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74 KiB
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<html><head><meta charset="utf-8"><title>Relation.Binary.Construct.NonStrictToStrict</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">-- Conversion of _≤_ to _<_</a>
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<a id="134" class="Comment">------------------------------------------------------------------------</a>
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<a id="208" class="Symbol">{-#</a> <a id="212" class="Keyword">OPTIONS</a> <a id="220" class="Pragma">--cubical-compatible</a> <a id="241" class="Pragma">--safe</a> <a id="248" class="Symbol">#-}</a>
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<a id="253" class="Keyword">open</a> <a id="258" class="Keyword">import</a> <a id="265" href="Relation.Binary.html" class="Module">Relation.Binary</a>
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<a id="282" class="Keyword">module</a> <a id="289" href="Relation.Binary.Construct.NonStrictToStrict.html" class="Module">Relation.Binary.Construct.NonStrictToStrict</a>
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<a id="335" class="Symbol">{</a><a id="336" href="Relation.Binary.Construct.NonStrictToStrict.html#336" class="Bound">a</a> <a id="338" href="Relation.Binary.Construct.NonStrictToStrict.html#338" class="Bound">ℓ₁</a> <a id="341" href="Relation.Binary.Construct.NonStrictToStrict.html#341" class="Bound">ℓ₂</a><a id="343" class="Symbol">}</a> <a id="345" class="Symbol">{</a><a id="346" href="Relation.Binary.Construct.NonStrictToStrict.html#346" class="Bound">A</a> <a id="348" class="Symbol">:</a> <a id="350" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="354" href="Relation.Binary.Construct.NonStrictToStrict.html#336" class="Bound">a</a><a id="355" class="Symbol">}</a> <a id="357" class="Symbol">(</a><a id="358" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="362" class="Symbol">:</a> <a id="364" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="368" href="Relation.Binary.Construct.NonStrictToStrict.html#346" class="Bound">A</a> <a id="370" href="Relation.Binary.Construct.NonStrictToStrict.html#338" class="Bound">ℓ₁</a><a id="372" class="Symbol">)</a> <a id="374" class="Symbol">(</a><a id="375" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="379" class="Symbol">:</a> <a id="381" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="385" href="Relation.Binary.Construct.NonStrictToStrict.html#346" class="Bound">A</a> <a id="387" href="Relation.Binary.Construct.NonStrictToStrict.html#341" class="Bound">ℓ₂</a><a id="389" class="Symbol">)</a> <a id="391" class="Keyword">where</a>
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<a id="398" class="Keyword">open</a> <a id="403" class="Keyword">import</a> <a id="410" href="Data.Product.html" class="Module">Data.Product</a> <a id="423" class="Keyword">using</a> <a id="429" class="Symbol">(</a><a id="430" href="Data.Product.html#1176" class="Function Operator">_×_</a><a id="433" class="Symbol">;</a> <a id="435" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="438" class="Symbol">;</a> <a id="440" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a><a id="445" class="Symbol">;</a> <a id="447" href="Agda.Builtin.Sigma.html#246" class="Field">proj₂</a><a id="452" class="Symbol">)</a>
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<a id="454" class="Keyword">open</a> <a id="459" class="Keyword">import</a> <a id="466" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="480" class="Keyword">using</a> <a id="486" class="Symbol">(</a><a id="487" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a><a id="491" class="Symbol">;</a> <a id="493" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a><a id="497" class="Symbol">)</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#1040" class="Function Operator">_∘_</a><a id="535" class="Symbol">;</a> <a id="537" href="Function.Base.html#1563" class="Function">flip</a><a id="541" class="Symbol">)</a>
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<a id="543" class="Keyword">open</a> <a id="548" class="Keyword">import</a> <a id="555" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="572" class="Keyword">using</a> <a id="578" class="Symbol">(</a><a id="579" href="Relation.Nullary.html#665" class="Function Operator">¬_</a><a id="581" class="Symbol">;</a> <a id="583" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a><a id="586" class="Symbol">;</a> <a id="588" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a><a id="590" class="Symbol">)</a>
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<a id="592" class="Keyword">open</a> <a id="597" class="Keyword">import</a> <a id="604" href="Relation.Nullary.Negation.html" class="Module">Relation.Nullary.Negation</a> <a id="630" class="Keyword">using</a> <a id="636" class="Symbol">(</a><a id="637" href="Relation.Nullary.Negation.Core.html#787" class="Function">contradiction</a><a id="650" class="Symbol">;</a> <a id="652" href="Relation.Nullary.Negation.Core.html#1364" class="Function">¬?</a><a id="654" class="Symbol">)</a>
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<a id="656" class="Keyword">open</a> <a id="661" class="Keyword">import</a> <a id="668" href="Relation.Nullary.Product.html" class="Module">Relation.Nullary.Product</a> <a id="693" class="Keyword">using</a> <a id="699" class="Symbol">(</a><a id="700" href="Relation.Nullary.Product.html#930" class="Function Operator">_×-dec_</a><a id="707" class="Symbol">)</a>
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<a id="710" class="Keyword">private</a>
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<a id="_≉_"></a><a id="720" href="Relation.Binary.Construct.NonStrictToStrict.html#720" class="Function Operator">_≉_</a> <a id="724" class="Symbol">:</a> <a id="726" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="730" href="Relation.Binary.Construct.NonStrictToStrict.html#346" class="Bound">A</a> <a id="732" href="Relation.Binary.Construct.NonStrictToStrict.html#338" class="Bound">ℓ₁</a>
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<a id="737" href="Relation.Binary.Construct.NonStrictToStrict.html#737" class="Bound">x</a> <a id="739" href="Relation.Binary.Construct.NonStrictToStrict.html#720" class="Function Operator">≉</a> <a id="741" href="Relation.Binary.Construct.NonStrictToStrict.html#741" class="Bound">y</a> <a id="743" class="Symbol">=</a> <a id="745" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="747" class="Symbol">(</a><a id="748" href="Relation.Binary.Construct.NonStrictToStrict.html#737" class="Bound">x</a> <a id="750" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">≈</a> <a id="752" href="Relation.Binary.Construct.NonStrictToStrict.html#741" class="Bound">y</a><a id="753" class="Symbol">)</a>
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<a id="756" class="Comment">------------------------------------------------------------------------</a>
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<a id="829" class="Comment">-- _≤_ can be turned into _<_ as follows:</a>
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<a id="872" class="Keyword">infix</a> <a id="878" class="Number">4</a> <a id="881" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
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<a id="_<_"></a><a id="886" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="890" class="Symbol">:</a> <a id="892" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="896" href="Relation.Binary.Construct.NonStrictToStrict.html#346" class="Bound">A</a> <a id="898" class="Symbol">_</a>
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<a id="900" href="Relation.Binary.Construct.NonStrictToStrict.html#900" class="Bound">x</a> <a id="902" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="904" href="Relation.Binary.Construct.NonStrictToStrict.html#904" class="Bound">y</a> <a id="906" class="Symbol">=</a> <a id="908" href="Relation.Binary.Construct.NonStrictToStrict.html#900" class="Bound">x</a> <a id="910" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="912" href="Relation.Binary.Construct.NonStrictToStrict.html#904" class="Bound">y</a> <a id="914" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="916" href="Relation.Binary.Construct.NonStrictToStrict.html#900" class="Bound">x</a> <a id="918" href="Relation.Binary.Construct.NonStrictToStrict.html#720" class="Function Operator">≉</a> <a id="920" href="Relation.Binary.Construct.NonStrictToStrict.html#904" class="Bound">y</a>
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<a id="923" class="Comment">------------------------------------------------------------------------</a>
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<a id="996" class="Comment">-- Relationship between relations</a>
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<a id="<⇒≤"></a><a id="1031" href="Relation.Binary.Construct.NonStrictToStrict.html#1031" class="Function"><⇒≤</a> <a id="1035" class="Symbol">:</a> <a id="1037" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="1041" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1043" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a>
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<a id="1047" href="Relation.Binary.Construct.NonStrictToStrict.html#1031" class="Function"><⇒≤</a> <a id="1051" class="Symbol">=</a> <a id="1053" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a>
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<a id="<⇒≉"></a><a id="1060" href="Relation.Binary.Construct.NonStrictToStrict.html#1060" class="Function"><⇒≉</a> <a id="1064" class="Symbol">:</a> <a id="1066" class="Symbol">∀</a> <a id="1068" class="Symbol">{</a><a id="1069" href="Relation.Binary.Construct.NonStrictToStrict.html#1069" class="Bound">x</a> <a id="1071" href="Relation.Binary.Construct.NonStrictToStrict.html#1071" class="Bound">y</a><a id="1072" class="Symbol">}</a> <a id="1074" class="Symbol">→</a> <a id="1076" href="Relation.Binary.Construct.NonStrictToStrict.html#1069" class="Bound">x</a> <a id="1078" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1080" href="Relation.Binary.Construct.NonStrictToStrict.html#1071" class="Bound">y</a> <a id="1082" class="Symbol">→</a> <a id="1084" href="Relation.Binary.Construct.NonStrictToStrict.html#1069" class="Bound">x</a> <a id="1086" href="Relation.Binary.Construct.NonStrictToStrict.html#720" class="Function Operator">≉</a> <a id="1088" href="Relation.Binary.Construct.NonStrictToStrict.html#1071" class="Bound">y</a>
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<a id="1090" href="Relation.Binary.Construct.NonStrictToStrict.html#1060" class="Function"><⇒≉</a> <a id="1094" class="Symbol">=</a> <a id="1096" href="Agda.Builtin.Sigma.html#246" class="Field">proj₂</a>
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<a id="≤∧≉⇒<"></a><a id="1103" href="Relation.Binary.Construct.NonStrictToStrict.html#1103" class="Function">≤∧≉⇒<</a> <a id="1109" class="Symbol">:</a> <a id="1111" class="Symbol">∀</a> <a id="1113" class="Symbol">{</a><a id="1114" href="Relation.Binary.Construct.NonStrictToStrict.html#1114" class="Bound">x</a> <a id="1116" href="Relation.Binary.Construct.NonStrictToStrict.html#1116" class="Bound">y</a><a id="1117" class="Symbol">}</a> <a id="1119" class="Symbol">→</a> <a id="1121" href="Relation.Binary.Construct.NonStrictToStrict.html#1114" class="Bound">x</a> <a id="1123" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="1125" href="Relation.Binary.Construct.NonStrictToStrict.html#1116" class="Bound">y</a> <a id="1127" class="Symbol">→</a> <a id="1129" href="Relation.Binary.Construct.NonStrictToStrict.html#1114" class="Bound">x</a> <a id="1131" href="Relation.Binary.Construct.NonStrictToStrict.html#720" class="Function Operator">≉</a> <a id="1133" href="Relation.Binary.Construct.NonStrictToStrict.html#1116" class="Bound">y</a> <a id="1135" class="Symbol">→</a> <a id="1137" href="Relation.Binary.Construct.NonStrictToStrict.html#1114" class="Bound">x</a> <a id="1139" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1141" href="Relation.Binary.Construct.NonStrictToStrict.html#1116" class="Bound">y</a>
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<a id="1143" href="Relation.Binary.Construct.NonStrictToStrict.html#1103" class="Function">≤∧≉⇒<</a> <a id="1149" class="Symbol">=</a> <a id="1151" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a>
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<a id="<⇒≱"></a><a id="1156" href="Relation.Binary.Construct.NonStrictToStrict.html#1156" class="Function"><⇒≱</a> <a id="1160" class="Symbol">:</a> <a id="1162" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1176" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1180" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="1184" class="Symbol">→</a> <a id="1186" class="Symbol">∀</a> <a id="1188" class="Symbol">{</a><a id="1189" href="Relation.Binary.Construct.NonStrictToStrict.html#1189" class="Bound">x</a> <a id="1191" href="Relation.Binary.Construct.NonStrictToStrict.html#1191" class="Bound">y</a><a id="1192" class="Symbol">}</a> <a id="1194" class="Symbol">→</a> <a id="1196" href="Relation.Binary.Construct.NonStrictToStrict.html#1189" class="Bound">x</a> <a id="1198" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1200" href="Relation.Binary.Construct.NonStrictToStrict.html#1191" class="Bound">y</a> <a id="1202" class="Symbol">→</a> <a id="1204" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="1206" class="Symbol">(</a><a id="1207" href="Relation.Binary.Construct.NonStrictToStrict.html#1191" class="Bound">y</a> <a id="1209" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="1211" href="Relation.Binary.Construct.NonStrictToStrict.html#1189" class="Bound">x</a><a id="1212" class="Symbol">)</a>
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<a id="1214" href="Relation.Binary.Construct.NonStrictToStrict.html#1156" class="Function"><⇒≱</a> <a id="1218" href="Relation.Binary.Construct.NonStrictToStrict.html#1218" class="Bound">antisym</a> <a id="1226" class="Symbol">(</a><a id="1227" href="Relation.Binary.Construct.NonStrictToStrict.html#1227" class="Bound">x≤y</a> <a id="1231" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1233" href="Relation.Binary.Construct.NonStrictToStrict.html#1233" class="Bound">x≉y</a><a id="1236" class="Symbol">)</a> <a id="1238" href="Relation.Binary.Construct.NonStrictToStrict.html#1238" class="Bound">y≤x</a> <a id="1242" class="Symbol">=</a> <a id="1244" href="Relation.Binary.Construct.NonStrictToStrict.html#1233" class="Bound">x≉y</a> <a id="1248" class="Symbol">(</a><a id="1249" href="Relation.Binary.Construct.NonStrictToStrict.html#1218" class="Bound">antisym</a> <a id="1257" href="Relation.Binary.Construct.NonStrictToStrict.html#1227" class="Bound">x≤y</a> <a id="1261" href="Relation.Binary.Construct.NonStrictToStrict.html#1238" class="Bound">y≤x</a><a id="1264" class="Symbol">)</a>
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<a id="≤⇒≯"></a><a id="1267" href="Relation.Binary.Construct.NonStrictToStrict.html#1267" class="Function">≤⇒≯</a> <a id="1271" class="Symbol">:</a> <a id="1273" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1287" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1291" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="1295" class="Symbol">→</a> <a id="1297" class="Symbol">∀</a> <a id="1299" class="Symbol">{</a><a id="1300" href="Relation.Binary.Construct.NonStrictToStrict.html#1300" class="Bound">x</a> <a id="1302" href="Relation.Binary.Construct.NonStrictToStrict.html#1302" class="Bound">y</a><a id="1303" class="Symbol">}</a> <a id="1305" class="Symbol">→</a> <a id="1307" href="Relation.Binary.Construct.NonStrictToStrict.html#1300" class="Bound">x</a> <a id="1309" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="1311" href="Relation.Binary.Construct.NonStrictToStrict.html#1302" class="Bound">y</a> <a id="1313" class="Symbol">→</a> <a id="1315" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="1317" class="Symbol">(</a><a id="1318" href="Relation.Binary.Construct.NonStrictToStrict.html#1302" class="Bound">y</a> <a id="1320" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1322" href="Relation.Binary.Construct.NonStrictToStrict.html#1300" class="Bound">x</a><a id="1323" class="Symbol">)</a>
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<a id="1325" href="Relation.Binary.Construct.NonStrictToStrict.html#1267" class="Function">≤⇒≯</a> <a id="1329" href="Relation.Binary.Construct.NonStrictToStrict.html#1329" class="Bound">antisym</a> <a id="1337" href="Relation.Binary.Construct.NonStrictToStrict.html#1337" class="Bound">x≤y</a> <a id="1341" href="Relation.Binary.Construct.NonStrictToStrict.html#1341" class="Bound">y<x</a> <a id="1345" class="Symbol">=</a> <a id="1347" href="Relation.Binary.Construct.NonStrictToStrict.html#1156" class="Function"><⇒≱</a> <a id="1351" href="Relation.Binary.Construct.NonStrictToStrict.html#1329" class="Bound">antisym</a> <a id="1359" href="Relation.Binary.Construct.NonStrictToStrict.html#1341" class="Bound">y<x</a> <a id="1363" href="Relation.Binary.Construct.NonStrictToStrict.html#1337" class="Bound">x≤y</a>
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<a id="≰⇒>"></a><a id="1368" href="Relation.Binary.Construct.NonStrictToStrict.html#1368" class="Function">≰⇒></a> <a id="1372" class="Symbol">:</a> <a id="1374" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1384" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1388" class="Symbol">→</a> <a id="1390" class="Symbol">(</a><a id="1391" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1395" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1397" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a><a id="1400" class="Symbol">)</a> <a id="1402" class="Symbol">→</a> <a id="1404" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="1410" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="1414" class="Symbol">→</a>
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<a id="1422" class="Symbol">∀</a> <a id="1424" class="Symbol">{</a><a id="1425" href="Relation.Binary.Construct.NonStrictToStrict.html#1425" class="Bound">x</a> <a id="1427" href="Relation.Binary.Construct.NonStrictToStrict.html#1427" class="Bound">y</a><a id="1428" class="Symbol">}</a> <a id="1430" class="Symbol">→</a> <a id="1432" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="1434" class="Symbol">(</a><a id="1435" href="Relation.Binary.Construct.NonStrictToStrict.html#1425" class="Bound">x</a> <a id="1437" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="1439" href="Relation.Binary.Construct.NonStrictToStrict.html#1427" class="Bound">y</a><a id="1440" class="Symbol">)</a> <a id="1442" class="Symbol">→</a> <a id="1444" href="Relation.Binary.Construct.NonStrictToStrict.html#1427" class="Bound">y</a> <a id="1446" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1448" href="Relation.Binary.Construct.NonStrictToStrict.html#1425" class="Bound">x</a>
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<a id="1450" href="Relation.Binary.Construct.NonStrictToStrict.html#1368" class="Function">≰⇒></a> <a id="1454" href="Relation.Binary.Construct.NonStrictToStrict.html#1454" class="Bound">sym</a> <a id="1458" href="Relation.Binary.Construct.NonStrictToStrict.html#1458" class="Bound">refl</a> <a id="1463" href="Relation.Binary.Construct.NonStrictToStrict.html#1463" class="Bound">total</a> <a id="1469" class="Symbol">{</a><a id="1470" href="Relation.Binary.Construct.NonStrictToStrict.html#1470" class="Bound">x</a><a id="1471" class="Symbol">}</a> <a id="1473" class="Symbol">{</a><a id="1474" href="Relation.Binary.Construct.NonStrictToStrict.html#1474" class="Bound">y</a><a id="1475" class="Symbol">}</a> <a id="1477" href="Relation.Binary.Construct.NonStrictToStrict.html#1477" class="Bound">x≰y</a> <a id="1481" class="Keyword">with</a> <a id="1486" href="Relation.Binary.Construct.NonStrictToStrict.html#1463" class="Bound">total</a> <a id="1492" href="Relation.Binary.Construct.NonStrictToStrict.html#1470" class="Bound">x</a> <a id="1494" href="Relation.Binary.Construct.NonStrictToStrict.html#1474" class="Bound">y</a>
|
||
<a id="1496" class="Symbol">...</a> <a id="1500" class="Symbol">|</a> <a id="1502" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="1507" href="Relation.Binary.Construct.NonStrictToStrict.html#1507" class="Bound">x≤y</a> <a id="1511" class="Symbol">=</a> <a id="1513" href="Relation.Nullary.Negation.Core.html#787" class="Function">contradiction</a> <a id="1527" href="Relation.Binary.Construct.NonStrictToStrict.html#1507" class="Bound">x≤y</a> <a id="1531" class="Bound">x≰y</a>
|
||
<a id="1535" class="Symbol">...</a> <a id="1539" class="Symbol">|</a> <a id="1541" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="1546" href="Relation.Binary.Construct.NonStrictToStrict.html#1546" class="Bound">y≤x</a> <a id="1550" class="Symbol">=</a> <a id="1552" href="Relation.Binary.Construct.NonStrictToStrict.html#1546" class="Bound">y≤x</a> <a id="1556" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1558" class="Bound">x≰y</a> <a id="1562" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="1564" class="Bound">refl</a> <a id="1569" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="1571" class="Bound">sym</a>
|
||
|
||
<a id="≮⇒≥"></a><a id="1576" href="Relation.Binary.Construct.NonStrictToStrict.html#1576" class="Function">≮⇒≥</a> <a id="1580" class="Symbol">:</a> <a id="1582" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="1592" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1596" class="Symbol">→</a> <a id="1598" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="1608" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1612" class="Symbol">→</a> <a id="1614" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1618" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1620" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="1624" class="Symbol">→</a> <a id="1626" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="1632" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="1636" class="Symbol">→</a>
|
||
<a id="1644" class="Symbol">∀</a> <a id="1646" class="Symbol">{</a><a id="1647" href="Relation.Binary.Construct.NonStrictToStrict.html#1647" class="Bound">x</a> <a id="1649" href="Relation.Binary.Construct.NonStrictToStrict.html#1649" class="Bound">y</a><a id="1650" class="Symbol">}</a> <a id="1652" class="Symbol">→</a> <a id="1654" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="1656" class="Symbol">(</a><a id="1657" href="Relation.Binary.Construct.NonStrictToStrict.html#1647" class="Bound">x</a> <a id="1659" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator"><</a> <a id="1661" href="Relation.Binary.Construct.NonStrictToStrict.html#1649" class="Bound">y</a><a id="1662" class="Symbol">)</a> <a id="1664" class="Symbol">→</a> <a id="1666" href="Relation.Binary.Construct.NonStrictToStrict.html#1649" class="Bound">y</a> <a id="1668" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">≤</a> <a id="1670" href="Relation.Binary.Construct.NonStrictToStrict.html#1647" class="Bound">x</a>
|
||
<a id="1672" href="Relation.Binary.Construct.NonStrictToStrict.html#1576" class="Function">≮⇒≥</a> <a id="1676" href="Relation.Binary.Construct.NonStrictToStrict.html#1676" class="Bound">sym</a> <a id="1680" href="Relation.Binary.Construct.NonStrictToStrict.html#1680" class="Bound Operator">_≟_</a> <a id="1684" href="Relation.Binary.Construct.NonStrictToStrict.html#1684" class="Bound">≤-refl</a> <a id="1691" href="Relation.Binary.Construct.NonStrictToStrict.html#1691" class="Bound Operator">_≤?_</a> <a id="1696" class="Symbol">{</a><a id="1697" href="Relation.Binary.Construct.NonStrictToStrict.html#1697" class="Bound">x</a><a id="1698" class="Symbol">}</a> <a id="1700" class="Symbol">{</a><a id="1701" href="Relation.Binary.Construct.NonStrictToStrict.html#1701" class="Bound">y</a><a id="1702" class="Symbol">}</a> <a id="1704" href="Relation.Binary.Construct.NonStrictToStrict.html#1704" class="Bound">x≮y</a> <a id="1708" class="Keyword">with</a> <a id="1713" href="Relation.Binary.Construct.NonStrictToStrict.html#1697" class="Bound">x</a> <a id="1715" href="Relation.Binary.Construct.NonStrictToStrict.html#1680" class="Bound Operator">≟</a> <a id="1717" href="Relation.Binary.Construct.NonStrictToStrict.html#1701" class="Bound">y</a> <a id="1719" class="Symbol">|</a> <a id="1721" href="Relation.Binary.Construct.NonStrictToStrict.html#1701" class="Bound">y</a> <a id="1723" href="Relation.Binary.Construct.NonStrictToStrict.html#1691" class="Bound Operator">≤?</a> <a id="1726" href="Relation.Binary.Construct.NonStrictToStrict.html#1697" class="Bound">x</a>
|
||
<a id="1728" class="Symbol">...</a> <a id="1732" class="Symbol">|</a> <a id="1734" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="1738" href="Relation.Binary.Construct.NonStrictToStrict.html#1738" class="Bound">x≈y</a> <a id="1743" class="Symbol">|</a> <a id="1745" class="Symbol">_</a> <a id="1754" class="Symbol">=</a> <a id="1756" class="Bound">≤-refl</a> <a id="1763" class="Symbol">(</a><a id="1764" class="Bound">sym</a> <a id="1768" href="Relation.Binary.Construct.NonStrictToStrict.html#1738" class="Bound">x≈y</a><a id="1771" class="Symbol">)</a>
|
||
<a id="1773" class="CatchallClause Symbol">...</a><a id="1776" class="CatchallClause"> </a><a id="1777" class="CatchallClause Symbol">|</a><a id="1778" class="CatchallClause"> </a><a id="1779" class="CatchallClause Symbol">_</a><a id="1780" class="CatchallClause"> </a><a id="1788" class="CatchallClause Symbol">|</a><a id="1789" class="CatchallClause"> </a><a id="1790" href="Data.Sum.Base.html#793" class="CatchallClause InductiveConstructor">inj₁</a><a id="1794" class="CatchallClause"> </a><a id="1795" href="Relation.Binary.Construct.NonStrictToStrict.html#1795" class="CatchallClause Bound">y≤x</a> <a id="1799" class="Symbol">=</a> <a id="1801" href="Relation.Binary.Construct.NonStrictToStrict.html#1795" class="Bound">y≤x</a>
|
||
<a id="1805" class="Symbol">...</a> <a id="1809" class="Symbol">|</a> <a id="1811" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="1815" href="Relation.Binary.Construct.NonStrictToStrict.html#1815" class="Bound">x≉y</a> <a id="1820" class="Symbol">|</a> <a id="1822" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="1827" href="Relation.Binary.Construct.NonStrictToStrict.html#1827" class="Bound">x≤y</a> <a id="1831" class="Symbol">=</a> <a id="1833" href="Relation.Nullary.Negation.Core.html#787" class="Function">contradiction</a> <a id="1847" class="Symbol">(</a><a id="1848" href="Relation.Binary.Construct.NonStrictToStrict.html#1827" class="Bound">x≤y</a> <a id="1852" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1854" href="Relation.Binary.Construct.NonStrictToStrict.html#1815" class="Bound">x≉y</a><a id="1857" class="Symbol">)</a> <a id="1859" class="Bound">x≮y</a>
|
||
|
||
<a id="1864" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="1937" class="Comment">-- Relational properties</a>
|
||
|
||
<a id="<-irrefl"></a><a id="1963" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a> <a id="1972" class="Symbol">:</a> <a id="1974" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="1986" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="1990" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="1994" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a> <a id="2003" href="Relation.Binary.Construct.NonStrictToStrict.html#2003" class="Bound">x≈y</a> <a id="2007" class="Symbol">(_</a> <a id="2010" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2012" href="Relation.Binary.Construct.NonStrictToStrict.html#2012" class="Bound">x≉y</a><a id="2015" class="Symbol">)</a> <a id="2017" class="Symbol">=</a> <a id="2019" href="Relation.Binary.Construct.NonStrictToStrict.html#2012" class="Bound">x≉y</a> <a id="2023" href="Relation.Binary.Construct.NonStrictToStrict.html#2003" class="Bound">x≈y</a>
|
||
|
||
<a id="<-trans"></a><a id="2028" href="Relation.Binary.Construct.NonStrictToStrict.html#2028" class="Function"><-trans</a> <a id="2036" class="Symbol">:</a> <a id="2038" href="Relation.Binary.Structures.html#3183" class="Record">IsPartialOrder</a> <a id="2053" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2057" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2061" class="Symbol">→</a> <a id="2063" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2074" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="2078" href="Relation.Binary.Construct.NonStrictToStrict.html#2028" class="Function"><-trans</a> <a id="2086" href="Relation.Binary.Construct.NonStrictToStrict.html#2086" class="Bound">po</a> <a id="2089" class="Symbol">(</a><a id="2090" href="Relation.Binary.Construct.NonStrictToStrict.html#2090" class="Bound">x≤y</a> <a id="2094" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2096" href="Relation.Binary.Construct.NonStrictToStrict.html#2096" class="Bound">x≉y</a><a id="2099" class="Symbol">)</a> <a id="2101" class="Symbol">(</a><a id="2102" href="Relation.Binary.Construct.NonStrictToStrict.html#2102" class="Bound">y≤z</a> <a id="2106" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2108" href="Relation.Binary.Construct.NonStrictToStrict.html#2108" class="Bound">y≉z</a><a id="2111" class="Symbol">)</a> <a id="2113" class="Symbol">=</a>
|
||
<a id="2117" class="Symbol">(</a><a id="2118" href="Relation.Binary.Structures.html#2370" class="Function">trans</a> <a id="2124" href="Relation.Binary.Construct.NonStrictToStrict.html#2090" class="Bound">x≤y</a> <a id="2128" href="Relation.Binary.Construct.NonStrictToStrict.html#2102" class="Bound">y≤z</a> <a id="2132" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2134" href="Relation.Binary.Construct.NonStrictToStrict.html#2096" class="Bound">x≉y</a> <a id="2138" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2140" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a> <a id="2148" href="Relation.Binary.Construct.NonStrictToStrict.html#2090" class="Bound">x≤y</a> <a id="2152" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2154" href="Relation.Binary.Structures.html#2370" class="Function">trans</a> <a id="2160" href="Relation.Binary.Construct.NonStrictToStrict.html#2102" class="Bound">y≤z</a> <a id="2164" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2166" href="Relation.Binary.Structures.html#2340" class="Function">reflexive</a> <a id="2176" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2178" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a><a id="2184" class="Symbol">)</a>
|
||
<a id="2188" class="Keyword">where</a> <a id="2194" class="Keyword">open</a> <a id="2199" href="Relation.Binary.Structures.html#3183" class="Module">IsPartialOrder</a> <a id="2214" href="Relation.Binary.Construct.NonStrictToStrict.html#2086" class="Bound">po</a>
|
||
|
||
<a id="<-≤-trans"></a><a id="2218" href="Relation.Binary.Construct.NonStrictToStrict.html#2218" class="Function"><-≤-trans</a> <a id="2228" class="Symbol">:</a> <a id="2230" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="2240" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2244" class="Symbol">→</a> <a id="2246" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2257" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2261" class="Symbol">→</a> <a id="2263" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="2277" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2281" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2285" class="Symbol">→</a>
|
||
<a id="2298" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2302" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="2312" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2316" class="Symbol">→</a> <a id="2318" href="Relation.Binary.Definitions.html#1594" class="Function">Trans</a> <a id="2324" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="2328" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2332" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="2336" href="Relation.Binary.Construct.NonStrictToStrict.html#2218" class="Function"><-≤-trans</a> <a id="2346" href="Relation.Binary.Construct.NonStrictToStrict.html#2346" class="Bound">sym</a> <a id="2350" href="Relation.Binary.Construct.NonStrictToStrict.html#2350" class="Bound">trans</a> <a id="2356" href="Relation.Binary.Construct.NonStrictToStrict.html#2356" class="Bound">antisym</a> <a id="2364" href="Relation.Binary.Construct.NonStrictToStrict.html#2364" class="Bound">respʳ</a> <a id="2370" class="Symbol">(</a><a id="2371" href="Relation.Binary.Construct.NonStrictToStrict.html#2371" class="Bound">x≤y</a> <a id="2375" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2377" href="Relation.Binary.Construct.NonStrictToStrict.html#2377" class="Bound">x≉y</a><a id="2380" class="Symbol">)</a> <a id="2382" href="Relation.Binary.Construct.NonStrictToStrict.html#2382" class="Bound">y≤z</a> <a id="2386" class="Symbol">=</a>
|
||
<a id="2390" href="Relation.Binary.Construct.NonStrictToStrict.html#2350" class="Bound">trans</a> <a id="2396" href="Relation.Binary.Construct.NonStrictToStrict.html#2371" class="Bound">x≤y</a> <a id="2400" href="Relation.Binary.Construct.NonStrictToStrict.html#2382" class="Bound">y≤z</a> <a id="2404" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2406" class="Symbol">(λ</a> <a id="2409" href="Relation.Binary.Construct.NonStrictToStrict.html#2409" class="Bound">x≈z</a> <a id="2413" class="Symbol">→</a> <a id="2415" href="Relation.Binary.Construct.NonStrictToStrict.html#2377" class="Bound">x≉y</a> <a id="2419" class="Symbol">(</a><a id="2420" href="Relation.Binary.Construct.NonStrictToStrict.html#2356" class="Bound">antisym</a> <a id="2428" href="Relation.Binary.Construct.NonStrictToStrict.html#2371" class="Bound">x≤y</a> <a id="2432" class="Symbol">(</a><a id="2433" href="Relation.Binary.Construct.NonStrictToStrict.html#2364" class="Bound">respʳ</a> <a id="2439" class="Symbol">(</a><a id="2440" href="Relation.Binary.Construct.NonStrictToStrict.html#2346" class="Bound">sym</a> <a id="2444" href="Relation.Binary.Construct.NonStrictToStrict.html#2409" class="Bound">x≈z</a><a id="2447" class="Symbol">)</a> <a id="2449" href="Relation.Binary.Construct.NonStrictToStrict.html#2382" class="Bound">y≤z</a><a id="2452" class="Symbol">)))</a>
|
||
|
||
<a id="≤-<-trans"></a><a id="2457" href="Relation.Binary.Construct.NonStrictToStrict.html#2457" class="Function">≤-<-trans</a> <a id="2467" class="Symbol">:</a> <a id="2469" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2480" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2484" class="Symbol">→</a> <a id="2486" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="2500" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2504" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2508" class="Symbol">→</a>
|
||
<a id="2521" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2525" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="2535" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2539" class="Symbol">→</a> <a id="2541" href="Relation.Binary.Definitions.html#1594" class="Function">Trans</a> <a id="2547" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2551" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="2555" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="2559" href="Relation.Binary.Construct.NonStrictToStrict.html#2457" class="Function">≤-<-trans</a> <a id="2569" href="Relation.Binary.Construct.NonStrictToStrict.html#2569" class="Bound">trans</a> <a id="2575" href="Relation.Binary.Construct.NonStrictToStrict.html#2575" class="Bound">antisym</a> <a id="2583" href="Relation.Binary.Construct.NonStrictToStrict.html#2583" class="Bound">respʳ</a> <a id="2589" href="Relation.Binary.Construct.NonStrictToStrict.html#2589" class="Bound">x≤y</a> <a id="2593" class="Symbol">(</a><a id="2594" href="Relation.Binary.Construct.NonStrictToStrict.html#2594" class="Bound">y≤z</a> <a id="2598" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2600" href="Relation.Binary.Construct.NonStrictToStrict.html#2600" class="Bound">y≉z</a><a id="2603" class="Symbol">)</a> <a id="2605" class="Symbol">=</a>
|
||
<a id="2609" href="Relation.Binary.Construct.NonStrictToStrict.html#2569" class="Bound">trans</a> <a id="2615" href="Relation.Binary.Construct.NonStrictToStrict.html#2589" class="Bound">x≤y</a> <a id="2619" href="Relation.Binary.Construct.NonStrictToStrict.html#2594" class="Bound">y≤z</a> <a id="2623" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2625" class="Symbol">(λ</a> <a id="2628" href="Relation.Binary.Construct.NonStrictToStrict.html#2628" class="Bound">x≈z</a> <a id="2632" class="Symbol">→</a> <a id="2634" href="Relation.Binary.Construct.NonStrictToStrict.html#2600" class="Bound">y≉z</a> <a id="2638" class="Symbol">(</a><a id="2639" href="Relation.Binary.Construct.NonStrictToStrict.html#2575" class="Bound">antisym</a> <a id="2647" href="Relation.Binary.Construct.NonStrictToStrict.html#2594" class="Bound">y≤z</a> <a id="2651" class="Symbol">(</a><a id="2652" href="Relation.Binary.Construct.NonStrictToStrict.html#2583" class="Bound">respʳ</a> <a id="2658" href="Relation.Binary.Construct.NonStrictToStrict.html#2628" class="Bound">x≈z</a> <a id="2662" href="Relation.Binary.Construct.NonStrictToStrict.html#2589" class="Bound">x≤y</a><a id="2665" class="Symbol">)))</a>
|
||
|
||
<a id="<-asym"></a><a id="2670" href="Relation.Binary.Construct.NonStrictToStrict.html#2670" class="Function"><-asym</a> <a id="2677" class="Symbol">:</a> <a id="2679" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="2693" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2697" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2701" class="Symbol">→</a> <a id="2703" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="2714" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="2718" href="Relation.Binary.Construct.NonStrictToStrict.html#2670" class="Function"><-asym</a> <a id="2725" href="Relation.Binary.Construct.NonStrictToStrict.html#2725" class="Bound">antisym</a> <a id="2733" class="Symbol">(</a><a id="2734" href="Relation.Binary.Construct.NonStrictToStrict.html#2734" class="Bound">x≤y</a> <a id="2738" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2740" href="Relation.Binary.Construct.NonStrictToStrict.html#2740" class="Bound">x≉y</a><a id="2743" class="Symbol">)</a> <a id="2745" class="Symbol">(</a><a id="2746" href="Relation.Binary.Construct.NonStrictToStrict.html#2746" class="Bound">y≤x</a> <a id="2750" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2752" class="Symbol">_)</a> <a id="2755" class="Symbol">=</a> <a id="2757" href="Relation.Binary.Construct.NonStrictToStrict.html#2740" class="Bound">x≉y</a> <a id="2761" class="Symbol">(</a><a id="2762" href="Relation.Binary.Construct.NonStrictToStrict.html#2725" class="Bound">antisym</a> <a id="2770" href="Relation.Binary.Construct.NonStrictToStrict.html#2734" class="Bound">x≤y</a> <a id="2774" href="Relation.Binary.Construct.NonStrictToStrict.html#2746" class="Bound">y≤x</a><a id="2777" class="Symbol">)</a>
|
||
|
||
<a id="<-respˡ-≈"></a><a id="2780" href="Relation.Binary.Construct.NonStrictToStrict.html#2780" class="Function"><-respˡ-≈</a> <a id="2790" class="Symbol">:</a> <a id="2792" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2803" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2807" class="Symbol">→</a> <a id="2809" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2813" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="2823" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2827" class="Symbol">→</a> <a id="2829" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="2833" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="2843" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a>
|
||
<a id="2847" href="Relation.Binary.Construct.NonStrictToStrict.html#2780" class="Function"><-respˡ-≈</a> <a id="2857" href="Relation.Binary.Construct.NonStrictToStrict.html#2857" class="Bound">trans</a> <a id="2863" href="Relation.Binary.Construct.NonStrictToStrict.html#2863" class="Bound">respˡ</a> <a id="2869" href="Relation.Binary.Construct.NonStrictToStrict.html#2869" class="Bound">y≈z</a> <a id="2873" class="Symbol">(</a><a id="2874" href="Relation.Binary.Construct.NonStrictToStrict.html#2874" class="Bound">y≤x</a> <a id="2878" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2880" href="Relation.Binary.Construct.NonStrictToStrict.html#2880" class="Bound">y≉x</a><a id="2883" class="Symbol">)</a> <a id="2885" class="Symbol">=</a>
|
||
<a id="2889" href="Relation.Binary.Construct.NonStrictToStrict.html#2863" class="Bound">respˡ</a> <a id="2895" href="Relation.Binary.Construct.NonStrictToStrict.html#2869" class="Bound">y≈z</a> <a id="2899" href="Relation.Binary.Construct.NonStrictToStrict.html#2874" class="Bound">y≤x</a> <a id="2903" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2905" href="Relation.Binary.Construct.NonStrictToStrict.html#2880" class="Bound">y≉x</a> <a id="2909" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="2911" href="Relation.Binary.Construct.NonStrictToStrict.html#2857" class="Bound">trans</a> <a id="2917" href="Relation.Binary.Construct.NonStrictToStrict.html#2869" class="Bound">y≈z</a>
|
||
|
||
<a id="<-respʳ-≈"></a><a id="2922" href="Relation.Binary.Construct.NonStrictToStrict.html#2922" class="Function"><-respʳ-≈</a> <a id="2932" class="Symbol">:</a> <a id="2934" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="2944" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2948" class="Symbol">→</a> <a id="2950" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="2961" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2965" class="Symbol">→</a>
|
||
<a id="2979" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="2983" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="2993" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="2997" class="Symbol">→</a> <a id="2999" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="3003" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="3013" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a>
|
||
<a id="3017" href="Relation.Binary.Construct.NonStrictToStrict.html#2922" class="Function"><-respʳ-≈</a> <a id="3027" href="Relation.Binary.Construct.NonStrictToStrict.html#3027" class="Bound">sym</a> <a id="3031" href="Relation.Binary.Construct.NonStrictToStrict.html#3031" class="Bound">trans</a> <a id="3037" href="Relation.Binary.Construct.NonStrictToStrict.html#3037" class="Bound">respʳ</a> <a id="3043" href="Relation.Binary.Construct.NonStrictToStrict.html#3043" class="Bound">y≈z</a> <a id="3047" class="Symbol">(</a><a id="3048" href="Relation.Binary.Construct.NonStrictToStrict.html#3048" class="Bound">x≤y</a> <a id="3052" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3054" href="Relation.Binary.Construct.NonStrictToStrict.html#3054" class="Bound">x≉y</a><a id="3057" class="Symbol">)</a> <a id="3059" class="Symbol">=</a>
|
||
<a id="3063" class="Symbol">(</a><a id="3064" href="Relation.Binary.Construct.NonStrictToStrict.html#3037" class="Bound">respʳ</a> <a id="3070" href="Relation.Binary.Construct.NonStrictToStrict.html#3043" class="Bound">y≈z</a> <a id="3074" href="Relation.Binary.Construct.NonStrictToStrict.html#3048" class="Bound">x≤y</a><a id="3077" class="Symbol">)</a> <a id="3079" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3081" class="Symbol">λ</a> <a id="3083" href="Relation.Binary.Construct.NonStrictToStrict.html#3083" class="Bound">x≈z</a> <a id="3087" class="Symbol">→</a> <a id="3089" href="Relation.Binary.Construct.NonStrictToStrict.html#3054" class="Bound">x≉y</a> <a id="3093" class="Symbol">(</a><a id="3094" href="Relation.Binary.Construct.NonStrictToStrict.html#3031" class="Bound">trans</a> <a id="3100" href="Relation.Binary.Construct.NonStrictToStrict.html#3083" class="Bound">x≈z</a> <a id="3104" class="Symbol">(</a><a id="3105" href="Relation.Binary.Construct.NonStrictToStrict.html#3027" class="Bound">sym</a> <a id="3109" href="Relation.Binary.Construct.NonStrictToStrict.html#3043" class="Bound">y≈z</a><a id="3112" class="Symbol">))</a>
|
||
|
||
<a id="<-resp-≈"></a><a id="3116" href="Relation.Binary.Construct.NonStrictToStrict.html#3116" class="Function"><-resp-≈</a> <a id="3125" class="Symbol">:</a> <a id="3127" href="Relation.Binary.Structures.html#1531" class="Record">IsEquivalence</a> <a id="3141" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3145" class="Symbol">→</a> <a id="3147" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="3151" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="3161" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3165" class="Symbol">→</a> <a id="3167" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a> <a id="3171" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="3181" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a>
|
||
<a id="3185" href="Relation.Binary.Construct.NonStrictToStrict.html#3116" class="Function"><-resp-≈</a> <a id="3194" href="Relation.Binary.Construct.NonStrictToStrict.html#3194" class="Bound">eq</a> <a id="3197" class="Symbol">(</a><a id="3198" href="Relation.Binary.Construct.NonStrictToStrict.html#3198" class="Bound">respʳ</a> <a id="3204" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3206" href="Relation.Binary.Construct.NonStrictToStrict.html#3206" class="Bound">respˡ</a><a id="3211" class="Symbol">)</a> <a id="3213" class="Symbol">=</a>
|
||
<a id="3217" href="Relation.Binary.Construct.NonStrictToStrict.html#2922" class="Function"><-respʳ-≈</a> <a id="3227" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="3231" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="3237" href="Relation.Binary.Construct.NonStrictToStrict.html#3198" class="Bound">respʳ</a> <a id="3243" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3245" href="Relation.Binary.Construct.NonStrictToStrict.html#2780" class="Function"><-respˡ-≈</a> <a id="3255" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="3261" href="Relation.Binary.Construct.NonStrictToStrict.html#3206" class="Bound">respˡ</a>
|
||
<a id="3269" class="Keyword">where</a> <a id="3275" class="Keyword">open</a> <a id="3280" href="Relation.Binary.Structures.html#1531" class="Module">IsEquivalence</a> <a id="3294" href="Relation.Binary.Construct.NonStrictToStrict.html#3194" class="Bound">eq</a>
|
||
|
||
<a id="<-trichotomous"></a><a id="3298" href="Relation.Binary.Construct.NonStrictToStrict.html#3298" class="Function"><-trichotomous</a> <a id="3313" class="Symbol">:</a> <a id="3315" href="Relation.Binary.Definitions.html#1507" class="Function">Symmetric</a> <a id="3325" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3329" class="Symbol">→</a> <a id="3331" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3341" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3345" class="Symbol">→</a>
|
||
<a id="3364" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="3378" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3382" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="3386" class="Symbol">→</a> <a id="3388" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="3394" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="3398" class="Symbol">→</a>
|
||
<a id="3417" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="3430" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3434" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="3438" href="Relation.Binary.Construct.NonStrictToStrict.html#3298" class="Function"><-trichotomous</a> <a id="3453" href="Relation.Binary.Construct.NonStrictToStrict.html#3453" class="Bound">≈-sym</a> <a id="3459" href="Relation.Binary.Construct.NonStrictToStrict.html#3459" class="Bound Operator">_≟_</a> <a id="3463" href="Relation.Binary.Construct.NonStrictToStrict.html#3463" class="Bound">antisym</a> <a id="3471" href="Relation.Binary.Construct.NonStrictToStrict.html#3471" class="Bound">total</a> <a id="3477" href="Relation.Binary.Construct.NonStrictToStrict.html#3477" class="Bound">x</a> <a id="3479" href="Relation.Binary.Construct.NonStrictToStrict.html#3479" class="Bound">y</a> <a id="3481" class="Keyword">with</a> <a id="3486" href="Relation.Binary.Construct.NonStrictToStrict.html#3477" class="Bound">x</a> <a id="3488" href="Relation.Binary.Construct.NonStrictToStrict.html#3459" class="Bound Operator">≟</a> <a id="3490" href="Relation.Binary.Construct.NonStrictToStrict.html#3479" class="Bound">y</a>
|
||
<a id="3492" class="Symbol">...</a> <a id="3496" class="Symbol">|</a> <a id="3498" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="3502" href="Relation.Binary.Construct.NonStrictToStrict.html#3502" class="Bound">x≈y</a> <a id="3506" class="Symbol">=</a> <a id="3508" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="3513" class="Symbol">(</a><a id="3514" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a> <a id="3523" href="Relation.Binary.Construct.NonStrictToStrict.html#3502" class="Bound">x≈y</a><a id="3526" class="Symbol">)</a> <a id="3528" href="Relation.Binary.Construct.NonStrictToStrict.html#3502" class="Bound">x≈y</a> <a id="3532" class="Symbol">(</a><a id="3533" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a> <a id="3542" class="Symbol">(</a><a id="3543" class="Bound">≈-sym</a> <a id="3549" href="Relation.Binary.Construct.NonStrictToStrict.html#3502" class="Bound">x≈y</a><a id="3552" class="Symbol">))</a>
|
||
<a id="3555" class="Symbol">...</a> <a id="3559" class="Symbol">|</a> <a id="3561" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="3565" href="Relation.Binary.Construct.NonStrictToStrict.html#3565" class="Bound">x≉y</a> <a id="3569" class="Keyword">with</a> <a id="3574" class="Bound">total</a> <a id="3580" class="Bound">x</a> <a id="3582" class="Bound">y</a>
|
||
<a id="3584" class="Symbol">...</a> <a id="3590" class="Symbol">|</a> <a id="3592" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="3597" href="Relation.Binary.Construct.NonStrictToStrict.html#3597" class="Bound">x≤y</a> <a id="3601" class="Symbol">=</a> <a id="3603" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="3608" class="Symbol">(</a><a id="3609" href="Relation.Binary.Construct.NonStrictToStrict.html#3597" class="Bound">x≤y</a> <a id="3613" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3615" class="Bound">x≉y</a><a id="3618" class="Symbol">)</a> <a id="3620" class="Bound">x≉y</a> <a id="3624" class="Symbol">(</a><a id="3625" class="Bound">x≉y</a> <a id="3629" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="3631" class="Bound">antisym</a> <a id="3639" href="Relation.Binary.Construct.NonStrictToStrict.html#3597" class="Bound">x≤y</a> <a id="3643" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="3645" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a><a id="3650" class="Symbol">)</a>
|
||
<a id="3652" class="Symbol">...</a> <a id="3658" class="Symbol">|</a> <a id="3660" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="3665" href="Relation.Binary.Construct.NonStrictToStrict.html#3665" class="Bound">y≤x</a> <a id="3669" class="Symbol">=</a> <a id="3671" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="3676" class="Symbol">(</a><a id="3677" class="Bound">x≉y</a> <a id="3681" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="3683" href="Function.Base.html#1563" class="Function">flip</a> <a id="3688" class="Bound">antisym</a> <a id="3696" href="Relation.Binary.Construct.NonStrictToStrict.html#3665" class="Bound">y≤x</a> <a id="3700" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="3702" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a><a id="3707" class="Symbol">)</a> <a id="3709" class="Bound">x≉y</a> <a id="3713" class="Symbol">(</a><a id="3714" href="Relation.Binary.Construct.NonStrictToStrict.html#3665" class="Bound">y≤x</a> <a id="3718" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3720" class="Bound">x≉y</a> <a id="3724" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="3726" class="Bound">≈-sym</a><a id="3731" class="Symbol">)</a>
|
||
|
||
<a id="<-decidable"></a><a id="3734" href="Relation.Binary.Construct.NonStrictToStrict.html#3734" class="Function"><-decidable</a> <a id="3746" class="Symbol">:</a> <a id="3748" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3758" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3762" class="Symbol">→</a> <a id="3764" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3774" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="3778" class="Symbol">→</a> <a id="3780" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3790" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="3794" href="Relation.Binary.Construct.NonStrictToStrict.html#3734" class="Function"><-decidable</a> <a id="3806" href="Relation.Binary.Construct.NonStrictToStrict.html#3806" class="Bound Operator">_≟_</a> <a id="3810" href="Relation.Binary.Construct.NonStrictToStrict.html#3810" class="Bound Operator">_≤?_</a> <a id="3815" href="Relation.Binary.Construct.NonStrictToStrict.html#3815" class="Bound">x</a> <a id="3817" href="Relation.Binary.Construct.NonStrictToStrict.html#3817" class="Bound">y</a> <a id="3819" class="Symbol">=</a> <a id="3821" href="Relation.Binary.Construct.NonStrictToStrict.html#3815" class="Bound">x</a> <a id="3823" href="Relation.Binary.Construct.NonStrictToStrict.html#3810" class="Bound Operator">≤?</a> <a id="3826" href="Relation.Binary.Construct.NonStrictToStrict.html#3817" class="Bound">y</a> <a id="3828" href="Relation.Nullary.Product.html#930" class="Function Operator">×-dec</a> <a id="3834" href="Relation.Nullary.Negation.Core.html#1364" class="Function">¬?</a> <a id="3837" class="Symbol">(</a><a id="3838" href="Relation.Binary.Construct.NonStrictToStrict.html#3815" class="Bound">x</a> <a id="3840" href="Relation.Binary.Construct.NonStrictToStrict.html#3806" class="Bound Operator">≟</a> <a id="3842" href="Relation.Binary.Construct.NonStrictToStrict.html#3817" class="Bound">y</a><a id="3843" class="Symbol">)</a>
|
||
|
||
<a id="3846" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="3919" class="Comment">-- Structures</a>
|
||
|
||
<a id="<-isStrictPartialOrder"></a><a id="3934" href="Relation.Binary.Construct.NonStrictToStrict.html#3934" class="Function"><-isStrictPartialOrder</a> <a id="3957" class="Symbol">:</a> <a id="3959" href="Relation.Binary.Structures.html#3183" class="Record">IsPartialOrder</a> <a id="3974" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="3978" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="3982" class="Symbol">→</a>
|
||
<a id="4009" href="Relation.Binary.Structures.html#3959" class="Record">IsStrictPartialOrder</a> <a id="4030" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4034" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="4038" href="Relation.Binary.Construct.NonStrictToStrict.html#3934" class="Function"><-isStrictPartialOrder</a> <a id="4061" href="Relation.Binary.Construct.NonStrictToStrict.html#4061" class="Bound">po</a> <a id="4064" class="Symbol">=</a> <a id="4066" class="Keyword">record</a>
|
||
<a id="4075" class="Symbol">{</a> <a id="4077" href="Relation.Binary.Structures.html#4034" class="Field">isEquivalence</a> <a id="4091" class="Symbol">=</a> <a id="4093" href="Relation.Binary.Structures.html#2237" class="Function">isEquivalence</a>
|
||
<a id="4109" class="Symbol">;</a> <a id="4111" href="Relation.Binary.Structures.html#4068" class="Field">irrefl</a> <a id="4125" class="Symbol">=</a> <a id="4127" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a>
|
||
<a id="4138" class="Symbol">;</a> <a id="4140" href="Relation.Binary.Structures.html#4108" class="Field">trans</a> <a id="4154" class="Symbol">=</a> <a id="4156" href="Relation.Binary.Construct.NonStrictToStrict.html#2028" class="Function"><-trans</a> <a id="4164" href="Relation.Binary.Construct.NonStrictToStrict.html#4061" class="Bound">po</a>
|
||
<a id="4169" class="Symbol">;</a> <a id="4171" href="Relation.Binary.Structures.html#4143" class="Field"><-resp-≈</a> <a id="4185" class="Symbol">=</a> <a id="4187" href="Relation.Binary.Construct.NonStrictToStrict.html#3116" class="Function"><-resp-≈</a> <a id="4196" href="Relation.Binary.Structures.html#2237" class="Function">isEquivalence</a> <a id="4210" href="Relation.Binary.Structures.html#3446" class="Function">≤-resp-≈</a>
|
||
<a id="4221" class="Symbol">}</a> <a id="4223" class="Keyword">where</a> <a id="4229" class="Keyword">open</a> <a id="4234" href="Relation.Binary.Structures.html#3183" class="Module">IsPartialOrder</a> <a id="4249" href="Relation.Binary.Construct.NonStrictToStrict.html#4061" class="Bound">po</a>
|
||
|
||
<a id="<-isDecStrictPartialOrder"></a><a id="4253" href="Relation.Binary.Construct.NonStrictToStrict.html#4253" class="Function"><-isDecStrictPartialOrder</a> <a id="4279" class="Symbol">:</a> <a id="4281" href="Relation.Binary.Structures.html#3470" class="Record">IsDecPartialOrder</a> <a id="4299" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4303" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="4307" class="Symbol">→</a>
|
||
<a id="4337" href="Relation.Binary.Structures.html#4580" class="Record">IsDecStrictPartialOrder</a> <a id="4361" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4365" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="4369" href="Relation.Binary.Construct.NonStrictToStrict.html#4253" class="Function"><-isDecStrictPartialOrder</a> <a id="4395" href="Relation.Binary.Construct.NonStrictToStrict.html#4395" class="Bound">dpo</a> <a id="4399" class="Symbol">=</a> <a id="4401" class="Keyword">record</a>
|
||
<a id="4410" class="Symbol">{</a> <a id="4412" href="Relation.Binary.Structures.html#4677" class="Field">isStrictPartialOrder</a> <a id="4433" class="Symbol">=</a> <a id="4435" href="Relation.Binary.Construct.NonStrictToStrict.html#3934" class="Function"><-isStrictPartialOrder</a> <a id="4458" href="Relation.Binary.Structures.html#3561" class="Field">isPartialOrder</a>
|
||
<a id="4475" class="Symbol">;</a> <a id="4477" href="Relation.Binary.Structures.html#4729" class="Field Operator">_≟_</a> <a id="4481" class="Symbol">=</a> <a id="4483" href="Relation.Binary.Structures.html#3601" class="Field Operator">_≟_</a>
|
||
<a id="4489" class="Symbol">;</a> <a id="4491" href="Relation.Binary.Structures.html#4770" class="Field Operator">_<?_</a> <a id="4496" class="Symbol">=</a> <a id="4498" href="Relation.Binary.Construct.NonStrictToStrict.html#3734" class="Function"><-decidable</a> <a id="4510" href="Relation.Binary.Structures.html#3601" class="Field Operator">_≟_</a> <a id="4514" href="Relation.Binary.Structures.html#3636" class="Field Operator">_≤?_</a>
|
||
<a id="4521" class="Symbol">}</a> <a id="4523" class="Keyword">where</a> <a id="4529" class="Keyword">open</a> <a id="4534" href="Relation.Binary.Structures.html#3470" class="Module">IsDecPartialOrder</a> <a id="4552" href="Relation.Binary.Construct.NonStrictToStrict.html#4395" class="Bound">dpo</a>
|
||
|
||
<a id="<-isStrictTotalOrder₁"></a><a id="4557" href="Relation.Binary.Construct.NonStrictToStrict.html#4557" class="Function"><-isStrictTotalOrder₁</a> <a id="4579" class="Symbol">:</a> <a id="4581" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="4591" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4595" class="Symbol">→</a> <a id="4597" href="Relation.Binary.Structures.html#5306" class="Record">IsTotalOrder</a> <a id="4610" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4614" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="4618" class="Symbol">→</a>
|
||
<a id="4644" href="Relation.Binary.Structures.html#6548" class="Record">IsStrictTotalOrder</a> <a id="4663" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4667" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="4671" href="Relation.Binary.Construct.NonStrictToStrict.html#4557" class="Function"><-isStrictTotalOrder₁</a> <a id="4693" href="Relation.Binary.Construct.NonStrictToStrict.html#4693" class="Bound">≟</a> <a id="4695" href="Relation.Binary.Construct.NonStrictToStrict.html#4695" class="Bound">tot</a> <a id="4699" class="Symbol">=</a> <a id="4701" class="Keyword">record</a>
|
||
<a id="4710" class="Symbol">{</a> <a id="4712" href="Relation.Binary.Structures.html#6621" class="Field">isEquivalence</a> <a id="4726" class="Symbol">=</a> <a id="4728" href="Relation.Binary.Structures.html#2237" class="Function">isEquivalence</a>
|
||
<a id="4744" class="Symbol">;</a> <a id="4746" href="Relation.Binary.Structures.html#6655" class="Field">trans</a> <a id="4760" class="Symbol">=</a> <a id="4762" href="Relation.Binary.Construct.NonStrictToStrict.html#2028" class="Function"><-trans</a> <a id="4770" href="Relation.Binary.Structures.html#5373" class="Field">isPartialOrder</a>
|
||
<a id="4787" class="Symbol">;</a> <a id="4789" href="Relation.Binary.Structures.html#6690" class="Field">compare</a> <a id="4803" class="Symbol">=</a> <a id="4805" href="Relation.Binary.Construct.NonStrictToStrict.html#3298" class="Function"><-trichotomous</a> <a id="4820" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a> <a id="4827" href="Relation.Binary.Construct.NonStrictToStrict.html#4693" class="Bound">≟</a> <a id="4829" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a> <a id="4837" href="Relation.Binary.Structures.html#5413" class="Field">total</a>
|
||
<a id="4845" class="Symbol">}</a> <a id="4847" class="Keyword">where</a> <a id="4853" class="Keyword">open</a> <a id="4858" href="Relation.Binary.Structures.html#5306" class="Module">IsTotalOrder</a> <a id="4871" href="Relation.Binary.Construct.NonStrictToStrict.html#4695" class="Bound">tot</a>
|
||
|
||
<a id="<-isStrictTotalOrder₂"></a><a id="4876" href="Relation.Binary.Construct.NonStrictToStrict.html#4876" class="Function"><-isStrictTotalOrder₂</a> <a id="4898" class="Symbol">:</a> <a id="4900" href="Relation.Binary.Structures.html#5623" class="Record">IsDecTotalOrder</a> <a id="4916" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4920" href="Relation.Binary.Construct.NonStrictToStrict.html#375" class="Bound Operator">_≤_</a> <a id="4924" class="Symbol">→</a>
|
||
<a id="4950" href="Relation.Binary.Structures.html#6548" class="Record">IsStrictTotalOrder</a> <a id="4969" href="Relation.Binary.Construct.NonStrictToStrict.html#358" class="Bound Operator">_≈_</a> <a id="4973" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">_<_</a>
|
||
<a id="4977" href="Relation.Binary.Construct.NonStrictToStrict.html#4876" class="Function"><-isStrictTotalOrder₂</a> <a id="4999" href="Relation.Binary.Construct.NonStrictToStrict.html#4999" class="Bound">dtot</a> <a id="5004" class="Symbol">=</a> <a id="5006" href="Relation.Binary.Construct.NonStrictToStrict.html#4557" class="Function"><-isStrictTotalOrder₁</a> <a id="5028" href="Relation.Binary.Structures.html#5748" class="Field Operator">_≟_</a> <a id="5032" href="Relation.Binary.Structures.html#5712" class="Field">isTotalOrder</a>
|
||
<a id="5047" class="Keyword">where</a> <a id="5053" class="Keyword">open</a> <a id="5058" href="Relation.Binary.Structures.html#5623" class="Module">IsDecTotalOrder</a> <a id="5074" href="Relation.Binary.Construct.NonStrictToStrict.html#4999" class="Bound">dtot</a>
|
||
|
||
<a id="5080" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="5153" class="Comment">-- DEPRECATED NAMES</a>
|
||
<a id="5173" class="Comment">------------------------------------------------------------------------</a>
|
||
<a id="5246" class="Comment">-- Please use the new names as continuing support for the old names is</a>
|
||
<a id="5317" class="Comment">-- not guaranteed.</a>
|
||
|
||
<a id="5337" class="Comment">-- Version 0.16</a>
|
||
|
||
<a id="irrefl"></a><a id="5354" href="Relation.Binary.Construct.NonStrictToStrict.html#5354" class="Function">irrefl</a> <a id="5369" class="Symbol">=</a> <a id="5371" href="Relation.Binary.Construct.NonStrictToStrict.html#1963" class="Function"><-irrefl</a>
|
||
<a id="5380" class="Symbol">{-#</a> <a id="5384" class="Keyword">WARNING_ON_USAGE</a> <a id="5401" class="Pragma">irrefl</a>
|
||
<a id="5408" class="String">"Warning: irrefl was deprecated in v0.16.
|
||
Please use <-irrefl instead."</a>
|
||
<a id="5480" class="Symbol">#-}</a>
|
||
<a id="trans"></a><a id="5484" href="Relation.Binary.Construct.NonStrictToStrict.html#5484" class="Function">trans</a> <a id="5499" class="Symbol">=</a> <a id="5501" href="Relation.Binary.Construct.NonStrictToStrict.html#2028" class="Function"><-trans</a>
|
||
<a id="5509" class="Symbol">{-#</a> <a id="5513" class="Keyword">WARNING_ON_USAGE</a> <a id="5530" class="Pragma">trans</a>
|
||
<a id="5536" class="String">"Warning: trans was deprecated in v0.16.
|
||
Please use <-trans instead."</a>
|
||
<a id="5606" class="Symbol">#-}</a>
|
||
<a id="antisym⟶asym"></a><a id="5610" href="Relation.Binary.Construct.NonStrictToStrict.html#5610" class="Function">antisym⟶asym</a> <a id="5623" class="Symbol">=</a> <a id="5625" href="Relation.Binary.Construct.NonStrictToStrict.html#2670" class="Function"><-asym</a>
|
||
<a id="5632" class="Symbol">{-#</a> <a id="5636" class="Keyword">WARNING_ON_USAGE</a> <a id="5653" class="Pragma">antisym⟶asym</a>
|
||
<a id="5666" class="String">"Warning: antisym⟶asym was deprecated in v0.16.
|
||
Please use <-asym instead."</a>
|
||
<a id="5742" class="Symbol">#-}</a>
|
||
<a id="decidable"></a><a id="5746" href="Relation.Binary.Construct.NonStrictToStrict.html#5746" class="Function">decidable</a> <a id="5761" class="Symbol">=</a> <a id="5763" href="Relation.Binary.Construct.NonStrictToStrict.html#3734" class="Function"><-decidable</a>
|
||
<a id="5775" class="Symbol">{-#</a> <a id="5779" class="Keyword">WARNING_ON_USAGE</a> <a id="5796" class="Pragma">decidable</a>
|
||
<a id="5806" class="String">"Warning: decidable was deprecated in v0.16.
|
||
Please use <-decidable instead."</a>
|
||
<a id="5884" class="Symbol">#-}</a>
|
||
<a id="trichotomous"></a><a id="5888" href="Relation.Binary.Construct.NonStrictToStrict.html#5888" class="Function">trichotomous</a> <a id="5903" class="Symbol">=</a> <a id="5905" href="Relation.Binary.Construct.NonStrictToStrict.html#3298" class="Function"><-trichotomous</a>
|
||
<a id="5920" class="Symbol">{-#</a> <a id="5924" class="Keyword">WARNING_ON_USAGE</a> <a id="5941" class="Pragma">trichotomous</a>
|
||
<a id="5954" class="String">"Warning: trichotomous was deprecated in v0.16.
|
||
Please use <-trichotomous instead."</a>
|
||
<a id="6038" class="Symbol">#-}</a>
|
||
<a id="isPartialOrder⟶isStrictPartialOrder"></a><a id="6042" href="Relation.Binary.Construct.NonStrictToStrict.html#6042" class="Function">isPartialOrder⟶isStrictPartialOrder</a> <a id="6078" class="Symbol">=</a> <a id="6080" href="Relation.Binary.Construct.NonStrictToStrict.html#3934" class="Function"><-isStrictPartialOrder</a>
|
||
<a id="6103" class="Symbol">{-#</a> <a id="6107" class="Keyword">WARNING_ON_USAGE</a> <a id="6124" class="Pragma">isPartialOrder⟶isStrictPartialOrder</a>
|
||
<a id="6160" class="String">"Warning: isPartialOrder⟶isStrictPartialOrder was deprecated in v0.16.
|
||
Please use <-isStrictPartialOrder instead."</a>
|
||
<a id="6275" class="Symbol">#-}</a>
|
||
<a id="isTotalOrder⟶isStrictTotalOrder"></a><a id="6279" href="Relation.Binary.Construct.NonStrictToStrict.html#6279" class="Function">isTotalOrder⟶isStrictTotalOrder</a> <a id="6315" class="Symbol">=</a> <a id="6317" href="Relation.Binary.Construct.NonStrictToStrict.html#4557" class="Function"><-isStrictTotalOrder₁</a>
|
||
<a id="6339" class="Symbol">{-#</a> <a id="6343" class="Keyword">WARNING_ON_USAGE</a> <a id="6360" class="Pragma">isTotalOrder⟶isStrictTotalOrder</a>
|
||
<a id="6392" class="String">"Warning: isTotalOrder⟶isStrictTotalOrder was deprecated in v0.16.
|
||
Please use <-isStrictTotalOrder₁ instead."</a>
|
||
<a id="6502" class="Symbol">#-}</a>
|
||
<a id="isDecTotalOrder⟶isStrictTotalOrder"></a><a id="6506" href="Relation.Binary.Construct.NonStrictToStrict.html#6506" class="Function">isDecTotalOrder⟶isStrictTotalOrder</a> <a id="6542" class="Symbol">=</a> <a id="6544" href="Relation.Binary.Construct.NonStrictToStrict.html#4876" class="Function"><-isStrictTotalOrder₂</a>
|
||
<a id="6566" class="Symbol">{-#</a> <a id="6570" class="Keyword">WARNING_ON_USAGE</a> <a id="6587" class="Pragma">isDecTotalOrder⟶isStrictTotalOrder</a>
|
||
<a id="6622" class="String">"Warning: isDecTotalOrder⟶isStrictTotalOrder was deprecated in v0.16.
|
||
Please use <-isStrictTotalOrder₂ instead."</a>
|
||
<a id="6735" class="Symbol">#-}</a>
|
||
</pre></body></html> |