mirror of
https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git
synced 2024-05-31 07:28:34 +02:00
207 lines
109 KiB
HTML
207 lines
109 KiB
HTML
|
<!DOCTYPE HTML>
|
|||
|
|
<html><head><meta charset="utf-8"><title>Relation.Binary.PropositionalEquality.Properties</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="74" class="Comment">-- The Agda standard library</a>
|
|||
|
|
<a id="103" class="Comment">--</a>
|
|||
|
|
<a id="106" class="Comment">-- Propositional equality</a>
|
|||
|
|
<a id="132" class="Comment">--</a>
|
|||
|
|
<a id="135" class="Comment">-- This file contains some core properies of propositional equality</a>
|
|||
|
|
<a id="203" class="Comment">-- which are re-exported by Relation.Binary.PropositionalEquality. They</a>
|
|||
|
|
<a id="275" class="Comment">-- are ``equality rearrangement'' lemmas.</a>
|
|||
|
|
<a id="317" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
|
|||
|
|
<a id="391" class="Symbol">{-#</a> <a id="395" class="Keyword">OPTIONS</a> <a id="403" class="Pragma">--cubical-compatible</a> <a id="424" class="Pragma">--safe</a> <a id="431" class="Symbol">#-}</a>
|
|||
|
|
|
|||
|
|
<a id="436" class="Keyword">module</a> <a id="443" href="Relation.Binary.PropositionalEquality.Properties.html" class="Module">Relation.Binary.PropositionalEquality.Properties</a> <a id="492" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="499" class="Keyword">open</a> <a id="504" class="Keyword">import</a> <a id="511" href="Function.Base.html" class="Module">Function.Base</a> <a id="525" class="Keyword">using</a> <a id="531" class="Symbol">(</a><a id="532" href="Function.Base.html#704" class="Function">id</a><a id="534" class="Symbol">;</a> <a id="536" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="539" class="Symbol">)</a>
|
|||
|
|
<a id="541" class="Keyword">open</a> <a id="546" class="Keyword">import</a> <a id="553" href="Level.html" class="Module">Level</a> <a id="559" class="Keyword">using</a> <a id="565" class="Symbol">(</a><a id="566" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="571" class="Symbol">)</a>
|
|||
|
|
<a id="573" class="Keyword">open</a> <a id="578" class="Keyword">import</a> <a id="585" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a>
|
|||
|
|
<a id="611" class="Keyword">using</a> <a id="617" class="Symbol">(</a><a id="618" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="624" class="Symbol">;</a> <a id="626" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a><a id="635" class="Symbol">;</a> <a id="637" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a><a id="645" class="Symbol">;</a> <a id="647" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="652" class="Symbol">)</a>
|
|||
|
|
<a id="654" class="Keyword">open</a> <a id="659" class="Keyword">import</a> <a id="666" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
|
|||
|
|
<a id="695" class="Keyword">using</a> <a id="701" class="Symbol">(</a><a id="702" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="715" class="Symbol">;</a> <a id="717" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a><a id="733" class="Symbol">;</a> <a id="735" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="745" class="Symbol">;</a> <a id="747" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="761" class="Symbol">)</a>
|
|||
|
|
<a id="763" class="Keyword">open</a> <a id="768" class="Keyword">import</a> <a id="775" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
|
|||
|
|
<a id="805" class="Keyword">using</a> <a id="811" class="Symbol">(</a><a id="812" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="821" class="Symbol">;</a> <a id="823" href="Relation.Binary.Definitions.html#6469" class="Function">DecidableEquality</a><a id="840" class="Symbol">)</a>
|
|||
|
|
<a id="842" class="Keyword">import</a> <a id="849" href="Relation.Binary.Properties.Setoid.html" class="Module">Relation.Binary.Properties.Setoid</a> <a id="883" class="Symbol">as</a> <a id="886" class="Module">Setoid</a>
|
|||
|
|
<a id="893" class="Keyword">open</a> <a id="898" class="Keyword">import</a> <a id="905" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a>
|
|||
|
|
<a id="948" class="Keyword">open</a> <a id="953" class="Keyword">import</a> <a id="960" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="975" class="Keyword">using</a> <a id="981" class="Symbol">(</a><a id="982" href="Relation.Unary.html#1232" class="Function">Pred</a><a id="986" class="Symbol">)</a>
|
|||
|
|
<a id="988" class="Keyword">open</a> <a id="993" class="Keyword">import</a> <a id="1000" href="Relation.Binary.Reasoning.Syntax.html" class="Module">Relation.Binary.Reasoning.Syntax</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
<a id="1035" class="Keyword">private</a>
|
|||
|
|
<a id="1045" class="Keyword">variable</a>
|
|||
|
|
<a id="1058" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a> <a id="1060" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a> <a id="1062" href="Relation.Binary.PropositionalEquality.Properties.html#1062" class="Generalizable">c</a> <a id="1064" href="Relation.Binary.PropositionalEquality.Properties.html#1064" class="Generalizable">p</a> <a id="1066" class="Symbol">:</a> <a id="1068" href="Agda.Primitive.html#742" class="Postulate">Level</a>
|
|||
|
|
<a id="1078" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="1080" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a> <a id="1082" href="Relation.Binary.PropositionalEquality.Properties.html#1082" class="Generalizable">C</a> <a id="1084" class="Symbol">:</a> <a id="1086" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1090" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a>
|
|||
|
|
|
|||
|
|
<a id="1093" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="1166" class="Comment">-- Standard eliminator for the propositional equality type</a>
|
|||
|
|
|
|||
|
|
<a id="J"></a><a id="1226" href="Relation.Binary.PropositionalEquality.Properties.html#1226" class="Function">J</a> <a id="1228" class="Symbol">:</a> <a id="1230" class="Symbol">{</a><a id="1231" href="Relation.Binary.PropositionalEquality.Properties.html#1231" class="Bound">A</a> <a id="1233" class="Symbol">:</a> <a id="1235" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1239" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a><a id="1240" class="Symbol">}</a> <a id="1242" class="Symbol">{</a><a id="1243" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Bound">x</a> <a id="1245" class="Symbol">:</a> <a id="1247" href="Relation.Binary.PropositionalEquality.Properties.html#1231" class="Bound">A</a><a id="1248" class="Symbol">}</a> <a id="1250" class="Symbol">(</a><a id="1251" href="Relation.Binary.PropositionalEquality.Properties.html#1251" class="Bound">B</a> <a id="1253" class="Symbol">:</a> <a id="1255" class="Symbol">(</a><a id="1256" href="Relation.Binary.PropositionalEquality.Properties.html#1256" class="Bound">y</a> <a id="1258" class="Symbol">:</a> <a id="1260" href="Relation.Binary.PropositionalEquality.Properties.html#1231" class="Bound">A</a><a id="1261" class="Symbol">)</a> <a id="1263" class="Symbol">→</a> <a id="1265" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Bound">x</a> <a id="1267" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1269" href="Relation.Binary.PropositionalEquality.Properties.html#1256" class="Bound">y</a> <a id="1271" class="Symbol">→</a> <a id="1273" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1277" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a><a id="1278" class="Symbol">)</a>
|
|||
|
|
<a id="1284" class="Symbol">{</a><a id="1285" href="Relation.Binary.PropositionalEquality.Properties.html#1285" class="Bound">y</a> <a id="1287" class="Symbol">:</a> <a id="1289" href="Relation.Binary.PropositionalEquality.Properties.html#1231" class="Bound">A</a><a id="1290" class="Symbol">}</a> <a id="1292" class="Symbol">(</a><a id="1293" href="Relation.Binary.PropositionalEquality.Properties.html#1293" class="Bound">p</a> <a id="1295" class="Symbol">:</a> <a id="1297" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Bound">x</a> <a id="1299" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1301" href="Relation.Binary.PropositionalEquality.Properties.html#1285" class="Bound">y</a><a id="1302" class="Symbol">)</a> <a id="1304" class="Symbol">→</a> <a id="1306" href="Relation.Binary.PropositionalEquality.Properties.html#1251" class="Bound">B</a> <a id="1308" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Bound">x</a> <a id="1310" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1315" class="Symbol">→</a> <a id="1317" href="Relation.Binary.PropositionalEquality.Properties.html#1251" class="Bound">B</a> <a id="1319" href="Relation.Binary.PropositionalEquality.Properties.html#1285" class="Bound">y</a> <a id="1321" href="Relation.Binary.PropositionalEquality.Properties.html#1293" class="Bound">p</a>
|
|||
|
|
<a id="1323" href="Relation.Binary.PropositionalEquality.Properties.html#1226" class="Function">J</a> <a id="1325" href="Relation.Binary.PropositionalEquality.Properties.html#1325" class="Bound">B</a> <a id="1327" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1332" href="Relation.Binary.PropositionalEquality.Properties.html#1332" class="Bound">b</a> <a id="1334" class="Symbol">=</a> <a id="1336" href="Relation.Binary.PropositionalEquality.Properties.html#1332" class="Bound">b</a>
|
|||
|
|
|
|||
|
|
<a id="1339" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="1412" class="Comment">-- Binary and/or dependent versions of standard operations on equality</a>
|
|||
|
|
|
|||
|
|
<a id="dcong"></a><a id="1484" href="Relation.Binary.PropositionalEquality.Properties.html#1484" class="Function">dcong</a> <a id="1490" class="Symbol">:</a> <a id="1492" class="Symbol">∀</a> <a id="1494" class="Symbol">{</a><a id="1495" href="Relation.Binary.PropositionalEquality.Properties.html#1495" class="Bound">A</a> <a id="1497" class="Symbol">:</a> <a id="1499" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1503" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a><a id="1504" class="Symbol">}</a> <a id="1506" class="Symbol">{</a><a id="1507" href="Relation.Binary.PropositionalEquality.Properties.html#1507" class="Bound">B</a> <a id="1509" class="Symbol">:</a> <a id="1511" href="Relation.Binary.PropositionalEquality.Properties.html#1495" class="Bound">A</a> <a id="1513" class="Symbol">→</a> <a id="1515" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1519" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a><a id="1520" class="Symbol">}</a> <a id="1522" class="Symbol">(</a><a id="1523" href="Relation.Binary.PropositionalEquality.Properties.html#1523" class="Bound">f</a> <a id="1525" class="Symbol">:</a> <a id="1527" class="Symbol">(</a><a id="1528" href="Relation.Binary.PropositionalEquality.Properties.html#1528" class="Bound">x</a> <a id="1530" class="Symbol">:</a> <a id="1532" href="Relation.Binary.PropositionalEquality.Properties.html#1495" class="Bound">A</a><a id="1533" class="Symbol">)</a> <a id="1535" class="Symbol">→</a> <a id="1537" href="Relation.Binary.PropositionalEquality.Properties.html#1507" class="Bound">B</a> <a id="1539" href="Relation.Binary.PropositionalEquality.Properties.html#1528" class="Bound">x</a><a id="1540" class="Symbol">)</a> <a id="1542" class="Symbol">{</a><a id="1543" href="Relation.Binary.PropositionalEquality.Properties.html#1543" class="Bound">x</a> <a id="1545" href="Relation.Binary.PropositionalEquality.Properties.html#1545" class="Bound">y</a><a id="1546" class="Symbol">}</a>
|
|||
|
|
<a id="1554" class="Symbol">→</a> <a id="1556" class="Symbol">(</a><a id="1557" href="Relation.Binary.PropositionalEquality.Properties.html#1557" class="Bound">p</a> <a id="1559" class="Symbol">:</a> <a id="1561" href="Relation.Binary.PropositionalEquality.Properties.html#1543" class="Bound">x</a> <a id="1563" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1565" href="Relation.Binary.PropositionalEquality.Properties.html#1545" class="Bound">y</a><a id="1566" class="Symbol">)</a> <a id="1568" class="Symbol">→</a> <a id="1570" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="1576" href="Relation.Binary.PropositionalEquality.Properties.html#1507" class="Bound">B</a> <a id="1578" href="Relation.Binary.PropositionalEquality.Properties.html#1557" class="Bound">p</a> <a id="1580" class="Symbol">(</a><a id="1581" href="Relation.Binary.PropositionalEquality.Properties.html#1523" class="Bound">f</a> <a id="1583" href="Relation.Binary.PropositionalEquality.Properties.html#1543" class="Bound">x</a><a id="1584" class="Symbol">)</a> <a id="1586" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1588" href="Relation.Binary.PropositionalEquality.Properties.html#1523" class="Bound">f</a> <a id="1590" href="Relation.Binary.PropositionalEquality.Properties.html#1545" class="Bound">y</a>
|
|||
|
|
<a id="1592" href="Relation.Binary.PropositionalEquality.Properties.html#1484" class="Function">dcong</a> <a id="1598" href="Relation.Binary.PropositionalEquality.Properties.html#1598" class="Bound">f</a> <a id="1600" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1605" class="Symbol">=</a> <a id="1607" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="dcong₂"></a><a id="1613" href="Relation.Binary.PropositionalEquality.Properties.html#1613" class="Function">dcong₂</a> <a id="1620" class="Symbol">:</a> <a id="1622" class="Symbol">∀</a> <a id="1624" class="Symbol">{</a><a id="1625" href="Relation.Binary.PropositionalEquality.Properties.html#1625" class="Bound">A</a> <a id="1627" class="Symbol">:</a> <a id="1629" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1633" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a><a id="1634" class="Symbol">}</a> <a id="1636" class="Symbol">{</a><a id="1637" href="Relation.Binary.PropositionalEquality.Properties.html#1637" class="Bound">B</a> <a id="1639" class="Symbol">:</a> <a id="1641" href="Relation.Binary.PropositionalEquality.Properties.html#1625" class="Bound">A</a> <a id="1643" class="Symbol">→</a> <a id="1645" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1649" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a><a id="1650" class="Symbol">}</a> <a id="1652" class="Symbol">{</a><a id="1653" href="Relation.Binary.PropositionalEquality.Properties.html#1653" class="Bound">C</a> <a id="1655" class="Symbol">:</a> <a id="1657" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1661" href="Relation.Binary.PropositionalEquality.Properties.html#1062" class="Generalizable">c</a><a id="1662" class="Symbol">}</a>
|
|||
|
|
<a id="1673" class="Symbol">(</a><a id="1674" href="Relation.Binary.PropositionalEquality.Properties.html#1674" class="Bound">f</a> <a id="1676" class="Symbol">:</a> <a id="1678" class="Symbol">(</a><a id="1679" href="Relation.Binary.PropositionalEquality.Properties.html#1679" class="Bound">x</a> <a id="1681" class="Symbol">:</a> <a id="1683" href="Relation.Binary.PropositionalEquality.Properties.html#1625" class="Bound">A</a><a id="1684" class="Symbol">)</a> <a id="1686" class="Symbol">→</a> <a id="1688" href="Relation.Binary.PropositionalEquality.Properties.html#1637" class="Bound">B</a> <a id="1690" href="Relation.Binary.PropositionalEquality.Properties.html#1679" class="Bound">x</a> <a id="1692" class="Symbol">→</a> <a id="1694" href="Relation.Binary.PropositionalEquality.Properties.html#1653" class="Bound">C</a><a id="1695" class="Symbol">)</a> <a id="1697" class="Symbol">{</a><a id="1698" href="Relation.Binary.PropositionalEquality.Properties.html#1698" class="Bound">x₁</a> <a id="1701" href="Relation.Binary.PropositionalEquality.Properties.html#1701" class="Bound">x₂</a> <a id="1704" href="Relation.Binary.PropositionalEquality.Properties.html#1704" class="Bound">y₁</a> <a id="1707" href="Relation.Binary.PropositionalEquality.Properties.html#1707" class="Bound">y₂</a><a id="1709" class="Symbol">}</a>
|
|||
|
|
<a id="1718" class="Symbol">→</a> <a id="1720" class="Symbol">(</a><a id="1721" href="Relation.Binary.PropositionalEquality.Properties.html#1721" class="Bound">p</a> <a id="1723" class="Symbol">:</a> <a id="1725" href="Relation.Binary.PropositionalEquality.Properties.html#1698" class="Bound">x₁</a> <a id="1728" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1730" href="Relation.Binary.PropositionalEquality.Properties.html#1701" class="Bound">x₂</a><a id="1732" class="Symbol">)</a> <a id="1734" class="Symbol">→</a> <a id="1736" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="1742" href="Relation.Binary.PropositionalEquality.Properties.html#1637" class="Bound">B</a> <a id="1744" href="Relation.Binary.PropositionalEquality.Properties.html#1721" class="Bound">p</a> <a id="1746" href="Relation.Binary.PropositionalEquality.Properties.html#1704" class="Bound">y₁</a> <a id="1749" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1751" href="Relation.Binary.PropositionalEquality.Properties.html#1707" class="Bound">y₂</a>
|
|||
|
|
<a id="1761" class="Symbol">→</a> <a id="1763" href="Relation.Binary.PropositionalEquality.Properties.html#1674" class="Bound">f</a> <a id="1765" href="Relation.Binary.PropositionalEquality.Properties.html#1698" class="Bound">x₁</a> <a id="1768" href="Relation.Binary.PropositionalEquality.Properties.html#1704" class="Bound">y₁</a> <a id="1771" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1773" href="Relation.Binary.PropositionalEquality.Properties.html#1674" class="Bound">f</a> <a id="1775" href="Relation.Binary.PropositionalEquality.Properties.html#1701" class="Bound">x₂</a> <a id="1778" href="Relation.Binary.PropositionalEquality.Properties.html#1707" class="Bound">y₂</a>
|
|||
|
|
<a id="1781" href="Relation.Binary.PropositionalEquality.Properties.html#1613" class="Function">dcong₂</a> <a id="1788" href="Relation.Binary.PropositionalEquality.Properties.html#1788" class="Bound">f</a> <a id="1790" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1795" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1800" class="Symbol">=</a> <a id="1802" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="dsubst₂"></a><a id="1808" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Function">dsubst₂</a> <a id="1816" class="Symbol">:</a> <a id="1818" class="Symbol">∀</a> <a id="1820" class="Symbol">{</a><a id="1821" href="Relation.Binary.PropositionalEquality.Properties.html#1821" class="Bound">A</a> <a id="1823" class="Symbol">:</a> <a id="1825" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1829" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a><a id="1830" class="Symbol">}</a> <a id="1832" class="Symbol">{</a><a id="1833" href="Relation.Binary.PropositionalEquality.Properties.html#1833" class="Bound">B</a> <a id="1835" class="Symbol">:</a> <a id="1837" href="Relation.Binary.PropositionalEquality.Properties.html#1821" class="Bound">A</a> <a id="1839" class="Symbol">→</a> <a id="1841" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1845" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a><a id="1846" class="Symbol">}</a> <a id="1848" class="Symbol">(</a><a id="1849" href="Relation.Binary.PropositionalEquality.Properties.html#1849" class="Bound">C</a> <a id="1851" class="Symbol">:</a> <a id="1853" class="Symbol">(</a><a id="1854" href="Relation.Binary.PropositionalEquality.Properties.html#1854" class="Bound">x</a> <a id="1856" class="Symbol">:</a> <a id="1858" href="Relation.Binary.PropositionalEquality.Properties.html#1821" class="Bound">A</a><a id="1859" class="Symbol">)</a> <a id="1861" class="Symbol">→</a> <a id="1863" href="Relation.Binary.PropositionalEquality.Properties.html#1833" class="Bound">B</a> <a id="1865" href="Relation.Binary.PropositionalEquality.Properties.html#1854" class="Bound">x</a> <a id="1867" class="Symbol">→</a> <a id="1869" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1873" href="Relation.Binary.PropositionalEquality.Properties.html#1062" class="Generalizable">c</a><a id="1874" class="Symbol">)</a>
|
|||
|
|
<a id="1886" class="Symbol">{</a><a id="1887" href="Relation.Binary.PropositionalEquality.Properties.html#1887" class="Bound">x₁</a> <a id="1890" href="Relation.Binary.PropositionalEquality.Properties.html#1890" class="Bound">x₂</a> <a id="1893" href="Relation.Binary.PropositionalEquality.Properties.html#1893" class="Bound">y₁</a> <a id="1896" href="Relation.Binary.PropositionalEquality.Properties.html#1896" class="Bound">y₂</a><a id="1898" class="Symbol">}</a> <a id="1900" class="Symbol">(</a><a id="1901" href="Relation.Binary.PropositionalEquality.Properties.html#1901" class="Bound">p</a> <a id="1903" class="Symbol">:</a> <a id="1905" href="Relation.Binary.PropositionalEquality.Properties.html#1887" class="Bound">x₁</a> <a id="1908" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1910" href="Relation.Binary.PropositionalEquality.Properties.html#1890" class="Bound">x₂</a><a id="1912" class="Symbol">)</a> <a id="1914" class="Symbol">→</a> <a id="1916" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="1922" href="Relation.Binary.PropositionalEquality.Properties.html#1833" class="Bound">B</a> <a id="1924" href="Relation.Binary.PropositionalEquality.Properties.html#1901" class="Bound">p</a> <a id="1926" href="Relation.Binary.PropositionalEquality.Properties.html#1893" class="Bound">y₁</a> <a id="1929" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="1931" href="Relation.Binary.PropositionalEquality.Properties.html#1896" class="Bound">y₂</a>
|
|||
|
|
<a id="1942" class="Symbol">→</a> <a id="1944" href="Relation.Binary.PropositionalEquality.Properties.html#1849" class="Bound">C</a> <a id="1946" href="Relation.Binary.PropositionalEquality.Properties.html#1887" class="Bound">x₁</a> <a id="1949" href="Relation.Binary.PropositionalEquality.Properties.html#1893" class="Bound">y₁</a> <a id="1952" class="Symbol">→</a> <a id="1954" href="Relation.Binary.PropositionalEquality.Properties.html#1849" class="Bound">C</a> <a id="1956" href="Relation.Binary.PropositionalEquality.Properties.html#1890" class="Bound">x₂</a> <a id="1959" href="Relation.Binary.PropositionalEquality.Properties.html#1896" class="Bound">y₂</a>
|
|||
|
|
<a id="1962" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Function">dsubst₂</a> <a id="1970" href="Relation.Binary.PropositionalEquality.Properties.html#1970" class="Bound">C</a> <a id="1972" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1977" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="1982" href="Relation.Binary.PropositionalEquality.Properties.html#1982" class="Bound">c</a> <a id="1984" class="Symbol">=</a> <a id="1986" href="Relation.Binary.PropositionalEquality.Properties.html#1982" class="Bound">c</a>
|
|||
|
|
|
|||
|
|
<a id="ddcong₂"></a><a id="1989" href="Relation.Binary.PropositionalEquality.Properties.html#1989" class="Function">ddcong₂</a> <a id="1997" class="Symbol">:</a> <a id="1999" class="Symbol">∀</a> <a id="2001" class="Symbol">{</a><a id="2002" href="Relation.Binary.PropositionalEquality.Properties.html#2002" class="Bound">A</a> <a id="2004" class="Symbol">:</a> <a id="2006" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2010" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a><a id="2011" class="Symbol">}</a> <a id="2013" class="Symbol">{</a><a id="2014" href="Relation.Binary.PropositionalEquality.Properties.html#2014" class="Bound">B</a> <a id="2016" class="Symbol">:</a> <a id="2018" href="Relation.Binary.PropositionalEquality.Properties.html#2002" class="Bound">A</a> <a id="2020" class="Symbol">→</a> <a id="2022" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2026" href="Relation.Binary.PropositionalEquality.Properties.html#1060" class="Generalizable">b</a><a id="2027" class="Symbol">}</a> <a id="2029" class="Symbol">{</a><a id="2030" href="Relation.Binary.PropositionalEquality.Properties.html#2030" class="Bound">C</a> <a id="2032" class="Symbol">:</a> <a id="2034" class="Symbol">(</a><a id="2035" href="Relation.Binary.PropositionalEquality.Properties.html#2035" class="Bound">x</a> <a id="2037" class="Symbol">:</a> <a id="2039" href="Relation.Binary.PropositionalEquality.Properties.html#2002" class="Bound">A</a><a id="2040" class="Symbol">)</a> <a id="2042" class="Symbol">→</a> <a id="2044" href="Relation.Binary.PropositionalEquality.Properties.html#2014" class="Bound">B</a> <a id="2046" href="Relation.Binary.PropositionalEquality.Properties.html#2035" class="Bound">x</a> <a id="2048" class="Symbol">→</a> <a id="2050" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2054" href="Relation.Binary.PropositionalEquality.Properties.html#1062" class="Generalizable">c</a><a id="2055" class="Symbol">}</a>
|
|||
|
|
<a id="2066" class="Symbol">(</a><a id="2067" href="Relation.Binary.PropositionalEquality.Properties.html#2067" class="Bound">f</a> <a id="2069" class="Symbol">:</a> <a id="2071" class="Symbol">(</a><a id="2072" href="Relation.Binary.PropositionalEquality.Properties.html#2072" class="Bound">x</a> <a id="2074" class="Symbol">:</a> <a id="2076" href="Relation.Binary.PropositionalEquality.Properties.html#2002" class="Bound">A</a><a id="2077" class="Symbol">)</a> <a id="2079" class="Symbol">(</a><a id="2080" href="Relation.Binary.PropositionalEquality.Properties.html#2080" class="Bound">y</a> <a id="2082" class="Symbol">:</a> <a id="2084" href="Relation.Binary.PropositionalEquality.Properties.html#2014" class="Bound">B</a> <a id="2086" href="Relation.Binary.PropositionalEquality.Properties.html#2072" class="Bound">x</a><a id="2087" class="Symbol">)</a> <a id="2089" class="Symbol">→</a> <a id="2091" href="Relation.Binary.PropositionalEquality.Properties.html#2030" class="Bound">C</a> <a id="2093" href="Relation.Binary.PropositionalEquality.Properties.html#2072" class="Bound">x</a> <a id="2095" href="Relation.Binary.PropositionalEquality.Properties.html#2080" class="Bound">y</a><a id="2096" class="Symbol">)</a> <a id="2098" class="Symbol">{</a><a id="2099" href="Relation.Binary.PropositionalEquality.Properties.html#2099" class="Bound">x₁</a> <a id="2102" href="Relation.Binary.PropositionalEquality.Properties.html#2102" class="Bound">x₂</a> <a id="2105" href="Relation.Binary.PropositionalEquality.Properties.html#2105" class="Bound">y₁</a> <a id="2108" href="Relation.Binary.PropositionalEquality.Properties.html#2108" class="Bound">y₂</a><a id="2110" class="Symbol">}</a>
|
|||
|
|
<a id="2121" class="Symbol">(</a><a id="2122" href="Relation.Binary.PropositionalEquality.Properties.html#2122" class="Bound">p</a> <a id="2124" class="Symbol">:</a> <a id="2126" href="Relation.Binary.PropositionalEquality.Properties.html#2099" class="Bound">x₁</a> <a id="2129" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2131" href="Relation.Binary.PropositionalEquality.Properties.html#2102" class="Bound">x₂</a><a id="2133" class="Symbol">)</a> <a id="2135" class="Symbol">(</a><a id="2136" href="Relation.Binary.PropositionalEquality.Properties.html#2136" class="Bound">q</a> <a id="2138" class="Symbol">:</a> <a id="2140" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="2146" href="Relation.Binary.PropositionalEquality.Properties.html#2014" class="Bound">B</a> <a id="2148" href="Relation.Binary.PropositionalEquality.Properties.html#2122" class="Bound">p</a> <a id="2150" href="Relation.Binary.PropositionalEquality.Properties.html#2105" class="Bound">y₁</a> <a id="2153" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2155" href="Relation.Binary.PropositionalEquality.Properties.html#2108" class="Bound">y₂</a><a id="2157" class="Symbol">)</a>
|
|||
|
|
<a id="2166" class="Symbol">→</a> <a id="2168" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Function">dsubst₂</a> <a id="2176" href="Relation.Binary.PropositionalEquality.Properties.html#2030" class="Bound">C</a> <a id="2178" href="Relation.Binary.PropositionalEquality.Properties.html#2122" class="Bound">p</a> <a id="2180" href="Relation.Binary.PropositionalEquality.Properties.html#2136" class="Bound">q</a> <a id="2182" class="Symbol">(</a><a id="2183" href="Relation.Binary.PropositionalEquality.Properties.html#2067" class="Bound">f</a> <a id="2185" href="Relation.Binary.PropositionalEquality.Properties.html#2099" class="Bound">x₁</a> <a id="2188" href="Relation.Binary.PropositionalEquality.Properties.html#2105" class="Bound">y₁</a><a id="2190" class="Symbol">)</a> <a id="2192" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2194" href="Relation.Binary.PropositionalEquality.Properties.html#2067" class="Bound">f</a> <a id="2196" href="Relation.Binary.PropositionalEquality.Properties.html#2102" class="Bound">x₂</a> <a id="2199" href="Relation.Binary.PropositionalEquality.Properties.html#2108" class="Bound">y₂</a>
|
|||
|
|
<a id="2202" href="Relation.Binary.PropositionalEquality.Properties.html#1989" class="Function">ddcong₂</a> <a id="2210" href="Relation.Binary.PropositionalEquality.Properties.html#2210" class="Bound">f</a> <a id="2212" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2217" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2222" class="Symbol">=</a> <a id="2224" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="2230" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="2303" class="Comment">-- Various equality rearrangement lemmas</a>
|
|||
|
|
|
|||
|
|
<a id="trans-reflʳ"></a><a id="2345" href="Relation.Binary.PropositionalEquality.Properties.html#2345" class="Function">trans-reflʳ</a> <a id="2357" class="Symbol">:</a> <a id="2359" class="Symbol">∀</a> <a id="2361" class="Symbol">{</a><a id="2362" href="Relation.Binary.PropositionalEquality.Properties.html#2362" class="Bound">x</a> <a id="2364" href="Relation.Binary.PropositionalEquality.Properties.html#2364" class="Bound">y</a> <a id="2366" class="Symbol">:</a> <a id="2368" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2369" class="Symbol">}</a> <a id="2371" class="Symbol">(</a><a id="2372" href="Relation.Binary.PropositionalEquality.Properties.html#2372" class="Bound">p</a> <a id="2374" class="Symbol">:</a> <a id="2376" href="Relation.Binary.PropositionalEquality.Properties.html#2362" class="Bound">x</a> <a id="2378" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2380" href="Relation.Binary.PropositionalEquality.Properties.html#2364" class="Bound">y</a><a id="2381" class="Symbol">)</a> <a id="2383" class="Symbol">→</a> <a id="2385" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2391" href="Relation.Binary.PropositionalEquality.Properties.html#2372" class="Bound">p</a> <a id="2393" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2398" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2400" href="Relation.Binary.PropositionalEquality.Properties.html#2372" class="Bound">p</a>
|
|||
|
|
<a id="2402" href="Relation.Binary.PropositionalEquality.Properties.html#2345" class="Function">trans-reflʳ</a> <a id="2414" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2419" class="Symbol">=</a> <a id="2421" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="trans-assoc"></a><a id="2427" href="Relation.Binary.PropositionalEquality.Properties.html#2427" class="Function">trans-assoc</a> <a id="2439" class="Symbol">:</a> <a id="2441" class="Symbol">∀</a> <a id="2443" class="Symbol">{</a><a id="2444" href="Relation.Binary.PropositionalEquality.Properties.html#2444" class="Bound">x</a> <a id="2446" href="Relation.Binary.PropositionalEquality.Properties.html#2446" class="Bound">y</a> <a id="2448" href="Relation.Binary.PropositionalEquality.Properties.html#2448" class="Bound">z</a> <a id="2450" href="Relation.Binary.PropositionalEquality.Properties.html#2450" class="Bound">u</a> <a id="2452" class="Symbol">:</a> <a id="2454" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2455" class="Symbol">}</a> <a id="2457" class="Symbol">(</a><a id="2458" href="Relation.Binary.PropositionalEquality.Properties.html#2458" class="Bound">p</a> <a id="2460" class="Symbol">:</a> <a id="2462" href="Relation.Binary.PropositionalEquality.Properties.html#2444" class="Bound">x</a> <a id="2464" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2466" href="Relation.Binary.PropositionalEquality.Properties.html#2446" class="Bound">y</a><a id="2467" class="Symbol">)</a> <a id="2469" class="Symbol">{</a><a id="2470" href="Relation.Binary.PropositionalEquality.Properties.html#2470" class="Bound">q</a> <a id="2472" class="Symbol">:</a> <a id="2474" href="Relation.Binary.PropositionalEquality.Properties.html#2446" class="Bound">y</a> <a id="2476" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2478" href="Relation.Binary.PropositionalEquality.Properties.html#2448" class="Bound">z</a><a id="2479" class="Symbol">}</a> <a id="2481" class="Symbol">{</a><a id="2482" href="Relation.Binary.PropositionalEquality.Properties.html#2482" class="Bound">r</a> <a id="2484" class="Symbol">:</a> <a id="2486" href="Relation.Binary.PropositionalEquality.Properties.html#2448" class="Bound">z</a> <a id="2488" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2490" href="Relation.Binary.PropositionalEquality.Properties.html#2450" class="Bound">u</a><a id="2491" class="Symbol">}</a> <a id="2493" class="Symbol">→</a>
|
|||
|
|
<a id="2497" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2503" class="Symbol">(</a><a id="2504" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2510" href="Relation.Binary.PropositionalEquality.Properties.html#2458" class="Bound">p</a> <a id="2512" href="Relation.Binary.PropositionalEquality.Properties.html#2470" class="Bound">q</a><a id="2513" class="Symbol">)</a> <a id="2515" href="Relation.Binary.PropositionalEquality.Properties.html#2482" class="Bound">r</a> <a id="2517" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2519" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2525" href="Relation.Binary.PropositionalEquality.Properties.html#2458" class="Bound">p</a> <a id="2527" class="Symbol">(</a><a id="2528" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2534" href="Relation.Binary.PropositionalEquality.Properties.html#2470" class="Bound">q</a> <a id="2536" href="Relation.Binary.PropositionalEquality.Properties.html#2482" class="Bound">r</a><a id="2537" class="Symbol">)</a>
|
|||
|
|
<a id="2539" href="Relation.Binary.PropositionalEquality.Properties.html#2427" class="Function">trans-assoc</a> <a id="2551" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2556" class="Symbol">=</a> <a id="2558" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="trans-symˡ"></a><a id="2564" href="Relation.Binary.PropositionalEquality.Properties.html#2564" class="Function">trans-symˡ</a> <a id="2575" class="Symbol">:</a> <a id="2577" class="Symbol">∀</a> <a id="2579" class="Symbol">{</a><a id="2580" href="Relation.Binary.PropositionalEquality.Properties.html#2580" class="Bound">x</a> <a id="2582" href="Relation.Binary.PropositionalEquality.Properties.html#2582" class="Bound">y</a> <a id="2584" class="Symbol">:</a> <a id="2586" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2587" class="Symbol">}</a> <a id="2589" class="Symbol">(</a><a id="2590" href="Relation.Binary.PropositionalEquality.Properties.html#2590" class="Bound">p</a> <a id="2592" class="Symbol">:</a> <a id="2594" href="Relation.Binary.PropositionalEquality.Properties.html#2580" class="Bound">x</a> <a id="2596" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2598" href="Relation.Binary.PropositionalEquality.Properties.html#2582" class="Bound">y</a><a id="2599" class="Symbol">)</a> <a id="2601" class="Symbol">→</a> <a id="2603" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2609" class="Symbol">(</a><a id="2610" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="2614" href="Relation.Binary.PropositionalEquality.Properties.html#2590" class="Bound">p</a><a id="2615" class="Symbol">)</a> <a id="2617" href="Relation.Binary.PropositionalEquality.Properties.html#2590" class="Bound">p</a> <a id="2619" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2621" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="2626" href="Relation.Binary.PropositionalEquality.Properties.html#2564" class="Function">trans-symˡ</a> <a id="2637" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2642" class="Symbol">=</a> <a id="2644" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="trans-symʳ"></a><a id="2650" href="Relation.Binary.PropositionalEquality.Properties.html#2650" class="Function">trans-symʳ</a> <a id="2661" class="Symbol">:</a> <a id="2663" class="Symbol">∀</a> <a id="2665" class="Symbol">{</a><a id="2666" href="Relation.Binary.PropositionalEquality.Properties.html#2666" class="Bound">x</a> <a id="2668" href="Relation.Binary.PropositionalEquality.Properties.html#2668" class="Bound">y</a> <a id="2670" class="Symbol">:</a> <a id="2672" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2673" class="Symbol">}</a> <a id="2675" class="Symbol">(</a><a id="2676" href="Relation.Binary.PropositionalEquality.Properties.html#2676" class="Bound">p</a> <a id="2678" class="Symbol">:</a> <a id="2680" href="Relation.Binary.PropositionalEquality.Properties.html#2666" class="Bound">x</a> <a id="2682" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2684" href="Relation.Binary.PropositionalEquality.Properties.html#2668" class="Bound">y</a><a id="2685" class="Symbol">)</a> <a id="2687" class="Symbol">→</a> <a id="2689" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2695" href="Relation.Binary.PropositionalEquality.Properties.html#2676" class="Bound">p</a> <a id="2697" class="Symbol">(</a><a id="2698" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="2702" href="Relation.Binary.PropositionalEquality.Properties.html#2676" class="Bound">p</a><a id="2703" class="Symbol">)</a> <a id="2705" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2707" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="2712" href="Relation.Binary.PropositionalEquality.Properties.html#2650" class="Function">trans-symʳ</a> <a id="2723" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2728" class="Symbol">=</a> <a id="2730" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="trans-injectiveˡ"></a><a id="2736" href="Relation.Binary.PropositionalEquality.Properties.html#2736" class="Function">trans-injectiveˡ</a> <a id="2753" class="Symbol">:</a> <a id="2755" class="Symbol">∀</a> <a id="2757" class="Symbol">{</a><a id="2758" href="Relation.Binary.PropositionalEquality.Properties.html#2758" class="Bound">x</a> <a id="2760" href="Relation.Binary.PropositionalEquality.Properties.html#2760" class="Bound">y</a> <a id="2762" href="Relation.Binary.PropositionalEquality.Properties.html#2762" class="Bound">z</a> <a id="2764" class="Symbol">:</a> <a id="2766" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2767" class="Symbol">}</a> <a id="2769" class="Symbol">{</a><a id="2770" href="Relation.Binary.PropositionalEquality.Properties.html#2770" class="Bound">p₁</a> <a id="2773" href="Relation.Binary.PropositionalEquality.Properties.html#2773" class="Bound">p₂</a> <a id="2776" class="Symbol">:</a> <a id="2778" href="Relation.Binary.PropositionalEquality.Properties.html#2758" class="Bound">x</a> <a id="2780" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2782" href="Relation.Binary.PropositionalEquality.Properties.html#2760" class="Bound">y</a><a id="2783" class="Symbol">}</a> <a id="2785" class="Symbol">(</a><a id="2786" href="Relation.Binary.PropositionalEquality.Properties.html#2786" class="Bound">q</a> <a id="2788" class="Symbol">:</a> <a id="2790" href="Relation.Binary.PropositionalEquality.Properties.html#2760" class="Bound">y</a> <a id="2792" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2794" href="Relation.Binary.PropositionalEquality.Properties.html#2762" class="Bound">z</a><a id="2795" class="Symbol">)</a> <a id="2797" class="Symbol">→</a>
|
|||
|
|
<a id="2818" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2824" href="Relation.Binary.PropositionalEquality.Properties.html#2770" class="Bound">p₁</a> <a id="2827" href="Relation.Binary.PropositionalEquality.Properties.html#2786" class="Bound">q</a> <a id="2829" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2831" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="2837" href="Relation.Binary.PropositionalEquality.Properties.html#2773" class="Bound">p₂</a> <a id="2840" href="Relation.Binary.PropositionalEquality.Properties.html#2786" class="Bound">q</a> <a id="2842" class="Symbol">→</a> <a id="2844" href="Relation.Binary.PropositionalEquality.Properties.html#2770" class="Bound">p₁</a> <a id="2847" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2849" href="Relation.Binary.PropositionalEquality.Properties.html#2773" class="Bound">p₂</a>
|
|||
|
|
<a id="2852" href="Relation.Binary.PropositionalEquality.Properties.html#2736" class="Function">trans-injectiveˡ</a> <a id="2869" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2874" class="Symbol">=</a> <a id="2876" href="Relation.Binary.PropositionalEquality.Core.html#1863" class="Function">subst₂</a> <a id="2883" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="2887" class="Symbol">(</a><a id="2888" href="Relation.Binary.PropositionalEquality.Properties.html#2345" class="Function">trans-reflʳ</a> <a id="2900" class="Symbol">_)</a> <a id="2903" class="Symbol">(</a><a id="2904" href="Relation.Binary.PropositionalEquality.Properties.html#2345" class="Function">trans-reflʳ</a> <a id="2916" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="trans-injectiveʳ"></a><a id="2920" href="Relation.Binary.PropositionalEquality.Properties.html#2920" class="Function">trans-injectiveʳ</a> <a id="2937" class="Symbol">:</a> <a id="2939" class="Symbol">∀</a> <a id="2941" class="Symbol">{</a><a id="2942" href="Relation.Binary.PropositionalEquality.Properties.html#2942" class="Bound">x</a> <a id="2944" href="Relation.Binary.PropositionalEquality.Properties.html#2944" class="Bound">y</a> <a id="2946" href="Relation.Binary.PropositionalEquality.Properties.html#2946" class="Bound">z</a> <a id="2948" class="Symbol">:</a> <a id="2950" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="2951" class="Symbol">}</a> <a id="2953" class="Symbol">(</a><a id="2954" href="Relation.Binary.PropositionalEquality.Properties.html#2954" class="Bound">p</a> <a id="2956" class="Symbol">:</a> <a id="2958" href="Relation.Binary.PropositionalEquality.Properties.html#2942" class="Bound">x</a> <a id="2960" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2962" href="Relation.Binary.PropositionalEquality.Properties.html#2944" class="Bound">y</a><a id="2963" class="Symbol">)</a> <a id="2965" class="Symbol">{</a><a id="2966" href="Relation.Binary.PropositionalEquality.Properties.html#2966" class="Bound">q₁</a> <a id="2969" href="Relation.Binary.PropositionalEquality.Properties.html#2969" class="Bound">q₂</a> <a id="2972" class="Symbol">:</a> <a id="2974" href="Relation.Binary.PropositionalEquality.Properties.html#2944" class="Bound">y</a> <a id="2976" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="2978" href="Relation.Binary.PropositionalEquality.Properties.html#2946" class="Bound">z</a><a id="2979" class="Symbol">}</a> <a id="2981" class="Symbol">→</a>
|
|||
|
|
<a id="3002" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="3008" href="Relation.Binary.PropositionalEquality.Properties.html#2954" class="Bound">p</a> <a id="3010" href="Relation.Binary.PropositionalEquality.Properties.html#2966" class="Bound">q₁</a> <a id="3013" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3015" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="3021" href="Relation.Binary.PropositionalEquality.Properties.html#2954" class="Bound">p</a> <a id="3023" href="Relation.Binary.PropositionalEquality.Properties.html#2969" class="Bound">q₂</a> <a id="3026" class="Symbol">→</a> <a id="3028" href="Relation.Binary.PropositionalEquality.Properties.html#2966" class="Bound">q₁</a> <a id="3031" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3033" href="Relation.Binary.PropositionalEquality.Properties.html#2969" class="Bound">q₂</a>
|
|||
|
|
<a id="3036" href="Relation.Binary.PropositionalEquality.Properties.html#2920" class="Function">trans-injectiveʳ</a> <a id="3053" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3058" href="Relation.Binary.PropositionalEquality.Properties.html#3058" class="Bound">eq</a> <a id="3061" class="Symbol">=</a> <a id="3063" href="Relation.Binary.PropositionalEquality.Properties.html#3058" class="Bound">eq</a>
|
|||
|
|
|
|||
|
|
<a id="cong-id"></a><a id="3067" href="Relation.Binary.PropositionalEquality.Properties.html#3067" class="Function">cong-id</a> <a id="3075" class="Symbol">:</a> <a id="3077" class="Symbol">∀</a> <a id="3079" class="Symbol">{</a><a id="3080" href="Relation.Binary.PropositionalEquality.Properties.html#3080" class="Bound">x</a> <a id="3082" href="Relation.Binary.PropositionalEquality.Properties.html#3082" class="Bound">y</a> <a id="3084" class="Symbol">:</a> <a id="3086" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="3087" class="Symbol">}</a> <a id="3089" class="Symbol">(</a><a id="3090" href="Relation.Binary.PropositionalEquality.Properties.html#3090" class="Bound">p</a> <a id="3092" class="Symbol">:</a> <a id="3094" href="Relation.Binary.PropositionalEquality.Properties.html#3080" class="Bound">x</a> <a id="3096" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3098" href="Relation.Binary.PropositionalEquality.Properties.html#3082" class="Bound">y</a><a id="3099" class="Symbol">)</a> <a id="3101" class="Symbol">→</a> <a id="3103" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3108" href="Function.Base.html#704" class="Function">id</a> <a id="3111" href="Relation.Binary.PropositionalEquality.Properties.html#3090" class="Bound">p</a> <a id="3113" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3115" href="Relation.Binary.PropositionalEquality.Properties.html#3090" class="Bound">p</a>
|
|||
|
|
<a id="3117" href="Relation.Binary.PropositionalEquality.Properties.html#3067" class="Function">cong-id</a> <a id="3125" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3130" class="Symbol">=</a> <a id="3132" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="cong-∘"></a><a id="3138" href="Relation.Binary.PropositionalEquality.Properties.html#3138" class="Function">cong-∘</a> <a id="3145" class="Symbol">:</a> <a id="3147" class="Symbol">∀</a> <a id="3149" class="Symbol">{</a><a id="3150" href="Relation.Binary.PropositionalEquality.Properties.html#3150" class="Bound">x</a> <a id="3152" href="Relation.Binary.PropositionalEquality.Properties.html#3152" class="Bound">y</a> <a id="3154" class="Symbol">:</a> <a id="3156" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="3157" class="Symbol">}</a> <a id="3159" class="Symbol">{</a><a id="3160" href="Relation.Binary.PropositionalEquality.Properties.html#3160" class="Bound">f</a> <a id="3162" class="Symbol">:</a> <a id="3164" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a> <a id="3166" class="Symbol">→</a> <a id="3168" href="Relation.Binary.PropositionalEquality.Properties.html#1082" class="Generalizable">C</a><a id="3169" class="Symbol">}</a> <a id="3171" class="Symbol">{</a><a id="3172" href="Relation.Binary.PropositionalEquality.Properties.html#3172" class="Bound">g</a> <a id="3174" class="Symbol">:</a> <a id="3176" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3178" class="Symbol">→</a> <a id="3180" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a><a id="3181" class="Symbol">}</a> <a id="3183" class="Symbol">(</a><a id="3184" href="Relation.Binary.PropositionalEquality.Properties.html#3184" class="Bound">p</a> <a id="3186" class="Symbol">:</a> <a id="3188" href="Relation.Binary.PropositionalEquality.Properties.html#3150" class="Bound">x</a> <a id="3190" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3192" href="Relation.Binary.PropositionalEquality.Properties.html#3152" class="Bound">y</a><a id="3193" class="Symbol">)</a> <a id="3195" class="Symbol">→</a>
|
|||
|
|
<a id="3206" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3211" class="Symbol">(</a><a id="3212" href="Relation.Binary.PropositionalEquality.Properties.html#3160" class="Bound">f</a> <a id="3214" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="3216" href="Relation.Binary.PropositionalEquality.Properties.html#3172" class="Bound">g</a><a id="3217" class="Symbol">)</a> <a id="3219" href="Relation.Binary.PropositionalEquality.Properties.html#3184" class="Bound">p</a> <a id="3221" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3223" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3228" href="Relation.Binary.PropositionalEquality.Properties.html#3160" class="Bound">f</a> <a id="3230" class="Symbol">(</a><a id="3231" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3236" href="Relation.Binary.PropositionalEquality.Properties.html#3172" class="Bound">g</a> <a id="3238" href="Relation.Binary.PropositionalEquality.Properties.html#3184" class="Bound">p</a><a id="3239" class="Symbol">)</a>
|
|||
|
|
<a id="3241" href="Relation.Binary.PropositionalEquality.Properties.html#3138" class="Function">cong-∘</a> <a id="3248" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3253" class="Symbol">=</a> <a id="3255" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="sym-cong"></a><a id="3261" href="Relation.Binary.PropositionalEquality.Properties.html#3261" class="Function">sym-cong</a> <a id="3270" class="Symbol">:</a> <a id="3272" class="Symbol">∀</a> <a id="3274" class="Symbol">{</a><a id="3275" href="Relation.Binary.PropositionalEquality.Properties.html#3275" class="Bound">x</a> <a id="3277" href="Relation.Binary.PropositionalEquality.Properties.html#3277" class="Bound">y</a> <a id="3279" class="Symbol">:</a> <a id="3281" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="3282" class="Symbol">}</a> <a id="3284" class="Symbol">{</a><a id="3285" href="Relation.Binary.PropositionalEquality.Properties.html#3285" class="Bound">f</a> <a id="3287" class="Symbol">:</a> <a id="3289" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3291" class="Symbol">→</a> <a id="3293" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a><a id="3294" class="Symbol">}</a> <a id="3296" class="Symbol">(</a><a id="3297" href="Relation.Binary.PropositionalEquality.Properties.html#3297" class="Bound">p</a> <a id="3299" class="Symbol">:</a> <a id="3301" href="Relation.Binary.PropositionalEquality.Properties.html#3275" class="Bound">x</a> <a id="3303" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3305" href="Relation.Binary.PropositionalEquality.Properties.html#3277" class="Bound">y</a><a id="3306" class="Symbol">)</a> <a id="3308" class="Symbol">→</a> <a id="3310" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="3314" class="Symbol">(</a><a id="3315" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3320" href="Relation.Binary.PropositionalEquality.Properties.html#3285" class="Bound">f</a> <a id="3322" href="Relation.Binary.PropositionalEquality.Properties.html#3297" class="Bound">p</a><a id="3323" class="Symbol">)</a> <a id="3325" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3327" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3332" href="Relation.Binary.PropositionalEquality.Properties.html#3285" class="Bound">f</a> <a id="3334" class="Symbol">(</a><a id="3335" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="3339" href="Relation.Binary.PropositionalEquality.Properties.html#3297" class="Bound">p</a><a id="3340" class="Symbol">)</a>
|
|||
|
|
<a id="3342" href="Relation.Binary.PropositionalEquality.Properties.html#3261" class="Function">sym-cong</a> <a id="3351" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3356" class="Symbol">=</a> <a id="3358" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="trans-cong"></a><a id="3364" href="Relation.Binary.PropositionalEquality.Properties.html#3364" class="Function">trans-cong</a> <a id="3375" class="Symbol">:</a> <a id="3377" class="Symbol">∀</a> <a id="3379" class="Symbol">{</a><a id="3380" href="Relation.Binary.PropositionalEquality.Properties.html#3380" class="Bound">x</a> <a id="3382" href="Relation.Binary.PropositionalEquality.Properties.html#3382" class="Bound">y</a> <a id="3384" href="Relation.Binary.PropositionalEquality.Properties.html#3384" class="Bound">z</a> <a id="3386" class="Symbol">:</a> <a id="3388" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="3389" class="Symbol">}</a> <a id="3391" class="Symbol">{</a><a id="3392" href="Relation.Binary.PropositionalEquality.Properties.html#3392" class="Bound">f</a> <a id="3394" class="Symbol">:</a> <a id="3396" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3398" class="Symbol">→</a> <a id="3400" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a><a id="3401" class="Symbol">}</a> <a id="3403" class="Symbol">(</a><a id="3404" href="Relation.Binary.PropositionalEquality.Properties.html#3404" class="Bound">p</a> <a id="3406" class="Symbol">:</a> <a id="3408" href="Relation.Binary.PropositionalEquality.Properties.html#3380" class="Bound">x</a> <a id="3410" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3412" href="Relation.Binary.PropositionalEquality.Properties.html#3382" class="Bound">y</a><a id="3413" class="Symbol">)</a> <a id="3415" class="Symbol">{</a><a id="3416" href="Relation.Binary.PropositionalEquality.Properties.html#3416" class="Bound">q</a> <a id="3418" class="Symbol">:</a> <a id="3420" href="Relation.Binary.PropositionalEquality.Properties.html#3382" class="Bound">y</a> <a id="3422" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3424" href="Relation.Binary.PropositionalEquality.Properties.html#3384" class="Bound">z</a><a id="3425" class="Symbol">}</a> <a id="3427" class="Symbol">→</a>
|
|||
|
|
<a id="3442" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="3448" class="Symbol">(</a><a id="3449" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3454" href="Relation.Binary.PropositionalEquality.Properties.html#3392" class="Bound">f</a> <a id="3456" href="Relation.Binary.PropositionalEquality.Properties.html#3404" class="Bound">p</a><a id="3457" class="Symbol">)</a> <a id="3459" class="Symbol">(</a><a id="3460" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3465" href="Relation.Binary.PropositionalEquality.Properties.html#3392" class="Bound">f</a> <a id="3467" href="Relation.Binary.PropositionalEquality.Properties.html#3416" class="Bound">q</a><a id="3468" class="Symbol">)</a> <a id="3470" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3472" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3477" href="Relation.Binary.PropositionalEquality.Properties.html#3392" class="Bound">f</a> <a id="3479" class="Symbol">(</a><a id="3480" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="3486" href="Relation.Binary.PropositionalEquality.Properties.html#3404" class="Bound">p</a> <a id="3488" href="Relation.Binary.PropositionalEquality.Properties.html#3416" class="Bound">q</a><a id="3489" class="Symbol">)</a>
|
|||
|
|
<a id="3491" href="Relation.Binary.PropositionalEquality.Properties.html#3364" class="Function">trans-cong</a> <a id="3502" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3507" class="Symbol">=</a> <a id="3509" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="cong₂-reflˡ"></a><a id="3515" href="Relation.Binary.PropositionalEquality.Properties.html#3515" class="Function">cong₂-reflˡ</a> <a id="3527" class="Symbol">:</a> <a id="3529" class="Symbol">∀</a> <a id="3531" class="Symbol">{</a><a id="3532" href="Relation.Binary.PropositionalEquality.Properties.html#3532" class="Bound Operator">_∙_</a> <a id="3536" class="Symbol">:</a> <a id="3538" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3540" class="Symbol">→</a> <a id="3542" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a> <a id="3544" class="Symbol">→</a> <a id="3546" href="Relation.Binary.PropositionalEquality.Properties.html#1082" class="Generalizable">C</a><a id="3547" class="Symbol">}</a> <a id="3549" class="Symbol">{</a><a id="3550" href="Relation.Binary.PropositionalEquality.Properties.html#3550" class="Bound">x</a> <a id="3552" href="Relation.Binary.PropositionalEquality.Properties.html#3552" class="Bound">u</a> <a id="3554" href="Relation.Binary.PropositionalEquality.Properties.html#3554" class="Bound">v</a><a id="3555" class="Symbol">}</a> <a id="3557" class="Symbol">→</a> <a id="3559" class="Symbol">(</a><a id="3560" href="Relation.Binary.PropositionalEquality.Properties.html#3560" class="Bound">p</a> <a id="3562" class="Symbol">:</a> <a id="3564" href="Relation.Binary.PropositionalEquality.Properties.html#3552" class="Bound">u</a> <a id="3566" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3568" href="Relation.Binary.PropositionalEquality.Properties.html#3554" class="Bound">v</a><a id="3569" class="Symbol">)</a> <a id="3571" class="Symbol">→</a>
|
|||
|
|
<a id="3587" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3593" href="Relation.Binary.PropositionalEquality.Properties.html#3532" class="Bound Operator">_∙_</a> <a id="3597" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3602" href="Relation.Binary.PropositionalEquality.Properties.html#3560" class="Bound">p</a> <a id="3604" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3606" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3611" class="Symbol">(</a><a id="3612" href="Relation.Binary.PropositionalEquality.Properties.html#3550" class="Bound">x</a> <a id="3614" href="Relation.Binary.PropositionalEquality.Properties.html#3532" class="Bound Operator">∙_</a><a id="3616" class="Symbol">)</a> <a id="3618" href="Relation.Binary.PropositionalEquality.Properties.html#3560" class="Bound">p</a>
|
|||
|
|
<a id="3620" href="Relation.Binary.PropositionalEquality.Properties.html#3515" class="Function">cong₂-reflˡ</a> <a id="3632" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3637" class="Symbol">=</a> <a id="3639" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="cong₂-reflʳ"></a><a id="3645" href="Relation.Binary.PropositionalEquality.Properties.html#3645" class="Function">cong₂-reflʳ</a> <a id="3657" class="Symbol">:</a> <a id="3659" class="Symbol">∀</a> <a id="3661" class="Symbol">{</a><a id="3662" href="Relation.Binary.PropositionalEquality.Properties.html#3662" class="Bound Operator">_∙_</a> <a id="3666" class="Symbol">:</a> <a id="3668" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3670" class="Symbol">→</a> <a id="3672" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a> <a id="3674" class="Symbol">→</a> <a id="3676" href="Relation.Binary.PropositionalEquality.Properties.html#1082" class="Generalizable">C</a><a id="3677" class="Symbol">}</a> <a id="3679" class="Symbol">{</a><a id="3680" href="Relation.Binary.PropositionalEquality.Properties.html#3680" class="Bound">x</a> <a id="3682" href="Relation.Binary.PropositionalEquality.Properties.html#3682" class="Bound">y</a> <a id="3684" href="Relation.Binary.PropositionalEquality.Properties.html#3684" class="Bound">u</a><a id="3685" class="Symbol">}</a> <a id="3687" class="Symbol">→</a> <a id="3689" class="Symbol">(</a><a id="3690" href="Relation.Binary.PropositionalEquality.Properties.html#3690" class="Bound">p</a> <a id="3692" class="Symbol">:</a> <a id="3694" href="Relation.Binary.PropositionalEquality.Properties.html#3680" class="Bound">x</a> <a id="3696" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3698" href="Relation.Binary.PropositionalEquality.Properties.html#3682" class="Bound">y</a><a id="3699" class="Symbol">)</a> <a id="3701" class="Symbol">→</a>
|
|||
|
|
<a id="3717" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="3723" href="Relation.Binary.PropositionalEquality.Properties.html#3662" class="Bound Operator">_∙_</a> <a id="3727" href="Relation.Binary.PropositionalEquality.Properties.html#3690" class="Bound">p</a> <a id="3729" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3734" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3736" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3741" class="Symbol">(</a><a id="3742" href="Relation.Binary.PropositionalEquality.Properties.html#3662" class="Bound Operator">_∙</a> <a id="3745" href="Relation.Binary.PropositionalEquality.Properties.html#3684" class="Bound">u</a><a id="3746" class="Symbol">)</a> <a id="3748" href="Relation.Binary.PropositionalEquality.Properties.html#3690" class="Bound">p</a>
|
|||
|
|
<a id="3750" href="Relation.Binary.PropositionalEquality.Properties.html#3645" class="Function">cong₂-reflʳ</a> <a id="3762" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3767" class="Symbol">=</a> <a id="3769" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="3775" class="Keyword">module</a> <a id="3782" href="Relation.Binary.PropositionalEquality.Properties.html#3782" class="Module">_</a> <a id="3784" class="Symbol">{</a><a id="3785" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="3787" class="Symbol">:</a> <a id="3789" href="Relation.Unary.html#1232" class="Function">Pred</a> <a id="3794" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="3796" href="Relation.Binary.PropositionalEquality.Properties.html#1064" class="Generalizable">p</a><a id="3797" class="Symbol">}</a> <a id="3799" class="Symbol">{</a><a id="3800" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a> <a id="3802" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a> <a id="3804" class="Symbol">:</a> <a id="3806" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="3807" class="Symbol">}</a> <a id="3809" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="3818" href="Relation.Binary.PropositionalEquality.Properties.html#3818" class="Function">subst-injective</a> <a id="3834" class="Symbol">:</a> <a id="3836" class="Symbol">∀</a> <a id="3838" class="Symbol">(</a><a id="3839" href="Relation.Binary.PropositionalEquality.Properties.html#3839" class="Bound">x≡y</a> <a id="3843" class="Symbol">:</a> <a id="3845" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a> <a id="3847" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3849" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a><a id="3850" class="Symbol">)</a> <a id="3852" class="Symbol">{</a><a id="3853" href="Relation.Binary.PropositionalEquality.Properties.html#3853" class="Bound">p</a> <a id="3855" href="Relation.Binary.PropositionalEquality.Properties.html#3855" class="Bound">q</a> <a id="3857" class="Symbol">:</a> <a id="3859" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="3861" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a><a id="3862" class="Symbol">}</a> <a id="3864" class="Symbol">→</a>
|
|||
|
|
<a id="3886" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="3892" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="3894" href="Relation.Binary.PropositionalEquality.Properties.html#3839" class="Bound">x≡y</a> <a id="3898" href="Relation.Binary.PropositionalEquality.Properties.html#3853" class="Bound">p</a> <a id="3900" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3902" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="3908" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="3910" href="Relation.Binary.PropositionalEquality.Properties.html#3839" class="Bound">x≡y</a> <a id="3914" href="Relation.Binary.PropositionalEquality.Properties.html#3855" class="Bound">q</a> <a id="3916" class="Symbol">→</a> <a id="3918" href="Relation.Binary.PropositionalEquality.Properties.html#3853" class="Bound">p</a> <a id="3920" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3922" href="Relation.Binary.PropositionalEquality.Properties.html#3855" class="Bound">q</a>
|
|||
|
|
<a id="3926" href="Relation.Binary.PropositionalEquality.Properties.html#3818" class="Function">subst-injective</a> <a id="3942" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="3947" href="Relation.Binary.PropositionalEquality.Properties.html#3947" class="Bound">p≡q</a> <a id="3951" class="Symbol">=</a> <a id="3953" href="Relation.Binary.PropositionalEquality.Properties.html#3947" class="Bound">p≡q</a>
|
|||
|
|
|
|||
|
|
<a id="3960" href="Relation.Binary.PropositionalEquality.Properties.html#3960" class="Function">subst-subst</a> <a id="3972" class="Symbol">:</a> <a id="3974" class="Symbol">∀</a> <a id="3976" class="Symbol">{</a><a id="3977" href="Relation.Binary.PropositionalEquality.Properties.html#3977" class="Bound">z</a><a id="3978" class="Symbol">}</a> <a id="3980" class="Symbol">(</a><a id="3981" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Bound">x≡y</a> <a id="3985" class="Symbol">:</a> <a id="3987" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a> <a id="3989" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="3991" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a><a id="3992" class="Symbol">)</a> <a id="3994" class="Symbol">{</a><a id="3995" href="Relation.Binary.PropositionalEquality.Properties.html#3995" class="Bound">y≡z</a> <a id="3999" class="Symbol">:</a> <a id="4001" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a> <a id="4003" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4005" href="Relation.Binary.PropositionalEquality.Properties.html#3977" class="Bound">z</a><a id="4006" class="Symbol">}</a> <a id="4008" class="Symbol">{</a><a id="4009" href="Relation.Binary.PropositionalEquality.Properties.html#4009" class="Bound">p</a> <a id="4011" class="Symbol">:</a> <a id="4013" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4015" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a><a id="4016" class="Symbol">}</a> <a id="4018" class="Symbol">→</a>
|
|||
|
|
<a id="4036" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4042" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4044" href="Relation.Binary.PropositionalEquality.Properties.html#3995" class="Bound">y≡z</a> <a id="4048" class="Symbol">(</a><a id="4049" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4055" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4057" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Bound">x≡y</a> <a id="4061" href="Relation.Binary.PropositionalEquality.Properties.html#4009" class="Bound">p</a><a id="4062" class="Symbol">)</a> <a id="4064" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4066" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4072" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4074" class="Symbol">(</a><a id="4075" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="4081" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Bound">x≡y</a> <a id="4085" href="Relation.Binary.PropositionalEquality.Properties.html#3995" class="Bound">y≡z</a><a id="4088" class="Symbol">)</a> <a id="4090" href="Relation.Binary.PropositionalEquality.Properties.html#4009" class="Bound">p</a>
|
|||
|
|
<a id="4094" href="Relation.Binary.PropositionalEquality.Properties.html#3960" class="Function">subst-subst</a> <a id="4106" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4111" class="Symbol">=</a> <a id="4113" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="4121" href="Relation.Binary.PropositionalEquality.Properties.html#4121" class="Function">subst-subst-sym</a> <a id="4137" class="Symbol">:</a> <a id="4139" class="Symbol">(</a><a id="4140" href="Relation.Binary.PropositionalEquality.Properties.html#4140" class="Bound">x≡y</a> <a id="4144" class="Symbol">:</a> <a id="4146" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a> <a id="4148" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4150" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a><a id="4151" class="Symbol">)</a> <a id="4153" class="Symbol">{</a><a id="4154" href="Relation.Binary.PropositionalEquality.Properties.html#4154" class="Bound">p</a> <a id="4156" class="Symbol">:</a> <a id="4158" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4160" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a><a id="4161" class="Symbol">}</a> <a id="4163" class="Symbol">→</a>
|
|||
|
|
<a id="4185" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4191" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4193" href="Relation.Binary.PropositionalEquality.Properties.html#4140" class="Bound">x≡y</a> <a id="4197" class="Symbol">(</a><a id="4198" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4204" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4206" class="Symbol">(</a><a id="4207" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="4211" href="Relation.Binary.PropositionalEquality.Properties.html#4140" class="Bound">x≡y</a><a id="4214" class="Symbol">)</a> <a id="4216" href="Relation.Binary.PropositionalEquality.Properties.html#4154" class="Bound">p</a><a id="4217" class="Symbol">)</a> <a id="4219" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4221" href="Relation.Binary.PropositionalEquality.Properties.html#4154" class="Bound">p</a>
|
|||
|
|
<a id="4225" href="Relation.Binary.PropositionalEquality.Properties.html#4121" class="Function">subst-subst-sym</a> <a id="4241" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4246" class="Symbol">=</a> <a id="4248" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="4256" href="Relation.Binary.PropositionalEquality.Properties.html#4256" class="Function">subst-sym-subst</a> <a id="4272" class="Symbol">:</a> <a id="4274" class="Symbol">(</a><a id="4275" href="Relation.Binary.PropositionalEquality.Properties.html#4275" class="Bound">x≡y</a> <a id="4279" class="Symbol">:</a> <a id="4281" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a> <a id="4283" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4285" href="Relation.Binary.PropositionalEquality.Properties.html#3802" class="Bound">y</a><a id="4286" class="Symbol">)</a> <a id="4288" class="Symbol">{</a><a id="4289" href="Relation.Binary.PropositionalEquality.Properties.html#4289" class="Bound">p</a> <a id="4291" class="Symbol">:</a> <a id="4293" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4295" href="Relation.Binary.PropositionalEquality.Properties.html#3800" class="Bound">x</a><a id="4296" class="Symbol">}</a> <a id="4298" class="Symbol">→</a>
|
|||
|
|
<a id="4320" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4326" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4328" class="Symbol">(</a><a id="4329" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="4333" href="Relation.Binary.PropositionalEquality.Properties.html#4275" class="Bound">x≡y</a><a id="4336" class="Symbol">)</a> <a id="4338" class="Symbol">(</a><a id="4339" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4345" href="Relation.Binary.PropositionalEquality.Properties.html#3785" class="Bound">P</a> <a id="4347" href="Relation.Binary.PropositionalEquality.Properties.html#4275" class="Bound">x≡y</a> <a id="4351" href="Relation.Binary.PropositionalEquality.Properties.html#4289" class="Bound">p</a><a id="4352" class="Symbol">)</a> <a id="4354" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4356" href="Relation.Binary.PropositionalEquality.Properties.html#4289" class="Bound">p</a>
|
|||
|
|
<a id="4360" href="Relation.Binary.PropositionalEquality.Properties.html#4256" class="Function">subst-sym-subst</a> <a id="4376" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4381" class="Symbol">=</a> <a id="4383" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="subst-∘"></a><a id="4389" href="Relation.Binary.PropositionalEquality.Properties.html#4389" class="Function">subst-∘</a> <a id="4397" class="Symbol">:</a> <a id="4399" class="Symbol">∀</a> <a id="4401" class="Symbol">{</a><a id="4402" href="Relation.Binary.PropositionalEquality.Properties.html#4402" class="Bound">x</a> <a id="4404" href="Relation.Binary.PropositionalEquality.Properties.html#4404" class="Bound">y</a> <a id="4406" class="Symbol">:</a> <a id="4408" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="4409" class="Symbol">}</a> <a id="4411" class="Symbol">{</a><a id="4412" href="Relation.Binary.PropositionalEquality.Properties.html#4412" class="Bound">P</a> <a id="4414" class="Symbol">:</a> <a id="4416" href="Relation.Unary.html#1232" class="Function">Pred</a> <a id="4421" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a> <a id="4423" href="Relation.Binary.PropositionalEquality.Properties.html#1064" class="Generalizable">p</a><a id="4424" class="Symbol">}</a> <a id="4426" class="Symbol">{</a><a id="4427" href="Relation.Binary.PropositionalEquality.Properties.html#4427" class="Bound">f</a> <a id="4429" class="Symbol">:</a> <a id="4431" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="4433" class="Symbol">→</a> <a id="4435" href="Relation.Binary.PropositionalEquality.Properties.html#1080" class="Generalizable">B</a><a id="4436" class="Symbol">}</a>
|
|||
|
|
<a id="4448" class="Symbol">(</a><a id="4449" href="Relation.Binary.PropositionalEquality.Properties.html#4449" class="Bound">x≡y</a> <a id="4453" class="Symbol">:</a> <a id="4455" href="Relation.Binary.PropositionalEquality.Properties.html#4402" class="Bound">x</a> <a id="4457" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4459" href="Relation.Binary.PropositionalEquality.Properties.html#4404" class="Bound">y</a><a id="4460" class="Symbol">)</a> <a id="4462" class="Symbol">{</a><a id="4463" href="Relation.Binary.PropositionalEquality.Properties.html#4463" class="Bound">p</a> <a id="4465" class="Symbol">:</a> <a id="4467" href="Relation.Binary.PropositionalEquality.Properties.html#4412" class="Bound">P</a> <a id="4469" class="Symbol">(</a><a id="4470" href="Relation.Binary.PropositionalEquality.Properties.html#4427" class="Bound">f</a> <a id="4472" href="Relation.Binary.PropositionalEquality.Properties.html#4402" class="Bound">x</a><a id="4473" class="Symbol">)}</a> <a id="4476" class="Symbol">→</a>
|
|||
|
|
<a id="4488" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4494" class="Symbol">(</a><a id="4495" href="Relation.Binary.PropositionalEquality.Properties.html#4412" class="Bound">P</a> <a id="4497" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4499" href="Relation.Binary.PropositionalEquality.Properties.html#4427" class="Bound">f</a><a id="4500" class="Symbol">)</a> <a id="4502" href="Relation.Binary.PropositionalEquality.Properties.html#4449" class="Bound">x≡y</a> <a id="4506" href="Relation.Binary.PropositionalEquality.Properties.html#4463" class="Bound">p</a> <a id="4508" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4510" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4516" href="Relation.Binary.PropositionalEquality.Properties.html#4412" class="Bound">P</a> <a id="4518" class="Symbol">(</a><a id="4519" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="4524" href="Relation.Binary.PropositionalEquality.Properties.html#4427" class="Bound">f</a> <a id="4526" href="Relation.Binary.PropositionalEquality.Properties.html#4449" class="Bound">x≡y</a><a id="4529" class="Symbol">)</a> <a id="4531" href="Relation.Binary.PropositionalEquality.Properties.html#4463" class="Bound">p</a>
|
|||
|
|
<a id="4533" href="Relation.Binary.PropositionalEquality.Properties.html#4389" class="Function">subst-∘</a> <a id="4541" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4546" class="Symbol">=</a> <a id="4548" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="4554" class="Comment">-- Lemma 2.3.11 in the HoTT book, and `transport_map` in the UniMath</a>
|
|||
|
|
<a id="4623" class="Comment">-- library</a>
|
|||
|
|
<a id="subst-application′"></a><a id="4634" href="Relation.Binary.PropositionalEquality.Properties.html#4634" class="Function">subst-application′</a> <a id="4653" class="Symbol">:</a> <a id="4655" class="Symbol">∀</a> <a id="4657" class="Symbol">{</a><a id="4658" href="Relation.Binary.PropositionalEquality.Properties.html#4658" class="Bound">a</a> <a id="4660" href="Relation.Binary.PropositionalEquality.Properties.html#4660" class="Bound">b₁</a> <a id="4663" href="Relation.Binary.PropositionalEquality.Properties.html#4663" class="Bound">b₂</a><a id="4665" class="Symbol">}</a> <a id="4667" class="Symbol">{</a><a id="4668" href="Relation.Binary.PropositionalEquality.Properties.html#4668" class="Bound">A</a> <a id="4670" class="Symbol">:</a> <a id="4672" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="4676" href="Relation.Binary.PropositionalEquality.Properties.html#4658" class="Bound">a</a><a id="4677" class="Symbol">}</a>
|
|||
|
|
<a id="4700" class="Symbol">(</a><a id="4701" href="Relation.Binary.PropositionalEquality.Properties.html#4701" class="Bound">B₁</a> <a id="4704" class="Symbol">:</a> <a id="4706" href="Relation.Binary.PropositionalEquality.Properties.html#4668" class="Bound">A</a> <a id="4708" class="Symbol">→</a> <a id="4710" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="4714" href="Relation.Binary.PropositionalEquality.Properties.html#4660" class="Bound">b₁</a><a id="4716" class="Symbol">)</a> <a id="4718" class="Symbol">{</a><a id="4719" href="Relation.Binary.PropositionalEquality.Properties.html#4719" class="Bound">B₂</a> <a id="4722" class="Symbol">:</a> <a id="4724" href="Relation.Binary.PropositionalEquality.Properties.html#4668" class="Bound">A</a> <a id="4726" class="Symbol">→</a> <a id="4728" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="4732" href="Relation.Binary.PropositionalEquality.Properties.html#4663" class="Bound">b₂</a><a id="4734" class="Symbol">}</a>
|
|||
|
|
<a id="4757" class="Symbol">{</a><a id="4758" href="Relation.Binary.PropositionalEquality.Properties.html#4758" class="Bound">x₁</a> <a id="4761" href="Relation.Binary.PropositionalEquality.Properties.html#4761" class="Bound">x₂</a> <a id="4764" class="Symbol">:</a> <a id="4766" href="Relation.Binary.PropositionalEquality.Properties.html#4668" class="Bound">A</a><a id="4767" class="Symbol">}</a> <a id="4769" class="Symbol">{</a><a id="4770" href="Relation.Binary.PropositionalEquality.Properties.html#4770" class="Bound">y</a> <a id="4772" class="Symbol">:</a> <a id="4774" href="Relation.Binary.PropositionalEquality.Properties.html#4701" class="Bound">B₁</a> <a id="4777" href="Relation.Binary.PropositionalEquality.Properties.html#4758" class="Bound">x₁</a><a id="4779" class="Symbol">}</a>
|
|||
|
|
<a id="4802" class="Symbol">(</a><a id="4803" href="Relation.Binary.PropositionalEquality.Properties.html#4803" class="Bound">g</a> <a id="4805" class="Symbol">:</a> <a id="4807" class="Symbol">∀</a> <a id="4809" href="Relation.Binary.PropositionalEquality.Properties.html#4809" class="Bound">x</a> <a id="4811" class="Symbol">→</a> <a id="4813" href="Relation.Binary.PropositionalEquality.Properties.html#4701" class="Bound">B₁</a> <a id="4816" href="Relation.Binary.PropositionalEquality.Properties.html#4809" class="Bound">x</a> <a id="4818" class="Symbol">→</a> <a id="4820" href="Relation.Binary.PropositionalEquality.Properties.html#4719" class="Bound">B₂</a> <a id="4823" href="Relation.Binary.PropositionalEquality.Properties.html#4809" class="Bound">x</a><a id="4824" class="Symbol">)</a> <a id="4826" class="Symbol">(</a><a id="4827" href="Relation.Binary.PropositionalEquality.Properties.html#4827" class="Bound">eq</a> <a id="4830" class="Symbol">:</a> <a id="4832" href="Relation.Binary.PropositionalEquality.Properties.html#4758" class="Bound">x₁</a> <a id="4835" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4837" href="Relation.Binary.PropositionalEquality.Properties.html#4761" class="Bound">x₂</a><a id="4839" class="Symbol">)</a> <a id="4841" class="Symbol">→</a>
|
|||
|
|
<a id="4864" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4870" href="Relation.Binary.PropositionalEquality.Properties.html#4719" class="Bound">B₂</a> <a id="4873" href="Relation.Binary.PropositionalEquality.Properties.html#4827" class="Bound">eq</a> <a id="4876" class="Symbol">(</a><a id="4877" href="Relation.Binary.PropositionalEquality.Properties.html#4803" class="Bound">g</a> <a id="4879" href="Relation.Binary.PropositionalEquality.Properties.html#4758" class="Bound">x₁</a> <a id="4882" href="Relation.Binary.PropositionalEquality.Properties.html#4770" class="Bound">y</a><a id="4883" class="Symbol">)</a> <a id="4885" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="4887" href="Relation.Binary.PropositionalEquality.Properties.html#4803" class="Bound">g</a> <a id="4889" href="Relation.Binary.PropositionalEquality.Properties.html#4761" class="Bound">x₂</a> <a id="4892" class="Symbol">(</a><a id="4893" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4899" href="Relation.Binary.PropositionalEquality.Properties.html#4701" class="Bound">B₁</a> <a id="4902" href="Relation.Binary.PropositionalEquality.Properties.html#4827" class="Bound">eq</a> <a id="4905" href="Relation.Binary.PropositionalEquality.Properties.html#4770" class="Bound">y</a><a id="4906" class="Symbol">)</a>
|
|||
|
|
<a id="4908" href="Relation.Binary.PropositionalEquality.Properties.html#4634" class="Function">subst-application′</a> <a id="4927" class="Symbol">_</a> <a id="4929" class="Symbol">_</a> <a id="4931" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4936" class="Symbol">=</a> <a id="4938" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="subst-application"></a><a id="4944" href="Relation.Binary.PropositionalEquality.Properties.html#4944" class="Function">subst-application</a> <a id="4962" class="Symbol">:</a> <a id="4964" class="Symbol">∀</a> <a id="4966" class="Symbol">{</a><a id="4967" href="Relation.Binary.PropositionalEquality.Properties.html#4967" class="Bound">a₁</a> <a id="4970" href="Relation.Binary.PropositionalEquality.Properties.html#4970" class="Bound">a₂</a> <a id="4973" href="Relation.Binary.PropositionalEquality.Properties.html#4973" class="Bound">b₁</a> <a id="4976" href="Relation.Binary.PropositionalEquality.Properties.html#4976" class="Bound">b₂</a><a id="4978" class="Symbol">}</a> <a id="4980" class="Symbol">{</a><a id="4981" href="Relation.Binary.PropositionalEquality.Properties.html#4981" class="Bound">A₁</a> <a id="4984" class="Symbol">:</a> <a id="4986" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="4990" href="Relation.Binary.PropositionalEquality.Properties.html#4967" class="Bound">a₁</a><a id="4992" class="Symbol">}</a> <a id="4994" class="Symbol">{</a><a id="4995" href="Relation.Binary.PropositionalEquality.Properties.html#4995" class="Bound">A₂</a> <a id="4998" class="Symbol">:</a> <a id="5000" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="5004" href="Relation.Binary.PropositionalEquality.Properties.html#4970" class="Bound">a₂</a><a id="5006" class="Symbol">}</a>
|
|||
|
|
<a id="5028" class="Symbol">(</a><a id="5029" href="Relation.Binary.PropositionalEquality.Properties.html#5029" class="Bound">B₁</a> <a id="5032" class="Symbol">:</a> <a id="5034" href="Relation.Binary.PropositionalEquality.Properties.html#4981" class="Bound">A₁</a> <a id="5037" class="Symbol">→</a> <a id="5039" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="5043" href="Relation.Binary.PropositionalEquality.Properties.html#4973" class="Bound">b₁</a><a id="5045" class="Symbol">)</a> <a id="5047" class="Symbol">{</a><a id="5048" href="Relation.Binary.PropositionalEquality.Properties.html#5048" class="Bound">B₂</a> <a id="5051" class="Symbol">:</a> <a id="5053" href="Relation.Binary.PropositionalEquality.Properties.html#4995" class="Bound">A₂</a> <a id="5056" class="Symbol">→</a> <a id="5058" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="5062" href="Relation.Binary.PropositionalEquality.Properties.html#4976" class="Bound">b₂</a><a id="5064" class="Symbol">}</a>
|
|||
|
|
<a id="5086" class="Symbol">{</a><a id="5087" href="Relation.Binary.PropositionalEquality.Properties.html#5087" class="Bound">f</a> <a id="5089" class="Symbol">:</a> <a id="5091" href="Relation.Binary.PropositionalEquality.Properties.html#4995" class="Bound">A₂</a> <a id="5094" class="Symbol">→</a> <a id="5096" href="Relation.Binary.PropositionalEquality.Properties.html#4981" class="Bound">A₁</a><a id="5098" class="Symbol">}</a> <a id="5100" class="Symbol">{</a><a id="5101" href="Relation.Binary.PropositionalEquality.Properties.html#5101" class="Bound">x₁</a> <a id="5104" href="Relation.Binary.PropositionalEquality.Properties.html#5104" class="Bound">x₂</a> <a id="5107" class="Symbol">:</a> <a id="5109" href="Relation.Binary.PropositionalEquality.Properties.html#4995" class="Bound">A₂</a><a id="5111" class="Symbol">}</a> <a id="5113" class="Symbol">{</a><a id="5114" href="Relation.Binary.PropositionalEquality.Properties.html#5114" class="Bound">y</a> <a id="5116" class="Symbol">:</a> <a id="5118" href="Relation.Binary.PropositionalEquality.Properties.html#5029" class="Bound">B₁</a> <a id="5121" class="Symbol">(</a><a id="5122" href="Relation.Binary.PropositionalEquality.Properties.html#5087" class="Bound">f</a> <a id="5124" href="Relation.Binary.PropositionalEquality.Properties.html#5101" class="Bound">x₁</a><a id="5126" class="Symbol">)}</a>
|
|||
|
|
<a id="5149" class="Symbol">(</a><a id="5150" href="Relation.Binary.PropositionalEquality.Properties.html#5150" class="Bound">g</a> <a id="5152" class="Symbol">:</a> <a id="5154" class="Symbol">∀</a> <a id="5156" href="Relation.Binary.PropositionalEquality.Properties.html#5156" class="Bound">x</a> <a id="5158" class="Symbol">→</a> <a id="5160" href="Relation.Binary.PropositionalEquality.Properties.html#5029" class="Bound">B₁</a> <a id="5163" class="Symbol">(</a><a id="5164" href="Relation.Binary.PropositionalEquality.Properties.html#5087" class="Bound">f</a> <a id="5166" href="Relation.Binary.PropositionalEquality.Properties.html#5156" class="Bound">x</a><a id="5167" class="Symbol">)</a> <a id="5169" class="Symbol">→</a> <a id="5171" href="Relation.Binary.PropositionalEquality.Properties.html#5048" class="Bound">B₂</a> <a id="5174" href="Relation.Binary.PropositionalEquality.Properties.html#5156" class="Bound">x</a><a id="5175" class="Symbol">)</a> <a id="5177" class="Symbol">(</a><a id="5178" href="Relation.Binary.PropositionalEquality.Properties.html#5178" class="Bound">eq</a> <a id="5181" class="Symbol">:</a> <a id="5183" href="Relation.Binary.PropositionalEquality.Properties.html#5101" class="Bound">x₁</a> <a id="5186" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="5188" href="Relation.Binary.PropositionalEquality.Properties.html#5104" class="Bound">x₂</a><a id="5190" class="Symbol">)</a> <a id="5192" class="Symbol">→</a>
|
|||
|
|
<a id="5214" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="5220" href="Relation.Binary.PropositionalEquality.Properties.html#5048" class="Bound">B₂</a> <a id="5223" href="Relation.Binary.PropositionalEquality.Properties.html#5178" class="Bound">eq</a> <a id="5226" class="Symbol">(</a><a id="5227" href="Relation.Binary.PropositionalEquality.Properties.html#5150" class="Bound">g</a> <a id="5229" href="Relation.Binary.PropositionalEquality.Properties.html#5101" class="Bound">x₁</a> <a id="5232" href="Relation.Binary.PropositionalEquality.Properties.html#5114" class="Bound">y</a><a id="5233" class="Symbol">)</a> <a id="5235" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="5237" href="Relation.Binary.PropositionalEquality.Properties.html#5150" class="Bound">g</a> <a id="5239" href="Relation.Binary.PropositionalEquality.Properties.html#5104" class="Bound">x₂</a> <a id="5242" class="Symbol">(</a><a id="5243" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="5249" href="Relation.Binary.PropositionalEquality.Properties.html#5029" class="Bound">B₁</a> <a id="5252" class="Symbol">(</a><a id="5253" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="5258" href="Relation.Binary.PropositionalEquality.Properties.html#5087" class="Bound">f</a> <a id="5260" href="Relation.Binary.PropositionalEquality.Properties.html#5178" class="Bound">eq</a><a id="5262" class="Symbol">)</a> <a id="5264" href="Relation.Binary.PropositionalEquality.Properties.html#5114" class="Bound">y</a><a id="5265" class="Symbol">)</a>
|
|||
|
|
<a id="5267" href="Relation.Binary.PropositionalEquality.Properties.html#4944" class="Function">subst-application</a> <a id="5285" class="Symbol">_</a> <a id="5287" class="Symbol">_</a> <a id="5289" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="5294" class="Symbol">=</a> <a id="5296" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="5302" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="5375" class="Comment">-- Structure of equality as a binary relation</a>
|
|||
|
|
|
|||
|
|
<a id="isEquivalence"></a><a id="5422" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a> <a id="5436" class="Symbol">:</a> <a id="5438" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="5452" class="Symbol">{</a><a id="5453" class="Argument">A</a> <a id="5455" class="Symbol">=</a> <a id="5457" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="5458" class="Symbol">}</a> <a id="5460" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="5464" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a> <a id="5478" class="Symbol">=</a> <a id="5480" class="Keyword">record</a>
|
|||
|
|
<a id="5489" class="Symbol">{</a> <a id="5491" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="5497" class="Symbol">=</a> <a id="5499" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5506" class="Symbol">;</a> <a id="5508" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="5514" class="Symbol">=</a> <a id="5516" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a>
|
|||
|
|
<a id="5522" class="Symbol">;</a> <a id="5524" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="5530" class="Symbol">=</a> <a id="5532" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a>
|
|||
|
|
<a id="5540" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="isDecEquivalence"></a><a id="5543" href="Relation.Binary.PropositionalEquality.Properties.html#5543" class="Function">isDecEquivalence</a> <a id="5560" class="Symbol">:</a> <a id="5562" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="5572" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="5576" class="Symbol">→</a> <a id="5578" href="Relation.Binary.Structures.html#1852" class="Record">IsDecEquivalence</a> <a id="5595" class="Symbol">{</a><a id="5596" class="Argument">A</a> <a id="5598" class="Symbol">=</a> <a id="5600" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="5601" class="Symbol">}</a> <a id="5603" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="5607" href="Relation.Binary.PropositionalEquality.Properties.html#5543" class="Function">isDecEquivalence</a> <a id="5624" href="Relation.Binary.PropositionalEquality.Properties.html#5624" class="Bound Operator">_≟_</a> <a id="5628" class="Symbol">=</a> <a id="5630" class="Keyword">record</a>
|
|||
|
|
<a id="5639" class="Symbol">{</a> <a id="5641" href="Relation.Binary.Structures.html#1915" class="Field">isEquivalence</a> <a id="5655" class="Symbol">=</a> <a id="5657" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
|
|||
|
|
<a id="5673" class="Symbol">;</a> <a id="5675" href="Relation.Binary.Structures.html#1949" class="Field Operator">_≟_</a> <a id="5689" class="Symbol">=</a> <a id="5691" href="Relation.Binary.PropositionalEquality.Properties.html#5624" class="Bound Operator">_≟_</a>
|
|||
|
|
<a id="5697" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="setoid"></a><a id="5700" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="5707" class="Symbol">:</a> <a id="5709" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="5713" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a> <a id="5715" class="Symbol">→</a> <a id="5717" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="5724" class="Symbol">_</a> <a id="5726" class="Symbol">_</a>
|
|||
|
|
<a id="5728" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="5735" href="Relation.Binary.PropositionalEquality.Properties.html#5735" class="Bound">A</a> <a id="5737" class="Symbol">=</a> <a id="5739" class="Keyword">record</a>
|
|||
|
|
<a id="5748" class="Symbol">{</a> <a id="5750" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="5764" class="Symbol">=</a> <a id="5766" href="Relation.Binary.PropositionalEquality.Properties.html#5735" class="Bound">A</a>
|
|||
|
|
<a id="5770" class="Symbol">;</a> <a id="5772" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="5786" class="Symbol">=</a> <a id="5788" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="5794" class="Symbol">;</a> <a id="5796" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="5810" class="Symbol">=</a> <a id="5812" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
|
|||
|
|
<a id="5828" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="decSetoid"></a><a id="5831" href="Relation.Binary.PropositionalEquality.Properties.html#5831" class="Function">decSetoid</a> <a id="5841" class="Symbol">:</a> <a id="5843" href="Relation.Binary.Definitions.html#6469" class="Function">DecidableEquality</a> <a id="5861" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a> <a id="5863" class="Symbol">→</a> <a id="5865" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="5875" class="Symbol">_</a> <a id="5877" class="Symbol">_</a>
|
|||
|
|
<a id="5879" href="Relation.Binary.PropositionalEquality.Properties.html#5831" class="Function">decSetoid</a> <a id="5889" href="Relation.Binary.PropositionalEquality.Properties.html#5889" class="Bound Operator">_≟_</a> <a id="5893" class="Symbol">=</a> <a id="5895" class="Keyword">record</a>
|
|||
|
|
<a id="5904" class="Symbol">{</a> <a id="5906" href="Relation.Binary.Bundles.html#1643" class="Field Operator">_≈_</a> <a id="5923" class="Symbol">=</a> <a id="5925" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="5931" class="Symbol">;</a> <a id="5933" href="Relation.Binary.Bundles.html#1680" class="Field">isDecEquivalence</a> <a id="5950" class="Symbol">=</a> <a id="5952" href="Relation.Binary.PropositionalEquality.Properties.html#5543" class="Function">isDecEquivalence</a> <a id="5969" href="Relation.Binary.PropositionalEquality.Properties.html#5889" class="Bound Operator">_≟_</a>
|
|||
|
|
<a id="5975" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="5978" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="6051" class="Comment">-- Bundles for equality as a binary relation</a>
|
|||
|
|
|
|||
|
|
<a id="isPreorder"></a><a id="6097" href="Relation.Binary.PropositionalEquality.Properties.html#6097" class="Function">isPreorder</a> <a id="6108" class="Symbol">:</a> <a id="6110" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="6121" class="Symbol">{</a><a id="6122" class="Argument">A</a> <a id="6124" class="Symbol">=</a> <a id="6126" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="6127" class="Symbol">}</a> <a id="6129" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="6133" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="6137" href="Relation.Binary.PropositionalEquality.Properties.html#6097" class="Function">isPreorder</a> <a id="6148" class="Symbol">=</a> <a id="6150" href="Relation.Binary.Properties.Setoid.html#1184" class="Function">Setoid.≈-isPreorder</a> <a id="6170" class="Symbol">(</a><a id="6171" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="6178" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="isPartialOrder"></a><a id="6182" href="Relation.Binary.PropositionalEquality.Properties.html#6182" class="Function">isPartialOrder</a> <a id="6197" class="Symbol">:</a> <a id="6199" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="6214" class="Symbol">{</a><a id="6215" class="Argument">A</a> <a id="6217" class="Symbol">=</a> <a id="6219" href="Relation.Binary.PropositionalEquality.Properties.html#1078" class="Generalizable">A</a><a id="6220" class="Symbol">}</a> <a id="6222" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="6226" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="6230" href="Relation.Binary.PropositionalEquality.Properties.html#6182" class="Function">isPartialOrder</a> <a id="6245" class="Symbol">=</a> <a id="6247" href="Relation.Binary.Properties.Setoid.html#1328" class="Function">Setoid.≈-isPartialOrder</a> <a id="6271" class="Symbol">(</a><a id="6272" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="6279" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="preorder"></a><a id="6283" href="Relation.Binary.PropositionalEquality.Properties.html#6283" class="Function">preorder</a> <a id="6292" class="Symbol">:</a> <a id="6294" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="6298" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a> <a id="6300" class="Symbol">→</a> <a id="6302" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="6311" class="Symbol">_</a> <a id="6313" class="Symbol">_</a> <a id="6315" class="Symbol">_</a>
|
|||
|
|
<a id="6317" href="Relation.Binary.PropositionalEquality.Properties.html#6283" class="Function">preorder</a> <a id="6326" href="Relation.Binary.PropositionalEquality.Properties.html#6326" class="Bound">A</a> <a id="6328" class="Symbol">=</a> <a id="6330" href="Relation.Binary.Properties.Setoid.html#1539" class="Function">Setoid.≈-preorder</a> <a id="6348" class="Symbol">(</a><a id="6349" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="6356" href="Relation.Binary.PropositionalEquality.Properties.html#6326" class="Bound">A</a><a id="6357" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="poset"></a><a id="6360" href="Relation.Binary.PropositionalEquality.Properties.html#6360" class="Function">poset</a> <a id="6366" class="Symbol">:</a> <a id="6368" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="6372" href="Relation.Binary.PropositionalEquality.Properties.html#1058" class="Generalizable">a</a> <a id="6374" class="Symbol">→</a> <a id="6376" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="6382" class="Symbol">_</a> <a id="6384" class="Symbol">_</a> <a id="6386" class="Symbol">_</a>
|
|||
|
|
<a id="6388" href="Relation.Binary.PropositionalEquality.Properties.html#6360" class="Function">poset</a> <a id="6394" href="Relation.Binary.PropositionalEquality.Properties.html#6394" class="Bound">A</a> <a id="6396" class="Symbol">=</a> <a id="6398" href="Relation.Binary.Properties.Setoid.html#1622" class="Function">Setoid.≈-poset</a> <a id="6413" class="Symbol">(</a><a id="6414" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="6421" href="Relation.Binary.PropositionalEquality.Properties.html#6394" class="Bound">A</a><a id="6422" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="6425" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="6498" class="Comment">-- Reasoning</a>
|
|||
|
|
|
|||
|
|
<a id="6512" class="Comment">-- This is a special instance of `Relation.Binary.Reasoning.Setoid`.</a>
|
|||
|
|
<a id="6581" class="Comment">-- Rather than instantiating the latter with (setoid A), we reimplement</a>
|
|||
|
|
<a id="6653" class="Comment">-- equation chains from scratch since then goals are printed much more</a>
|
|||
|
|
<a id="6724" class="Comment">-- readably.</a>
|
|||
|
|
<a id="6737" class="Keyword">module</a> <a id="≡-Reasoning"></a><a id="6744" href="Relation.Binary.PropositionalEquality.Properties.html#6744" class="Module">≡-Reasoning</a> <a id="6756" class="Symbol">{</a><a id="6757" href="Relation.Binary.PropositionalEquality.Properties.html#6757" class="Bound">a</a><a id="6758" class="Symbol">}</a> <a id="6760" class="Symbol">{</a><a id="6761" href="Relation.Binary.PropositionalEquality.Properties.html#6761" class="Bound">A</a> <a id="6763" class="Symbol">:</a> <a id="6765" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="6769" href="Relation.Binary.PropositionalEquality.Properties.html#6757" class="Bound">a</a><a id="6770" class="Symbol">}</a> <a id="6772" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="6781" class="Keyword">open</a> <a id="6786" href="Relation.Binary.Reasoning.Syntax.html#1408" class="Module">begin-syntax</a> <a id="6799" class="Symbol">{</a><a id="6800" class="Argument">A</a> <a id="6802" class="Symbol">=</a> <a id="6804" href="Relation.Binary.PropositionalEquality.Properties.html#6761" class="Bound">A</a><a id="6805" class="Symbol">}</a> <a id="6807" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="6811" href="Function.Base.html#704" class="Function">id</a> <a id="6814" class="Keyword">public</a>
|
|||
|
|
<a id="6823" class="Keyword">open</a> <a id="6828" href="Relation.Binary.Reasoning.Syntax.html#10883" class="Module">≡-syntax</a> <a id="6837" class="Symbol">{</a><a id="6838" class="Argument">A</a> <a id="6840" class="Symbol">=</a> <a id="6842" href="Relation.Binary.PropositionalEquality.Properties.html#6761" class="Bound">A</a><a id="6843" class="Symbol">}</a> <a id="6845" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="6849" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="6855" class="Keyword">public</a>
|
|||
|
|
<a id="6864" class="Keyword">open</a> <a id="6869" href="Relation.Binary.Reasoning.Syntax.html#12202" class="Module">end-syntax</a> <a id="6880" class="Symbol">{</a><a id="6881" class="Argument">A</a> <a id="6883" class="Symbol">=</a> <a id="6885" href="Relation.Binary.PropositionalEquality.Properties.html#6761" class="Bound">A</a><a id="6886" class="Symbol">}</a> <a id="6888" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="6892" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="6897" class="Keyword">public</a>
|
|||
|
|
</pre></body></html>
|