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<!DOCTYPE HTML>
<html><head><meta charset="utf-8"><title>Function.Metric.Definitions</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">-- Definitions of properties over distance functions</a>
<a id="159" class="Comment">------------------------------------------------------------------------</a>
<a id="233" class="Comment">-- The contents of this module should be accessed via `Function.Metric`.</a>
<a id="307" class="Symbol">{-#</a> <a id="311" class="Keyword">OPTIONS</a> <a id="319" class="Pragma">--cubical-compatible</a> <a id="340" class="Pragma">--safe</a> <a id="347" class="Symbol">#-}</a>
<a id="352" class="Keyword">module</a> <a id="359" href="Function.Metric.Definitions.html" class="Module">Function.Metric.Definitions</a> <a id="387" class="Keyword">where</a>
<a id="394" class="Keyword">open</a> <a id="399" class="Keyword">import</a> <a id="406" href="Algebra.Core.html" class="Module">Algebra.Core</a> <a id="419" class="Keyword">using</a> <a id="425" class="Symbol">(</a><a id="426" href="Algebra.Core.html#527" class="Function">Op₂</a><a id="429" class="Symbol">)</a>
<a id="431" class="Keyword">open</a> <a id="436" class="Keyword">import</a> <a id="443" href="Data.Product.html" class="Module">Data.Product</a> <a id="456" class="Keyword">using</a> <a id="462" class="Symbol">(</a><a id="463" href="Data.Product.html#1378" class="Function">∃</a><a id="464" class="Symbol">)</a>
<a id="466" class="Keyword">open</a> <a id="471" class="Keyword">import</a> <a id="478" href="Function.Metric.Core.html" class="Module">Function.Metric.Core</a> <a id="499" class="Keyword">using</a> <a id="505" class="Symbol">(</a><a id="506" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a><a id="522" class="Symbol">)</a>
<a id="524" class="Keyword">open</a> <a id="529" class="Keyword">import</a> <a id="536" href="Level.html" class="Module">Level</a> <a id="542" class="Keyword">using</a> <a id="548" class="Symbol">(</a><a id="549" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="554" class="Symbol">)</a>
<a id="556" class="Keyword">open</a> <a id="561" class="Keyword">import</a> <a id="568" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="589" class="Keyword">using</a> <a id="595" class="Symbol">(</a><a id="596" href="Relation.Binary.Core.html#891" class="Function">Rel</a><a id="599" class="Symbol">;</a> <a id="601" href="Relation.Binary.Core.html#1698" class="Function Operator">_Preserves₂_⟶_⟶_</a><a id="617" class="Symbol">)</a>
<a id="619" class="Keyword">open</a> <a id="624" class="Keyword">import</a> <a id="631" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="648" class="Keyword">using</a> <a id="654" class="Symbol">(</a><a id="655" href="Relation.Nullary.html#665" class="Function Operator">¬_</a><a id="657" class="Symbol">)</a>
<a id="660" class="Keyword">private</a>
<a id="670" class="Keyword">variable</a>
<a id="683" href="Function.Metric.Definitions.html#683" class="Generalizable">a</a> <a id="685" href="Function.Metric.Definitions.html#685" class="Generalizable">i</a> <a id="687" href="Function.Metric.Definitions.html#687" class="Generalizable">ℓ</a> <a id="689" href="Function.Metric.Definitions.html#689" class="Generalizable">ℓ₁</a> <a id="692" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="695" class="Symbol">:</a> <a id="697" href="Agda.Primitive.html#591" class="Postulate">Level</a>
<a id="707" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="709" class="Symbol">:</a> <a id="711" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="715" href="Function.Metric.Definitions.html#683" class="Generalizable">a</a>
<a id="721" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="723" class="Symbol">:</a> <a id="725" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="729" href="Function.Metric.Definitions.html#685" class="Generalizable">i</a>
<a id="732" class="Comment">-----------------------------------------------------------------------</a>
<a id="804" class="Comment">-- Properties</a>
<a id="Congruent"></a><a id="819" href="Function.Metric.Definitions.html#819" class="Function">Congruent</a> <a id="829" class="Symbol">:</a> <a id="831" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="835" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="837" href="Function.Metric.Definitions.html#689" class="Generalizable">ℓ₁</a> <a id="840" class="Symbol">→</a> <a id="842" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="846" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="848" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="851" class="Symbol">→</a> <a id="853" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="870" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="872" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="874" class="Symbol">→</a> <a id="876" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="880" class="Symbol">_</a>
<a id="882" href="Function.Metric.Definitions.html#819" class="Function">Congruent</a> <a id="892" href="Function.Metric.Definitions.html#892" class="Bound Operator">_≈ₐ_</a> <a id="897" href="Function.Metric.Definitions.html#897" class="Bound Operator">_≈ᵢ_</a> <a id="902" href="Function.Metric.Definitions.html#902" class="Bound">d</a> <a id="904" class="Symbol">=</a> <a id="906" href="Function.Metric.Definitions.html#902" class="Bound">d</a> <a id="908" href="Relation.Binary.Core.html#1698" class="Function Operator">Preserves₂</a> <a id="919" href="Function.Metric.Definitions.html#892" class="Bound Operator">_≈ₐ_</a> <a id="924" href="Relation.Binary.Core.html#1698" class="Function Operator">⟶</a> <a id="926" href="Function.Metric.Definitions.html#892" class="Bound Operator">_≈ₐ_</a> <a id="931" href="Relation.Binary.Core.html#1698" class="Function Operator">⟶</a> <a id="933" href="Function.Metric.Definitions.html#897" class="Bound Operator">_≈ᵢ_</a>
<a id="Indiscernable"></a><a id="939" href="Function.Metric.Definitions.html#939" class="Function">Indiscernable</a> <a id="953" class="Symbol">:</a> <a id="955" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="959" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="961" href="Function.Metric.Definitions.html#689" class="Generalizable">ℓ₁</a> <a id="964" class="Symbol">→</a> <a id="966" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="970" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="972" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="975" class="Symbol">→</a> <a id="977" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="994" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="996" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="998" class="Symbol">→</a> <a id="1000" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1002" class="Symbol">→</a> <a id="1004" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1008" class="Symbol">_</a>
<a id="1010" href="Function.Metric.Definitions.html#939" class="Function">Indiscernable</a> <a id="1024" href="Function.Metric.Definitions.html#1024" class="Bound Operator">_≈ₐ_</a> <a id="1029" href="Function.Metric.Definitions.html#1029" class="Bound Operator">_≈ᵢ_</a> <a id="1034" href="Function.Metric.Definitions.html#1034" class="Bound">d</a> <a id="1036" href="Function.Metric.Definitions.html#1036" class="Bound">0#</a> <a id="1039" class="Symbol">=</a> <a id="1041" class="Symbol">∀</a> <a id="1043" class="Symbol">{</a><a id="1044" href="Function.Metric.Definitions.html#1044" class="Bound">x</a> <a id="1046" href="Function.Metric.Definitions.html#1046" class="Bound">y</a><a id="1047" class="Symbol">}</a> <a id="1049" class="Symbol">→</a> <a id="1051" href="Function.Metric.Definitions.html#1034" class="Bound">d</a> <a id="1053" href="Function.Metric.Definitions.html#1044" class="Bound">x</a> <a id="1055" href="Function.Metric.Definitions.html#1046" class="Bound">y</a> <a id="1057" href="Function.Metric.Definitions.html#1029" class="Bound Operator">≈ᵢ</a> <a id="1060" href="Function.Metric.Definitions.html#1036" class="Bound">0#</a> <a id="1063" class="Symbol">→</a> <a id="1065" href="Function.Metric.Definitions.html#1044" class="Bound">x</a> <a id="1067" href="Function.Metric.Definitions.html#1024" class="Bound Operator">≈ₐ</a> <a id="1070" href="Function.Metric.Definitions.html#1046" class="Bound">y</a>
<a id="Definite"></a><a id="1073" href="Function.Metric.Definitions.html#1073" class="Function">Definite</a> <a id="1082" class="Symbol">:</a> <a id="1084" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1088" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1090" href="Function.Metric.Definitions.html#689" class="Generalizable">ℓ₁</a> <a id="1093" class="Symbol">→</a> <a id="1095" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1099" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1101" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="1104" class="Symbol">→</a> <a id="1106" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1123" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1125" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1127" class="Symbol">→</a> <a id="1129" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1131" class="Symbol">→</a> <a id="1133" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1137" class="Symbol">_</a>
<a id="1139" href="Function.Metric.Definitions.html#1073" class="Function">Definite</a> <a id="1148" href="Function.Metric.Definitions.html#1148" class="Bound Operator">_≈ₐ_</a> <a id="1153" href="Function.Metric.Definitions.html#1153" class="Bound Operator">_≈ᵢ_</a> <a id="1158" href="Function.Metric.Definitions.html#1158" class="Bound">d</a> <a id="1160" href="Function.Metric.Definitions.html#1160" class="Bound">0#</a> <a id="1163" class="Symbol">=</a> <a id="1165" class="Symbol">∀</a> <a id="1167" class="Symbol">{</a><a id="1168" href="Function.Metric.Definitions.html#1168" class="Bound">x</a> <a id="1170" href="Function.Metric.Definitions.html#1170" class="Bound">y</a><a id="1171" class="Symbol">}</a> <a id="1173" class="Symbol">→</a> <a id="1175" href="Function.Metric.Definitions.html#1168" class="Bound">x</a> <a id="1177" href="Function.Metric.Definitions.html#1148" class="Bound Operator">≈ₐ</a> <a id="1180" href="Function.Metric.Definitions.html#1170" class="Bound">y</a> <a id="1182" class="Symbol">→</a> <a id="1184" href="Function.Metric.Definitions.html#1158" class="Bound">d</a> <a id="1186" href="Function.Metric.Definitions.html#1168" class="Bound">x</a> <a id="1188" href="Function.Metric.Definitions.html#1170" class="Bound">y</a> <a id="1190" href="Function.Metric.Definitions.html#1153" class="Bound Operator">≈ᵢ</a> <a id="1193" href="Function.Metric.Definitions.html#1160" class="Bound">0#</a>
<a id="NonNegative"></a><a id="1197" href="Function.Metric.Definitions.html#1197" class="Function">NonNegative</a> <a id="1209" class="Symbol">:</a> <a id="1211" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1215" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1217" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="1220" class="Symbol">→</a> <a id="1222" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1239" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1241" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1243" class="Symbol">→</a> <a id="1245" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1247" class="Symbol">→</a> <a id="1249" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1253" class="Symbol">_</a>
<a id="1255" href="Function.Metric.Definitions.html#1197" class="Function">NonNegative</a> <a id="1267" href="Function.Metric.Definitions.html#1267" class="Bound Operator">_≤_</a> <a id="1271" href="Function.Metric.Definitions.html#1271" class="Bound">d</a> <a id="1273" href="Function.Metric.Definitions.html#1273" class="Bound">0#</a> <a id="1276" class="Symbol">=</a> <a id="1278" class="Symbol">∀</a> <a id="1280" class="Symbol">{</a><a id="1281" href="Function.Metric.Definitions.html#1281" class="Bound">x</a> <a id="1283" href="Function.Metric.Definitions.html#1283" class="Bound">y</a><a id="1284" class="Symbol">}</a> <a id="1286" class="Symbol">→</a> <a id="1288" href="Function.Metric.Definitions.html#1273" class="Bound">0#</a> <a id="1291" href="Function.Metric.Definitions.html#1267" class="Bound Operator">≤</a> <a id="1293" href="Function.Metric.Definitions.html#1271" class="Bound">d</a> <a id="1295" href="Function.Metric.Definitions.html#1281" class="Bound">x</a> <a id="1297" href="Function.Metric.Definitions.html#1283" class="Bound">y</a>
<a id="Symmetric"></a><a id="1300" href="Function.Metric.Definitions.html#1300" class="Function">Symmetric</a> <a id="1310" class="Symbol">:</a> <a id="1312" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1316" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1318" href="Function.Metric.Definitions.html#687" class="Generalizable">ℓ</a> <a id="1320" class="Symbol">→</a> <a id="1322" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1339" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1341" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1343" class="Symbol">→</a> <a id="1345" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1349" class="Symbol">_</a>
<a id="1351" href="Function.Metric.Definitions.html#1300" class="Function">Symmetric</a> <a id="1361" href="Function.Metric.Definitions.html#1361" class="Bound Operator">_≈_</a> <a id="1365" href="Function.Metric.Definitions.html#1365" class="Bound">d</a> <a id="1367" class="Symbol">=</a> <a id="1369" class="Symbol">∀</a> <a id="1371" href="Function.Metric.Definitions.html#1371" class="Bound">x</a> <a id="1373" href="Function.Metric.Definitions.html#1373" class="Bound">y</a> <a id="1375" class="Symbol">→</a> <a id="1377" href="Function.Metric.Definitions.html#1365" class="Bound">d</a> <a id="1379" href="Function.Metric.Definitions.html#1371" class="Bound">x</a> <a id="1381" href="Function.Metric.Definitions.html#1373" class="Bound">y</a> <a id="1383" href="Function.Metric.Definitions.html#1361" class="Bound Operator">≈</a> <a id="1385" href="Function.Metric.Definitions.html#1365" class="Bound">d</a> <a id="1387" href="Function.Metric.Definitions.html#1373" class="Bound">y</a> <a id="1389" href="Function.Metric.Definitions.html#1371" class="Bound">x</a>
<a id="TriangleInequality"></a><a id="1392" href="Function.Metric.Definitions.html#1392" class="Function">TriangleInequality</a> <a id="1411" class="Symbol">:</a> <a id="1413" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1417" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1419" href="Function.Metric.Definitions.html#687" class="Generalizable">ℓ</a> <a id="1421" class="Symbol">→</a> <a id="1423" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1427" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1429" class="Symbol">→</a> <a id="1431" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1448" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1450" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1452" class="Symbol">→</a> <a id="1454" class="Symbol">_</a>
<a id="1456" href="Function.Metric.Definitions.html#1392" class="Function">TriangleInequality</a> <a id="1475" href="Function.Metric.Definitions.html#1475" class="Bound Operator">_≤_</a> <a id="1479" href="Function.Metric.Definitions.html#1479" class="Bound Operator">_∙_</a> <a id="1483" href="Function.Metric.Definitions.html#1483" class="Bound">d</a> <a id="1485" class="Symbol">=</a> <a id="1487" class="Symbol">∀</a> <a id="1489" href="Function.Metric.Definitions.html#1489" class="Bound">x</a> <a id="1491" href="Function.Metric.Definitions.html#1491" class="Bound">y</a> <a id="1493" href="Function.Metric.Definitions.html#1493" class="Bound">z</a> <a id="1495" class="Symbol">→</a> <a id="1497" href="Function.Metric.Definitions.html#1483" class="Bound">d</a> <a id="1499" href="Function.Metric.Definitions.html#1489" class="Bound">x</a> <a id="1501" href="Function.Metric.Definitions.html#1493" class="Bound">z</a> <a id="1503" href="Function.Metric.Definitions.html#1475" class="Bound Operator">≤</a> <a id="1505" class="Symbol">(</a><a id="1506" href="Function.Metric.Definitions.html#1483" class="Bound">d</a> <a id="1508" href="Function.Metric.Definitions.html#1489" class="Bound">x</a> <a id="1510" href="Function.Metric.Definitions.html#1491" class="Bound">y</a> <a id="1512" href="Function.Metric.Definitions.html#1479" class="Bound Operator">∙</a> <a id="1514" href="Function.Metric.Definitions.html#1483" class="Bound">d</a> <a id="1516" href="Function.Metric.Definitions.html#1491" class="Bound">y</a> <a id="1518" href="Function.Metric.Definitions.html#1493" class="Bound">z</a><a id="1519" class="Symbol">)</a>
<a id="Bounded"></a><a id="1522" href="Function.Metric.Definitions.html#1522" class="Function">Bounded</a> <a id="1530" class="Symbol">:</a> <a id="1532" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1536" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1538" href="Function.Metric.Definitions.html#687" class="Generalizable">ℓ</a> <a id="1540" class="Symbol">→</a> <a id="1542" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1559" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1561" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1563" class="Symbol">→</a> <a id="1565" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1569" class="Symbol">_</a>
<a id="1571" href="Function.Metric.Definitions.html#1522" class="Function">Bounded</a> <a id="1579" href="Function.Metric.Definitions.html#1579" class="Bound Operator">_≤_</a> <a id="1583" href="Function.Metric.Definitions.html#1583" class="Bound">d</a> <a id="1585" class="Symbol">=</a> <a id="1587" href="Data.Product.html#1378" class="Function">∃</a> <a id="1589" class="Symbol">λ</a> <a id="1591" href="Function.Metric.Definitions.html#1591" class="Bound">n</a> <a id="1593" class="Symbol">→</a> <a id="1595" class="Symbol">∀</a> <a id="1597" href="Function.Metric.Definitions.html#1597" class="Bound">x</a> <a id="1599" href="Function.Metric.Definitions.html#1599" class="Bound">y</a> <a id="1601" class="Symbol">→</a> <a id="1603" href="Function.Metric.Definitions.html#1583" class="Bound">d</a> <a id="1605" href="Function.Metric.Definitions.html#1597" class="Bound">x</a> <a id="1607" href="Function.Metric.Definitions.html#1599" class="Bound">y</a> <a id="1609" href="Function.Metric.Definitions.html#1579" class="Bound Operator">≤</a> <a id="1611" href="Function.Metric.Definitions.html#1591" class="Bound">n</a>
<a id="TranslationInvariant"></a><a id="1614" href="Function.Metric.Definitions.html#1614" class="Function">TranslationInvariant</a> <a id="1635" class="Symbol">:</a> <a id="1637" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1641" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1643" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="1646" class="Symbol">→</a> <a id="1648" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1652" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1654" class="Symbol">→</a> <a id="1656" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1673" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1675" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1677" class="Symbol">→</a> <a id="1679" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1683" class="Symbol">_</a>
<a id="1685" href="Function.Metric.Definitions.html#1614" class="Function">TranslationInvariant</a> <a id="1706" href="Function.Metric.Definitions.html#1706" class="Bound Operator">_≈_</a> <a id="1710" href="Function.Metric.Definitions.html#1710" class="Bound Operator">_∙_</a> <a id="1714" href="Function.Metric.Definitions.html#1714" class="Bound">d</a> <a id="1716" class="Symbol">=</a> <a id="1718" class="Symbol">∀</a> <a id="1720" class="Symbol">{</a><a id="1721" href="Function.Metric.Definitions.html#1721" class="Bound">x</a> <a id="1723" href="Function.Metric.Definitions.html#1723" class="Bound">y</a> <a id="1725" href="Function.Metric.Definitions.html#1725" class="Bound">a</a><a id="1726" class="Symbol">}</a> <a id="1728" class="Symbol">→</a> <a id="1730" href="Function.Metric.Definitions.html#1714" class="Bound">d</a> <a id="1732" class="Symbol">(</a><a id="1733" href="Function.Metric.Definitions.html#1721" class="Bound">x</a> <a id="1735" href="Function.Metric.Definitions.html#1710" class="Bound Operator">∙</a> <a id="1737" href="Function.Metric.Definitions.html#1725" class="Bound">a</a><a id="1738" class="Symbol">)</a> <a id="1740" class="Symbol">(</a><a id="1741" href="Function.Metric.Definitions.html#1723" class="Bound">y</a> <a id="1743" href="Function.Metric.Definitions.html#1710" class="Bound Operator">∙</a> <a id="1745" href="Function.Metric.Definitions.html#1725" class="Bound">a</a><a id="1746" class="Symbol">)</a> <a id="1748" href="Function.Metric.Definitions.html#1706" class="Bound Operator">≈</a> <a id="1750" href="Function.Metric.Definitions.html#1714" class="Bound">d</a> <a id="1752" href="Function.Metric.Definitions.html#1721" class="Bound">x</a> <a id="1754" href="Function.Metric.Definitions.html#1723" class="Bound">y</a>
<a id="Contracting"></a><a id="1757" href="Function.Metric.Definitions.html#1757" class="Function">Contracting</a> <a id="1769" class="Symbol">:</a> <a id="1771" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1775" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1777" href="Function.Metric.Definitions.html#687" class="Generalizable">ℓ</a> <a id="1779" class="Symbol">→</a> <a id="1781" class="Symbol">(</a><a id="1782" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1784" class="Symbol">→</a> <a id="1786" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a><a id="1787" class="Symbol">)</a> <a id="1789" class="Symbol">→</a> <a id="1791" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1808" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="1810" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="1812" class="Symbol">→</a> <a id="1814" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1818" class="Symbol">_</a>
<a id="1820" href="Function.Metric.Definitions.html#1757" class="Function">Contracting</a> <a id="1832" href="Function.Metric.Definitions.html#1832" class="Bound Operator">_≤_</a> <a id="1836" href="Function.Metric.Definitions.html#1836" class="Bound">f</a> <a id="1838" href="Function.Metric.Definitions.html#1838" class="Bound">d</a> <a id="1840" class="Symbol">=</a> <a id="1842" class="Symbol">∀</a> <a id="1844" href="Function.Metric.Definitions.html#1844" class="Bound">x</a> <a id="1846" href="Function.Metric.Definitions.html#1846" class="Bound">y</a> <a id="1848" class="Symbol">→</a> <a id="1850" href="Function.Metric.Definitions.html#1838" class="Bound">d</a> <a id="1852" class="Symbol">(</a><a id="1853" href="Function.Metric.Definitions.html#1836" class="Bound">f</a> <a id="1855" href="Function.Metric.Definitions.html#1844" class="Bound">x</a><a id="1856" class="Symbol">)</a> <a id="1858" class="Symbol">(</a><a id="1859" href="Function.Metric.Definitions.html#1836" class="Bound">f</a> <a id="1861" href="Function.Metric.Definitions.html#1846" class="Bound">y</a><a id="1862" class="Symbol">)</a> <a id="1864" href="Function.Metric.Definitions.html#1832" class="Bound Operator">≤</a> <a id="1866" href="Function.Metric.Definitions.html#1838" class="Bound">d</a> <a id="1868" href="Function.Metric.Definitions.html#1844" class="Bound">x</a> <a id="1870" href="Function.Metric.Definitions.html#1846" class="Bound">y</a>
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<a id="1944" href="Function.Metric.Definitions.html#1873" class="Function">ContractingOnOrbits</a> <a id="1964" href="Function.Metric.Definitions.html#1964" class="Bound Operator">_≤_</a> <a id="1968" href="Function.Metric.Definitions.html#1968" class="Bound">f</a> <a id="1970" href="Function.Metric.Definitions.html#1970" class="Bound">d</a> <a id="1972" class="Symbol">=</a> <a id="1974" class="Symbol">∀</a> <a id="1976" href="Function.Metric.Definitions.html#1976" class="Bound">x</a> <a id="1978" class="Symbol">→</a> <a id="1980" href="Function.Metric.Definitions.html#1970" class="Bound">d</a> <a id="1982" class="Symbol">(</a><a id="1983" href="Function.Metric.Definitions.html#1968" class="Bound">f</a> <a id="1985" href="Function.Metric.Definitions.html#1976" class="Bound">x</a><a id="1986" class="Symbol">)</a> <a id="1988" class="Symbol">(</a><a id="1989" href="Function.Metric.Definitions.html#1968" class="Bound">f</a> <a id="1991" class="Symbol">(</a><a id="1992" href="Function.Metric.Definitions.html#1968" class="Bound">f</a> <a id="1994" href="Function.Metric.Definitions.html#1976" class="Bound">x</a><a id="1995" class="Symbol">))</a> <a id="1998" href="Function.Metric.Definitions.html#1964" class="Bound Operator">≤</a> <a id="2000" href="Function.Metric.Definitions.html#1970" class="Bound">d</a> <a id="2002" href="Function.Metric.Definitions.html#1976" class="Bound">x</a> <a id="2004" class="Symbol">(</a><a id="2005" href="Function.Metric.Definitions.html#1968" class="Bound">f</a> <a id="2007" href="Function.Metric.Definitions.html#1976" class="Bound">x</a><a id="2008" class="Symbol">)</a>
<a id="StrictlyContracting"></a><a id="2011" href="Function.Metric.Definitions.html#2011" class="Function">StrictlyContracting</a> <a id="2031" class="Symbol">:</a> <a id="2033" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2037" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="2039" href="Function.Metric.Definitions.html#689" class="Generalizable">ℓ₁</a> <a id="2042" class="Symbol">→</a> <a id="2044" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="2048" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="2050" href="Function.Metric.Definitions.html#692" class="Generalizable">ℓ₂</a> <a id="2053" class="Symbol">→</a> <a id="2055" class="Symbol">(</a><a id="2056" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="2058" class="Symbol">→</a> <a id="2060" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a><a id="2061" class="Symbol">)</a> <a id="2063" class="Symbol">→</a> <a id="2065" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="2082" href="Function.Metric.Definitions.html#707" class="Generalizable">A</a> <a id="2084" href="Function.Metric.Definitions.html#721" class="Generalizable">I</a> <a id="2086" class="Symbol">→</a> <a id="2088" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="2092" class="Symbol">_</a>
<a id="2094" href="Function.Metric.Definitions.html#2011" class="Function">StrictlyContracting</a> <a id="2114" href="Function.Metric.Definitions.html#2114" class="Bound Operator">_≈_</a> <a id="2118" href="Function.Metric.Definitions.html#2118" class="Bound Operator">_&lt;_</a> <a id="2122" href="Function.Metric.Definitions.html#2122" class="Bound">f</a> <a id="2124" href="Function.Metric.Definitions.html#2124" class="Bound">d</a> <a id="2126" class="Symbol">=</a> <a id="2128" class="Symbol">∀</a> <a id="2130" class="Symbol">{</a><a id="2131" href="Function.Metric.Definitions.html#2131" class="Bound">x</a> <a id="2133" href="Function.Metric.Definitions.html#2133" class="Bound">y</a><a id="2134" class="Symbol">}</a> <a id="2136" class="Symbol">→</a> <a id="2138" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="2140" class="Symbol">(</a><a id="2141" href="Function.Metric.Definitions.html#2133" class="Bound">y</a> <a id="2143" href="Function.Metric.Definitions.html#2114" class="Bound Operator">≈</a> <a id="2145" href="Function.Metric.Definitions.html#2131" class="Bound">x</a><a id="2146" class="Symbol">)</a> <a id="2148" class="Symbol">→</a> <a id="2150" href="Function.Metric.Definitions.html#2124" class="Bound">d</a> <a id="2152" class="Symbol">(</a><a id="2153" href="Function.Metric.Definitions.html#2122" class="Bound">f</a> <a id="2155" href="Function.Metric.Definitions.html#2131" class="Bound">x</a><a id="2156" class="Symbol">)</a> <a id="2158" class="Symbol">(</a><a id="2159" href="Function.Metric.Definitions.html#2122" class="Bound">f</a> <a id="2161" href="Function.Metric.Definitions.html#2133" class="Bound">y</a><a id="2162" class="Symbol">)</a> <a id="2164" href="Function.Metric.Definitions.html#2118" class="Bound Operator">&lt;</a> <a id="2166" href="Function.Metric.Definitions.html#2124" class="Bound">d</a> <a id="2168" href="Function.Metric.Definitions.html#2131" class="Bound">x</a> <a id="2170" href="Function.Metric.Definitions.html#2133" class="Bound">y</a>
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