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<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>
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<a id="74" class="Comment">-- The Agda standard library</a>
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<a id="103" class="Comment">--</a>
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<a id="106" class="Comment">-- Propositional equality</a>
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<a id="132" class="Comment">--</a>
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<a id="135" class="Comment">-- This file contains some core properies of propositional equality which</a>
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<a id="209" class="Comment">-- are re-exported by Relation.Binary.PropositionalEquality. They are</a>
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<a id="279" class="Comment">-- ``equality rearrangement'' lemmas.</a>
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<a id="317" class="Comment">------------------------------------------------------------------------</a>
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<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>
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<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>
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<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#624" class="Function">id</a><a id="534" class="Symbol">;</a> <a id="536" href="Function.Base.html#1040" class="Function Operator">_∘_</a><a id="539" class="Symbol">)</a>
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<a id="541" class="Keyword">open</a> <a id="546" class="Keyword">import</a> <a id="553" href="Level.html" class="Module">Level</a>
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<a id="559" class="Keyword">open</a> <a id="564" class="Keyword">import</a> <a id="571" href="Relation.Binary.html" class="Module">Relation.Binary</a>
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<a id="587" class="Keyword">open</a> <a id="592" class="Keyword">import</a> <a id="599" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a>
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<a id="642" class="Keyword">open</a> <a id="647" class="Keyword">import</a> <a id="654" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="669" class="Keyword">using</a> <a id="675" class="Symbol">(</a><a id="676" href="Relation.Unary.html#1110" class="Function">Pred</a><a id="680" class="Symbol">)</a>
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<a id="683" class="Keyword">private</a>
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<a id="693" class="Keyword">variable</a>
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<a id="706" href="Relation.Binary.PropositionalEquality.Properties.html#706" class="Generalizable">a</a> <a id="708" href="Relation.Binary.PropositionalEquality.Properties.html#708" class="Generalizable">p</a> <a id="710" class="Symbol">:</a> <a id="712" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="722" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="724" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a> <a id="726" href="Relation.Binary.PropositionalEquality.Properties.html#726" class="Generalizable">C</a> <a id="728" class="Symbol">:</a> <a id="730" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="734" href="Relation.Binary.PropositionalEquality.Properties.html#706" class="Generalizable">a</a>
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<a id="737" class="Comment">------------------------------------------------------------------------</a>
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<a id="810" class="Comment">-- Various equality rearrangement lemmas</a>
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<a id="trans-reflʳ"></a><a id="852" href="Relation.Binary.PropositionalEquality.Properties.html#852" class="Function">trans-reflʳ</a> <a id="864" class="Symbol">:</a> <a id="866" class="Symbol">∀</a> <a id="868" class="Symbol">{</a><a id="869" href="Relation.Binary.PropositionalEquality.Properties.html#869" class="Bound">x</a> <a id="871" href="Relation.Binary.PropositionalEquality.Properties.html#871" class="Bound">y</a> <a id="873" class="Symbol">:</a> <a id="875" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="876" class="Symbol">}</a> <a id="878" class="Symbol">(</a><a id="879" href="Relation.Binary.PropositionalEquality.Properties.html#879" class="Bound">p</a> <a id="881" class="Symbol">:</a> <a id="883" href="Relation.Binary.PropositionalEquality.Properties.html#869" class="Bound">x</a> <a id="885" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="887" href="Relation.Binary.PropositionalEquality.Properties.html#871" class="Bound">y</a><a id="888" class="Symbol">)</a> <a id="890" class="Symbol">→</a> <a id="892" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="898" href="Relation.Binary.PropositionalEquality.Properties.html#879" class="Bound">p</a> <a id="900" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="905" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="907" href="Relation.Binary.PropositionalEquality.Properties.html#879" class="Bound">p</a>
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<a id="909" href="Relation.Binary.PropositionalEquality.Properties.html#852" class="Function">trans-reflʳ</a> <a id="921" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="926" class="Symbol">=</a> <a id="928" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="trans-assoc"></a><a id="934" href="Relation.Binary.PropositionalEquality.Properties.html#934" class="Function">trans-assoc</a> <a id="946" class="Symbol">:</a> <a id="948" class="Symbol">∀</a> <a id="950" class="Symbol">{</a><a id="951" href="Relation.Binary.PropositionalEquality.Properties.html#951" class="Bound">x</a> <a id="953" href="Relation.Binary.PropositionalEquality.Properties.html#953" class="Bound">y</a> <a id="955" href="Relation.Binary.PropositionalEquality.Properties.html#955" class="Bound">z</a> <a id="957" href="Relation.Binary.PropositionalEquality.Properties.html#957" class="Bound">u</a> <a id="959" class="Symbol">:</a> <a id="961" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="962" class="Symbol">}</a> <a id="964" class="Symbol">(</a><a id="965" href="Relation.Binary.PropositionalEquality.Properties.html#965" class="Bound">p</a> <a id="967" class="Symbol">:</a> <a id="969" href="Relation.Binary.PropositionalEquality.Properties.html#951" class="Bound">x</a> <a id="971" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="973" href="Relation.Binary.PropositionalEquality.Properties.html#953" class="Bound">y</a><a id="974" class="Symbol">)</a> <a id="976" class="Symbol">{</a><a id="977" href="Relation.Binary.PropositionalEquality.Properties.html#977" class="Bound">q</a> <a id="979" class="Symbol">:</a> <a id="981" href="Relation.Binary.PropositionalEquality.Properties.html#953" class="Bound">y</a> <a id="983" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="985" href="Relation.Binary.PropositionalEquality.Properties.html#955" class="Bound">z</a><a id="986" class="Symbol">}</a> <a id="988" class="Symbol">{</a><a id="989" href="Relation.Binary.PropositionalEquality.Properties.html#989" class="Bound">r</a> <a id="991" class="Symbol">:</a> <a id="993" href="Relation.Binary.PropositionalEquality.Properties.html#955" class="Bound">z</a> <a id="995" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="997" href="Relation.Binary.PropositionalEquality.Properties.html#957" class="Bound">u</a><a id="998" class="Symbol">}</a> <a id="1000" class="Symbol">→</a>
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<a id="1004" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1010" class="Symbol">(</a><a id="1011" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1017" href="Relation.Binary.PropositionalEquality.Properties.html#965" class="Bound">p</a> <a id="1019" href="Relation.Binary.PropositionalEquality.Properties.html#977" class="Bound">q</a><a id="1020" class="Symbol">)</a> <a id="1022" href="Relation.Binary.PropositionalEquality.Properties.html#989" class="Bound">r</a> <a id="1024" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1026" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1032" href="Relation.Binary.PropositionalEquality.Properties.html#965" class="Bound">p</a> <a id="1034" class="Symbol">(</a><a id="1035" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1041" href="Relation.Binary.PropositionalEquality.Properties.html#977" class="Bound">q</a> <a id="1043" href="Relation.Binary.PropositionalEquality.Properties.html#989" class="Bound">r</a><a id="1044" class="Symbol">)</a>
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<a id="1046" href="Relation.Binary.PropositionalEquality.Properties.html#934" class="Function">trans-assoc</a> <a id="1058" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1063" class="Symbol">=</a> <a id="1065" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="trans-symˡ"></a><a id="1071" href="Relation.Binary.PropositionalEquality.Properties.html#1071" class="Function">trans-symˡ</a> <a id="1082" class="Symbol">:</a> <a id="1084" class="Symbol">∀</a> <a id="1086" class="Symbol">{</a><a id="1087" href="Relation.Binary.PropositionalEquality.Properties.html#1087" class="Bound">x</a> <a id="1089" href="Relation.Binary.PropositionalEquality.Properties.html#1089" class="Bound">y</a> <a id="1091" class="Symbol">:</a> <a id="1093" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1094" class="Symbol">}</a> <a id="1096" class="Symbol">(</a><a id="1097" href="Relation.Binary.PropositionalEquality.Properties.html#1097" class="Bound">p</a> <a id="1099" class="Symbol">:</a> <a id="1101" href="Relation.Binary.PropositionalEquality.Properties.html#1087" class="Bound">x</a> <a id="1103" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1105" href="Relation.Binary.PropositionalEquality.Properties.html#1089" class="Bound">y</a><a id="1106" class="Symbol">)</a> <a id="1108" class="Symbol">→</a> <a id="1110" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1116" class="Symbol">(</a><a id="1117" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="1121" href="Relation.Binary.PropositionalEquality.Properties.html#1097" class="Bound">p</a><a id="1122" class="Symbol">)</a> <a id="1124" href="Relation.Binary.PropositionalEquality.Properties.html#1097" class="Bound">p</a> <a id="1126" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1128" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="1133" href="Relation.Binary.PropositionalEquality.Properties.html#1071" class="Function">trans-symˡ</a> <a id="1144" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1149" class="Symbol">=</a> <a id="1151" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="trans-symʳ"></a><a id="1157" href="Relation.Binary.PropositionalEquality.Properties.html#1157" class="Function">trans-symʳ</a> <a id="1168" class="Symbol">:</a> <a id="1170" class="Symbol">∀</a> <a id="1172" class="Symbol">{</a><a id="1173" href="Relation.Binary.PropositionalEquality.Properties.html#1173" class="Bound">x</a> <a id="1175" href="Relation.Binary.PropositionalEquality.Properties.html#1175" class="Bound">y</a> <a id="1177" class="Symbol">:</a> <a id="1179" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1180" class="Symbol">}</a> <a id="1182" class="Symbol">(</a><a id="1183" href="Relation.Binary.PropositionalEquality.Properties.html#1183" class="Bound">p</a> <a id="1185" class="Symbol">:</a> <a id="1187" href="Relation.Binary.PropositionalEquality.Properties.html#1173" class="Bound">x</a> <a id="1189" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1191" href="Relation.Binary.PropositionalEquality.Properties.html#1175" class="Bound">y</a><a id="1192" class="Symbol">)</a> <a id="1194" class="Symbol">→</a> <a id="1196" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1202" href="Relation.Binary.PropositionalEquality.Properties.html#1183" class="Bound">p</a> <a id="1204" class="Symbol">(</a><a id="1205" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="1209" href="Relation.Binary.PropositionalEquality.Properties.html#1183" class="Bound">p</a><a id="1210" class="Symbol">)</a> <a id="1212" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1214" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="1219" href="Relation.Binary.PropositionalEquality.Properties.html#1157" class="Function">trans-symʳ</a> <a id="1230" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1235" class="Symbol">=</a> <a id="1237" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="trans-injectiveˡ"></a><a id="1243" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Function">trans-injectiveˡ</a> <a id="1260" class="Symbol">:</a> <a id="1262" class="Symbol">∀</a> <a id="1264" class="Symbol">{</a><a id="1265" href="Relation.Binary.PropositionalEquality.Properties.html#1265" class="Bound">x</a> <a id="1267" href="Relation.Binary.PropositionalEquality.Properties.html#1267" class="Bound">y</a> <a id="1269" href="Relation.Binary.PropositionalEquality.Properties.html#1269" class="Bound">z</a> <a id="1271" class="Symbol">:</a> <a id="1273" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1274" class="Symbol">}</a> <a id="1276" class="Symbol">{</a><a id="1277" href="Relation.Binary.PropositionalEquality.Properties.html#1277" class="Bound">p₁</a> <a id="1280" href="Relation.Binary.PropositionalEquality.Properties.html#1280" class="Bound">p₂</a> <a id="1283" class="Symbol">:</a> <a id="1285" href="Relation.Binary.PropositionalEquality.Properties.html#1265" class="Bound">x</a> <a id="1287" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1289" href="Relation.Binary.PropositionalEquality.Properties.html#1267" class="Bound">y</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">q</a> <a id="1295" class="Symbol">:</a> <a id="1297" href="Relation.Binary.PropositionalEquality.Properties.html#1267" class="Bound">y</a> <a id="1299" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1301" href="Relation.Binary.PropositionalEquality.Properties.html#1269" class="Bound">z</a><a id="1302" class="Symbol">)</a> <a id="1304" class="Symbol">→</a>
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<a id="1325" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1331" href="Relation.Binary.PropositionalEquality.Properties.html#1277" class="Bound">p₁</a> <a id="1334" href="Relation.Binary.PropositionalEquality.Properties.html#1293" class="Bound">q</a> <a id="1336" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1338" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1344" href="Relation.Binary.PropositionalEquality.Properties.html#1280" class="Bound">p₂</a> <a id="1347" href="Relation.Binary.PropositionalEquality.Properties.html#1293" class="Bound">q</a> <a id="1349" class="Symbol">→</a> <a id="1351" href="Relation.Binary.PropositionalEquality.Properties.html#1277" class="Bound">p₁</a> <a id="1354" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1356" href="Relation.Binary.PropositionalEquality.Properties.html#1280" class="Bound">p₂</a>
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<a id="1359" href="Relation.Binary.PropositionalEquality.Properties.html#1243" class="Function">trans-injectiveˡ</a> <a id="1376" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1381" class="Symbol">=</a> <a id="1383" href="Relation.Binary.PropositionalEquality.Core.html#1844" class="Function">subst₂</a> <a id="1390" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="1394" class="Symbol">(</a><a id="1395" href="Relation.Binary.PropositionalEquality.Properties.html#852" class="Function">trans-reflʳ</a> <a id="1407" class="Symbol">_)</a> <a id="1410" class="Symbol">(</a><a id="1411" href="Relation.Binary.PropositionalEquality.Properties.html#852" class="Function">trans-reflʳ</a> <a id="1423" class="Symbol">_)</a>
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<a id="trans-injectiveʳ"></a><a id="1427" href="Relation.Binary.PropositionalEquality.Properties.html#1427" class="Function">trans-injectiveʳ</a> <a id="1444" class="Symbol">:</a> <a id="1446" class="Symbol">∀</a> <a id="1448" class="Symbol">{</a><a id="1449" href="Relation.Binary.PropositionalEquality.Properties.html#1449" class="Bound">x</a> <a id="1451" href="Relation.Binary.PropositionalEquality.Properties.html#1451" class="Bound">y</a> <a id="1453" href="Relation.Binary.PropositionalEquality.Properties.html#1453" class="Bound">z</a> <a id="1455" class="Symbol">:</a> <a id="1457" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1458" class="Symbol">}</a> <a id="1460" class="Symbol">(</a><a id="1461" href="Relation.Binary.PropositionalEquality.Properties.html#1461" class="Bound">p</a> <a id="1463" class="Symbol">:</a> <a id="1465" href="Relation.Binary.PropositionalEquality.Properties.html#1449" class="Bound">x</a> <a id="1467" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1469" href="Relation.Binary.PropositionalEquality.Properties.html#1451" class="Bound">y</a><a id="1470" class="Symbol">)</a> <a id="1472" class="Symbol">{</a><a id="1473" href="Relation.Binary.PropositionalEquality.Properties.html#1473" class="Bound">q₁</a> <a id="1476" href="Relation.Binary.PropositionalEquality.Properties.html#1476" class="Bound">q₂</a> <a id="1479" class="Symbol">:</a> <a id="1481" href="Relation.Binary.PropositionalEquality.Properties.html#1451" class="Bound">y</a> <a id="1483" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1485" href="Relation.Binary.PropositionalEquality.Properties.html#1453" class="Bound">z</a><a id="1486" class="Symbol">}</a> <a id="1488" class="Symbol">→</a>
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||
<a id="1509" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1515" href="Relation.Binary.PropositionalEquality.Properties.html#1461" class="Bound">p</a> <a id="1517" href="Relation.Binary.PropositionalEquality.Properties.html#1473" class="Bound">q₁</a> <a id="1520" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1522" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1528" href="Relation.Binary.PropositionalEquality.Properties.html#1461" class="Bound">p</a> <a id="1530" href="Relation.Binary.PropositionalEquality.Properties.html#1476" class="Bound">q₂</a> <a id="1533" class="Symbol">→</a> <a id="1535" href="Relation.Binary.PropositionalEquality.Properties.html#1473" class="Bound">q₁</a> <a id="1538" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1540" href="Relation.Binary.PropositionalEquality.Properties.html#1476" class="Bound">q₂</a>
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||
<a id="1543" href="Relation.Binary.PropositionalEquality.Properties.html#1427" class="Function">trans-injectiveʳ</a> <a id="1560" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1565" href="Relation.Binary.PropositionalEquality.Properties.html#1565" class="Bound">eq</a> <a id="1568" class="Symbol">=</a> <a id="1570" href="Relation.Binary.PropositionalEquality.Properties.html#1565" class="Bound">eq</a>
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<a id="cong-id"></a><a id="1574" href="Relation.Binary.PropositionalEquality.Properties.html#1574" class="Function">cong-id</a> <a id="1582" class="Symbol">:</a> <a id="1584" class="Symbol">∀</a> <a id="1586" class="Symbol">{</a><a id="1587" href="Relation.Binary.PropositionalEquality.Properties.html#1587" class="Bound">x</a> <a id="1589" href="Relation.Binary.PropositionalEquality.Properties.html#1589" class="Bound">y</a> <a id="1591" class="Symbol">:</a> <a id="1593" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1594" class="Symbol">}</a> <a id="1596" class="Symbol">(</a><a id="1597" href="Relation.Binary.PropositionalEquality.Properties.html#1597" class="Bound">p</a> <a id="1599" class="Symbol">:</a> <a id="1601" href="Relation.Binary.PropositionalEquality.Properties.html#1587" class="Bound">x</a> <a id="1603" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1605" href="Relation.Binary.PropositionalEquality.Properties.html#1589" class="Bound">y</a><a id="1606" class="Symbol">)</a> <a id="1608" class="Symbol">→</a> <a id="1610" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1615" href="Function.Base.html#624" class="Function">id</a> <a id="1618" href="Relation.Binary.PropositionalEquality.Properties.html#1597" class="Bound">p</a> <a id="1620" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1622" href="Relation.Binary.PropositionalEquality.Properties.html#1597" class="Bound">p</a>
|
||
<a id="1624" href="Relation.Binary.PropositionalEquality.Properties.html#1574" class="Function">cong-id</a> <a id="1632" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1637" class="Symbol">=</a> <a id="1639" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="cong-∘"></a><a id="1645" href="Relation.Binary.PropositionalEquality.Properties.html#1645" class="Function">cong-∘</a> <a id="1652" class="Symbol">:</a> <a id="1654" class="Symbol">∀</a> <a id="1656" class="Symbol">{</a><a id="1657" href="Relation.Binary.PropositionalEquality.Properties.html#1657" class="Bound">x</a> <a id="1659" href="Relation.Binary.PropositionalEquality.Properties.html#1659" class="Bound">y</a> <a id="1661" class="Symbol">:</a> <a id="1663" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1664" class="Symbol">}</a> <a id="1666" class="Symbol">{</a><a id="1667" href="Relation.Binary.PropositionalEquality.Properties.html#1667" class="Bound">f</a> <a id="1669" class="Symbol">:</a> <a id="1671" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a> <a id="1673" class="Symbol">→</a> <a id="1675" href="Relation.Binary.PropositionalEquality.Properties.html#726" class="Generalizable">C</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">g</a> <a id="1681" class="Symbol">:</a> <a id="1683" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="1685" class="Symbol">→</a> <a id="1687" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a><a id="1688" class="Symbol">}</a> <a id="1690" class="Symbol">(</a><a id="1691" href="Relation.Binary.PropositionalEquality.Properties.html#1691" class="Bound">p</a> <a id="1693" class="Symbol">:</a> <a id="1695" href="Relation.Binary.PropositionalEquality.Properties.html#1657" class="Bound">x</a> <a id="1697" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1699" href="Relation.Binary.PropositionalEquality.Properties.html#1659" class="Bound">y</a><a id="1700" class="Symbol">)</a> <a id="1702" class="Symbol">→</a>
|
||
<a id="1713" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1718" class="Symbol">(</a><a id="1719" href="Relation.Binary.PropositionalEquality.Properties.html#1667" class="Bound">f</a> <a id="1721" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="1723" href="Relation.Binary.PropositionalEquality.Properties.html#1679" class="Bound">g</a><a id="1724" class="Symbol">)</a> <a id="1726" href="Relation.Binary.PropositionalEquality.Properties.html#1691" class="Bound">p</a> <a id="1728" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1730" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1735" href="Relation.Binary.PropositionalEquality.Properties.html#1667" class="Bound">f</a> <a id="1737" class="Symbol">(</a><a id="1738" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1743" href="Relation.Binary.PropositionalEquality.Properties.html#1679" class="Bound">g</a> <a id="1745" href="Relation.Binary.PropositionalEquality.Properties.html#1691" class="Bound">p</a><a id="1746" class="Symbol">)</a>
|
||
<a id="1748" href="Relation.Binary.PropositionalEquality.Properties.html#1645" class="Function">cong-∘</a> <a id="1755" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1760" class="Symbol">=</a> <a id="1762" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="trans-cong"></a><a id="1768" href="Relation.Binary.PropositionalEquality.Properties.html#1768" class="Function">trans-cong</a> <a id="1779" class="Symbol">:</a> <a id="1781" class="Symbol">∀</a> <a id="1783" class="Symbol">{</a><a id="1784" href="Relation.Binary.PropositionalEquality.Properties.html#1784" class="Bound">x</a> <a id="1786" href="Relation.Binary.PropositionalEquality.Properties.html#1786" class="Bound">y</a> <a id="1788" href="Relation.Binary.PropositionalEquality.Properties.html#1788" class="Bound">z</a> <a id="1790" class="Symbol">:</a> <a id="1792" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="1793" class="Symbol">}</a> <a id="1795" class="Symbol">{</a><a id="1796" href="Relation.Binary.PropositionalEquality.Properties.html#1796" class="Bound">f</a> <a id="1798" class="Symbol">:</a> <a id="1800" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="1802" class="Symbol">→</a> <a id="1804" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a><a id="1805" class="Symbol">}</a> <a id="1807" class="Symbol">(</a><a id="1808" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Bound">p</a> <a id="1810" class="Symbol">:</a> <a id="1812" href="Relation.Binary.PropositionalEquality.Properties.html#1784" class="Bound">x</a> <a id="1814" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1816" href="Relation.Binary.PropositionalEquality.Properties.html#1786" class="Bound">y</a><a id="1817" class="Symbol">)</a> <a id="1819" class="Symbol">{</a><a id="1820" href="Relation.Binary.PropositionalEquality.Properties.html#1820" class="Bound">q</a> <a id="1822" class="Symbol">:</a> <a id="1824" href="Relation.Binary.PropositionalEquality.Properties.html#1786" class="Bound">y</a> <a id="1826" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1828" href="Relation.Binary.PropositionalEquality.Properties.html#1788" class="Bound">z</a><a id="1829" class="Symbol">}</a> <a id="1831" class="Symbol">→</a>
|
||
<a id="1846" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1852" class="Symbol">(</a><a id="1853" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1858" href="Relation.Binary.PropositionalEquality.Properties.html#1796" class="Bound">f</a> <a id="1860" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Bound">p</a><a id="1861" class="Symbol">)</a> <a id="1863" class="Symbol">(</a><a id="1864" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1869" href="Relation.Binary.PropositionalEquality.Properties.html#1796" class="Bound">f</a> <a id="1871" href="Relation.Binary.PropositionalEquality.Properties.html#1820" class="Bound">q</a><a id="1872" class="Symbol">)</a> <a id="1874" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1876" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="1881" href="Relation.Binary.PropositionalEquality.Properties.html#1796" class="Bound">f</a> <a id="1883" class="Symbol">(</a><a id="1884" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="1890" href="Relation.Binary.PropositionalEquality.Properties.html#1808" class="Bound">p</a> <a id="1892" href="Relation.Binary.PropositionalEquality.Properties.html#1820" class="Bound">q</a><a id="1893" class="Symbol">)</a>
|
||
<a id="1895" href="Relation.Binary.PropositionalEquality.Properties.html#1768" class="Function">trans-cong</a> <a id="1906" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1911" class="Symbol">=</a> <a id="1913" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="cong₂-reflˡ"></a><a id="1919" href="Relation.Binary.PropositionalEquality.Properties.html#1919" class="Function">cong₂-reflˡ</a> <a id="1931" class="Symbol">:</a> <a id="1933" class="Symbol">∀</a> <a id="1935" class="Symbol">{</a><a id="1936" href="Relation.Binary.PropositionalEquality.Properties.html#1936" class="Bound Operator">_∙_</a> <a id="1940" class="Symbol">:</a> <a id="1942" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="1944" class="Symbol">→</a> <a id="1946" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a> <a id="1948" class="Symbol">→</a> <a id="1950" href="Relation.Binary.PropositionalEquality.Properties.html#726" class="Generalizable">C</a><a id="1951" class="Symbol">}</a> <a id="1953" class="Symbol">{</a><a id="1954" href="Relation.Binary.PropositionalEquality.Properties.html#1954" class="Bound">x</a> <a id="1956" href="Relation.Binary.PropositionalEquality.Properties.html#1956" class="Bound">u</a> <a id="1958" href="Relation.Binary.PropositionalEquality.Properties.html#1958" class="Bound">v</a><a id="1959" class="Symbol">}</a> <a id="1961" class="Symbol">→</a> <a id="1963" class="Symbol">(</a><a id="1964" href="Relation.Binary.PropositionalEquality.Properties.html#1964" class="Bound">p</a> <a id="1966" class="Symbol">:</a> <a id="1968" href="Relation.Binary.PropositionalEquality.Properties.html#1956" class="Bound">u</a> <a id="1970" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="1972" href="Relation.Binary.PropositionalEquality.Properties.html#1958" class="Bound">v</a><a id="1973" class="Symbol">)</a> <a id="1975" class="Symbol">→</a>
|
||
<a id="1991" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="1997" href="Relation.Binary.PropositionalEquality.Properties.html#1936" class="Bound Operator">_∙_</a> <a id="2001" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2006" href="Relation.Binary.PropositionalEquality.Properties.html#1964" class="Bound">p</a> <a id="2008" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2010" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="2015" class="Symbol">(</a><a id="2016" href="Relation.Binary.PropositionalEquality.Properties.html#1954" class="Bound">x</a> <a id="2018" href="Relation.Binary.PropositionalEquality.Properties.html#1936" class="Bound Operator">∙_</a><a id="2020" class="Symbol">)</a> <a id="2022" href="Relation.Binary.PropositionalEquality.Properties.html#1964" class="Bound">p</a>
|
||
<a id="2024" href="Relation.Binary.PropositionalEquality.Properties.html#1919" class="Function">cong₂-reflˡ</a> <a id="2036" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2041" class="Symbol">=</a> <a id="2043" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="cong₂-reflʳ"></a><a id="2049" href="Relation.Binary.PropositionalEquality.Properties.html#2049" class="Function">cong₂-reflʳ</a> <a id="2061" class="Symbol">:</a> <a id="2063" class="Symbol">∀</a> <a id="2065" class="Symbol">{</a><a id="2066" href="Relation.Binary.PropositionalEquality.Properties.html#2066" class="Bound Operator">_∙_</a> <a id="2070" class="Symbol">:</a> <a id="2072" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="2074" class="Symbol">→</a> <a id="2076" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a> <a id="2078" class="Symbol">→</a> <a id="2080" href="Relation.Binary.PropositionalEquality.Properties.html#726" class="Generalizable">C</a><a id="2081" class="Symbol">}</a> <a id="2083" class="Symbol">{</a><a id="2084" href="Relation.Binary.PropositionalEquality.Properties.html#2084" class="Bound">x</a> <a id="2086" href="Relation.Binary.PropositionalEquality.Properties.html#2086" class="Bound">y</a> <a id="2088" href="Relation.Binary.PropositionalEquality.Properties.html#2088" class="Bound">u</a><a id="2089" class="Symbol">}</a> <a id="2091" class="Symbol">→</a> <a id="2093" class="Symbol">(</a><a id="2094" href="Relation.Binary.PropositionalEquality.Properties.html#2094" class="Bound">p</a> <a id="2096" class="Symbol">:</a> <a id="2098" href="Relation.Binary.PropositionalEquality.Properties.html#2084" class="Bound">x</a> <a id="2100" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2102" href="Relation.Binary.PropositionalEquality.Properties.html#2086" class="Bound">y</a><a id="2103" class="Symbol">)</a> <a id="2105" class="Symbol">→</a>
|
||
<a id="2121" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="2127" href="Relation.Binary.PropositionalEquality.Properties.html#2066" class="Bound Operator">_∙_</a> <a id="2131" href="Relation.Binary.PropositionalEquality.Properties.html#2094" class="Bound">p</a> <a id="2133" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2138" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2140" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="2145" class="Symbol">(</a><a id="2146" href="Relation.Binary.PropositionalEquality.Properties.html#2066" class="Bound Operator">_∙</a> <a id="2149" href="Relation.Binary.PropositionalEquality.Properties.html#2088" class="Bound">u</a><a id="2150" class="Symbol">)</a> <a id="2152" href="Relation.Binary.PropositionalEquality.Properties.html#2094" class="Bound">p</a>
|
||
<a id="2154" href="Relation.Binary.PropositionalEquality.Properties.html#2049" class="Function">cong₂-reflʳ</a> <a id="2166" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2171" class="Symbol">=</a> <a id="2173" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="2179" class="Keyword">module</a> <a id="2186" href="Relation.Binary.PropositionalEquality.Properties.html#2186" class="Module">_</a> <a id="2188" class="Symbol">{</a><a id="2189" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2191" class="Symbol">:</a> <a id="2193" href="Relation.Unary.html#1110" class="Function">Pred</a> <a id="2198" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="2200" href="Relation.Binary.PropositionalEquality.Properties.html#708" class="Generalizable">p</a><a id="2201" class="Symbol">}</a> <a id="2203" class="Symbol">{</a><a id="2204" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a> <a id="2206" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a> <a id="2208" class="Symbol">:</a> <a id="2210" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="2211" class="Symbol">}</a> <a id="2213" class="Keyword">where</a>
|
||
|
||
<a id="2222" href="Relation.Binary.PropositionalEquality.Properties.html#2222" class="Function">subst-injective</a> <a id="2238" class="Symbol">:</a> <a id="2240" class="Symbol">∀</a> <a id="2242" class="Symbol">(</a><a id="2243" href="Relation.Binary.PropositionalEquality.Properties.html#2243" class="Bound">x≡y</a> <a id="2247" class="Symbol">:</a> <a id="2249" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a> <a id="2251" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2253" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a><a id="2254" class="Symbol">)</a> <a id="2256" class="Symbol">{</a><a id="2257" href="Relation.Binary.PropositionalEquality.Properties.html#2257" class="Bound">p</a> <a id="2259" href="Relation.Binary.PropositionalEquality.Properties.html#2259" class="Bound">q</a> <a id="2261" class="Symbol">:</a> <a id="2263" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2265" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a><a id="2266" class="Symbol">}</a> <a id="2268" class="Symbol">→</a>
|
||
<a id="2290" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2296" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2298" href="Relation.Binary.PropositionalEquality.Properties.html#2243" class="Bound">x≡y</a> <a id="2302" href="Relation.Binary.PropositionalEquality.Properties.html#2257" class="Bound">p</a> <a id="2304" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2306" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2312" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2314" href="Relation.Binary.PropositionalEquality.Properties.html#2243" class="Bound">x≡y</a> <a id="2318" href="Relation.Binary.PropositionalEquality.Properties.html#2259" class="Bound">q</a> <a id="2320" class="Symbol">→</a> <a id="2322" href="Relation.Binary.PropositionalEquality.Properties.html#2257" class="Bound">p</a> <a id="2324" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2326" href="Relation.Binary.PropositionalEquality.Properties.html#2259" class="Bound">q</a>
|
||
<a id="2330" href="Relation.Binary.PropositionalEquality.Properties.html#2222" class="Function">subst-injective</a> <a id="2346" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2351" href="Relation.Binary.PropositionalEquality.Properties.html#2351" class="Bound">p≡q</a> <a id="2355" class="Symbol">=</a> <a id="2357" href="Relation.Binary.PropositionalEquality.Properties.html#2351" class="Bound">p≡q</a>
|
||
|
||
<a id="2364" href="Relation.Binary.PropositionalEquality.Properties.html#2364" class="Function">subst-subst</a> <a id="2376" class="Symbol">:</a> <a id="2378" class="Symbol">∀</a> <a id="2380" class="Symbol">{</a><a id="2381" href="Relation.Binary.PropositionalEquality.Properties.html#2381" class="Bound">z</a><a id="2382" class="Symbol">}</a> <a id="2384" class="Symbol">(</a><a id="2385" href="Relation.Binary.PropositionalEquality.Properties.html#2385" class="Bound">x≡y</a> <a id="2389" class="Symbol">:</a> <a id="2391" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a> <a id="2393" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2395" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a><a id="2396" class="Symbol">)</a> <a id="2398" class="Symbol">{</a><a id="2399" href="Relation.Binary.PropositionalEquality.Properties.html#2399" class="Bound">y≡z</a> <a id="2403" class="Symbol">:</a> <a id="2405" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a> <a id="2407" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2409" href="Relation.Binary.PropositionalEquality.Properties.html#2381" class="Bound">z</a><a id="2410" class="Symbol">}</a> <a id="2412" class="Symbol">{</a><a id="2413" href="Relation.Binary.PropositionalEquality.Properties.html#2413" class="Bound">p</a> <a id="2415" class="Symbol">:</a> <a id="2417" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2419" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a><a id="2420" class="Symbol">}</a> <a id="2422" class="Symbol">→</a>
|
||
<a id="2440" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2446" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2448" href="Relation.Binary.PropositionalEquality.Properties.html#2399" class="Bound">y≡z</a> <a id="2452" class="Symbol">(</a><a id="2453" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2459" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2461" href="Relation.Binary.PropositionalEquality.Properties.html#2385" class="Bound">x≡y</a> <a id="2465" href="Relation.Binary.PropositionalEquality.Properties.html#2413" class="Bound">p</a><a id="2466" class="Symbol">)</a> <a id="2468" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2470" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2476" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2478" class="Symbol">(</a><a id="2479" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="2485" href="Relation.Binary.PropositionalEquality.Properties.html#2385" class="Bound">x≡y</a> <a id="2489" href="Relation.Binary.PropositionalEquality.Properties.html#2399" class="Bound">y≡z</a><a id="2492" class="Symbol">)</a> <a id="2494" href="Relation.Binary.PropositionalEquality.Properties.html#2413" class="Bound">p</a>
|
||
<a id="2498" href="Relation.Binary.PropositionalEquality.Properties.html#2364" class="Function">subst-subst</a> <a id="2510" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2515" class="Symbol">=</a> <a id="2517" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="2525" href="Relation.Binary.PropositionalEquality.Properties.html#2525" class="Function">subst-subst-sym</a> <a id="2541" class="Symbol">:</a> <a id="2543" class="Symbol">(</a><a id="2544" href="Relation.Binary.PropositionalEquality.Properties.html#2544" class="Bound">x≡y</a> <a id="2548" class="Symbol">:</a> <a id="2550" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a> <a id="2552" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2554" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a><a id="2555" class="Symbol">)</a> <a id="2557" class="Symbol">{</a><a id="2558" href="Relation.Binary.PropositionalEquality.Properties.html#2558" class="Bound">p</a> <a id="2560" class="Symbol">:</a> <a id="2562" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2564" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a><a id="2565" class="Symbol">}</a> <a id="2567" class="Symbol">→</a>
|
||
<a id="2589" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2595" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2597" href="Relation.Binary.PropositionalEquality.Properties.html#2544" class="Bound">x≡y</a> <a id="2601" class="Symbol">(</a><a id="2602" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2608" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2610" class="Symbol">(</a><a id="2611" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="2615" href="Relation.Binary.PropositionalEquality.Properties.html#2544" class="Bound">x≡y</a><a id="2618" class="Symbol">)</a> <a id="2620" href="Relation.Binary.PropositionalEquality.Properties.html#2558" class="Bound">p</a><a id="2621" class="Symbol">)</a> <a id="2623" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2625" href="Relation.Binary.PropositionalEquality.Properties.html#2558" class="Bound">p</a>
|
||
<a id="2629" href="Relation.Binary.PropositionalEquality.Properties.html#2525" class="Function">subst-subst-sym</a> <a id="2645" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2650" class="Symbol">=</a> <a id="2652" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="2660" href="Relation.Binary.PropositionalEquality.Properties.html#2660" class="Function">subst-sym-subst</a> <a id="2676" class="Symbol">:</a> <a id="2678" class="Symbol">(</a><a id="2679" href="Relation.Binary.PropositionalEquality.Properties.html#2679" class="Bound">x≡y</a> <a id="2683" class="Symbol">:</a> <a id="2685" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a> <a id="2687" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2689" href="Relation.Binary.PropositionalEquality.Properties.html#2206" class="Bound">y</a><a id="2690" class="Symbol">)</a> <a id="2692" class="Symbol">{</a><a id="2693" href="Relation.Binary.PropositionalEquality.Properties.html#2693" class="Bound">p</a> <a id="2695" class="Symbol">:</a> <a id="2697" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2699" href="Relation.Binary.PropositionalEquality.Properties.html#2204" class="Bound">x</a><a id="2700" class="Symbol">}</a> <a id="2702" class="Symbol">→</a>
|
||
<a id="2724" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2730" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2732" class="Symbol">(</a><a id="2733" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="2737" href="Relation.Binary.PropositionalEquality.Properties.html#2679" class="Bound">x≡y</a><a id="2740" class="Symbol">)</a> <a id="2742" class="Symbol">(</a><a id="2743" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2749" href="Relation.Binary.PropositionalEquality.Properties.html#2189" class="Bound">P</a> <a id="2751" href="Relation.Binary.PropositionalEquality.Properties.html#2679" class="Bound">x≡y</a> <a id="2755" href="Relation.Binary.PropositionalEquality.Properties.html#2693" class="Bound">p</a><a id="2756" class="Symbol">)</a> <a id="2758" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2760" href="Relation.Binary.PropositionalEquality.Properties.html#2693" class="Bound">p</a>
|
||
<a id="2764" href="Relation.Binary.PropositionalEquality.Properties.html#2660" class="Function">subst-sym-subst</a> <a id="2780" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2785" class="Symbol">=</a> <a id="2787" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="subst-∘"></a><a id="2793" href="Relation.Binary.PropositionalEquality.Properties.html#2793" class="Function">subst-∘</a> <a id="2801" class="Symbol">:</a> <a id="2803" class="Symbol">∀</a> <a id="2805" class="Symbol">{</a><a id="2806" href="Relation.Binary.PropositionalEquality.Properties.html#2806" class="Bound">x</a> <a id="2808" href="Relation.Binary.PropositionalEquality.Properties.html#2808" class="Bound">y</a> <a id="2810" class="Symbol">:</a> <a id="2812" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="2813" class="Symbol">}</a> <a id="2815" class="Symbol">{</a><a id="2816" href="Relation.Binary.PropositionalEquality.Properties.html#2816" class="Bound">P</a> <a id="2818" class="Symbol">:</a> <a id="2820" href="Relation.Unary.html#1110" class="Function">Pred</a> <a id="2825" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a> <a id="2827" href="Relation.Binary.PropositionalEquality.Properties.html#708" class="Generalizable">p</a><a id="2828" class="Symbol">}</a> <a id="2830" class="Symbol">{</a><a id="2831" href="Relation.Binary.PropositionalEquality.Properties.html#2831" class="Bound">f</a> <a id="2833" class="Symbol">:</a> <a id="2835" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a> <a id="2837" class="Symbol">→</a> <a id="2839" href="Relation.Binary.PropositionalEquality.Properties.html#724" class="Generalizable">B</a><a id="2840" class="Symbol">}</a>
|
||
<a id="2852" class="Symbol">(</a><a id="2853" href="Relation.Binary.PropositionalEquality.Properties.html#2853" class="Bound">x≡y</a> <a id="2857" class="Symbol">:</a> <a id="2859" href="Relation.Binary.PropositionalEquality.Properties.html#2806" class="Bound">x</a> <a id="2861" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2863" href="Relation.Binary.PropositionalEquality.Properties.html#2808" class="Bound">y</a><a id="2864" class="Symbol">)</a> <a id="2866" class="Symbol">{</a><a id="2867" href="Relation.Binary.PropositionalEquality.Properties.html#2867" class="Bound">p</a> <a id="2869" class="Symbol">:</a> <a id="2871" href="Relation.Binary.PropositionalEquality.Properties.html#2816" class="Bound">P</a> <a id="2873" class="Symbol">(</a><a id="2874" href="Relation.Binary.PropositionalEquality.Properties.html#2831" class="Bound">f</a> <a id="2876" href="Relation.Binary.PropositionalEquality.Properties.html#2806" class="Bound">x</a><a id="2877" class="Symbol">)}</a> <a id="2880" class="Symbol">→</a>
|
||
<a id="2892" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2898" class="Symbol">(</a><a id="2899" href="Relation.Binary.PropositionalEquality.Properties.html#2816" class="Bound">P</a> <a id="2901" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2903" href="Relation.Binary.PropositionalEquality.Properties.html#2831" class="Bound">f</a><a id="2904" class="Symbol">)</a> <a id="2906" href="Relation.Binary.PropositionalEquality.Properties.html#2853" class="Bound">x≡y</a> <a id="2910" href="Relation.Binary.PropositionalEquality.Properties.html#2867" class="Bound">p</a> <a id="2912" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="2914" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="2920" href="Relation.Binary.PropositionalEquality.Properties.html#2816" class="Bound">P</a> <a id="2922" class="Symbol">(</a><a id="2923" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="2928" href="Relation.Binary.PropositionalEquality.Properties.html#2831" class="Bound">f</a> <a id="2930" href="Relation.Binary.PropositionalEquality.Properties.html#2853" class="Bound">x≡y</a><a id="2933" class="Symbol">)</a> <a id="2935" href="Relation.Binary.PropositionalEquality.Properties.html#2867" class="Bound">p</a>
|
||
<a id="2937" href="Relation.Binary.PropositionalEquality.Properties.html#2793" class="Function">subst-∘</a> <a id="2945" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="2950" class="Symbol">=</a> <a id="2952" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="subst-application"></a><a id="2958" href="Relation.Binary.PropositionalEquality.Properties.html#2958" class="Function">subst-application</a> <a id="2976" class="Symbol">:</a> <a id="2978" class="Symbol">∀</a> <a id="2980" class="Symbol">{</a><a id="2981" href="Relation.Binary.PropositionalEquality.Properties.html#2981" class="Bound">a₁</a> <a id="2984" href="Relation.Binary.PropositionalEquality.Properties.html#2984" class="Bound">a₂</a> <a id="2987" href="Relation.Binary.PropositionalEquality.Properties.html#2987" class="Bound">b₁</a> <a id="2990" href="Relation.Binary.PropositionalEquality.Properties.html#2990" class="Bound">b₂</a><a id="2992" class="Symbol">}</a> <a id="2994" class="Symbol">{</a><a id="2995" href="Relation.Binary.PropositionalEquality.Properties.html#2995" class="Bound">A₁</a> <a id="2998" class="Symbol">:</a> <a id="3000" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3004" href="Relation.Binary.PropositionalEquality.Properties.html#2981" class="Bound">a₁</a><a id="3006" class="Symbol">}</a> <a id="3008" class="Symbol">{</a><a id="3009" href="Relation.Binary.PropositionalEquality.Properties.html#3009" class="Bound">A₂</a> <a id="3012" class="Symbol">:</a> <a id="3014" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3018" href="Relation.Binary.PropositionalEquality.Properties.html#2984" class="Bound">a₂</a><a id="3020" class="Symbol">}</a>
|
||
<a id="3042" class="Symbol">(</a><a id="3043" href="Relation.Binary.PropositionalEquality.Properties.html#3043" class="Bound">B₁</a> <a id="3046" class="Symbol">:</a> <a id="3048" href="Relation.Binary.PropositionalEquality.Properties.html#2995" class="Bound">A₁</a> <a id="3051" class="Symbol">→</a> <a id="3053" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3057" href="Relation.Binary.PropositionalEquality.Properties.html#2987" class="Bound">b₁</a><a id="3059" class="Symbol">)</a> <a id="3061" class="Symbol">{</a><a id="3062" href="Relation.Binary.PropositionalEquality.Properties.html#3062" class="Bound">B₂</a> <a id="3065" class="Symbol">:</a> <a id="3067" href="Relation.Binary.PropositionalEquality.Properties.html#3009" class="Bound">A₂</a> <a id="3070" class="Symbol">→</a> <a id="3072" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3076" href="Relation.Binary.PropositionalEquality.Properties.html#2990" class="Bound">b₂</a><a id="3078" class="Symbol">}</a>
|
||
<a id="3100" class="Symbol">{</a><a id="3101" href="Relation.Binary.PropositionalEquality.Properties.html#3101" class="Bound">f</a> <a id="3103" class="Symbol">:</a> <a id="3105" href="Relation.Binary.PropositionalEquality.Properties.html#3009" class="Bound">A₂</a> <a id="3108" class="Symbol">→</a> <a id="3110" href="Relation.Binary.PropositionalEquality.Properties.html#2995" class="Bound">A₁</a><a id="3112" class="Symbol">}</a> <a id="3114" class="Symbol">{</a><a id="3115" href="Relation.Binary.PropositionalEquality.Properties.html#3115" class="Bound">x₁</a> <a id="3118" href="Relation.Binary.PropositionalEquality.Properties.html#3118" class="Bound">x₂</a> <a id="3121" class="Symbol">:</a> <a id="3123" href="Relation.Binary.PropositionalEquality.Properties.html#3009" class="Bound">A₂</a><a id="3125" class="Symbol">}</a> <a id="3127" class="Symbol">{</a><a id="3128" href="Relation.Binary.PropositionalEquality.Properties.html#3128" class="Bound">y</a> <a id="3130" class="Symbol">:</a> <a id="3132" href="Relation.Binary.PropositionalEquality.Properties.html#3043" class="Bound">B₁</a> <a id="3135" class="Symbol">(</a><a id="3136" href="Relation.Binary.PropositionalEquality.Properties.html#3101" class="Bound">f</a> <a id="3138" href="Relation.Binary.PropositionalEquality.Properties.html#3115" class="Bound">x₁</a><a id="3140" class="Symbol">)}</a>
|
||
<a id="3163" class="Symbol">(</a><a id="3164" href="Relation.Binary.PropositionalEquality.Properties.html#3164" class="Bound">g</a> <a id="3166" class="Symbol">:</a> <a id="3168" class="Symbol">∀</a> <a id="3170" href="Relation.Binary.PropositionalEquality.Properties.html#3170" class="Bound">x</a> <a id="3172" class="Symbol">→</a> <a id="3174" href="Relation.Binary.PropositionalEquality.Properties.html#3043" class="Bound">B₁</a> <a id="3177" class="Symbol">(</a><a id="3178" href="Relation.Binary.PropositionalEquality.Properties.html#3101" class="Bound">f</a> <a id="3180" href="Relation.Binary.PropositionalEquality.Properties.html#3170" class="Bound">x</a><a id="3181" class="Symbol">)</a> <a id="3183" class="Symbol">→</a> <a id="3185" href="Relation.Binary.PropositionalEquality.Properties.html#3062" class="Bound">B₂</a> <a id="3188" href="Relation.Binary.PropositionalEquality.Properties.html#3170" class="Bound">x</a><a id="3189" class="Symbol">)</a> <a id="3191" class="Symbol">(</a><a id="3192" href="Relation.Binary.PropositionalEquality.Properties.html#3192" class="Bound">eq</a> <a id="3195" class="Symbol">:</a> <a id="3197" href="Relation.Binary.PropositionalEquality.Properties.html#3115" class="Bound">x₁</a> <a id="3200" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="3202" href="Relation.Binary.PropositionalEquality.Properties.html#3118" class="Bound">x₂</a><a id="3204" class="Symbol">)</a> <a id="3206" class="Symbol">→</a>
|
||
<a id="3228" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="3234" href="Relation.Binary.PropositionalEquality.Properties.html#3062" class="Bound">B₂</a> <a id="3237" href="Relation.Binary.PropositionalEquality.Properties.html#3192" class="Bound">eq</a> <a id="3240" class="Symbol">(</a><a id="3241" href="Relation.Binary.PropositionalEquality.Properties.html#3164" class="Bound">g</a> <a id="3243" href="Relation.Binary.PropositionalEquality.Properties.html#3115" class="Bound">x₁</a> <a id="3246" href="Relation.Binary.PropositionalEquality.Properties.html#3128" class="Bound">y</a><a id="3247" class="Symbol">)</a> <a id="3249" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="3251" href="Relation.Binary.PropositionalEquality.Properties.html#3164" class="Bound">g</a> <a id="3253" href="Relation.Binary.PropositionalEquality.Properties.html#3118" class="Bound">x₂</a> <a id="3256" class="Symbol">(</a><a id="3257" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="3263" href="Relation.Binary.PropositionalEquality.Properties.html#3043" class="Bound">B₁</a> <a id="3266" class="Symbol">(</a><a id="3267" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="3272" href="Relation.Binary.PropositionalEquality.Properties.html#3101" class="Bound">f</a> <a id="3274" href="Relation.Binary.PropositionalEquality.Properties.html#3192" class="Bound">eq</a><a id="3276" class="Symbol">)</a> <a id="3278" href="Relation.Binary.PropositionalEquality.Properties.html#3128" class="Bound">y</a><a id="3279" class="Symbol">)</a>
|
||
<a id="3281" href="Relation.Binary.PropositionalEquality.Properties.html#2958" class="Function">subst-application</a> <a id="3299" class="Symbol">_</a> <a id="3301" class="Symbol">_</a> <a id="3303" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="3308" class="Symbol">=</a> <a id="3310" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
|
||
<a id="3316" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="3389" class="Comment">-- Structure of equality as a binary relation</a>
|
||
|
||
<a id="isEquivalence"></a><a id="3436" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a> <a id="3450" class="Symbol">:</a> <a id="3452" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a> <a id="3466" class="Symbol">{</a><a id="3467" class="Argument">A</a> <a id="3469" class="Symbol">=</a> <a id="3471" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="3472" class="Symbol">}</a> <a id="3474" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="3478" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a> <a id="3492" class="Symbol">=</a> <a id="3494" class="Keyword">record</a>
|
||
<a id="3503" class="Symbol">{</a> <a id="3505" href="Relation.Binary.Structures.html#1577" class="Field">refl</a> <a id="3511" class="Symbol">=</a> <a id="3513" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
||
<a id="3520" class="Symbol">;</a> <a id="3522" href="Relation.Binary.Structures.html#1603" class="Field">sym</a> <a id="3528" class="Symbol">=</a> <a id="3530" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a>
|
||
<a id="3536" class="Symbol">;</a> <a id="3538" href="Relation.Binary.Structures.html#1629" class="Field">trans</a> <a id="3544" class="Symbol">=</a> <a id="3546" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a>
|
||
<a id="3554" class="Symbol">}</a>
|
||
|
||
<a id="isDecEquivalence"></a><a id="3557" href="Relation.Binary.PropositionalEquality.Properties.html#3557" class="Function">isDecEquivalence</a> <a id="3574" class="Symbol">:</a> <a id="3576" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3586" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="3590" class="Symbol">→</a> <a id="3592" href="Relation.Binary.Structures.html#1833" class="Record">IsDecEquivalence</a> <a id="3609" class="Symbol">{</a><a id="3610" class="Argument">A</a> <a id="3612" class="Symbol">=</a> <a id="3614" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="3615" class="Symbol">}</a> <a id="3617" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="3621" href="Relation.Binary.PropositionalEquality.Properties.html#3557" class="Function">isDecEquivalence</a> <a id="3638" href="Relation.Binary.PropositionalEquality.Properties.html#3638" class="Bound Operator">_≟_</a> <a id="3642" class="Symbol">=</a> <a id="3644" class="Keyword">record</a>
|
||
<a id="3653" class="Symbol">{</a> <a id="3655" href="Relation.Binary.Structures.html#1896" class="Field">isEquivalence</a> <a id="3669" class="Symbol">=</a> <a id="3671" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
||
<a id="3687" class="Symbol">;</a> <a id="3689" href="Relation.Binary.Structures.html#1930" class="Field Operator">_≟_</a> <a id="3703" class="Symbol">=</a> <a id="3705" href="Relation.Binary.PropositionalEquality.Properties.html#3638" class="Bound Operator">_≟_</a>
|
||
<a id="3711" class="Symbol">}</a>
|
||
|
||
<a id="isPreorder"></a><a id="3714" href="Relation.Binary.PropositionalEquality.Properties.html#3714" class="Function">isPreorder</a> <a id="3725" class="Symbol">:</a> <a id="3727" href="Relation.Binary.Structures.html#2172" class="Record">IsPreorder</a> <a id="3738" class="Symbol">{</a><a id="3739" class="Argument">A</a> <a id="3741" class="Symbol">=</a> <a id="3743" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="3744" class="Symbol">}</a> <a id="3746" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="3750" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="3754" href="Relation.Binary.PropositionalEquality.Properties.html#3714" class="Function">isPreorder</a> <a id="3765" class="Symbol">=</a> <a id="3767" class="Keyword">record</a>
|
||
<a id="3776" class="Symbol">{</a> <a id="3778" href="Relation.Binary.Structures.html#2237" class="Field">isEquivalence</a> <a id="3792" class="Symbol">=</a> <a id="3794" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
||
<a id="3810" class="Symbol">;</a> <a id="3812" href="Relation.Binary.Structures.html#2340" class="Field">reflexive</a> <a id="3826" class="Symbol">=</a> <a id="3828" href="Function.Base.html#624" class="Function">id</a>
|
||
<a id="3833" class="Symbol">;</a> <a id="3835" href="Relation.Binary.Structures.html#2370" class="Field">trans</a> <a id="3849" class="Symbol">=</a> <a id="3851" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a>
|
||
<a id="3859" class="Symbol">}</a>
|
||
|
||
<a id="3862" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="3935" class="Comment">-- Bundles for equality as a binary relation</a>
|
||
|
||
<a id="setoid"></a><a id="3981" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Function">setoid</a> <a id="3988" class="Symbol">:</a> <a id="3990" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="3994" href="Relation.Binary.PropositionalEquality.Properties.html#706" class="Generalizable">a</a> <a id="3996" class="Symbol">→</a> <a id="3998" href="Relation.Binary.Bundles.html#1018" class="Record">Setoid</a> <a id="4005" class="Symbol">_</a> <a id="4007" class="Symbol">_</a>
|
||
<a id="4009" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Function">setoid</a> <a id="4016" href="Relation.Binary.PropositionalEquality.Properties.html#4016" class="Bound">A</a> <a id="4018" class="Symbol">=</a> <a id="4020" class="Keyword">record</a>
|
||
<a id="4029" class="Symbol">{</a> <a id="4031" href="Relation.Binary.Bundles.html#1081" class="Field">Carrier</a> <a id="4045" class="Symbol">=</a> <a id="4047" href="Relation.Binary.PropositionalEquality.Properties.html#4016" class="Bound">A</a>
|
||
<a id="4051" class="Symbol">;</a> <a id="4053" href="Relation.Binary.Bundles.html#1107" class="Field Operator">_≈_</a> <a id="4067" class="Symbol">=</a> <a id="4069" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="4075" class="Symbol">;</a> <a id="4077" href="Relation.Binary.Bundles.html#1141" class="Field">isEquivalence</a> <a id="4091" class="Symbol">=</a> <a id="4093" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
||
<a id="4109" class="Symbol">}</a>
|
||
|
||
<a id="decSetoid"></a><a id="4112" href="Relation.Binary.PropositionalEquality.Properties.html#4112" class="Function">decSetoid</a> <a id="4122" class="Symbol">:</a> <a id="4124" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="4134" class="Symbol">{</a><a id="4135" class="Argument">A</a> <a id="4137" class="Symbol">=</a> <a id="4139" href="Relation.Binary.PropositionalEquality.Properties.html#722" class="Generalizable">A</a><a id="4140" class="Symbol">}</a> <a id="4142" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="4146" class="Symbol">→</a> <a id="4148" href="Relation.Binary.Bundles.html#1400" class="Record">DecSetoid</a> <a id="4158" class="Symbol">_</a> <a id="4160" class="Symbol">_</a>
|
||
<a id="4162" href="Relation.Binary.PropositionalEquality.Properties.html#4112" class="Function">decSetoid</a> <a id="4172" href="Relation.Binary.PropositionalEquality.Properties.html#4172" class="Bound Operator">_≟_</a> <a id="4176" class="Symbol">=</a> <a id="4178" class="Keyword">record</a>
|
||
<a id="4187" class="Symbol">{</a> <a id="4189" href="Relation.Binary.Bundles.html#1495" class="Field Operator">_≈_</a> <a id="4206" class="Symbol">=</a> <a id="4208" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="4214" class="Symbol">;</a> <a id="4216" href="Relation.Binary.Bundles.html#1532" class="Field">isDecEquivalence</a> <a id="4233" class="Symbol">=</a> <a id="4235" href="Relation.Binary.PropositionalEquality.Properties.html#3557" class="Function">isDecEquivalence</a> <a id="4252" href="Relation.Binary.PropositionalEquality.Properties.html#4172" class="Bound Operator">_≟_</a>
|
||
<a id="4258" class="Symbol">}</a>
|
||
|
||
<a id="preorder"></a><a id="4261" href="Relation.Binary.PropositionalEquality.Properties.html#4261" class="Function">preorder</a> <a id="4270" class="Symbol">:</a> <a id="4272" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="4276" href="Relation.Binary.PropositionalEquality.Properties.html#706" class="Generalizable">a</a> <a id="4278" class="Symbol">→</a> <a id="4280" href="Relation.Binary.Bundles.html#1929" class="Record">Preorder</a> <a id="4289" class="Symbol">_</a> <a id="4291" class="Symbol">_</a> <a id="4293" class="Symbol">_</a>
|
||
<a id="4295" href="Relation.Binary.PropositionalEquality.Properties.html#4261" class="Function">preorder</a> <a id="4304" href="Relation.Binary.PropositionalEquality.Properties.html#4304" class="Bound">A</a> <a id="4306" class="Symbol">=</a> <a id="4308" class="Keyword">record</a>
|
||
<a id="4317" class="Symbol">{</a> <a id="4319" href="Relation.Binary.Bundles.html#2008" class="Field">Carrier</a> <a id="4330" class="Symbol">=</a> <a id="4332" href="Relation.Binary.PropositionalEquality.Properties.html#4304" class="Bound">A</a>
|
||
<a id="4336" class="Symbol">;</a> <a id="4338" href="Relation.Binary.Bundles.html#2031" class="Field Operator">_≈_</a> <a id="4349" class="Symbol">=</a> <a id="4351" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="4357" class="Symbol">;</a> <a id="4359" href="Relation.Binary.Bundles.html#2092" class="Field Operator">_∼_</a> <a id="4370" class="Symbol">=</a> <a id="4372" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
||
<a id="4378" class="Symbol">;</a> <a id="4380" href="Relation.Binary.Bundles.html#2142" class="Field">isPreorder</a> <a id="4391" class="Symbol">=</a> <a id="4393" href="Relation.Binary.PropositionalEquality.Properties.html#3714" class="Function">isPreorder</a>
|
||
<a id="4406" class="Symbol">}</a>
|
||
</pre></body></html> |