical.Data.Sigma.Properties.html#5044" class="Bound">A'</a> <a id="5096" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="5098" href="Cubical.Data.Sigma.Properties.html#5031" class="Bound">A</a><a id="5099" class="Symbol">)</a>
<a id="5103" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="5107" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a> <a id="5118" class="Symbol">(</a><a id="5119" href="Cubical.Data.Sigma.Properties.html#5119" class="Bound">x</a> <a id="5121" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5123" href="Cubical.Data.Sigma.Properties.html#5123" class="Bound">y</a><a id="5124" class="Symbol">)</a> <a id="5126" class="Symbol">=</a> <a id="5128" class="Symbol">(</a><a id="5129" href="Cubical.Data.Sigma.Properties.html#5123" class="Bound">y</a> <a id="5131" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5133" href="Cubical.Data.Sigma.Properties.html#5119" class="Bound">x</a><a id="5134" class="Symbol">)</a>
<a id="5138" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="5142" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a> <a id="5153" class="Symbol">(</a><a id="5154" href="Cubical.Data.Sigma.Properties.html#5154" class="Bound">x</a> <a id="5156" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5158" href="Cubical.Data.Sigma.Properties.html#5158" class="Bound">y</a><a id="5159" class="Symbol">)</a> <a id="5161" class="Symbol">=</a> <a id="5163" class="Symbol">(</a><a id="5164" href="Cubical.Data.Sigma.Properties.html#5158" class="Bound">y</a> <a id="5166" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5168" href="Cubical.Data.Sigma.Properties.html#5154" class="Bound">x</a><a id="5169" class="Symbol">)</a>
<a id="5173" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="5182" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a> <a id="5193" class="Symbol">_</a> <a id="5195" class="Symbol">=</a> <a id="5197" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5204" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="5212" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a> <a id="5223" class="Symbol">_</a> <a id="5226" class="Symbol">=</a> <a id="5228" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5236" class="Keyword">unquoteDecl</a> <a id="5248" href="Cubical.Data.Sigma.Properties.html#5248" class="Function">Σ-swap-≃</a> <a id="5257" class="Symbol">=</a> <a id="5259" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="5280" href="Cubical.Data.Sigma.Properties.html#5248" class="Function">Σ-swap-≃</a> <a id="5289" href="Cubical.Data.Sigma.Properties.html#5066" class="Function">Σ-swap-Iso</a>
<a id="5301" class="Keyword">module</a> <a id="5308" href="Cubical.Data.Sigma.Properties.html#5308" class="Module">_</a> <a id="5310" class="Symbol">{</a><a id="5311" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a> <a id="5313" class="Symbol">:</a> <a id="5315" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="5320" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="5321" class="Symbol">}</a> <a id="5323" class="Symbol">{</a><a id="5324" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5326" class="Symbol">:</a> <a id="5328" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a> <a id="5330" class="Symbol">→</a> <a id="5332" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="5337" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="5339" class="Symbol">}</a> <a id="5341" class="Symbol">{</a><a id="5342" href="Cubical.Data.Sigma.Properties.html#5342" class="Bound">C</a> <a id="5344" class="Symbol">:</a> <a id="5346" class="Symbol">∀</a> <a id="5348" href="Cubical.Data.Sigma.Properties.html#5348" class="Bound">a</a> <a id="5350" class="Symbol">→</a> <a id="5352" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5354" href="Cubical.Data.Sigma.Properties.html#5348" class="Bound">a</a> <a id="5356" class="Symbol">→</a> <a id="5358" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="5363" href="Cubical.Data.Sigma.Properties.html#1302" class="Generalizable">ℓ''</a><a id="5366" class="Symbol">}</a> <a id="5368" class="Keyword">where</a>
<a id="5376" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a> <a id="5388" class="Symbol">:</a> <a id="5390" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="5394" class="Symbol">(</a><a id="5395" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="5398" href="Cubical.Data.Sigma.Properties.html#5398" class="Bound">a</a> <a id="5400" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="5402" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="5404" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a> <a id="5406" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5408" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="5410" href="Cubical.Data.Sigma.Properties.html#5342" class="Bound">C</a> <a id="5412" class="Symbol">(</a><a id="5413" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="5417" href="Cubical.Data.Sigma.Properties.html#5398" class="Bound">a</a><a id="5418" class="Symbol">)</a> <a id="5420" class="Symbol">(</a><a id="5421" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="5425" href="Cubical.Data.Sigma.Properties.html#5398" class="Bound">a</a><a id="5426" class="Symbol">))</a> <a id="5429" class="Symbol">(</a><a id="5430" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="5433" href="Cubical.Data.Sigma.Properties.html#5433" class="Bound">a</a> <a id="5435" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="5437" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a> <a id="5439" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="5441" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="5444" href="Cubical.Data.Sigma.Properties.html#5444" class="Bound">b</a> <a id="5446" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="5448" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5450" href="Cubical.Data.Sigma.Properties.html#5433" class="Bound">a</a> <a id="5452" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="5454" href="Cubical.Data.Sigma.Properties.html#5342" class="Bound">C</a> <a id="5456" href="Cubical.Data.Sigma.Properties.html#5433" class="Bound">a</a> <a id="5458" href="Cubical.Data.Sigma.Properties.html#5444" class="Bound">b</a><a id="5459" class="Symbol">)</a>
<a id="5463" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="5467" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a> <a id="5479" class="Symbol">((</a><a id="5481" href="Cubical.Data.Sigma.Properties.html#5481" class="Bound">x</a> <a id="5483" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5485" href="Cubical.Data.Sigma.Properties.html#5485" class="Bound">y</a><a id="5486" class="Symbol">)</a> <a id="5488" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5490" href="Cubical.Data.Sigma.Properties.html#5490" class="Bound">z</a><a id="5491" class="Symbol">)</a> <a id="5493" class="Symbol">=</a> <a id="5495" class="Symbol">(</a><a id="5496" href="Cubical.Data.Sigma.Properties.html#5481" class="Bound">x</a> <a id="5498" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5500" class="Symbol">(</a><a id="5501" href="Cubical.Data.Sigma.Properties.html#5485" class="Bound">y</a> <a id="5503" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5505" href="Cubical.Data.Sigma.Properties.html#5490" class="Bound">z</a><a id="5506" class="Symbol">))</a>
<a id="5511" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="5515" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a> <a id="5527" class="Symbol">(</a><a id="5528" href="Cubical.Data.Sigma.Properties.html#5528" class="Bound">x</a> <a id="5530" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5532" class="Symbol">(</a><a id="5533" href="Cubical.Data.Sigma.Properties.html#5533" class="Bound">y</a> <a id="5535" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5537" href="Cubical.Data.Sigma.Properties.html#5537" class="Bound">z</a><a id="5538" class="Symbol">))</a> <a id="5541" class="Symbol">=</a> <a id="5543" class="Symbol">((</a><a id="5545" href="Cubical.Data.Sigma.Properties.html#5528" class="Bound">x</a> <a id="5547" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5549" href="Cubical.Data.Sigma.Properties.html#5533" class="Bound">y</a><a id="5550" class="Symbol">)</a> <a id="5552" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5554" href="Cubical.Data.Sigma.Properties.html#5537" class="Bound">z</a><a id="5555" class="Symbol">)</a>
<a id="5559" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="5568" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a> <a id="5580" class="Symbol">_</a> <a id="5582" class="Symbol">=</a> <a id="5584" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5591" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="5599" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a> <a id="5611" class="Symbol">_</a> <a id="5614" class="Symbol">=</a> <a id="5616" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5624" class="Keyword">unquoteDecl</a> <a id="5636" href="Cubical.Data.Sigma.Properties.html#5636" class="Function">Σ-assoc-≃</a> <a id="5646" class="Symbol">=</a> <a id="5648" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="5669" href="Cubical.Data.Sigma.Properties.html#5636" class="Function">Σ-assoc-≃</a> <a id="5679" href="Cubical.Data.Sigma.Properties.html#5376" class="Function">Σ-assoc-Iso</a>
<a id="5694" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a> <a id="5702" class="Symbol">:</a> <a id="5704" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="5708" class="Symbol">((</a><a id="5710" href="Cubical.Data.Sigma.Properties.html#5710" class="Bound">a</a> <a id="5712" class="Symbol">:</a> <a id="5714" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a><a id="5715" class="Symbol">)</a> <a id="5717" class="Symbol">→</a> <a id="5719" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="5722" href="Cubical.Data.Sigma.Properties.html#5722" class="Bound">b</a> <a id="5724" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="5726" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5728" href="Cubical.Data.Sigma.Properties.html#5710" class="Bound">a</a> <a id="5730" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="5732" href="Cubical.Data.Sigma.Properties.html#5342" class="Bound">C</a> <a id="5734" href="Cubical.Data.Sigma.Properties.html#5710" class="Bound">a</a> <a id="5736" href="Cubical.Data.Sigma.Properties.html#5722" class="Bound">b</a><a id="5737" class="Symbol">)</a> <a id="5739" class="Symbol">(</a><a id="5740" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="5743" href="Cubical.Data.Sigma.Properties.html#5743" class="Bound">f</a> <a id="5745" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="5747" class="Symbol">((</a><a id="5749" href="Cubical.Data.Sigma.Properties.html#5749" class="Bound">a</a> <a id="5751" class="Symbol">:</a> <a id="5753" href="Cubical.Data.Sigma.Properties.html#5311" class="Bound">A</a><a id="5754" class="Symbol">)</a> <a id="5756" class="Symbol">→</a> <a id="5758" href="Cubical.Data.Sigma.Properties.html#5324" class="Bound">B</a> <a id="5760" href="Cubical.Data.Sigma.Properties.html#5749" class="Bound">a</a><a id="5761" class="Symbol">)</a> <a id="5763" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="5765" class="Symbol">∀</a> <a id="5767" href="Cubical.Data.Sigma.Properties.html#5767" class="Bound">a</a> <a id="5769" class="Symbol">→</a> <a id="5771" href="Cubical.Data.Sigma.Properties.html#5342" class="Bound">C</a> <a id="5773" href="Cubical.Data.Sigma.Properties.html#5767" class="Bound">a</a> <a id="5775" class="Symbol">(</a><a id="5776" href="Cubical.Data.Sigma.Properties.html#5743" class="Bound">f</a> <a id="5778" href="Cubical.Data.Sigma.Properties.html#5767" class="Bound">a</a><a id="5779" class="Symbol">))</a>
<a id="5784" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="5788" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a> <a id="5796" href="Cubical.Data.Sigma.Properties.html#5796" class="Bound">f</a> <a id="5806" class="Symbol">=</a> <a id="5808" class="Symbol">(</a><a id="5809" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="5813" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="5815" href="Cubical.Data.Sigma.Properties.html#5796" class="Bound">f</a> <a id="5817" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5819" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="5823" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="5825" href="Cubical.Data.Sigma.Properties.html#5796" class="Bound">f</a><a id="5826" class="Symbol">)</a>
<a id="5830" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="5834" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a> <a id="5842" class="Symbol">(</a><a id="5843" href="Cubical.Data.Sigma.Properties.html#5843" class="Bound">f</a> <a id="5845" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5847" href="Cubical.Data.Sigma.Properties.html#5847" class="Bound">g</a><a id="5848" class="Symbol">)</a> <a id="5850" href="Cubical.Data.Sigma.Properties.html#5850" class="Bound">x</a> <a id="5852" class="Symbol">=</a> <a id="5854" class="Symbol">(</a><a id="5855" href="Cubical.Data.Sigma.Properties.html#5843" class="Bound">f</a> <a id="5857" href="Cubical.Data.Sigma.Properties.html#5850" class="Bound">x</a> <a id="5859" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="5861" href="Cubical.Data.Sigma.Properties.html#5847" class="Bound">g</a> <a id="5863" href="Cubical.Data.Sigma.Properties.html#5850" class="Bound">x</a><a id="5864" class="Symbol">)</a>
<a id="5868" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="5877" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a> <a id="5885" class="Symbol">_</a> <a id="5890" class="Symbol">=</a> <a id="5892" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5899" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="5907" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a> <a id="5915" class="Symbol">_</a> <a id="5921" class="Symbol">=</a> <a id="5923" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="5931" class="Keyword">unquoteDecl</a> <a id="5943" href="Cubical.Data.Sigma.Properties.html#5943" class="Function">Σ-Π-≃</a> <a id="5949" class="Symbol">=</a> <a id="5951" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="5972" href="Cubical.Data.Sigma.Properties.html#5943" class="Function">Σ-Π-≃</a> <a id="5978" href="Cubical.Data.Sigma.Properties.html#5694" class="Function">Σ-Π-Iso</a>
<a id="Σ-cong-iso-fst"></a><a id="5987" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="6002" class="Symbol">:</a> <a id="6004" class="Symbol">(</a><a id="6005" href="Cubical.Data.Sigma.Properties.html#6005" class="Bound">isom</a> <a id="6010" class="Symbol">:</a> <a id="6012" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="6016" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="6018" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="6020" class="Symbol">)</a> <a id="6022" class="Symbol">→</a> <a id="6024" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="6028" class="Symbol">(</a><a id="6029" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="6031" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="6033" class="Symbol">(</a><a id="6034" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="6036" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="6038" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6042" href="Cubical.Data.Sigma.Properties.html#6005" class="Bound">isom</a><a id="6046" class="Symbol">))</a> <a id="6049" class="Symbol">(</a><a id="6050" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="6052" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="6055" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="6056" class="Symbol">)</a>
<a id="6058" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6062" class="Symbol">(</a><a id="6063" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="6078" href="Cubical.Data.Sigma.Properties.html#6078" class="Bound">isom</a><a id="6082" class="Symbol">)</a> <a id="6084" href="Cubical.Data.Sigma.Properties.html#6084" class="Bound">x</a> <a id="6086" class="Symbol">=</a> <a id="6088" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6092" href="Cubical.Data.Sigma.Properties.html#6078" class="Bound">isom</a> <a id="6097" class="Symbol">(</a><a id="6098" href="Cubical.Data.Sigma.Properties.html#6084" class="Bound">x</a> <a id="6100" class="Symbol">.</a><a id="6101" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="6104" class="Symbol">)</a> <a id="6106" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6108" href="Cubical.Data.Sigma.Properties.html#6084" class="Bound">x</a> <a id="6110" class="Symbol">.</a><a id="6111" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a>
<a id="6115" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="6119" class="Symbol">(</a><a id="6120" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="6135" class="Symbol">{</a><a id="6136" class="Argument">B</a> <a id="6138" class="Symbol">=</a> <a id="6140" href="Cubical.Data.Sigma.Properties.html#6140" class="Bound">B</a><a id="6141" class="Symbol">}</a> <a id="6143" href="Cubical.Data.Sigma.Properties.html#6143" class="Bound">isom</a><a id="6147" class="Symbol">)</a> <a id="6149" href="Cubical.Data.Sigma.Properties.html#6149" class="Bound">x</a> <a id="6151" class="Symbol">=</a> <a id="6153" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="6157" href="Cubical.Data.Sigma.Properties.html#6143" class="Bound">isom</a> <a id="6162" class="Symbol">(</a><a id="6163" href="Cubical.Data.Sigma.Properties.html#6149" class="Bound">x</a> <a id="6165" class="Symbol">.</a><a id="6166" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="6169" class="Symbol">)</a> <a id="6171" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6173" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6179" href="Cubical.Data.Sigma.Properties.html#6140" class="Bound">B</a> <a id="6181" class="Symbol">(</a><a id="6182" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6186" class="Symbol">(</a><a id="6187" href="Cubical.Data.Sigma.Properties.html#6219" class="Function">ε</a> <a id="6189" class="Symbol">(</a><a id="6190" href="Cubical.Data.Sigma.Properties.html#6149" class="Bound">x</a> <a id="6192" class="Symbol">.</a><a id="6193" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="6196" class="Symbol">)))</a> <a id="6200" class="Symbol">(</a><a id="6201" href="Cubical.Data.Sigma.Properties.html#6149" class="Bound">x</a> <a id="6203" class="Symbol">.</a><a id="6204" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="6207" class="Symbol">)</a>
<a id="6211" class="Keyword">where</a>
<a id="6219" href="Cubical.Data.Sigma.Properties.html#6219" class="Function">ε</a> <a id="6221" class="Symbol">=</a> <a id="6223" href="Cubical.Foundations.Equiv.HalfAdjoint.html#612" class="Field">isHAEquiv.rinv</a> <a id="6238" class="Symbol">(</a><a id="6239" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="6243" class="Symbol">(</a><a id="6244" href="Cubical.Foundations.Equiv.HalfAdjoint.html#2344" class="Function">iso→HAEquiv</a> <a id="6256" href="Cubical.Data.Sigma.Properties.html#6143" class="Bound">isom</a><a id="6260" class="Symbol">))</a>
<a id="6263" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="6272" class="Symbol">(</a><a id="6273" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="6288" class="Symbol">{</a><a id="6289" class="Argument">B</a> <a id="6291" class="Symbol">=</a> <a id="6293" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a><a id="6294" class="Symbol">}</a> <a id="6296" href="Cubical.Data.Sigma.Properties.html#6296" class="Bound">isom</a><a id="6300" class="Symbol">)</a> <a id="6302" class="Symbol">(</a><a id="6303" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a> <a id="6305" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6307" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a><a id="6308" class="Symbol">)</a> <a id="6310" class="Symbol">=</a> <a id="6312" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="6319" class="Symbol">(</a><a id="6320" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6322" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a> <a id="6324" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6326" href="Cubical.Foundations.Prelude.html#12746" class="Function">toPathP</a> <a id="6334" href="Cubical.Data.Sigma.Properties.html#6396" class="Function">goal</a><a id="6338" class="Symbol">)</a>
<a id="6342" class="Keyword">where</a>
<a id="6350" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6352" class="Symbol">=</a> <a id="6354" href="Cubical.Foundations.Equiv.HalfAdjoint.html#612" class="Field">isHAEquiv.rinv</a> <a id="6369" class="Symbol">(</a><a id="6370" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="6374" class="Symbol">(</a><a id="6375" href="Cubical.Foundations.Equiv.HalfAdjoint.html#2344" class="Function">iso→HAEquiv</a> <a id="6387" href="Cubical.Data.Sigma.Properties.html#6296" class="Bound">isom</a><a id="6391" class="Symbol">))</a>
<a id="6396" href="Cubical.Data.Sigma.Properties.html#6396" class="Function">goal</a> <a id="6401" class="Symbol">:</a> <a id="6403" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6409" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a> <a id="6411" class="Symbol">(</a><a id="6412" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6414" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a><a id="6415" class="Symbol">)</a> <a id="6417" class="Symbol">(</a><a id="6418" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6424" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a> <a id="6426" class="Symbol">(</a><a id="6427" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6431" class="Symbol">(</a><a id="6432" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6434" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a><a id="6435" class="Symbol">))</a> <a id="6438" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a><a id="6439" class="Symbol">)</a> <a id="6441" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="6443" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a>
<a id="6447" href="Cubical.Data.Sigma.Properties.html#6396" class="Function">goal</a> <a id="6452" class="Symbol">=</a> <a id="6454" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6458" class="Symbol">(</a><a id="6459" href="Cubical.Foundations.Transport.html#5268" class="Function">substComposite</a> <a id="6474" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a> <a id="6476" class="Symbol">(</a><a id="6477" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6481" class="Symbol">(</a><a id="6482" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6484" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a><a id="6485" class="Symbol">))</a> <a id="6488" class="Symbol">(</a><a id="6489" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6491" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a><a id="6492" class="Symbol">)</a> <a id="6494" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a><a id="6495" class="Symbol">)</a>
<a id="6503" href="Cubical.Foundations.Prelude.html#3247" class="Function Operator">∙∙</a> <a id="6506" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="6511" class="Symbol">(λ</a> <a id="6514" href="Cubical.Data.Sigma.Properties.html#6514" class="Bound">x</a> <a id="6516" class="Symbol">→</a> <a id="6518" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6524" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a> <a id="6526" href="Cubical.Data.Sigma.Properties.html#6514" class="Bound">x</a> <a id="6528" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a><a id="6529" class="Symbol">)</a> <a id="6531" class="Symbol">(</a><a id="6532" href="Cubical.Foundations.GroupoidLaws.html#2147" class="Function">lCancel</a> <a id="6540" class="Symbol">(</a><a id="6541" href="Cubical.Data.Sigma.Properties.html#6350" class="Function">ε</a> <a id="6543" href="Cubical.Data.Sigma.Properties.html#6303" class="Bound">x</a><a id="6544" class="Symbol">))</a>
<a id="6553" href="Cubical.Foundations.Prelude.html#3247" class="Function Operator">∙∙</a> <a id="6556" href="Cubical.Foundations.Prelude.html#9390" class="Function">substRefl</a> <a id="6566" class="Symbol">{</a><a id="6567" class="Argument">B</a> <a id="6569" class="Symbol">=</a> <a id="6571" href="Cubical.Data.Sigma.Properties.html#6293" class="Bound">B</a><a id="6572" class="Symbol">}</a> <a id="6574" href="Cubical.Data.Sigma.Properties.html#6307" class="Bound">y</a>
<a id="6576" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="6584" class="Symbol">(</a><a id="6585" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="6600" class="Symbol">{</a><a id="6601" class="Argument">A</a> <a id="6603" class="Symbol">=</a> <a id="6605" href="Cubical.Data.Sigma.Properties.html#6605" class="Bound">A</a><a id="6606" class="Symbol">}</a> <a id="6608" class="Symbol">{</a><a id="6609" class="Argument">B</a> <a id="6611" class="Symbol">=</a> <a id="6613" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a><a id="6614" class="Symbol">}</a> <a id="6616" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="6620" class="Symbol">)</a> <a id="6622" class="Symbol">(</a><a id="6623" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a> <a id="6625" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6627" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a><a id="6628" class="Symbol">)</a> <a id="6630" class="Symbol">=</a> <a id="6632" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="6639" class="Symbol">(</a><a id="6640" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="6648" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6653" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a> <a id="6655" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="6657" href="Cubical.Foundations.Prelude.html#12746" class="Function">toPathP</a> <a id="6665" href="Cubical.Data.Sigma.Properties.html#6936" class="Function">goal</a><a id="6669" class="Symbol">)</a>
<a id="6673" class="Keyword">where</a>
<a id="6681" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="6683" class="Symbol">=</a> <a id="6685" href="Cubical.Foundations.Equiv.HalfAdjoint.html#612" class="Field">isHAEquiv.rinv</a> <a id="6700" class="Symbol">(</a><a id="6701" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="6705" class="Symbol">(</a><a id="6706" href="Cubical.Foundations.Equiv.HalfAdjoint.html#2344" class="Function">iso→HAEquiv</a> <a id="6718" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="6722" class="Symbol">))</a>
<a id="6727" href="Cubical.Data.Sigma.Properties.html#6727" class="Function">γ</a> <a id="6729" class="Symbol">=</a> <a id="6731" href="Cubical.Foundations.Equiv.HalfAdjoint.html#641" class="Field">isHAEquiv.com</a> <a id="6745" class="Symbol">(</a><a id="6746" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="6750" class="Symbol">(</a><a id="6751" href="Cubical.Foundations.Equiv.HalfAdjoint.html#2344" class="Function">iso→HAEquiv</a> <a id="6763" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="6767" class="Symbol">))</a>
<a id="6773" href="Cubical.Data.Sigma.Properties.html#6773" class="Function">lem</a> <a id="6777" class="Symbol">:</a> <a id="6779" class="Symbol">(</a><a id="6780" href="Cubical.Data.Sigma.Properties.html#6780" class="Bound">x</a> <a id="6782" class="Symbol">:</a> <a id="6784" href="Cubical.Data.Sigma.Properties.html#6605" class="Bound">A</a><a id="6785" class="Symbol">)</a> <a id="6787" class="Symbol">→</a> <a id="6789" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6793" class="Symbol">(</a><a id="6794" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="6796" class="Symbol">(</a><a id="6797" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6801" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6806" href="Cubical.Data.Sigma.Properties.html#6780" class="Bound">x</a><a id="6807" class="Symbol">))</a> <a id="6810" href="Cubical.Foundations.Prelude.html#4447" class="Function Operator">∙</a> <a id="6812" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="6817" class="Symbol">(</a><a id="6818" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6822" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="6826" class="Symbol">)</a> <a id="6828" class="Symbol">(</a><a id="6829" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="6837" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6842" href="Cubical.Data.Sigma.Properties.html#6780" class="Bound">x</a><a id="6843" class="Symbol">)</a> <a id="6845" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="6847" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="6854" href="Cubical.Data.Sigma.Properties.html#6773" class="Function">lem</a> <a id="6858" href="Cubical.Data.Sigma.Properties.html#6858" class="Bound">x</a> <a id="6860" class="Symbol">=</a> <a id="6862" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="6867" class="Symbol">(λ</a> <a id="6870" href="Cubical.Data.Sigma.Properties.html#6870" class="Bound">a</a> <a id="6872" class="Symbol">→</a> <a id="6874" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="6878" class="Symbol">(</a><a id="6879" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="6881" class="Symbol">(</a><a id="6882" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6886" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6891" href="Cubical.Data.Sigma.Properties.html#6858" class="Bound">x</a><a id="6892" class="Symbol">))</a> <a id="6895" href="Cubical.Foundations.Prelude.html#4447" class="Function Operator">∙</a> <a id="6897" href="Cubical.Data.Sigma.Properties.html#6870" class="Bound">a</a><a id="6898" class="Symbol">)</a> <a id="6900" class="Symbol">(</a><a id="6901" href="Cubical.Data.Sigma.Properties.html#6727" class="Function">γ</a> <a id="6903" href="Cubical.Data.Sigma.Properties.html#6858" class="Bound">x</a><a id="6904" class="Symbol">)</a> <a id="6906" href="Cubical.Foundations.Prelude.html#4447" class="Function Operator">∙</a> <a id="6908" href="Cubical.Foundations.GroupoidLaws.html#2147" class="Function">lCancel</a> <a id="6916" class="Symbol">(</a><a id="6917" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="6919" class="Symbol">(</a><a id="6920" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6924" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6929" href="Cubical.Data.Sigma.Properties.html#6858" class="Bound">x</a><a id="6930" class="Symbol">))</a>
<a id="6936" href="Cubical.Data.Sigma.Properties.html#6936" class="Function">goal</a> <a id="6941" class="Symbol">:</a> <a id="6943" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6949" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a> <a id="6951" class="Symbol">(</a><a id="6952" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="6957" class="Symbol">(</a><a id="6958" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="6962" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="6966" class="Symbol">)</a> <a id="6968" class="Symbol">(</a><a id="6969" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="6977" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="6982" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a><a id="6983" class="Symbol">))</a> <a id="6986" class="Symbol">(</a><a id="6987" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="6993" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a> <a id="6995" class="Symbol">(</a><a id="6996" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="7000" class="Symbol">(</a><a id="7001" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="7003" class="Symbol">(</a><a id="7004" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="7008" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="7013" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a><a id="7014" class="Symbol">)))</a> <a id="7018" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a><a id="7019" class="Symbol">)</a> <a id="7021" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="7023" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a>
<a id="7027" href="Cubical.Data.Sigma.Properties.html#6936" class="Function">goal</a> <a id="7032" class="Symbol">=</a> <a id="7034" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="7038" class="Symbol">(</a><a id="7039" href="Cubical.Foundations.Transport.html#5268" class="Function">substComposite</a> <a id="7054" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a> <a id="7056" class="Symbol">(</a><a id="7057" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="7061" class="Symbol">(</a><a id="7062" href="Cubical.Data.Sigma.Properties.html#6681" class="Function">ε</a> <a id="7064" class="Symbol">(</a><a id="7065" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="7069" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="7074" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a><a id="7075" class="Symbol">)))</a> <a id="7079" class="Symbol">(</a><a id="7080" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="7085" class="Symbol">(</a><a id="7086" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="7090" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a><a id="7094" class="Symbol">)</a> <a id="7096" class="Symbol">(</a><a id="7097" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="7105" href="Cubical.Data.Sigma.Properties.html#6616" class="Bound">isom</a> <a id="7110" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a><a id="7111" class="Symbol">))</a> <a id="7114" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a><a id="7115" class="Symbol">)</a>
<a id="7123" href="Cubical.Foundations.Prelude.html#3247" class="Function Operator">∙∙</a> <a id="7126" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="7131" class="Symbol">(λ</a> <a id="7134" href="Cubical.Data.Sigma.Properties.html#7134" class="Bound">a</a> <a id="7136" class="Symbol">→</a> <a id="7138" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="7144" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a> <a id="7146" href="Cubical.Data.Sigma.Properties.html#7134" class="Bound">a</a> <a id="7148" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a><a id="7149" class="Symbol">)</a> <a id="7151" class="Symbol">(</a><a id="7152" href="Cubical.Data.Sigma.Properties.html#6773" class="Function">lem</a> <a id="7156" href="Cubical.Data.Sigma.Properties.html#6623" class="Bound">x</a><a id="7157" class="Symbol">)</a>
<a id="7165" href="Cubical.Foundations.Prelude.html#3247" class="Function Operator">∙∙</a> <a id="7168" href="Cubical.Foundations.Prelude.html#9390" class="Function">substRefl</a> <a id="7178" class="Symbol">{</a><a id="7179" class="Argument">B</a> <a id="7181" class="Symbol">=</a> <a id="7183" href="Cubical.Data.Sigma.Properties.html#6613" class="Bound">B</a><a id="7184" class="Symbol">}</a> <a id="7186" href="Cubical.Data.Sigma.Properties.html#6627" class="Bound">y</a>
<a id="Σ-cong-equiv-fst"></a><a id="7189" href="Cubical.Data.Sigma.Properties.html#7189" class="Function">Σ-cong-equiv-fst</a> <a id="7206" class="Symbol">:</a> <a id="7208" class="Symbol">(</a><a id="7209" href="Cubical.Data.Sigma.Properties.html#7209" class="Bound">e</a> <a id="7211" class="Symbol">:</a> <a id="7213" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="7215" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="7217" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="7219" class="Symbol">)</a> <a id="7221" class="Symbol">→</a> <a id="7223" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="7225" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="7227" class="Symbol">(</a><a id="7228" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="7230" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="7232" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="7241" href="Cubical.Data.Sigma.Properties.html#7209" class="Bound">e</a><a id="7242" class="Symbol">)</a> <a id="7244" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="7246" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="7248" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="7251" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a>
<a id="7253" class="Comment">-- we could just do this:</a>
<a id="7279" class="Comment">-- Σ-cong-equiv-fst e = isoToEquiv (Σ-cong-iso-fst (equivToIso e))</a>
<a id="7346" class="Comment">-- but the following reduces slightly better</a>
<a id="7391" href="Cubical.Data.Sigma.Properties.html#7189" class="Function">Σ-cong-equiv-fst</a> <a id="7408" class="Symbol">{</a><a id="7409" class="Argument">A</a> <a id="7411" class="Symbol">=</a> <a id="7413" href="Cubical.Data.Sigma.Properties.html#7413" class="Bound">A</a><a id="7414" class="Symbol">}</a> <a id="7416" class="Symbol">{</a><a id="7417" class="Argument">A'</a> <a id="7420" class="Symbol">=</a> <a id="7422" href="Cubical.Data.Sigma.Properties.html#7422" class="Bound">A'</a><a id="7424" class="Symbol">}</a> <a id="7426" class="Symbol">{</a><a id="7427" class="Argument">B</a> <a id="7429" class="Symbol">=</a> <a id="7431" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a><a id="7432" class="Symbol">}</a> <a id="7434" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a> <a id="7436" class="Symbol">=</a> <a id="7438" href="Cubical.Data.Sigma.Properties.html#7465" class="Function">intro</a> <a id="7444" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7446" href="Cubical.Data.Sigma.Properties.html#7540" class="Function">isEqIntro</a>
<a id="7457" class="Keyword">where</a>
<a id="7465" href="Cubical.Data.Sigma.Properties.html#7465" class="Function">intro</a> <a id="7471" class="Symbol">:</a> <a id="7473" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="7475" href="Cubical.Data.Sigma.Properties.html#7413" class="Bound">A</a> <a id="7477" class="Symbol">(</a><a id="7478" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7480" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="7482" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="7491" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a><a id="7492" class="Symbol">)</a> <a id="7494" class="Symbol">→</a> <a id="7496" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="7498" href="Cubical.Data.Sigma.Properties.html#7422" class="Bound">A'</a> <a id="7501" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a>
<a id="7505" href="Cubical.Data.Sigma.Properties.html#7465" class="Function">intro</a> <a id="7511" class="Symbol">(</a><a id="7512" href="Cubical.Data.Sigma.Properties.html#7512" class="Bound">a</a> <a id="7514" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7516" href="Cubical.Data.Sigma.Properties.html#7516" class="Bound">b</a><a id="7517" class="Symbol">)</a> <a id="7519" class="Symbol">=</a> <a id="7521" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="7530" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a> <a id="7532" href="Cubical.Data.Sigma.Properties.html#7512" class="Bound">a</a> <a id="7534" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7536" href="Cubical.Data.Sigma.Properties.html#7516" class="Bound">b</a>
<a id="7540" href="Cubical.Data.Sigma.Properties.html#7540" class="Function">isEqIntro</a> <a id="7550" class="Symbol">:</a> <a id="7552" href="Agda.Builtin.Cubical.Glue.html#868" class="Record">isEquiv</a> <a id="7560" href="Cubical.Data.Sigma.Properties.html#7465" class="Function">intro</a>
<a id="7568" href="Cubical.Data.Sigma.Properties.html#7540" class="Function">isEqIntro</a> <a id="7578" class="Symbol">.</a><a id="7579" href="Agda.Builtin.Cubical.Glue.html#971" class="Field">equiv-proof</a> <a id="7591" href="Cubical.Data.Sigma.Properties.html#7591" class="Bound">x</a> <a id="7593" class="Symbol">=</a> <a id="7595" href="Cubical.Data.Sigma.Properties.html#7941" class="Function">ctr</a> <a id="7599" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7601" href="Cubical.Data.Sigma.Properties.html#8010" class="Function">isCtr</a> <a id="7607" class="Keyword">where</a>
<a id="7617" href="Cubical.Data.Sigma.Properties.html#7617" class="Function">PB</a> <a id="7620" class="Symbol">:</a> <a id="7622" class="Symbol">∀</a> <a id="7624" class="Symbol">{</a><a id="7625" href="Cubical.Data.Sigma.Properties.html#7625" class="Bound">x</a> <a id="7627" href="Cubical.Data.Sigma.Properties.html#7627" class="Bound">y</a><a id="7628" class="Symbol">}</a> <a id="7630" class="Symbol">→</a> <a id="7632" href="Cubical.Data.Sigma.Properties.html#7625" class="Bound">x</a> <a id="7634" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="7636" href="Cubical.Data.Sigma.Properties.html#7627" class="Bound">y</a> <a id="7638" class="Symbol">→</a> <a id="7640" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7642" href="Cubical.Data.Sigma.Properties.html#7625" class="Bound">x</a> <a id="7644" class="Symbol">→</a> <a id="7646" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7648" href="Cubical.Data.Sigma.Properties.html#7627" class="Bound">y</a> <a id="7650" class="Symbol">→</a> <a id="7652" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="7657" class="Symbol">_</a>
<a id="7663" href="Cubical.Data.Sigma.Properties.html#7617" class="Function">PB</a> <a id="7666" href="Cubical.Data.Sigma.Properties.html#7666" class="Bound">p</a> <a id="7668" class="Symbol">=</a> <a id="7670" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="7676" class="Symbol">(λ</a> <a id="7679" href="Cubical.Data.Sigma.Properties.html#7679" class="Bound">i</a> <a id="7681" class="Symbol">→</a> <a id="7683" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7685" class="Symbol">(</a><a id="7686" href="Cubical.Data.Sigma.Properties.html#7666" class="Bound">p</a> <a id="7688" href="Cubical.Data.Sigma.Properties.html#7679" class="Bound">i</a><a id="7689" class="Symbol">))</a>
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<a id="7801" href="Cubical.Data.Sigma.Properties.html#7801" class="Function">ctrB</a> <a id="7806" class="Symbol">:</a> <a id="7808" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7810" class="Symbol">(</a><a id="7811" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="7820" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a> <a id="7822" href="Cubical.Data.Sigma.Properties.html#7781" class="Function">ctrA</a><a id="7826" class="Symbol">)</a>
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<a id="7884" href="Cubical.Data.Sigma.Properties.html#7861" class="Function">ctrP</a> <a id="7889" class="Symbol">=</a> <a id="7891" href="Cubical.Foundations.Prelude.html#1094" class="Function">symP</a> <a id="7896" class="Symbol">(</a><a id="7897" href="Cubical.Foundations.Prelude.html#8915" class="Function">transport-filler</a> <a id="7914" class="Symbol">(λ</a> <a id="7917" href="Cubical.Data.Sigma.Properties.html#7917" class="Bound">i</a> <a id="7919" class="Symbol">→</a> <a id="7921" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="7923" class="Symbol">(</a><a id="7924" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="7928" href="Cubical.Data.Sigma.Properties.html#7794" class="Function">α</a> <a id="7930" href="Cubical.Data.Sigma.Properties.html#7917" class="Bound">i</a><a id="7931" class="Symbol">))</a> <a id="7934" href="Cubical.Data.Sigma.Properties.html#7734" class="Field">b</a><a id="7935" class="Symbol">)</a>
<a id="7941" href="Cubical.Data.Sigma.Properties.html#7941" class="Function">ctr</a> <a id="7945" class="Symbol">:</a> <a id="7947" href="Cubical.Foundations.Equiv.Base.html#253" class="Function">fiber</a> <a id="7953" href="Cubical.Data.Sigma.Properties.html#7465" class="Function">intro</a> <a id="7959" href="Cubical.Data.Sigma.Properties.html#7591" class="Bound">x</a>
<a id="7965" href="Cubical.Data.Sigma.Properties.html#7941" class="Function">ctr</a> <a id="7969" class="Symbol">=</a> <a id="7971" class="Symbol">(</a><a id="7972" href="Cubical.Data.Sigma.Properties.html#7781" class="Function">ctrA</a> <a id="7977" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7979" href="Cubical.Data.Sigma.Properties.html#7801" class="Function">ctrB</a><a id="7983" class="Symbol">)</a> <a id="7985" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7987" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="7994" class="Symbol">(</a><a id="7995" href="Cubical.Data.Sigma.Properties.html#7794" class="Function">α</a> <a id="7997" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="7999" href="Cubical.Data.Sigma.Properties.html#7861" class="Function">ctrP</a><a id="8003" class="Symbol">)</a>
<a id="8010" href="Cubical.Data.Sigma.Properties.html#8010" class="Function">isCtr</a> <a id="8016" class="Symbol">:</a> <a id="8018" class="Symbol">∀</a> <a id="8020" href="Cubical.Data.Sigma.Properties.html#8020" class="Bound">y</a> <a id="8022" class="Symbol">→</a> <a id="8024" href="Cubical.Data.Sigma.Properties.html#7941" class="Function">ctr</a> <a id="8028" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="8030" href="Cubical.Data.Sigma.Properties.html#8020" class="Bound">y</a>
<a id="8036" href="Cubical.Data.Sigma.Properties.html#8010" class="Function">isCtr</a> <a id="8042" class="Symbol">((</a><a id="8044" href="Cubical.Data.Sigma.Properties.html#8044" class="Bound">r</a> <a id="8046" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8048" href="Cubical.Data.Sigma.Properties.html#8048" class="Bound">s</a><a id="8049" class="Symbol">)</a> <a id="8051" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8053" href="Cubical.Data.Sigma.Properties.html#8053" class="Bound">p</a><a id="8054" class="Symbol">)</a> <a id="8056" class="Symbol">=</a> <a id="8058" class="Symbol">λ</a> <a id="8060" href="Cubical.Data.Sigma.Properties.html#8060" class="Bound">i</a> <a id="8062" class="Symbol">→</a> <a id="8064" class="Symbol">(</a><a id="8065" href="Cubical.Data.Sigma.Properties.html#8225" class="Function">a≡r</a> <a id="8069" href="Cubical.Data.Sigma.Properties.html#8060" class="Bound">i</a> <a id="8071" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8073" href="Cubical.Data.Sigma.Properties.html#8249" class="Function">b!≡s</a> <a id="8078" href="Cubical.Data.Sigma.Properties.html#8060" class="Bound">i</a><a id="8079" class="Symbol">)</a> <a id="8081" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8083" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="8090" class="Symbol">(</a><a id="8091" href="Cubical.Data.Sigma.Properties.html#8237" class="Function">α≡ρ</a> <a id="8095" href="Cubical.Data.Sigma.Properties.html#8060" class="Bound">i</a> <a id="8097" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8099" href="Cubical.Data.Sigma.Properties.html#8426" class="Function">coh</a> <a id="8103" href="Cubical.Data.Sigma.Properties.html#8060" class="Bound">i</a><a id="8104" class="Symbol">)</a> <a id="8106" class="Keyword">where</a>
<a id="8118" class="Keyword">open</a> <a id="8123" href="Cubical.Data.Sigma.Properties.html#2384" class="Module">PathPΣ</a> <a id="8130" href="Cubical.Data.Sigma.Properties.html#8053" class="Bound">p</a> <a id="8132" class="Keyword">renaming</a> <a id="8141" class="Symbol">(</a><a id="8142" href="Agda.Builtin.Sigma.html#252" class="Function">fst</a> <a id="8146" class="Symbol">to</a> <a id="8149" class="Function">ρ</a><a id="8150" class="Symbol">;</a> <a id="8152" href="Agda.Builtin.Sigma.html#264" class="Function">snd</a> <a id="8156" class="Symbol">to</a> <a id="8159" class="Function">σ</a><a id="8160" class="Symbol">)</a>
<a id="8168" class="Keyword">open</a> <a id="8173" href="Cubical.Data.Sigma.Properties.html#2384" class="Module">PathPΣ</a> <a id="8180" class="Symbol">(</a><a id="8181" href="Cubical.Foundations.Equiv.html#995" class="Function">equivCtrPath</a> <a id="8194" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a> <a id="8196" href="Cubical.Data.Sigma.Properties.html#7723" class="Field">a'</a> <a id="8199" class="Symbol">(</a><a id="8200" href="Cubical.Data.Sigma.Properties.html#8044" class="Bound">r</a> <a id="8202" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8204" href="Cubical.Data.Sigma.Properties.html#8149" class="Function">ρ</a><a id="8205" class="Symbol">))</a> <a id="8208" class="Keyword">renaming</a> <a id="8217" class="Symbol">(</a><a id="8218" href="Agda.Builtin.Sigma.html#252" class="Function">fst</a> <a id="8222" class="Symbol">to</a> <a id="8225" class="Function">a≡r</a><a id="8228" class="Symbol">;</a> <a id="8230" href="Agda.Builtin.Sigma.html#264" class="Function">snd</a> <a id="8234" class="Symbol">to</a> <a id="8237" class="Function">α≡ρ</a><a id="8240" class="Symbol">)</a>
<a id="8249" href="Cubical.Data.Sigma.Properties.html#8249" class="Function">b!≡s</a> <a id="8254" class="Symbol">:</a> <a id="8256" href="Cubical.Data.Sigma.Properties.html#7617" class="Function">PB</a> <a id="8259" class="Symbol">(</a><a id="8260" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="8265" class="Symbol">(</a><a id="8266" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="8275" href="Cubical.Data.Sigma.Properties.html#7434" class="Bound">e</a><a id="8276" class="Symbol">)</a> <a id="8278" href="Cubical.Data.Sigma.Properties.html#8225" class="Function">a≡r</a><a id="8281" class="Symbol">)</a> <a id="8283" href="Cubical.Data.Sigma.Properties.html#7801" class="Function">ctrB</a> <a id="8288" href="Cubical.Data.Sigma.Properties.html#8048" class="Bound">s</a>
<a id="8296" href="Cubical.Data.Sigma.Properties.html#8249" class="Function">b!≡s</a> <a id="8301" href="Cubical.Data.Sigma.Properties.html#8301" class="Bound">i</a> <a id="8303" class="Symbol">=</a> <a id="8305" href="Cubical.Core.Primitives.html#621" class="Primitive">comp</a> <a id="8310" class="Symbol">(λ</a> <a id="8313" href="Cubical.Data.Sigma.Properties.html#8313" class="Bound">k</a> <a id="8315" class="Symbol">→</a> <a id="8317" href="Cubical.Data.Sigma.Properties.html#7431" class="Bound">B</a> <a id="8319" class="Symbol">(</a><a id="8320" href="Cubical.Data.Sigma.Properties.html#8237" class="Function">α≡ρ</a> <a id="8324" href="Cubical.Data.Sigma.Properties.html#8301" class="Bound">i</a> <a id="8326" class="Symbol">(</a><a id="8327" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="8329" href="Cubical.Data.Sigma.Properties.html#8313" class="Bound">k</a><a id="8330" class="Symbol">)))</a> <a id="8334" class="Symbol">(λ</a> <a id="8337" href="Cubical.Data.Sigma.Properties.html#8337" class="Bound">k</a> <a id="8339" class="Symbol">→</a> <a id="8341" class="Symbol">(λ</a>
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<a id="8384" class="Symbol">;</a> <a id="8386" class="Symbol">(</a><a id="8387" href="Cubical.Data.Sigma.Properties.html#8301" class="Bound">i</a> <a id="8389" class="Symbol">=</a> <a id="8391" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="8393" class="Symbol">)</a> <a id="8395" class="Symbol">→</a> <a id="8397" href="Cubical.Data.Sigma.Properties.html#8159" class="Function">σ</a> <a id="8399" class="Symbol">(</a><a id="8400" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="8402" href="Cubical.Data.Sigma.Properties.html#8337" class="Bound">k</a><a id="8403" class="Symbol">)</a>
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<a id="8538" class="Symbol">{</a> <a id="8540" class="Symbol">(</a><a id="8541" href="Cubical.Data.Sigma.Properties.html#8485" class="Bound">i</a> <a id="8543" class="Symbol">=</a> <a id="8545" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="8547" class="Symbol">)</a> <a id="8549" class="Symbol">→</a> <a id="8551" href="Cubical.Data.Sigma.Properties.html#7861" class="Function">ctrP</a> <a id="8556" class="Symbol">(</a><a id="8557" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="8559" href="Cubical.Data.Sigma.Properties.html#8523" class="Bound">k</a><a id="8560" class="Symbol">)</a>
<a id="8570" class="Symbol">;</a> <a id="8572" class="Symbol">(</a><a id="8573" href="Cubical.Data.Sigma.Properties.html#8485" class="Bound">i</a> <a id="8575" class="Symbol">=</a> <a id="8577" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="8579" class="Symbol">)</a> <a id="8581" class="Symbol">→</a> <a id="8583" href="Cubical.Data.Sigma.Properties.html#8159" class="Function">σ</a> <a id="8585" class="Symbol">(</a><a id="8586" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="8588" href="Cubical.Data.Sigma.Properties.html#8523" class="Bound">k</a><a id="8589" class="Symbol">)</a>
<a id="8599" class="Symbol">}))</a> <a id="8603" class="Symbol">(</a><a id="8604" href="Agda.Builtin.Cubical.Sub.html#223" class="Postulate">inS</a> <a id="8608" href="Cubical.Data.Sigma.Properties.html#7734" class="Field">b</a><a id="8609" class="Symbol">)</a> <a id="8611" class="Symbol">(</a><a id="8612" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="8614" href="Cubical.Data.Sigma.Properties.html#8487" class="Bound">j</a><a id="8615" class="Symbol">)</a>
<a id="Σ-cong-fst"></a><a id="8618" href="Cubical.Data.Sigma.Properties.html#8618" class="Function">Σ-cong-fst</a> <a id="8629" class="Symbol">:</a> <a id="8631" class="Symbol">(</a><a id="8632" href="Cubical.Data.Sigma.Properties.html#8632" class="Bound">p</a> <a id="8634" class="Symbol">:</a> <a id="8636" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="8638" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="8640" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="8642" class="Symbol">)</a> <a id="8644" class="Symbol">→</a> <a id="8646" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="8648" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="8650" class="Symbol">(</a><a id="8651" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="8653" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="8655" href="Cubical.Foundations.Prelude.html#8597" class="Function">transport</a> <a id="8665" href="Cubical.Data.Sigma.Properties.html#8632" class="Bound">p</a><a id="8666" class="Symbol">)</a> <a id="8668" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="8670" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="8672" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="8675" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a>
<a id="8677" href="Cubical.Data.Sigma.Properties.html#8618" class="Function">Σ-cong-fst</a> <a id="8688" class="Symbol">{</a><a id="8689" class="Argument">B</a> <a id="8691" class="Symbol">=</a> <a id="8693" href="Cubical.Data.Sigma.Properties.html#8693" class="Bound">B</a><a id="8694" class="Symbol">}</a> <a id="8696" href="Cubical.Data.Sigma.Properties.html#8696" class="Bound">p</a> <a id="8698" href="Cubical.Data.Sigma.Properties.html#8698" class="Bound">i</a> <a id="8700" class="Symbol">=</a> <a id="8702" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="8704" class="Symbol">(</a><a id="8705" href="Cubical.Data.Sigma.Properties.html#8696" class="Bound">p</a> <a id="8707" href="Cubical.Data.Sigma.Properties.html#8698" class="Bound">i</a><a id="8708" class="Symbol">)</a> <a id="8710" class="Symbol">(</a><a id="8711" href="Cubical.Data.Sigma.Properties.html#8693" class="Bound">B</a> <a id="8713" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="8715" href="Cubical.Core.Primitives.html#694" class="Primitive">transp</a> <a id="8722" class="Symbol">(λ</a> <a id="8725" href="Cubical.Data.Sigma.Properties.html#8725" class="Bound">j</a> <a id="8727" class="Symbol">→</a> <a id="8729" href="Cubical.Data.Sigma.Properties.html#8696" class="Bound">p</a> <a id="8731" class="Symbol">(</a><a id="8732" href="Cubical.Data.Sigma.Properties.html#8698" class="Bound">i</a> <a id="8734" href="Cubical.Core.Primitives.html#490" class="Primitive Operator">∨</a> <a id="8736" href="Cubical.Data.Sigma.Properties.html#8725" class="Bound">j</a><a id="8737" class="Symbol">))</a> <a id="8740" href="Cubical.Data.Sigma.Properties.html#8698" class="Bound">i</a><a id="8741" class="Symbol">)</a>
<a id="Σ-cong-iso-snd"></a><a id="8744" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="8759" class="Symbol">:</a> <a id="8761" class="Symbol">((</a><a id="8763" href="Cubical.Data.Sigma.Properties.html#8763" class="Bound">x</a> <a id="8765" class="Symbol">:</a> <a id="8767" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="8768" class="Symbol">)</a> <a id="8770" class="Symbol">→</a> <a id="8772" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="8776" class="Symbol">(</a><a id="8777" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="8779" href="Cubical.Data.Sigma.Properties.html#8763" class="Bound">x</a><a id="8780" class="Symbol">)</a> <a id="8782" class="Symbol">(</a><a id="8783" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="8786" href="Cubical.Data.Sigma.Properties.html#8763" class="Bound">x</a><a id="8787" class="Symbol">))</a> <a id="8790" class="Symbol">→</a> <a id="8792" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="8796" class="Symbol">(</a><a id="8797" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="8799" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="8801" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="8802" class="Symbol">)</a> <a id="8804" class="Symbol">(</a><a id="8805" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="8807" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="8809" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a><a id="8811" class="Symbol">)</a>
<a id="8813" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="8817" class="Symbol">(</a><a id="8818" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="8833" href="Cubical.Data.Sigma.Properties.html#8833" class="Bound">isom</a><a id="8837" class="Symbol">)</a> <a id="8839" class="Symbol">(</a><a id="8840" href="Cubical.Data.Sigma.Properties.html#8840" class="Bound">x</a> <a id="8842" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8844" href="Cubical.Data.Sigma.Properties.html#8844" class="Bound">y</a><a id="8845" class="Symbol">)</a> <a id="8847" class="Symbol">=</a> <a id="8849" href="Cubical.Data.Sigma.Properties.html#8840" class="Bound">x</a> <a id="8851" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8853" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="8857" class="Symbol">(</a><a id="8858" href="Cubical.Data.Sigma.Properties.html#8833" class="Bound">isom</a> <a id="8863" href="Cubical.Data.Sigma.Properties.html#8840" class="Bound">x</a><a id="8864" class="Symbol">)</a> <a id="8866" href="Cubical.Data.Sigma.Properties.html#8844" class="Bound">y</a>
<a id="8868" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="8872" class="Symbol">(</a><a id="8873" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="8888" href="Cubical.Data.Sigma.Properties.html#8888" class="Bound">isom</a><a id="8892" class="Symbol">)</a> <a id="8894" class="Symbol">(</a><a id="8895" href="Cubical.Data.Sigma.Properties.html#8895" class="Bound">x</a> <a id="8897" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8899" href="Cubical.Data.Sigma.Properties.html#8899" class="Bound">y'</a><a id="8901" class="Symbol">)</a> <a id="8903" class="Symbol">=</a> <a id="8905" href="Cubical.Data.Sigma.Properties.html#8895" class="Bound">x</a> <a id="8907" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8909" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="8913" class="Symbol">(</a><a id="8914" href="Cubical.Data.Sigma.Properties.html#8888" class="Bound">isom</a> <a id="8919" href="Cubical.Data.Sigma.Properties.html#8895" class="Bound">x</a><a id="8920" class="Symbol">)</a> <a id="8922" href="Cubical.Data.Sigma.Properties.html#8899" class="Bound">y'</a>
<a id="8925" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="8934" class="Symbol">(</a><a id="8935" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="8950" href="Cubical.Data.Sigma.Properties.html#8950" class="Bound">isom</a><a id="8954" class="Symbol">)</a> <a id="8956" class="Symbol">(</a><a id="8957" href="Cubical.Data.Sigma.Properties.html#8957" class="Bound">x</a> <a id="8959" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8961" href="Cubical.Data.Sigma.Properties.html#8961" class="Bound">y</a><a id="8962" class="Symbol">)</a> <a id="8964" class="Symbol">=</a> <a id="8966" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="8973" class="Symbol">(</a><a id="8974" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a> <a id="8979" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="8981" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="8990" class="Symbol">(</a><a id="8991" href="Cubical.Data.Sigma.Properties.html#8950" class="Bound">isom</a> <a id="8996" href="Cubical.Data.Sigma.Properties.html#8957" class="Bound">x</a><a id="8997" class="Symbol">)</a> <a id="8999" href="Cubical.Data.Sigma.Properties.html#8961" class="Bound">y</a><a id="9000" class="Symbol">)</a>
<a id="9002" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="9010" class="Symbol">(</a><a id="9011" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="9026" href="Cubical.Data.Sigma.Properties.html#9026" class="Bound">isom</a><a id="9030" class="Symbol">)</a> <a id="9032" class="Symbol">(</a><a id="9033" href="Cubical.Data.Sigma.Properties.html#9033" class="Bound">x</a> <a id="9035" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="9037" href="Cubical.Data.Sigma.Properties.html#9037" class="Bound">y'</a><a id="9039" class="Symbol">)</a> <a id="9041" class="Symbol">=</a> <a id="9043" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="9050" class="Symbol">(</a><a id="9051" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a> <a id="9056" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="9058" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="9066" class="Symbol">(</a><a id="9067" href="Cubical.Data.Sigma.Properties.html#9026" class="Bound">isom</a> <a id="9072" href="Cubical.Data.Sigma.Properties.html#9033" class="Bound">x</a><a id="9073" class="Symbol">)</a> <a id="9075" href="Cubical.Data.Sigma.Properties.html#9037" class="Bound">y'</a><a id="9077" class="Symbol">)</a>
<a id="Σ-cong-equiv-snd"></a><a id="9080" href="Cubical.Data.Sigma.Properties.html#9080" class="Function">Σ-cong-equiv-snd</a> <a id="9097" class="Symbol">:</a> <a id="9099" class="Symbol">(∀</a> <a id="9102" href="Cubical.Data.Sigma.Properties.html#9102" class="Bound">a</a> <a id="9104" class="Symbol">→</a> <a id="9106" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9108" href="Cubical.Data.Sigma.Properties.html#9102" class="Bound">a</a> <a id="9110" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9112" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9115" href="Cubical.Data.Sigma.Properties.html#9102" class="Bound">a</a><a id="9116" class="Symbol">)</a> <a id="9118" class="Symbol">→</a> <a id="9120" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9122" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9124" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9126" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9128" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9130" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9132" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="9135" href="Cubical.Data.Sigma.Properties.html#9080" class="Function">Σ-cong-equiv-snd</a> <a id="9152" href="Cubical.Data.Sigma.Properties.html#9152" class="Bound">h</a> <a id="9154" class="Symbol">=</a> <a id="9156" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="9167" class="Symbol">(</a><a id="9168" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="9183" class="Symbol">(</a><a id="9184" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="9195" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="9197" href="Cubical.Data.Sigma.Properties.html#9152" class="Bound">h</a><a id="9198" class="Symbol">))</a>
<a id="Σ-cong-snd"></a><a id="9202" href="Cubical.Data.Sigma.Properties.html#9202" class="Function">Σ-cong-snd</a> <a id="9213" class="Symbol">:</a> <a id="9215" class="Symbol">((</a><a id="9217" href="Cubical.Data.Sigma.Properties.html#9217" class="Bound">x</a> <a id="9219" class="Symbol">:</a> <a id="9221" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="9222" class="Symbol">)</a> <a id="9224" class="Symbol">→</a> <a id="9226" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9228" href="Cubical.Data.Sigma.Properties.html#9217" class="Bound">x</a> <a id="9230" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="9232" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9235" href="Cubical.Data.Sigma.Properties.html#9217" class="Bound">x</a><a id="9236" class="Symbol">)</a> <a id="9238" class="Symbol">→</a> <a id="9240" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9242" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9244" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9246" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="9248" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9250" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9252" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="9255" href="Cubical.Data.Sigma.Properties.html#9202" class="Function">Σ-cong-snd</a> <a id="9266" class="Symbol">{</a><a id="9267" class="Argument">A</a> <a id="9269" class="Symbol">=</a> <a id="9271" href="Cubical.Data.Sigma.Properties.html#9271" class="Bound">A</a><a id="9272" class="Symbol">}</a> <a id="9274" href="Cubical.Data.Sigma.Properties.html#9274" class="Bound">p</a> <a id="9276" href="Cubical.Data.Sigma.Properties.html#9276" class="Bound">i</a> <a id="9278" class="Symbol">=</a> <a id="9280" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="9283" href="Cubical.Data.Sigma.Properties.html#9283" class="Bound">x</a> <a id="9285" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="9287" href="Cubical.Data.Sigma.Properties.html#9271" class="Bound">A</a> <a id="9289" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="9291" class="Symbol">(</a><a id="9292" href="Cubical.Data.Sigma.Properties.html#9274" class="Bound">p</a> <a id="9294" href="Cubical.Data.Sigma.Properties.html#9283" class="Bound">x</a> <a id="9296" href="Cubical.Data.Sigma.Properties.html#9276" class="Bound">i</a><a id="9297" class="Symbol">)</a>
<a id="Σ-cong-iso"></a><a id="9300" href="Cubical.Data.Sigma.Properties.html#9300" class="Function">Σ-cong-iso</a> <a id="9311" class="Symbol">:</a> <a id="9313" class="Symbol">(</a><a id="9314" href="Cubical.Data.Sigma.Properties.html#9314" class="Bound">isom</a> <a id="9319" class="Symbol">:</a> <a id="9321" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="9325" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9327" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="9329" class="Symbol">)</a>
<a id="9342" class="Symbol">→</a> <a id="9344" class="Symbol">((</a><a id="9346" href="Cubical.Data.Sigma.Properties.html#9346" class="Bound">x</a> <a id="9348" class="Symbol">:</a> <a id="9350" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="9351" class="Symbol">)</a> <a id="9353" class="Symbol">→</a> <a id="9355" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="9359" class="Symbol">(</a><a id="9360" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9362" href="Cubical.Data.Sigma.Properties.html#9346" class="Bound">x</a><a id="9363" class="Symbol">)</a> <a id="9365" class="Symbol">(</a><a id="9366" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9369" class="Symbol">(</a><a id="9370" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="9374" href="Cubical.Data.Sigma.Properties.html#9314" class="Bound">isom</a> <a id="9379" href="Cubical.Data.Sigma.Properties.html#9346" class="Bound">x</a><a id="9380" class="Symbol">)))</a>
<a id="9395" class="Symbol">→</a> <a id="9397" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="9401" class="Symbol">(</a><a id="9402" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9404" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9406" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="9407" class="Symbol">)</a> <a id="9409" class="Symbol">(</a><a id="9410" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9412" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="9415" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a><a id="9417" class="Symbol">)</a>
<a id="9419" href="Cubical.Data.Sigma.Properties.html#9300" class="Function">Σ-cong-iso</a> <a id="9430" href="Cubical.Data.Sigma.Properties.html#9430" class="Bound">isom</a> <a id="9435" href="Cubical.Data.Sigma.Properties.html#9435" class="Bound">isom'</a> <a id="9441" class="Symbol">=</a> <a id="9443" href="Cubical.Foundations.Isomorphism.html#3521" class="Function">compIso</a> <a id="9451" class="Symbol">(</a><a id="9452" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="9467" href="Cubical.Data.Sigma.Properties.html#9435" class="Bound">isom'</a><a id="9472" class="Symbol">)</a> <a id="9474" class="Symbol">(</a><a id="9475" href="Cubical.Data.Sigma.Properties.html#5987" class="Function">Σ-cong-iso-fst</a> <a id="9490" href="Cubical.Data.Sigma.Properties.html#9430" class="Bound">isom</a><a id="9494" class="Symbol">)</a>
<a id="Σ-cong-equiv"></a><a id="9497" href="Cubical.Data.Sigma.Properties.html#9497" class="Function">Σ-cong-equiv</a> <a id="9510" class="Symbol">:</a> <a id="9512" class="Symbol">(</a><a id="9513" href="Cubical.Data.Sigma.Properties.html#9513" class="Bound">e</a> <a id="9515" class="Symbol">:</a> <a id="9517" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9519" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9521" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="9523" class="Symbol">)</a>
<a id="9538" class="Symbol">→</a> <a id="9540" class="Symbol">((</a><a id="9542" href="Cubical.Data.Sigma.Properties.html#9542" class="Bound">x</a> <a id="9544" class="Symbol">:</a> <a id="9546" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="9547" class="Symbol">)</a> <a id="9549" class="Symbol">→</a> <a id="9551" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9553" href="Cubical.Data.Sigma.Properties.html#9542" class="Bound">x</a> <a id="9555" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9557" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9560" class="Symbol">(</a><a id="9561" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="9570" href="Cubical.Data.Sigma.Properties.html#9513" class="Bound">e</a> <a id="9572" href="Cubical.Data.Sigma.Properties.html#9542" class="Bound">x</a><a id="9573" class="Symbol">))</a>
<a id="9589" class="Symbol">→</a> <a id="9591" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9593" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9595" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9597" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9599" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9601" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="9604" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="9607" href="Cubical.Data.Sigma.Properties.html#9497" class="Function">Σ-cong-equiv</a> <a id="9620" href="Cubical.Data.Sigma.Properties.html#9620" class="Bound">e</a> <a id="9622" href="Cubical.Data.Sigma.Properties.html#9622" class="Bound">e'</a> <a id="9625" class="Symbol">=</a> <a id="9627" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="9638" class="Symbol">(</a><a id="9639" href="Cubical.Data.Sigma.Properties.html#9300" class="Function">Σ-cong-iso</a> <a id="9650" class="Symbol">(</a><a id="9651" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="9662" href="Cubical.Data.Sigma.Properties.html#9620" class="Bound">e</a><a id="9663" class="Symbol">)</a> <a id="9665" class="Symbol">(</a><a id="9666" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="9677" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="9679" href="Cubical.Data.Sigma.Properties.html#9622" class="Bound">e'</a><a id="9681" class="Symbol">))</a>
<a id="Σ-cong'"></a><a id="9685" href="Cubical.Data.Sigma.Properties.html#9685" class="Function">Σ-cong'</a> <a id="9693" class="Symbol">:</a> <a id="9695" class="Symbol">(</a><a id="9696" href="Cubical.Data.Sigma.Properties.html#9696" class="Bound">p</a> <a id="9698" class="Symbol">:</a> <a id="9700" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9702" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="9704" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="9706" class="Symbol">)</a> <a id="9708" class="Symbol">→</a> <a id="9710" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="9716" class="Symbol">(λ</a> <a id="9719" href="Cubical.Data.Sigma.Properties.html#9719" class="Bound">i</a> <a id="9721" class="Symbol">→</a> <a id="9723" href="Cubical.Data.Sigma.Properties.html#9696" class="Bound">p</a> <a id="9725" href="Cubical.Data.Sigma.Properties.html#9719" class="Bound">i</a> <a id="9727" class="Symbol">→</a> <a id="9729" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="9734" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="9736" class="Symbol">)</a> <a id="9738" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9740" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9743" class="Symbol">→</a> <a id="9745" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9747" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9749" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9751" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="9753" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9755" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="9758" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="9761" href="Cubical.Data.Sigma.Properties.html#9685" class="Function">Σ-cong'</a> <a id="9769" href="Cubical.Data.Sigma.Properties.html#9769" class="Bound">p</a> <a id="9771" href="Cubical.Data.Sigma.Properties.html#9771" class="Bound">p'</a> <a id="9774" class="Symbol">=</a> <a id="9776" href="Cubical.Foundations.Prelude.html#2025" class="Function">cong₂</a> <a id="9782" class="Symbol">(λ</a> <a id="9785" class="Symbol">(</a><a id="9786" href="Cubical.Data.Sigma.Properties.html#9786" class="Bound">A</a> <a id="9788" class="Symbol">:</a> <a id="9790" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="9795" class="Symbol">_)</a> <a id="9798" class="Symbol">(</a><a id="9799" href="Cubical.Data.Sigma.Properties.html#9799" class="Bound">B</a> <a id="9801" class="Symbol">:</a> <a id="9803" href="Cubical.Data.Sigma.Properties.html#9786" class="Bound">A</a> <a id="9805" class="Symbol">→</a> <a id="9807" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="9812" class="Symbol">_)</a> <a id="9815" class="Symbol">→</a> <a id="9817" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="9819" href="Cubical.Data.Sigma.Properties.html#9786" class="Bound">A</a> <a id="9821" href="Cubical.Data.Sigma.Properties.html#9799" class="Bound">B</a><a id="9822" class="Symbol">)</a> <a id="9824" href="Cubical.Data.Sigma.Properties.html#9769" class="Bound">p</a> <a id="9826" href="Cubical.Data.Sigma.Properties.html#9771" class="Bound">p'</a>
<a id="Σ-cong-equiv-prop"></a><a id="9830" href="Cubical.Data.Sigma.Properties.html#9830" class="Function">Σ-cong-equiv-prop</a> <a id="9848" class="Symbol">:</a>
<a id="9854" class="Symbol">(</a><a id="9855" href="Cubical.Data.Sigma.Properties.html#9855" class="Bound">e</a> <a id="9857" class="Symbol">:</a> <a id="9859" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9861" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="9863" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="9865" class="Symbol">)</a>
<a id="9869" class="Symbol">→</a> <a id="9871" class="Symbol">((</a><a id="9873" href="Cubical.Data.Sigma.Properties.html#9873" class="Bound">x</a> <a id="9875" class="Symbol">:</a> <a id="9877" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="9879" class="Symbol">)</a> <a id="9881" class="Symbol">→</a> <a id="9883" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="9890" class="Symbol">(</a><a id="9891" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9894" href="Cubical.Data.Sigma.Properties.html#9873" class="Bound">x</a><a id="9895" class="Symbol">))</a>
<a id="9900" class="Symbol">→</a> <a id="9902" class="Symbol">((</a><a id="9904" href="Cubical.Data.Sigma.Properties.html#9904" class="Bound">x</a> <a id="9906" class="Symbol">:</a> <a id="9908" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a><a id="9910" class="Symbol">)</a> <a id="9912" class="Symbol">→</a> <a id="9914" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="9921" class="Symbol">(</a><a id="9922" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9925" href="Cubical.Data.Sigma.Properties.html#9904" class="Bound">x</a><a id="9926" class="Symbol">))</a>
<a id="9931" class="Symbol">→</a> <a id="9933" class="Symbol">((</a><a id="9935" href="Cubical.Data.Sigma.Properties.html#9935" class="Bound">x</a> <a id="9937" class="Symbol">:</a> <a id="9939" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="9940" class="Symbol">)</a> <a id="9942" class="Symbol">→</a> <a id="9944" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="9946" href="Cubical.Data.Sigma.Properties.html#9935" class="Bound">x</a> <a id="9948" class="Symbol">→</a> <a id="9950" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9953" class="Symbol">(</a><a id="9954" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="9963" href="Cubical.Data.Sigma.Properties.html#9855" class="Bound">e</a> <a id="9965" href="Cubical.Data.Sigma.Properties.html#9935" class="Bound">x</a><a id="9966" class="Symbol">))</a>
<a id="9971" class="Symbol">→</a> <a id="9973" class="Symbol">((</a><a id="9975" href="Cubical.Data.Sigma.Properties.html#9975" class="Bound">x</a> <a id="9977" class="Symbol">:</a> <a id="9979" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="9980" class="Symbol">)</a> <a id="9982" class="Symbol">→</a> <a id="9984" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a> <a id="9987" class="Symbol">(</a><a id="9988" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="9997" href="Cubical.Data.Sigma.Properties.html#9855" class="Bound">e</a> <a id="9999" href="Cubical.Data.Sigma.Properties.html#9975" class="Bound">x</a><a id="10000" class="Symbol">)</a> <a id="10002" class="Symbol">→</a> <a id="10004" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="10006" href="Cubical.Data.Sigma.Properties.html#9975" class="Bound">x</a><a id="10007" class="Symbol">)</a>
<a id="10011" class="Symbol">→</a> <a id="10013" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10015" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10017" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="10019" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="10021" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10023" href="Cubical.Data.Sigma.Properties.html#1320" class="Generalizable">A'</a> <a id="10026" href="Cubical.Data.Sigma.Properties.html#1338" class="Generalizable">B'</a>
<a id="10029" href="Cubical.Data.Sigma.Properties.html#9830" class="Function">Σ-cong-equiv-prop</a> <a id="10047" href="Cubical.Data.Sigma.Properties.html#10047" class="Bound">e</a> <a id="10049" href="Cubical.Data.Sigma.Properties.html#10049" class="Bound">prop</a> <a id="10054" href="Cubical.Data.Sigma.Properties.html#10054" class="Bound">prop'</a> <a id="10060" href="Cubical.Data.Sigma.Properties.html#10060" class="Bound">prop→</a> <a id="10066" href="Cubical.Data.Sigma.Properties.html#10066" class="Bound">prop←</a> <a id="10072" class="Symbol">=</a>
<a id="10076" href="Cubical.Data.Sigma.Properties.html#9497" class="Function">Σ-cong-equiv</a> <a id="10089" href="Cubical.Data.Sigma.Properties.html#10047" class="Bound">e</a> <a id="10091" class="Symbol">(λ</a> <a id="10094" href="Cubical.Data.Sigma.Properties.html#10094" class="Bound">x</a> <a id="10096" class="Symbol">→</a> <a id="10098" href="Cubical.Foundations.Equiv.html#6072" class="Function">propBiimpl→Equiv</a> <a id="10115" class="Symbol">(</a><a id="10116" href="Cubical.Data.Sigma.Properties.html#10049" class="Bound">prop</a> <a id="10121" href="Cubical.Data.Sigma.Properties.html#10094" class="Bound">x</a><a id="10122" class="Symbol">)</a> <a id="10124" class="Symbol">(</a><a id="10125" href="Cubical.Data.Sigma.Properties.html#10054" class="Bound">prop'</a> <a id="10131" class="Symbol">(</a><a id="10132" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="10141" href="Cubical.Data.Sigma.Properties.html#10047" class="Bound">e</a> <a id="10143" href="Cubical.Data.Sigma.Properties.html#10094" class="Bound">x</a><a id="10144" class="Symbol">))</a> <a id="10147" class="Symbol">(</a><a id="10148" href="Cubical.Data.Sigma.Properties.html#10060" class="Bound">prop→</a> <a id="10154" href="Cubical.Data.Sigma.Properties.html#10094" class="Bound">x</a><a id="10155" class="Symbol">)</a> <a id="10157" class="Symbol">(</a><a id="10158" href="Cubical.Data.Sigma.Properties.html#10066" class="Bound">prop←</a> <a id="10164" href="Cubical.Data.Sigma.Properties.html#10094" class="Bound">x</a><a id="10165" class="Symbol">))</a>
<a id="10169" class="Comment">-- Alternative version for path in Σ-types, as in the HoTT book</a>
<a id="ΣPathTransport"></a><a id="10234" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10249" class="Symbol">:</a> <a id="10251" class="Symbol">(</a><a id="10252" href="Cubical.Data.Sigma.Properties.html#10252" class="Bound">a</a> <a id="10254" href="Cubical.Data.Sigma.Properties.html#10254" class="Bound">b</a> <a id="10256" class="Symbol">:</a> <a id="10258" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10260" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10262" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="10263" class="Symbol">)</a> <a id="10265" class="Symbol">→</a> <a id="10267" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="10272" class="Symbol">_</a>
<a id="10274" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10289" class="Symbol">{</a><a id="10290" class="Argument">B</a> <a id="10292" class="Symbol">=</a> <a id="10294" href="Cubical.Data.Sigma.Properties.html#10294" class="Bound">B</a><a id="10295" class="Symbol">}</a> <a id="10297" href="Cubical.Data.Sigma.Properties.html#10297" class="Bound">a</a> <a id="10299" href="Cubical.Data.Sigma.Properties.html#10299" class="Bound">b</a> <a id="10301" class="Symbol">=</a> <a id="10303" href="Cubical.Core.Primitives.html#6306" class="Function">Σ[</a> <a id="10306" href="Cubical.Data.Sigma.Properties.html#10306" class="Bound">p</a> <a id="10308" href="Cubical.Core.Primitives.html#6306" class="Function">∈</a> <a id="10310" class="Symbol">(</a><a id="10311" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="10315" href="Cubical.Data.Sigma.Properties.html#10297" class="Bound">a</a> <a id="10317" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10319" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="10323" href="Cubical.Data.Sigma.Properties.html#10299" class="Bound">b</a><a id="10324" class="Symbol">)</a> <a id="10326" href="Cubical.Core.Primitives.html#6306" class="Function">]</a> <a id="10328" href="Cubical.Foundations.Prelude.html#8597" class="Function">transport</a> <a id="10338" class="Symbol">(λ</a> <a id="10341" href="Cubical.Data.Sigma.Properties.html#10341" class="Bound">i</a> <a id="10343" class="Symbol">→</a> <a id="10345" href="Cubical.Data.Sigma.Properties.html#10294" class="Bound">B</a> <a id="10347" class="Symbol">(</a><a id="10348" href="Cubical.Data.Sigma.Properties.html#10306" class="Bound">p</a> <a id="10350" href="Cubical.Data.Sigma.Properties.html#10341" class="Bound">i</a><a id="10351" class="Symbol">))</a> <a id="10354" class="Symbol">(</a><a id="10355" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="10359" href="Cubical.Data.Sigma.Properties.html#10297" class="Bound">a</a><a id="10360" class="Symbol">)</a> <a id="10362" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10364" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="10368" href="Cubical.Data.Sigma.Properties.html#10299" class="Bound">b</a>
<a id="IsoΣPathTransportPathΣ"></a><a id="10371" href="Cubical.Data.Sigma.Properties.html#10371" class="Function">IsoΣPathTransportPathΣ</a> <a id="10394" class="Symbol">:</a> <a id="10396" class="Symbol">(</a><a id="10397" href="Cubical.Data.Sigma.Properties.html#10397" class="Bound">a</a> <a id="10399" href="Cubical.Data.Sigma.Properties.html#10399" class="Bound">b</a> <a id="10401" class="Symbol">:</a> <a id="10403" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10405" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10407" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="10408" class="Symbol">)</a> <a id="10410" class="Symbol">→</a> <a id="10412" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="10416" class="Symbol">(</a><a id="10417" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10432" href="Cubical.Data.Sigma.Properties.html#10397" class="Bound">a</a> <a id="10434" href="Cubical.Data.Sigma.Properties.html#10399" class="Bound">b</a><a id="10435" class="Symbol">)</a> <a id="10437" class="Symbol">(</a><a id="10438" href="Cubical.Data.Sigma.Properties.html#10397" class="Bound">a</a> <a id="10440" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10442" href="Cubical.Data.Sigma.Properties.html#10399" class="Bound">b</a><a id="10443" class="Symbol">)</a>
<a id="10445" href="Cubical.Data.Sigma.Properties.html#10371" class="Function">IsoΣPathTransportPathΣ</a> <a id="10468" class="Symbol">{</a><a id="10469" class="Argument">B</a> <a id="10471" class="Symbol">=</a> <a id="10473" href="Cubical.Data.Sigma.Properties.html#10473" class="Bound">B</a><a id="10474" class="Symbol">}</a> <a id="10476" href="Cubical.Data.Sigma.Properties.html#10476" class="Bound">a</a> <a id="10478" href="Cubical.Data.Sigma.Properties.html#10478" class="Bound">b</a> <a id="10480" class="Symbol">=</a>
<a id="10484" href="Cubical.Foundations.Isomorphism.html#3521" class="Function">compIso</a> <a id="10492" class="Symbol">(</a><a id="10493" href="Cubical.Data.Sigma.Properties.html#8744" class="Function">Σ-cong-iso-snd</a> <a id="10508" class="Symbol">(λ</a> <a id="10511" href="Cubical.Data.Sigma.Properties.html#10511" class="Bound">p</a> <a id="10513" class="Symbol">→</a> <a id="10515" href="Cubical.Foundations.Isomorphism.html#3382" class="Function">invIso</a> <a id="10522" class="Symbol">(</a><a id="10523" href="Cubical.Foundations.Path.html#1335" class="Function">PathPIsoPath</a> <a id="10536" class="Symbol">(λ</a> <a id="10539" href="Cubical.Data.Sigma.Properties.html#10539" class="Bound">i</a> <a id="10541" class="Symbol">→</a> <a id="10543" href="Cubical.Data.Sigma.Properties.html#10473" class="Bound">B</a> <a id="10545" class="Symbol">(</a><a id="10546" href="Cubical.Data.Sigma.Properties.html#10511" class="Bound">p</a> <a id="10548" href="Cubical.Data.Sigma.Properties.html#10539" class="Bound">i</a><a id="10549" class="Symbol">))</a> <a id="10552" class="Symbol">_</a> <a id="10554" class="Symbol">_)))</a>
<a id="10569" href="Cubical.Data.Sigma.Properties.html#2467" class="Function">ΣPathIsoPathΣ</a>
<a id="ΣPathTransport≃PathΣ"></a><a id="10584" href="Cubical.Data.Sigma.Properties.html#10584" class="Function">ΣPathTransport≃PathΣ</a> <a id="10605" class="Symbol">:</a> <a id="10607" class="Symbol">(</a><a id="10608" href="Cubical.Data.Sigma.Properties.html#10608" class="Bound">a</a> <a id="10610" href="Cubical.Data.Sigma.Properties.html#10610" class="Bound">b</a> <a id="10612" class="Symbol">:</a> <a id="10614" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10616" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10618" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="10619" class="Symbol">)</a> <a id="10621" class="Symbol">→</a> <a id="10623" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10638" href="Cubical.Data.Sigma.Properties.html#10608" class="Bound">a</a> <a id="10640" href="Cubical.Data.Sigma.Properties.html#10610" class="Bound">b</a> <a id="10642" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="10644" class="Symbol">(</a><a id="10645" href="Cubical.Data.Sigma.Properties.html#10608" class="Bound">a</a> <a id="10647" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10649" href="Cubical.Data.Sigma.Properties.html#10610" class="Bound">b</a><a id="10650" class="Symbol">)</a>
<a id="10652" href="Cubical.Data.Sigma.Properties.html#10584" class="Function">ΣPathTransport≃PathΣ</a> <a id="10673" class="Symbol">{</a><a id="10674" class="Argument">B</a> <a id="10676" class="Symbol">=</a> <a id="10678" href="Cubical.Data.Sigma.Properties.html#10678" class="Bound">B</a><a id="10679" class="Symbol">}</a> <a id="10681" href="Cubical.Data.Sigma.Properties.html#10681" class="Bound">a</a> <a id="10683" href="Cubical.Data.Sigma.Properties.html#10683" class="Bound">b</a> <a id="10685" class="Symbol">=</a> <a id="10687" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="10698" class="Symbol">(</a><a id="10699" href="Cubical.Data.Sigma.Properties.html#10371" class="Function">IsoΣPathTransportPathΣ</a> <a id="10722" href="Cubical.Data.Sigma.Properties.html#10681" class="Bound">a</a> <a id="10724" href="Cubical.Data.Sigma.Properties.html#10683" class="Bound">b</a><a id="10725" class="Symbol">)</a>
<a id="ΣPathTransport→PathΣ"></a><a id="10728" href="Cubical.Data.Sigma.Properties.html#10728" class="Function">ΣPathTransport→PathΣ</a> <a id="10749" class="Symbol">:</a> <a id="10751" class="Symbol">(</a><a id="10752" href="Cubical.Data.Sigma.Properties.html#10752" class="Bound">a</a> <a id="10754" href="Cubical.Data.Sigma.Properties.html#10754" class="Bound">b</a> <a id="10756" class="Symbol">:</a> <a id="10758" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10760" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10762" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="10763" class="Symbol">)</a> <a id="10765" class="Symbol">→</a> <a id="10767" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10782" href="Cubical.Data.Sigma.Properties.html#10752" class="Bound">a</a> <a id="10784" href="Cubical.Data.Sigma.Properties.html#10754" class="Bound">b</a> <a id="10786" class="Symbol">→</a> <a id="10788" class="Symbol">(</a><a id="10789" href="Cubical.Data.Sigma.Properties.html#10752" class="Bound">a</a> <a id="10791" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10793" href="Cubical.Data.Sigma.Properties.html#10754" class="Bound">b</a><a id="10794" class="Symbol">)</a>
<a id="10796" href="Cubical.Data.Sigma.Properties.html#10728" class="Function">ΣPathTransport→PathΣ</a> <a id="10817" href="Cubical.Data.Sigma.Properties.html#10817" class="Bound">a</a> <a id="10819" href="Cubical.Data.Sigma.Properties.html#10819" class="Bound">b</a> <a id="10821" class="Symbol">=</a> <a id="10823" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="10831" class="Symbol">(</a><a id="10832" href="Cubical.Data.Sigma.Properties.html#10371" class="Function">IsoΣPathTransportPathΣ</a> <a id="10855" href="Cubical.Data.Sigma.Properties.html#10817" class="Bound">a</a> <a id="10857" href="Cubical.Data.Sigma.Properties.html#10819" class="Bound">b</a><a id="10858" class="Symbol">)</a>
<a id="PathΣ→ΣPathTransport"></a><a id="10861" href="Cubical.Data.Sigma.Properties.html#10861" class="Function">PathΣ→ΣPathTransport</a> <a id="10882" class="Symbol">:</a> <a id="10884" class="Symbol">(</a><a id="10885" href="Cubical.Data.Sigma.Properties.html#10885" class="Bound">a</a> <a id="10887" href="Cubical.Data.Sigma.Properties.html#10887" class="Bound">b</a> <a id="10889" class="Symbol">:</a> <a id="10891" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="10893" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="10895" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="10896" class="Symbol">)</a> <a id="10898" class="Symbol">→</a> <a id="10900" class="Symbol">(</a><a id="10901" href="Cubical.Data.Sigma.Properties.html#10885" class="Bound">a</a> <a id="10903" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="10905" href="Cubical.Data.Sigma.Properties.html#10887" class="Bound">b</a><a id="10906" class="Symbol">)</a> <a id="10908" class="Symbol">→</a> <a id="10910" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="10925" href="Cubical.Data.Sigma.Properties.html#10885" class="Bound">a</a> <a id="10927" href="Cubical.Data.Sigma.Properties.html#10887" class="Bound">b</a>
<a id="10929" href="Cubical.Data.Sigma.Properties.html#10861" class="Function">PathΣ→ΣPathTransport</a> <a id="10950" href="Cubical.Data.Sigma.Properties.html#10950" class="Bound">a</a> <a id="10952" href="Cubical.Data.Sigma.Properties.html#10952" class="Bound">b</a> <a id="10954" class="Symbol">=</a> <a id="10956" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="10964" class="Symbol">(</a><a id="10965" href="Cubical.Data.Sigma.Properties.html#10371" class="Function">IsoΣPathTransportPathΣ</a> <a id="10988" href="Cubical.Data.Sigma.Properties.html#10950" class="Bound">a</a> <a id="10990" href="Cubical.Data.Sigma.Properties.html#10952" class="Bound">b</a><a id="10991" class="Symbol">)</a>
<a id="ΣPathTransport≡PathΣ"></a><a id="10994" href="Cubical.Data.Sigma.Properties.html#10994" class="Function">ΣPathTransport≡PathΣ</a> <a id="11015" class="Symbol">:</a> <a id="11017" class="Symbol">(</a><a id="11018" href="Cubical.Data.Sigma.Properties.html#11018" class="Bound">a</a> <a id="11020" href="Cubical.Data.Sigma.Properties.html#11020" class="Bound">b</a> <a id="11022" class="Symbol">:</a> <a id="11024" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="11026" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="11028" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="11029" class="Symbol">)</a> <a id="11031" class="Symbol">→</a> <a id="11033" href="Cubical.Data.Sigma.Properties.html#10234" class="Function">ΣPathTransport</a> <a id="11048" href="Cubical.Data.Sigma.Properties.html#11018" class="Bound">a</a> <a id="11050" href="Cubical.Data.Sigma.Properties.html#11020" class="Bound">b</a> <a id="11052" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="11054" class="Symbol">(</a><a id="11055" href="Cubical.Data.Sigma.Properties.html#11018" class="Bound">a</a> <a id="11057" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="11059" href="Cubical.Data.Sigma.Properties.html#11020" class="Bound">b</a><a id="11060" class="Symbol">)</a>
<a id="11062" href="Cubical.Data.Sigma.Properties.html#10994" class="Function">ΣPathTransport≡PathΣ</a> <a id="11083" href="Cubical.Data.Sigma.Properties.html#11083" class="Bound">a</a> <a id="11085" href="Cubical.Data.Sigma.Properties.html#11085" class="Bound">b</a> <a id="11087" class="Symbol">=</a> <a id="11089" href="Cubical.Foundations.Univalence.html#928" class="Function">ua</a> <a id="11092" class="Symbol">(</a><a id="11093" href="Cubical.Data.Sigma.Properties.html#10584" class="Function">ΣPathTransport≃PathΣ</a> <a id="11114" href="Cubical.Data.Sigma.Properties.html#11083" class="Bound">a</a> <a id="11116" href="Cubical.Data.Sigma.Properties.html#11085" class="Bound">b</a><a id="11117" class="Symbol">)</a>
<a id="Σ-contractFstIso"></a><a id="11120" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11137" class="Symbol">:</a> <a id="11139" class="Symbol">(</a><a id="11140" href="Cubical.Data.Sigma.Properties.html#11140" class="Bound">c</a> <a id="11142" class="Symbol">:</a> <a id="11144" href="Cubical.Foundations.Prelude.html#13975" class="Function">isContr</a> <a id="11152" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="11153" class="Symbol">)</a> <a id="11155" class="Symbol">→</a> <a id="11157" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="11161" class="Symbol">(</a><a id="11162" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="11164" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="11166" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="11167" class="Symbol">)</a> <a id="11169" class="Symbol">(</a><a id="11170" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="11172" class="Symbol">(</a><a id="11173" href="Cubical.Data.Sigma.Properties.html#11140" class="Bound">c</a> <a id="11175" class="Symbol">.</a><a id="11176" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="11179" class="Symbol">))</a>
<a id="11182" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="11186" class="Symbol">(</a><a id="11187" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11204" class="Symbol">{</a><a id="11205" class="Argument">B</a> <a id="11207" class="Symbol">=</a> <a id="11209" href="Cubical.Data.Sigma.Properties.html#11209" class="Bound">B</a><a id="11210" class="Symbol">}</a> <a id="11212" href="Cubical.Data.Sigma.Properties.html#11212" class="Bound">c</a><a id="11213" class="Symbol">)</a> <a id="11215" href="Cubical.Data.Sigma.Properties.html#11215" class="Bound">p</a> <a id="11217" class="Symbol">=</a> <a id="11219" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="11225" href="Cubical.Data.Sigma.Properties.html#11209" class="Bound">B</a> <a id="11227" class="Symbol">(</a><a id="11228" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="11232" class="Symbol">(</a><a id="11233" href="Cubical.Data.Sigma.Properties.html#11212" class="Bound">c</a> <a id="11235" class="Symbol">.</a><a id="11236" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11240" class="Symbol">(</a><a id="11241" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="11245" href="Cubical.Data.Sigma.Properties.html#11215" class="Bound">p</a><a id="11246" class="Symbol">)))</a> <a id="11250" class="Symbol">(</a><a id="11251" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11255" href="Cubical.Data.Sigma.Properties.html#11215" class="Bound">p</a><a id="11256" class="Symbol">)</a>
<a id="11258" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="11262" class="Symbol">(</a><a id="11263" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11280" class="Symbol">{</a><a id="11281" class="Argument">B</a> <a id="11283" class="Symbol">=</a> <a id="11285" href="Cubical.Data.Sigma.Properties.html#11285" class="Bound">B</a><a id="11286" class="Symbol">}</a> <a id="11288" href="Cubical.Data.Sigma.Properties.html#11288" class="Bound">c</a><a id="11289" class="Symbol">)</a> <a id="11291" href="Cubical.Data.Sigma.Properties.html#11291" class="Bound">b</a> <a id="11293" class="Symbol">=</a> <a id="11295" class="Symbol">_</a> <a id="11297" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="11299" href="Cubical.Data.Sigma.Properties.html#11291" class="Bound">b</a>
<a id="11301" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="11310" class="Symbol">(</a><a id="11311" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11328" class="Symbol">{</a><a id="11329" class="Argument">B</a> <a id="11331" class="Symbol">=</a> <a id="11333" href="Cubical.Data.Sigma.Properties.html#11333" class="Bound">B</a><a id="11334" class="Symbol">}</a> <a id="11336" href="Cubical.Data.Sigma.Properties.html#11336" class="Bound">c</a><a id="11337" class="Symbol">)</a> <a id="11339" href="Cubical.Data.Sigma.Properties.html#11339" class="Bound">b</a> <a id="11341" class="Symbol">=</a>
<a id="11345" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="11350" class="Symbol">(λ</a> <a id="11353" href="Cubical.Data.Sigma.Properties.html#11353" class="Bound">p</a> <a id="11355" class="Symbol">→</a> <a id="11357" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="11363" href="Cubical.Data.Sigma.Properties.html#11333" class="Bound">B</a> <a id="11365" href="Cubical.Data.Sigma.Properties.html#11353" class="Bound">p</a> <a id="11367" href="Cubical.Data.Sigma.Properties.html#11339" class="Bound">b</a><a id="11368" class="Symbol">)</a> <a id="11370" class="Symbol">(</a><a id="11371" href="Cubical.Foundations.Prelude.html#18316" class="Function">isProp→isSet</a> <a id="11384" class="Symbol">(</a><a id="11385" href="Cubical.Foundations.Prelude.html#18232" class="Function">isContr→isProp</a> <a id="11400" href="Cubical.Data.Sigma.Properties.html#11336" class="Bound">c</a><a id="11401" class="Symbol">)</a> <a id="11403" class="Symbol">_</a> <a id="11405" class="Symbol">_</a> <a id="11407" class="Symbol">_</a> <a id="11409" class="Symbol">_)</a> <a id="11412" href="Cubical.Foundations.Prelude.html#4447" class="Function Operator">∙</a> <a id="11414" href="Cubical.Foundations.Prelude.html#8818" class="Function">transportRefl</a> <a id="11428" class="Symbol">_</a>
<a id="11430" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="11434" class="Symbol">(</a><a id="11435" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="11443" class="Symbol">(</a><a id="11444" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11461" class="Symbol">{</a><a id="11462" class="Argument">B</a> <a id="11464" class="Symbol">=</a> <a id="11466" href="Cubical.Data.Sigma.Properties.html#11466" class="Bound">B</a><a id="11467" class="Symbol">}</a> <a id="11469" href="Cubical.Data.Sigma.Properties.html#11469" class="Bound">c</a><a id="11470" class="Symbol">)</a> <a id="11472" href="Cubical.Data.Sigma.Properties.html#11472" class="Bound">p</a> <a id="11474" href="Cubical.Data.Sigma.Properties.html#11474" class="Bound">j</a><a id="11475" class="Symbol">)</a> <a id="11477" class="Symbol">=</a> <a id="11479" href="Cubical.Data.Sigma.Properties.html#11469" class="Bound">c</a> <a id="11481" class="Symbol">.</a><a id="11482" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11486" class="Symbol">(</a><a id="11487" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="11491" href="Cubical.Data.Sigma.Properties.html#11472" class="Bound">p</a><a id="11492" class="Symbol">)</a> <a id="11494" href="Cubical.Data.Sigma.Properties.html#11474" class="Bound">j</a>
<a id="11496" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11500" class="Symbol">(</a><a id="11501" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="11509" class="Symbol">(</a><a id="11510" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11527" class="Symbol">{</a><a id="11528" class="Argument">B</a> <a id="11530" class="Symbol">=</a> <a id="11532" href="Cubical.Data.Sigma.Properties.html#11532" class="Bound">B</a><a id="11533" class="Symbol">}</a> <a id="11535" href="Cubical.Data.Sigma.Properties.html#11535" class="Bound">c</a><a id="11536" class="Symbol">)</a> <a id="11538" href="Cubical.Data.Sigma.Properties.html#11538" class="Bound">p</a> <a id="11540" href="Cubical.Data.Sigma.Properties.html#11540" class="Bound">j</a><a id="11541" class="Symbol">)</a> <a id="11543" class="Symbol">=</a>
<a id="11547" href="Cubical.Core.Primitives.html#694" class="Primitive">transp</a> <a id="11554" class="Symbol">(λ</a> <a id="11557" href="Cubical.Data.Sigma.Properties.html#11557" class="Bound">i</a> <a id="11559" class="Symbol">→</a> <a id="11561" href="Cubical.Data.Sigma.Properties.html#11532" class="Bound">B</a> <a id="11563" class="Symbol">(</a><a id="11564" href="Cubical.Data.Sigma.Properties.html#11535" class="Bound">c</a> <a id="11566" class="Symbol">.</a><a id="11567" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11571" class="Symbol">(</a><a id="11572" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="11576" href="Cubical.Data.Sigma.Properties.html#11538" class="Bound">p</a><a id="11577" class="Symbol">)</a> <a id="11579" class="Symbol">(</a><a id="11580" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="11582" href="Cubical.Data.Sigma.Properties.html#11557" class="Bound">i</a> <a id="11584" href="Cubical.Core.Primitives.html#490" class="Primitive Operator">∨</a> <a id="11586" href="Cubical.Data.Sigma.Properties.html#11540" class="Bound">j</a><a id="11587" class="Symbol">)))</a> <a id="11591" href="Cubical.Data.Sigma.Properties.html#11540" class="Bound">j</a> <a id="11593" class="Symbol">(</a><a id="11594" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11598" href="Cubical.Data.Sigma.Properties.html#11538" class="Bound">p</a><a id="11599" class="Symbol">)</a>
<a id="Σ-contractFst"></a><a id="11602" href="Cubical.Data.Sigma.Properties.html#11602" class="Function">Σ-contractFst</a> <a id="11616" class="Symbol">:</a> <a id="11618" class="Symbol">(</a><a id="11619" href="Cubical.Data.Sigma.Properties.html#11619" class="Bound">c</a> <a id="11621" class="Symbol">:</a> <a id="11623" href="Cubical.Foundations.Prelude.html#13975" class="Function">isContr</a> <a id="11631" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="11632" class="Symbol">)</a> <a id="11634" class="Symbol">→</a> <a id="11636" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="11638" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="11640" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="11642" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="11644" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="11646" class="Symbol">(</a><a id="11647" href="Cubical.Data.Sigma.Properties.html#11619" class="Bound">c</a> <a id="11649" class="Symbol">.</a><a id="11650" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="11653" class="Symbol">)</a>
<a id="11655" href="Cubical.Data.Sigma.Properties.html#11602" class="Function">Σ-contractFst</a> <a id="11669" class="Symbol">{</a><a id="11670" class="Argument">B</a> <a id="11672" class="Symbol">=</a> <a id="11674" href="Cubical.Data.Sigma.Properties.html#11674" class="Bound">B</a><a id="11675" class="Symbol">}</a> <a id="11677" href="Cubical.Data.Sigma.Properties.html#11677" class="Bound">c</a> <a id="11679" class="Symbol">=</a> <a id="11681" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="11692" class="Symbol">(</a><a id="11693" href="Cubical.Data.Sigma.Properties.html#11120" class="Function">Σ-contractFstIso</a> <a id="11710" href="Cubical.Data.Sigma.Properties.html#11677" class="Bound">c</a><a id="11711" class="Symbol">)</a>
<a id="11714" class="Comment">-- a special case of the above</a>
<a id="11745" class="Keyword">module</a> <a id="11752" href="Cubical.Data.Sigma.Properties.html#11752" class="Module">_</a> <a id="11754" class="Symbol">(</a><a id="11755" href="Cubical.Data.Sigma.Properties.html#11755" class="Bound">A</a> <a id="11757" class="Symbol">:</a> <a id="11759" href="Cubical.Data.Unit.Base.html#171" class="Record">Unit</a> <a id="11764" class="Symbol">→</a> <a id="11766" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="11771" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="11772" class="Symbol">)</a> <a id="11774" class="Keyword">where</a>
<a id="11782" href="Cubical.Data.Sigma.Properties.html#11782" class="Function">ΣUnit</a> <a id="11788" class="Symbol">:</a> <a id="11790" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="11792" href="Cubical.Data.Unit.Base.html#171" class="Record">Unit</a> <a id="11797" href="Cubical.Data.Sigma.Properties.html#11755" class="Bound">A</a> <a id="11799" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="11801" href="Cubical.Data.Sigma.Properties.html#11755" class="Bound">A</a> <a id="11803" href="Agda.Builtin.Unit.html#201" class="InductiveConstructor">tt</a>
<a id="11808" class="Keyword">unquoteDef</a> <a id="11819" href="Cubical.Data.Sigma.Properties.html#11782" class="Function">ΣUnit</a> <a id="11825" class="Symbol">=</a> <a id="11827" href="Cubical.Reflection.StrictEquiv.html#1246" class="Function">defStrictEquiv</a> <a id="11842" href="Cubical.Data.Sigma.Properties.html#11782" class="Function">ΣUnit</a> <a id="11848" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="11852" class="Symbol">(λ</a> <a id="11855" class="Symbol">{</a> <a id="11857" href="Cubical.Data.Sigma.Properties.html#11857" class="Bound">x</a> <a id="11859" class="Symbol">→</a> <a id="11861" class="Symbol">(</a><a id="11862" href="Agda.Builtin.Unit.html#201" class="InductiveConstructor">tt</a> <a id="11865" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="11867" href="Cubical.Data.Sigma.Properties.html#11857" class="Bound">x</a><a id="11868" class="Symbol">)</a> <a id="11870" class="Symbol">})</a>
<a id="Σ-contractSnd"></a><a id="11874" href="Cubical.Data.Sigma.Properties.html#11874" class="Function">Σ-contractSnd</a> <a id="11888" class="Symbol">:</a> <a id="11890" class="Symbol">((</a><a id="11892" href="Cubical.Data.Sigma.Properties.html#11892" class="Bound">a</a> <a id="11894" class="Symbol">:</a> <a id="11896" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="11897" class="Symbol">)</a> <a id="11899" class="Symbol">→</a> <a id="11901" href="Cubical.Foundations.Prelude.html#13975" class="Function">isContr</a> <a id="11909" class="Symbol">(</a><a id="11910" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="11912" href="Cubical.Data.Sigma.Properties.html#11892" class="Bound">a</a><a id="11913" class="Symbol">))</a> <a id="11916" class="Symbol">→</a> <a id="11918" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="11920" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="11922" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="11924" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="11926" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a>
<a id="11928" href="Cubical.Data.Sigma.Properties.html#11874" class="Function">Σ-contractSnd</a> <a id="11942" href="Cubical.Data.Sigma.Properties.html#11942" class="Bound">c</a> <a id="11944" class="Symbol">=</a> <a id="11946" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="11957" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a>
<a id="11964" class="Keyword">where</a>
<a id="11972" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a> <a id="11977" class="Symbol">:</a> <a id="11979" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="11983" class="Symbol">_</a> <a id="11985" class="Symbol">_</a>
<a id="11989" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a> <a id="11994" class="Symbol">.</a><a id="11995" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="11999" class="Symbol">=</a> <a id="12001" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a>
<a id="12007" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a> <a id="12012" class="Symbol">.</a><a id="12013" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="12017" href="Cubical.Data.Sigma.Properties.html#12017" class="Bound">a</a> <a id="12019" class="Symbol">=</a> <a id="12021" href="Cubical.Data.Sigma.Properties.html#12017" class="Bound">a</a> <a id="12023" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12025" href="Cubical.Data.Sigma.Properties.html#11942" class="Bound">c</a> <a id="12027" href="Cubical.Data.Sigma.Properties.html#12017" class="Bound">a</a> <a id="12029" class="Symbol">.</a><a id="12030" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a>
<a id="12036" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a> <a id="12041" class="Symbol">.</a><a id="12042" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="12051" class="Symbol">_</a> <a id="12053" class="Symbol">=</a> <a id="12055" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="12062" href="Cubical.Data.Sigma.Properties.html#11972" class="Function">isom</a> <a id="12067" class="Symbol">.</a><a id="12068" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="12076" class="Symbol">(</a><a id="12077" href="Cubical.Data.Sigma.Properties.html#12077" class="Bound">a</a> <a id="12079" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12081" href="Cubical.Data.Sigma.Properties.html#12081" class="Bound">b</a><a id="12082" class="Symbol">)</a> <a id="12084" class="Symbol">=</a> <a id="12086" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12091" class="Symbol">(</a><a id="12092" href="Cubical.Data.Sigma.Properties.html#12077" class="Bound">a</a> <a id="12094" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,_</a><a id="12096" class="Symbol">)</a> <a id="12098" class="Symbol">(</a><a id="12099" href="Cubical.Data.Sigma.Properties.html#11942" class="Bound">c</a> <a id="12101" href="Cubical.Data.Sigma.Properties.html#12077" class="Bound">a</a> <a id="12103" class="Symbol">.</a><a id="12104" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="12108" href="Cubical.Data.Sigma.Properties.html#12081" class="Bound">b</a><a id="12109" class="Symbol">)</a>
<a id="isEmbeddingFstΣProp"></a><a id="12112" href="Cubical.Data.Sigma.Properties.html#12112" class="Function">isEmbeddingFstΣProp</a> <a id="12132" class="Symbol">:</a> <a id="12134" class="Symbol">((</a><a id="12136" href="Cubical.Data.Sigma.Properties.html#12136" class="Bound">x</a> <a id="12138" class="Symbol">:</a> <a id="12140" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="12141" class="Symbol">)</a> <a id="12143" class="Symbol">→</a> <a id="12145" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="12152" class="Symbol">(</a><a id="12153" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="12155" href="Cubical.Data.Sigma.Properties.html#12136" class="Bound">x</a><a id="12156" class="Symbol">))</a>
<a id="12179" class="Symbol">→</a> <a id="12181" class="Symbol">{</a><a id="12182" href="Cubical.Data.Sigma.Properties.html#12182" class="Bound">u</a> <a id="12184" href="Cubical.Data.Sigma.Properties.html#12184" class="Bound">v</a> <a id="12186" class="Symbol">:</a> <a id="12188" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="12190" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="12192" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="12193" class="Symbol">}</a>
<a id="12215" class="Symbol">→</a> <a id="12217" href="Agda.Builtin.Cubical.Glue.html#868" class="Record">isEquiv</a> <a id="12225" class="Symbol">(λ</a> <a id="12228" class="Symbol">(</a><a id="12229" href="Cubical.Data.Sigma.Properties.html#12229" class="Bound">p</a> <a id="12231" class="Symbol">:</a> <a id="12233" href="Cubical.Data.Sigma.Properties.html#12182" class="Bound">u</a> <a id="12235" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12237" href="Cubical.Data.Sigma.Properties.html#12184" class="Bound">v</a><a id="12238" class="Symbol">)</a> <a id="12240" class="Symbol">→</a> <a id="12242" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12247" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12251" href="Cubical.Data.Sigma.Properties.html#12229" class="Bound">p</a><a id="12252" class="Symbol">)</a>
<a id="12254" href="Cubical.Data.Sigma.Properties.html#12112" class="Function">isEmbeddingFstΣProp</a> <a id="12274" class="Symbol">{</a><a id="12275" class="Argument">B</a> <a id="12277" class="Symbol">=</a> <a id="12279" href="Cubical.Data.Sigma.Properties.html#12279" class="Bound">B</a><a id="12280" class="Symbol">}</a> <a id="12282" href="Cubical.Data.Sigma.Properties.html#12282" class="Bound">pB</a> <a id="12285" class="Symbol">{</a><a id="12286" class="Argument">u</a> <a id="12288" class="Symbol">=</a> <a id="12290" href="Cubical.Data.Sigma.Properties.html#12290" class="Bound">u</a><a id="12291" class="Symbol">}</a> <a id="12293" class="Symbol">{</a><a id="12294" class="Argument">v</a> <a id="12296" class="Symbol">=</a> <a id="12298" href="Cubical.Data.Sigma.Properties.html#12298" class="Bound">v</a><a id="12299" class="Symbol">}</a> <a id="12301" class="Symbol">.</a><a id="12302" href="Agda.Builtin.Cubical.Glue.html#971" class="Field">equiv-proof</a> <a id="12314" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a> <a id="12316" class="Symbol">=</a> <a id="12318" href="Cubical.Data.Sigma.Properties.html#12407" class="Function">ctr</a> <a id="12322" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12324" href="Cubical.Data.Sigma.Properties.html#12475" class="Function">isCtr</a>
<a id="12332" class="Keyword">where</a>
<a id="12340" href="Cubical.Data.Sigma.Properties.html#12340" class="Function">ctrP</a> <a id="12345" class="Symbol">:</a> <a id="12347" href="Cubical.Data.Sigma.Properties.html#12290" class="Bound">u</a> <a id="12349" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12351" href="Cubical.Data.Sigma.Properties.html#12298" class="Bound">v</a>
<a id="12355" href="Cubical.Data.Sigma.Properties.html#12340" class="Function">ctrP</a> <a id="12360" class="Symbol">=</a> <a id="12362" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="12369" class="Symbol">(</a><a id="12370" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a> <a id="12372" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12374" href="Cubical.Foundations.Prelude.html#17730" class="Function">isProp→PathP</a> <a id="12387" class="Symbol">(λ</a> <a id="12390" href="Cubical.Data.Sigma.Properties.html#12390" class="Bound">_</a> <a id="12392" class="Symbol">→</a> <a id="12394" href="Cubical.Data.Sigma.Properties.html#12282" class="Bound">pB</a> <a id="12397" class="Symbol">_)</a> <a id="12400" class="Symbol">_</a> <a id="12402" class="Symbol">_)</a>
<a id="12407" href="Cubical.Data.Sigma.Properties.html#12407" class="Function">ctr</a> <a id="12412" class="Symbol">:</a> <a id="12414" href="Cubical.Foundations.Equiv.Base.html#253" class="Function">fiber</a> <a id="12420" class="Symbol">(λ</a> <a id="12423" class="Symbol">(</a><a id="12424" href="Cubical.Data.Sigma.Properties.html#12424" class="Bound">p</a> <a id="12426" class="Symbol">:</a> <a id="12428" href="Cubical.Data.Sigma.Properties.html#12290" class="Bound">u</a> <a id="12430" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12432" href="Cubical.Data.Sigma.Properties.html#12298" class="Bound">v</a><a id="12433" class="Symbol">)</a> <a id="12435" class="Symbol">→</a> <a id="12437" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12442" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12446" href="Cubical.Data.Sigma.Properties.html#12424" class="Bound">p</a><a id="12447" class="Symbol">)</a> <a id="12449" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a>
<a id="12453" href="Cubical.Data.Sigma.Properties.html#12407" class="Function">ctr</a> <a id="12458" class="Symbol">=</a> <a id="12460" href="Cubical.Data.Sigma.Properties.html#12340" class="Function">ctrP</a> <a id="12465" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12467" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="12475" href="Cubical.Data.Sigma.Properties.html#12475" class="Function">isCtr</a> <a id="12481" class="Symbol">:</a> <a id="12483" class="Symbol">∀</a> <a id="12485" href="Cubical.Data.Sigma.Properties.html#12485" class="Bound">z</a> <a id="12487" class="Symbol">→</a> <a id="12489" href="Cubical.Data.Sigma.Properties.html#12407" class="Function">ctr</a> <a id="12493" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12495" href="Cubical.Data.Sigma.Properties.html#12485" class="Bound">z</a>
<a id="12499" href="Cubical.Data.Sigma.Properties.html#12475" class="Function">isCtr</a> <a id="12505" class="Symbol">(</a><a id="12506" href="Cubical.Data.Sigma.Properties.html#12506" class="Bound">z</a> <a id="12508" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12510" href="Cubical.Data.Sigma.Properties.html#12510" class="Bound">p</a><a id="12511" class="Symbol">)</a> <a id="12513" class="Symbol">=</a> <a id="12515" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="12522" class="Symbol">(</a><a id="12523" href="Cubical.Data.Sigma.Properties.html#12819" class="Function">ctrP≡</a> <a id="12529" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12531" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12536" class="Symbol">(</a><a id="12537" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="12541" href="Cubical.Foundations.Function.html#526" class="Function Operator">∘</a> <a id="12543" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="12546" class="Symbol">)</a> <a id="12548" href="Cubical.Data.Sigma.Properties.html#12567" class="Function">fzsingl</a><a id="12555" class="Symbol">)</a> <a id="12557" class="Keyword">where</a>
<a id="12567" href="Cubical.Data.Sigma.Properties.html#12567" class="Function">fzsingl</a> <a id="12575" class="Symbol">:</a> <a id="12577" href="Cubical.Core.Primitives.html#1866" class="Function">Path</a> <a id="12582" class="Symbol">(</a><a id="12583" href="Cubical.Foundations.Prelude.html#14427" class="Function">singl</a> <a id="12589" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a><a id="12590" class="Symbol">)</a> <a id="12592" class="Symbol">(</a><a id="12593" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a> <a id="12595" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12597" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a><a id="12601" class="Symbol">)</a> <a id="12603" class="Symbol">(</a><a id="12604" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12609" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12613" href="Cubical.Data.Sigma.Properties.html#12506" class="Bound">z</a> <a id="12615" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12617" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="12621" href="Cubical.Data.Sigma.Properties.html#12510" class="Bound">p</a><a id="12622" class="Symbol">)</a>
<a id="12628" href="Cubical.Data.Sigma.Properties.html#12567" class="Function">fzsingl</a> <a id="12636" class="Symbol">=</a> <a id="12638" href="Cubical.Foundations.Prelude.html#14490" class="Function">isContrSingl</a> <a id="12651" href="Cubical.Data.Sigma.Properties.html#12314" class="Bound">x</a> <a id="12653" class="Symbol">.</a><a id="12654" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="12658" class="Symbol">(</a><a id="12659" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12664" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12668" href="Cubical.Data.Sigma.Properties.html#12506" class="Bound">z</a> <a id="12670" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12672" href="Cubical.Foundations.Prelude.html#968" class="Function">sym</a> <a id="12676" href="Cubical.Data.Sigma.Properties.html#12510" class="Bound">p</a><a id="12677" class="Symbol">)</a>
<a id="12683" href="Cubical.Data.Sigma.Properties.html#12683" class="Function">ctrSnd</a> <a id="12690" class="Symbol">:</a> <a id="12692" href="Cubical.Foundations.Prelude.html#14941" class="Function">SquareP</a> <a id="12700" class="Symbol">(λ</a> <a id="12703" href="Cubical.Data.Sigma.Properties.html#12703" class="Bound">i</a> <a id="12705" href="Cubical.Data.Sigma.Properties.html#12705" class="Bound">j</a> <a id="12707" class="Symbol">→</a> <a id="12709" href="Cubical.Data.Sigma.Properties.html#12279" class="Bound">B</a> <a id="12711" class="Symbol">(</a><a id="12712" href="Cubical.Data.Sigma.Properties.html#12567" class="Function">fzsingl</a> <a id="12720" href="Cubical.Data.Sigma.Properties.html#12703" class="Bound">i</a> <a id="12722" class="Symbol">.</a><a id="12723" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12727" href="Cubical.Data.Sigma.Properties.html#12705" class="Bound">j</a><a id="12728" class="Symbol">))</a> <a id="12731" class="Symbol">(</a><a id="12732" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12737" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="12741" href="Cubical.Data.Sigma.Properties.html#12340" class="Function">ctrP</a><a id="12745" class="Symbol">)</a> <a id="12747" class="Symbol">(</a><a id="12748" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="12753" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="12757" href="Cubical.Data.Sigma.Properties.html#12506" class="Bound">z</a><a id="12758" class="Symbol">)</a> <a id="12760" class="Symbol">_</a> <a id="12762" class="Symbol">_</a>
<a id="12768" href="Cubical.Data.Sigma.Properties.html#12683" class="Function">ctrSnd</a> <a id="12775" class="Symbol">=</a> <a id="12777" href="Cubical.Foundations.Path.html#5314" class="Function">isProp→SquareP</a> <a id="12792" class="Symbol">(λ</a> <a id="12795" href="Cubical.Data.Sigma.Properties.html#12795" class="Bound">_</a> <a id="12797" href="Cubical.Data.Sigma.Properties.html#12797" class="Bound">_</a> <a id="12799" class="Symbol">→</a> <a id="12801" href="Cubical.Data.Sigma.Properties.html#12282" class="Bound">pB</a> <a id="12804" class="Symbol">_)</a> <a id="12807" class="Symbol">_</a> <a id="12809" class="Symbol">_</a> <a id="12811" class="Symbol">_</a> <a id="12813" class="Symbol">_</a>
<a id="12819" href="Cubical.Data.Sigma.Properties.html#12819" class="Function">ctrP≡</a> <a id="12825" class="Symbol">:</a> <a id="12827" href="Cubical.Data.Sigma.Properties.html#12340" class="Function">ctrP</a> <a id="12832" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12834" href="Cubical.Data.Sigma.Properties.html#12506" class="Bound">z</a>
<a id="12840" href="Cubical.Data.Sigma.Properties.html#12819" class="Function">ctrP≡</a> <a id="12846" href="Cubical.Data.Sigma.Properties.html#12846" class="Bound">i</a> <a id="12848" class="Symbol">=</a> <a id="12850" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="12857" class="Symbol">(</a><a id="12858" href="Cubical.Data.Sigma.Properties.html#12567" class="Function">fzsingl</a> <a id="12866" href="Cubical.Data.Sigma.Properties.html#12846" class="Bound">i</a> <a id="12868" class="Symbol">.</a><a id="12869" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12873" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="12875" href="Cubical.Data.Sigma.Properties.html#12683" class="Function">ctrSnd</a> <a id="12882" href="Cubical.Data.Sigma.Properties.html#12846" class="Bound">i</a><a id="12883" class="Symbol">)</a>
<a id="Σ≡PropEquiv"></a><a id="12886" href="Cubical.Data.Sigma.Properties.html#12886" class="Function">Σ≡PropEquiv</a> <a id="12898" class="Symbol">:</a> <a id="12900" class="Symbol">((</a><a id="12902" href="Cubical.Data.Sigma.Properties.html#12902" class="Bound">x</a> <a id="12904" class="Symbol">:</a> <a id="12906" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="12907" class="Symbol">)</a> <a id="12909" class="Symbol">→</a> <a id="12911" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="12918" class="Symbol">(</a><a id="12919" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="12921" href="Cubical.Data.Sigma.Properties.html#12902" class="Bound">x</a><a id="12922" class="Symbol">))</a> <a id="12925" class="Symbol">→</a> <a id="12927" class="Symbol">{</a><a id="12928" href="Cubical.Data.Sigma.Properties.html#12928" class="Bound">u</a> <a id="12930" href="Cubical.Data.Sigma.Properties.html#12930" class="Bound">v</a> <a id="12932" class="Symbol">:</a> <a id="12934" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="12936" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="12938" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="12939" class="Symbol">}</a>
<a id="12953" class="Symbol">→</a> <a id="12955" class="Symbol">(</a><a id="12956" href="Cubical.Data.Sigma.Properties.html#12928" class="Bound">u</a> <a id="12958" class="Symbol">.</a><a id="12959" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="12963" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12965" href="Cubical.Data.Sigma.Properties.html#12930" class="Bound">v</a> <a id="12967" class="Symbol">.</a><a id="12968" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="12971" class="Symbol">)</a> <a id="12973" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="12975" class="Symbol">(</a><a id="12976" href="Cubical.Data.Sigma.Properties.html#12928" class="Bound">u</a> <a id="12978" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="12980" href="Cubical.Data.Sigma.Properties.html#12930" class="Bound">v</a><a id="12981" class="Symbol">)</a>
<a id="12983" href="Cubical.Data.Sigma.Properties.html#12886" class="Function">Σ≡PropEquiv</a> <a id="12995" href="Cubical.Data.Sigma.Properties.html#12995" class="Bound">pB</a> <a id="12998" class="Symbol">=</a> <a id="13000" href="Cubical.Foundations.Equiv.html#3368" class="Function">invEquiv</a> <a id="13009" class="Symbol">(_</a> <a id="13012" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="13014" href="Cubical.Data.Sigma.Properties.html#12112" class="Function">isEmbeddingFstΣProp</a> <a id="13034" href="Cubical.Data.Sigma.Properties.html#12995" class="Bound">pB</a><a id="13036" class="Symbol">)</a>
<a id="Σ≡Prop"></a><a id="13039" href="Cubical.Data.Sigma.Properties.html#13039" class="Function">Σ≡Prop</a> <a id="13046" class="Symbol">:</a> <a id="13048" class="Symbol">((</a><a id="13050" href="Cubical.Data.Sigma.Properties.html#13050" class="Bound">x</a> <a id="13052" class="Symbol">:</a> <a id="13054" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="13055" class="Symbol">)</a> <a id="13057" class="Symbol">→</a> <a id="13059" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="13066" class="Symbol">(</a><a id="13067" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a> <a id="13069" href="Cubical.Data.Sigma.Properties.html#13050" class="Bound">x</a><a id="13070" class="Symbol">))</a> <a id="13073" class="Symbol">→</a> <a id="13075" class="Symbol">{</a><a id="13076" href="Cubical.Data.Sigma.Properties.html#13076" class="Bound">u</a> <a id="13078" href="Cubical.Data.Sigma.Properties.html#13078" class="Bound">v</a> <a id="13080" class="Symbol">:</a> <a id="13082" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="13084" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="13086" href="Cubical.Data.Sigma.Properties.html#1336" class="Generalizable">B</a><a id="13087" class="Symbol">}</a>
<a id="13096" class="Symbol">→</a> <a id="13098" class="Symbol">(</a><a id="13099" href="Cubical.Data.Sigma.Properties.html#13099" class="Bound">p</a> <a id="13101" class="Symbol">:</a> <a id="13103" href="Cubical.Data.Sigma.Properties.html#13076" class="Bound">u</a> <a id="13105" class="Symbol">.</a><a id="13106" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13110" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="13112" href="Cubical.Data.Sigma.Properties.html#13078" class="Bound">v</a> <a id="13114" class="Symbol">.</a><a id="13115" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="13118" class="Symbol">)</a> <a id="13120" class="Symbol">→</a> <a id="13122" href="Cubical.Data.Sigma.Properties.html#13076" class="Bound">u</a> <a id="13124" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="13126" href="Cubical.Data.Sigma.Properties.html#13078" class="Bound">v</a>
<a id="13128" href="Cubical.Data.Sigma.Properties.html#13039" class="Function">Σ≡Prop</a> <a id="13135" href="Cubical.Data.Sigma.Properties.html#13135" class="Bound">pB</a> <a id="13138" href="Cubical.Data.Sigma.Properties.html#13138" class="Bound">p</a> <a id="13140" class="Symbol">=</a> <a id="13142" href="Agda.Builtin.Cubical.Glue.html#1143" class="Function">equivFun</a> <a id="13151" class="Symbol">(</a><a id="13152" href="Cubical.Data.Sigma.Properties.html#12886" class="Function">Σ≡PropEquiv</a> <a id="13164" href="Cubical.Data.Sigma.Properties.html#13135" class="Bound">pB</a><a id="13166" class="Symbol">)</a> <a id="13168" href="Cubical.Data.Sigma.Properties.html#13138" class="Bound">p</a>
<a id="13171" class="Comment">-- dependent version</a>
<a id="ΣPathPProp"></a><a id="13192" href="Cubical.Data.Sigma.Properties.html#13192" class="Function">ΣPathPProp</a> <a id="13203" class="Symbol">:</a> <a id="13205" class="Symbol">∀</a> <a id="13207" class="Symbol">{</a><a id="13208" href="Cubical.Data.Sigma.Properties.html#13208" class="Bound">ℓ</a> <a id="13210" href="Cubical.Data.Sigma.Properties.html#13210" class="Bound">ℓ'</a><a id="13212" class="Symbol">}</a> <a id="13214" class="Symbol">{</a><a id="13215" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13217" class="Symbol">:</a> <a id="13219" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a> <a id="13221" class="Symbol">→</a> <a id="13223" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13228" href="Cubical.Data.Sigma.Properties.html#13208" class="Bound">ℓ</a><a id="13229" class="Symbol">}</a> <a id="13231" class="Symbol">{</a><a id="13232" href="Cubical.Data.Sigma.Properties.html#13232" class="Bound">B</a> <a id="13234" class="Symbol">:</a> <a id="13236" class="Symbol">(</a><a id="13237" href="Cubical.Data.Sigma.Properties.html#13237" class="Bound">i</a> <a id="13239" class="Symbol">:</a> <a id="13241" href="Agda.Primitive.Cubical.html#108" class="Datatype">I</a><a id="13242" class="Symbol">)</a> <a id="13244" class="Symbol">→</a> <a id="13246" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13248" href="Cubical.Data.Sigma.Properties.html#13237" class="Bound">i</a> <a id="13250" class="Symbol">→</a> <a id="13252" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13257" href="Cubical.Data.Sigma.Properties.html#13210" class="Bound">ℓ'</a><a id="13259" class="Symbol">}</a>
<a id="13272" class="Symbol">→</a> <a id="13274" class="Symbol">{</a><a id="13275" href="Cubical.Data.Sigma.Properties.html#13275" class="Bound">u</a> <a id="13277" class="Symbol">:</a> <a id="13279" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="13281" class="Symbol">(</a><a id="13282" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13284" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="13286" class="Symbol">)</a> <a id="13288" class="Symbol">(</a><a id="13289" href="Cubical.Data.Sigma.Properties.html#13232" class="Bound">B</a> <a id="13291" href="Agda.Primitive.Cubical.html#150" class="InductiveConstructor">i0</a><a id="13293" class="Symbol">)}</a> <a id="13296" class="Symbol">{</a><a id="13297" href="Cubical.Data.Sigma.Properties.html#13297" class="Bound">v</a> <a id="13299" class="Symbol">:</a> <a id="13301" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="13303" class="Symbol">(</a><a id="13304" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13306" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="13308" class="Symbol">)</a> <a id="13310" class="Symbol">(</a><a id="13311" href="Cubical.Data.Sigma.Properties.html#13232" class="Bound">B</a> <a id="13313" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="13315" class="Symbol">)}</a>
<a id="13329" class="Symbol">→</a> <a id="13331" class="Symbol">((</a><a id="13333" href="Cubical.Data.Sigma.Properties.html#13333" class="Bound">a</a> <a id="13335" class="Symbol">:</a> <a id="13337" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13339" class="Symbol">(</a><a id="13340" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a><a id="13342" class="Symbol">))</a> <a id="13345" class="Symbol">→</a> <a id="13347" href="Cubical.Foundations.Prelude.html#14039" class="Function">isProp</a> <a id="13354" class="Symbol">(</a><a id="13355" href="Cubical.Data.Sigma.Properties.html#13232" class="Bound">B</a> <a id="13357" href="Agda.Primitive.Cubical.html#178" class="InductiveConstructor">i1</a> <a id="13360" href="Cubical.Data.Sigma.Properties.html#13333" class="Bound">a</a><a id="13361" class="Symbol">))</a>
<a id="13375" class="Symbol">→</a> <a id="13377" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="13383" class="Symbol">(λ</a> <a id="13386" href="Cubical.Data.Sigma.Properties.html#13386" class="Bound">i</a> <a id="13388" class="Symbol">→</a> <a id="13390" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13392" href="Cubical.Data.Sigma.Properties.html#13386" class="Bound">i</a><a id="13393" class="Symbol">)</a> <a id="13395" class="Symbol">(</a><a id="13396" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13400" href="Cubical.Data.Sigma.Properties.html#13275" class="Bound">u</a><a id="13401" class="Symbol">)</a> <a id="13403" class="Symbol">(</a><a id="13404" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13408" href="Cubical.Data.Sigma.Properties.html#13297" class="Bound">v</a><a id="13409" class="Symbol">)</a>
<a id="13422" class="Symbol">→</a> <a id="13424" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="13430" class="Symbol">(λ</a> <a id="13433" href="Cubical.Data.Sigma.Properties.html#13433" class="Bound">i</a> <a id="13435" class="Symbol">→</a> <a id="13437" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="13439" class="Symbol">(</a><a id="13440" href="Cubical.Data.Sigma.Properties.html#13215" class="Bound">A</a> <a id="13442" href="Cubical.Data.Sigma.Properties.html#13433" class="Bound">i</a><a id="13443" class="Symbol">)</a> <a id="13445" class="Symbol">(</a><a id="13446" href="Cubical.Data.Sigma.Properties.html#13232" class="Bound">B</a> <a id="13448" href="Cubical.Data.Sigma.Properties.html#13433" class="Bound">i</a><a id="13449" class="Symbol">))</a> <a id="13452" href="Cubical.Data.Sigma.Properties.html#13275" class="Bound">u</a> <a id="13454" href="Cubical.Data.Sigma.Properties.html#13297" class="Bound">v</a>
<a id="13456" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13460" class="Symbol">(</a><a id="13461" href="Cubical.Data.Sigma.Properties.html#13192" class="Function">ΣPathPProp</a> <a id="13472" class="Symbol">{</a><a id="13473" class="Argument">u</a> <a id="13475" class="Symbol">=</a> <a id="13477" href="Cubical.Data.Sigma.Properties.html#13477" class="Bound">u</a><a id="13478" class="Symbol">}</a> <a id="13480" class="Symbol">{</a><a id="13481" class="Argument">v</a> <a id="13483" class="Symbol">=</a> <a id="13485" href="Cubical.Data.Sigma.Properties.html#13485" class="Bound">v</a><a id="13486" class="Symbol">}</a> <a id="13488" href="Cubical.Data.Sigma.Properties.html#13488" class="Bound">pB</a> <a id="13491" href="Cubical.Data.Sigma.Properties.html#13491" class="Bound">p</a> <a id="13493" href="Cubical.Data.Sigma.Properties.html#13493" class="Bound">i</a><a id="13494" class="Symbol">)</a> <a id="13496" class="Symbol">=</a> <a id="13498" href="Cubical.Data.Sigma.Properties.html#13491" class="Bound">p</a> <a id="13500" href="Cubical.Data.Sigma.Properties.html#13493" class="Bound">i</a>
<a id="13502" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="13506" class="Symbol">(</a><a id="13507" href="Cubical.Data.Sigma.Properties.html#13192" class="Function">ΣPathPProp</a> <a id="13518" class="Symbol">{</a><a id="13519" class="Argument">B</a> <a id="13521" class="Symbol">=</a> <a id="13523" href="Cubical.Data.Sigma.Properties.html#13523" class="Bound">B</a><a id="13524" class="Symbol">}</a> <a id="13526" class="Symbol">{</a><a id="13527" class="Argument">u</a> <a id="13529" class="Symbol">=</a> <a id="13531" href="Cubical.Data.Sigma.Properties.html#13531" class="Bound">u</a><a id="13532" class="Symbol">}</a> <a id="13534" class="Symbol">{</a><a id="13535" class="Argument">v</a> <a id="13537" class="Symbol">=</a> <a id="13539" href="Cubical.Data.Sigma.Properties.html#13539" class="Bound">v</a><a id="13540" class="Symbol">}</a> <a id="13542" href="Cubical.Data.Sigma.Properties.html#13542" class="Bound">pB</a> <a id="13545" href="Cubical.Data.Sigma.Properties.html#13545" class="Bound">p</a> <a id="13547" href="Cubical.Data.Sigma.Properties.html#13547" class="Bound">i</a><a id="13548" class="Symbol">)</a> <a id="13550" class="Symbol">=</a> <a id="13552" href="Cubical.Data.Sigma.Properties.html#13568" class="Function">lem</a> <a id="13556" href="Cubical.Data.Sigma.Properties.html#13547" class="Bound">i</a>
<a id="13560" class="Keyword">where</a>
<a id="13568" href="Cubical.Data.Sigma.Properties.html#13568" class="Function">lem</a> <a id="13572" class="Symbol">:</a> <a id="13574" href="Agda.Builtin.Cubical.Path.html#197" class="Postulate">PathP</a> <a id="13580" class="Symbol">(λ</a> <a id="13583" href="Cubical.Data.Sigma.Properties.html#13583" class="Bound">i</a> <a id="13585" class="Symbol">→</a> <a id="13587" href="Cubical.Data.Sigma.Properties.html#13523" class="Bound">B</a> <a id="13589" href="Cubical.Data.Sigma.Properties.html#13583" class="Bound">i</a> <a id="13591" class="Symbol">(</a><a id="13592" href="Cubical.Data.Sigma.Properties.html#13545" class="Bound">p</a> <a id="13594" href="Cubical.Data.Sigma.Properties.html#13583" class="Bound">i</a><a id="13595" class="Symbol">))</a> <a id="13598" class="Symbol">(</a><a id="13599" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="13603" href="Cubical.Data.Sigma.Properties.html#13531" class="Bound">u</a><a id="13604" class="Symbol">)</a> <a id="13606" class="Symbol">(</a><a id="13607" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="13611" href="Cubical.Data.Sigma.Properties.html#13539" class="Bound">v</a><a id="13612" class="Symbol">)</a>
<a id="13616" href="Cubical.Data.Sigma.Properties.html#13568" class="Function">lem</a> <a id="13620" class="Symbol">=</a> <a id="13622" href="Cubical.Foundations.Prelude.html#12746" class="Function">toPathP</a> <a id="13630" class="Symbol">(</a><a id="13631" href="Cubical.Data.Sigma.Properties.html#13542" class="Bound">pB</a> <a id="13634" class="Symbol">_</a> <a id="13636" class="Symbol">_</a> <a id="13638" class="Symbol">_)</a>
<a id="≃-×"></a><a id="13642" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="13646" class="Symbol">:</a> <a id="13648" class="Symbol">∀</a> <a id="13650" class="Symbol">{</a><a id="13651" href="Cubical.Data.Sigma.Properties.html#13651" class="Bound">ℓ''</a> <a id="13655" href="Cubical.Data.Sigma.Properties.html#13655" class="Bound">ℓ'''</a><a id="13659" class="Symbol">}</a> <a id="13661" class="Symbol">{</a><a id="13662" href="Cubical.Data.Sigma.Properties.html#13662" class="Bound">A</a> <a id="13664" class="Symbol">:</a> <a id="13666" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13671" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="13672" class="Symbol">}</a> <a id="13674" class="Symbol">{</a><a id="13675" href="Cubical.Data.Sigma.Properties.html#13675" class="Bound">B</a> <a id="13677" class="Symbol">:</a> <a id="13679" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13684" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="13686" class="Symbol">}</a> <a id="13688" class="Symbol">{</a><a id="13689" href="Cubical.Data.Sigma.Properties.html#13689" class="Bound">C</a> <a id="13691" class="Symbol">:</a> <a id="13693" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13698" href="Cubical.Data.Sigma.Properties.html#13651" class="Bound">ℓ''</a><a id="13701" class="Symbol">}</a> <a id="13703" class="Symbol">{</a><a id="13704" href="Cubical.Data.Sigma.Properties.html#13704" class="Bound">D</a> <a id="13706" class="Symbol">:</a> <a id="13708" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="13713" href="Cubical.Data.Sigma.Properties.html#13655" class="Bound">ℓ'''</a><a id="13717" class="Symbol">}</a> <a id="13719" class="Symbol">→</a> <a id="13721" href="Cubical.Data.Sigma.Properties.html#13662" class="Bound">A</a> <a id="13723" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="13725" href="Cubical.Data.Sigma.Properties.html#13689" class="Bound">C</a> <a id="13727" class="Symbol">→</a> <a id="13729" href="Cubical.Data.Sigma.Properties.html#13675" class="Bound">B</a> <a id="13731" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="13733" href="Cubical.Data.Sigma.Properties.html#13704" class="Bound">D</a> <a id="13735" class="Symbol">→</a> <a id="13737" href="Cubical.Data.Sigma.Properties.html#13662" class="Bound">A</a> <a id="13739" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="13741" href="Cubical.Data.Sigma.Properties.html#13675" class="Bound">B</a> <a id="13743" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="13745" href="Cubical.Data.Sigma.Properties.html#13689" class="Bound">C</a> <a id="13747" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="13749" href="Cubical.Data.Sigma.Properties.html#13704" class="Bound">D</a>
<a id="13751" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="13755" href="Cubical.Data.Sigma.Properties.html#13755" class="Bound">eq1</a> <a id="13759" href="Cubical.Data.Sigma.Properties.html#13759" class="Bound">eq2</a> <a id="13763" class="Symbol">=</a>
<a id="13769" href="Cubical.Data.Sigma.Properties.html#1561" class="Function">map-×</a> <a id="13775" class="Symbol">(</a><a id="13776" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13780" href="Cubical.Data.Sigma.Properties.html#13755" class="Bound">eq1</a><a id="13783" class="Symbol">)</a> <a id="13785" class="Symbol">(</a><a id="13786" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13790" href="Cubical.Data.Sigma.Properties.html#13759" class="Bound">eq2</a><a id="13793" class="Symbol">)</a>
<a id="13797" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="13799" class="Keyword">record</a>
<a id="13811" class="Symbol">{</a> <a id="13813" href="Agda.Builtin.Cubical.Glue.html#971" class="Field">equiv-proof</a>
<a id="13832" class="Symbol">=</a> <a id="13834" class="Symbol">λ</a> <a id="13836" class="Symbol">{(</a><a id="13838" href="Cubical.Data.Sigma.Properties.html#13838" class="Bound">c</a> <a id="13840" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="13842" href="Cubical.Data.Sigma.Properties.html#13842" class="Bound">d</a><a id="13843" class="Symbol">)</a> <a id="13845" class="Symbol">→</a> <a id="13847" class="Symbol">((</a><a id="13849" href="Cubical.Data.Sigma.Properties.html#14423" class="Function">eq1⁻</a> <a id="13854" href="Cubical.Data.Sigma.Properties.html#13838" class="Bound">c</a> <a id="13856" class="Symbol">.</a><a id="13857" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13861" class="Symbol">.</a><a id="13862" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a>
<a id="13890" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="13892" href="Cubical.Data.Sigma.Properties.html#14455" class="Function">eq2⁻</a> <a id="13897" href="Cubical.Data.Sigma.Properties.html#13842" class="Bound">d</a> <a id="13899" class="Symbol">.</a><a id="13900" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13904" class="Symbol">.</a><a id="13905" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a><a id="13908" class="Symbol">)</a>
<a id="13936" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="13938" href="Cubical.Data.Sigma.Properties.html#1670" class="Function">≡-×</a> <a id="13942" class="Symbol">(</a><a id="13943" href="Cubical.Data.Sigma.Properties.html#14423" class="Function">eq1⁻</a> <a id="13948" href="Cubical.Data.Sigma.Properties.html#13838" class="Bound">c</a> <a id="13950" class="Symbol">.</a><a id="13951" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="13955" class="Symbol">.</a><a id="13956" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="13959" class="Symbol">)</a>
<a id="13993" class="Symbol">(</a><a id="13994" href="Cubical.Data.Sigma.Properties.html#14455" class="Function">eq2⁻</a> <a id="13999" href="Cubical.Data.Sigma.Properties.html#13842" class="Bound">d</a> <a id="14001" class="Symbol">.</a><a id="14002" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="14006" class="Symbol">.</a><a id="14007" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="14010" class="Symbol">))</a>
<a id="14034" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14036" class="Symbol">λ</a> <a id="14038" class="Symbol">{((</a><a id="14041" href="Cubical.Data.Sigma.Properties.html#14041" class="Bound">a</a> <a id="14043" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14045" href="Cubical.Data.Sigma.Properties.html#14045" class="Bound">b</a><a id="14046" class="Symbol">)</a> <a id="14048" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14050" href="Cubical.Data.Sigma.Properties.html#14050" class="Bound">p</a><a id="14051" class="Symbol">)</a> <a id="14053" class="Symbol">→</a> <a id="14055" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="14062" class="Symbol">(</a><a id="14063" href="Cubical.Data.Sigma.Properties.html#1670" class="Function">≡-×</a> <a id="14067" class="Symbol">(</a><a id="14068" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14073" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="14077" class="Symbol">(</a><a id="14078" href="Cubical.Data.Sigma.Properties.html#14423" class="Function">eq1⁻</a> <a id="14083" href="Cubical.Data.Sigma.Properties.html#13838" class="Bound">c</a> <a id="14085" class="Symbol">.</a><a id="14086" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14090" class="Symbol">(</a><a id="14091" href="Cubical.Data.Sigma.Properties.html#14041" class="Bound">a</a> <a id="14093" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14095" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14100" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="14104" href="Cubical.Data.Sigma.Properties.html#14050" class="Bound">p</a><a id="14105" class="Symbol">)))</a>
<a id="14164" class="Symbol">(</a><a id="14165" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14170" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="14174" class="Symbol">(</a><a id="14175" href="Cubical.Data.Sigma.Properties.html#14455" class="Function">eq2⁻</a> <a id="14180" href="Cubical.Data.Sigma.Properties.html#13842" class="Bound">d</a> <a id="14182" class="Symbol">.</a><a id="14183" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14187" class="Symbol">(</a><a id="14188" href="Cubical.Data.Sigma.Properties.html#14045" class="Bound">b</a> <a id="14190" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14192" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14197" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14201" href="Cubical.Data.Sigma.Properties.html#14050" class="Bound">p</a><a id="14202" class="Symbol">)))</a>
<a id="14254" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14256" class="Symbol">λ</a> <a id="14258" href="Cubical.Data.Sigma.Properties.html#14258" class="Bound">i</a> <a id="14260" class="Symbol">→</a> <a id="14262" href="Cubical.Data.Sigma.Properties.html#1670" class="Function">≡-×</a> <a id="14266" class="Symbol">(</a><a id="14267" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14271" class="Symbol">((</a><a id="14273" href="Cubical.Data.Sigma.Properties.html#14423" class="Function">eq1⁻</a> <a id="14278" href="Cubical.Data.Sigma.Properties.html#13838" class="Bound">c</a> <a id="14280" class="Symbol">.</a><a id="14281" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14285" class="Symbol">(</a><a id="14286" href="Cubical.Data.Sigma.Properties.html#14041" class="Bound">a</a> <a id="14288" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14290" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14295" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="14299" href="Cubical.Data.Sigma.Properties.html#14050" class="Bound">p</a><a id="14300" class="Symbol">))</a> <a id="14303" href="Cubical.Data.Sigma.Properties.html#14258" class="Bound">i</a><a id="14304" class="Symbol">))</a>
<a id="14368" class="Symbol">(</a><a id="14369" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14373" class="Symbol">((</a><a id="14375" href="Cubical.Data.Sigma.Properties.html#14455" class="Function">eq2⁻</a> <a id="14380" href="Cubical.Data.Sigma.Properties.html#13842" class="Bound">d</a> <a id="14382" class="Symbol">.</a><a id="14383" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14387" class="Symbol">(</a><a id="14388" href="Cubical.Data.Sigma.Properties.html#14045" class="Bound">b</a> <a id="14390" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14392" href="Cubical.Foundations.Prelude.html#1430" class="Function">cong</a> <a id="14397" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="14401" href="Cubical.Data.Sigma.Properties.html#14050" class="Bound">p</a><a id="14402" class="Symbol">))</a> <a id="14405" href="Cubical.Data.Sigma.Properties.html#14258" class="Bound">i</a><a id="14406" class="Symbol">)))}}}</a>
<a id="14415" class="Keyword">where</a>
<a id="14423" href="Cubical.Data.Sigma.Properties.html#14423" class="Function">eq1⁻</a> <a id="14428" class="Symbol">=</a> <a id="14430" href="Agda.Builtin.Cubical.Glue.html#971" class="Field">equiv-proof</a> <a id="14442" class="Symbol">(</a><a id="14443" href="Cubical.Data.Sigma.Properties.html#13755" class="Bound">eq1</a> <a id="14447" class="Symbol">.</a><a id="14448" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="14451" class="Symbol">)</a>
<a id="14455" href="Cubical.Data.Sigma.Properties.html#14455" class="Function">eq2⁻</a> <a id="14460" class="Symbol">=</a> <a id="14462" href="Agda.Builtin.Cubical.Glue.html#971" class="Field">equiv-proof</a> <a id="14474" class="Symbol">(</a><a id="14475" href="Cubical.Data.Sigma.Properties.html#13759" class="Bound">eq2</a> <a id="14479" class="Symbol">.</a><a id="14480" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a><a id="14483" class="Symbol">)</a>
<a id="14486" class="Comment">{- Some simple ismorphisms -}</a>
<a id="prodIso"></a><a id="14517" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="14525" class="Symbol">:</a> <a id="14527" class="Symbol">∀</a> <a id="14529" class="Symbol">{</a><a id="14530" href="Cubical.Data.Sigma.Properties.html#14530" class="Bound">ℓ</a> <a id="14532" href="Cubical.Data.Sigma.Properties.html#14532" class="Bound">ℓ'</a> <a id="14535" href="Cubical.Data.Sigma.Properties.html#14535" class="Bound">ℓ''</a> <a id="14539" href="Cubical.Data.Sigma.Properties.html#14539" class="Bound">ℓ'''</a><a id="14543" class="Symbol">}</a> <a id="14545" class="Symbol">{</a><a id="14546" href="Cubical.Data.Sigma.Properties.html#14546" class="Bound">A</a> <a id="14548" class="Symbol">:</a> <a id="14550" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="14555" href="Cubical.Data.Sigma.Properties.html#14530" class="Bound">ℓ</a><a id="14556" class="Symbol">}</a> <a id="14558" class="Symbol">{</a><a id="14559" href="Cubical.Data.Sigma.Properties.html#14559" class="Bound">B</a> <a id="14561" class="Symbol">:</a> <a id="14563" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="14568" href="Cubical.Data.Sigma.Properties.html#14532" class="Bound">ℓ'</a><a id="14570" class="Symbol">}</a> <a id="14572" class="Symbol">{</a><a id="14573" href="Cubical.Data.Sigma.Properties.html#14573" class="Bound">C</a> <a id="14575" class="Symbol">:</a> <a id="14577" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="14582" href="Cubical.Data.Sigma.Properties.html#14535" class="Bound">ℓ''</a><a id="14585" class="Symbol">}</a> <a id="14587" class="Symbol">{</a><a id="14588" href="Cubical.Data.Sigma.Properties.html#14588" class="Bound">D</a> <a id="14590" class="Symbol">:</a> <a id="14592" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="14597" href="Cubical.Data.Sigma.Properties.html#14539" class="Bound">ℓ'''</a><a id="14601" class="Symbol">}</a>
<a id="14610" class="Symbol">→</a> <a id="14612" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="14616" href="Cubical.Data.Sigma.Properties.html#14546" class="Bound">A</a> <a id="14618" href="Cubical.Data.Sigma.Properties.html#14573" class="Bound">C</a>
<a id="14627" class="Symbol">→</a> <a id="14629" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="14633" href="Cubical.Data.Sigma.Properties.html#14559" class="Bound">B</a> <a id="14635" href="Cubical.Data.Sigma.Properties.html#14588" class="Bound">D</a>
<a id="14644" class="Symbol">→</a> <a id="14646" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="14650" class="Symbol">(</a><a id="14651" href="Cubical.Data.Sigma.Properties.html#14546" class="Bound">A</a> <a id="14653" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="14655" href="Cubical.Data.Sigma.Properties.html#14559" class="Bound">B</a><a id="14656" class="Symbol">)</a> <a id="14658" class="Symbol">(</a><a id="14659" href="Cubical.Data.Sigma.Properties.html#14573" class="Bound">C</a> <a id="14661" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="14663" href="Cubical.Data.Sigma.Properties.html#14588" class="Bound">D</a><a id="14664" class="Symbol">)</a>
<a id="14666" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="14674" class="Symbol">(</a><a id="14675" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="14683" href="Cubical.Data.Sigma.Properties.html#14683" class="Bound">iAC</a> <a id="14687" href="Cubical.Data.Sigma.Properties.html#14687" class="Bound">iBD</a><a id="14690" class="Symbol">)</a> <a id="14692" class="Symbol">(</a><a id="14693" href="Cubical.Data.Sigma.Properties.html#14693" class="Bound">a</a> <a id="14695" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14697" href="Cubical.Data.Sigma.Properties.html#14697" class="Bound">b</a><a id="14698" class="Symbol">)</a> <a id="14700" class="Symbol">=</a> <a id="14702" class="Symbol">(</a><a id="14703" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="14711" href="Cubical.Data.Sigma.Properties.html#14683" class="Bound">iAC</a> <a id="14715" href="Cubical.Data.Sigma.Properties.html#14693" class="Bound">a</a><a id="14716" class="Symbol">)</a> <a id="14718" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14720" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="14728" href="Cubical.Data.Sigma.Properties.html#14687" class="Bound">iBD</a> <a id="14732" href="Cubical.Data.Sigma.Properties.html#14697" class="Bound">b</a>
<a id="14734" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="14742" class="Symbol">(</a><a id="14743" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="14751" href="Cubical.Data.Sigma.Properties.html#14751" class="Bound">iAC</a> <a id="14755" href="Cubical.Data.Sigma.Properties.html#14755" class="Bound">iBD</a><a id="14758" class="Symbol">)</a> <a id="14760" class="Symbol">(</a><a id="14761" href="Cubical.Data.Sigma.Properties.html#14761" class="Bound">c</a> <a id="14763" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14765" href="Cubical.Data.Sigma.Properties.html#14765" class="Bound">d</a><a id="14766" class="Symbol">)</a> <a id="14768" class="Symbol">=</a> <a id="14770" class="Symbol">(</a><a id="14771" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="14779" href="Cubical.Data.Sigma.Properties.html#14751" class="Bound">iAC</a> <a id="14783" href="Cubical.Data.Sigma.Properties.html#14761" class="Bound">c</a><a id="14784" class="Symbol">)</a> <a id="14786" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14788" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="14796" href="Cubical.Data.Sigma.Properties.html#14755" class="Bound">iBD</a> <a id="14800" href="Cubical.Data.Sigma.Properties.html#14765" class="Bound">d</a>
<a id="14802" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="14815" class="Symbol">(</a><a id="14816" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="14824" href="Cubical.Data.Sigma.Properties.html#14824" class="Bound">iAC</a> <a id="14828" href="Cubical.Data.Sigma.Properties.html#14828" class="Bound">iBD</a><a id="14831" class="Symbol">)</a> <a id="14833" class="Symbol">(</a><a id="14834" href="Cubical.Data.Sigma.Properties.html#14834" class="Bound">c</a> <a id="14836" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14838" href="Cubical.Data.Sigma.Properties.html#14838" class="Bound">d</a><a id="14839" class="Symbol">)</a> <a id="14841" class="Symbol">=</a> <a id="14843" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="14850" class="Symbol">((</a><a id="14852" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="14865" href="Cubical.Data.Sigma.Properties.html#14824" class="Bound">iAC</a> <a id="14869" href="Cubical.Data.Sigma.Properties.html#14834" class="Bound">c</a><a id="14870" class="Symbol">)</a> <a id="14872" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14874" class="Symbol">(</a><a id="14875" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="14888" href="Cubical.Data.Sigma.Properties.html#14828" class="Bound">iBD</a> <a id="14892" href="Cubical.Data.Sigma.Properties.html#14838" class="Bound">d</a><a id="14893" class="Symbol">))</a>
<a id="14896" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="14908" class="Symbol">(</a><a id="14909" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="14917" href="Cubical.Data.Sigma.Properties.html#14917" class="Bound">iAC</a> <a id="14921" href="Cubical.Data.Sigma.Properties.html#14921" class="Bound">iBD</a><a id="14924" class="Symbol">)</a> <a id="14926" class="Symbol">(</a><a id="14927" href="Cubical.Data.Sigma.Properties.html#14927" class="Bound">a</a> <a id="14929" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14931" href="Cubical.Data.Sigma.Properties.html#14931" class="Bound">b</a><a id="14932" class="Symbol">)</a> <a id="14934" class="Symbol">=</a> <a id="14936" href="Cubical.Data.Sigma.Properties.html#1955" class="Function">ΣPathP</a> <a id="14943" class="Symbol">((</a><a id="14945" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="14957" href="Cubical.Data.Sigma.Properties.html#14917" class="Bound">iAC</a> <a id="14961" href="Cubical.Data.Sigma.Properties.html#14927" class="Bound">a</a><a id="14962" class="Symbol">)</a> <a id="14964" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="14966" class="Symbol">(</a><a id="14967" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="14979" href="Cubical.Data.Sigma.Properties.html#14921" class="Bound">iBD</a> <a id="14983" href="Cubical.Data.Sigma.Properties.html#14931" class="Bound">b</a><a id="14984" class="Symbol">))</a>
<a id="prodEquivToIso"></a><a id="14988" href="Cubical.Data.Sigma.Properties.html#14988" class="Function">prodEquivToIso</a> <a id="15003" class="Symbol">:</a> <a id="15005" class="Symbol">∀</a> <a id="15007" class="Symbol">{</a><a id="15008" href="Cubical.Data.Sigma.Properties.html#15008" class="Bound">ℓ''</a> <a id="15012" href="Cubical.Data.Sigma.Properties.html#15012" class="Bound">ℓ'''</a><a id="15016" class="Symbol">}</a> <a id="15018" class="Symbol">{</a><a id="15019" href="Cubical.Data.Sigma.Properties.html#15019" class="Bound">A</a> <a id="15021" class="Symbol">:</a> <a id="15023" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15028" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="15029" class="Symbol">}</a> <a id="15031" class="Symbol">{</a><a id="15032" href="Cubical.Data.Sigma.Properties.html#15032" class="Bound">B</a> <a id="15034" class="Symbol">:</a> <a id="15036" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15041" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="15043" class="Symbol">}</a> <a id="15045" class="Symbol">{</a><a id="15046" href="Cubical.Data.Sigma.Properties.html#15046" class="Bound">C</a> <a id="15048" class="Symbol">:</a> <a id="15050" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15055" href="Cubical.Data.Sigma.Properties.html#15008" class="Bound">ℓ''</a><a id="15058" class="Symbol">}</a> <a id="15060" class="Symbol">{</a><a id="15061" href="Cubical.Data.Sigma.Properties.html#15061" class="Bound">D</a> <a id="15063" class="Symbol">:</a> <a id="15065" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15070" href="Cubical.Data.Sigma.Properties.html#15012" class="Bound">ℓ'''</a><a id="15074" class="Symbol">}</a>
<a id="15078" class="Symbol">→</a> <a id="15080" class="Symbol">(</a><a id="15081" href="Cubical.Data.Sigma.Properties.html#15081" class="Bound">e</a> <a id="15083" class="Symbol">:</a> <a id="15085" href="Cubical.Data.Sigma.Properties.html#15019" class="Bound">A</a> <a id="15087" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="15089" href="Cubical.Data.Sigma.Properties.html#15046" class="Bound">C</a><a id="15090" class="Symbol">)(</a><a id="15092" href="Cubical.Data.Sigma.Properties.html#15092" class="Bound">e'</a> <a id="15095" class="Symbol">:</a> <a id="15097" href="Cubical.Data.Sigma.Properties.html#15032" class="Bound">B</a> <a id="15099" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="15101" href="Cubical.Data.Sigma.Properties.html#15061" class="Bound">D</a><a id="15102" class="Symbol">)</a>
<a id="15106" class="Symbol">→</a> <a id="15108" href="Cubical.Data.Sigma.Properties.html#14517" class="Function">prodIso</a> <a id="15116" class="Symbol">(</a><a id="15117" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15128" href="Cubical.Data.Sigma.Properties.html#15081" class="Bound">e</a><a id="15129" class="Symbol">)</a> <a id="15131" class="Symbol">(</a><a id="15132" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15143" href="Cubical.Data.Sigma.Properties.html#15092" class="Bound">e'</a><a id="15145" class="Symbol">)</a> <a id="15147" href="Agda.Builtin.Cubical.Path.html#388" class="Function Operator">≡</a> <a id="15149" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15160" class="Symbol">(</a><a id="15161" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="15165" href="Cubical.Data.Sigma.Properties.html#15081" class="Bound">e</a> <a id="15167" href="Cubical.Data.Sigma.Properties.html#15092" class="Bound">e'</a><a id="15169" class="Symbol">)</a>
<a id="15171" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="15179" class="Symbol">(</a><a id="15180" href="Cubical.Data.Sigma.Properties.html#14988" class="Function">prodEquivToIso</a> <a id="15195" href="Cubical.Data.Sigma.Properties.html#15195" class="Bound">e</a> <a id="15197" href="Cubical.Data.Sigma.Properties.html#15197" class="Bound">e'</a> <a id="15200" href="Cubical.Data.Sigma.Properties.html#15200" class="Bound">i</a><a id="15201" class="Symbol">)</a> <a id="15203" class="Symbol">=</a> <a id="15205" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="15213" class="Symbol">(</a><a id="15214" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15225" class="Symbol">(</a><a id="15226" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="15230" href="Cubical.Data.Sigma.Properties.html#15195" class="Bound">e</a> <a id="15232" href="Cubical.Data.Sigma.Properties.html#15197" class="Bound">e'</a><a id="15234" class="Symbol">))</a>
<a id="15237" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="15245" class="Symbol">(</a><a id="15246" href="Cubical.Data.Sigma.Properties.html#14988" class="Function">prodEquivToIso</a> <a id="15261" href="Cubical.Data.Sigma.Properties.html#15261" class="Bound">e</a> <a id="15263" href="Cubical.Data.Sigma.Properties.html#15263" class="Bound">e'</a> <a id="15266" href="Cubical.Data.Sigma.Properties.html#15266" class="Bound">i</a><a id="15267" class="Symbol">)</a> <a id="15269" class="Symbol">=</a> <a id="15271" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="15279" class="Symbol">(</a><a id="15280" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15291" class="Symbol">(</a><a id="15292" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="15296" href="Cubical.Data.Sigma.Properties.html#15261" class="Bound">e</a> <a id="15298" href="Cubical.Data.Sigma.Properties.html#15263" class="Bound">e'</a><a id="15300" class="Symbol">))</a>
<a id="15303" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="15316" class="Symbol">(</a><a id="15317" href="Cubical.Data.Sigma.Properties.html#14988" class="Function">prodEquivToIso</a> <a id="15332" href="Cubical.Data.Sigma.Properties.html#15332" class="Bound">e</a> <a id="15334" href="Cubical.Data.Sigma.Properties.html#15334" class="Bound">e'</a> <a id="15337" href="Cubical.Data.Sigma.Properties.html#15337" class="Bound">i</a><a id="15338" class="Symbol">)</a> <a id="15340" class="Symbol">=</a> <a id="15342" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="15355" class="Symbol">(</a><a id="15356" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15367" class="Symbol">(</a><a id="15368" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="15372" href="Cubical.Data.Sigma.Properties.html#15332" class="Bound">e</a> <a id="15374" href="Cubical.Data.Sigma.Properties.html#15334" class="Bound">e'</a><a id="15376" class="Symbol">))</a>
<a id="15379" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="15391" class="Symbol">(</a><a id="15392" href="Cubical.Data.Sigma.Properties.html#14988" class="Function">prodEquivToIso</a> <a id="15407" href="Cubical.Data.Sigma.Properties.html#15407" class="Bound">e</a> <a id="15409" href="Cubical.Data.Sigma.Properties.html#15409" class="Bound">e'</a> <a id="15412" href="Cubical.Data.Sigma.Properties.html#15412" class="Bound">i</a><a id="15413" class="Symbol">)</a> <a id="15415" class="Symbol">=</a> <a id="15417" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="15429" class="Symbol">(</a><a id="15430" href="Cubical.Foundations.Equiv.html#3099" class="Function">equivToIso</a> <a id="15441" class="Symbol">(</a><a id="15442" href="Cubical.Data.Sigma.Properties.html#13642" class="Function">≃-×</a> <a id="15446" href="Cubical.Data.Sigma.Properties.html#15407" class="Bound">e</a> <a id="15448" href="Cubical.Data.Sigma.Properties.html#15409" class="Bound">e'</a><a id="15450" class="Symbol">))</a>
<a id="toProdIso"></a><a id="15454" href="Cubical.Data.Sigma.Properties.html#15454" class="Function">toProdIso</a> <a id="15464" class="Symbol">:</a> <a id="15466" class="Symbol">{</a><a id="15467" href="Cubical.Data.Sigma.Properties.html#15467" class="Bound">B</a> <a id="15469" href="Cubical.Data.Sigma.Properties.html#15469" class="Bound">C</a> <a id="15471" class="Symbol">:</a> <a id="15473" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a> <a id="15475" class="Symbol">→</a> <a id="15477" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15482" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="15483" class="Symbol">}</a>
<a id="15495" class="Symbol">→</a> <a id="15497" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="15501" class="Symbol">((</a><a id="15503" href="Cubical.Data.Sigma.Properties.html#15503" class="Bound">a</a> <a id="15505" class="Symbol">:</a> <a id="15507" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="15508" class="Symbol">)</a> <a id="15510" class="Symbol">→</a> <a id="15512" href="Cubical.Data.Sigma.Properties.html#15467" class="Bound">B</a> <a id="15514" href="Cubical.Data.Sigma.Properties.html#15503" class="Bound">a</a> <a id="15516" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="15518" href="Cubical.Data.Sigma.Properties.html#15469" class="Bound">C</a> <a id="15520" href="Cubical.Data.Sigma.Properties.html#15503" class="Bound">a</a><a id="15521" class="Symbol">)</a> <a id="15523" class="Symbol">(((</a><a id="15526" href="Cubical.Data.Sigma.Properties.html#15526" class="Bound">a</a> <a id="15528" class="Symbol">:</a> <a id="15530" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="15531" class="Symbol">)</a> <a id="15533" class="Symbol">→</a> <a id="15535" href="Cubical.Data.Sigma.Properties.html#15467" class="Bound">B</a> <a id="15537" href="Cubical.Data.Sigma.Properties.html#15526" class="Bound">a</a><a id="15538" class="Symbol">)</a> <a id="15540" href="Cubical.Data.Sigma.Base.html#461" class="Function Operator">×</a> <a id="15542" class="Symbol">((</a><a id="15544" href="Cubical.Data.Sigma.Properties.html#15544" class="Bound">a</a> <a id="15546" class="Symbol">:</a> <a id="15548" href="Cubical.Data.Sigma.Properties.html#1318" class="Generalizable">A</a><a id="15549" class="Symbol">)</a> <a id="15551" class="Symbol">→</a> <a id="15553" href="Cubical.Data.Sigma.Properties.html#15469" class="Bound">C</a> <a id="15555" href="Cubical.Data.Sigma.Properties.html#15544" class="Bound">a</a><a id="15556" class="Symbol">))</a>
<a id="15559" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="15567" href="Cubical.Data.Sigma.Properties.html#15454" class="Function">toProdIso</a> <a id="15577" class="Symbol">=</a> <a id="15579" class="Symbol">λ</a> <a id="15581" href="Cubical.Data.Sigma.Properties.html#15581" class="Bound">f</a> <a id="15583" class="Symbol">→</a> <a id="15585" class="Symbol">(λ</a> <a id="15588" href="Cubical.Data.Sigma.Properties.html#15588" class="Bound">a</a> <a id="15590" class="Symbol">→</a> <a id="15592" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="15596" class="Symbol">(</a><a id="15597" href="Cubical.Data.Sigma.Properties.html#15581" class="Bound">f</a> <a id="15599" href="Cubical.Data.Sigma.Properties.html#15588" class="Bound">a</a><a id="15600" class="Symbol">))</a> <a id="15603" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15605" class="Symbol">(λ</a> <a id="15608" href="Cubical.Data.Sigma.Properties.html#15608" class="Bound">a</a> <a id="15610" class="Symbol">→</a> <a id="15612" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="15616" class="Symbol">(</a><a id="15617" href="Cubical.Data.Sigma.Properties.html#15581" class="Bound">f</a> <a id="15619" href="Cubical.Data.Sigma.Properties.html#15608" class="Bound">a</a><a id="15620" class="Symbol">))</a>
<a id="15623" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="15631" href="Cubical.Data.Sigma.Properties.html#15454" class="Function">toProdIso</a> <a id="15641" class="Symbol">(</a><a id="15642" href="Cubical.Data.Sigma.Properties.html#15642" class="Bound">f</a> <a id="15644" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15646" href="Cubical.Data.Sigma.Properties.html#15646" class="Bound">g</a><a id="15647" class="Symbol">)</a> <a id="15649" class="Symbol">=</a> <a id="15651" class="Symbol">λ</a> <a id="15653" href="Cubical.Data.Sigma.Properties.html#15653" class="Bound">a</a> <a id="15655" class="Symbol">→</a> <a id="15657" class="Symbol">(</a><a id="15658" href="Cubical.Data.Sigma.Properties.html#15642" class="Bound">f</a> <a id="15660" href="Cubical.Data.Sigma.Properties.html#15653" class="Bound">a</a><a id="15661" class="Symbol">)</a> <a id="15663" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15665" class="Symbol">(</a><a id="15666" href="Cubical.Data.Sigma.Properties.html#15646" class="Bound">g</a> <a id="15668" href="Cubical.Data.Sigma.Properties.html#15653" class="Bound">a</a><a id="15669" class="Symbol">)</a>
<a id="15671" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="15684" href="Cubical.Data.Sigma.Properties.html#15454" class="Function">toProdIso</a> <a id="15694" class="Symbol">(</a><a id="15695" href="Cubical.Data.Sigma.Properties.html#15695" class="Bound">f</a> <a id="15697" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15699" href="Cubical.Data.Sigma.Properties.html#15699" class="Bound">g</a><a id="15700" class="Symbol">)</a> <a id="15702" class="Symbol">=</a> <a id="15704" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="15709" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="15721" href="Cubical.Data.Sigma.Properties.html#15454" class="Function">toProdIso</a> <a id="15731" href="Cubical.Data.Sigma.Properties.html#15731" class="Bound">b</a> <a id="15733" class="Symbol">=</a> <a id="15735" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="15741" class="Keyword">module</a> <a id="15748" href="Cubical.Data.Sigma.Properties.html#15748" class="Module">_</a> <a id="15750" class="Symbol">{</a><a id="15751" href="Cubical.Data.Sigma.Properties.html#15751" class="Bound">A</a> <a id="15753" class="Symbol">:</a> <a id="15755" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15760" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="15761" class="Symbol">}</a> <a id="15763" class="Symbol">{</a><a id="15764" href="Cubical.Data.Sigma.Properties.html#15764" class="Bound">B</a> <a id="15766" class="Symbol">:</a> <a id="15768" href="Cubical.Data.Sigma.Properties.html#15751" class="Bound">A</a> <a id="15770" class="Symbol">→</a> <a id="15772" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15777" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="15779" class="Symbol">}</a> <a id="15781" class="Symbol">{</a><a id="15782" href="Cubical.Data.Sigma.Properties.html#15782" class="Bound">C</a> <a id="15784" class="Symbol">:</a> <a id="15786" class="Symbol">∀</a> <a id="15788" href="Cubical.Data.Sigma.Properties.html#15788" class="Bound">a</a> <a id="15790" class="Symbol">→</a> <a id="15792" href="Cubical.Data.Sigma.Properties.html#15764" class="Bound">B</a> <a id="15794" href="Cubical.Data.Sigma.Properties.html#15788" class="Bound">a</a> <a id="15796" class="Symbol">→</a> <a id="15798" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="15803" href="Cubical.Data.Sigma.Properties.html#1302" class="Generalizable">ℓ''</a><a id="15806" class="Symbol">}</a> <a id="15808" class="Keyword">where</a>
<a id="15816" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a> <a id="15825" class="Symbol">:</a> <a id="15827" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="15831" class="Symbol">((</a><a id="15833" href="Cubical.Data.Sigma.Properties.html#15833" class="Symbol">(</a><a id="15834" href="Cubical.Data.Sigma.Properties.html#15834" class="Bound">a</a> <a id="15836" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15838" href="Cubical.Data.Sigma.Properties.html#15838" class="Bound">b</a><a id="15839" href="Cubical.Data.Sigma.Properties.html#15833" class="Symbol">)</a> <a id="15841" class="Symbol">:</a> <a id="15843" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="15845" href="Cubical.Data.Sigma.Properties.html#15751" class="Bound">A</a> <a id="15847" href="Cubical.Data.Sigma.Properties.html#15764" class="Bound">B</a><a id="15848" class="Symbol">)</a> <a id="15850" class="Symbol">→</a> <a id="15852" href="Cubical.Data.Sigma.Properties.html#15782" class="Bound">C</a> <a id="15854" href="Cubical.Data.Sigma.Properties.html#15834" class="Bound">a</a> <a id="15856" href="Cubical.Data.Sigma.Properties.html#15838" class="Bound">b</a><a id="15857" class="Symbol">)</a> <a id="15859" class="Symbol">((</a><a id="15861" href="Cubical.Data.Sigma.Properties.html#15861" class="Bound">a</a> <a id="15863" class="Symbol">:</a> <a id="15865" href="Cubical.Data.Sigma.Properties.html#15751" class="Bound">A</a><a id="15866" class="Symbol">)</a> <a id="15868" class="Symbol">→</a> <a id="15870" class="Symbol">(</a><a id="15871" href="Cubical.Data.Sigma.Properties.html#15871" class="Bound">b</a> <a id="15873" class="Symbol">:</a> <a id="15875" href="Cubical.Data.Sigma.Properties.html#15764" class="Bound">B</a> <a id="15877" href="Cubical.Data.Sigma.Properties.html#15861" class="Bound">a</a><a id="15878" class="Symbol">)</a> <a id="15880" class="Symbol">→</a> <a id="15882" href="Cubical.Data.Sigma.Properties.html#15782" class="Bound">C</a> <a id="15884" href="Cubical.Data.Sigma.Properties.html#15861" class="Bound">a</a> <a id="15886" href="Cubical.Data.Sigma.Properties.html#15871" class="Bound">b</a><a id="15887" class="Symbol">)</a>
<a id="15891" href="Cubical.Foundations.Isomorphism.html#885" class="Field">Iso.fun</a> <a id="15899" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a> <a id="15908" href="Cubical.Data.Sigma.Properties.html#15908" class="Bound">f</a> <a id="15910" href="Cubical.Data.Sigma.Properties.html#15910" class="Bound">a</a> <a id="15912" href="Cubical.Data.Sigma.Properties.html#15912" class="Bound">b</a> <a id="15914" class="Symbol">=</a> <a id="15916" href="Cubical.Data.Sigma.Properties.html#15908" class="Bound">f</a> <a id="15918" class="Symbol">(</a><a id="15919" href="Cubical.Data.Sigma.Properties.html#15910" class="Bound">a</a> <a id="15921" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="15923" href="Cubical.Data.Sigma.Properties.html#15912" class="Bound">b</a><a id="15924" class="Symbol">)</a>
<a id="15928" href="Cubical.Foundations.Isomorphism.html#901" class="Field">Iso.inv</a> <a id="15936" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a> <a id="15945" href="Cubical.Data.Sigma.Properties.html#15945" class="Bound">f</a> <a id="15947" href="Cubical.Data.Sigma.Properties.html#15947" class="Bound">a</a> <a id="15949" class="Symbol">=</a> <a id="15951" href="Cubical.Data.Sigma.Properties.html#15945" class="Bound">f</a> <a id="15953" class="Symbol">(</a><a id="15954" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="15958" href="Cubical.Data.Sigma.Properties.html#15947" class="Bound">a</a><a id="15959" class="Symbol">)</a> <a id="15961" class="Symbol">(</a><a id="15962" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="15966" href="Cubical.Data.Sigma.Properties.html#15947" class="Bound">a</a><a id="15967" class="Symbol">)</a>
<a id="15971" href="Cubical.Foundations.Isomorphism.html#917" class="Field">Iso.rightInv</a> <a id="15984" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a> <a id="15993" href="Cubical.Data.Sigma.Properties.html#15993" class="Bound">a</a> <a id="15995" class="Symbol">=</a> <a id="15997" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="16004" href="Cubical.Foundations.Isomorphism.html#948" class="Field">Iso.leftInv</a> <a id="16016" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a> <a id="16025" href="Cubical.Data.Sigma.Properties.html#16025" class="Bound">f</a> <a id="16027" class="Symbol">=</a> <a id="16029" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="16037" class="Keyword">unquoteDecl</a> <a id="16049" href="Cubical.Data.Sigma.Properties.html#16049" class="Function">curryEquiv</a> <a id="16060" class="Symbol">=</a> <a id="16062" href="Cubical.Reflection.StrictEquiv.html#1918" class="Function">declStrictIsoToEquiv</a> <a id="16083" href="Cubical.Data.Sigma.Properties.html#16049" class="Function">curryEquiv</a> <a id="16094" href="Cubical.Data.Sigma.Properties.html#15816" class="Function">curryIso</a>
<a id="16104" class="Comment">-- Sigma type with empty base</a>
<a id="16135" class="Keyword">module</a> <a id="16142" href="Cubical.Data.Sigma.Properties.html#16142" class="Module">_</a> <a id="16144" class="Symbol">(</a><a id="16145" href="Cubical.Data.Sigma.Properties.html#16145" class="Bound">A</a> <a id="16147" class="Symbol">:</a> <a id="16149" href="Cubical.Data.Empty.Base.html#145" class="Datatype">⊥</a> <a id="16151" class="Symbol">→</a> <a id="16153" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="16158" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="16159" class="Symbol">)</a> <a id="16161" class="Keyword">where</a>
<a id="16170" class="Keyword">open</a> <a id="16175" href="Cubical.Foundations.Isomorphism.html#773" class="Module">Iso</a>
<a id="16182" href="Cubical.Data.Sigma.Properties.html#16182" class="Function">ΣEmptyIso</a> <a id="16192" class="Symbol">:</a> <a id="16194" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="16198" class="Symbol">(</a><a id="16199" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="16201" href="Cubical.Data.Empty.Base.html#145" class="Datatype">⊥</a> <a id="16203" href="Cubical.Data.Sigma.Properties.html#16145" class="Bound">A</a><a id="16204" class="Symbol">)</a> <a id="16206" href="Cubical.Data.Empty.Base.html#145" class="Datatype">⊥</a>
<a id="16210" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="16214" href="Cubical.Data.Sigma.Properties.html#16182" class="Function">ΣEmptyIso</a> <a id="16224" class="Symbol">(</a><a id="16225" href="Cubical.Data.Sigma.Properties.html#16225" class="Bound">*</a> <a id="16227" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16229" class="Symbol">_)</a> <a id="16232" class="Symbol">=</a> <a id="16234" href="Cubical.Data.Sigma.Properties.html#16225" class="Bound">*</a>
<a id="16239" href="Cubical.Data.Sigma.Properties.html#16239" class="Function">ΣEmpty</a> <a id="16246" class="Symbol">:</a> <a id="16248" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="16250" href="Cubical.Data.Empty.Base.html#145" class="Datatype">⊥</a> <a id="16252" href="Cubical.Data.Sigma.Properties.html#16145" class="Bound">A</a> <a id="16254" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="16256" href="Cubical.Data.Empty.Base.html#145" class="Datatype">⊥</a>
<a id="16260" href="Cubical.Data.Sigma.Properties.html#16239" class="Function">ΣEmpty</a> <a id="16267" class="Symbol">=</a> <a id="16269" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="16280" href="Cubical.Data.Sigma.Properties.html#16182" class="Function">ΣEmptyIso</a>
<a id="16291" class="Comment">-- fiber of projection map</a>
<a id="16319" class="Keyword">module</a> <a id="16326" href="Cubical.Data.Sigma.Properties.html#16326" class="Module">_</a>
<a id="16330" class="Symbol">(</a><a id="16331" href="Cubical.Data.Sigma.Properties.html#16331" class="Bound">A</a> <a id="16333" class="Symbol">:</a> <a id="16335" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="16340" href="Cubical.Data.Sigma.Properties.html#1297" class="Generalizable">ℓ</a><a id="16341" class="Symbol">)</a>
<a id="16345" class="Symbol">(</a><a id="16346" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16348" class="Symbol">:</a> <a id="16350" href="Cubical.Data.Sigma.Properties.html#16331" class="Bound">A</a> <a id="16352" class="Symbol">→</a> <a id="16354" href="Agda.Primitive.html#326" class="Primitive">Type</a> <a id="16359" href="Cubical.Data.Sigma.Properties.html#1299" class="Generalizable">ℓ'</a><a id="16361" class="Symbol">)</a> <a id="16363" class="Keyword">where</a>
<a id="16372" class="Keyword">private</a>
<a id="16384" href="Cubical.Data.Sigma.Properties.html#16384" class="Function">proj</a> <a id="16389" class="Symbol">:</a> <a id="16391" href="Agda.Builtin.Sigma.html#166" class="Record">Σ</a> <a id="16393" href="Cubical.Data.Sigma.Properties.html#16331" class="Bound">A</a> <a id="16395" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16397" class="Symbol">→</a> <a id="16399" href="Cubical.Data.Sigma.Properties.html#16331" class="Bound">A</a>
<a id="16405" href="Cubical.Data.Sigma.Properties.html#16384" class="Function">proj</a> <a id="16410" class="Symbol">(</a><a id="16411" href="Cubical.Data.Sigma.Properties.html#16411" class="Bound">a</a> <a id="16413" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16415" href="Cubical.Data.Sigma.Properties.html#16415" class="Bound">b</a><a id="16416" class="Symbol">)</a> <a id="16418" class="Symbol">=</a> <a id="16420" href="Cubical.Data.Sigma.Properties.html#16411" class="Bound">a</a>
<a id="16425" class="Keyword">module</a> <a id="16432" href="Cubical.Data.Sigma.Properties.html#16432" class="Module">_</a>
<a id="16438" class="Symbol">(</a><a id="16439" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a> <a id="16441" class="Symbol">:</a> <a id="16443" href="Cubical.Data.Sigma.Properties.html#16331" class="Bound">A</a><a id="16444" class="Symbol">)</a> <a id="16446" class="Keyword">where</a>
<a id="16457" class="Keyword">open</a> <a id="16462" href="Cubical.Foundations.Isomorphism.html#773" class="Module">Iso</a>
<a id="16471" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16484" class="Symbol">:</a> <a id="16486" href="Cubical.Foundations.Isomorphism.html#773" class="Record">Iso</a> <a id="16490" class="Symbol">(</a><a id="16491" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16493" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a><a id="16494" class="Symbol">)</a> <a id="16496" class="Symbol">(</a><a id="16497" href="Cubical.Foundations.Equiv.Base.html#253" class="Function">fiber</a> <a id="16503" href="Cubical.Data.Sigma.Properties.html#16384" class="Function">proj</a> <a id="16508" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a><a id="16509" class="Symbol">)</a>
<a id="16515" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16528" class="Symbol">.</a><a id="16529" href="Cubical.Foundations.Isomorphism.html#885" class="Field">fun</a> <a id="16533" href="Cubical.Data.Sigma.Properties.html#16533" class="Bound">b</a> <a id="16535" class="Symbol">=</a> <a id="16537" class="Symbol">(</a><a id="16538" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a> <a id="16540" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16542" href="Cubical.Data.Sigma.Properties.html#16533" class="Bound">b</a><a id="16543" class="Symbol">)</a> <a id="16545" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16547" href="Cubical.Foundations.Prelude.html#915" class="Function">refl</a>
<a id="16556" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16569" class="Symbol">.</a><a id="16570" href="Cubical.Foundations.Isomorphism.html#901" class="Field">inv</a> <a id="16574" class="Symbol">((</a><a id="16576" href="Cubical.Data.Sigma.Properties.html#16576" class="Bound">a'</a> <a id="16579" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16581" href="Cubical.Data.Sigma.Properties.html#16581" class="Bound">b'</a><a id="16583" class="Symbol">)</a> <a id="16585" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16587" href="Cubical.Data.Sigma.Properties.html#16587" class="Bound">p</a><a id="16588" class="Symbol">)</a> <a id="16590" class="Symbol">=</a> <a id="16592" href="Cubical.Foundations.Prelude.html#9128" class="Function">subst</a> <a id="16598" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16600" href="Cubical.Data.Sigma.Properties.html#16587" class="Bound">p</a> <a id="16602" href="Cubical.Data.Sigma.Properties.html#16581" class="Bound">b'</a>
<a id="16609" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16622" class="Symbol">.</a><a id="16623" href="Cubical.Foundations.Isomorphism.html#948" class="Field">leftInv</a> <a id="16631" href="Cubical.Data.Sigma.Properties.html#16631" class="Bound">b</a> <a id="16633" href="Cubical.Data.Sigma.Properties.html#16633" class="Bound">i</a> <a id="16635" class="Symbol">=</a> <a id="16637" href="Cubical.Foundations.Prelude.html#9390" class="Function">substRefl</a> <a id="16647" class="Symbol">{</a><a id="16648" class="Argument">B</a> <a id="16650" class="Symbol">=</a> <a id="16652" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a><a id="16653" class="Symbol">}</a> <a id="16655" href="Cubical.Data.Sigma.Properties.html#16631" class="Bound">b</a> <a id="16657" href="Cubical.Data.Sigma.Properties.html#16633" class="Bound">i</a>
<a id="16663" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16676" class="Symbol">.</a><a id="16677" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="16686" class="Symbol">(_</a> <a id="16689" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16691" href="Cubical.Data.Sigma.Properties.html#16691" class="Bound">p</a><a id="16692" class="Symbol">)</a> <a id="16694" href="Cubical.Data.Sigma.Properties.html#16694" class="Bound">i</a> <a id="16696" class="Symbol">.</a><a id="16697" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="16701" class="Symbol">.</a><a id="16702" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="16706" class="Symbol">=</a> <a id="16708" href="Cubical.Data.Sigma.Properties.html#16691" class="Bound">p</a> <a id="16710" class="Symbol">(</a><a id="16711" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="16713" href="Cubical.Data.Sigma.Properties.html#16694" class="Bound">i</a><a id="16714" class="Symbol">)</a>
<a id="16720" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16733" class="Symbol">.</a><a id="16734" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="16743" class="Symbol">((_</a> <a id="16747" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16749" href="Cubical.Data.Sigma.Properties.html#16749" class="Bound">b'</a><a id="16751" class="Symbol">)</a> <a id="16753" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16755" href="Cubical.Data.Sigma.Properties.html#16755" class="Bound">p</a><a id="16756" class="Symbol">)</a> <a id="16758" href="Cubical.Data.Sigma.Properties.html#16758" class="Bound">i</a> <a id="16760" class="Symbol">.</a><a id="16761" href="Agda.Builtin.Sigma.html#252" class="Field">fst</a> <a id="16765" class="Symbol">.</a><a id="16766" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="16770" class="Symbol">=</a> <a id="16772" href="Cubical.Foundations.Prelude.html#9494" class="Function">subst-filler</a> <a id="16785" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16787" href="Cubical.Data.Sigma.Properties.html#16755" class="Bound">p</a> <a id="16789" href="Cubical.Data.Sigma.Properties.html#16749" class="Bound">b'</a> <a id="16792" class="Symbol">(</a><a id="16793" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="16795" href="Cubical.Data.Sigma.Properties.html#16758" class="Bound">i</a><a id="16796" class="Symbol">)</a>
<a id="16802" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a> <a id="16815" class="Symbol">.</a><a id="16816" href="Cubical.Foundations.Isomorphism.html#917" class="Field">rightInv</a> <a id="16825" class="Symbol">(_</a> <a id="16828" href="Agda.Builtin.Sigma.html#236" class="InductiveConstructor Operator">,</a> <a id="16830" href="Cubical.Data.Sigma.Properties.html#16830" class="Bound">p</a><a id="16831" class="Symbol">)</a> <a id="16833" href="Cubical.Data.Sigma.Properties.html#16833" class="Bound">i</a> <a id="16835" class="Symbol">.</a><a id="16836" href="Agda.Builtin.Sigma.html#264" class="Field">snd</a> <a id="16840" href="Cubical.Data.Sigma.Properties.html#16840" class="Bound">j</a> <a id="16842" class="Symbol">=</a> <a id="16844" href="Cubical.Data.Sigma.Properties.html#16830" class="Bound">p</a> <a id="16846" class="Symbol">(</a><a id="16847" href="Cubical.Core.Primitives.html#539" class="Primitive Operator">~</a> <a id="16849" href="Cubical.Data.Sigma.Properties.html#16833" class="Bound">i</a> <a id="16851" href="Cubical.Core.Primitives.html#490" class="Primitive Operator">∨</a> <a id="16853" href="Cubical.Data.Sigma.Properties.html#16840" class="Bound">j</a><a id="16854" class="Symbol">)</a>
<a id="16861" href="Cubical.Data.Sigma.Properties.html#16861" class="Function">fiberProjEquiv</a> <a id="16876" class="Symbol">:</a> <a id="16878" href="Cubical.Data.Sigma.Properties.html#16346" class="Bound">B</a> <a id="16880" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a> <a id="16882" href="Agda.Builtin.Cubical.Glue.html#1051" class="Function Operator">≃</a> <a id="16884" href="Cubical.Foundations.Equiv.Base.html#253" class="Function">fiber</a> <a id="16890" href="Cubical.Data.Sigma.Properties.html#16384" class="Function">proj</a> <a id="16895" href="Cubical.Data.Sigma.Properties.html#16439" class="Bound">a</a>
<a id="16901" href="Cubical.Data.Sigma.Properties.html#16861" class="Function">fiberProjEquiv</a> <a id="16916" class="Symbol">=</a> <a id="16918" href="Cubical.Foundations.Isomorphism.html#3127" class="Function">isoToEquiv</a> <a id="16929" href="Cubical.Data.Sigma.Properties.html#16471" class="Function">fiberProjIso</a>
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