mirror of
https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git
synced 2024-05-31 07:28:34 +02:00
150 lines
73 KiB
HTML
150 lines
73 KiB
HTML
|
<!DOCTYPE HTML>
|
|||
|
|
<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>
|
|||
|
|
<a id="74" class="Comment">-- The Agda standard library</a>
|
|||
|
|
<a id="103" class="Comment">--</a>
|
|||
|
|
<a id="106" class="Comment">-- Conversion of _≤_ to _<_</a>
|
|||
|
|
<a id="134" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
|
|||
|
|
<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>
|
|||
|
|
|
|||
|
|
<a id="253" class="Keyword">open</a> <a id="258" class="Keyword">import</a> <a id="265" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="286" class="Keyword">using</a> <a id="292" class="Symbol">(</a><a id="293" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="296" class="Symbol">;</a> <a id="298" href="Relation.Binary.Core.html#1268" class="Function Operator">_⇒_</a><a id="301" class="Symbol">)</a>
|
|||
|
|
<a id="303" class="Keyword">open</a> <a id="308" class="Keyword">import</a> <a id="315" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
|
|||
|
|
<a id="344" class="Keyword">using</a> <a id="350" class="Symbol">(</a><a id="351" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="365" class="Symbol">;</a> <a id="367" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a><a id="380" class="Symbol">;</a> <a id="382" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a><a id="402" class="Symbol">;</a> <a id="404" href="Relation.Binary.Structures.html#3809" class="Record">IsDecPartialOrder</a><a id="421" class="Symbol">;</a> <a id="423" href="Relation.Binary.Structures.html#4782" class="Record">IsDecStrictPartialOrder</a><a id="446" class="Symbol">;</a> <a id="448" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a><a id="460" class="Symbol">;</a> <a id="462" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a><a id="480" class="Symbol">;</a> <a id="482" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a><a id="497" class="Symbol">)</a>
|
|||
|
|
<a id="499" class="Keyword">open</a> <a id="504" class="Keyword">import</a> <a id="511" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
|
|||
|
|
<a id="541" class="Keyword">using</a> <a id="547" class="Symbol">(</a><a id="548" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a><a id="560" class="Symbol">;</a> <a id="562" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a><a id="575" class="Symbol">;</a> <a id="577" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a><a id="586" class="Symbol">;</a> <a id="588" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="593" class="Symbol">;</a> <a id="595" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="604" class="Symbol">;</a> <a id="606" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a><a id="617" class="Symbol">;</a> <a id="619" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a><a id="629" class="Symbol">;</a> <a id="631" href="Relation.Binary.Definitions.html#5335" class="Function Operator">_Respectsʳ_</a><a id="642" class="Symbol">;</a> <a id="644" href="Relation.Binary.Definitions.html#5500" class="Function Operator">_Respectsˡ_</a><a id="655" class="Symbol">;</a> <a id="657" href="Relation.Binary.Definitions.html#5663" class="Function Operator">_Respects₂_</a><a id="668" class="Symbol">;</a> <a id="670" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a><a id="675" class="Symbol">;</a> <a id="677" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a><a id="687" class="Symbol">;</a> <a id="689" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a><a id="693" class="Symbol">;</a> <a id="695" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri<</a><a id="699" class="Symbol">;</a> <a id="701" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri></a><a id="705" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="708" class="Keyword">module</a> <a id="715" href="Relation.Binary.Construct.NonStrictToStrict.html" class="Module">Relation.Binary.Construct.NonStrictToStrict</a>
|
|||
|
|
<a id="761" class="Symbol">{</a><a id="762" href="Relation.Binary.Construct.NonStrictToStrict.html#762" class="Bound">a</a> <a id="764" href="Relation.Binary.Construct.NonStrictToStrict.html#764" class="Bound">ℓ₁</a> <a id="767" href="Relation.Binary.Construct.NonStrictToStrict.html#767" class="Bound">ℓ₂</a><a id="769" class="Symbol">}</a> <a id="771" class="Symbol">{</a><a id="772" href="Relation.Binary.Construct.NonStrictToStrict.html#772" class="Bound">A</a> <a id="774" class="Symbol">:</a> <a id="776" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="780" href="Relation.Binary.Construct.NonStrictToStrict.html#762" class="Bound">a</a><a id="781" class="Symbol">}</a> <a id="783" class="Symbol">(</a><a id="784" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="788" class="Symbol">:</a> <a id="790" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="794" href="Relation.Binary.Construct.NonStrictToStrict.html#772" class="Bound">A</a> <a id="796" href="Relation.Binary.Construct.NonStrictToStrict.html#764" class="Bound">ℓ₁</a><a id="798" class="Symbol">)</a> <a id="800" class="Symbol">(</a><a id="801" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="805" class="Symbol">:</a> <a id="807" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="811" href="Relation.Binary.Construct.NonStrictToStrict.html#772" class="Bound">A</a> <a id="813" href="Relation.Binary.Construct.NonStrictToStrict.html#767" class="Bound">ℓ₂</a><a id="815" class="Symbol">)</a> <a id="817" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="824" class="Keyword">open</a> <a id="829" class="Keyword">import</a> <a id="836" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="854" class="Keyword">using</a> <a id="860" class="Symbol">(</a><a id="861" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="864" class="Symbol">;</a> <a id="866" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="869" class="Symbol">;</a> <a id="871" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="876" class="Symbol">;</a> <a id="878" href="Data.Product.Base.html#650" class="Field">proj₂</a><a id="883" class="Symbol">)</a>
|
|||
|
|
<a id="885" class="Keyword">open</a> <a id="890" class="Keyword">import</a> <a id="897" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="911" class="Keyword">using</a> <a id="917" class="Symbol">(</a><a id="918" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="922" class="Symbol">;</a> <a id="924" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="928" class="Symbol">)</a>
|
|||
|
|
<a id="930" class="Keyword">open</a> <a id="935" class="Keyword">import</a> <a id="942" href="Function.Base.html" class="Module">Function.Base</a> <a id="956" class="Keyword">using</a> <a id="962" class="Symbol">(</a><a id="963" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="966" class="Symbol">;</a> <a id="968" href="Function.Base.html#1638" class="Function">flip</a><a id="972" class="Symbol">)</a>
|
|||
|
|
<a id="974" class="Keyword">open</a> <a id="979" class="Keyword">import</a> <a id="986" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="1003" class="Keyword">using</a> <a id="1009" class="Symbol">(</a><a id="1010" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬_</a><a id="1012" class="Symbol">;</a> <a id="1014" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a><a id="1017" class="Symbol">;</a> <a id="1019" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a><a id="1021" class="Symbol">)</a>
|
|||
|
|
<a id="1023" class="Keyword">open</a> <a id="1028" class="Keyword">import</a> <a id="1035" href="Relation.Nullary.Negation.html" class="Module">Relation.Nullary.Negation</a> <a id="1061" class="Keyword">using</a> <a id="1067" class="Symbol">(</a><a id="1068" href="Relation.Nullary.Negation.Core.html#1226" class="Function">contradiction</a><a id="1081" class="Symbol">)</a>
|
|||
|
|
<a id="1083" class="Keyword">open</a> <a id="1088" class="Keyword">import</a> <a id="1095" href="Relation.Nullary.Decidable.html" class="Module">Relation.Nullary.Decidable</a> <a id="1122" class="Keyword">using</a> <a id="1128" class="Symbol">(</a><a id="1129" href="Relation.Nullary.Decidable.Core.html#2517" class="Function">¬?</a><a id="1131" class="Symbol">;</a> <a id="1133" href="Relation.Nullary.Decidable.Core.html#2609" class="Function Operator">_×-dec_</a><a id="1140" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="1143" class="Keyword">private</a>
|
|||
|
|
<a id="_≉_"></a><a id="1153" href="Relation.Binary.Construct.NonStrictToStrict.html#1153" class="Function Operator">_≉_</a> <a id="1157" class="Symbol">:</a> <a id="1159" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1163" href="Relation.Binary.Construct.NonStrictToStrict.html#772" class="Bound">A</a> <a id="1165" href="Relation.Binary.Construct.NonStrictToStrict.html#764" class="Bound">ℓ₁</a>
|
|||
|
|
<a id="1170" href="Relation.Binary.Construct.NonStrictToStrict.html#1170" class="Bound">x</a> <a id="1172" href="Relation.Binary.Construct.NonStrictToStrict.html#1153" class="Function Operator">≉</a> <a id="1174" href="Relation.Binary.Construct.NonStrictToStrict.html#1174" class="Bound">y</a> <a id="1176" class="Symbol">=</a> <a id="1178" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="1180" class="Symbol">(</a><a id="1181" href="Relation.Binary.Construct.NonStrictToStrict.html#1170" class="Bound">x</a> <a id="1183" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">≈</a> <a id="1185" href="Relation.Binary.Construct.NonStrictToStrict.html#1174" class="Bound">y</a><a id="1186" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="1189" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="1262" class="Comment">-- _≤_ can be turned into _<_ as follows:</a>
|
|||
|
|
|
|||
|
|
<a id="1305" class="Keyword">infix</a> <a id="1311" class="Number">4</a> <a id="1314" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
|
|||
|
|
<a id="_<_"></a><a id="1319" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="1323" class="Symbol">:</a> <a id="1325" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="1329" href="Relation.Binary.Construct.NonStrictToStrict.html#772" class="Bound">A</a> <a id="1331" class="Symbol">_</a>
|
|||
|
|
<a id="1333" href="Relation.Binary.Construct.NonStrictToStrict.html#1333" class="Bound">x</a> <a id="1335" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1337" href="Relation.Binary.Construct.NonStrictToStrict.html#1337" class="Bound">y</a> <a id="1339" class="Symbol">=</a> <a id="1341" href="Relation.Binary.Construct.NonStrictToStrict.html#1333" class="Bound">x</a> <a id="1343" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="1345" href="Relation.Binary.Construct.NonStrictToStrict.html#1337" class="Bound">y</a> <a id="1347" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="1349" href="Relation.Binary.Construct.NonStrictToStrict.html#1333" class="Bound">x</a> <a id="1351" href="Relation.Binary.Construct.NonStrictToStrict.html#1153" class="Function Operator">≉</a> <a id="1353" href="Relation.Binary.Construct.NonStrictToStrict.html#1337" class="Bound">y</a>
|
|||
|
|
|
|||
|
|
<a id="1356" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="1429" class="Comment">-- Relationship between relations</a>
|
|||
|
|
|
|||
|
|
<a id="<⇒≤"></a><a id="1464" href="Relation.Binary.Construct.NonStrictToStrict.html#1464" class="Function"><⇒≤</a> <a id="1468" class="Symbol">:</a> <a id="1470" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="1474" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1476" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a>
|
|||
|
|
<a id="1480" href="Relation.Binary.Construct.NonStrictToStrict.html#1464" class="Function"><⇒≤</a> <a id="1484" class="Symbol">=</a> <a id="1486" href="Data.Product.Base.html#636" class="Field">proj₁</a>
|
|||
|
|
|
|||
|
|
<a id="<⇒≉"></a><a id="1493" href="Relation.Binary.Construct.NonStrictToStrict.html#1493" class="Function"><⇒≉</a> <a id="1497" class="Symbol">:</a> <a id="1499" class="Symbol">∀</a> <a id="1501" class="Symbol">{</a><a id="1502" href="Relation.Binary.Construct.NonStrictToStrict.html#1502" class="Bound">x</a> <a id="1504" href="Relation.Binary.Construct.NonStrictToStrict.html#1504" class="Bound">y</a><a id="1505" class="Symbol">}</a> <a id="1507" class="Symbol">→</a> <a id="1509" href="Relation.Binary.Construct.NonStrictToStrict.html#1502" class="Bound">x</a> <a id="1511" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1513" href="Relation.Binary.Construct.NonStrictToStrict.html#1504" class="Bound">y</a> <a id="1515" class="Symbol">→</a> <a id="1517" href="Relation.Binary.Construct.NonStrictToStrict.html#1502" class="Bound">x</a> <a id="1519" href="Relation.Binary.Construct.NonStrictToStrict.html#1153" class="Function Operator">≉</a> <a id="1521" href="Relation.Binary.Construct.NonStrictToStrict.html#1504" class="Bound">y</a>
|
|||
|
|
<a id="1523" href="Relation.Binary.Construct.NonStrictToStrict.html#1493" class="Function"><⇒≉</a> <a id="1527" class="Symbol">=</a> <a id="1529" href="Data.Product.Base.html#650" class="Field">proj₂</a>
|
|||
|
|
|
|||
|
|
<a id="≤∧≉⇒<"></a><a id="1536" href="Relation.Binary.Construct.NonStrictToStrict.html#1536" class="Function">≤∧≉⇒<</a> <a id="1542" class="Symbol">:</a> <a id="1544" class="Symbol">∀</a> <a id="1546" class="Symbol">{</a><a id="1547" href="Relation.Binary.Construct.NonStrictToStrict.html#1547" class="Bound">x</a> <a id="1549" href="Relation.Binary.Construct.NonStrictToStrict.html#1549" class="Bound">y</a><a id="1550" class="Symbol">}</a> <a id="1552" class="Symbol">→</a> <a id="1554" href="Relation.Binary.Construct.NonStrictToStrict.html#1547" class="Bound">x</a> <a id="1556" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="1558" href="Relation.Binary.Construct.NonStrictToStrict.html#1549" class="Bound">y</a> <a id="1560" class="Symbol">→</a> <a id="1562" href="Relation.Binary.Construct.NonStrictToStrict.html#1547" class="Bound">x</a> <a id="1564" href="Relation.Binary.Construct.NonStrictToStrict.html#1153" class="Function Operator">≉</a> <a id="1566" href="Relation.Binary.Construct.NonStrictToStrict.html#1549" class="Bound">y</a> <a id="1568" class="Symbol">→</a> <a id="1570" href="Relation.Binary.Construct.NonStrictToStrict.html#1547" class="Bound">x</a> <a id="1572" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1574" href="Relation.Binary.Construct.NonStrictToStrict.html#1549" class="Bound">y</a>
|
|||
|
|
<a id="1576" href="Relation.Binary.Construct.NonStrictToStrict.html#1536" class="Function">≤∧≉⇒<</a> <a id="1582" class="Symbol">=</a> <a id="1584" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a>
|
|||
|
|
|
|||
|
|
<a id="<⇒≱"></a><a id="1589" href="Relation.Binary.Construct.NonStrictToStrict.html#1589" class="Function"><⇒≱</a> <a id="1593" class="Symbol">:</a> <a id="1595" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1609" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="1613" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="1617" class="Symbol">→</a> <a id="1619" class="Symbol">∀</a> <a id="1621" class="Symbol">{</a><a id="1622" href="Relation.Binary.Construct.NonStrictToStrict.html#1622" class="Bound">x</a> <a id="1624" href="Relation.Binary.Construct.NonStrictToStrict.html#1624" class="Bound">y</a><a id="1625" class="Symbol">}</a> <a id="1627" class="Symbol">→</a> <a id="1629" href="Relation.Binary.Construct.NonStrictToStrict.html#1622" class="Bound">x</a> <a id="1631" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1633" href="Relation.Binary.Construct.NonStrictToStrict.html#1624" class="Bound">y</a> <a id="1635" class="Symbol">→</a> <a id="1637" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="1639" class="Symbol">(</a><a id="1640" href="Relation.Binary.Construct.NonStrictToStrict.html#1624" class="Bound">y</a> <a id="1642" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="1644" href="Relation.Binary.Construct.NonStrictToStrict.html#1622" class="Bound">x</a><a id="1645" class="Symbol">)</a>
|
|||
|
|
<a id="1647" href="Relation.Binary.Construct.NonStrictToStrict.html#1589" class="Function"><⇒≱</a> <a id="1651" href="Relation.Binary.Construct.NonStrictToStrict.html#1651" class="Bound">antisym</a> <a id="1659" class="Symbol">(</a><a id="1660" href="Relation.Binary.Construct.NonStrictToStrict.html#1660" class="Bound">x≤y</a> <a id="1664" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1666" href="Relation.Binary.Construct.NonStrictToStrict.html#1666" class="Bound">x≉y</a><a id="1669" class="Symbol">)</a> <a id="1671" href="Relation.Binary.Construct.NonStrictToStrict.html#1671" class="Bound">y≤x</a> <a id="1675" class="Symbol">=</a> <a id="1677" href="Relation.Binary.Construct.NonStrictToStrict.html#1666" class="Bound">x≉y</a> <a id="1681" class="Symbol">(</a><a id="1682" href="Relation.Binary.Construct.NonStrictToStrict.html#1651" class="Bound">antisym</a> <a id="1690" href="Relation.Binary.Construct.NonStrictToStrict.html#1660" class="Bound">x≤y</a> <a id="1694" href="Relation.Binary.Construct.NonStrictToStrict.html#1671" class="Bound">y≤x</a><a id="1697" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="≤⇒≯"></a><a id="1700" href="Relation.Binary.Construct.NonStrictToStrict.html#1700" class="Function">≤⇒≯</a> <a id="1704" class="Symbol">:</a> <a id="1706" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1720" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="1724" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="1728" class="Symbol">→</a> <a id="1730" class="Symbol">∀</a> <a id="1732" class="Symbol">{</a><a id="1733" href="Relation.Binary.Construct.NonStrictToStrict.html#1733" class="Bound">x</a> <a id="1735" href="Relation.Binary.Construct.NonStrictToStrict.html#1735" class="Bound">y</a><a id="1736" class="Symbol">}</a> <a id="1738" class="Symbol">→</a> <a id="1740" href="Relation.Binary.Construct.NonStrictToStrict.html#1733" class="Bound">x</a> <a id="1742" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="1744" href="Relation.Binary.Construct.NonStrictToStrict.html#1735" class="Bound">y</a> <a id="1746" class="Symbol">→</a> <a id="1748" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="1750" class="Symbol">(</a><a id="1751" href="Relation.Binary.Construct.NonStrictToStrict.html#1735" class="Bound">y</a> <a id="1753" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1755" href="Relation.Binary.Construct.NonStrictToStrict.html#1733" class="Bound">x</a><a id="1756" class="Symbol">)</a>
|
|||
|
|
<a id="1758" href="Relation.Binary.Construct.NonStrictToStrict.html#1700" class="Function">≤⇒≯</a> <a id="1762" href="Relation.Binary.Construct.NonStrictToStrict.html#1762" class="Bound">antisym</a> <a id="1770" href="Relation.Binary.Construct.NonStrictToStrict.html#1770" class="Bound">x≤y</a> <a id="1774" href="Relation.Binary.Construct.NonStrictToStrict.html#1774" class="Bound">y<x</a> <a id="1778" class="Symbol">=</a> <a id="1780" href="Relation.Binary.Construct.NonStrictToStrict.html#1589" class="Function"><⇒≱</a> <a id="1784" href="Relation.Binary.Construct.NonStrictToStrict.html#1762" class="Bound">antisym</a> <a id="1792" href="Relation.Binary.Construct.NonStrictToStrict.html#1774" class="Bound">y<x</a> <a id="1796" href="Relation.Binary.Construct.NonStrictToStrict.html#1770" class="Bound">x≤y</a>
|
|||
|
|
|
|||
|
|
<a id="≰⇒>"></a><a id="1801" href="Relation.Binary.Construct.NonStrictToStrict.html#1801" class="Function">≰⇒></a> <a id="1805" class="Symbol">:</a> <a id="1807" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="1817" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="1821" class="Symbol">→</a> <a id="1823" class="Symbol">(</a><a id="1824" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="1828" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="1830" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a><a id="1833" class="Symbol">)</a> <a id="1835" class="Symbol">→</a> <a id="1837" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="1843" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="1847" class="Symbol">→</a>
|
|||
|
|
<a id="1855" class="Symbol">∀</a> <a id="1857" class="Symbol">{</a><a id="1858" href="Relation.Binary.Construct.NonStrictToStrict.html#1858" class="Bound">x</a> <a id="1860" href="Relation.Binary.Construct.NonStrictToStrict.html#1860" class="Bound">y</a><a id="1861" class="Symbol">}</a> <a id="1863" class="Symbol">→</a> <a id="1865" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="1867" class="Symbol">(</a><a id="1868" href="Relation.Binary.Construct.NonStrictToStrict.html#1858" class="Bound">x</a> <a id="1870" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="1872" href="Relation.Binary.Construct.NonStrictToStrict.html#1860" class="Bound">y</a><a id="1873" class="Symbol">)</a> <a id="1875" class="Symbol">→</a> <a id="1877" href="Relation.Binary.Construct.NonStrictToStrict.html#1860" class="Bound">y</a> <a id="1879" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="1881" href="Relation.Binary.Construct.NonStrictToStrict.html#1858" class="Bound">x</a>
|
|||
|
|
<a id="1883" href="Relation.Binary.Construct.NonStrictToStrict.html#1801" class="Function">≰⇒></a> <a id="1887" href="Relation.Binary.Construct.NonStrictToStrict.html#1887" class="Bound">sym</a> <a id="1891" href="Relation.Binary.Construct.NonStrictToStrict.html#1891" class="Bound">refl</a> <a id="1896" href="Relation.Binary.Construct.NonStrictToStrict.html#1896" class="Bound">total</a> <a id="1902" class="Symbol">{</a><a id="1903" href="Relation.Binary.Construct.NonStrictToStrict.html#1903" class="Bound">x</a><a id="1904" class="Symbol">}</a> <a id="1906" class="Symbol">{</a><a id="1907" href="Relation.Binary.Construct.NonStrictToStrict.html#1907" class="Bound">y</a><a id="1908" class="Symbol">}</a> <a id="1910" href="Relation.Binary.Construct.NonStrictToStrict.html#1910" class="Bound">x≰y</a> <a id="1914" class="Keyword">with</a> <a id="1919" href="Relation.Binary.Construct.NonStrictToStrict.html#1896" class="Bound">total</a> <a id="1925" href="Relation.Binary.Construct.NonStrictToStrict.html#1903" class="Bound">x</a> <a id="1927" href="Relation.Binary.Construct.NonStrictToStrict.html#1907" class="Bound">y</a>
|
|||
|
|
<a id="1929" class="Symbol">...</a> <a id="1933" class="Symbol">|</a> <a id="1935" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1940" href="Relation.Binary.Construct.NonStrictToStrict.html#1940" class="Bound">x≤y</a> <a id="1944" class="Symbol">=</a> <a id="1946" href="Relation.Nullary.Negation.Core.html#1226" class="Function">contradiction</a> <a id="1960" href="Relation.Binary.Construct.NonStrictToStrict.html#1940" class="Bound">x≤y</a> <a id="1964" class="Bound">x≰y</a>
|
|||
|
|
<a id="1968" class="Symbol">...</a> <a id="1972" class="Symbol">|</a> <a id="1974" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1979" href="Relation.Binary.Construct.NonStrictToStrict.html#1979" class="Bound">y≤x</a> <a id="1983" class="Symbol">=</a> <a id="1985" href="Relation.Binary.Construct.NonStrictToStrict.html#1979" class="Bound">y≤x</a> <a id="1989" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1991" class="Bound">x≰y</a> <a id="1995" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="1997" class="Bound">refl</a> <a id="2002" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2004" class="Bound">sym</a>
|
|||
|
|
|
|||
|
|
<a id="≮⇒≥"></a><a id="2009" href="Relation.Binary.Construct.NonStrictToStrict.html#2009" class="Function">≮⇒≥</a> <a id="2013" class="Symbol">:</a> <a id="2015" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2025" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2029" class="Symbol">→</a> <a id="2031" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2041" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2045" class="Symbol">→</a> <a id="2047" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2051" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="2053" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2057" class="Symbol">→</a> <a id="2059" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2065" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2069" class="Symbol">→</a>
|
|||
|
|
<a id="2077" class="Symbol">∀</a> <a id="2079" class="Symbol">{</a><a id="2080" href="Relation.Binary.Construct.NonStrictToStrict.html#2080" class="Bound">x</a> <a id="2082" href="Relation.Binary.Construct.NonStrictToStrict.html#2082" class="Bound">y</a><a id="2083" class="Symbol">}</a> <a id="2085" class="Symbol">→</a> <a id="2087" href="Relation.Nullary.Negation.Core.html#698" class="Function Operator">¬</a> <a id="2089" class="Symbol">(</a><a id="2090" href="Relation.Binary.Construct.NonStrictToStrict.html#2080" class="Bound">x</a> <a id="2092" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator"><</a> <a id="2094" href="Relation.Binary.Construct.NonStrictToStrict.html#2082" class="Bound">y</a><a id="2095" class="Symbol">)</a> <a id="2097" class="Symbol">→</a> <a id="2099" href="Relation.Binary.Construct.NonStrictToStrict.html#2082" class="Bound">y</a> <a id="2101" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">≤</a> <a id="2103" href="Relation.Binary.Construct.NonStrictToStrict.html#2080" class="Bound">x</a>
|
|||
|
|
<a id="2105" href="Relation.Binary.Construct.NonStrictToStrict.html#2009" class="Function">≮⇒≥</a> <a id="2109" href="Relation.Binary.Construct.NonStrictToStrict.html#2109" class="Bound">sym</a> <a id="2113" href="Relation.Binary.Construct.NonStrictToStrict.html#2113" class="Bound Operator">_≟_</a> <a id="2117" href="Relation.Binary.Construct.NonStrictToStrict.html#2117" class="Bound">≤-refl</a> <a id="2124" href="Relation.Binary.Construct.NonStrictToStrict.html#2124" class="Bound Operator">_≤?_</a> <a id="2129" class="Symbol">{</a><a id="2130" href="Relation.Binary.Construct.NonStrictToStrict.html#2130" class="Bound">x</a><a id="2131" class="Symbol">}</a> <a id="2133" class="Symbol">{</a><a id="2134" href="Relation.Binary.Construct.NonStrictToStrict.html#2134" class="Bound">y</a><a id="2135" class="Symbol">}</a> <a id="2137" href="Relation.Binary.Construct.NonStrictToStrict.html#2137" class="Bound">x≮y</a> <a id="2141" class="Keyword">with</a> <a id="2146" href="Relation.Binary.Construct.NonStrictToStrict.html#2130" class="Bound">x</a> <a id="2148" href="Relation.Binary.Construct.NonStrictToStrict.html#2113" class="Bound Operator">≟</a> <a id="2150" href="Relation.Binary.Construct.NonStrictToStrict.html#2134" class="Bound">y</a> <a id="2152" class="Symbol">|</a> <a id="2154" href="Relation.Binary.Construct.NonStrictToStrict.html#2134" class="Bound">y</a> <a id="2156" href="Relation.Binary.Construct.NonStrictToStrict.html#2124" class="Bound Operator">≤?</a> <a id="2159" href="Relation.Binary.Construct.NonStrictToStrict.html#2130" class="Bound">x</a>
|
|||
|
|
<a id="2161" class="Symbol">...</a> <a id="2165" class="Symbol">|</a> <a id="2167" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="2171" href="Relation.Binary.Construct.NonStrictToStrict.html#2171" class="Bound">x≈y</a> <a id="2176" class="Symbol">|</a> <a id="2178" class="Symbol">_</a> <a id="2187" class="Symbol">=</a> <a id="2189" class="Bound">≤-refl</a> <a id="2196" class="Symbol">(</a><a id="2197" class="Bound">sym</a> <a id="2201" href="Relation.Binary.Construct.NonStrictToStrict.html#2171" class="Bound">x≈y</a><a id="2204" class="Symbol">)</a>
|
|||
|
|
<a id="2206" class="CatchallClause Symbol">...</a><a id="2209" class="CatchallClause"> </a><a id="2210" class="CatchallClause Symbol">|</a><a id="2211" class="CatchallClause"> </a><a id="2212" class="CatchallClause Symbol">_</a><a id="2213" class="CatchallClause"> </a><a id="2221" class="CatchallClause Symbol">|</a><a id="2222" class="CatchallClause"> </a><a id="2223" href="Data.Sum.Base.html#675" class="CatchallClause InductiveConstructor">inj₁</a><a id="2227" class="CatchallClause"> </a><a id="2228" href="Relation.Binary.Construct.NonStrictToStrict.html#2228" class="CatchallClause Bound">y≤x</a> <a id="2232" class="Symbol">=</a> <a id="2234" href="Relation.Binary.Construct.NonStrictToStrict.html#2228" class="Bound">y≤x</a>
|
|||
|
|
<a id="2238" class="Symbol">...</a> <a id="2242" class="Symbol">|</a> <a id="2244" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="2248" href="Relation.Binary.Construct.NonStrictToStrict.html#2248" class="Bound">x≉y</a> <a id="2253" class="Symbol">|</a> <a id="2255" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2260" href="Relation.Binary.Construct.NonStrictToStrict.html#2260" class="Bound">x≤y</a> <a id="2264" class="Symbol">=</a> <a id="2266" href="Relation.Nullary.Negation.Core.html#1226" class="Function">contradiction</a> <a id="2280" class="Symbol">(</a><a id="2281" href="Relation.Binary.Construct.NonStrictToStrict.html#2260" class="Bound">x≤y</a> <a id="2285" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2287" href="Relation.Binary.Construct.NonStrictToStrict.html#2248" class="Bound">x≉y</a><a id="2290" class="Symbol">)</a> <a id="2292" class="Bound">x≮y</a>
|
|||
|
|
|
|||
|
|
<a id="2297" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="2370" class="Comment">-- Relational properties</a>
|
|||
|
|
|
|||
|
|
<a id="<-irrefl"></a><a id="2396" href="Relation.Binary.Construct.NonStrictToStrict.html#2396" class="Function"><-irrefl</a> <a id="2405" class="Symbol">:</a> <a id="2407" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="2419" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2423" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="2427" href="Relation.Binary.Construct.NonStrictToStrict.html#2396" class="Function"><-irrefl</a> <a id="2436" href="Relation.Binary.Construct.NonStrictToStrict.html#2436" class="Bound">x≈y</a> <a id="2440" class="Symbol">(_</a> <a id="2443" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2445" href="Relation.Binary.Construct.NonStrictToStrict.html#2445" class="Bound">x≉y</a><a id="2448" class="Symbol">)</a> <a id="2450" class="Symbol">=</a> <a id="2452" href="Relation.Binary.Construct.NonStrictToStrict.html#2445" class="Bound">x≉y</a> <a id="2456" href="Relation.Binary.Construct.NonStrictToStrict.html#2436" class="Bound">x≈y</a>
|
|||
|
|
|
|||
|
|
<a id="<-trans"></a><a id="2461" href="Relation.Binary.Construct.NonStrictToStrict.html#2461" class="Function"><-trans</a> <a id="2469" class="Symbol">:</a> <a id="2471" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="2486" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2490" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2494" class="Symbol">→</a> <a id="2496" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2507" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="2511" href="Relation.Binary.Construct.NonStrictToStrict.html#2461" class="Function"><-trans</a> <a id="2519" href="Relation.Binary.Construct.NonStrictToStrict.html#2519" class="Bound">po</a> <a id="2522" class="Symbol">(</a><a id="2523" href="Relation.Binary.Construct.NonStrictToStrict.html#2523" class="Bound">x≤y</a> <a id="2527" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2529" href="Relation.Binary.Construct.NonStrictToStrict.html#2529" class="Bound">x≉y</a><a id="2532" class="Symbol">)</a> <a id="2534" class="Symbol">(</a><a id="2535" href="Relation.Binary.Construct.NonStrictToStrict.html#2535" class="Bound">y≤z</a> <a id="2539" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2541" href="Relation.Binary.Construct.NonStrictToStrict.html#2541" class="Bound">y≉z</a><a id="2544" class="Symbol">)</a> <a id="2546" class="Symbol">=</a>
|
|||
|
|
<a id="2550" class="Symbol">(</a><a id="2551" href="Relation.Binary.Structures.html#2389" class="Function">trans</a> <a id="2557" href="Relation.Binary.Construct.NonStrictToStrict.html#2523" class="Bound">x≤y</a> <a id="2561" href="Relation.Binary.Construct.NonStrictToStrict.html#2535" class="Bound">y≤z</a> <a id="2565" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2567" href="Relation.Binary.Construct.NonStrictToStrict.html#2529" class="Bound">x≉y</a> <a id="2571" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2573" href="Relation.Binary.Structures.html#3623" class="Function">antisym</a> <a id="2581" href="Relation.Binary.Construct.NonStrictToStrict.html#2523" class="Bound">x≤y</a> <a id="2585" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2587" href="Relation.Binary.Structures.html#2389" class="Function">trans</a> <a id="2593" href="Relation.Binary.Construct.NonStrictToStrict.html#2535" class="Bound">y≤z</a> <a id="2597" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2599" href="Relation.Binary.Structures.html#2359" class="Function">reflexive</a> <a id="2609" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2611" href="Relation.Binary.Structures.html#1622" class="Function">Eq.sym</a><a id="2617" class="Symbol">)</a>
|
|||
|
|
<a id="2621" class="Keyword">where</a> <a id="2627" class="Keyword">open</a> <a id="2632" href="Relation.Binary.Structures.html#3522" class="Module">IsPartialOrder</a> <a id="2647" href="Relation.Binary.Construct.NonStrictToStrict.html#2519" class="Bound">po</a>
|
|||
|
|
|
|||
|
|
<a id="<-≤-trans"></a><a id="2651" href="Relation.Binary.Construct.NonStrictToStrict.html#2651" class="Function"><-≤-trans</a> <a id="2661" class="Symbol">:</a> <a id="2663" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="2673" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2677" class="Symbol">→</a> <a id="2679" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2690" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2694" class="Symbol">→</a> <a id="2696" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2710" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2714" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2718" class="Symbol">→</a>
|
|||
|
|
<a id="2731" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2735" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="2745" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2749" class="Symbol">→</a> <a id="2751" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a> <a id="2757" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="2761" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2765" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="2769" href="Relation.Binary.Construct.NonStrictToStrict.html#2651" class="Function"><-≤-trans</a> <a id="2779" href="Relation.Binary.Construct.NonStrictToStrict.html#2779" class="Bound">sym</a> <a id="2783" href="Relation.Binary.Construct.NonStrictToStrict.html#2783" class="Bound">trans</a> <a id="2789" href="Relation.Binary.Construct.NonStrictToStrict.html#2789" class="Bound">antisym</a> <a id="2797" href="Relation.Binary.Construct.NonStrictToStrict.html#2797" class="Bound">respʳ</a> <a id="2803" class="Symbol">(</a><a id="2804" href="Relation.Binary.Construct.NonStrictToStrict.html#2804" class="Bound">x≤y</a> <a id="2808" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2810" href="Relation.Binary.Construct.NonStrictToStrict.html#2810" class="Bound">x≉y</a><a id="2813" class="Symbol">)</a> <a id="2815" href="Relation.Binary.Construct.NonStrictToStrict.html#2815" class="Bound">y≤z</a> <a id="2819" class="Symbol">=</a>
|
|||
|
|
<a id="2823" href="Relation.Binary.Construct.NonStrictToStrict.html#2783" class="Bound">trans</a> <a id="2829" href="Relation.Binary.Construct.NonStrictToStrict.html#2804" class="Bound">x≤y</a> <a id="2833" href="Relation.Binary.Construct.NonStrictToStrict.html#2815" class="Bound">y≤z</a> <a id="2837" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2839" class="Symbol">(λ</a> <a id="2842" href="Relation.Binary.Construct.NonStrictToStrict.html#2842" class="Bound">x≈z</a> <a id="2846" class="Symbol">→</a> <a id="2848" href="Relation.Binary.Construct.NonStrictToStrict.html#2810" class="Bound">x≉y</a> <a id="2852" class="Symbol">(</a><a id="2853" href="Relation.Binary.Construct.NonStrictToStrict.html#2789" class="Bound">antisym</a> <a id="2861" href="Relation.Binary.Construct.NonStrictToStrict.html#2804" class="Bound">x≤y</a> <a id="2865" class="Symbol">(</a><a id="2866" href="Relation.Binary.Construct.NonStrictToStrict.html#2797" class="Bound">respʳ</a> <a id="2872" class="Symbol">(</a><a id="2873" href="Relation.Binary.Construct.NonStrictToStrict.html#2779" class="Bound">sym</a> <a id="2877" href="Relation.Binary.Construct.NonStrictToStrict.html#2842" class="Bound">x≈z</a><a id="2880" class="Symbol">)</a> <a id="2882" href="Relation.Binary.Construct.NonStrictToStrict.html#2815" class="Bound">y≤z</a><a id="2885" class="Symbol">)))</a>
|
|||
|
|
|
|||
|
|
<a id="≤-<-trans"></a><a id="2890" href="Relation.Binary.Construct.NonStrictToStrict.html#2890" class="Function">≤-<-trans</a> <a id="2900" class="Symbol">:</a> <a id="2902" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2913" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2917" class="Symbol">→</a> <a id="2919" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2933" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2937" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2941" class="Symbol">→</a>
|
|||
|
|
<a id="2954" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2958" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="2968" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="2972" class="Symbol">→</a> <a id="2974" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a> <a id="2980" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="2984" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="2988" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="2992" href="Relation.Binary.Construct.NonStrictToStrict.html#2890" class="Function">≤-<-trans</a> <a id="3002" href="Relation.Binary.Construct.NonStrictToStrict.html#3002" class="Bound">trans</a> <a id="3008" href="Relation.Binary.Construct.NonStrictToStrict.html#3008" class="Bound">antisym</a> <a id="3016" href="Relation.Binary.Construct.NonStrictToStrict.html#3016" class="Bound">respʳ</a> <a id="3022" href="Relation.Binary.Construct.NonStrictToStrict.html#3022" class="Bound">x≤y</a> <a id="3026" class="Symbol">(</a><a id="3027" href="Relation.Binary.Construct.NonStrictToStrict.html#3027" class="Bound">y≤z</a> <a id="3031" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3033" href="Relation.Binary.Construct.NonStrictToStrict.html#3033" class="Bound">y≉z</a><a id="3036" class="Symbol">)</a> <a id="3038" class="Symbol">=</a>
|
|||
|
|
<a id="3042" href="Relation.Binary.Construct.NonStrictToStrict.html#3002" class="Bound">trans</a> <a id="3048" href="Relation.Binary.Construct.NonStrictToStrict.html#3022" class="Bound">x≤y</a> <a id="3052" href="Relation.Binary.Construct.NonStrictToStrict.html#3027" class="Bound">y≤z</a> <a id="3056" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3058" class="Symbol">(λ</a> <a id="3061" href="Relation.Binary.Construct.NonStrictToStrict.html#3061" class="Bound">x≈z</a> <a id="3065" class="Symbol">→</a> <a id="3067" href="Relation.Binary.Construct.NonStrictToStrict.html#3033" class="Bound">y≉z</a> <a id="3071" class="Symbol">(</a><a id="3072" href="Relation.Binary.Construct.NonStrictToStrict.html#3008" class="Bound">antisym</a> <a id="3080" href="Relation.Binary.Construct.NonStrictToStrict.html#3027" class="Bound">y≤z</a> <a id="3084" class="Symbol">(</a><a id="3085" href="Relation.Binary.Construct.NonStrictToStrict.html#3016" class="Bound">respʳ</a> <a id="3091" href="Relation.Binary.Construct.NonStrictToStrict.html#3061" class="Bound">x≈z</a> <a id="3095" href="Relation.Binary.Construct.NonStrictToStrict.html#3022" class="Bound">x≤y</a><a id="3098" class="Symbol">)))</a>
|
|||
|
|
|
|||
|
|
<a id="<-asym"></a><a id="3103" href="Relation.Binary.Construct.NonStrictToStrict.html#3103" class="Function"><-asym</a> <a id="3110" class="Symbol">:</a> <a id="3112" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="3126" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3130" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3134" class="Symbol">→</a> <a id="3136" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="3147" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="3151" href="Relation.Binary.Construct.NonStrictToStrict.html#3103" class="Function"><-asym</a> <a id="3158" href="Relation.Binary.Construct.NonStrictToStrict.html#3158" class="Bound">antisym</a> <a id="3166" class="Symbol">(</a><a id="3167" href="Relation.Binary.Construct.NonStrictToStrict.html#3167" class="Bound">x≤y</a> <a id="3171" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3173" href="Relation.Binary.Construct.NonStrictToStrict.html#3173" class="Bound">x≉y</a><a id="3176" class="Symbol">)</a> <a id="3178" class="Symbol">(</a><a id="3179" href="Relation.Binary.Construct.NonStrictToStrict.html#3179" class="Bound">y≤x</a> <a id="3183" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3185" class="Symbol">_)</a> <a id="3188" class="Symbol">=</a> <a id="3190" href="Relation.Binary.Construct.NonStrictToStrict.html#3173" class="Bound">x≉y</a> <a id="3194" class="Symbol">(</a><a id="3195" href="Relation.Binary.Construct.NonStrictToStrict.html#3158" class="Bound">antisym</a> <a id="3203" href="Relation.Binary.Construct.NonStrictToStrict.html#3167" class="Bound">x≤y</a> <a id="3207" href="Relation.Binary.Construct.NonStrictToStrict.html#3179" class="Bound">y≤x</a><a id="3210" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="<-respˡ-≈"></a><a id="3213" href="Relation.Binary.Construct.NonStrictToStrict.html#3213" class="Function"><-respˡ-≈</a> <a id="3223" class="Symbol">:</a> <a id="3225" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="3236" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3240" class="Symbol">→</a> <a id="3242" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3246" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="3256" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3260" class="Symbol">→</a> <a id="3262" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="3266" href="Relation.Binary.Definitions.html#5500" class="Function Operator">Respectsˡ</a> <a id="3276" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="3280" href="Relation.Binary.Construct.NonStrictToStrict.html#3213" class="Function"><-respˡ-≈</a> <a id="3290" href="Relation.Binary.Construct.NonStrictToStrict.html#3290" class="Bound">trans</a> <a id="3296" href="Relation.Binary.Construct.NonStrictToStrict.html#3296" class="Bound">respˡ</a> <a id="3302" href="Relation.Binary.Construct.NonStrictToStrict.html#3302" class="Bound">y≈z</a> <a id="3306" class="Symbol">(</a><a id="3307" href="Relation.Binary.Construct.NonStrictToStrict.html#3307" class="Bound">y≤x</a> <a id="3311" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3313" href="Relation.Binary.Construct.NonStrictToStrict.html#3313" class="Bound">y≉x</a><a id="3316" class="Symbol">)</a> <a id="3318" class="Symbol">=</a>
|
|||
|
|
<a id="3322" href="Relation.Binary.Construct.NonStrictToStrict.html#3296" class="Bound">respˡ</a> <a id="3328" href="Relation.Binary.Construct.NonStrictToStrict.html#3302" class="Bound">y≈z</a> <a id="3332" href="Relation.Binary.Construct.NonStrictToStrict.html#3307" class="Bound">y≤x</a> <a id="3336" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3338" href="Relation.Binary.Construct.NonStrictToStrict.html#3313" class="Bound">y≉x</a> <a id="3342" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="3344" href="Relation.Binary.Construct.NonStrictToStrict.html#3290" class="Bound">trans</a> <a id="3350" href="Relation.Binary.Construct.NonStrictToStrict.html#3302" class="Bound">y≈z</a>
|
|||
|
|
|
|||
|
|
<a id="<-respʳ-≈"></a><a id="3355" href="Relation.Binary.Construct.NonStrictToStrict.html#3355" class="Function"><-respʳ-≈</a> <a id="3365" class="Symbol">:</a> <a id="3367" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="3377" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3381" class="Symbol">→</a> <a id="3383" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="3394" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3398" class="Symbol">→</a>
|
|||
|
|
<a id="3412" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3416" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="3426" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3430" class="Symbol">→</a> <a id="3432" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="3436" href="Relation.Binary.Definitions.html#5335" class="Function Operator">Respectsʳ</a> <a id="3446" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="3450" href="Relation.Binary.Construct.NonStrictToStrict.html#3355" class="Function"><-respʳ-≈</a> <a id="3460" href="Relation.Binary.Construct.NonStrictToStrict.html#3460" class="Bound">sym</a> <a id="3464" href="Relation.Binary.Construct.NonStrictToStrict.html#3464" class="Bound">trans</a> <a id="3470" href="Relation.Binary.Construct.NonStrictToStrict.html#3470" class="Bound">respʳ</a> <a id="3476" href="Relation.Binary.Construct.NonStrictToStrict.html#3476" class="Bound">y≈z</a> <a id="3480" class="Symbol">(</a><a id="3481" href="Relation.Binary.Construct.NonStrictToStrict.html#3481" class="Bound">x≤y</a> <a id="3485" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3487" href="Relation.Binary.Construct.NonStrictToStrict.html#3487" class="Bound">x≉y</a><a id="3490" class="Symbol">)</a> <a id="3492" class="Symbol">=</a>
|
|||
|
|
<a id="3496" class="Symbol">(</a><a id="3497" href="Relation.Binary.Construct.NonStrictToStrict.html#3470" class="Bound">respʳ</a> <a id="3503" href="Relation.Binary.Construct.NonStrictToStrict.html#3476" class="Bound">y≈z</a> <a id="3507" href="Relation.Binary.Construct.NonStrictToStrict.html#3481" class="Bound">x≤y</a><a id="3510" class="Symbol">)</a> <a id="3512" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3514" class="Symbol">λ</a> <a id="3516" href="Relation.Binary.Construct.NonStrictToStrict.html#3516" class="Bound">x≈z</a> <a id="3520" class="Symbol">→</a> <a id="3522" href="Relation.Binary.Construct.NonStrictToStrict.html#3487" class="Bound">x≉y</a> <a id="3526" class="Symbol">(</a><a id="3527" href="Relation.Binary.Construct.NonStrictToStrict.html#3464" class="Bound">trans</a> <a id="3533" href="Relation.Binary.Construct.NonStrictToStrict.html#3516" class="Bound">x≈z</a> <a id="3537" class="Symbol">(</a><a id="3538" href="Relation.Binary.Construct.NonStrictToStrict.html#3460" class="Bound">sym</a> <a id="3542" href="Relation.Binary.Construct.NonStrictToStrict.html#3476" class="Bound">y≈z</a><a id="3545" class="Symbol">))</a>
|
|||
|
|
|
|||
|
|
<a id="<-resp-≈"></a><a id="3549" href="Relation.Binary.Construct.NonStrictToStrict.html#3549" class="Function"><-resp-≈</a> <a id="3558" class="Symbol">:</a> <a id="3560" href="Relation.Binary.Structures.html#1550" class="Record">IsEquivalence</a> <a id="3574" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3578" class="Symbol">→</a> <a id="3580" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3584" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="3594" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3598" class="Symbol">→</a> <a id="3600" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a> <a id="3604" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="3614" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a>
|
|||
|
|
<a id="3618" href="Relation.Binary.Construct.NonStrictToStrict.html#3549" class="Function"><-resp-≈</a> <a id="3627" href="Relation.Binary.Construct.NonStrictToStrict.html#3627" class="Bound">eq</a> <a id="3630" class="Symbol">(</a><a id="3631" href="Relation.Binary.Construct.NonStrictToStrict.html#3631" class="Bound">respʳ</a> <a id="3637" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3639" href="Relation.Binary.Construct.NonStrictToStrict.html#3639" class="Bound">respˡ</a><a id="3644" class="Symbol">)</a> <a id="3646" class="Symbol">=</a>
|
|||
|
|
<a id="3650" href="Relation.Binary.Construct.NonStrictToStrict.html#3355" class="Function"><-respʳ-≈</a> <a id="3660" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="3664" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="3670" href="Relation.Binary.Construct.NonStrictToStrict.html#3631" class="Bound">respʳ</a> <a id="3676" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="3678" href="Relation.Binary.Construct.NonStrictToStrict.html#3213" class="Function"><-respˡ-≈</a> <a id="3688" href="Relation.Binary.Structures.html#1648" class="Function">trans</a> <a id="3694" href="Relation.Binary.Construct.NonStrictToStrict.html#3639" class="Bound">respˡ</a>
|
|||
|
|
<a id="3702" class="Keyword">where</a> <a id="3708" class="Keyword">open</a> <a id="3713" href="Relation.Binary.Structures.html#1550" class="Module">IsEquivalence</a> <a id="3727" href="Relation.Binary.Construct.NonStrictToStrict.html#3627" class="Bound">eq</a>
|
|||
|
|
|
|||
|
|
<a id="<-trichotomous"></a><a id="3731" href="Relation.Binary.Construct.NonStrictToStrict.html#3731" class="Function"><-trichotomous</a> <a id="3746" class="Symbol">:</a> <a id="3748" href="Relation.Binary.Definitions.html#1587" class="Function">Symmetric</a> <a id="3758" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3762" class="Symbol">→</a> <a id="3764" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="3774" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3778" class="Symbol">→</a>
|
|||
|
|
<a id="3797" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="3811" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3815" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3819" class="Symbol">→</a> <a id="3821" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="3827" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="3831" class="Symbol">→</a>
|
|||
|
|
<a id="3850" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="3863" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="3867" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="3871" href="Relation.Binary.Construct.NonStrictToStrict.html#3731" class="Function"><-trichotomous</a> <a id="3886" href="Relation.Binary.Construct.NonStrictToStrict.html#3886" class="Bound">≈-sym</a> <a id="3892" href="Relation.Binary.Construct.NonStrictToStrict.html#3892" class="Bound Operator">_≟_</a> <a id="3896" href="Relation.Binary.Construct.NonStrictToStrict.html#3896" class="Bound">antisym</a> <a id="3904" href="Relation.Binary.Construct.NonStrictToStrict.html#3904" class="Bound">total</a> <a id="3910" href="Relation.Binary.Construct.NonStrictToStrict.html#3910" class="Bound">x</a> <a id="3912" href="Relation.Binary.Construct.NonStrictToStrict.html#3912" class="Bound">y</a> <a id="3914" class="Keyword">with</a> <a id="3919" href="Relation.Binary.Construct.NonStrictToStrict.html#3910" class="Bound">x</a> <a id="3921" href="Relation.Binary.Construct.NonStrictToStrict.html#3892" class="Bound Operator">≟</a> <a id="3923" href="Relation.Binary.Construct.NonStrictToStrict.html#3912" class="Bound">y</a>
|
|||
|
|
<a id="3925" class="Symbol">...</a> <a id="3929" class="Symbol">|</a> <a id="3931" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="3935" href="Relation.Binary.Construct.NonStrictToStrict.html#3935" class="Bound">x≈y</a> <a id="3939" class="Symbol">=</a> <a id="3941" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a> <a id="3946" class="Symbol">(</a><a id="3947" href="Relation.Binary.Construct.NonStrictToStrict.html#2396" class="Function"><-irrefl</a> <a id="3956" href="Relation.Binary.Construct.NonStrictToStrict.html#3935" class="Bound">x≈y</a><a id="3959" class="Symbol">)</a> <a id="3961" href="Relation.Binary.Construct.NonStrictToStrict.html#3935" class="Bound">x≈y</a> <a id="3965" class="Symbol">(</a><a id="3966" href="Relation.Binary.Construct.NonStrictToStrict.html#2396" class="Function"><-irrefl</a> <a id="3975" class="Symbol">(</a><a id="3976" class="Bound">≈-sym</a> <a id="3982" href="Relation.Binary.Construct.NonStrictToStrict.html#3935" class="Bound">x≈y</a><a id="3985" class="Symbol">))</a>
|
|||
|
|
<a id="3988" class="Symbol">...</a> <a id="3992" class="Symbol">|</a> <a id="3994" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="3998" href="Relation.Binary.Construct.NonStrictToStrict.html#3998" class="Bound">x≉y</a> <a id="4002" class="Keyword">with</a> <a id="4007" class="Bound">total</a> <a id="4013" class="Bound">x</a> <a id="4015" class="Bound">y</a>
|
|||
|
|
<a id="4017" class="Symbol">...</a> <a id="4023" class="Symbol">|</a> <a id="4025" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="4030" href="Relation.Binary.Construct.NonStrictToStrict.html#4030" class="Bound">x≤y</a> <a id="4034" class="Symbol">=</a> <a id="4036" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri<</a> <a id="4041" class="Symbol">(</a><a id="4042" href="Relation.Binary.Construct.NonStrictToStrict.html#4030" class="Bound">x≤y</a> <a id="4046" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4048" class="Bound">x≉y</a><a id="4051" class="Symbol">)</a> <a id="4053" class="Bound">x≉y</a> <a id="4057" class="Symbol">(</a><a id="4058" class="Bound">x≉y</a> <a id="4062" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4064" class="Bound">antisym</a> <a id="4072" href="Relation.Binary.Construct.NonStrictToStrict.html#4030" class="Bound">x≤y</a> <a id="4076" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4078" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="4083" class="Symbol">)</a>
|
|||
|
|
<a id="4085" class="Symbol">...</a> <a id="4091" class="Symbol">|</a> <a id="4093" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="4098" href="Relation.Binary.Construct.NonStrictToStrict.html#4098" class="Bound">y≤x</a> <a id="4102" class="Symbol">=</a> <a id="4104" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri></a> <a id="4109" class="Symbol">(</a><a id="4110" class="Bound">x≉y</a> <a id="4114" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4116" href="Function.Base.html#1638" class="Function">flip</a> <a id="4121" class="Bound">antisym</a> <a id="4129" href="Relation.Binary.Construct.NonStrictToStrict.html#4098" class="Bound">y≤x</a> <a id="4133" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4135" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="4140" class="Symbol">)</a> <a id="4142" class="Bound">x≉y</a> <a id="4146" class="Symbol">(</a><a id="4147" href="Relation.Binary.Construct.NonStrictToStrict.html#4098" class="Bound">y≤x</a> <a id="4151" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4153" class="Bound">x≉y</a> <a id="4157" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="4159" class="Bound">≈-sym</a><a id="4164" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="<-decidable"></a><a id="4167" href="Relation.Binary.Construct.NonStrictToStrict.html#4167" class="Function"><-decidable</a> <a id="4179" class="Symbol">:</a> <a id="4181" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="4191" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="4195" class="Symbol">→</a> <a id="4197" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="4207" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="4211" class="Symbol">→</a> <a id="4213" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="4223" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="4227" href="Relation.Binary.Construct.NonStrictToStrict.html#4167" class="Function"><-decidable</a> <a id="4239" href="Relation.Binary.Construct.NonStrictToStrict.html#4239" class="Bound Operator">_≟_</a> <a id="4243" href="Relation.Binary.Construct.NonStrictToStrict.html#4243" class="Bound Operator">_≤?_</a> <a id="4248" href="Relation.Binary.Construct.NonStrictToStrict.html#4248" class="Bound">x</a> <a id="4250" href="Relation.Binary.Construct.NonStrictToStrict.html#4250" class="Bound">y</a> <a id="4252" class="Symbol">=</a> <a id="4254" href="Relation.Binary.Construct.NonStrictToStrict.html#4248" class="Bound">x</a> <a id="4256" href="Relation.Binary.Construct.NonStrictToStrict.html#4243" class="Bound Operator">≤?</a> <a id="4259" href="Relation.Binary.Construct.NonStrictToStrict.html#4250" class="Bound">y</a> <a id="4261" href="Relation.Nullary.Decidable.Core.html#2609" class="Function Operator">×-dec</a> <a id="4267" href="Relation.Nullary.Decidable.Core.html#2517" class="Function">¬?</a> <a id="4270" class="Symbol">(</a><a id="4271" href="Relation.Binary.Construct.NonStrictToStrict.html#4248" class="Bound">x</a> <a id="4273" href="Relation.Binary.Construct.NonStrictToStrict.html#4239" class="Bound Operator">≟</a> <a id="4275" href="Relation.Binary.Construct.NonStrictToStrict.html#4250" class="Bound">y</a><a id="4276" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="4279" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="4352" class="Comment">-- Structures</a>
|
|||
|
|
|
|||
|
|
<a id="<-isStrictPartialOrder"></a><a id="4367" href="Relation.Binary.Construct.NonStrictToStrict.html#4367" class="Function"><-isStrictPartialOrder</a> <a id="4390" class="Symbol">:</a> <a id="4392" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="4407" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="4411" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="4415" class="Symbol">→</a>
|
|||
|
|
<a id="4442" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="4463" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="4467" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="4471" href="Relation.Binary.Construct.NonStrictToStrict.html#4367" class="Function"><-isStrictPartialOrder</a> <a id="4494" href="Relation.Binary.Construct.NonStrictToStrict.html#4494" class="Bound">po</a> <a id="4497" class="Symbol">=</a> <a id="4499" class="Keyword">record</a>
|
|||
|
|
<a id="4508" class="Symbol">{</a> <a id="4510" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="4524" class="Symbol">=</a> <a id="4526" href="Relation.Binary.Structures.html#2256" class="Function">isEquivalence</a>
|
|||
|
|
<a id="4542" class="Symbol">;</a> <a id="4544" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="4558" class="Symbol">=</a> <a id="4560" href="Relation.Binary.Construct.NonStrictToStrict.html#2396" class="Function"><-irrefl</a>
|
|||
|
|
<a id="4571" class="Symbol">;</a> <a id="4573" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="4587" class="Symbol">=</a> <a id="4589" href="Relation.Binary.Construct.NonStrictToStrict.html#2461" class="Function"><-trans</a> <a id="4597" href="Relation.Binary.Construct.NonStrictToStrict.html#4494" class="Bound">po</a>
|
|||
|
|
<a id="4602" class="Symbol">;</a> <a id="4604" href="Relation.Binary.Structures.html#4482" class="Field"><-resp-≈</a> <a id="4618" class="Symbol">=</a> <a id="4620" href="Relation.Binary.Construct.NonStrictToStrict.html#3549" class="Function"><-resp-≈</a> <a id="4629" href="Relation.Binary.Structures.html#2256" class="Function">isEquivalence</a> <a id="4643" href="Relation.Binary.Structures.html#3785" class="Function">≤-resp-≈</a>
|
|||
|
|
<a id="4654" class="Symbol">}</a> <a id="4656" class="Keyword">where</a> <a id="4662" class="Keyword">open</a> <a id="4667" href="Relation.Binary.Structures.html#3522" class="Module">IsPartialOrder</a> <a id="4682" href="Relation.Binary.Construct.NonStrictToStrict.html#4494" class="Bound">po</a>
|
|||
|
|
|
|||
|
|
<a id="<-isDecStrictPartialOrder"></a><a id="4686" href="Relation.Binary.Construct.NonStrictToStrict.html#4686" class="Function"><-isDecStrictPartialOrder</a> <a id="4712" class="Symbol">:</a> <a id="4714" href="Relation.Binary.Structures.html#3809" class="Record">IsDecPartialOrder</a> <a id="4732" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="4736" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="4740" class="Symbol">→</a>
|
|||
|
|
<a id="4770" href="Relation.Binary.Structures.html#4782" class="Record">IsDecStrictPartialOrder</a> <a id="4794" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="4798" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="4802" href="Relation.Binary.Construct.NonStrictToStrict.html#4686" class="Function"><-isDecStrictPartialOrder</a> <a id="4828" href="Relation.Binary.Construct.NonStrictToStrict.html#4828" class="Bound">dpo</a> <a id="4832" class="Symbol">=</a> <a id="4834" class="Keyword">record</a>
|
|||
|
|
<a id="4843" class="Symbol">{</a> <a id="4845" href="Relation.Binary.Structures.html#4879" class="Field">isStrictPartialOrder</a> <a id="4866" class="Symbol">=</a> <a id="4868" href="Relation.Binary.Construct.NonStrictToStrict.html#4367" class="Function"><-isStrictPartialOrder</a> <a id="4891" href="Relation.Binary.Structures.html#3900" class="Field">isPartialOrder</a>
|
|||
|
|
<a id="4908" class="Symbol">;</a> <a id="4910" href="Relation.Binary.Structures.html#4931" class="Field Operator">_≟_</a> <a id="4914" class="Symbol">=</a> <a id="4916" href="Relation.Binary.Structures.html#3940" class="Field Operator">_≟_</a>
|
|||
|
|
<a id="4922" class="Symbol">;</a> <a id="4924" href="Relation.Binary.Structures.html#4972" class="Field Operator">_<?_</a> <a id="4929" class="Symbol">=</a> <a id="4931" href="Relation.Binary.Construct.NonStrictToStrict.html#4167" class="Function"><-decidable</a> <a id="4943" href="Relation.Binary.Structures.html#3940" class="Field Operator">_≟_</a> <a id="4947" href="Relation.Binary.Structures.html#3975" class="Field Operator">_≤?_</a>
|
|||
|
|
<a id="4954" class="Symbol">}</a> <a id="4956" class="Keyword">where</a> <a id="4962" class="Keyword">open</a> <a id="4967" href="Relation.Binary.Structures.html#3809" class="Module">IsDecPartialOrder</a> <a id="4985" href="Relation.Binary.Construct.NonStrictToStrict.html#4828" class="Bound">dpo</a>
|
|||
|
|
|
|||
|
|
<a id="<-isStrictTotalOrder₁"></a><a id="4990" href="Relation.Binary.Construct.NonStrictToStrict.html#4990" class="Function"><-isStrictTotalOrder₁</a> <a id="5012" class="Symbol">:</a> <a id="5014" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="5024" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="5028" class="Symbol">→</a> <a id="5030" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="5043" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="5047" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="5051" class="Symbol">→</a>
|
|||
|
|
<a id="5077" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="5096" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="5100" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="5104" href="Relation.Binary.Construct.NonStrictToStrict.html#4990" class="Function"><-isStrictTotalOrder₁</a> <a id="5126" href="Relation.Binary.Construct.NonStrictToStrict.html#5126" class="Bound">≟</a> <a id="5128" href="Relation.Binary.Construct.NonStrictToStrict.html#5128" class="Bound">tot</a> <a id="5132" class="Symbol">=</a> <a id="5134" class="Keyword">record</a>
|
|||
|
|
<a id="5143" class="Symbol">{</a> <a id="5145" href="Relation.Binary.Structures.html#6805" class="Field">isStrictPartialOrder</a> <a id="5166" class="Symbol">=</a> <a id="5168" href="Relation.Binary.Construct.NonStrictToStrict.html#4367" class="Function"><-isStrictPartialOrder</a> <a id="5191" href="Relation.Binary.Structures.html#5575" class="Field">isPartialOrder</a>
|
|||
|
|
<a id="5208" class="Symbol">;</a> <a id="5210" href="Relation.Binary.Structures.html#6857" class="Field">compare</a> <a id="5231" class="Symbol">=</a> <a id="5233" href="Relation.Binary.Construct.NonStrictToStrict.html#3731" class="Function"><-trichotomous</a> <a id="5248" href="Relation.Binary.Structures.html#1622" class="Function">Eq.sym</a> <a id="5255" href="Relation.Binary.Construct.NonStrictToStrict.html#5126" class="Bound">≟</a> <a id="5257" href="Relation.Binary.Structures.html#3623" class="Function">antisym</a> <a id="5265" href="Relation.Binary.Structures.html#5615" class="Field">total</a>
|
|||
|
|
<a id="5273" class="Symbol">}</a> <a id="5275" class="Keyword">where</a> <a id="5281" class="Keyword">open</a> <a id="5286" href="Relation.Binary.Structures.html#5508" class="Module">IsTotalOrder</a> <a id="5299" href="Relation.Binary.Construct.NonStrictToStrict.html#5128" class="Bound">tot</a>
|
|||
|
|
|
|||
|
|
<a id="<-isStrictTotalOrder₂"></a><a id="5304" href="Relation.Binary.Construct.NonStrictToStrict.html#5304" class="Function"><-isStrictTotalOrder₂</a> <a id="5326" class="Symbol">:</a> <a id="5328" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="5344" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="5348" href="Relation.Binary.Construct.NonStrictToStrict.html#801" class="Bound Operator">_≤_</a> <a id="5352" class="Symbol">→</a>
|
|||
|
|
<a id="5378" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="5397" href="Relation.Binary.Construct.NonStrictToStrict.html#784" class="Bound Operator">_≈_</a> <a id="5401" href="Relation.Binary.Construct.NonStrictToStrict.html#1319" class="Function Operator">_<_</a>
|
|||
|
|
<a id="5405" href="Relation.Binary.Construct.NonStrictToStrict.html#5304" class="Function"><-isStrictTotalOrder₂</a> <a id="5427" href="Relation.Binary.Construct.NonStrictToStrict.html#5427" class="Bound">dtot</a> <a id="5432" class="Symbol">=</a> <a id="5434" href="Relation.Binary.Construct.NonStrictToStrict.html#4990" class="Function"><-isStrictTotalOrder₁</a> <a id="5456" href="Relation.Binary.Structures.html#5950" class="Field Operator">_≟_</a> <a id="5460" href="Relation.Binary.Structures.html#5914" class="Field">isTotalOrder</a>
|
|||
|
|
<a id="5475" class="Keyword">where</a> <a id="5481" class="Keyword">open</a> <a id="5486" href="Relation.Binary.Structures.html#5825" class="Module">IsDecTotalOrder</a> <a id="5502" href="Relation.Binary.Construct.NonStrictToStrict.html#5427" class="Bound">dtot</a>
|
|||
|
|
</pre></body></html>
|