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<html><head><meta charset="utf-8"><title>Data.Unit.Properties</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 of the unit type</a>
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<a id="137" class="Comment">------------------------------------------------------------------------</a>
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<a id="211" class="Symbol">{-#</a> <a id="215" class="Keyword">OPTIONS</a> <a id="223" class="Pragma">--cubical-compatible</a> <a id="244" class="Pragma">--safe</a> <a id="251" class="Symbol">#-}</a>
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<a id="256" class="Keyword">module</a> <a id="263" href="Data.Unit.Properties.html" class="Module">Data.Unit.Properties</a> <a id="284" class="Keyword">where</a>
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<a id="291" class="Keyword">open</a> <a id="296" class="Keyword">import</a> <a id="303" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
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<a id="317" class="Keyword">open</a> <a id="322" class="Keyword">import</a> <a id="329" href="Data.Unit.Base.html" class="Module">Data.Unit.Base</a>
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<a id="344" class="Keyword">open</a> <a id="349" class="Keyword">import</a> <a id="356" href="Level.html" class="Module">Level</a> <a id="362" class="Keyword">using</a> <a id="368" class="Symbol">(</a><a id="369" href="Level.html#521" class="Function">0ℓ</a><a id="371" class="Symbol">)</a>
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<a id="373" class="Keyword">open</a> <a id="378" class="Keyword">import</a> <a id="385" href="Relation.Nullary.html" class="Module">Relation.Nullary</a>
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<a id="402" class="Keyword">open</a> <a id="407" class="Keyword">import</a> <a id="414" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a>
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<a id="440" class="Keyword">using</a> <a id="446" class="Symbol">(</a><a id="447" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="453" class="Symbol">;</a> <a id="455" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a><a id="464" class="Symbol">;</a> <a id="466" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="471" class="Symbol">;</a> <a id="473" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a><a id="486" class="Symbol">)</a>
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<a id="488" class="Keyword">open</a> <a id="493" class="Keyword">import</a> <a id="500" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
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<a id="529" class="Keyword">using</a> <a id="535" class="Symbol">(</a><a id="536" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="546" class="Symbol">;</a> <a id="548" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="562" class="Symbol">;</a> <a id="564" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a><a id="576" class="Symbol">;</a> <a id="578" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a><a id="593" class="Symbol">)</a>
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<a id="595" class="Keyword">open</a> <a id="600" class="Keyword">import</a> <a id="607" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a> <a id="635" class="Keyword">using</a> <a id="641" class="Symbol">(</a><a id="642" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="651" class="Symbol">;</a> <a id="653" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="658" class="Symbol">;</a> <a id="660" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a><a id="673" class="Symbol">)</a>
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<a id="675" class="Keyword">open</a> <a id="680" class="Keyword">import</a> <a id="687" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a>
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<a id="726" class="Comment">------------------------------------------------------------------------</a>
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<a id="799" class="Comment">-- Irrelevancy</a>
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<a id="⊤-irrelevant"></a><a id="815" href="Data.Unit.Properties.html#815" class="Function">⊤-irrelevant</a> <a id="828" class="Symbol">:</a> <a id="830" href="Relation.Nullary.html#755" class="Function">Irrelevant</a> <a id="841" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a>
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<a id="843" href="Data.Unit.Properties.html#815" class="Function">⊤-irrelevant</a> <a id="856" class="Symbol">_</a> <a id="858" class="Symbol">_</a> <a id="860" class="Symbol">=</a> <a id="862" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
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<a id="868" class="Comment">------------------------------------------------------------------------</a>
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<a id="941" class="Comment">-- Equality</a>
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<a id="954" class="Keyword">infix</a> <a id="960" class="Number">4</a> <a id="962" href="Data.Unit.Properties.html#967" class="Function Operator">_≟_</a>
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<a id="_≟_"></a><a id="967" href="Data.Unit.Properties.html#967" class="Function Operator">_≟_</a> <a id="971" class="Symbol">:</a> <a id="973" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="983" class="Symbol">{</a><a id="984" class="Argument">A</a> <a id="986" class="Symbol">=</a> <a id="988" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a><a id="989" class="Symbol">}</a> <a id="991" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="995" class="Symbol">_</a> <a id="997" href="Data.Unit.Properties.html#967" class="Function Operator">≟</a> <a id="999" class="Symbol">_</a> <a id="1001" class="Symbol">=</a> <a id="1003" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="1007" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
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<a id="≡-setoid"></a><a id="1013" href="Data.Unit.Properties.html#1013" class="Function">≡-setoid</a> <a id="1022" class="Symbol">:</a> <a id="1024" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="1031" href="Level.html#521" class="Function">0ℓ</a> <a id="1034" href="Level.html#521" class="Function">0ℓ</a>
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<a id="1037" href="Data.Unit.Properties.html#1013" class="Function">≡-setoid</a> <a id="1046" class="Symbol">=</a> <a id="1048" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="1055" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a>
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<a id="≡-decSetoid"></a><a id="1058" href="Data.Unit.Properties.html#1058" class="Function">≡-decSetoid</a> <a id="1070" class="Symbol">:</a> <a id="1072" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="1082" href="Level.html#521" class="Function">0ℓ</a> <a id="1085" href="Level.html#521" class="Function">0ℓ</a>
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<a id="1088" href="Data.Unit.Properties.html#1058" class="Function">≡-decSetoid</a> <a id="1100" class="Symbol">=</a> <a id="1102" href="Relation.Binary.PropositionalEquality.Properties.html#5831" class="Function">decSetoid</a> <a id="1112" href="Data.Unit.Properties.html#967" class="Function Operator">_≟_</a>
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<a id="1117" class="Comment">------------------------------------------------------------------------</a>
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<a id="1190" class="Comment">-- Relational properties</a>
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<a id="≡-total"></a><a id="1216" href="Data.Unit.Properties.html#1216" class="Function">≡-total</a> <a id="1224" class="Symbol">:</a> <a id="1226" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="1232" class="Symbol">{</a><a id="1233" class="Argument">A</a> <a id="1235" class="Symbol">=</a> <a id="1237" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a><a id="1238" class="Symbol">}</a> <a id="1240" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1244" href="Data.Unit.Properties.html#1216" class="Function">≡-total</a> <a id="1252" class="Symbol">_</a> <a id="1254" class="Symbol">_</a> <a id="1256" class="Symbol">=</a> <a id="1258" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1263" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
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<a id="≡-antisym"></a><a id="1269" href="Data.Unit.Properties.html#1269" class="Function">≡-antisym</a> <a id="1279" class="Symbol">:</a> <a id="1281" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="1295" class="Symbol">{</a><a id="1296" class="Argument">A</a> <a id="1298" class="Symbol">=</a> <a id="1300" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a><a id="1301" class="Symbol">}</a> <a id="1303" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1307" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1311" href="Data.Unit.Properties.html#1269" class="Function">≡-antisym</a> <a id="1321" href="Data.Unit.Properties.html#1321" class="Bound">eq</a> <a id="1324" class="Symbol">_</a> <a id="1326" class="Symbol">=</a> <a id="1328" href="Data.Unit.Properties.html#1321" class="Bound">eq</a>
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<a id="1332" class="Comment">------------------------------------------------------------------------</a>
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<a id="1405" class="Comment">-- Structures</a>
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<a id="≡-isPreorder"></a><a id="1420" href="Data.Unit.Properties.html#1420" class="Function">≡-isPreorder</a> <a id="1433" class="Symbol">:</a> <a id="1435" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="1446" class="Symbol">{</a><a id="1447" class="Argument">A</a> <a id="1449" class="Symbol">=</a> <a id="1451" href="Agda.Builtin.Unit.html#175" class="Record">⊤</a><a id="1452" class="Symbol">}</a> <a id="1454" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1458" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1462" href="Data.Unit.Properties.html#1420" class="Function">≡-isPreorder</a> <a id="1475" class="Symbol">=</a> <a id="1477" class="Keyword">record</a>
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<a id="1486" class="Symbol">{</a> <a id="1488" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="1502" class="Symbol">=</a> <a id="1504" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
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<a id="1520" class="Symbol">;</a> <a id="1522" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="1536" class="Symbol">=</a> <a id="1538" class="Symbol">λ</a> <a id="1540" href="Data.Unit.Properties.html#1540" class="Bound">x</a> <a id="1542" class="Symbol">→</a> <a id="1544" href="Data.Unit.Properties.html#1540" class="Bound">x</a>
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<a id="1548" class="Symbol">;</a> <a id="1550" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="1564" class="Symbol">=</a> <a id="1566" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a>
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<a id="1574" class="Symbol">}</a>
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<a id="≡-isPartialOrder"></a><a id="1577" href="Data.Unit.Properties.html#1577" class="Function">≡-isPartialOrder</a> <a id="1594" class="Symbol">:</a> <a id="1596" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="1611" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1615" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1619" href="Data.Unit.Properties.html#1577" class="Function">≡-isPartialOrder</a> <a id="1636" class="Symbol">=</a> <a id="1638" class="Keyword">record</a>
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<a id="1647" class="Symbol">{</a> <a id="1649" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="1660" class="Symbol">=</a> <a id="1662" href="Data.Unit.Properties.html#1420" class="Function">≡-isPreorder</a>
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<a id="1677" class="Symbol">;</a> <a id="1679" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="1690" class="Symbol">=</a> <a id="1692" href="Data.Unit.Properties.html#1269" class="Function">≡-antisym</a>
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<a id="1704" class="Symbol">}</a>
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<a id="≡-isTotalOrder"></a><a id="1707" href="Data.Unit.Properties.html#1707" class="Function">≡-isTotalOrder</a> <a id="1722" class="Symbol">:</a> <a id="1724" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="1737" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1741" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1745" href="Data.Unit.Properties.html#1707" class="Function">≡-isTotalOrder</a> <a id="1760" class="Symbol">=</a> <a id="1762" class="Keyword">record</a>
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<a id="1771" class="Symbol">{</a> <a id="1773" href="Relation.Binary.Structures.html#5575" class="Field">isPartialOrder</a> <a id="1788" class="Symbol">=</a> <a id="1790" href="Data.Unit.Properties.html#1577" class="Function">≡-isPartialOrder</a>
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<a id="1809" class="Symbol">;</a> <a id="1811" href="Relation.Binary.Structures.html#5615" class="Field">total</a> <a id="1826" class="Symbol">=</a> <a id="1828" href="Data.Unit.Properties.html#1216" class="Function">≡-total</a>
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<a id="1838" class="Symbol">}</a>
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<a id="≡-isDecTotalOrder"></a><a id="1841" href="Data.Unit.Properties.html#1841" class="Function">≡-isDecTotalOrder</a> <a id="1859" class="Symbol">:</a> <a id="1861" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="1877" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="1881" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1885" href="Data.Unit.Properties.html#1841" class="Function">≡-isDecTotalOrder</a> <a id="1903" class="Symbol">=</a> <a id="1905" class="Keyword">record</a>
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<a id="1914" class="Symbol">{</a> <a id="1916" href="Relation.Binary.Structures.html#5914" class="Field">isTotalOrder</a> <a id="1929" class="Symbol">=</a> <a id="1931" href="Data.Unit.Properties.html#1707" class="Function">≡-isTotalOrder</a>
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<a id="1948" class="Symbol">;</a> <a id="1950" href="Relation.Binary.Structures.html#5950" class="Field Operator">_≟_</a> <a id="1963" class="Symbol">=</a> <a id="1965" href="Data.Unit.Properties.html#967" class="Function Operator">_≟_</a>
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<a id="1971" class="Symbol">;</a> <a id="1973" href="Relation.Binary.Structures.html#5983" class="Field Operator">_≤?_</a> <a id="1986" class="Symbol">=</a> <a id="1988" href="Data.Unit.Properties.html#967" class="Function Operator">_≟_</a>
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<a id="1994" class="Symbol">}</a>
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<a id="1997" class="Comment">------------------------------------------------------------------------</a>
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<a id="2070" class="Comment">-- Bundles</a>
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<a id="≡-poset"></a><a id="2082" href="Data.Unit.Properties.html#2082" class="Function">≡-poset</a> <a id="2090" class="Symbol">:</a> <a id="2092" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="2098" href="Level.html#521" class="Function">0ℓ</a> <a id="2101" href="Level.html#521" class="Function">0ℓ</a> <a id="2104" href="Level.html#521" class="Function">0ℓ</a>
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<a id="2107" href="Data.Unit.Properties.html#2082" class="Function">≡-poset</a> <a id="2115" class="Symbol">=</a> <a id="2117" class="Keyword">record</a>
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<a id="2126" class="Symbol">{</a> <a id="2128" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="2143" class="Symbol">=</a> <a id="2145" href="Data.Unit.Properties.html#1577" class="Function">≡-isPartialOrder</a>
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<a id="2164" class="Symbol">}</a>
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<a id="≡-decTotalOrder"></a><a id="2167" href="Data.Unit.Properties.html#2167" class="Function">≡-decTotalOrder</a> <a id="2183" class="Symbol">:</a> <a id="2185" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a> <a id="2199" href="Level.html#521" class="Function">0ℓ</a> <a id="2202" href="Level.html#521" class="Function">0ℓ</a> <a id="2205" href="Level.html#521" class="Function">0ℓ</a>
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<a id="2208" href="Data.Unit.Properties.html#2167" class="Function">≡-decTotalOrder</a> <a id="2224" class="Symbol">=</a> <a id="2226" class="Keyword">record</a>
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<a id="2235" class="Symbol">{</a> <a id="2237" href="Relation.Binary.Bundles.html#7227" class="Field">isDecTotalOrder</a> <a id="2253" class="Symbol">=</a> <a id="2255" href="Data.Unit.Properties.html#1841" class="Function">≡-isDecTotalOrder</a>
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<a id="2275" class="Symbol">}</a>
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</pre></body></html> |