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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.Base.html" class="Module">Data.Product.Base</a> <a id="461" class="Keyword">using</a> <a id="467" class="Symbol">(</a><a id="468" href="Data.Product.Base.html#852" class="Function">∃</a><a id="469" class="Symbol">)</a>
<a id="471" class="Keyword">open</a> <a id="476" class="Keyword">import</a> <a id="483" href="Function.Metric.Core.html" class="Module">Function.Metric.Core</a> <a id="504" class="Keyword">using</a> <a id="510" class="Symbol">(</a><a id="511" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a><a id="527" class="Symbol">)</a>
<a id="529" class="Keyword">open</a> <a id="534" class="Keyword">import</a> <a id="541" href="Level.html" class="Module">Level</a> <a id="547" class="Keyword">using</a> <a id="553" class="Symbol">(</a><a id="554" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="559" class="Symbol">)</a>
<a id="561" class="Keyword">open</a> <a id="566" class="Keyword">import</a> <a id="573" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="594" class="Keyword">using</a> <a id="600" class="Symbol">(</a><a id="601" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="604" class="Symbol">;</a> <a id="606" href="Relation.Binary.Core.html#1703" class="Function Operator">_Preserves₂_⟶_⟶_</a><a id="622" class="Symbol">)</a>
<a id="624" class="Keyword">open</a> <a id="629" class="Keyword">import</a> <a id="636" href="Relation.Nullary.Negation.html" class="Module">Relation.Nullary.Negation</a> <a id="662" class="Keyword">using</a> <a id="668" class="Symbol">(</a><a id="669" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬_</a><a id="671" class="Symbol">)</a>
<a id="674" class="Keyword">private</a>
<a id="684" class="Keyword">variable</a>
<a id="697" href="Function.Metric.Definitions.html#697" class="Generalizable">a</a> <a id="699" href="Function.Metric.Definitions.html#699" class="Generalizable">i</a> <a id="701" href="Function.Metric.Definitions.html#701" class="Generalizable">ℓ</a> <a id="703" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="706" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="709" class="Symbol">:</a> <a id="711" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="721" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="723" class="Symbol">:</a> <a id="725" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="729" href="Function.Metric.Definitions.html#697" class="Generalizable">a</a>
<a id="735" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="737" class="Symbol">:</a> <a id="739" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="743" href="Function.Metric.Definitions.html#699" class="Generalizable">i</a>
<a id="746" class="Comment">------------------------------------------------------------------------</a>
<a id="819" class="Comment">-- Properties</a>
<a id="Congruent"></a><a id="834" href="Function.Metric.Definitions.html#834" class="Function">Congruent</a> <a id="844" class="Symbol">:</a> <a id="846" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="850" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="852" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="855" class="Symbol">→</a> <a id="857" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="861" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="863" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="866" class="Symbol">→</a> <a id="868" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="885" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="887" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="889" class="Symbol">→</a> <a id="891" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="895" class="Symbol">_</a>
<a id="897" href="Function.Metric.Definitions.html#834" class="Function">Congruent</a> <a id="907" href="Function.Metric.Definitions.html#907" class="Bound Operator">_≈ₐ_</a> <a id="912" href="Function.Metric.Definitions.html#912" class="Bound Operator">_≈ᵢ_</a> <a id="917" href="Function.Metric.Definitions.html#917" class="Bound">d</a> <a id="919" class="Symbol">=</a> <a id="921" href="Function.Metric.Definitions.html#917" class="Bound">d</a> <a id="923" href="Relation.Binary.Core.html#1703" class="Function Operator">Preserves₂</a> <a id="934" href="Function.Metric.Definitions.html#907" class="Bound Operator">_≈ₐ_</a> <a id="939" href="Relation.Binary.Core.html#1703" class="Function Operator">⟶</a> <a id="941" href="Function.Metric.Definitions.html#907" class="Bound Operator">_≈ₐ_</a> <a id="946" href="Relation.Binary.Core.html#1703" class="Function Operator">⟶</a> <a id="948" href="Function.Metric.Definitions.html#912" class="Bound Operator">_≈ᵢ_</a>
<a id="Indiscernable"></a><a id="954" href="Function.Metric.Definitions.html#954" class="Function">Indiscernable</a> <a id="968" class="Symbol">:</a> <a id="970" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="974" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="976" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="979" class="Symbol">→</a> <a id="981" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="985" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="987" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="990" class="Symbol">→</a> <a id="992" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1009" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1011" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1013" class="Symbol">→</a> <a id="1015" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1017" class="Symbol">→</a> <a id="1019" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1023" class="Symbol">_</a>
<a id="1025" href="Function.Metric.Definitions.html#954" class="Function">Indiscernable</a> <a id="1039" href="Function.Metric.Definitions.html#1039" class="Bound Operator">_≈ₐ_</a> <a id="1044" href="Function.Metric.Definitions.html#1044" class="Bound Operator">_≈ᵢ_</a> <a id="1049" href="Function.Metric.Definitions.html#1049" class="Bound">d</a> <a id="1051" href="Function.Metric.Definitions.html#1051" class="Bound">0#</a> <a id="1054" class="Symbol">=</a> <a id="1056" class="Symbol">∀</a> <a id="1058" class="Symbol">{</a><a id="1059" href="Function.Metric.Definitions.html#1059" class="Bound">x</a> <a id="1061" href="Function.Metric.Definitions.html#1061" class="Bound">y</a><a id="1062" class="Symbol">}</a> <a id="1064" class="Symbol">→</a> <a id="1066" href="Function.Metric.Definitions.html#1049" class="Bound">d</a> <a id="1068" href="Function.Metric.Definitions.html#1059" class="Bound">x</a> <a id="1070" href="Function.Metric.Definitions.html#1061" class="Bound">y</a> <a id="1072" href="Function.Metric.Definitions.html#1044" class="Bound Operator">≈ᵢ</a> <a id="1075" href="Function.Metric.Definitions.html#1051" class="Bound">0#</a> <a id="1078" class="Symbol">→</a> <a id="1080" href="Function.Metric.Definitions.html#1059" class="Bound">x</a> <a id="1082" href="Function.Metric.Definitions.html#1039" class="Bound Operator">≈ₐ</a> <a id="1085" href="Function.Metric.Definitions.html#1061" class="Bound">y</a>
<a id="Definite"></a><a id="1088" href="Function.Metric.Definitions.html#1088" class="Function">Definite</a> <a id="1097" class="Symbol">:</a> <a id="1099" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1103" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1105" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="1108" class="Symbol">→</a> <a id="1110" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1114" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1116" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="1119" class="Symbol">→</a> <a id="1121" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1138" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1140" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1142" class="Symbol">→</a> <a id="1144" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1146" class="Symbol">→</a> <a id="1148" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1152" class="Symbol">_</a>
<a id="1154" href="Function.Metric.Definitions.html#1088" class="Function">Definite</a> <a id="1163" href="Function.Metric.Definitions.html#1163" class="Bound Operator">_≈ₐ_</a> <a id="1168" href="Function.Metric.Definitions.html#1168" class="Bound Operator">_≈ᵢ_</a> <a id="1173" href="Function.Metric.Definitions.html#1173" class="Bound">d</a> <a id="1175" href="Function.Metric.Definitions.html#1175" class="Bound">0#</a> <a id="1178" class="Symbol">=</a> <a id="1180" class="Symbol">∀</a> <a id="1182" class="Symbol">{</a><a id="1183" href="Function.Metric.Definitions.html#1183" class="Bound">x</a> <a id="1185" href="Function.Metric.Definitions.html#1185" class="Bound">y</a><a id="1186" class="Symbol">}</a> <a id="1188" class="Symbol">→</a> <a id="1190" href="Function.Metric.Definitions.html#1183" class="Bound">x</a> <a id="1192" href="Function.Metric.Definitions.html#1163" class="Bound Operator">≈ₐ</a> <a id="1195" href="Function.Metric.Definitions.html#1185" class="Bound">y</a> <a id="1197" class="Symbol">→</a> <a id="1199" href="Function.Metric.Definitions.html#1173" class="Bound">d</a> <a id="1201" href="Function.Metric.Definitions.html#1183" class="Bound">x</a> <a id="1203" href="Function.Metric.Definitions.html#1185" class="Bound">y</a> <a id="1205" href="Function.Metric.Definitions.html#1168" class="Bound Operator">≈ᵢ</a> <a id="1208" href="Function.Metric.Definitions.html#1175" class="Bound">0#</a>
<a id="NonNegative"></a><a id="1212" href="Function.Metric.Definitions.html#1212" class="Function">NonNegative</a> <a id="1224" class="Symbol">:</a> <a id="1226" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1230" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1232" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="1235" class="Symbol">→</a> <a id="1237" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1254" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1256" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1258" class="Symbol">→</a> <a id="1260" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1262" class="Symbol">→</a> <a id="1264" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1268" class="Symbol">_</a>
<a id="1270" href="Function.Metric.Definitions.html#1212" class="Function">NonNegative</a> <a id="1282" href="Function.Metric.Definitions.html#1282" class="Bound Operator">_≤_</a> <a id="1286" href="Function.Metric.Definitions.html#1286" class="Bound">d</a> <a id="1288" href="Function.Metric.Definitions.html#1288" class="Bound">0#</a> <a id="1291" class="Symbol">=</a> <a id="1293" class="Symbol">∀</a> <a id="1295" class="Symbol">{</a><a id="1296" href="Function.Metric.Definitions.html#1296" class="Bound">x</a> <a id="1298" href="Function.Metric.Definitions.html#1298" class="Bound">y</a><a id="1299" class="Symbol">}</a> <a id="1301" class="Symbol">→</a> <a id="1303" href="Function.Metric.Definitions.html#1288" class="Bound">0#</a> <a id="1306" href="Function.Metric.Definitions.html#1282" class="Bound Operator">≤</a> <a id="1308" href="Function.Metric.Definitions.html#1286" class="Bound">d</a> <a id="1310" href="Function.Metric.Definitions.html#1296" class="Bound">x</a> <a id="1312" href="Function.Metric.Definitions.html#1298" class="Bound">y</a>
<a id="Symmetric"></a><a id="1315" href="Function.Metric.Definitions.html#1315" class="Function">Symmetric</a> <a id="1325" class="Symbol">:</a> <a id="1327" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1331" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1333" href="Function.Metric.Definitions.html#701" class="Generalizable">ℓ</a> <a id="1335" class="Symbol">→</a> <a id="1337" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1354" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1356" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1358" class="Symbol">→</a> <a id="1360" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1364" class="Symbol">_</a>
<a id="1366" href="Function.Metric.Definitions.html#1315" class="Function">Symmetric</a> <a id="1376" href="Function.Metric.Definitions.html#1376" class="Bound Operator">_≈_</a> <a id="1380" href="Function.Metric.Definitions.html#1380" class="Bound">d</a> <a id="1382" class="Symbol">=</a> <a id="1384" class="Symbol">∀</a> <a id="1386" href="Function.Metric.Definitions.html#1386" class="Bound">x</a> <a id="1388" href="Function.Metric.Definitions.html#1388" class="Bound">y</a> <a id="1390" class="Symbol">→</a> <a id="1392" href="Function.Metric.Definitions.html#1380" class="Bound">d</a> <a id="1394" href="Function.Metric.Definitions.html#1386" class="Bound">x</a> <a id="1396" href="Function.Metric.Definitions.html#1388" class="Bound">y</a> <a id="1398" href="Function.Metric.Definitions.html#1376" class="Bound Operator">≈</a> <a id="1400" href="Function.Metric.Definitions.html#1380" class="Bound">d</a> <a id="1402" href="Function.Metric.Definitions.html#1388" class="Bound">y</a> <a id="1404" href="Function.Metric.Definitions.html#1386" class="Bound">x</a>
<a id="TriangleInequality"></a><a id="1407" href="Function.Metric.Definitions.html#1407" class="Function">TriangleInequality</a> <a id="1426" class="Symbol">:</a> <a id="1428" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1432" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1434" href="Function.Metric.Definitions.html#701" class="Generalizable">ℓ</a> <a id="1436" class="Symbol">→</a> <a id="1438" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1442" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1444" class="Symbol">→</a> <a id="1446" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1463" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1465" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1467" class="Symbol">→</a> <a id="1469" class="Symbol">_</a>
<a id="1471" href="Function.Metric.Definitions.html#1407" class="Function">TriangleInequality</a> <a id="1490" href="Function.Metric.Definitions.html#1490" class="Bound Operator">_≤_</a> <a id="1494" href="Function.Metric.Definitions.html#1494" class="Bound Operator">_∙_</a> <a id="1498" href="Function.Metric.Definitions.html#1498" class="Bound">d</a> <a id="1500" class="Symbol">=</a> <a id="1502" class="Symbol">∀</a> <a id="1504" href="Function.Metric.Definitions.html#1504" class="Bound">x</a> <a id="1506" href="Function.Metric.Definitions.html#1506" class="Bound">y</a> <a id="1508" href="Function.Metric.Definitions.html#1508" class="Bound">z</a> <a id="1510" class="Symbol">→</a> <a id="1512" href="Function.Metric.Definitions.html#1498" class="Bound">d</a> <a id="1514" href="Function.Metric.Definitions.html#1504" class="Bound">x</a> <a id="1516" href="Function.Metric.Definitions.html#1508" class="Bound">z</a> <a id="1518" href="Function.Metric.Definitions.html#1490" class="Bound Operator">≤</a> <a id="1520" class="Symbol">(</a><a id="1521" href="Function.Metric.Definitions.html#1498" class="Bound">d</a> <a id="1523" href="Function.Metric.Definitions.html#1504" class="Bound">x</a> <a id="1525" href="Function.Metric.Definitions.html#1506" class="Bound">y</a> <a id="1527" href="Function.Metric.Definitions.html#1494" class="Bound Operator">∙</a> <a id="1529" href="Function.Metric.Definitions.html#1498" class="Bound">d</a> <a id="1531" href="Function.Metric.Definitions.html#1506" class="Bound">y</a> <a id="1533" href="Function.Metric.Definitions.html#1508" class="Bound">z</a><a id="1534" class="Symbol">)</a>
<a id="Bounded"></a><a id="1537" href="Function.Metric.Definitions.html#1537" class="Function">Bounded</a> <a id="1545" class="Symbol">:</a> <a id="1547" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1551" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1553" href="Function.Metric.Definitions.html#701" class="Generalizable">ℓ</a> <a id="1555" class="Symbol">→</a> <a id="1557" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1574" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1576" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1578" class="Symbol">→</a> <a id="1580" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1584" class="Symbol">_</a>
<a id="1586" href="Function.Metric.Definitions.html#1537" class="Function">Bounded</a> <a id="1594" href="Function.Metric.Definitions.html#1594" class="Bound Operator">_≤_</a> <a id="1598" href="Function.Metric.Definitions.html#1598" class="Bound">d</a> <a id="1600" class="Symbol">=</a> <a id="1602" href="Data.Product.Base.html#852" class="Function">∃</a> <a id="1604" class="Symbol">λ</a> <a id="1606" href="Function.Metric.Definitions.html#1606" class="Bound">n</a> <a id="1608" class="Symbol">→</a> <a id="1610" class="Symbol">∀</a> <a id="1612" href="Function.Metric.Definitions.html#1612" class="Bound">x</a> <a id="1614" href="Function.Metric.Definitions.html#1614" class="Bound">y</a> <a id="1616" class="Symbol">→</a> <a id="1618" href="Function.Metric.Definitions.html#1598" class="Bound">d</a> <a id="1620" href="Function.Metric.Definitions.html#1612" class="Bound">x</a> <a id="1622" href="Function.Metric.Definitions.html#1614" class="Bound">y</a> <a id="1624" href="Function.Metric.Definitions.html#1594" class="Bound Operator">≤</a> <a id="1626" href="Function.Metric.Definitions.html#1606" class="Bound">n</a>
<a id="TranslationInvariant"></a><a id="1629" href="Function.Metric.Definitions.html#1629" class="Function">TranslationInvariant</a> <a id="1650" class="Symbol">:</a> <a id="1652" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1656" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1658" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="1661" class="Symbol">→</a> <a id="1663" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="1667" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1669" class="Symbol">→</a> <a id="1671" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1688" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1690" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1692" class="Symbol">→</a> <a id="1694" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1698" class="Symbol">_</a>
<a id="1700" href="Function.Metric.Definitions.html#1629" class="Function">TranslationInvariant</a> <a id="1721" href="Function.Metric.Definitions.html#1721" class="Bound Operator">_≈_</a> <a id="1725" href="Function.Metric.Definitions.html#1725" class="Bound Operator">_∙_</a> <a id="1729" href="Function.Metric.Definitions.html#1729" class="Bound">d</a> <a id="1731" class="Symbol">=</a> <a id="1733" class="Symbol">∀</a> <a id="1735" class="Symbol">{</a><a id="1736" href="Function.Metric.Definitions.html#1736" class="Bound">x</a> <a id="1738" href="Function.Metric.Definitions.html#1738" class="Bound">y</a> <a id="1740" href="Function.Metric.Definitions.html#1740" class="Bound">a</a><a id="1741" class="Symbol">}</a> <a id="1743" class="Symbol">→</a> <a id="1745" href="Function.Metric.Definitions.html#1729" class="Bound">d</a> <a id="1747" class="Symbol">(</a><a id="1748" href="Function.Metric.Definitions.html#1736" class="Bound">x</a> <a id="1750" href="Function.Metric.Definitions.html#1725" class="Bound Operator">∙</a> <a id="1752" href="Function.Metric.Definitions.html#1740" class="Bound">a</a><a id="1753" class="Symbol">)</a> <a id="1755" class="Symbol">(</a><a id="1756" href="Function.Metric.Definitions.html#1738" class="Bound">y</a> <a id="1758" href="Function.Metric.Definitions.html#1725" class="Bound Operator">∙</a> <a id="1760" href="Function.Metric.Definitions.html#1740" class="Bound">a</a><a id="1761" class="Symbol">)</a> <a id="1763" href="Function.Metric.Definitions.html#1721" class="Bound Operator">≈</a> <a id="1765" href="Function.Metric.Definitions.html#1729" class="Bound">d</a> <a id="1767" href="Function.Metric.Definitions.html#1736" class="Bound">x</a> <a id="1769" href="Function.Metric.Definitions.html#1738" class="Bound">y</a>
<a id="Contracting"></a><a id="1772" href="Function.Metric.Definitions.html#1772" class="Function">Contracting</a> <a id="1784" class="Symbol">:</a> <a id="1786" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1790" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1792" href="Function.Metric.Definitions.html#701" class="Generalizable">ℓ</a> <a id="1794" class="Symbol">→</a> <a id="1796" class="Symbol">(</a><a id="1797" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1799" class="Symbol">→</a> <a id="1801" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a><a id="1802" class="Symbol">)</a> <a id="1804" class="Symbol">→</a> <a id="1806" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="1823" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="1825" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="1827" class="Symbol">→</a> <a id="1829" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1833" class="Symbol">_</a>
<a id="1835" href="Function.Metric.Definitions.html#1772" class="Function">Contracting</a> <a id="1847" href="Function.Metric.Definitions.html#1847" class="Bound Operator">_≤_</a> <a id="1851" href="Function.Metric.Definitions.html#1851" class="Bound">f</a> <a id="1853" href="Function.Metric.Definitions.html#1853" class="Bound">d</a> <a id="1855" class="Symbol">=</a> <a id="1857" class="Symbol">∀</a> <a id="1859" href="Function.Metric.Definitions.html#1859" class="Bound">x</a> <a id="1861" href="Function.Metric.Definitions.html#1861" class="Bound">y</a> <a id="1863" class="Symbol">→</a> <a id="1865" href="Function.Metric.Definitions.html#1853" class="Bound">d</a> <a id="1867" class="Symbol">(</a><a id="1868" href="Function.Metric.Definitions.html#1851" class="Bound">f</a> <a id="1870" href="Function.Metric.Definitions.html#1859" class="Bound">x</a><a id="1871" class="Symbol">)</a> <a id="1873" class="Symbol">(</a><a id="1874" href="Function.Metric.Definitions.html#1851" class="Bound">f</a> <a id="1876" href="Function.Metric.Definitions.html#1861" class="Bound">y</a><a id="1877" class="Symbol">)</a> <a id="1879" href="Function.Metric.Definitions.html#1847" class="Bound Operator">≤</a> <a id="1881" href="Function.Metric.Definitions.html#1853" class="Bound">d</a> <a id="1883" href="Function.Metric.Definitions.html#1859" class="Bound">x</a> <a id="1885" href="Function.Metric.Definitions.html#1861" class="Bound">y</a>
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<a id="1959" href="Function.Metric.Definitions.html#1888" class="Function">ContractingOnOrbits</a> <a id="1979" href="Function.Metric.Definitions.html#1979" class="Bound Operator">_≤_</a> <a id="1983" href="Function.Metric.Definitions.html#1983" class="Bound">f</a> <a id="1985" href="Function.Metric.Definitions.html#1985" class="Bound">d</a> <a id="1987" class="Symbol">=</a> <a id="1989" class="Symbol">∀</a> <a id="1991" href="Function.Metric.Definitions.html#1991" class="Bound">x</a> <a id="1993" class="Symbol">→</a> <a id="1995" href="Function.Metric.Definitions.html#1985" class="Bound">d</a> <a id="1997" class="Symbol">(</a><a id="1998" href="Function.Metric.Definitions.html#1983" class="Bound">f</a> <a id="2000" href="Function.Metric.Definitions.html#1991" class="Bound">x</a><a id="2001" class="Symbol">)</a> <a id="2003" class="Symbol">(</a><a id="2004" href="Function.Metric.Definitions.html#1983" class="Bound">f</a> <a id="2006" class="Symbol">(</a><a id="2007" href="Function.Metric.Definitions.html#1983" class="Bound">f</a> <a id="2009" href="Function.Metric.Definitions.html#1991" class="Bound">x</a><a id="2010" class="Symbol">))</a> <a id="2013" href="Function.Metric.Definitions.html#1979" class="Bound Operator">≤</a> <a id="2015" href="Function.Metric.Definitions.html#1985" class="Bound">d</a> <a id="2017" href="Function.Metric.Definitions.html#1991" class="Bound">x</a> <a id="2019" class="Symbol">(</a><a id="2020" href="Function.Metric.Definitions.html#1983" class="Bound">f</a> <a id="2022" href="Function.Metric.Definitions.html#1991" class="Bound">x</a><a id="2023" class="Symbol">)</a>
<a id="StrictlyContracting"></a><a id="2026" href="Function.Metric.Definitions.html#2026" class="Function">StrictlyContracting</a> <a id="2046" class="Symbol">:</a> <a id="2048" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2052" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2054" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="2057" class="Symbol">→</a> <a id="2059" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2063" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="2065" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="2068" class="Symbol">→</a> <a id="2070" class="Symbol">(</a><a id="2071" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2073" class="Symbol">→</a> <a id="2075" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a><a id="2076" class="Symbol">)</a> <a id="2078" class="Symbol">→</a> <a id="2080" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="2097" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2099" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="2101" class="Symbol">→</a> <a id="2103" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2107" class="Symbol">_</a>
<a id="2109" href="Function.Metric.Definitions.html#2026" class="Function">StrictlyContracting</a> <a id="2129" href="Function.Metric.Definitions.html#2129" class="Bound Operator">_≈_</a> <a id="2133" href="Function.Metric.Definitions.html#2133" class="Bound Operator">_&lt;_</a> <a id="2137" href="Function.Metric.Definitions.html#2137" class="Bound">f</a> <a id="2139" href="Function.Metric.Definitions.html#2139" class="Bound">d</a> <a id="2141" class="Symbol">=</a> <a id="2143" class="Symbol">∀</a> <a id="2145" class="Symbol">{</a><a id="2146" href="Function.Metric.Definitions.html#2146" class="Bound">x</a> <a id="2148" href="Function.Metric.Definitions.html#2148" class="Bound">y</a><a id="2149" class="Symbol">}</a> <a id="2151" class="Symbol">→</a> <a id="2153" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="2155" class="Symbol">(</a><a id="2156" href="Function.Metric.Definitions.html#2148" class="Bound">y</a> <a id="2158" href="Function.Metric.Definitions.html#2129" class="Bound Operator">≈</a> <a id="2160" href="Function.Metric.Definitions.html#2146" class="Bound">x</a><a id="2161" class="Symbol">)</a> <a id="2163" class="Symbol">→</a> <a id="2165" href="Function.Metric.Definitions.html#2139" class="Bound">d</a> <a id="2167" class="Symbol">(</a><a id="2168" href="Function.Metric.Definitions.html#2137" class="Bound">f</a> <a id="2170" href="Function.Metric.Definitions.html#2146" class="Bound">x</a><a id="2171" class="Symbol">)</a> <a id="2173" class="Symbol">(</a><a id="2174" href="Function.Metric.Definitions.html#2137" class="Bound">f</a> <a id="2176" href="Function.Metric.Definitions.html#2148" class="Bound">y</a><a id="2177" class="Symbol">)</a> <a id="2179" href="Function.Metric.Definitions.html#2133" class="Bound Operator">&lt;</a> <a id="2181" href="Function.Metric.Definitions.html#2139" class="Bound">d</a> <a id="2183" href="Function.Metric.Definitions.html#2146" class="Bound">x</a> <a id="2185" href="Function.Metric.Definitions.html#2148" class="Bound">y</a>
<a id="StrictlyContractingOnOrbits"></a><a id="2188" href="Function.Metric.Definitions.html#2188" class="Function">StrictlyContractingOnOrbits</a> <a id="2216" class="Symbol">:</a> <a id="2218" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2222" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2224" href="Function.Metric.Definitions.html#703" class="Generalizable">ℓ₁</a> <a id="2227" class="Symbol">→</a> <a id="2229" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="2233" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="2235" href="Function.Metric.Definitions.html#706" class="Generalizable">ℓ₂</a> <a id="2238" class="Symbol">→</a> <a id="2240" class="Symbol">(</a><a id="2241" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2243" class="Symbol">→</a> <a id="2245" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a><a id="2246" class="Symbol">)</a> <a id="2248" class="Symbol">→</a> <a id="2250" href="Function.Metric.Core.html#451" class="Function">DistanceFunction</a> <a id="2267" href="Function.Metric.Definitions.html#721" class="Generalizable">A</a> <a id="2269" href="Function.Metric.Definitions.html#735" class="Generalizable">I</a> <a id="2271" class="Symbol">→</a> <a id="2273" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2277" class="Symbol">_</a>
<a id="2279" href="Function.Metric.Definitions.html#2188" class="Function">StrictlyContractingOnOrbits</a> <a id="2307" href="Function.Metric.Definitions.html#2307" class="Bound Operator">_≈_</a> <a id="2311" href="Function.Metric.Definitions.html#2311" class="Bound Operator">_&lt;_</a> <a id="2315" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2317" href="Function.Metric.Definitions.html#2317" class="Bound">d</a> <a id="2319" class="Symbol">=</a> <a id="2321" class="Symbol">∀</a> <a id="2323" class="Symbol">{</a><a id="2324" href="Function.Metric.Definitions.html#2324" class="Bound">x</a><a id="2325" class="Symbol">}</a> <a id="2327" class="Symbol">→</a> <a id="2329" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="2331" class="Symbol">(</a><a id="2332" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2334" href="Function.Metric.Definitions.html#2324" class="Bound">x</a> <a id="2336" href="Function.Metric.Definitions.html#2307" class="Bound Operator">≈</a> <a id="2338" href="Function.Metric.Definitions.html#2324" class="Bound">x</a><a id="2339" class="Symbol">)</a> <a id="2341" class="Symbol">→</a> <a id="2343" href="Function.Metric.Definitions.html#2317" class="Bound">d</a> <a id="2345" class="Symbol">(</a><a id="2346" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2348" href="Function.Metric.Definitions.html#2324" class="Bound">x</a><a id="2349" class="Symbol">)</a> <a id="2351" class="Symbol">(</a><a id="2352" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2354" class="Symbol">(</a><a id="2355" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2357" href="Function.Metric.Definitions.html#2324" class="Bound">x</a><a id="2358" class="Symbol">))</a> <a id="2361" href="Function.Metric.Definitions.html#2311" class="Bound Operator">&lt;</a> <a id="2363" href="Function.Metric.Definitions.html#2317" class="Bound">d</a> <a id="2365" href="Function.Metric.Definitions.html#2324" class="Bound">x</a> <a id="2367" class="Symbol">(</a><a id="2368" href="Function.Metric.Definitions.html#2315" class="Bound">f</a> <a id="2370" href="Function.Metric.Definitions.html#2324" class="Bound">x</a><a id="2371" class="Symbol">)</a>
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