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143 lines
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29 KiB
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<html><head><meta charset="utf-8"><title>Relation.Binary.Properties.Poset</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">-- Properties satisfied by posets</a>
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<a id="140" class="Comment">------------------------------------------------------------------------</a>
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<a id="214" class="Symbol">{-#</a> <a id="218" class="Keyword">OPTIONS</a> <a id="226" class="Pragma">--cubical-compatible</a> <a id="247" class="Pragma">--safe</a> <a id="254" class="Symbol">#-}</a>
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<a id="259" class="Keyword">open</a> <a id="264" class="Keyword">import</a> <a id="271" href="Function.Base.html" class="Module">Function.Base</a> <a id="285" class="Keyword">using</a> <a id="291" class="Symbol">(</a><a id="292" href="Function.Base.html#1563" class="Function">flip</a><a id="296" class="Symbol">;</a> <a id="298" href="Function.Base.html#1040" class="Function Operator">_∘_</a><a id="301" class="Symbol">)</a>
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<a id="303" class="Keyword">open</a> <a id="308" class="Keyword">import</a> <a id="315" href="Relation.Binary.html" class="Module">Relation.Binary</a>
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<a id="331" class="Keyword">import</a> <a id="338" href="Relation.Binary.Consequences.html" class="Module">Relation.Binary.Consequences</a> <a id="367" class="Symbol">as</a> <a id="370" class="Module">Consequences</a>
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<a id="383" class="Keyword">open</a> <a id="388" class="Keyword">import</a> <a id="395" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="412" class="Keyword">using</a> <a id="418" class="Symbol">(</a><a id="419" href="Relation.Nullary.html#665" class="Function Operator">¬_</a><a id="421" class="Symbol">)</a>
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<a id="424" class="Keyword">module</a> <a id="431" href="Relation.Binary.Properties.Poset.html" class="Module">Relation.Binary.Properties.Poset</a>
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<a id="467" class="Symbol">{</a><a id="468" href="Relation.Binary.Properties.Poset.html#468" class="Bound">p₁</a> <a id="471" href="Relation.Binary.Properties.Poset.html#471" class="Bound">p₂</a> <a id="474" href="Relation.Binary.Properties.Poset.html#474" class="Bound">p₃</a><a id="476" class="Symbol">}</a> <a id="478" class="Symbol">(</a><a id="479" href="Relation.Binary.Properties.Poset.html#479" class="Bound">P</a> <a id="481" class="Symbol">:</a> <a id="483" href="Relation.Binary.Bundles.html#3037" class="Record">Poset</a> <a id="489" href="Relation.Binary.Properties.Poset.html#468" class="Bound">p₁</a> <a id="492" href="Relation.Binary.Properties.Poset.html#471" class="Bound">p₂</a> <a id="495" href="Relation.Binary.Properties.Poset.html#474" class="Bound">p₃</a><a id="497" class="Symbol">)</a> <a id="499" class="Keyword">where</a>
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<a id="506" class="Keyword">open</a> <a id="511" href="Relation.Binary.Bundles.html#3037" class="Module">Poset</a> <a id="517" href="Relation.Binary.Properties.Poset.html#479" class="Bound">P</a> <a id="519" class="Keyword">renaming</a> <a id="528" class="Symbol">(</a><a id="529" href="Relation.Binary.Bundles.html#3113" class="Field">Carrier</a> <a id="537" class="Symbol">to</a> <a id="540" class="Field">A</a><a id="541" class="Symbol">)</a>
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<a id="544" class="Keyword">import</a> <a id="551" href="Relation.Binary.Construct.NonStrictToStrict.html" class="Module">Relation.Binary.Construct.NonStrictToStrict</a> <a id="595" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="599" href="Relation.Binary.Bundles.html#3176" class="Field Operator">_≤_</a> as <a id="ToStrict"></a><a id="606" href="Relation.Binary.Properties.Poset.html#606" class="Module">ToStrict</a>
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<a id="615" class="Keyword">import</a> <a id="622" href="Relation.Binary.Properties.Preorder.html" class="Module">Relation.Binary.Properties.Preorder</a> <a id="658" href="Relation.Binary.Bundles.html#3323" class="Function">preorder</a> as <a id="PreorderProperties"></a><a id="670" href="Relation.Binary.Properties.Poset.html#670" class="Module">PreorderProperties</a>
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<a id="689" class="Keyword">open</a> <a id="694" href="Relation.Binary.Bundles.html#2244" class="Module">Eq</a> <a id="697" class="Keyword">using</a> <a id="703" class="Symbol">(</a><a id="704" href="Relation.Binary.Bundles.html#969" class="Function Operator">_≉_</a><a id="707" class="Symbol">)</a>
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<a id="710" class="Comment">------------------------------------------------------------------------</a>
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<a id="783" class="Comment">-- The _≥_ relation is also a poset.</a>
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<a id="821" class="Keyword">infix</a> <a id="827" class="Number">4</a> <a id="829" href="Relation.Binary.Properties.Poset.html#834" class="Function Operator">_≥_</a>
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<a id="_≥_"></a><a id="834" href="Relation.Binary.Properties.Poset.html#834" class="Function Operator">_≥_</a> <a id="838" class="Symbol">:</a> <a id="840" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="844" href="Relation.Binary.Properties.Poset.html#540" class="Field">A</a> <a id="846" href="Relation.Binary.Properties.Poset.html#474" class="Bound">p₃</a>
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<a id="849" href="Relation.Binary.Properties.Poset.html#849" class="Bound">x</a> <a id="851" href="Relation.Binary.Properties.Poset.html#834" class="Function Operator">≥</a> <a id="853" href="Relation.Binary.Properties.Poset.html#853" class="Bound">y</a> <a id="855" class="Symbol">=</a> <a id="857" href="Relation.Binary.Properties.Poset.html#853" class="Bound">y</a> <a id="859" href="Relation.Binary.Bundles.html#3176" class="Field Operator">≤</a> <a id="861" href="Relation.Binary.Properties.Poset.html#849" class="Bound">x</a>
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<a id="864" class="Keyword">open</a> <a id="869" href="Relation.Binary.Properties.Poset.html#670" class="Module">PreorderProperties</a> <a id="888" class="Keyword">public</a>
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<a id="897" class="Keyword">using</a> <a id="903" class="Symbol">()</a>
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<a id="908" class="Keyword">renaming</a>
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<a id="919" class="Symbol">(</a> <a id="921" href="Relation.Binary.Properties.Preorder.html#568" class="Function">invIsPreorder</a> <a id="935" class="Symbol">to</a> <a id="938" class="Function">≥-isPreorder</a>
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<a id="953" class="Symbol">;</a> <a id="955" href="Relation.Binary.Properties.Preorder.html#742" class="Function">invPreorder</a> <a id="969" class="Symbol">to</a> <a id="972" class="Function">≥-preorder</a>
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<a id="985" class="Symbol">)</a>
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<a id="≥-isPartialOrder"></a><a id="988" href="Relation.Binary.Properties.Poset.html#988" class="Function">≥-isPartialOrder</a> <a id="1005" class="Symbol">:</a> <a id="1007" href="Relation.Binary.Structures.html#3183" class="Record">IsPartialOrder</a> <a id="1022" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="1026" href="Relation.Binary.Properties.Poset.html#834" class="Function Operator">_≥_</a>
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<a id="1030" href="Relation.Binary.Properties.Poset.html#988" class="Function">≥-isPartialOrder</a> <a id="1047" class="Symbol">=</a> <a id="1049" class="Keyword">record</a>
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<a id="1058" class="Symbol">{</a> <a id="1060" href="Relation.Binary.Structures.html#3252" class="Field">isPreorder</a> <a id="1073" class="Symbol">=</a> <a id="1075" href="Relation.Binary.Properties.Preorder.html#568" class="Function">PreorderProperties.invIsPreorder</a>
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<a id="1110" class="Symbol">;</a> <a id="1112" href="Relation.Binary.Structures.html#3284" class="Field">antisym</a> <a id="1125" class="Symbol">=</a> <a id="1127" href="Function.Base.html#1563" class="Function">flip</a> <a id="1132" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a>
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<a id="1142" class="Symbol">}</a>
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<a id="≥-poset"></a><a id="1145" href="Relation.Binary.Properties.Poset.html#1145" class="Function">≥-poset</a> <a id="1153" class="Symbol">:</a> <a id="1155" href="Relation.Binary.Bundles.html#3037" class="Record">Poset</a> <a id="1161" href="Relation.Binary.Properties.Poset.html#468" class="Bound">p₁</a> <a id="1164" href="Relation.Binary.Properties.Poset.html#471" class="Bound">p₂</a> <a id="1167" href="Relation.Binary.Properties.Poset.html#474" class="Bound">p₃</a>
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<a id="1170" href="Relation.Binary.Properties.Poset.html#1145" class="Function">≥-poset</a> <a id="1178" class="Symbol">=</a> <a id="1180" class="Keyword">record</a>
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<a id="1189" class="Symbol">{</a> <a id="1191" href="Relation.Binary.Bundles.html#3212" class="Field">isPartialOrder</a> <a id="1206" class="Symbol">=</a> <a id="1208" href="Relation.Binary.Properties.Poset.html#988" class="Function">≥-isPartialOrder</a>
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<a id="1227" class="Symbol">}</a>
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<a id="1230" class="Keyword">open</a> <a id="1235" href="Relation.Binary.Bundles.html#3037" class="Module">Poset</a> <a id="1241" href="Relation.Binary.Properties.Poset.html#1145" class="Function">≥-poset</a> <a id="1249" class="Keyword">public</a>
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<a id="1258" class="Keyword">using</a> <a id="1264" class="Symbol">()</a>
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<a id="1269" class="Keyword">renaming</a>
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<a id="1280" class="Symbol">(</a> <a id="1282" href="Relation.Binary.Structures.html#2447" class="Function">refl</a> <a id="1292" class="Symbol">to</a> <a id="1295" class="Function">≥-refl</a>
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<a id="1304" class="Symbol">;</a> <a id="1306" href="Relation.Binary.Structures.html#2340" class="Function">reflexive</a> <a id="1316" class="Symbol">to</a> <a id="1319" class="Function">≥-reflexive</a>
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<a id="1333" class="Symbol">;</a> <a id="1335" href="Relation.Binary.Structures.html#2370" class="Function">trans</a> <a id="1345" class="Symbol">to</a> <a id="1348" class="Function">≥-trans</a>
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<a id="1358" class="Symbol">;</a> <a id="1360" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a> <a id="1370" class="Symbol">to</a> <a id="1373" class="Function">≥-antisym</a>
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<a id="1385" class="Symbol">)</a>
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<a id="1388" class="Comment">------------------------------------------------------------------------</a>
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<a id="1461" class="Comment">-- Negated order</a>
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<a id="1479" class="Keyword">infix</a> <a id="1485" class="Number">4</a> <a id="1487" href="Relation.Binary.Properties.Poset.html#1492" class="Function Operator">_≰_</a>
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<a id="_≰_"></a><a id="1492" href="Relation.Binary.Properties.Poset.html#1492" class="Function Operator">_≰_</a> <a id="1496" class="Symbol">:</a> <a id="1498" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1502" href="Relation.Binary.Properties.Poset.html#540" class="Field">A</a> <a id="1504" href="Relation.Binary.Properties.Poset.html#474" class="Bound">p₃</a>
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<a id="1507" href="Relation.Binary.Properties.Poset.html#1507" class="Bound">x</a> <a id="1509" href="Relation.Binary.Properties.Poset.html#1492" class="Function Operator">≰</a> <a id="1511" href="Relation.Binary.Properties.Poset.html#1511" class="Bound">y</a> <a id="1513" class="Symbol">=</a> <a id="1515" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="1517" class="Symbol">(</a><a id="1518" href="Relation.Binary.Properties.Poset.html#1507" class="Bound">x</a> <a id="1520" href="Relation.Binary.Bundles.html#3176" class="Field Operator">≤</a> <a id="1522" href="Relation.Binary.Properties.Poset.html#1511" class="Bound">y</a><a id="1523" class="Symbol">)</a>
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<a id="≰-respˡ-≈"></a><a id="1526" href="Relation.Binary.Properties.Poset.html#1526" class="Function">≰-respˡ-≈</a> <a id="1536" class="Symbol">:</a> <a id="1538" href="Relation.Binary.Properties.Poset.html#1492" class="Function Operator">_≰_</a> <a id="1542" href="Relation.Binary.Definitions.html#3923" class="Function Operator">Respectsˡ</a> <a id="1552" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a>
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<a id="1556" href="Relation.Binary.Properties.Poset.html#1526" class="Function">≰-respˡ-≈</a> <a id="1566" href="Relation.Binary.Properties.Poset.html#1566" class="Bound">x≈y</a> <a id="1570" class="Symbol">=</a> <a id="1572" href="Function.Base.html#1040" class="Function Operator">_∘</a> <a id="1575" href="Relation.Binary.Structures.html#3388" class="Function">≤-respˡ-≈</a> <a id="1585" class="Symbol">(</a><a id="1586" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a> <a id="1593" href="Relation.Binary.Properties.Poset.html#1566" class="Bound">x≈y</a><a id="1596" class="Symbol">)</a>
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<a id="≰-respʳ-≈"></a><a id="1599" href="Relation.Binary.Properties.Poset.html#1599" class="Function">≰-respʳ-≈</a> <a id="1609" class="Symbol">:</a> <a id="1611" href="Relation.Binary.Properties.Poset.html#1492" class="Function Operator">_≰_</a> <a id="1615" href="Relation.Binary.Definitions.html#3758" class="Function Operator">Respectsʳ</a> <a id="1625" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a>
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<a id="1629" href="Relation.Binary.Properties.Poset.html#1599" class="Function">≰-respʳ-≈</a> <a id="1639" href="Relation.Binary.Properties.Poset.html#1639" class="Bound">x≈y</a> <a id="1643" class="Symbol">=</a> <a id="1645" href="Function.Base.html#1040" class="Function Operator">_∘</a> <a id="1648" href="Relation.Binary.Structures.html#3417" class="Function">≤-respʳ-≈</a> <a id="1658" class="Symbol">(</a><a id="1659" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a> <a id="1666" href="Relation.Binary.Properties.Poset.html#1639" class="Bound">x≈y</a><a id="1669" class="Symbol">)</a>
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<a id="1672" class="Comment">------------------------------------------------------------------------</a>
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<a id="1745" class="Comment">-- Partial orders can be turned into strict partial orders</a>
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<a id="1805" class="Keyword">infix</a> <a id="1811" class="Number">4</a> <a id="1813" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator">_<_</a>
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<a id="_<_"></a><a id="1818" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator">_<_</a> <a id="1822" class="Symbol">:</a> <a id="1824" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="1828" href="Relation.Binary.Properties.Poset.html#540" class="Field">A</a> <a id="1830" class="Symbol">_</a>
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<a id="1832" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator">_<_</a> <a id="1836" class="Symbol">=</a> <a id="1838" href="Relation.Binary.Construct.NonStrictToStrict.html#886" class="Function Operator">ToStrict._<_</a>
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<a id="<-isStrictPartialOrder"></a><a id="1852" href="Relation.Binary.Properties.Poset.html#1852" class="Function"><-isStrictPartialOrder</a> <a id="1875" class="Symbol">:</a> <a id="1877" href="Relation.Binary.Structures.html#3959" class="Record">IsStrictPartialOrder</a> <a id="1898" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="1902" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator">_<_</a>
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<a id="1906" href="Relation.Binary.Properties.Poset.html#1852" class="Function"><-isStrictPartialOrder</a> <a id="1929" class="Symbol">=</a> <a id="1931" href="Relation.Binary.Construct.NonStrictToStrict.html#3934" class="Function">ToStrict.<-isStrictPartialOrder</a> <a id="1963" href="Relation.Binary.Bundles.html#3212" class="Field">isPartialOrder</a>
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<a id="<-strictPartialOrder"></a><a id="1979" href="Relation.Binary.Properties.Poset.html#1979" class="Function"><-strictPartialOrder</a> <a id="2000" class="Symbol">:</a> <a id="2002" href="Relation.Binary.Bundles.html#4117" class="Record">StrictPartialOrder</a> <a id="2021" class="Symbol">_</a> <a id="2023" class="Symbol">_</a> <a id="2025" class="Symbol">_</a>
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<a id="2027" href="Relation.Binary.Properties.Poset.html#1979" class="Function"><-strictPartialOrder</a> <a id="2048" class="Symbol">=</a> <a id="2050" class="Keyword">record</a>
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<a id="2059" class="Symbol">{</a> <a id="2061" href="Relation.Binary.Bundles.html#4323" class="Field">isStrictPartialOrder</a> <a id="2082" class="Symbol">=</a> <a id="2084" href="Relation.Binary.Properties.Poset.html#1852" class="Function"><-isStrictPartialOrder</a>
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<a id="2109" class="Symbol">}</a>
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<a id="2112" class="Keyword">open</a> <a id="2117" href="Relation.Binary.Bundles.html#4117" class="Module">StrictPartialOrder</a> <a id="2136" href="Relation.Binary.Properties.Poset.html#1979" class="Function"><-strictPartialOrder</a> <a id="2157" class="Keyword">public</a>
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<a id="2166" class="Keyword">using</a> <a id="2172" class="Symbol">(</a> <a id="2174" href="Relation.Binary.Structures.html#4143" class="Function"><-resp-≈</a><a id="2182" class="Symbol">;</a> <a id="2184" href="Relation.Binary.Structures.html#4313" class="Function"><-respʳ-≈</a><a id="2193" class="Symbol">;</a> <a id="2195" href="Relation.Binary.Structures.html#4375" class="Function"><-respˡ-≈</a><a id="2204" class="Symbol">)</a>
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<a id="2208" class="Keyword">renaming</a>
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<a id="2219" class="Symbol">(</a> <a id="2221" href="Relation.Binary.Structures.html#4068" class="Function">irrefl</a> <a id="2228" class="Symbol">to</a> <a id="2231" class="Function"><-irrefl</a>
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<a id="2242" class="Symbol">;</a> <a id="2244" href="Relation.Binary.Structures.html#4223" class="Function">asym</a> <a id="2251" class="Symbol">to</a> <a id="2254" class="Function"><-asym</a>
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<a id="2263" class="Symbol">;</a> <a id="2265" href="Relation.Binary.Structures.html#4108" class="Function">trans</a> <a id="2272" class="Symbol">to</a> <a id="2275" class="Function"><-trans</a>
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<a id="2285" class="Symbol">)</a>
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<a id="<⇒≉"></a><a id="2288" href="Relation.Binary.Properties.Poset.html#2288" class="Function"><⇒≉</a> <a id="2292" class="Symbol">:</a> <a id="2294" class="Symbol">∀</a> <a id="2296" class="Symbol">{</a><a id="2297" href="Relation.Binary.Properties.Poset.html#2297" class="Bound">x</a> <a id="2299" href="Relation.Binary.Properties.Poset.html#2299" class="Bound">y</a><a id="2300" class="Symbol">}</a> <a id="2302" class="Symbol">→</a> <a id="2304" href="Relation.Binary.Properties.Poset.html#2297" class="Bound">x</a> <a id="2306" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator"><</a> <a id="2308" href="Relation.Binary.Properties.Poset.html#2299" class="Bound">y</a> <a id="2310" class="Symbol">→</a> <a id="2312" href="Relation.Binary.Properties.Poset.html#2297" class="Bound">x</a> <a id="2314" href="Relation.Binary.Bundles.html#969" class="Function Operator">≉</a> <a id="2316" href="Relation.Binary.Properties.Poset.html#2299" class="Bound">y</a>
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<a id="2318" href="Relation.Binary.Properties.Poset.html#2288" class="Function"><⇒≉</a> <a id="2322" class="Symbol">=</a> <a id="2324" href="Relation.Binary.Construct.NonStrictToStrict.html#1060" class="Function">ToStrict.<⇒≉</a>
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<a id="≤∧≉⇒<"></a><a id="2338" href="Relation.Binary.Properties.Poset.html#2338" class="Function">≤∧≉⇒<</a> <a id="2344" class="Symbol">:</a> <a id="2346" class="Symbol">∀</a> <a id="2348" class="Symbol">{</a><a id="2349" href="Relation.Binary.Properties.Poset.html#2349" class="Bound">x</a> <a id="2351" href="Relation.Binary.Properties.Poset.html#2351" class="Bound">y</a><a id="2352" class="Symbol">}</a> <a id="2354" class="Symbol">→</a> <a id="2356" href="Relation.Binary.Properties.Poset.html#2349" class="Bound">x</a> <a id="2358" href="Relation.Binary.Bundles.html#3176" class="Field Operator">≤</a> <a id="2360" href="Relation.Binary.Properties.Poset.html#2351" class="Bound">y</a> <a id="2362" class="Symbol">→</a> <a id="2364" href="Relation.Binary.Properties.Poset.html#2349" class="Bound">x</a> <a id="2366" href="Relation.Binary.Bundles.html#969" class="Function Operator">≉</a> <a id="2368" href="Relation.Binary.Properties.Poset.html#2351" class="Bound">y</a> <a id="2370" class="Symbol">→</a> <a id="2372" href="Relation.Binary.Properties.Poset.html#2349" class="Bound">x</a> <a id="2374" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator"><</a> <a id="2376" href="Relation.Binary.Properties.Poset.html#2351" class="Bound">y</a>
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<a id="2378" href="Relation.Binary.Properties.Poset.html#2338" class="Function">≤∧≉⇒<</a> <a id="2384" class="Symbol">=</a> <a id="2386" href="Relation.Binary.Construct.NonStrictToStrict.html#1103" class="Function">ToStrict.≤∧≉⇒<</a>
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<a id="<⇒≱"></a><a id="2402" href="Relation.Binary.Properties.Poset.html#2402" class="Function"><⇒≱</a> <a id="2406" class="Symbol">:</a> <a id="2408" class="Symbol">∀</a> <a id="2410" class="Symbol">{</a><a id="2411" href="Relation.Binary.Properties.Poset.html#2411" class="Bound">x</a> <a id="2413" href="Relation.Binary.Properties.Poset.html#2413" class="Bound">y</a><a id="2414" class="Symbol">}</a> <a id="2416" class="Symbol">→</a> <a id="2418" href="Relation.Binary.Properties.Poset.html#2411" class="Bound">x</a> <a id="2420" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator"><</a> <a id="2422" href="Relation.Binary.Properties.Poset.html#2413" class="Bound">y</a> <a id="2424" class="Symbol">→</a> <a id="2426" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="2428" class="Symbol">(</a><a id="2429" href="Relation.Binary.Properties.Poset.html#2413" class="Bound">y</a> <a id="2431" href="Relation.Binary.Bundles.html#3176" class="Field Operator">≤</a> <a id="2433" href="Relation.Binary.Properties.Poset.html#2411" class="Bound">x</a><a id="2434" class="Symbol">)</a>
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<a id="2436" href="Relation.Binary.Properties.Poset.html#2402" class="Function"><⇒≱</a> <a id="2440" class="Symbol">=</a> <a id="2442" href="Relation.Binary.Construct.NonStrictToStrict.html#1156" class="Function">ToStrict.<⇒≱</a> <a id="2455" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a>
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<a id="≤⇒≯"></a><a id="2464" href="Relation.Binary.Properties.Poset.html#2464" class="Function">≤⇒≯</a> <a id="2468" class="Symbol">:</a> <a id="2470" class="Symbol">∀</a> <a id="2472" class="Symbol">{</a><a id="2473" href="Relation.Binary.Properties.Poset.html#2473" class="Bound">x</a> <a id="2475" href="Relation.Binary.Properties.Poset.html#2475" class="Bound">y</a><a id="2476" class="Symbol">}</a> <a id="2478" class="Symbol">→</a> <a id="2480" href="Relation.Binary.Properties.Poset.html#2473" class="Bound">x</a> <a id="2482" href="Relation.Binary.Bundles.html#3176" class="Field Operator">≤</a> <a id="2484" href="Relation.Binary.Properties.Poset.html#2475" class="Bound">y</a> <a id="2486" class="Symbol">→</a> <a id="2488" href="Relation.Nullary.html#665" class="Function Operator">¬</a> <a id="2490" class="Symbol">(</a><a id="2491" href="Relation.Binary.Properties.Poset.html#2475" class="Bound">y</a> <a id="2493" href="Relation.Binary.Properties.Poset.html#1818" class="Function Operator"><</a> <a id="2495" href="Relation.Binary.Properties.Poset.html#2473" class="Bound">x</a><a id="2496" class="Symbol">)</a>
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<a id="2498" href="Relation.Binary.Properties.Poset.html#2464" class="Function">≤⇒≯</a> <a id="2502" class="Symbol">=</a> <a id="2504" href="Relation.Binary.Construct.NonStrictToStrict.html#1267" class="Function">ToStrict.≤⇒≯</a> <a id="2517" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a>
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<a id="2526" class="Comment">------------------------------------------------------------------------</a>
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<a id="2599" class="Comment">-- Other properties</a>
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<a id="mono⇒cong"></a><a id="2620" href="Relation.Binary.Properties.Poset.html#2620" class="Function">mono⇒cong</a> <a id="2630" class="Symbol">:</a> <a id="2632" class="Symbol">∀</a> <a id="2634" class="Symbol">{</a><a id="2635" href="Relation.Binary.Properties.Poset.html#2635" class="Bound">f</a><a id="2636" class="Symbol">}</a> <a id="2638" class="Symbol">→</a> <a id="2640" href="Relation.Binary.Properties.Poset.html#2635" class="Bound">f</a> <a id="2642" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2652" href="Relation.Binary.Bundles.html#3176" class="Field Operator">_≤_</a> <a id="2656" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2658" href="Relation.Binary.Bundles.html#3176" class="Field Operator">_≤_</a> <a id="2662" class="Symbol">→</a> <a id="2664" href="Relation.Binary.Properties.Poset.html#2635" class="Bound">f</a> <a id="2666" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2676" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2680" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2682" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a>
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<a id="2686" href="Relation.Binary.Properties.Poset.html#2620" class="Function">mono⇒cong</a> <a id="2696" class="Symbol">=</a> <a id="2698" href="Relation.Binary.Consequences.html#2266" class="Function">Consequences.mono⇒cong</a> <a id="2721" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2725" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2729" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a> <a id="2736" href="Relation.Binary.Structures.html#2340" class="Function">reflexive</a> <a id="2746" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a>
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<a id="antimono⇒cong"></a><a id="2755" href="Relation.Binary.Properties.Poset.html#2755" class="Function">antimono⇒cong</a> <a id="2769" class="Symbol">:</a> <a id="2771" class="Symbol">∀</a> <a id="2773" class="Symbol">{</a><a id="2774" href="Relation.Binary.Properties.Poset.html#2774" class="Bound">f</a><a id="2775" class="Symbol">}</a> <a id="2777" class="Symbol">→</a> <a id="2779" href="Relation.Binary.Properties.Poset.html#2774" class="Bound">f</a> <a id="2781" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2791" href="Relation.Binary.Bundles.html#3176" class="Field Operator">_≤_</a> <a id="2795" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2797" href="Relation.Binary.Properties.Poset.html#834" class="Function Operator">_≥_</a> <a id="2801" class="Symbol">→</a> <a id="2803" href="Relation.Binary.Properties.Poset.html#2774" class="Bound">f</a> <a id="2805" href="Relation.Binary.Core.html#1572" class="Function Operator">Preserves</a> <a id="2815" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2819" href="Relation.Binary.Core.html#1572" class="Function Operator">⟶</a> <a id="2821" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a>
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<a id="2825" href="Relation.Binary.Properties.Poset.html#2755" class="Function">antimono⇒cong</a> <a id="2839" class="Symbol">=</a> <a id="2841" href="Relation.Binary.Consequences.html#2505" class="Function">Consequences.antimono⇒cong</a> <a id="2868" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2872" href="Relation.Binary.Bundles.html#3140" class="Field Operator">_≈_</a> <a id="2876" href="Relation.Binary.Structures.html#1603" class="Function">Eq.sym</a> <a id="2883" href="Relation.Binary.Structures.html#2340" class="Function">reflexive</a> <a id="2893" href="Relation.Binary.Structures.html#3284" class="Function">antisym</a>
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<a id="2902" class="Comment">------------------------------------------------------------------------</a>
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<a id="2975" class="Comment">-- DEPRECATED NAMES</a>
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<a id="2995" class="Comment">------------------------------------------------------------------------</a>
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<a id="3068" class="Comment">-- Please use the new names as continuing support for the old names is</a>
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<a id="3139" class="Comment">-- not guaranteed.</a>
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<a id="3159" class="Comment">-- Version 1.2</a>
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<a id="invIsPartialOrder"></a><a id="3175" href="Relation.Binary.Properties.Poset.html#3175" class="Function">invIsPartialOrder</a> <a id="3193" class="Symbol">=</a> <a id="3195" href="Relation.Binary.Properties.Poset.html#988" class="Function">≥-isPartialOrder</a>
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<a id="3212" class="Symbol">{-#</a> <a id="3216" class="Keyword">WARNING_ON_USAGE</a> <a id="3233" class="Pragma">invIsPartialOrder</a>
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<a id="3251" class="String">"Warning: invIsPartialOrder was deprecated in v1.2.
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Please use ≥-isPartialOrder instead."</a>
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<a id="3341" class="Symbol">#-}</a>
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<a id="invPoset"></a><a id="3346" href="Relation.Binary.Properties.Poset.html#3346" class="Function">invPoset</a> <a id="3355" class="Symbol">=</a> <a id="3357" href="Relation.Binary.Properties.Poset.html#1145" class="Function">≥-poset</a>
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<a id="3365" class="Symbol">{-#</a> <a id="3369" class="Keyword">WARNING_ON_USAGE</a> <a id="3386" class="Pragma">invPoset</a>
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<a id="3395" class="String">"Warning: invPoset was deprecated in v1.2.
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Please use ≥-poset instead."</a>
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<a id="3467" class="Symbol">#-}</a>
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<a id="strictPartialOrder"></a><a id="3472" href="Relation.Binary.Properties.Poset.html#3472" class="Function">strictPartialOrder</a> <a id="3491" class="Symbol">=</a> <a id="3493" href="Relation.Binary.Properties.Poset.html#1979" class="Function"><-strictPartialOrder</a>
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<a id="3514" class="Symbol">{-#</a> <a id="3518" class="Keyword">WARNING_ON_USAGE</a> <a id="3535" class="Pragma">strictPartialOrder</a>
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<a id="3554" class="String">"Warning: strictPartialOrder was deprecated in v1.2.
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Please use <-strictPartialOrder instead."</a>
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<a id="3649" class="Symbol">#-}</a>
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</pre></body></html> |