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798 lines
178 KiB
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<html><head><meta charset="utf-8"><title>Data.Bool.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">-- A bunch of properties</a>
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<a id="131" class="Comment">------------------------------------------------------------------------</a>
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<a id="205" class="Symbol">{-#</a> <a id="209" class="Keyword">OPTIONS</a> <a id="217" class="Pragma">--cubical-compatible</a> <a id="238" class="Pragma">--safe</a> <a id="245" class="Symbol">#-}</a>
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<a id="250" class="Keyword">module</a> <a id="257" href="Data.Bool.Properties.html" class="Module">Data.Bool.Properties</a> <a id="278" class="Keyword">where</a>
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<a id="285" class="Keyword">open</a> <a id="290" class="Keyword">import</a> <a id="297" href="Algebra.Bundles.html" class="Module">Algebra.Bundles</a>
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<a id="313" class="Keyword">open</a> <a id="318" class="Keyword">import</a> <a id="325" href="Data.Bool.Base.html" class="Module">Data.Bool.Base</a>
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<a id="340" class="Keyword">open</a> <a id="345" class="Keyword">import</a> <a id="352" href="Data.Empty.html" class="Module">Data.Empty</a>
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<a id="363" class="Keyword">open</a> <a id="368" class="Keyword">import</a> <a id="375" href="Data.Product.html" class="Module">Data.Product</a>
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<a id="388" class="Keyword">open</a> <a id="393" class="Keyword">import</a> <a id="400" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
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<a id="414" class="Keyword">open</a> <a id="419" class="Keyword">import</a> <a id="426" href="Function.Base.html" class="Module">Function.Base</a>
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<a id="440" class="Keyword">open</a> <a id="445" class="Keyword">import</a> <a id="452" href="Function.Equality.html" class="Module">Function.Equality</a> <a id="470" class="Keyword">using</a> <a id="476" class="Symbol">(</a><a id="477" href="Function.Equality.html#1073" class="Field Operator">_⟨$⟩_</a><a id="482" class="Symbol">)</a>
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<a id="484" class="Keyword">open</a> <a id="489" class="Keyword">import</a> <a id="496" href="Function.Equivalence.html" class="Module">Function.Equivalence</a>
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<a id="519" class="Keyword">using</a> <a id="525" class="Symbol">(</a><a id="526" href="Function.Equivalence.html#1213" class="Function Operator">_⇔_</a><a id="529" class="Symbol">;</a> <a id="531" href="Function.Equivalence.html#1306" class="Function">equivalence</a><a id="542" class="Symbol">;</a> <a id="544" class="Keyword">module</a> <a id="551" href="Function.Equivalence.html#838" class="Module">Equivalence</a><a id="562" class="Symbol">)</a>
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<a id="564" class="Keyword">open</a> <a id="569" class="Keyword">import</a> <a id="576" href="Level.html" class="Module">Level</a> <a id="582" class="Keyword">using</a> <a id="588" class="Symbol">(</a><a id="589" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="594" class="Symbol">;</a> <a id="596" href="Level.html#521" class="Function">0ℓ</a><a id="598" class="Symbol">)</a>
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<a id="600" class="Keyword">open</a> <a id="605" class="Keyword">import</a> <a id="612" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="628" class="Keyword">hiding</a> <a id="635" class="Symbol">(</a><a id="636" href="Relation.Binary.Core.html#1334" class="Function Operator">_⇔_</a><a id="639" class="Symbol">)</a>
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<a id="641" class="Keyword">open</a> <a id="646" class="Keyword">import</a> <a id="653" href="Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a> <a id="691" class="Keyword">hiding</a> <a id="698" class="Symbol">(</a><a id="699" href="Relation.Binary.PropositionalEquality.html#2365" class="InductiveConstructor Operator">[_]</a><a id="702" class="Symbol">)</a>
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<a id="704" class="Keyword">open</a> <a id="709" class="Keyword">import</a> <a id="716" href="Relation.Nullary.html" class="Module">Relation.Nullary</a> <a id="733" class="Keyword">using</a> <a id="739" class="Symbol">(</a><a id="740" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a><a id="743" class="Symbol">;</a> <a id="745" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a><a id="748" class="Symbol">;</a> <a id="750" href="Relation.Nullary.html#1590" class="Field">does</a><a id="754" class="Symbol">;</a> <a id="756" href="Relation.Nullary.html#1607" class="Field">proof</a><a id="761" class="Symbol">;</a> <a id="763" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a><a id="766" class="Symbol">;</a> <a id="768" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a><a id="770" class="Symbol">)</a>
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<a id="772" class="Keyword">open</a> <a id="777" class="Keyword">import</a> <a id="784" href="Relation.Nullary.Decidable.html" class="Module">Relation.Nullary.Decidable</a> <a id="811" class="Keyword">using</a> <a id="817" class="Symbol">(</a><a id="818" href="Relation.Nullary.Decidable.Core.html#1374" class="Function">True</a><a id="822" class="Symbol">)</a>
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<a id="824" class="Keyword">import</a> <a id="831" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="846" class="Symbol">as</a> <a id="849" class="Module">U</a>
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<a id="852" class="Keyword">open</a> <a id="857" class="Keyword">import</a> <a id="864" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="884" class="Symbol">{</a><a id="885" class="Argument">A</a> <a id="887" class="Symbol">=</a> <a id="889" href="Agda.Builtin.Bool.html#156" class="Datatype">Bool</a><a id="893" class="Symbol">}</a> <a id="895" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
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<a id="899" class="Keyword">open</a> <a id="904" class="Keyword">import</a> <a id="911" href="Algebra.Structures.html" class="Module">Algebra.Structures</a> <a id="930" class="Symbol">{</a><a id="931" class="Argument">A</a> <a id="933" class="Symbol">=</a> <a id="935" href="Agda.Builtin.Bool.html#156" class="Datatype">Bool</a><a id="939" class="Symbol">}</a> <a id="941" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
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<a id="945" class="Keyword">open</a> <a id="950" href="Relation.Binary.PropositionalEquality.Core.html#2717" class="Module">≡-Reasoning</a>
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<a id="963" class="Keyword">private</a>
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<a id="973" class="Keyword">variable</a>
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<a id="986" href="Data.Bool.Properties.html#986" class="Generalizable">a</a> <a id="988" href="Data.Bool.Properties.html#988" class="Generalizable">b</a> <a id="990" class="Symbol">:</a> <a id="992" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="1002" href="Data.Bool.Properties.html#1002" class="Generalizable">A</a> <a id="1004" class="Symbol">:</a> <a id="1006" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1010" href="Data.Bool.Properties.html#986" class="Generalizable">a</a>
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<a id="1016" href="Data.Bool.Properties.html#1016" class="Generalizable">B</a> <a id="1018" class="Symbol">:</a> <a id="1020" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="1024" href="Data.Bool.Properties.html#988" class="Generalizable">b</a>
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<a id="1027" class="Comment">------------------------------------------------------------------------</a>
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<a id="1100" class="Comment">-- Properties of _≡_</a>
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<a id="1122" class="Keyword">infix</a> <a id="1128" class="Number">4</a> <a id="1130" href="Data.Bool.Properties.html#1135" class="Function Operator">_≟_</a>
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<a id="_≟_"></a><a id="1135" href="Data.Bool.Properties.html#1135" class="Function Operator">_≟_</a> <a id="1139" class="Symbol">:</a> <a id="1141" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="1151" class="Symbol">{</a><a id="1152" class="Argument">A</a> <a id="1154" class="Symbol">=</a> <a id="1156" href="Agda.Builtin.Bool.html#156" class="Datatype">Bool</a><a id="1160" class="Symbol">}</a> <a id="1162" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
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<a id="1166" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1172" href="Data.Bool.Properties.html#1135" class="Function Operator">≟</a> <a id="1174" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1180" class="Symbol">=</a> <a id="1182" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="1186" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="1191" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1197" href="Data.Bool.Properties.html#1135" class="Function Operator">≟</a> <a id="1199" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1205" class="Symbol">=</a> <a id="1207" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="1211" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="1216" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1222" href="Data.Bool.Properties.html#1135" class="Function Operator">≟</a> <a id="1224" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1230" class="Symbol">=</a> <a id="1232" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="1235" class="Symbol">λ()</a>
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<a id="1239" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1245" href="Data.Bool.Properties.html#1135" class="Function Operator">≟</a> <a id="1247" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1253" class="Symbol">=</a> <a id="1255" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="1258" class="Symbol">λ()</a>
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<a id="≡-setoid"></a><a id="1263" href="Data.Bool.Properties.html#1263" class="Function">≡-setoid</a> <a id="1272" class="Symbol">:</a> <a id="1274" href="Relation.Binary.Bundles.html#1018" class="Record">Setoid</a> <a id="1281" href="Level.html#521" class="Function">0ℓ</a> <a id="1284" href="Level.html#521" class="Function">0ℓ</a>
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<a id="1287" href="Data.Bool.Properties.html#1263" class="Function">≡-setoid</a> <a id="1296" class="Symbol">=</a> <a id="1298" href="Relation.Binary.PropositionalEquality.Properties.html#3981" class="Function">setoid</a> <a id="1305" href="Agda.Builtin.Bool.html#156" class="Datatype">Bool</a>
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<a id="≡-decSetoid"></a><a id="1311" href="Data.Bool.Properties.html#1311" class="Function">≡-decSetoid</a> <a id="1323" class="Symbol">:</a> <a id="1325" href="Relation.Binary.Bundles.html#1400" class="Record">DecSetoid</a> <a id="1335" href="Level.html#521" class="Function">0ℓ</a> <a id="1338" href="Level.html#521" class="Function">0ℓ</a>
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<a id="1341" href="Data.Bool.Properties.html#1311" class="Function">≡-decSetoid</a> <a id="1353" class="Symbol">=</a> <a id="1355" href="Relation.Binary.PropositionalEquality.Properties.html#4112" class="Function">decSetoid</a> <a id="1365" href="Data.Bool.Properties.html#1135" class="Function Operator">_≟_</a>
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<a id="1370" class="Comment">------------------------------------------------------------------------</a>
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<a id="1443" class="Comment">-- Properties of _≤_</a>
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<a id="1465" class="Comment">-- Relational properties</a>
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<a id="≤-reflexive"></a><a id="1491" href="Data.Bool.Properties.html#1491" class="Function">≤-reflexive</a> <a id="1503" class="Symbol">:</a> <a id="1505" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="1509" href="Relation.Binary.Core.html#1263" class="Function Operator">⇒</a> <a id="1511" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1515" href="Data.Bool.Properties.html#1491" class="Function">≤-reflexive</a> <a id="1527" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="1532" class="Symbol">=</a> <a id="1534" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
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<a id="≤-refl"></a><a id="1539" href="Data.Bool.Properties.html#1539" class="Function">≤-refl</a> <a id="1546" class="Symbol">:</a> <a id="1548" href="Relation.Binary.Definitions.html#1348" class="Function">Reflexive</a> <a id="1558" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1562" href="Data.Bool.Properties.html#1539" class="Function">≤-refl</a> <a id="1569" class="Symbol">=</a> <a id="1571" href="Data.Bool.Properties.html#1491" class="Function">≤-reflexive</a> <a id="1583" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="≤-trans"></a><a id="1589" href="Data.Bool.Properties.html#1589" class="Function">≤-trans</a> <a id="1597" class="Symbol">:</a> <a id="1599" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="1610" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1614" href="Data.Bool.Properties.html#1589" class="Function">≤-trans</a> <a id="1622" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="1626" href="Data.Bool.Properties.html#1626" class="Bound">p</a> <a id="1630" class="Symbol">=</a> <a id="1632" href="Data.Bool.Properties.html#1626" class="Bound">p</a>
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<a id="1634" href="Data.Bool.Properties.html#1589" class="Function">≤-trans</a> <a id="1642" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="1646" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="1650" class="Symbol">=</a> <a id="1652" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
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<a id="≤-antisym"></a><a id="1657" href="Data.Bool.Properties.html#1657" class="Function">≤-antisym</a> <a id="1667" class="Symbol">:</a> <a id="1669" href="Relation.Binary.Definitions.html#2091" class="Function">Antisymmetric</a> <a id="1683" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="1687" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1691" href="Data.Bool.Properties.html#1657" class="Function">≤-antisym</a> <a id="1701" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="1705" class="Symbol">_</a> <a id="1707" class="Symbol">=</a> <a id="1709" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
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<a id="≤-minimum"></a><a id="1715" href="Data.Bool.Properties.html#1715" class="Function">≤-minimum</a> <a id="1725" class="Symbol">:</a> <a id="1727" href="Relation.Binary.Definitions.html#3346" class="Function">Minimum</a> <a id="1735" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="1739" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
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<a id="1745" href="Data.Bool.Properties.html#1715" class="Function">≤-minimum</a> <a id="1755" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1761" class="Symbol">=</a> <a id="1763" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
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<a id="1767" href="Data.Bool.Properties.html#1715" class="Function">≤-minimum</a> <a id="1777" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1783" class="Symbol">=</a> <a id="1785" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
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<a id="≤-maximum"></a><a id="1790" href="Data.Bool.Properties.html#1790" class="Function">≤-maximum</a> <a id="1800" class="Symbol">:</a> <a id="1802" href="Relation.Binary.Definitions.html#3197" class="Function">Maximum</a> <a id="1810" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="1814" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
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<a id="1819" href="Data.Bool.Properties.html#1790" class="Function">≤-maximum</a> <a id="1829" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1835" class="Symbol">=</a> <a id="1837" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
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<a id="1841" href="Data.Bool.Properties.html#1790" class="Function">≤-maximum</a> <a id="1851" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1857" class="Symbol">=</a> <a id="1859" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
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<a id="≤-total"></a><a id="1864" href="Data.Bool.Properties.html#1864" class="Function">≤-total</a> <a id="1872" class="Symbol">:</a> <a id="1874" href="Relation.Binary.Definitions.html#2593" class="Function">Total</a> <a id="1880" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1884" href="Data.Bool.Properties.html#1864" class="Function">≤-total</a> <a id="1892" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="1898" href="Data.Bool.Properties.html#1898" class="Bound">b</a> <a id="1900" class="Symbol">=</a> <a id="1902" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="1907" class="Symbol">(</a><a id="1908" href="Data.Bool.Properties.html#1715" class="Function">≤-minimum</a> <a id="1918" href="Data.Bool.Properties.html#1898" class="Bound">b</a><a id="1919" class="Symbol">)</a>
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<a id="1921" href="Data.Bool.Properties.html#1864" class="Function">≤-total</a> <a id="1929" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="1935" href="Data.Bool.Properties.html#1935" class="Bound">b</a> <a id="1937" class="Symbol">=</a> <a id="1939" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="1944" class="Symbol">(</a><a id="1945" href="Data.Bool.Properties.html#1790" class="Function">≤-maximum</a> <a id="1955" href="Data.Bool.Properties.html#1935" class="Bound">b</a><a id="1956" class="Symbol">)</a>
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<a id="1959" class="Keyword">infix</a> <a id="1965" class="Number">4</a> <a id="1967" href="Data.Bool.Properties.html#1973" class="Function Operator">_≤?_</a>
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<a id="_≤?_"></a><a id="1973" href="Data.Bool.Properties.html#1973" class="Function Operator">_≤?_</a> <a id="1978" class="Symbol">:</a> <a id="1980" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="1990" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
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<a id="1994" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="2000" href="Data.Bool.Properties.html#1973" class="Function Operator">≤?</a> <a id="2003" href="Data.Bool.Properties.html#2003" class="Bound">b</a> <a id="2009" class="Symbol">=</a> <a id="2011" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="2015" class="Symbol">(</a><a id="2016" href="Data.Bool.Properties.html#1715" class="Function">≤-minimum</a> <a id="2026" href="Data.Bool.Properties.html#2003" class="Bound">b</a><a id="2027" class="Symbol">)</a>
|
|||
|
|
<a id="2029" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="2035" href="Data.Bool.Properties.html#1973" class="Function Operator">≤?</a> <a id="2038" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="2044" class="Symbol">=</a> <a id="2046" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="2049" class="Symbol">λ</a> <a id="2051" class="Symbol">()</a>
|
|||
|
|
<a id="2054" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="2060" href="Data.Bool.Properties.html#1973" class="Function Operator">≤?</a> <a id="2063" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="2069" class="Symbol">=</a> <a id="2071" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="2075" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
|
|||
|
|
|
|||
|
|
<a id="≤-irrelevant"></a><a id="2080" href="Data.Bool.Properties.html#2080" class="Function">≤-irrelevant</a> <a id="2093" class="Symbol">:</a> <a id="2095" href="Relation.Binary.Definitions.html#5074" class="Function">Irrelevant</a> <a id="2106" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
|
|||
|
|
<a id="2110" href="Data.Bool.Properties.html#2080" class="Function">≤-irrelevant</a> <a id="2123" class="Symbol">{_}</a> <a id="2131" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="2135" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="2139" class="Symbol">=</a> <a id="2141" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="2146" href="Data.Bool.Properties.html#2080" class="Function">≤-irrelevant</a> <a id="2159" class="Symbol">{</a><a id="2160" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="2165" class="Symbol">}</a> <a id="2167" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2171" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2175" class="Symbol">=</a> <a id="2177" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="2182" href="Data.Bool.Properties.html#2080" class="Function">≤-irrelevant</a> <a id="2195" class="Symbol">{</a><a id="2196" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="2200" class="Symbol">}</a> <a id="2203" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2207" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2211" class="Symbol">=</a> <a id="2213" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="2219" class="Comment">-- Structures</a>
|
|||
|
|
|
|||
|
|
<a id="≤-isPreorder"></a><a id="2234" href="Data.Bool.Properties.html#2234" class="Function">≤-isPreorder</a> <a id="2247" class="Symbol">:</a> <a id="2249" href="Relation.Binary.Structures.html#2172" class="Record">IsPreorder</a> <a id="2260" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="2264" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
|
|||
|
|
<a id="2268" href="Data.Bool.Properties.html#2234" class="Function">≤-isPreorder</a> <a id="2281" class="Symbol">=</a> <a id="2283" class="Keyword">record</a>
|
|||
|
|
<a id="2292" class="Symbol">{</a> <a id="2294" href="Relation.Binary.Structures.html#2237" class="Field">isEquivalence</a> <a id="2308" class="Symbol">=</a> <a id="2310" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="2326" class="Symbol">;</a> <a id="2328" href="Relation.Binary.Structures.html#2340" class="Field">reflexive</a> <a id="2342" class="Symbol">=</a> <a id="2344" href="Data.Bool.Properties.html#1491" class="Function">≤-reflexive</a>
|
|||
|
|
<a id="2358" class="Symbol">;</a> <a id="2360" href="Relation.Binary.Structures.html#2370" class="Field">trans</a> <a id="2374" class="Symbol">=</a> <a id="2376" href="Data.Bool.Properties.html#1589" class="Function">≤-trans</a>
|
|||
|
|
<a id="2386" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-isPartialOrder"></a><a id="2389" href="Data.Bool.Properties.html#2389" class="Function">≤-isPartialOrder</a> <a id="2406" class="Symbol">:</a> <a id="2408" href="Relation.Binary.Structures.html#3183" class="Record">IsPartialOrder</a> <a id="2423" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="2427" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
|
|||
|
|
<a id="2431" href="Data.Bool.Properties.html#2389" class="Function">≤-isPartialOrder</a> <a id="2448" class="Symbol">=</a> <a id="2450" class="Keyword">record</a>
|
|||
|
|
<a id="2459" class="Symbol">{</a> <a id="2461" href="Relation.Binary.Structures.html#3252" class="Field">isPreorder</a> <a id="2472" class="Symbol">=</a> <a id="2474" href="Data.Bool.Properties.html#2234" class="Function">≤-isPreorder</a>
|
|||
|
|
<a id="2489" class="Symbol">;</a> <a id="2491" href="Relation.Binary.Structures.html#3284" class="Field">antisym</a> <a id="2502" class="Symbol">=</a> <a id="2504" href="Data.Bool.Properties.html#1657" class="Function">≤-antisym</a>
|
|||
|
|
<a id="2516" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-isTotalOrder"></a><a id="2519" href="Data.Bool.Properties.html#2519" class="Function">≤-isTotalOrder</a> <a id="2534" class="Symbol">:</a> <a id="2536" href="Relation.Binary.Structures.html#5306" class="Record">IsTotalOrder</a> <a id="2549" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="2553" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
|
|||
|
|
<a id="2557" href="Data.Bool.Properties.html#2519" class="Function">≤-isTotalOrder</a> <a id="2572" class="Symbol">=</a> <a id="2574" class="Keyword">record</a>
|
|||
|
|
<a id="2583" class="Symbol">{</a> <a id="2585" href="Relation.Binary.Structures.html#5373" class="Field">isPartialOrder</a> <a id="2600" class="Symbol">=</a> <a id="2602" href="Data.Bool.Properties.html#2389" class="Function">≤-isPartialOrder</a>
|
|||
|
|
<a id="2621" class="Symbol">;</a> <a id="2623" href="Relation.Binary.Structures.html#5413" class="Field">total</a> <a id="2638" class="Symbol">=</a> <a id="2640" href="Data.Bool.Properties.html#1864" class="Function">≤-total</a>
|
|||
|
|
<a id="2650" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-isDecTotalOrder"></a><a id="2653" href="Data.Bool.Properties.html#2653" class="Function">≤-isDecTotalOrder</a> <a id="2671" class="Symbol">:</a> <a id="2673" href="Relation.Binary.Structures.html#5623" class="Record">IsDecTotalOrder</a> <a id="2689" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="2693" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
|
|||
|
|
<a id="2697" href="Data.Bool.Properties.html#2653" class="Function">≤-isDecTotalOrder</a> <a id="2715" class="Symbol">=</a> <a id="2717" class="Keyword">record</a>
|
|||
|
|
<a id="2726" class="Symbol">{</a> <a id="2728" href="Relation.Binary.Structures.html#5712" class="Field">isTotalOrder</a> <a id="2741" class="Symbol">=</a> <a id="2743" href="Data.Bool.Properties.html#2519" class="Function">≤-isTotalOrder</a>
|
|||
|
|
<a id="2760" class="Symbol">;</a> <a id="2762" href="Relation.Binary.Structures.html#5748" class="Field Operator">_≟_</a> <a id="2775" class="Symbol">=</a> <a id="2777" href="Data.Bool.Properties.html#1135" class="Function Operator">_≟_</a>
|
|||
|
|
<a id="2783" class="Symbol">;</a> <a id="2785" href="Relation.Binary.Structures.html#5781" class="Field Operator">_≤?_</a> <a id="2798" class="Symbol">=</a> <a id="2800" href="Data.Bool.Properties.html#1973" class="Function Operator">_≤?_</a>
|
|||
|
|
<a id="2807" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="2810" class="Comment">-- Bundles</a>
|
|||
|
|
|
|||
|
|
<a id="≤-poset"></a><a id="2822" href="Data.Bool.Properties.html#2822" class="Function">≤-poset</a> <a id="2830" class="Symbol">:</a> <a id="2832" href="Relation.Binary.Bundles.html#3037" class="Record">Poset</a> <a id="2838" href="Level.html#521" class="Function">0ℓ</a> <a id="2841" href="Level.html#521" class="Function">0ℓ</a> <a id="2844" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="2847" href="Data.Bool.Properties.html#2822" class="Function">≤-poset</a> <a id="2855" class="Symbol">=</a> <a id="2857" class="Keyword">record</a>
|
|||
|
|
<a id="2866" class="Symbol">{</a> <a id="2868" href="Relation.Binary.Bundles.html#3212" class="Field">isPartialOrder</a> <a id="2883" class="Symbol">=</a> <a id="2885" href="Data.Bool.Properties.html#2389" class="Function">≤-isPartialOrder</a>
|
|||
|
|
<a id="2904" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-preorder"></a><a id="2907" href="Data.Bool.Properties.html#2907" class="Function">≤-preorder</a> <a id="2918" class="Symbol">:</a> <a id="2920" href="Relation.Binary.Bundles.html#1929" class="Record">Preorder</a> <a id="2929" href="Level.html#521" class="Function">0ℓ</a> <a id="2932" href="Level.html#521" class="Function">0ℓ</a> <a id="2935" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="2938" href="Data.Bool.Properties.html#2907" class="Function">≤-preorder</a> <a id="2949" class="Symbol">=</a> <a id="2951" class="Keyword">record</a>
|
|||
|
|
<a id="2960" class="Symbol">{</a> <a id="2962" href="Relation.Binary.Bundles.html#2142" class="Field">isPreorder</a> <a id="2973" class="Symbol">=</a> <a id="2975" href="Data.Bool.Properties.html#2234" class="Function">≤-isPreorder</a>
|
|||
|
|
<a id="2990" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-totalOrder"></a><a id="2993" href="Data.Bool.Properties.html#2993" class="Function">≤-totalOrder</a> <a id="3006" class="Symbol">:</a> <a id="3008" href="Relation.Binary.Bundles.html#5476" class="Record">TotalOrder</a> <a id="3019" href="Level.html#521" class="Function">0ℓ</a> <a id="3022" href="Level.html#521" class="Function">0ℓ</a> <a id="3025" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="3028" href="Data.Bool.Properties.html#2993" class="Function">≤-totalOrder</a> <a id="3041" class="Symbol">=</a> <a id="3043" class="Keyword">record</a>
|
|||
|
|
<a id="3052" class="Symbol">{</a> <a id="3054" href="Relation.Binary.Bundles.html#5650" class="Field">isTotalOrder</a> <a id="3067" class="Symbol">=</a> <a id="3069" href="Data.Bool.Properties.html#2519" class="Function">≤-isTotalOrder</a>
|
|||
|
|
<a id="3086" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="≤-decTotalOrder"></a><a id="3089" href="Data.Bool.Properties.html#3089" class="Function">≤-decTotalOrder</a> <a id="3105" class="Symbol">:</a> <a id="3107" href="Relation.Binary.Bundles.html#6016" class="Record">DecTotalOrder</a> <a id="3121" href="Level.html#521" class="Function">0ℓ</a> <a id="3124" href="Level.html#521" class="Function">0ℓ</a> <a id="3127" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="3130" href="Data.Bool.Properties.html#3089" class="Function">≤-decTotalOrder</a> <a id="3146" class="Symbol">=</a> <a id="3148" class="Keyword">record</a>
|
|||
|
|
<a id="3157" class="Symbol">{</a> <a id="3159" href="Relation.Binary.Bundles.html#6202" class="Field">isDecTotalOrder</a> <a id="3175" class="Symbol">=</a> <a id="3177" href="Data.Bool.Properties.html#2653" class="Function">≤-isDecTotalOrder</a>
|
|||
|
|
<a id="3197" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="3200" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="3273" class="Comment">-- Properties of _<_</a>
|
|||
|
|
|
|||
|
|
<a id="3295" class="Comment">-- Relational properties</a>
|
|||
|
|
|
|||
|
|
<a id="<-irrefl"></a><a id="3321" href="Data.Bool.Properties.html#3321" class="Function"><-irrefl</a> <a id="3330" class="Symbol">:</a> <a id="3332" href="Relation.Binary.Definitions.html#2250" class="Function">Irreflexive</a> <a id="3344" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="3348" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3352" href="Data.Bool.Properties.html#3321" class="Function"><-irrefl</a> <a id="3361" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="3366" class="Symbol">()</a>
|
|||
|
|
|
|||
|
|
<a id="<-asym"></a><a id="3370" href="Data.Bool.Properties.html#3370" class="Function"><-asym</a> <a id="3377" class="Symbol">:</a> <a id="3379" href="Relation.Binary.Definitions.html#2362" class="Function">Asymmetric</a> <a id="3390" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3394" href="Data.Bool.Properties.html#3370" class="Function"><-asym</a> <a id="3401" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3405" class="Symbol">()</a>
|
|||
|
|
|
|||
|
|
<a id="<-trans"></a><a id="3409" href="Data.Bool.Properties.html#3409" class="Function"><-trans</a> <a id="3417" class="Symbol">:</a> <a id="3419" href="Relation.Binary.Definitions.html#1875" class="Function">Transitive</a> <a id="3430" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3434" href="Data.Bool.Properties.html#3409" class="Function"><-trans</a> <a id="3442" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3446" class="Symbol">()</a>
|
|||
|
|
|
|||
|
|
<a id="<-transʳ"></a><a id="3450" href="Data.Bool.Properties.html#3450" class="Function"><-transʳ</a> <a id="3459" class="Symbol">:</a> <a id="3461" href="Relation.Binary.Definitions.html#1594" class="Function">Trans</a> <a id="3467" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="3471" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a> <a id="3475" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3479" href="Data.Bool.Properties.html#3450" class="Function"><-transʳ</a> <a id="3488" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="3492" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3496" class="Symbol">=</a> <a id="3498" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a>
|
|||
|
|
|
|||
|
|
<a id="<-transˡ"></a><a id="3503" href="Data.Bool.Properties.html#3503" class="Function"><-transˡ</a> <a id="3512" class="Symbol">:</a> <a id="3514" href="Relation.Binary.Definitions.html#1594" class="Function">Trans</a> <a id="3520" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a> <a id="3524" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="3528" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3532" href="Data.Bool.Properties.html#3503" class="Function"><-transˡ</a> <a id="3541" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3545" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="3549" class="Symbol">=</a> <a id="3551" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a>
|
|||
|
|
|
|||
|
|
<a id="<-cmp"></a><a id="3556" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a> <a id="3562" class="Symbol">:</a> <a id="3564" href="Relation.Binary.Definitions.html#2964" class="Function">Trichotomous</a> <a id="3577" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="3581" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3585" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a> <a id="3591" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3597" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3603" class="Symbol">=</a> <a id="3605" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="3610" class="Symbol">(λ())</a> <a id="3616" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="3622" class="Symbol">(λ())</a>
|
|||
|
|
<a id="3628" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a> <a id="3634" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3640" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3646" class="Symbol">=</a> <a id="3648" href="Relation.Binary.Definitions.html#2787" class="InductiveConstructor">tri<</a> <a id="3653" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3659" class="Symbol">(λ())</a> <a id="3665" class="Symbol">(λ())</a>
|
|||
|
|
<a id="3671" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a> <a id="3677" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3683" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3689" class="Symbol">=</a> <a id="3691" href="Relation.Binary.Definitions.html#2895" class="InductiveConstructor">tri></a> <a id="3696" class="Symbol">(λ())</a> <a id="3702" class="Symbol">(λ())</a> <a id="3708" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a>
|
|||
|
|
<a id="3712" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a> <a id="3718" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3724" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3730" class="Symbol">=</a> <a id="3732" href="Relation.Binary.Definitions.html#2841" class="InductiveConstructor">tri≈</a> <a id="3737" class="Symbol">(λ())</a> <a id="3743" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="3749" class="Symbol">(λ())</a>
|
|||
|
|
|
|||
|
|
<a id="3756" class="Keyword">infix</a> <a id="3762" class="Number">4</a> <a id="3764" href="Data.Bool.Properties.html#3770" class="Function Operator">_<?_</a>
|
|||
|
|
|
|||
|
|
<a id="_<?_"></a><a id="3770" href="Data.Bool.Properties.html#3770" class="Function Operator">_<?_</a> <a id="3775" class="Symbol">:</a> <a id="3777" href="Relation.Binary.Definitions.html#4584" class="Function">Decidable</a> <a id="3787" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3791" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3797" href="Data.Bool.Properties.html#3770" class="Function Operator"><?</a> <a id="3800" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3806" class="Symbol">=</a> <a id="3808" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="3812" class="Symbol">(λ())</a>
|
|||
|
|
<a id="3818" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="3824" href="Data.Bool.Properties.html#3770" class="Function Operator"><?</a> <a id="3827" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3833" class="Symbol">=</a> <a id="3835" href="Relation.Nullary.html#1657" class="InductiveConstructor">yes</a> <a id="3839" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a>
|
|||
|
|
<a id="3843" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="3849" href="Data.Bool.Properties.html#3770" class="Function Operator"><?</a> <a id="3852" class="Symbol">_</a> <a id="3858" class="Symbol">=</a> <a id="3860" href="Relation.Nullary.html#1694" class="InductiveConstructor">no</a> <a id="3864" class="Symbol">(λ())</a>
|
|||
|
|
|
|||
|
|
<a id="<-resp₂-≡"></a><a id="3871" href="Data.Bool.Properties.html#3871" class="Function"><-resp₂-≡</a> <a id="3881" class="Symbol">:</a> <a id="3883" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a> <a id="3887" href="Relation.Binary.Definitions.html#4086" class="Function Operator">Respects₂</a> <a id="3897" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a>
|
|||
|
|
<a id="3901" href="Data.Bool.Properties.html#3871" class="Function"><-resp₂-≡</a> <a id="3911" class="Symbol">=</a> <a id="3913" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="3919" class="Symbol">(_</a> <a id="3922" href="Data.Bool.Base.html#760" class="Datatype Operator"><_</a><a id="3924" class="Symbol">)</a> <a id="3926" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3928" href="Relation.Binary.PropositionalEquality.Core.html#1789" class="Function">subst</a> <a id="3934" class="Symbol">(</a><a id="3935" href="Data.Bool.Base.html#760" class="Datatype Operator">_<</a> <a id="3938" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="<-irrelevant"></a><a id="3942" href="Data.Bool.Properties.html#3942" class="Function"><-irrelevant</a> <a id="3955" class="Symbol">:</a> <a id="3957" href="Relation.Binary.Definitions.html#5074" class="Function">Irrelevant</a> <a id="3968" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="3972" href="Data.Bool.Properties.html#3942" class="Function"><-irrelevant</a> <a id="3985" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3989" href="Data.Bool.Base.html#792" class="InductiveConstructor">f<t</a> <a id="3993" class="Symbol">=</a> <a id="3995" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="4001" class="Comment">-- Structures</a>
|
|||
|
|
|
|||
|
|
<a id="<-isStrictPartialOrder"></a><a id="4016" href="Data.Bool.Properties.html#4016" class="Function"><-isStrictPartialOrder</a> <a id="4039" class="Symbol">:</a> <a id="4041" href="Relation.Binary.Structures.html#3959" class="Record">IsStrictPartialOrder</a> <a id="4062" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="4066" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="4070" href="Data.Bool.Properties.html#4016" class="Function"><-isStrictPartialOrder</a> <a id="4093" class="Symbol">=</a> <a id="4095" class="Keyword">record</a>
|
|||
|
|
<a id="4104" class="Symbol">{</a> <a id="4106" href="Relation.Binary.Structures.html#4034" class="Field">isEquivalence</a> <a id="4120" class="Symbol">=</a> <a id="4122" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="4138" class="Symbol">;</a> <a id="4140" href="Relation.Binary.Structures.html#4068" class="Field">irrefl</a> <a id="4154" class="Symbol">=</a> <a id="4156" href="Data.Bool.Properties.html#3321" class="Function"><-irrefl</a>
|
|||
|
|
<a id="4167" class="Symbol">;</a> <a id="4169" href="Relation.Binary.Structures.html#4108" class="Field">trans</a> <a id="4183" class="Symbol">=</a> <a id="4185" href="Data.Bool.Properties.html#3409" class="Function"><-trans</a>
|
|||
|
|
<a id="4195" class="Symbol">;</a> <a id="4197" href="Relation.Binary.Structures.html#4143" class="Field"><-resp-≈</a> <a id="4211" class="Symbol">=</a> <a id="4213" href="Data.Bool.Properties.html#3871" class="Function"><-resp₂-≡</a>
|
|||
|
|
<a id="4225" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="<-isStrictTotalOrder"></a><a id="4228" href="Data.Bool.Properties.html#4228" class="Function"><-isStrictTotalOrder</a> <a id="4249" class="Symbol">:</a> <a id="4251" href="Relation.Binary.Structures.html#6548" class="Record">IsStrictTotalOrder</a> <a id="4270" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">_≡_</a> <a id="4274" href="Data.Bool.Base.html#760" class="Datatype Operator">_<_</a>
|
|||
|
|
<a id="4278" href="Data.Bool.Properties.html#4228" class="Function"><-isStrictTotalOrder</a> <a id="4299" class="Symbol">=</a> <a id="4301" class="Keyword">record</a>
|
|||
|
|
<a id="4310" class="Symbol">{</a> <a id="4312" href="Relation.Binary.Structures.html#6621" class="Field">isEquivalence</a> <a id="4326" class="Symbol">=</a> <a id="4328" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="4344" class="Symbol">;</a> <a id="4346" href="Relation.Binary.Structures.html#6655" class="Field">trans</a> <a id="4360" class="Symbol">=</a> <a id="4362" href="Data.Bool.Properties.html#3409" class="Function"><-trans</a>
|
|||
|
|
<a id="4372" class="Symbol">;</a> <a id="4374" href="Relation.Binary.Structures.html#6690" class="Field">compare</a> <a id="4388" class="Symbol">=</a> <a id="4390" href="Data.Bool.Properties.html#3556" class="Function"><-cmp</a>
|
|||
|
|
<a id="4398" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="4401" class="Comment">-- Bundles</a>
|
|||
|
|
|
|||
|
|
<a id="<-strictPartialOrder"></a><a id="4413" href="Data.Bool.Properties.html#4413" class="Function"><-strictPartialOrder</a> <a id="4434" class="Symbol">:</a> <a id="4436" href="Relation.Binary.Bundles.html#4117" class="Record">StrictPartialOrder</a> <a id="4455" href="Level.html#521" class="Function">0ℓ</a> <a id="4458" href="Level.html#521" class="Function">0ℓ</a> <a id="4461" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="4464" href="Data.Bool.Properties.html#4413" class="Function"><-strictPartialOrder</a> <a id="4485" class="Symbol">=</a> <a id="4487" class="Keyword">record</a>
|
|||
|
|
<a id="4496" class="Symbol">{</a> <a id="4498" href="Relation.Binary.Bundles.html#4323" class="Field">isStrictPartialOrder</a> <a id="4519" class="Symbol">=</a> <a id="4521" href="Data.Bool.Properties.html#4016" class="Function"><-isStrictPartialOrder</a>
|
|||
|
|
<a id="4546" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="<-strictTotalOrder"></a><a id="4549" href="Data.Bool.Properties.html#4549" class="Function"><-strictTotalOrder</a> <a id="4568" class="Symbol">:</a> <a id="4570" href="Relation.Binary.Bundles.html#6937" class="Record">StrictTotalOrder</a> <a id="4587" href="Level.html#521" class="Function">0ℓ</a> <a id="4590" href="Level.html#521" class="Function">0ℓ</a> <a id="4593" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="4596" href="Data.Bool.Properties.html#4549" class="Function"><-strictTotalOrder</a> <a id="4615" class="Symbol">=</a> <a id="4617" class="Keyword">record</a>
|
|||
|
|
<a id="4626" class="Symbol">{</a> <a id="4628" href="Relation.Binary.Bundles.html#7135" class="Field">isStrictTotalOrder</a> <a id="4647" class="Symbol">=</a> <a id="4649" href="Data.Bool.Properties.html#4228" class="Function"><-isStrictTotalOrder</a>
|
|||
|
|
<a id="4672" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="4675" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="4748" class="Comment">-- Properties of _∨_</a>
|
|||
|
|
|
|||
|
|
<a id="∨-assoc"></a><a id="4770" href="Data.Bool.Properties.html#4770" class="Function">∨-assoc</a> <a id="4778" class="Symbol">:</a> <a id="4780" href="Algebra.Definitions.html#1117" class="Function">Associative</a> <a id="4792" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="4796" href="Data.Bool.Properties.html#4770" class="Function">∨-assoc</a> <a id="4804" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="4810" href="Data.Bool.Properties.html#4810" class="Bound">y</a> <a id="4812" href="Data.Bool.Properties.html#4812" class="Bound">z</a> <a id="4814" class="Symbol">=</a> <a id="4816" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="4821" href="Data.Bool.Properties.html#4770" class="Function">∨-assoc</a> <a id="4829" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="4835" href="Data.Bool.Properties.html#4835" class="Bound">y</a> <a id="4837" href="Data.Bool.Properties.html#4837" class="Bound">z</a> <a id="4839" class="Symbol">=</a> <a id="4841" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-comm"></a><a id="4847" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="4854" class="Symbol">:</a> <a id="4856" href="Algebra.Definitions.html#1204" class="Function">Commutative</a> <a id="4868" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="4872" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="4879" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="4885" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="4891" class="Symbol">=</a> <a id="4893" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="4898" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="4905" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="4911" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="4917" class="Symbol">=</a> <a id="4919" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="4924" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="4931" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="4937" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="4943" class="Symbol">=</a> <a id="4945" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="4950" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="4957" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="4963" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="4969" class="Symbol">=</a> <a id="4971" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-identityˡ"></a><a id="4977" href="Data.Bool.Properties.html#4977" class="Function">∨-identityˡ</a> <a id="4989" class="Symbol">:</a> <a id="4991" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="5004" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5010" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5014" href="Data.Bool.Properties.html#4977" class="Function">∨-identityˡ</a> <a id="5026" class="Symbol">_</a> <a id="5028" class="Symbol">=</a> <a id="5030" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-identityʳ"></a><a id="5036" href="Data.Bool.Properties.html#5036" class="Function">∨-identityʳ</a> <a id="5048" class="Symbol">:</a> <a id="5050" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="5064" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5070" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5074" href="Data.Bool.Properties.html#5036" class="Function">∨-identityʳ</a> <a id="5086" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5092" class="Symbol">=</a> <a id="5094" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5099" href="Data.Bool.Properties.html#5036" class="Function">∨-identityʳ</a> <a id="5111" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5117" class="Symbol">=</a> <a id="5119" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-identity"></a><a id="5125" href="Data.Bool.Properties.html#5125" class="Function">∨-identity</a> <a id="5136" class="Symbol">:</a> <a id="5138" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="5147" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5153" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5157" href="Data.Bool.Properties.html#5125" class="Function">∨-identity</a> <a id="5168" class="Symbol">=</a> <a id="5170" href="Data.Bool.Properties.html#4977" class="Function">∨-identityˡ</a> <a id="5182" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5184" href="Data.Bool.Properties.html#5036" class="Function">∨-identityʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zeroˡ"></a><a id="5197" href="Data.Bool.Properties.html#5197" class="Function">∨-zeroˡ</a> <a id="5205" class="Symbol">:</a> <a id="5207" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="5216" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5221" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5225" href="Data.Bool.Properties.html#5197" class="Function">∨-zeroˡ</a> <a id="5233" class="Symbol">_</a> <a id="5235" class="Symbol">=</a> <a id="5237" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zeroʳ"></a><a id="5243" href="Data.Bool.Properties.html#5243" class="Function">∨-zeroʳ</a> <a id="5251" class="Symbol">:</a> <a id="5253" href="Algebra.Definitions.html#1576" class="Function">RightZero</a> <a id="5263" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5268" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5272" href="Data.Bool.Properties.html#5243" class="Function">∨-zeroʳ</a> <a id="5280" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5286" class="Symbol">=</a> <a id="5288" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5293" href="Data.Bool.Properties.html#5243" class="Function">∨-zeroʳ</a> <a id="5301" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5307" class="Symbol">=</a> <a id="5309" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zero"></a><a id="5315" href="Data.Bool.Properties.html#5315" class="Function">∨-zero</a> <a id="5322" class="Symbol">:</a> <a id="5324" href="Algebra.Definitions.html#1643" class="Function">Zero</a> <a id="5329" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5334" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5338" href="Data.Bool.Properties.html#5315" class="Function">∨-zero</a> <a id="5345" class="Symbol">=</a> <a id="5347" href="Data.Bool.Properties.html#5197" class="Function">∨-zeroˡ</a> <a id="5355" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5357" href="Data.Bool.Properties.html#5243" class="Function">∨-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-inverseˡ"></a><a id="5366" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a> <a id="5377" class="Symbol">:</a> <a id="5379" href="Algebra.Definitions.html#1713" class="Function">LeftInverse</a> <a id="5391" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5396" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="5400" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5404" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a> <a id="5415" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5421" class="Symbol">=</a> <a id="5423" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5428" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a> <a id="5439" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5445" class="Symbol">=</a> <a id="5447" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-inverseʳ"></a><a id="5453" href="Data.Bool.Properties.html#5453" class="Function">∨-inverseʳ</a> <a id="5464" class="Symbol">:</a> <a id="5466" href="Algebra.Definitions.html#1801" class="Function">RightInverse</a> <a id="5479" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5484" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="5488" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5492" href="Data.Bool.Properties.html#5453" class="Function">∨-inverseʳ</a> <a id="5503" href="Data.Bool.Properties.html#5503" class="Bound">x</a> <a id="5505" class="Symbol">=</a> <a id="5507" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="5514" href="Data.Bool.Properties.html#5503" class="Bound">x</a> <a id="5516" class="Symbol">(</a><a id="5517" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="5521" href="Data.Bool.Properties.html#5503" class="Bound">x</a><a id="5522" class="Symbol">)</a> <a id="5524" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="5526" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="5532" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="5534" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a> <a id="5545" href="Data.Bool.Properties.html#5503" class="Bound">x</a>
|
|||
|
|
|
|||
|
|
<a id="∨-inverse"></a><a id="5548" href="Data.Bool.Properties.html#5548" class="Function">∨-inverse</a> <a id="5558" class="Symbol">:</a> <a id="5560" href="Algebra.Definitions.html#1891" class="Function">Inverse</a> <a id="5568" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5573" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="5577" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5581" href="Data.Bool.Properties.html#5548" class="Function">∨-inverse</a> <a id="5591" class="Symbol">=</a> <a id="5593" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a> <a id="5604" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="5606" href="Data.Bool.Properties.html#5453" class="Function">∨-inverseʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-idem"></a><a id="5618" href="Data.Bool.Properties.html#5618" class="Function">∨-idem</a> <a id="5625" class="Symbol">:</a> <a id="5627" href="Algebra.Definitions.html#2656" class="Function">Idempotent</a> <a id="5638" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5642" href="Data.Bool.Properties.html#5618" class="Function">∨-idem</a> <a id="5649" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5655" class="Symbol">=</a> <a id="5657" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5662" href="Data.Bool.Properties.html#5618" class="Function">∨-idem</a> <a id="5669" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5675" class="Symbol">=</a> <a id="5677" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-sel"></a><a id="5683" href="Data.Bool.Properties.html#5683" class="Function">∨-sel</a> <a id="5689" class="Symbol">:</a> <a id="5691" href="Algebra.Definitions.html#2791" class="Function">Selective</a> <a id="5701" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5705" href="Data.Bool.Properties.html#5683" class="Function">∨-sel</a> <a id="5711" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="5717" href="Data.Bool.Properties.html#5717" class="Bound">y</a> <a id="5719" class="Symbol">=</a> <a id="5721" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="5726" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="5731" href="Data.Bool.Properties.html#5683" class="Function">∨-sel</a> <a id="5737" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="5742" href="Data.Bool.Properties.html#5742" class="Bound">y</a> <a id="5745" class="Symbol">=</a> <a id="5747" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="5752" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isMagma"></a><a id="5758" href="Data.Bool.Properties.html#5758" class="Function">∨-isMagma</a> <a id="5768" class="Symbol">:</a> <a id="5770" href="Algebra.Structures.html#1129" class="Record">IsMagma</a> <a id="5778" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5782" href="Data.Bool.Properties.html#5758" class="Function">∨-isMagma</a> <a id="5792" class="Symbol">=</a> <a id="5794" class="Keyword">record</a>
|
|||
|
|
<a id="5803" class="Symbol">{</a> <a id="5805" href="Algebra.Structures.html#1181" class="Field">isEquivalence</a> <a id="5819" class="Symbol">=</a> <a id="5821" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="5837" class="Symbol">;</a> <a id="5839" href="Algebra.Structures.html#1219" class="Field">∙-cong</a> <a id="5853" class="Symbol">=</a> <a id="5855" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="5861" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5867" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-magma"></a><a id="5870" href="Data.Bool.Properties.html#5870" class="Function">∨-magma</a> <a id="5878" class="Symbol">:</a> <a id="5880" href="Algebra.Bundles.html#1009" class="Record">Magma</a> <a id="5886" href="Level.html#521" class="Function">0ℓ</a> <a id="5889" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="5892" href="Data.Bool.Properties.html#5870" class="Function">∨-magma</a> <a id="5900" class="Symbol">=</a> <a id="5902" class="Keyword">record</a>
|
|||
|
|
<a id="5911" class="Symbol">{</a> <a id="5913" href="Algebra.Bundles.html#1161" class="Field">isMagma</a> <a id="5921" class="Symbol">=</a> <a id="5923" href="Data.Bool.Properties.html#5758" class="Function">∨-isMagma</a>
|
|||
|
|
<a id="5935" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isSemigroup"></a><a id="5938" href="Data.Bool.Properties.html#5938" class="Function">∨-isSemigroup</a> <a id="5952" class="Symbol">:</a> <a id="5954" href="Algebra.Structures.html#1796" class="Record">IsSemigroup</a> <a id="5966" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="5970" href="Data.Bool.Properties.html#5938" class="Function">∨-isSemigroup</a> <a id="5984" class="Symbol">=</a> <a id="5986" class="Keyword">record</a>
|
|||
|
|
<a id="5995" class="Symbol">{</a> <a id="5997" href="Algebra.Structures.html#1852" class="Field">isMagma</a> <a id="6005" class="Symbol">=</a> <a id="6007" href="Data.Bool.Properties.html#5758" class="Function">∨-isMagma</a>
|
|||
|
|
<a id="6019" class="Symbol">;</a> <a id="6021" href="Algebra.Structures.html#1876" class="Field">assoc</a> <a id="6029" class="Symbol">=</a> <a id="6031" href="Data.Bool.Properties.html#4770" class="Function">∨-assoc</a>
|
|||
|
|
<a id="6041" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-semigroup"></a><a id="6044" href="Data.Bool.Properties.html#6044" class="Function">∨-semigroup</a> <a id="6056" class="Symbol">:</a> <a id="6058" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a> <a id="6068" href="Level.html#521" class="Function">0ℓ</a> <a id="6071" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="6074" href="Data.Bool.Properties.html#6044" class="Function">∨-semigroup</a> <a id="6086" class="Symbol">=</a> <a id="6088" class="Keyword">record</a>
|
|||
|
|
<a id="6097" class="Symbol">{</a> <a id="6099" href="Algebra.Bundles.html#2320" class="Field">isSemigroup</a> <a id="6111" class="Symbol">=</a> <a id="6113" href="Data.Bool.Properties.html#5938" class="Function">∨-isSemigroup</a>
|
|||
|
|
<a id="6129" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isBand"></a><a id="6132" href="Data.Bool.Properties.html#6132" class="Function">∨-isBand</a> <a id="6141" class="Symbol">:</a> <a id="6143" href="Algebra.Structures.html#1940" class="Record">IsBand</a> <a id="6150" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="6154" href="Data.Bool.Properties.html#6132" class="Function">∨-isBand</a> <a id="6163" class="Symbol">=</a> <a id="6165" class="Keyword">record</a>
|
|||
|
|
<a id="6174" class="Symbol">{</a> <a id="6176" href="Algebra.Structures.html#1991" class="Field">isSemigroup</a> <a id="6188" class="Symbol">=</a> <a id="6190" href="Data.Bool.Properties.html#5938" class="Function">∨-isSemigroup</a>
|
|||
|
|
<a id="6206" class="Symbol">;</a> <a id="6208" href="Algebra.Structures.html#2023" class="Field">idem</a> <a id="6220" class="Symbol">=</a> <a id="6222" href="Data.Bool.Properties.html#5618" class="Function">∨-idem</a>
|
|||
|
|
<a id="6231" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-band"></a><a id="6234" href="Data.Bool.Properties.html#6234" class="Function">∨-band</a> <a id="6241" class="Symbol">:</a> <a id="6243" href="Algebra.Bundles.html#2515" class="Record">Band</a> <a id="6248" href="Level.html#521" class="Function">0ℓ</a> <a id="6251" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="6254" href="Data.Bool.Properties.html#6234" class="Function">∨-band</a> <a id="6261" class="Symbol">=</a> <a id="6263" class="Keyword">record</a>
|
|||
|
|
<a id="6272" class="Symbol">{</a> <a id="6274" href="Algebra.Bundles.html#2666" class="Field">isBand</a> <a id="6281" class="Symbol">=</a> <a id="6283" href="Data.Bool.Properties.html#6132" class="Function">∨-isBand</a>
|
|||
|
|
<a id="6294" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isSemilattice"></a><a id="6297" href="Data.Bool.Properties.html#6297" class="Function">∨-isSemilattice</a> <a id="6313" class="Symbol">:</a> <a id="6315" href="Algebra.Structures.html#2399" class="Record">IsSemilattice</a> <a id="6329" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="6333" href="Data.Bool.Properties.html#6297" class="Function">∨-isSemilattice</a> <a id="6349" class="Symbol">=</a> <a id="6351" class="Keyword">record</a>
|
|||
|
|
<a id="6360" class="Symbol">{</a> <a id="6362" href="Algebra.Structures.html#2457" class="Field">isBand</a> <a id="6369" class="Symbol">=</a> <a id="6371" href="Data.Bool.Properties.html#6132" class="Function">∨-isBand</a>
|
|||
|
|
<a id="6382" class="Symbol">;</a> <a id="6384" href="Algebra.Structures.html#2479" class="Field">comm</a> <a id="6391" class="Symbol">=</a> <a id="6393" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a>
|
|||
|
|
<a id="6402" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-semilattice"></a><a id="6405" href="Data.Bool.Properties.html#6405" class="Function">∨-semilattice</a> <a id="6419" class="Symbol">:</a> <a id="6421" href="Algebra.Bundles.html#3482" class="Record">Semilattice</a> <a id="6433" href="Level.html#521" class="Function">0ℓ</a> <a id="6436" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="6439" href="Data.Bool.Properties.html#6405" class="Function">∨-semilattice</a> <a id="6453" class="Symbol">=</a> <a id="6455" class="Keyword">record</a>
|
|||
|
|
<a id="6464" class="Symbol">{</a> <a id="6466" href="Algebra.Bundles.html#3658" class="Field">isSemilattice</a> <a id="6480" class="Symbol">=</a> <a id="6482" href="Data.Bool.Properties.html#6297" class="Function">∨-isSemilattice</a>
|
|||
|
|
<a id="6500" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isMonoid"></a><a id="6503" href="Data.Bool.Properties.html#6503" class="Function">∨-isMonoid</a> <a id="6514" class="Symbol">:</a> <a id="6516" href="Algebra.Structures.html#2810" class="Record">IsMonoid</a> <a id="6525" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="6529" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="6535" href="Data.Bool.Properties.html#6503" class="Function">∨-isMonoid</a> <a id="6546" class="Symbol">=</a> <a id="6548" class="Keyword">record</a>
|
|||
|
|
<a id="6557" class="Symbol">{</a> <a id="6559" href="Algebra.Structures.html#2871" class="Field">isSemigroup</a> <a id="6571" class="Symbol">=</a> <a id="6573" href="Data.Bool.Properties.html#5938" class="Function">∨-isSemigroup</a>
|
|||
|
|
<a id="6589" class="Symbol">;</a> <a id="6591" href="Algebra.Structures.html#2903" class="Field">identity</a> <a id="6600" class="Symbol">=</a> <a id="6602" href="Data.Bool.Properties.html#5125" class="Function">∨-identity</a>
|
|||
|
|
<a id="6615" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isCommutativeMonoid"></a><a id="6618" href="Data.Bool.Properties.html#6618" class="Function">∨-isCommutativeMonoid</a> <a id="6640" class="Symbol">:</a> <a id="6642" href="Algebra.Structures.html#3101" class="Record">IsCommutativeMonoid</a> <a id="6662" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="6666" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="6672" href="Data.Bool.Properties.html#6618" class="Function">∨-isCommutativeMonoid</a> <a id="6694" class="Symbol">=</a> <a id="6696" class="Keyword">record</a>
|
|||
|
|
<a id="6705" class="Symbol">{</a> <a id="6707" href="Algebra.Structures.html#3173" class="Field">isMonoid</a> <a id="6716" class="Symbol">=</a> <a id="6718" href="Data.Bool.Properties.html#6503" class="Function">∨-isMonoid</a>
|
|||
|
|
<a id="6731" class="Symbol">;</a> <a id="6733" href="Algebra.Structures.html#3201" class="Field">comm</a> <a id="6738" class="Symbol">=</a> <a id="6740" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a>
|
|||
|
|
<a id="6749" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-commutativeMonoid"></a><a id="6752" href="Data.Bool.Properties.html#6752" class="Function">∨-commutativeMonoid</a> <a id="6772" class="Symbol">:</a> <a id="6774" href="Algebra.Bundles.html#4910" class="Record">CommutativeMonoid</a> <a id="6792" href="Level.html#521" class="Function">0ℓ</a> <a id="6795" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="6798" href="Data.Bool.Properties.html#6752" class="Function">∨-commutativeMonoid</a> <a id="6818" class="Symbol">=</a> <a id="6820" class="Keyword">record</a>
|
|||
|
|
<a id="6829" class="Symbol">{</a> <a id="6831" href="Algebra.Bundles.html#5144" class="Field">isCommutativeMonoid</a> <a id="6851" class="Symbol">=</a> <a id="6853" href="Data.Bool.Properties.html#6618" class="Function">∨-isCommutativeMonoid</a>
|
|||
|
|
<a id="6877" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-isIdempotentCommutativeMonoid"></a><a id="6880" href="Data.Bool.Properties.html#6880" class="Function">∨-isIdempotentCommutativeMonoid</a> <a id="6912" class="Symbol">:</a>
|
|||
|
|
<a id="6916" href="Algebra.Structures.html#3510" class="Record">IsIdempotentCommutativeMonoid</a> <a id="6946" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="6950" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="6956" href="Data.Bool.Properties.html#6880" class="Function">∨-isIdempotentCommutativeMonoid</a> <a id="6988" class="Symbol">=</a> <a id="6990" class="Keyword">record</a>
|
|||
|
|
<a id="6999" class="Symbol">{</a> <a id="7001" href="Algebra.Structures.html#3629" class="Field">isCommutativeMonoid</a> <a id="7021" class="Symbol">=</a> <a id="7023" href="Data.Bool.Properties.html#6618" class="Function">∨-isCommutativeMonoid</a>
|
|||
|
|
<a id="7047" class="Symbol">;</a> <a id="7049" href="Algebra.Structures.html#3679" class="Field">idem</a> <a id="7069" class="Symbol">=</a> <a id="7071" href="Data.Bool.Properties.html#5618" class="Function">∨-idem</a>
|
|||
|
|
<a id="7080" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-idempotentCommutativeMonoid"></a><a id="7083" href="Data.Bool.Properties.html#7083" class="Function">∨-idempotentCommutativeMonoid</a> <a id="7113" class="Symbol">:</a> <a id="7115" href="Algebra.Bundles.html#5630" class="Record">IdempotentCommutativeMonoid</a> <a id="7143" href="Level.html#521" class="Function">0ℓ</a> <a id="7146" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="7149" href="Data.Bool.Properties.html#7083" class="Function">∨-idempotentCommutativeMonoid</a> <a id="7179" class="Symbol">=</a> <a id="7181" class="Keyword">record</a>
|
|||
|
|
<a id="7190" class="Symbol">{</a> <a id="7192" href="Algebra.Bundles.html#5914" class="Field">isIdempotentCommutativeMonoid</a> <a id="7222" class="Symbol">=</a> <a id="7224" href="Data.Bool.Properties.html#6880" class="Function">∨-isIdempotentCommutativeMonoid</a>
|
|||
|
|
<a id="7258" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="7261" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="7334" class="Comment">-- Properties of _∧_</a>
|
|||
|
|
|
|||
|
|
<a id="∧-assoc"></a><a id="7356" href="Data.Bool.Properties.html#7356" class="Function">∧-assoc</a> <a id="7364" class="Symbol">:</a> <a id="7366" href="Algebra.Definitions.html#1117" class="Function">Associative</a> <a id="7378" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7382" href="Data.Bool.Properties.html#7356" class="Function">∧-assoc</a> <a id="7390" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7396" href="Data.Bool.Properties.html#7396" class="Bound">y</a> <a id="7398" href="Data.Bool.Properties.html#7398" class="Bound">z</a> <a id="7400" class="Symbol">=</a> <a id="7402" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7407" href="Data.Bool.Properties.html#7356" class="Function">∧-assoc</a> <a id="7415" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7421" href="Data.Bool.Properties.html#7421" class="Bound">y</a> <a id="7423" href="Data.Bool.Properties.html#7423" class="Bound">z</a> <a id="7425" class="Symbol">=</a> <a id="7427" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-comm"></a><a id="7433" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="7440" class="Symbol">:</a> <a id="7442" href="Algebra.Definitions.html#1204" class="Function">Commutative</a> <a id="7454" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7458" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="7465" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7471" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7477" class="Symbol">=</a> <a id="7479" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7484" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="7491" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7497" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7503" class="Symbol">=</a> <a id="7505" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7510" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="7517" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7523" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7529" class="Symbol">=</a> <a id="7531" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7536" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="7543" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7549" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7555" class="Symbol">=</a> <a id="7557" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-identityˡ"></a><a id="7563" href="Data.Bool.Properties.html#7563" class="Function">∧-identityˡ</a> <a id="7575" class="Symbol">:</a> <a id="7577" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="7590" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7595" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7599" href="Data.Bool.Properties.html#7563" class="Function">∧-identityˡ</a> <a id="7611" class="Symbol">_</a> <a id="7613" class="Symbol">=</a> <a id="7615" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-identityʳ"></a><a id="7621" href="Data.Bool.Properties.html#7621" class="Function">∧-identityʳ</a> <a id="7633" class="Symbol">:</a> <a id="7635" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="7649" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7654" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7658" href="Data.Bool.Properties.html#7621" class="Function">∧-identityʳ</a> <a id="7670" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7676" class="Symbol">=</a> <a id="7678" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7683" href="Data.Bool.Properties.html#7621" class="Function">∧-identityʳ</a> <a id="7695" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7701" class="Symbol">=</a> <a id="7703" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-identity"></a><a id="7709" href="Data.Bool.Properties.html#7709" class="Function">∧-identity</a> <a id="7720" class="Symbol">:</a> <a id="7722" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="7731" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7736" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7740" href="Data.Bool.Properties.html#7709" class="Function">∧-identity</a> <a id="7751" class="Symbol">=</a> <a id="7753" href="Data.Bool.Properties.html#7563" class="Function">∧-identityˡ</a> <a id="7765" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="7767" href="Data.Bool.Properties.html#7621" class="Function">∧-identityʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zeroˡ"></a><a id="7780" href="Data.Bool.Properties.html#7780" class="Function">∧-zeroˡ</a> <a id="7788" class="Symbol">:</a> <a id="7790" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="7799" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7805" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7809" href="Data.Bool.Properties.html#7780" class="Function">∧-zeroˡ</a> <a id="7817" class="Symbol">_</a> <a id="7819" class="Symbol">=</a> <a id="7821" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zeroʳ"></a><a id="7827" href="Data.Bool.Properties.html#7827" class="Function">∧-zeroʳ</a> <a id="7835" class="Symbol">:</a> <a id="7837" href="Algebra.Definitions.html#1576" class="Function">RightZero</a> <a id="7847" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7853" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7857" href="Data.Bool.Properties.html#7827" class="Function">∧-zeroʳ</a> <a id="7865" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7871" class="Symbol">=</a> <a id="7873" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="7878" href="Data.Bool.Properties.html#7827" class="Function">∧-zeroʳ</a> <a id="7886" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="7892" class="Symbol">=</a> <a id="7894" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zero"></a><a id="7900" href="Data.Bool.Properties.html#7900" class="Function">∧-zero</a> <a id="7907" class="Symbol">:</a> <a id="7909" href="Algebra.Definitions.html#1643" class="Function">Zero</a> <a id="7914" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7920" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7924" href="Data.Bool.Properties.html#7900" class="Function">∧-zero</a> <a id="7931" class="Symbol">=</a> <a id="7933" href="Data.Bool.Properties.html#7780" class="Function">∧-zeroˡ</a> <a id="7941" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="7943" href="Data.Bool.Properties.html#7827" class="Function">∧-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∧-inverseˡ"></a><a id="7952" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a> <a id="7963" class="Symbol">:</a> <a id="7965" href="Algebra.Definitions.html#1713" class="Function">LeftInverse</a> <a id="7977" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="7983" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="7987" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="7991" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a> <a id="8002" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8008" class="Symbol">=</a> <a id="8010" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="8015" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a> <a id="8026" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="8031" class="Symbol">=</a> <a id="8033" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-inverseʳ"></a><a id="8039" href="Data.Bool.Properties.html#8039" class="Function">∧-inverseʳ</a> <a id="8050" class="Symbol">:</a> <a id="8052" href="Algebra.Definitions.html#1801" class="Function">RightInverse</a> <a id="8065" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8071" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="8075" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8079" href="Data.Bool.Properties.html#8039" class="Function">∧-inverseʳ</a> <a id="8090" href="Data.Bool.Properties.html#8090" class="Bound">x</a> <a id="8092" class="Symbol">=</a> <a id="8094" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="8101" href="Data.Bool.Properties.html#8090" class="Bound">x</a> <a id="8103" class="Symbol">(</a><a id="8104" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="8108" href="Data.Bool.Properties.html#8090" class="Bound">x</a><a id="8109" class="Symbol">)</a> <a id="8111" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="8113" href="Relation.Binary.PropositionalEquality.Core.html#1738" class="Function">trans</a> <a id="8119" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="8121" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a> <a id="8132" href="Data.Bool.Properties.html#8090" class="Bound">x</a>
|
|||
|
|
|
|||
|
|
<a id="∧-inverse"></a><a id="8135" href="Data.Bool.Properties.html#8135" class="Function">∧-inverse</a> <a id="8145" class="Symbol">:</a> <a id="8147" href="Algebra.Definitions.html#1891" class="Function">Inverse</a> <a id="8155" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8161" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="8165" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8169" href="Data.Bool.Properties.html#8135" class="Function">∧-inverse</a> <a id="8179" class="Symbol">=</a> <a id="8181" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a> <a id="8192" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="8194" href="Data.Bool.Properties.html#8039" class="Function">∧-inverseʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∧-idem"></a><a id="8206" href="Data.Bool.Properties.html#8206" class="Function">∧-idem</a> <a id="8213" class="Symbol">:</a> <a id="8215" href="Algebra.Definitions.html#2656" class="Function">Idempotent</a> <a id="8226" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8230" href="Data.Bool.Properties.html#8206" class="Function">∧-idem</a> <a id="8237" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8243" class="Symbol">=</a> <a id="8245" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="8250" href="Data.Bool.Properties.html#8206" class="Function">∧-idem</a> <a id="8257" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="8263" class="Symbol">=</a> <a id="8265" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-sel"></a><a id="8271" href="Data.Bool.Properties.html#8271" class="Function">∧-sel</a> <a id="8277" class="Symbol">:</a> <a id="8279" href="Algebra.Definitions.html#2791" class="Function">Selective</a> <a id="8289" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8293" href="Data.Bool.Properties.html#8271" class="Function">∧-sel</a> <a id="8299" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8305" href="Data.Bool.Properties.html#8305" class="Bound">y</a> <a id="8307" class="Symbol">=</a> <a id="8309" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="8314" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="8319" href="Data.Bool.Properties.html#8271" class="Function">∧-sel</a> <a id="8325" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="8330" href="Data.Bool.Properties.html#8330" class="Bound">y</a> <a id="8333" class="Symbol">=</a> <a id="8335" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="8340" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-distribˡ-∨"></a><a id="8346" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a> <a id="8359" class="Symbol">:</a> <a id="8361" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="8365" href="Algebra.Definitions.html#2236" class="Function Operator">DistributesOverˡ</a> <a id="8382" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="8386" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a> <a id="8399" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="8405" href="Data.Bool.Properties.html#8405" class="Bound">y</a> <a id="8407" href="Data.Bool.Properties.html#8407" class="Bound">z</a> <a id="8409" class="Symbol">=</a> <a id="8411" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="8416" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a> <a id="8429" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8435" href="Data.Bool.Properties.html#8435" class="Bound">y</a> <a id="8437" href="Data.Bool.Properties.html#8437" class="Bound">z</a> <a id="8439" class="Symbol">=</a> <a id="8441" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∧-distribʳ-∨"></a><a id="8447" href="Data.Bool.Properties.html#8447" class="Function">∧-distribʳ-∨</a> <a id="8460" class="Symbol">:</a> <a id="8462" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="8466" href="Algebra.Definitions.html#2355" class="Function Operator">DistributesOverʳ</a> <a id="8483" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="8487" href="Data.Bool.Properties.html#8447" class="Function">∧-distribʳ-∨</a> <a id="8500" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8502" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8504" href="Data.Bool.Properties.html#8504" class="Bound">z</a> <a id="8506" class="Symbol">=</a> <a id="8508" href="Relation.Binary.PropositionalEquality.Core.html#2815" class="Function Operator">begin</a>
|
|||
|
|
<a id="8516" class="Symbol">(</a><a id="8517" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8519" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8521" href="Data.Bool.Properties.html#8504" class="Bound">z</a><a id="8522" class="Symbol">)</a> <a id="8524" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8526" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8532" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="8535" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="8542" class="Symbol">(</a><a id="8543" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8545" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8547" href="Data.Bool.Properties.html#8504" class="Bound">z</a><a id="8548" class="Symbol">)</a> <a id="8550" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8552" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="8556" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8558" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8560" class="Symbol">(</a><a id="8561" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8563" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8565" href="Data.Bool.Properties.html#8504" class="Bound">z</a><a id="8566" class="Symbol">)</a> <a id="8572" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="8575" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a> <a id="8588" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8590" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8592" href="Data.Bool.Properties.html#8504" class="Bound">z</a> <a id="8594" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="8598" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8600" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8602" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8604" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8606" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8608" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8610" href="Data.Bool.Properties.html#8504" class="Bound">z</a> <a id="8614" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="8617" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="8623" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="8627" class="Symbol">(</a><a id="8628" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="8635" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8637" href="Data.Bool.Properties.html#8502" class="Bound">y</a><a id="8638" class="Symbol">)</a> <a id="8640" class="Symbol">(</a><a id="8641" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a> <a id="8648" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8650" href="Data.Bool.Properties.html#8504" class="Bound">z</a><a id="8651" class="Symbol">)</a> <a id="8653" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="8657" href="Data.Bool.Properties.html#8502" class="Bound">y</a> <a id="8659" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8661" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8663" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8665" href="Data.Bool.Properties.html#8504" class="Bound">z</a> <a id="8667" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8669" href="Data.Bool.Properties.html#8500" class="Bound">x</a> <a id="8673" href="Relation.Binary.PropositionalEquality.Core.html#3114" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="∧-distrib-∨"></a><a id="8676" href="Data.Bool.Properties.html#8676" class="Function">∧-distrib-∨</a> <a id="8688" class="Symbol">:</a> <a id="8690" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="8694" href="Algebra.Definitions.html#2474" class="Function Operator">DistributesOver</a> <a id="8710" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="8714" href="Data.Bool.Properties.html#8676" class="Function">∧-distrib-∨</a> <a id="8726" class="Symbol">=</a> <a id="8728" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a> <a id="8741" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="8743" href="Data.Bool.Properties.html#8447" class="Function">∧-distribʳ-∨</a>
|
|||
|
|
|
|||
|
|
<a id="∨-distribˡ-∧"></a><a id="8757" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a> <a id="8770" class="Symbol">:</a> <a id="8772" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="8776" href="Algebra.Definitions.html#2236" class="Function Operator">DistributesOverˡ</a> <a id="8793" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8797" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a> <a id="8810" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="8816" href="Data.Bool.Properties.html#8816" class="Bound">y</a> <a id="8818" href="Data.Bool.Properties.html#8818" class="Bound">z</a> <a id="8820" class="Symbol">=</a> <a id="8822" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="8827" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a> <a id="8840" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="8846" href="Data.Bool.Properties.html#8846" class="Bound">y</a> <a id="8848" href="Data.Bool.Properties.html#8848" class="Bound">z</a> <a id="8850" class="Symbol">=</a> <a id="8852" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-distribʳ-∧"></a><a id="8858" href="Data.Bool.Properties.html#8858" class="Function">∨-distribʳ-∧</a> <a id="8871" class="Symbol">:</a> <a id="8873" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="8877" href="Algebra.Definitions.html#2355" class="Function Operator">DistributesOverʳ</a> <a id="8894" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="8898" href="Data.Bool.Properties.html#8858" class="Function">∨-distribʳ-∧</a> <a id="8911" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="8913" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="8915" href="Data.Bool.Properties.html#8915" class="Bound">z</a> <a id="8917" class="Symbol">=</a> <a id="8919" href="Relation.Binary.PropositionalEquality.Core.html#2815" class="Function Operator">begin</a>
|
|||
|
|
<a id="8927" class="Symbol">(</a><a id="8928" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="8930" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8932" href="Data.Bool.Properties.html#8915" class="Bound">z</a><a id="8933" class="Symbol">)</a> <a id="8935" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8937" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="8946" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="8949" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="8956" class="Symbol">(</a><a id="8957" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="8959" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8961" href="Data.Bool.Properties.html#8915" class="Bound">z</a><a id="8962" class="Symbol">)</a> <a id="8964" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="8966" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="8970" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="8972" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="8974" class="Symbol">(</a><a id="8975" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="8977" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="8979" href="Data.Bool.Properties.html#8915" class="Bound">z</a><a id="8980" class="Symbol">)</a> <a id="8989" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="8992" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a> <a id="9005" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="9007" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="9009" href="Data.Bool.Properties.html#8915" class="Bound">z</a> <a id="9011" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="9015" class="Symbol">(</a><a id="9016" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="9018" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9020" href="Data.Bool.Properties.html#8913" class="Bound">y</a><a id="9021" class="Symbol">)</a> <a id="9023" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9025" class="Symbol">(</a><a id="9026" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="9028" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9030" href="Data.Bool.Properties.html#8915" class="Bound">z</a><a id="9031" class="Symbol">)</a> <a id="9034" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">≡⟨</a> <a id="9037" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="9043" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="9047" class="Symbol">(</a><a id="9048" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="9055" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="9057" href="Data.Bool.Properties.html#8913" class="Bound">y</a><a id="9058" class="Symbol">)</a> <a id="9060" class="Symbol">(</a><a id="9061" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a> <a id="9068" href="Data.Bool.Properties.html#8911" class="Bound">x</a> <a id="9070" href="Data.Bool.Properties.html#8915" class="Bound">z</a><a id="9071" class="Symbol">)</a> <a id="9073" href="Relation.Binary.PropositionalEquality.Core.html#2932" class="Function">⟩</a>
|
|||
|
|
<a id="9077" class="Symbol">(</a><a id="9078" href="Data.Bool.Properties.html#8913" class="Bound">y</a> <a id="9080" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9082" href="Data.Bool.Properties.html#8911" class="Bound">x</a><a id="9083" class="Symbol">)</a> <a id="9085" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9087" class="Symbol">(</a><a id="9088" href="Data.Bool.Properties.html#8915" class="Bound">z</a> <a id="9090" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9092" href="Data.Bool.Properties.html#8911" class="Bound">x</a><a id="9093" class="Symbol">)</a> <a id="9096" href="Relation.Binary.PropositionalEquality.Core.html#3114" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="∨-distrib-∧"></a><a id="9099" href="Data.Bool.Properties.html#9099" class="Function">∨-distrib-∧</a> <a id="9111" class="Symbol">:</a> <a id="9113" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="9117" href="Algebra.Definitions.html#2474" class="Function Operator">DistributesOver</a> <a id="9133" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9137" href="Data.Bool.Properties.html#9099" class="Function">∨-distrib-∧</a> <a id="9149" class="Symbol">=</a> <a id="9151" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a> <a id="9164" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="9166" href="Data.Bool.Properties.html#8858" class="Function">∨-distribʳ-∧</a>
|
|||
|
|
|
|||
|
|
<a id="∧-abs-∨"></a><a id="9180" href="Data.Bool.Properties.html#9180" class="Function">∧-abs-∨</a> <a id="9188" class="Symbol">:</a> <a id="9190" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="9194" href="Algebra.Definitions.html#2868" class="Function Operator">Absorbs</a> <a id="9202" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="9206" href="Data.Bool.Properties.html#9180" class="Function">∧-abs-∨</a> <a id="9214" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="9220" href="Data.Bool.Properties.html#9220" class="Bound">y</a> <a id="9222" class="Symbol">=</a> <a id="9224" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="9229" href="Data.Bool.Properties.html#9180" class="Function">∧-abs-∨</a> <a id="9237" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="9243" href="Data.Bool.Properties.html#9243" class="Bound">y</a> <a id="9245" class="Symbol">=</a> <a id="9247" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-abs-∧"></a><a id="9253" href="Data.Bool.Properties.html#9253" class="Function">∨-abs-∧</a> <a id="9261" class="Symbol">:</a> <a id="9263" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="9267" href="Algebra.Definitions.html#2868" class="Function Operator">Absorbs</a> <a id="9275" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9279" href="Data.Bool.Properties.html#9253" class="Function">∨-abs-∧</a> <a id="9287" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="9293" href="Data.Bool.Properties.html#9293" class="Bound">y</a> <a id="9295" class="Symbol">=</a> <a id="9297" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="9302" href="Data.Bool.Properties.html#9253" class="Function">∨-abs-∧</a> <a id="9310" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="9316" href="Data.Bool.Properties.html#9316" class="Bound">y</a> <a id="9318" class="Symbol">=</a> <a id="9320" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-absorptive"></a><a id="9326" href="Data.Bool.Properties.html#9326" class="Function">∨-∧-absorptive</a> <a id="9341" class="Symbol">:</a> <a id="9343" href="Algebra.Definitions.html#2947" class="Function">Absorptive</a> <a id="9354" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="9358" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9362" href="Data.Bool.Properties.html#9326" class="Function">∨-∧-absorptive</a> <a id="9377" class="Symbol">=</a> <a id="9379" href="Data.Bool.Properties.html#9253" class="Function">∨-abs-∧</a> <a id="9387" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="9389" href="Data.Bool.Properties.html#9180" class="Function">∧-abs-∨</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isMagma"></a><a id="9398" href="Data.Bool.Properties.html#9398" class="Function">∧-isMagma</a> <a id="9408" class="Symbol">:</a> <a id="9410" href="Algebra.Structures.html#1129" class="Record">IsMagma</a> <a id="9418" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9422" href="Data.Bool.Properties.html#9398" class="Function">∧-isMagma</a> <a id="9432" class="Symbol">=</a> <a id="9434" class="Keyword">record</a>
|
|||
|
|
<a id="9443" class="Symbol">{</a> <a id="9445" href="Algebra.Structures.html#1181" class="Field">isEquivalence</a> <a id="9459" class="Symbol">=</a> <a id="9461" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="9477" class="Symbol">;</a> <a id="9479" href="Algebra.Structures.html#1219" class="Field">∙-cong</a> <a id="9493" class="Symbol">=</a> <a id="9495" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="9501" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9507" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-magma"></a><a id="9510" href="Data.Bool.Properties.html#9510" class="Function">∧-magma</a> <a id="9518" class="Symbol">:</a> <a id="9520" href="Algebra.Bundles.html#1009" class="Record">Magma</a> <a id="9526" href="Level.html#521" class="Function">0ℓ</a> <a id="9529" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="9532" href="Data.Bool.Properties.html#9510" class="Function">∧-magma</a> <a id="9540" class="Symbol">=</a> <a id="9542" class="Keyword">record</a>
|
|||
|
|
<a id="9551" class="Symbol">{</a> <a id="9553" href="Algebra.Bundles.html#1161" class="Field">isMagma</a> <a id="9561" class="Symbol">=</a> <a id="9563" href="Data.Bool.Properties.html#9398" class="Function">∧-isMagma</a>
|
|||
|
|
<a id="9575" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isSemigroup"></a><a id="9578" href="Data.Bool.Properties.html#9578" class="Function">∧-isSemigroup</a> <a id="9592" class="Symbol">:</a> <a id="9594" href="Algebra.Structures.html#1796" class="Record">IsSemigroup</a> <a id="9606" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9610" href="Data.Bool.Properties.html#9578" class="Function">∧-isSemigroup</a> <a id="9624" class="Symbol">=</a> <a id="9626" class="Keyword">record</a>
|
|||
|
|
<a id="9635" class="Symbol">{</a> <a id="9637" href="Algebra.Structures.html#1852" class="Field">isMagma</a> <a id="9645" class="Symbol">=</a> <a id="9647" href="Data.Bool.Properties.html#9398" class="Function">∧-isMagma</a>
|
|||
|
|
<a id="9659" class="Symbol">;</a> <a id="9661" href="Algebra.Structures.html#1876" class="Field">assoc</a> <a id="9669" class="Symbol">=</a> <a id="9671" href="Data.Bool.Properties.html#7356" class="Function">∧-assoc</a>
|
|||
|
|
<a id="9681" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-semigroup"></a><a id="9684" href="Data.Bool.Properties.html#9684" class="Function">∧-semigroup</a> <a id="9696" class="Symbol">:</a> <a id="9698" href="Algebra.Bundles.html#2152" class="Record">Semigroup</a> <a id="9708" href="Level.html#521" class="Function">0ℓ</a> <a id="9711" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="9714" href="Data.Bool.Properties.html#9684" class="Function">∧-semigroup</a> <a id="9726" class="Symbol">=</a> <a id="9728" class="Keyword">record</a>
|
|||
|
|
<a id="9737" class="Symbol">{</a> <a id="9739" href="Algebra.Bundles.html#2320" class="Field">isSemigroup</a> <a id="9751" class="Symbol">=</a> <a id="9753" href="Data.Bool.Properties.html#9578" class="Function">∧-isSemigroup</a>
|
|||
|
|
<a id="9769" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isBand"></a><a id="9772" href="Data.Bool.Properties.html#9772" class="Function">∧-isBand</a> <a id="9781" class="Symbol">:</a> <a id="9783" href="Algebra.Structures.html#1940" class="Record">IsBand</a> <a id="9790" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9794" href="Data.Bool.Properties.html#9772" class="Function">∧-isBand</a> <a id="9803" class="Symbol">=</a> <a id="9805" class="Keyword">record</a>
|
|||
|
|
<a id="9814" class="Symbol">{</a> <a id="9816" href="Algebra.Structures.html#1991" class="Field">isSemigroup</a> <a id="9828" class="Symbol">=</a> <a id="9830" href="Data.Bool.Properties.html#9578" class="Function">∧-isSemigroup</a>
|
|||
|
|
<a id="9846" class="Symbol">;</a> <a id="9848" href="Algebra.Structures.html#2023" class="Field">idem</a> <a id="9860" class="Symbol">=</a> <a id="9862" href="Data.Bool.Properties.html#8206" class="Function">∧-idem</a>
|
|||
|
|
<a id="9871" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-band"></a><a id="9874" href="Data.Bool.Properties.html#9874" class="Function">∧-band</a> <a id="9881" class="Symbol">:</a> <a id="9883" href="Algebra.Bundles.html#2515" class="Record">Band</a> <a id="9888" href="Level.html#521" class="Function">0ℓ</a> <a id="9891" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="9894" href="Data.Bool.Properties.html#9874" class="Function">∧-band</a> <a id="9901" class="Symbol">=</a> <a id="9903" class="Keyword">record</a>
|
|||
|
|
<a id="9912" class="Symbol">{</a> <a id="9914" href="Algebra.Bundles.html#2666" class="Field">isBand</a> <a id="9921" class="Symbol">=</a> <a id="9923" href="Data.Bool.Properties.html#9772" class="Function">∧-isBand</a>
|
|||
|
|
<a id="9934" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isSemilattice"></a><a id="9937" href="Data.Bool.Properties.html#9937" class="Function">∧-isSemilattice</a> <a id="9953" class="Symbol">:</a> <a id="9955" href="Algebra.Structures.html#2399" class="Record">IsSemilattice</a> <a id="9969" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="9973" href="Data.Bool.Properties.html#9937" class="Function">∧-isSemilattice</a> <a id="9989" class="Symbol">=</a> <a id="9991" class="Keyword">record</a>
|
|||
|
|
<a id="10000" class="Symbol">{</a> <a id="10002" href="Algebra.Structures.html#2457" class="Field">isBand</a> <a id="10009" class="Symbol">=</a> <a id="10011" href="Data.Bool.Properties.html#9772" class="Function">∧-isBand</a>
|
|||
|
|
<a id="10022" class="Symbol">;</a> <a id="10024" href="Algebra.Structures.html#2479" class="Field">comm</a> <a id="10031" class="Symbol">=</a> <a id="10033" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a>
|
|||
|
|
<a id="10042" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-semilattice"></a><a id="10045" href="Data.Bool.Properties.html#10045" class="Function">∧-semilattice</a> <a id="10059" class="Symbol">:</a> <a id="10061" href="Algebra.Bundles.html#3482" class="Record">Semilattice</a> <a id="10073" href="Level.html#521" class="Function">0ℓ</a> <a id="10076" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="10079" href="Data.Bool.Properties.html#10045" class="Function">∧-semilattice</a> <a id="10093" class="Symbol">=</a> <a id="10095" class="Keyword">record</a>
|
|||
|
|
<a id="10104" class="Symbol">{</a> <a id="10106" href="Algebra.Bundles.html#3658" class="Field">isSemilattice</a> <a id="10120" class="Symbol">=</a> <a id="10122" href="Data.Bool.Properties.html#9937" class="Function">∧-isSemilattice</a>
|
|||
|
|
<a id="10140" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isMonoid"></a><a id="10143" href="Data.Bool.Properties.html#10143" class="Function">∧-isMonoid</a> <a id="10154" class="Symbol">:</a> <a id="10156" href="Algebra.Structures.html#2810" class="Record">IsMonoid</a> <a id="10165" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10169" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="10174" href="Data.Bool.Properties.html#10143" class="Function">∧-isMonoid</a> <a id="10185" class="Symbol">=</a> <a id="10187" class="Keyword">record</a>
|
|||
|
|
<a id="10196" class="Symbol">{</a> <a id="10198" href="Algebra.Structures.html#2871" class="Field">isSemigroup</a> <a id="10210" class="Symbol">=</a> <a id="10212" href="Data.Bool.Properties.html#9578" class="Function">∧-isSemigroup</a>
|
|||
|
|
<a id="10228" class="Symbol">;</a> <a id="10230" href="Algebra.Structures.html#2903" class="Field">identity</a> <a id="10239" class="Symbol">=</a> <a id="10241" href="Data.Bool.Properties.html#7709" class="Function">∧-identity</a>
|
|||
|
|
<a id="10254" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isCommutativeMonoid"></a><a id="10257" href="Data.Bool.Properties.html#10257" class="Function">∧-isCommutativeMonoid</a> <a id="10279" class="Symbol">:</a> <a id="10281" href="Algebra.Structures.html#3101" class="Record">IsCommutativeMonoid</a> <a id="10301" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10305" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="10310" href="Data.Bool.Properties.html#10257" class="Function">∧-isCommutativeMonoid</a> <a id="10332" class="Symbol">=</a> <a id="10334" class="Keyword">record</a>
|
|||
|
|
<a id="10343" class="Symbol">{</a> <a id="10345" href="Algebra.Structures.html#3173" class="Field">isMonoid</a> <a id="10354" class="Symbol">=</a> <a id="10356" href="Data.Bool.Properties.html#10143" class="Function">∧-isMonoid</a>
|
|||
|
|
<a id="10369" class="Symbol">;</a> <a id="10371" href="Algebra.Structures.html#3201" class="Field">comm</a> <a id="10376" class="Symbol">=</a> <a id="10378" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a>
|
|||
|
|
<a id="10387" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-commutativeMonoid"></a><a id="10390" href="Data.Bool.Properties.html#10390" class="Function">∧-commutativeMonoid</a> <a id="10410" class="Symbol">:</a> <a id="10412" href="Algebra.Bundles.html#4910" class="Record">CommutativeMonoid</a> <a id="10430" href="Level.html#521" class="Function">0ℓ</a> <a id="10433" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="10436" href="Data.Bool.Properties.html#10390" class="Function">∧-commutativeMonoid</a> <a id="10456" class="Symbol">=</a> <a id="10458" class="Keyword">record</a>
|
|||
|
|
<a id="10467" class="Symbol">{</a> <a id="10469" href="Algebra.Bundles.html#5144" class="Field">isCommutativeMonoid</a> <a id="10489" class="Symbol">=</a> <a id="10491" href="Data.Bool.Properties.html#10257" class="Function">∧-isCommutativeMonoid</a>
|
|||
|
|
<a id="10515" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-isIdempotentCommutativeMonoid"></a><a id="10518" href="Data.Bool.Properties.html#10518" class="Function">∧-isIdempotentCommutativeMonoid</a> <a id="10550" class="Symbol">:</a>
|
|||
|
|
<a id="10554" href="Algebra.Structures.html#3510" class="Record">IsIdempotentCommutativeMonoid</a> <a id="10584" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10588" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="10593" href="Data.Bool.Properties.html#10518" class="Function">∧-isIdempotentCommutativeMonoid</a> <a id="10625" class="Symbol">=</a> <a id="10627" class="Keyword">record</a>
|
|||
|
|
<a id="10636" class="Symbol">{</a> <a id="10638" href="Algebra.Structures.html#3629" class="Field">isCommutativeMonoid</a> <a id="10658" class="Symbol">=</a> <a id="10660" href="Data.Bool.Properties.html#10257" class="Function">∧-isCommutativeMonoid</a>
|
|||
|
|
<a id="10684" class="Symbol">;</a> <a id="10686" href="Algebra.Structures.html#3679" class="Field">idem</a> <a id="10691" class="Symbol">=</a> <a id="10693" href="Data.Bool.Properties.html#8206" class="Function">∧-idem</a>
|
|||
|
|
<a id="10702" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-idempotentCommutativeMonoid"></a><a id="10705" href="Data.Bool.Properties.html#10705" class="Function">∧-idempotentCommutativeMonoid</a> <a id="10735" class="Symbol">:</a> <a id="10737" href="Algebra.Bundles.html#5630" class="Record">IdempotentCommutativeMonoid</a> <a id="10765" href="Level.html#521" class="Function">0ℓ</a> <a id="10768" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="10771" href="Data.Bool.Properties.html#10705" class="Function">∧-idempotentCommutativeMonoid</a> <a id="10801" class="Symbol">=</a> <a id="10803" class="Keyword">record</a>
|
|||
|
|
<a id="10812" class="Symbol">{</a> <a id="10814" href="Algebra.Bundles.html#5914" class="Field">isIdempotentCommutativeMonoid</a> <a id="10844" class="Symbol">=</a> <a id="10846" href="Data.Bool.Properties.html#10518" class="Function">∧-isIdempotentCommutativeMonoid</a>
|
|||
|
|
<a id="10880" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isSemiring"></a><a id="10883" href="Data.Bool.Properties.html#10883" class="Function">∨-∧-isSemiring</a> <a id="10898" class="Symbol">:</a> <a id="10900" href="Algebra.Structures.html#11370" class="Record">IsSemiring</a> <a id="10911" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10915" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10919" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="10925" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="10930" href="Data.Bool.Properties.html#10883" class="Function">∨-∧-isSemiring</a> <a id="10945" class="Symbol">=</a> <a id="10947" class="Keyword">record</a>
|
|||
|
|
<a id="10956" class="Symbol">{</a> <a id="10958" href="Algebra.Structures.html#11439" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="10992" class="Symbol">=</a> <a id="10994" class="Keyword">record</a>
|
|||
|
|
<a id="11005" class="Symbol">{</a> <a id="11007" href="Algebra.Structures.html#10106" class="Field">+-isCommutativeMonoid</a> <a id="11029" class="Symbol">=</a> <a id="11031" href="Data.Bool.Properties.html#6618" class="Function">∨-isCommutativeMonoid</a>
|
|||
|
|
<a id="11057" class="Symbol">;</a> <a id="11059" href="Algebra.Structures.html#10159" class="Field">*-isMonoid</a> <a id="11070" class="Symbol">=</a> <a id="11072" href="Data.Bool.Properties.html#10143" class="Function">∧-isMonoid</a>
|
|||
|
|
<a id="11087" class="Symbol">;</a> <a id="11089" href="Algebra.Structures.html#10201" class="Field">distrib</a> <a id="11097" class="Symbol">=</a> <a id="11099" href="Data.Bool.Properties.html#8676" class="Function">∧-distrib-∨</a>
|
|||
|
|
<a id="11115" class="Symbol">}</a>
|
|||
|
|
<a id="11119" class="Symbol">;</a> <a id="11121" href="Algebra.Structures.html#11529" class="Field">zero</a> <a id="11126" class="Symbol">=</a> <a id="11128" href="Data.Bool.Properties.html#7900" class="Function">∧-zero</a>
|
|||
|
|
<a id="11137" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isCommutativeSemiring"></a><a id="11140" href="Data.Bool.Properties.html#11140" class="Function">∨-∧-isCommutativeSemiring</a>
|
|||
|
|
<a id="11168" class="Symbol">:</a> <a id="11170" href="Algebra.Structures.html#12026" class="Record">IsCommutativeSemiring</a> <a id="11192" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="11196" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11200" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="11206" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="11211" href="Data.Bool.Properties.html#11140" class="Function">∨-∧-isCommutativeSemiring</a> <a id="11237" class="Symbol">=</a> <a id="11239" class="Keyword">record</a>
|
|||
|
|
<a id="11248" class="Symbol">{</a> <a id="11250" href="Algebra.Structures.html#12106" class="Field">isSemiring</a> <a id="11261" class="Symbol">=</a> <a id="11263" href="Data.Bool.Properties.html#10883" class="Function">∨-∧-isSemiring</a>
|
|||
|
|
<a id="11280" class="Symbol">;</a> <a id="11282" href="Algebra.Structures.html#12144" class="Field">*-comm</a> <a id="11289" class="Symbol">=</a> <a id="11291" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a>
|
|||
|
|
<a id="11300" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-commutativeSemiring"></a><a id="11303" href="Data.Bool.Properties.html#11303" class="Function">∨-∧-commutativeSemiring</a> <a id="11327" class="Symbol">:</a> <a id="11329" href="Algebra.Bundles.html#17360" class="Record">CommutativeSemiring</a> <a id="11349" href="Level.html#521" class="Function">0ℓ</a> <a id="11352" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="11355" href="Data.Bool.Properties.html#11303" class="Function">∨-∧-commutativeSemiring</a> <a id="11379" class="Symbol">=</a> <a id="11381" class="Keyword">record</a>
|
|||
|
|
<a id="11390" class="Symbol">{</a> <a id="11392" href="Algebra.Bundles.html#17543" class="Field Operator">_+_</a> <a id="11414" class="Symbol">=</a> <a id="11416" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="11422" class="Symbol">;</a> <a id="11424" href="Algebra.Bundles.html#17583" class="Field Operator">_*_</a> <a id="11446" class="Symbol">=</a> <a id="11448" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="11454" class="Symbol">;</a> <a id="11456" href="Algebra.Bundles.html#17623" class="Field">0#</a> <a id="11478" class="Symbol">=</a> <a id="11480" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="11488" class="Symbol">;</a> <a id="11490" href="Algebra.Bundles.html#17659" class="Field">1#</a> <a id="11512" class="Symbol">=</a> <a id="11514" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="11521" class="Symbol">;</a> <a id="11523" href="Algebra.Bundles.html#17695" class="Field">isCommutativeSemiring</a> <a id="11545" class="Symbol">=</a> <a id="11547" href="Data.Bool.Properties.html#11140" class="Function">∨-∧-isCommutativeSemiring</a>
|
|||
|
|
<a id="11575" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-isSemiring"></a><a id="11578" href="Data.Bool.Properties.html#11578" class="Function">∧-∨-isSemiring</a> <a id="11593" class="Symbol">:</a> <a id="11595" href="Algebra.Structures.html#11370" class="Record">IsSemiring</a> <a id="11606" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11610" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="11614" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="11619" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="11625" href="Data.Bool.Properties.html#11578" class="Function">∧-∨-isSemiring</a> <a id="11640" class="Symbol">=</a> <a id="11642" class="Keyword">record</a>
|
|||
|
|
<a id="11651" class="Symbol">{</a> <a id="11653" href="Algebra.Structures.html#11439" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="11687" class="Symbol">=</a> <a id="11689" class="Keyword">record</a>
|
|||
|
|
<a id="11700" class="Symbol">{</a> <a id="11702" href="Algebra.Structures.html#10106" class="Field">+-isCommutativeMonoid</a> <a id="11724" class="Symbol">=</a> <a id="11726" href="Data.Bool.Properties.html#10257" class="Function">∧-isCommutativeMonoid</a>
|
|||
|
|
<a id="11752" class="Symbol">;</a> <a id="11754" href="Algebra.Structures.html#10159" class="Field">*-isMonoid</a> <a id="11765" class="Symbol">=</a> <a id="11767" href="Data.Bool.Properties.html#6503" class="Function">∨-isMonoid</a>
|
|||
|
|
<a id="11782" class="Symbol">;</a> <a id="11784" href="Algebra.Structures.html#10201" class="Field">distrib</a> <a id="11792" class="Symbol">=</a> <a id="11794" href="Data.Bool.Properties.html#9099" class="Function">∨-distrib-∧</a>
|
|||
|
|
<a id="11810" class="Symbol">}</a>
|
|||
|
|
<a id="11814" class="Symbol">;</a> <a id="11816" href="Algebra.Structures.html#11529" class="Field">zero</a> <a id="11821" class="Symbol">=</a> <a id="11823" href="Data.Bool.Properties.html#5315" class="Function">∨-zero</a>
|
|||
|
|
<a id="11832" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-isCommutativeSemiring"></a><a id="11835" href="Data.Bool.Properties.html#11835" class="Function">∧-∨-isCommutativeSemiring</a>
|
|||
|
|
<a id="11863" class="Symbol">:</a> <a id="11865" href="Algebra.Structures.html#12026" class="Record">IsCommutativeSemiring</a> <a id="11887" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11891" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="11895" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="11900" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="11906" href="Data.Bool.Properties.html#11835" class="Function">∧-∨-isCommutativeSemiring</a> <a id="11932" class="Symbol">=</a> <a id="11934" class="Keyword">record</a>
|
|||
|
|
<a id="11943" class="Symbol">{</a> <a id="11945" href="Algebra.Structures.html#12106" class="Field">isSemiring</a> <a id="11956" class="Symbol">=</a> <a id="11958" href="Data.Bool.Properties.html#11578" class="Function">∧-∨-isSemiring</a>
|
|||
|
|
<a id="11975" class="Symbol">;</a> <a id="11977" href="Algebra.Structures.html#12144" class="Field">*-comm</a> <a id="11984" class="Symbol">=</a> <a id="11986" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a>
|
|||
|
|
<a id="11995" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-commutativeSemiring"></a><a id="11998" href="Data.Bool.Properties.html#11998" class="Function">∧-∨-commutativeSemiring</a> <a id="12022" class="Symbol">:</a> <a id="12024" href="Algebra.Bundles.html#17360" class="Record">CommutativeSemiring</a> <a id="12044" href="Level.html#521" class="Function">0ℓ</a> <a id="12047" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="12050" href="Data.Bool.Properties.html#11998" class="Function">∧-∨-commutativeSemiring</a> <a id="12074" class="Symbol">=</a> <a id="12076" class="Keyword">record</a>
|
|||
|
|
<a id="12085" class="Symbol">{</a> <a id="12087" href="Algebra.Bundles.html#17543" class="Field Operator">_+_</a> <a id="12109" class="Symbol">=</a> <a id="12111" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="12117" class="Symbol">;</a> <a id="12119" href="Algebra.Bundles.html#17583" class="Field Operator">_*_</a> <a id="12141" class="Symbol">=</a> <a id="12143" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="12149" class="Symbol">;</a> <a id="12151" href="Algebra.Bundles.html#17623" class="Field">0#</a> <a id="12173" class="Symbol">=</a> <a id="12175" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="12182" class="Symbol">;</a> <a id="12184" href="Algebra.Bundles.html#17659" class="Field">1#</a> <a id="12206" class="Symbol">=</a> <a id="12208" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="12216" class="Symbol">;</a> <a id="12218" href="Algebra.Bundles.html#17695" class="Field">isCommutativeSemiring</a> <a id="12240" class="Symbol">=</a> <a id="12242" href="Data.Bool.Properties.html#11835" class="Function">∧-∨-isCommutativeSemiring</a>
|
|||
|
|
<a id="12270" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isLattice"></a><a id="12273" href="Data.Bool.Properties.html#12273" class="Function">∨-∧-isLattice</a> <a id="12287" class="Symbol">:</a> <a id="12289" href="Algebra.Structures.html#6130" class="Record">IsLattice</a> <a id="12299" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="12303" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="12307" href="Data.Bool.Properties.html#12273" class="Function">∨-∧-isLattice</a> <a id="12321" class="Symbol">=</a> <a id="12323" class="Keyword">record</a>
|
|||
|
|
<a id="12332" class="Symbol">{</a> <a id="12334" href="Algebra.Structures.html#6186" class="Field">isEquivalence</a> <a id="12348" class="Symbol">=</a> <a id="12350" href="Relation.Binary.PropositionalEquality.Properties.html#3436" class="Function">isEquivalence</a>
|
|||
|
|
<a id="12366" class="Symbol">;</a> <a id="12368" href="Algebra.Structures.html#6224" class="Field">∨-comm</a> <a id="12382" class="Symbol">=</a> <a id="12384" href="Data.Bool.Properties.html#4847" class="Function">∨-comm</a>
|
|||
|
|
<a id="12393" class="Symbol">;</a> <a id="12395" href="Algebra.Structures.html#6258" class="Field">∨-assoc</a> <a id="12409" class="Symbol">=</a> <a id="12411" href="Data.Bool.Properties.html#4770" class="Function">∨-assoc</a>
|
|||
|
|
<a id="12421" class="Symbol">;</a> <a id="12423" href="Algebra.Structures.html#6292" class="Field">∨-cong</a> <a id="12437" class="Symbol">=</a> <a id="12439" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="12445" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
|
|||
|
|
<a id="12451" class="Symbol">;</a> <a id="12453" href="Algebra.Structures.html#6325" class="Field">∧-comm</a> <a id="12467" class="Symbol">=</a> <a id="12469" href="Data.Bool.Properties.html#7433" class="Function">∧-comm</a>
|
|||
|
|
<a id="12478" class="Symbol">;</a> <a id="12480" href="Algebra.Structures.html#6359" class="Field">∧-assoc</a> <a id="12494" class="Symbol">=</a> <a id="12496" href="Data.Bool.Properties.html#7356" class="Function">∧-assoc</a>
|
|||
|
|
<a id="12506" class="Symbol">;</a> <a id="12508" href="Algebra.Structures.html#6393" class="Field">∧-cong</a> <a id="12522" class="Symbol">=</a> <a id="12524" href="Relation.Binary.PropositionalEquality.Core.html#1376" class="Function">cong₂</a> <a id="12530" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="12536" class="Symbol">;</a> <a id="12538" href="Algebra.Structures.html#6426" class="Field">absorptive</a> <a id="12552" class="Symbol">=</a> <a id="12554" href="Data.Bool.Properties.html#9326" class="Function">∨-∧-absorptive</a>
|
|||
|
|
<a id="12571" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-lattice"></a><a id="12574" href="Data.Bool.Properties.html#12574" class="Function">∨-∧-lattice</a> <a id="12586" class="Symbol">:</a> <a id="12588" href="Algebra.Bundles.html#9170" class="Record">Lattice</a> <a id="12596" href="Level.html#521" class="Function">0ℓ</a> <a id="12599" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="12602" href="Data.Bool.Properties.html#12574" class="Function">∨-∧-lattice</a> <a id="12614" class="Symbol">=</a> <a id="12616" class="Keyword">record</a>
|
|||
|
|
<a id="12625" class="Symbol">{</a> <a id="12627" href="Algebra.Bundles.html#9373" class="Field">isLattice</a> <a id="12637" class="Symbol">=</a> <a id="12639" href="Data.Bool.Properties.html#12273" class="Function">∨-∧-isLattice</a>
|
|||
|
|
<a id="12655" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isDistributiveLattice"></a><a id="12658" href="Data.Bool.Properties.html#12658" class="Function">∨-∧-isDistributiveLattice</a> <a id="12684" class="Symbol">:</a> <a id="12686" href="Algebra.Structures.html#6879" class="Record">IsDistributiveLattice</a> <a id="12708" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="12712" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
|
|||
|
|
<a id="12716" href="Data.Bool.Properties.html#12658" class="Function">∨-∧-isDistributiveLattice</a> <a id="12742" class="Symbol">=</a> <a id="12744" class="Keyword">record</a>
|
|||
|
|
<a id="12753" class="Symbol">{</a> <a id="12755" href="Algebra.Structures.html#6947" class="Field">isLattice</a> <a id="12768" class="Symbol">=</a> <a id="12770" href="Data.Bool.Properties.html#12273" class="Function">∨-∧-isLattice</a>
|
|||
|
|
<a id="12786" class="Symbol">;</a> <a id="12788" href="Algebra.Structures.html#6980" class="Field">∨-distribʳ-∧</a> <a id="12801" class="Symbol">=</a> <a id="12803" href="Data.Bool.Properties.html#8858" class="Function">∨-distribʳ-∧</a>
|
|||
|
|
<a id="12818" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-distributiveLattice"></a><a id="12821" href="Data.Bool.Properties.html#12821" class="Function">∨-∧-distributiveLattice</a> <a id="12845" class="Symbol">:</a> <a id="12847" href="Algebra.Bundles.html#9746" class="Record">DistributiveLattice</a> <a id="12867" href="Level.html#521" class="Function">0ℓ</a> <a id="12870" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="12873" href="Data.Bool.Properties.html#12821" class="Function">∨-∧-distributiveLattice</a> <a id="12897" class="Symbol">=</a> <a id="12899" class="Keyword">record</a>
|
|||
|
|
<a id="12908" class="Symbol">{</a> <a id="12910" href="Algebra.Bundles.html#10009" class="Field">isDistributiveLattice</a> <a id="12932" class="Symbol">=</a> <a id="12934" href="Data.Bool.Properties.html#12658" class="Function">∨-∧-isDistributiveLattice</a>
|
|||
|
|
<a id="12962" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isBooleanAlgebra"></a><a id="12965" href="Data.Bool.Properties.html#12965" class="Function">∨-∧-isBooleanAlgebra</a> <a id="12986" class="Symbol">:</a> <a id="12988" href="Algebra.Structures.html#15866" class="Record">IsBooleanAlgebra</a> <a id="13005" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="13009" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="13013" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="13017" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="13022" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="13028" href="Data.Bool.Properties.html#12965" class="Function">∨-∧-isBooleanAlgebra</a> <a id="13049" class="Symbol">=</a> <a id="13051" class="Keyword">record</a>
|
|||
|
|
<a id="13060" class="Symbol">{</a> <a id="13062" href="Algebra.Structures.html#15960" class="Field">isDistributiveLattice</a> <a id="13084" class="Symbol">=</a> <a id="13086" href="Data.Bool.Properties.html#12658" class="Function">∨-∧-isDistributiveLattice</a>
|
|||
|
|
<a id="13114" class="Symbol">;</a> <a id="13116" href="Algebra.Structures.html#16014" class="Field">∨-complementʳ</a> <a id="13130" class="Symbol">=</a> <a id="13132" href="Data.Bool.Properties.html#5453" class="Function">∨-inverseʳ</a>
|
|||
|
|
<a id="13145" class="Symbol">;</a> <a id="13147" href="Algebra.Structures.html#16061" class="Field">∧-complementʳ</a> <a id="13161" class="Symbol">=</a> <a id="13163" href="Data.Bool.Properties.html#8039" class="Function">∧-inverseʳ</a>
|
|||
|
|
<a id="13176" class="Symbol">;</a> <a id="13178" href="Algebra.Structures.html#16108" class="Field">¬-cong</a> <a id="13192" class="Symbol">=</a> <a id="13194" href="Relation.Binary.PropositionalEquality.Core.html#1139" class="Function">cong</a> <a id="13199" href="Data.Bool.Base.html#941" class="Function">not</a>
|
|||
|
|
<a id="13205" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-booleanAlgebra"></a><a id="13208" href="Data.Bool.Properties.html#13208" class="Function">∨-∧-booleanAlgebra</a> <a id="13227" class="Symbol">:</a> <a id="13229" href="Algebra.Bundles.html#23152" class="Record">BooleanAlgebra</a> <a id="13244" href="Level.html#521" class="Function">0ℓ</a> <a id="13247" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="13250" href="Data.Bool.Properties.html#13208" class="Function">∨-∧-booleanAlgebra</a> <a id="13269" class="Symbol">=</a> <a id="13271" class="Keyword">record</a>
|
|||
|
|
<a id="13280" class="Symbol">{</a> <a id="13282" href="Algebra.Bundles.html#23501" class="Field">isBooleanAlgebra</a> <a id="13299" class="Symbol">=</a> <a id="13301" href="Data.Bool.Properties.html#12965" class="Function">∨-∧-isBooleanAlgebra</a>
|
|||
|
|
<a id="13324" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="13327" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="13400" class="Comment">-- Properties of _xor_</a>
|
|||
|
|
|
|||
|
|
<a id="xor-is-ok"></a><a id="13424" href="Data.Bool.Properties.html#13424" class="Function">xor-is-ok</a> <a id="13434" class="Symbol">:</a> <a id="13436" class="Symbol">∀</a> <a id="13438" href="Data.Bool.Properties.html#13438" class="Bound">x</a> <a id="13440" href="Data.Bool.Properties.html#13440" class="Bound">y</a> <a id="13442" class="Symbol">→</a> <a id="13444" href="Data.Bool.Properties.html#13438" class="Bound">x</a> <a id="13446" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="13450" href="Data.Bool.Properties.html#13440" class="Bound">y</a> <a id="13452" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="13454" class="Symbol">(</a><a id="13455" href="Data.Bool.Properties.html#13438" class="Bound">x</a> <a id="13457" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="13459" href="Data.Bool.Properties.html#13440" class="Bound">y</a><a id="13460" class="Symbol">)</a> <a id="13462" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="13464" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="13468" class="Symbol">(</a><a id="13469" href="Data.Bool.Properties.html#13438" class="Bound">x</a> <a id="13471" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="13473" href="Data.Bool.Properties.html#13440" class="Bound">y</a><a id="13474" class="Symbol">)</a>
|
|||
|
|
<a id="13476" href="Data.Bool.Properties.html#13424" class="Function">xor-is-ok</a> <a id="13486" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="13492" href="Data.Bool.Properties.html#13492" class="Bound">y</a> <a id="13494" class="Symbol">=</a> <a id="13496" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="13501" href="Data.Bool.Properties.html#13424" class="Function">xor-is-ok</a> <a id="13511" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="13517" href="Data.Bool.Properties.html#13517" class="Bound">y</a> <a id="13519" class="Symbol">=</a> <a id="13521" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="13525" class="Symbol">(</a><a id="13526" href="Data.Bool.Properties.html#7621" class="Function">∧-identityʳ</a> <a id="13538" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="xor-∧-commutativeRing"></a><a id="13542" href="Data.Bool.Properties.html#13542" class="Function">xor-∧-commutativeRing</a> <a id="13564" class="Symbol">:</a> <a id="13566" href="Algebra.Bundles.html#21958" class="Record">CommutativeRing</a> <a id="13582" href="Level.html#521" class="Function">0ℓ</a> <a id="13585" href="Level.html#521" class="Function">0ℓ</a>
|
|||
|
|
<a id="13588" href="Data.Bool.Properties.html#13542" class="Function">xor-∧-commutativeRing</a> <a id="13610" class="Symbol">=</a> <a id="13612" href="Algebra.Properties.BooleanAlgebra.html#18566" class="Function">commutativeRing</a>
|
|||
|
|
<a id="13630" class="Keyword">where</a>
|
|||
|
|
<a id="13638" class="Keyword">import</a> <a id="13645" href="Algebra.Properties.BooleanAlgebra.html" class="Module">Algebra.Properties.BooleanAlgebra</a> <a id="13679" class="Symbol">as</a> <a id="13682" class="Module">BA</a>
|
|||
|
|
<a id="13687" class="Keyword">open</a> <a id="13692" href="Algebra.Properties.BooleanAlgebra.html" class="Module">BA</a> <a id="13695" href="Data.Bool.Properties.html#13208" class="Function">∨-∧-booleanAlgebra</a>
|
|||
|
|
<a id="13716" class="Keyword">open</a> <a id="13721" href="Algebra.Properties.BooleanAlgebra.html#7758" class="Module">XorRing</a> <a id="13729" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a> <a id="13735" href="Data.Bool.Properties.html#13424" class="Function">xor-is-ok</a>
|
|||
|
|
|
|||
|
|
<a id="13746" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="13819" class="Comment">-- Miscellaneous other properties</a>
|
|||
|
|
|
|||
|
|
<a id="not-involutive"></a><a id="13854" href="Data.Bool.Properties.html#13854" class="Function">not-involutive</a> <a id="13869" class="Symbol">:</a> <a id="13871" href="Algebra.Definitions.html#3030" class="Function">Involutive</a> <a id="13882" href="Data.Bool.Base.html#941" class="Function">not</a>
|
|||
|
|
<a id="13886" href="Data.Bool.Properties.html#13854" class="Function">not-involutive</a> <a id="13901" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="13907" class="Symbol">=</a> <a id="13909" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="13914" href="Data.Bool.Properties.html#13854" class="Function">not-involutive</a> <a id="13929" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="13935" class="Symbol">=</a> <a id="13937" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="not-injective"></a><a id="13943" href="Data.Bool.Properties.html#13943" class="Function">not-injective</a> <a id="13957" class="Symbol">:</a> <a id="13959" class="Symbol">∀</a> <a id="13961" class="Symbol">{</a><a id="13962" href="Data.Bool.Properties.html#13962" class="Bound">x</a> <a id="13964" href="Data.Bool.Properties.html#13964" class="Bound">y</a><a id="13965" class="Symbol">}</a> <a id="13967" class="Symbol">→</a> <a id="13969" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="13973" href="Data.Bool.Properties.html#13962" class="Bound">x</a> <a id="13975" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="13977" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="13981" href="Data.Bool.Properties.html#13964" class="Bound">y</a> <a id="13983" class="Symbol">→</a> <a id="13985" href="Data.Bool.Properties.html#13962" class="Bound">x</a> <a id="13987" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="13989" href="Data.Bool.Properties.html#13964" class="Bound">y</a>
|
|||
|
|
<a id="13991" href="Data.Bool.Properties.html#13943" class="Function">not-injective</a> <a id="14005" class="Symbol">{</a><a id="14006" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14011" class="Symbol">}</a> <a id="14013" class="Symbol">{</a><a id="14014" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14019" class="Symbol">}</a> <a id="14021" href="Data.Bool.Properties.html#14021" class="Bound">nx≢ny</a> <a id="14027" class="Symbol">=</a> <a id="14029" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14034" href="Data.Bool.Properties.html#13943" class="Function">not-injective</a> <a id="14048" class="Symbol">{</a><a id="14049" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14053" class="Symbol">}</a> <a id="14056" class="Symbol">{</a><a id="14057" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14061" class="Symbol">}</a> <a id="14064" href="Data.Bool.Properties.html#14064" class="Bound">nx≢ny</a> <a id="14070" class="Symbol">=</a> <a id="14072" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="not-¬"></a><a id="14078" href="Data.Bool.Properties.html#14078" class="Function">not-¬</a> <a id="14084" class="Symbol">:</a> <a id="14086" class="Symbol">∀</a> <a id="14088" class="Symbol">{</a><a id="14089" href="Data.Bool.Properties.html#14089" class="Bound">x</a> <a id="14091" href="Data.Bool.Properties.html#14091" class="Bound">y</a><a id="14092" class="Symbol">}</a> <a id="14094" class="Symbol">→</a> <a id="14096" href="Data.Bool.Properties.html#14089" class="Bound">x</a> <a id="14098" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14100" href="Data.Bool.Properties.html#14091" class="Bound">y</a> <a id="14102" class="Symbol">→</a> <a id="14104" href="Data.Bool.Properties.html#14089" class="Bound">x</a> <a id="14106" href="Relation.Binary.PropositionalEquality.Core.html#839" class="Function Operator">≢</a> <a id="14108" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="14112" href="Data.Bool.Properties.html#14091" class="Bound">y</a>
|
|||
|
|
<a id="14114" href="Data.Bool.Properties.html#14078" class="Function">not-¬</a> <a id="14120" class="Symbol">{</a><a id="14121" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14125" class="Symbol">}</a> <a id="14128" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="14133" class="Symbol">()</a>
|
|||
|
|
<a id="14136" href="Data.Bool.Properties.html#14078" class="Function">not-¬</a> <a id="14142" class="Symbol">{</a><a id="14143" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14148" class="Symbol">}</a> <a id="14150" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a> <a id="14155" class="Symbol">()</a>
|
|||
|
|
|
|||
|
|
<a id="¬-not"></a><a id="14159" href="Data.Bool.Properties.html#14159" class="Function">¬-not</a> <a id="14165" class="Symbol">:</a> <a id="14167" class="Symbol">∀</a> <a id="14169" class="Symbol">{</a><a id="14170" href="Data.Bool.Properties.html#14170" class="Bound">x</a> <a id="14172" href="Data.Bool.Properties.html#14172" class="Bound">y</a><a id="14173" class="Symbol">}</a> <a id="14175" class="Symbol">→</a> <a id="14177" href="Data.Bool.Properties.html#14170" class="Bound">x</a> <a id="14179" href="Relation.Binary.PropositionalEquality.Core.html#839" class="Function Operator">≢</a> <a id="14181" href="Data.Bool.Properties.html#14172" class="Bound">y</a> <a id="14183" class="Symbol">→</a> <a id="14185" href="Data.Bool.Properties.html#14170" class="Bound">x</a> <a id="14187" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14189" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="14193" href="Data.Bool.Properties.html#14172" class="Bound">y</a>
|
|||
|
|
<a id="14195" href="Data.Bool.Properties.html#14159" class="Function">¬-not</a> <a id="14201" class="Symbol">{</a><a id="14202" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14206" class="Symbol">}</a> <a id="14209" class="Symbol">{</a><a id="14210" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14214" class="Symbol">}</a> <a id="14217" href="Data.Bool.Properties.html#14217" class="Bound">x≢y</a> <a id="14221" class="Symbol">=</a> <a id="14223" href="Data.Empty.html#637" class="Function">⊥-elim</a> <a id="14230" class="Symbol">(</a><a id="14231" href="Data.Bool.Properties.html#14217" class="Bound">x≢y</a> <a id="14235" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14239" class="Symbol">)</a>
|
|||
|
|
<a id="14241" href="Data.Bool.Properties.html#14159" class="Function">¬-not</a> <a id="14247" class="Symbol">{</a><a id="14248" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14252" class="Symbol">}</a> <a id="14255" class="Symbol">{</a><a id="14256" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14261" class="Symbol">}</a> <a id="14263" class="Symbol">_</a> <a id="14267" class="Symbol">=</a> <a id="14269" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14274" href="Data.Bool.Properties.html#14159" class="Function">¬-not</a> <a id="14280" class="Symbol">{</a><a id="14281" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14286" class="Symbol">}</a> <a id="14288" class="Symbol">{</a><a id="14289" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14293" class="Symbol">}</a> <a id="14296" class="Symbol">_</a> <a id="14300" class="Symbol">=</a> <a id="14302" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14307" href="Data.Bool.Properties.html#14159" class="Function">¬-not</a> <a id="14313" class="Symbol">{</a><a id="14314" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14319" class="Symbol">}</a> <a id="14321" class="Symbol">{</a><a id="14322" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14327" class="Symbol">}</a> <a id="14329" href="Data.Bool.Properties.html#14329" class="Bound">x≢y</a> <a id="14333" class="Symbol">=</a> <a id="14335" href="Data.Empty.html#637" class="Function">⊥-elim</a> <a id="14342" class="Symbol">(</a><a id="14343" href="Data.Bool.Properties.html#14329" class="Bound">x≢y</a> <a id="14347" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14351" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="⇔→≡"></a><a id="14354" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14358" class="Symbol">:</a> <a id="14360" class="Symbol">{</a><a id="14361" href="Data.Bool.Properties.html#14361" class="Bound">x</a> <a id="14363" href="Data.Bool.Properties.html#14363" class="Bound">y</a> <a id="14365" href="Data.Bool.Properties.html#14365" class="Bound">z</a> <a id="14367" class="Symbol">:</a> <a id="14369" href="Agda.Builtin.Bool.html#156" class="Datatype">Bool</a><a id="14373" class="Symbol">}</a> <a id="14375" class="Symbol">→</a> <a id="14377" href="Data.Bool.Properties.html#14361" class="Bound">x</a> <a id="14379" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14381" href="Data.Bool.Properties.html#14365" class="Bound">z</a> <a id="14383" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="14385" href="Data.Bool.Properties.html#14363" class="Bound">y</a> <a id="14387" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14389" href="Data.Bool.Properties.html#14365" class="Bound">z</a> <a id="14391" class="Symbol">→</a> <a id="14393" href="Data.Bool.Properties.html#14361" class="Bound">x</a> <a id="14395" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14397" href="Data.Bool.Properties.html#14363" class="Bound">y</a>
|
|||
|
|
<a id="14399" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14403" class="Symbol">{</a><a id="14404" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14409" class="Symbol">}</a> <a id="14411" class="Symbol">{</a><a id="14412" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14417" class="Symbol">}</a> <a id="14427" href="Data.Bool.Properties.html#14427" class="Bound">hyp</a> <a id="14431" class="Symbol">=</a> <a id="14433" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14438" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14442" class="Symbol">{</a><a id="14443" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14448" class="Symbol">}</a> <a id="14450" class="Symbol">{</a><a id="14451" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14456" class="Symbol">}</a> <a id="14458" class="Symbol">{</a><a id="14459" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14464" class="Symbol">}</a> <a id="14466" href="Data.Bool.Properties.html#14466" class="Bound">hyp</a> <a id="14470" class="Symbol">=</a> <a id="14472" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="14476" class="Symbol">(</a><a id="14477" href="Function.Equivalence.html#988" class="Field">Equivalence.to</a> <a id="14492" href="Data.Bool.Properties.html#14466" class="Bound">hyp</a> <a id="14496" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="14500" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14504" class="Symbol">)</a>
|
|||
|
|
<a id="14506" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14510" class="Symbol">{</a><a id="14511" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14516" class="Symbol">}</a> <a id="14518" class="Symbol">{</a><a id="14519" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14524" class="Symbol">}</a> <a id="14526" class="Symbol">{</a><a id="14527" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14532" class="Symbol">}</a> <a id="14534" href="Data.Bool.Properties.html#14534" class="Bound">hyp</a> <a id="14538" class="Symbol">=</a> <a id="14540" href="Function.Equivalence.html#1009" class="Field">Equivalence.from</a> <a id="14557" href="Data.Bool.Properties.html#14534" class="Bound">hyp</a> <a id="14561" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="14565" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14570" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14574" class="Symbol">{</a><a id="14575" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14580" class="Symbol">}</a> <a id="14582" class="Symbol">{</a><a id="14583" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14588" class="Symbol">}</a> <a id="14590" class="Symbol">{</a><a id="14591" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14596" class="Symbol">}</a> <a id="14598" href="Data.Bool.Properties.html#14598" class="Bound">hyp</a> <a id="14602" class="Symbol">=</a> <a id="14604" href="Function.Equivalence.html#1009" class="Field">Equivalence.from</a> <a id="14621" href="Data.Bool.Properties.html#14598" class="Bound">hyp</a> <a id="14625" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="14629" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="14634" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14638" class="Symbol">{</a><a id="14639" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14644" class="Symbol">}</a> <a id="14646" class="Symbol">{</a><a id="14647" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="14652" class="Symbol">}</a> <a id="14654" class="Symbol">{</a><a id="14655" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14660" class="Symbol">}</a> <a id="14662" href="Data.Bool.Properties.html#14662" class="Bound">hyp</a> <a id="14666" class="Symbol">=</a> <a id="14668" href="Relation.Binary.PropositionalEquality.Core.html#1693" class="Function">sym</a> <a id="14672" class="Symbol">(</a><a id="14673" href="Function.Equivalence.html#988" class="Field">Equivalence.to</a> <a id="14688" href="Data.Bool.Properties.html#14662" class="Bound">hyp</a> <a id="14692" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="14696" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14700" class="Symbol">)</a>
|
|||
|
|
<a id="14702" href="Data.Bool.Properties.html#14354" class="Function">⇔→≡</a> <a id="14706" class="Symbol">{</a><a id="14707" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14712" class="Symbol">}</a> <a id="14714" class="Symbol">{</a><a id="14715" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14720" class="Symbol">}</a> <a id="14730" href="Data.Bool.Properties.html#14730" class="Bound">hyp</a> <a id="14734" class="Symbol">=</a> <a id="14736" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="T-≡"></a><a id="14742" href="Data.Bool.Properties.html#14742" class="Function">T-≡</a> <a id="14746" class="Symbol">:</a> <a id="14748" class="Symbol">∀</a> <a id="14750" class="Symbol">{</a><a id="14751" href="Data.Bool.Properties.html#14751" class="Bound">x</a><a id="14752" class="Symbol">}</a> <a id="14754" class="Symbol">→</a> <a id="14756" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="14758" href="Data.Bool.Properties.html#14751" class="Bound">x</a> <a id="14760" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="14762" href="Data.Bool.Properties.html#14751" class="Bound">x</a> <a id="14764" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14766" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a>
|
|||
|
|
<a id="14771" href="Data.Bool.Properties.html#14742" class="Function">T-≡</a> <a id="14775" class="Symbol">{</a><a id="14776" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14781" class="Symbol">}</a> <a id="14783" class="Symbol">=</a> <a id="14785" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="14797" class="Symbol">(λ</a> <a id="14800" class="Symbol">())</a> <a id="14810" class="Symbol">(λ</a> <a id="14813" class="Symbol">())</a>
|
|||
|
|
<a id="14817" href="Data.Bool.Properties.html#14742" class="Function">T-≡</a> <a id="14821" class="Symbol">{</a><a id="14822" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14826" class="Symbol">}</a> <a id="14829" class="Symbol">=</a> <a id="14831" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="14843" class="Symbol">(</a><a id="14844" href="Function.Base.html#645" class="Function">const</a> <a id="14850" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14854" class="Symbol">)</a> <a id="14856" class="Symbol">(</a><a id="14857" href="Function.Base.html#645" class="Function">const</a> <a id="14863" class="Symbol">_)</a>
|
|||
|
|
|
|||
|
|
<a id="T-not-≡"></a><a id="14867" href="Data.Bool.Properties.html#14867" class="Function">T-not-≡</a> <a id="14875" class="Symbol">:</a> <a id="14877" class="Symbol">∀</a> <a id="14879" class="Symbol">{</a><a id="14880" href="Data.Bool.Properties.html#14880" class="Bound">x</a><a id="14881" class="Symbol">}</a> <a id="14883" class="Symbol">→</a> <a id="14885" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="14887" class="Symbol">(</a><a id="14888" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="14892" href="Data.Bool.Properties.html#14880" class="Bound">x</a><a id="14893" class="Symbol">)</a> <a id="14895" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="14897" href="Data.Bool.Properties.html#14880" class="Bound">x</a> <a id="14899" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="14901" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a>
|
|||
|
|
<a id="14907" href="Data.Bool.Properties.html#14867" class="Function">T-not-≡</a> <a id="14915" class="Symbol">{</a><a id="14916" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="14921" class="Symbol">}</a> <a id="14923" class="Symbol">=</a> <a id="14925" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="14937" class="Symbol">(</a><a id="14938" href="Function.Base.html#645" class="Function">const</a> <a id="14944" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a><a id="14948" class="Symbol">)</a> <a id="14950" class="Symbol">(</a><a id="14951" href="Function.Base.html#645" class="Function">const</a> <a id="14957" class="Symbol">_)</a>
|
|||
|
|
<a id="14960" href="Data.Bool.Properties.html#14867" class="Function">T-not-≡</a> <a id="14968" class="Symbol">{</a><a id="14969" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="14973" class="Symbol">}</a> <a id="14976" class="Symbol">=</a> <a id="14978" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="14990" class="Symbol">(λ</a> <a id="14993" class="Symbol">())</a> <a id="15003" class="Symbol">(λ</a> <a id="15006" class="Symbol">())</a>
|
|||
|
|
|
|||
|
|
<a id="T-∧"></a><a id="15011" href="Data.Bool.Properties.html#15011" class="Function">T-∧</a> <a id="15015" class="Symbol">:</a> <a id="15017" class="Symbol">∀</a> <a id="15019" class="Symbol">{</a><a id="15020" href="Data.Bool.Properties.html#15020" class="Bound">x</a> <a id="15022" href="Data.Bool.Properties.html#15022" class="Bound">y</a><a id="15023" class="Symbol">}</a> <a id="15025" class="Symbol">→</a> <a id="15027" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15029" class="Symbol">(</a><a id="15030" href="Data.Bool.Properties.html#15020" class="Bound">x</a> <a id="15032" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="15034" href="Data.Bool.Properties.html#15022" class="Bound">y</a><a id="15035" class="Symbol">)</a> <a id="15037" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="15039" class="Symbol">(</a><a id="15040" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15042" href="Data.Bool.Properties.html#15020" class="Bound">x</a> <a id="15044" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="15046" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15048" href="Data.Bool.Properties.html#15022" class="Bound">y</a><a id="15049" class="Symbol">)</a>
|
|||
|
|
<a id="15051" href="Data.Bool.Properties.html#15011" class="Function">T-∧</a> <a id="15055" class="Symbol">{</a><a id="15056" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15060" class="Symbol">}</a> <a id="15063" class="Symbol">{</a><a id="15064" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15068" class="Symbol">}</a> <a id="15071" class="Symbol">=</a> <a id="15073" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15085" class="Symbol">(</a><a id="15086" href="Function.Base.html#645" class="Function">const</a> <a id="15092" class="Symbol">(_</a> <a id="15095" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="15097" class="Symbol">_))</a> <a id="15101" class="Symbol">(</a><a id="15102" href="Function.Base.html#645" class="Function">const</a> <a id="15108" class="Symbol">_)</a>
|
|||
|
|
<a id="15111" href="Data.Bool.Properties.html#15011" class="Function">T-∧</a> <a id="15115" class="Symbol">{</a><a id="15116" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15120" class="Symbol">}</a> <a id="15123" class="Symbol">{</a><a id="15124" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15129" class="Symbol">}</a> <a id="15131" class="Symbol">=</a> <a id="15133" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15145" class="Symbol">(λ</a> <a id="15148" class="Symbol">())</a> <a id="15161" href="Agda.Builtin.Sigma.html#246" class="Field">proj₂</a>
|
|||
|
|
<a id="15167" href="Data.Bool.Properties.html#15011" class="Function">T-∧</a> <a id="15171" class="Symbol">{</a><a id="15172" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15177" class="Symbol">}</a> <a id="15179" class="Symbol">{_}</a> <a id="15187" class="Symbol">=</a> <a id="15189" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15201" class="Symbol">(λ</a> <a id="15204" class="Symbol">())</a> <a id="15217" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a>
|
|||
|
|
|
|||
|
|
<a id="T-∨"></a><a id="15224" href="Data.Bool.Properties.html#15224" class="Function">T-∨</a> <a id="15228" class="Symbol">:</a> <a id="15230" class="Symbol">∀</a> <a id="15232" class="Symbol">{</a><a id="15233" href="Data.Bool.Properties.html#15233" class="Bound">x</a> <a id="15235" href="Data.Bool.Properties.html#15235" class="Bound">y</a><a id="15236" class="Symbol">}</a> <a id="15238" class="Symbol">→</a> <a id="15240" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15242" class="Symbol">(</a><a id="15243" href="Data.Bool.Properties.html#15233" class="Bound">x</a> <a id="15245" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="15247" href="Data.Bool.Properties.html#15235" class="Bound">y</a><a id="15248" class="Symbol">)</a> <a id="15250" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="15252" class="Symbol">(</a><a id="15253" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15255" href="Data.Bool.Properties.html#15233" class="Bound">x</a> <a id="15257" href="Data.Sum.Base.html#743" class="Datatype Operator">⊎</a> <a id="15259" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15261" href="Data.Bool.Properties.html#15235" class="Bound">y</a><a id="15262" class="Symbol">)</a>
|
|||
|
|
<a id="15264" href="Data.Bool.Properties.html#15224" class="Function">T-∨</a> <a id="15268" class="Symbol">{</a><a id="15269" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15273" class="Symbol">}</a> <a id="15276" class="Symbol">{_}</a> <a id="15284" class="Symbol">=</a> <a id="15286" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15298" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="15303" class="Symbol">(</a><a id="15304" href="Function.Base.html#645" class="Function">const</a> <a id="15310" class="Symbol">_)</a>
|
|||
|
|
<a id="15313" href="Data.Bool.Properties.html#15224" class="Function">T-∨</a> <a id="15317" class="Symbol">{</a><a id="15318" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15323" class="Symbol">}</a> <a id="15325" class="Symbol">{</a><a id="15326" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15330" class="Symbol">}</a> <a id="15333" class="Symbol">=</a> <a id="15335" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15347" href="Data.Sum.Base.html#818" class="InductiveConstructor">inj₂</a> <a id="15352" class="Symbol">(</a><a id="15353" href="Function.Base.html#645" class="Function">const</a> <a id="15359" class="Symbol">_)</a>
|
|||
|
|
<a id="15362" href="Data.Bool.Properties.html#15224" class="Function">T-∨</a> <a id="15366" class="Symbol">{</a><a id="15367" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15372" class="Symbol">}</a> <a id="15374" class="Symbol">{</a><a id="15375" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15380" class="Symbol">}</a> <a id="15382" class="Symbol">=</a> <a id="15384" href="Function.Equivalence.html#1306" class="Function">equivalence</a> <a id="15396" href="Data.Sum.Base.html#793" class="InductiveConstructor">inj₁</a> <a id="15401" href="Data.Sum.Base.html#929" class="Function Operator">[</a> <a id="15403" href="Function.Base.html#624" class="Function">id</a> <a id="15406" href="Data.Sum.Base.html#929" class="Function Operator">,</a> <a id="15408" href="Function.Base.html#624" class="Function">id</a> <a id="15411" href="Data.Sum.Base.html#929" class="Function Operator">]</a>
|
|||
|
|
|
|||
|
|
<a id="T-irrelevant"></a><a id="15414" href="Data.Bool.Properties.html#15414" class="Function">T-irrelevant</a> <a id="15427" class="Symbol">:</a> <a id="15429" href="Relation.Unary.html#3864" class="Function">U.Irrelevant</a> <a id="15442" href="Data.Bool.Base.html#1460" class="Function">T</a>
|
|||
|
|
<a id="15444" href="Data.Bool.Properties.html#15414" class="Function">T-irrelevant</a> <a id="15457" class="Symbol">{</a><a id="15458" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a><a id="15462" class="Symbol">}</a> <a id="15465" class="Symbol">_</a> <a id="15468" class="Symbol">_</a> <a id="15471" class="Symbol">=</a> <a id="15473" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="T?"></a><a id="15479" href="Data.Bool.Properties.html#15479" class="Function">T?</a> <a id="15482" class="Symbol">:</a> <a id="15484" href="Relation.Unary.html#3545" class="Function">U.Decidable</a> <a id="15496" href="Data.Bool.Base.html#1460" class="Function">T</a>
|
|||
|
|
<a id="15498" href="Relation.Nullary.html#1590" class="Field">does</a> <a id="15504" class="Symbol">(</a><a id="15505" href="Data.Bool.Properties.html#15479" class="Function">T?</a> <a id="15508" href="Data.Bool.Properties.html#15508" class="Bound">b</a><a id="15509" class="Symbol">)</a> <a id="15511" class="Symbol">=</a> <a id="15513" href="Data.Bool.Properties.html#15508" class="Bound">b</a>
|
|||
|
|
<a id="15515" href="Relation.Nullary.html#1607" class="Field">proof</a> <a id="15521" class="Symbol">(</a><a id="15522" href="Data.Bool.Properties.html#15479" class="Function">T?</a> <a id="15525" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="15530" class="Symbol">)</a> <a id="15532" class="Symbol">=</a> <a id="15534" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a> <a id="15538" class="Symbol">_</a>
|
|||
|
|
<a id="15540" href="Relation.Nullary.html#1607" class="Field">proof</a> <a id="15546" class="Symbol">(</a><a id="15547" href="Data.Bool.Properties.html#15479" class="Function">T?</a> <a id="15550" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a><a id="15555" class="Symbol">)</a> <a id="15557" class="Symbol">=</a> <a id="15559" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a> <a id="15563" class="Symbol">λ()</a>
|
|||
|
|
|
|||
|
|
<a id="T?-diag"></a><a id="15568" href="Data.Bool.Properties.html#15568" class="Function">T?-diag</a> <a id="15576" class="Symbol">:</a> <a id="15578" class="Symbol">∀</a> <a id="15580" href="Data.Bool.Properties.html#15580" class="Bound">b</a> <a id="15582" class="Symbol">→</a> <a id="15584" href="Data.Bool.Base.html#1460" class="Function">T</a> <a id="15586" href="Data.Bool.Properties.html#15580" class="Bound">b</a> <a id="15588" class="Symbol">→</a> <a id="15590" href="Relation.Nullary.Decidable.Core.html#1374" class="Function">True</a> <a id="15595" class="Symbol">(</a><a id="15596" href="Data.Bool.Properties.html#15479" class="Function">T?</a> <a id="15599" href="Data.Bool.Properties.html#15580" class="Bound">b</a><a id="15600" class="Symbol">)</a>
|
|||
|
|
<a id="15602" href="Data.Bool.Properties.html#15568" class="Function">T?-diag</a> <a id="15610" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="15616" class="Symbol">_</a> <a id="15618" class="Symbol">=</a> <a id="15620" class="Symbol">_</a>
|
|||
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|
|
<a id="push-function-into-if"></a><a id="15623" href="Data.Bool.Properties.html#15623" class="Function">push-function-into-if</a> <a id="15645" class="Symbol">:</a> <a id="15647" class="Symbol">∀</a> <a id="15649" class="Symbol">(</a><a id="15650" href="Data.Bool.Properties.html#15650" class="Bound">f</a> <a id="15652" class="Symbol">:</a> <a id="15654" href="Data.Bool.Properties.html#1002" class="Generalizable">A</a> <a id="15656" class="Symbol">→</a> <a id="15658" href="Data.Bool.Properties.html#1016" class="Generalizable">B</a><a id="15659" class="Symbol">)</a> <a id="15661" href="Data.Bool.Properties.html#15661" class="Bound">x</a> <a id="15663" class="Symbol">{</a><a id="15664" href="Data.Bool.Properties.html#15664" class="Bound">y</a> <a id="15666" href="Data.Bool.Properties.html#15666" class="Bound">z</a><a id="15667" class="Symbol">}</a> <a id="15669" class="Symbol">→</a>
|
|||
|
|
<a id="15695" href="Data.Bool.Properties.html#15650" class="Bound">f</a> <a id="15697" class="Symbol">(</a><a id="15698" href="Data.Bool.Base.html#1292" class="Function Operator">if</a> <a id="15701" href="Data.Bool.Properties.html#15661" class="Bound">x</a> <a id="15703" href="Data.Bool.Base.html#1292" class="Function Operator">then</a> <a id="15708" href="Data.Bool.Properties.html#15664" class="Bound">y</a> <a id="15710" href="Data.Bool.Base.html#1292" class="Function Operator">else</a> <a id="15715" href="Data.Bool.Properties.html#15666" class="Bound">z</a><a id="15716" class="Symbol">)</a> <a id="15718" href="Agda.Builtin.Equality.html#133" class="Datatype Operator">≡</a> <a id="15720" class="Symbol">(</a><a id="15721" href="Data.Bool.Base.html#1292" class="Function Operator">if</a> <a id="15724" href="Data.Bool.Properties.html#15661" class="Bound">x</a> <a id="15726" href="Data.Bool.Base.html#1292" class="Function Operator">then</a> <a id="15731" href="Data.Bool.Properties.html#15650" class="Bound">f</a> <a id="15733" href="Data.Bool.Properties.html#15664" class="Bound">y</a> <a id="15735" href="Data.Bool.Base.html#1292" class="Function Operator">else</a> <a id="15740" href="Data.Bool.Properties.html#15650" class="Bound">f</a> <a id="15742" href="Data.Bool.Properties.html#15666" class="Bound">z</a><a id="15743" class="Symbol">)</a>
|
|||
|
|
<a id="15745" href="Data.Bool.Properties.html#15623" class="Function">push-function-into-if</a> <a id="15767" class="Symbol">_</a> <a id="15769" href="Agda.Builtin.Bool.html#181" class="InductiveConstructor">true</a> <a id="15775" class="Symbol">=</a> <a id="15777" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
<a id="15782" href="Data.Bool.Properties.html#15623" class="Function">push-function-into-if</a> <a id="15804" class="Symbol">_</a> <a id="15806" href="Agda.Builtin.Bool.html#175" class="InductiveConstructor">false</a> <a id="15812" class="Symbol">=</a> <a id="15814" href="Agda.Builtin.Equality.html#190" class="InductiveConstructor">refl</a>
|
|||
|
|
|
|||
|
|
<a id="15820" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="15893" class="Comment">-- DEPRECATED NAMES</a>
|
|||
|
|
<a id="15913" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="15986" class="Comment">-- Please use the new names as continuing support for the old names is</a>
|
|||
|
|
<a id="16057" class="Comment">-- not guaranteed.</a>
|
|||
|
|
|
|||
|
|
<a id="16077" class="Comment">-- Version 0.15</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-distˡ"></a><a id="16094" href="Data.Bool.Properties.html#16094" class="Function">∧-∨-distˡ</a> <a id="16106" class="Symbol">=</a> <a id="16108" href="Data.Bool.Properties.html#8346" class="Function">∧-distribˡ-∨</a>
|
|||
|
|
<a id="16121" class="Symbol">{-#</a> <a id="16125" class="Keyword">WARNING_ON_USAGE</a> <a id="16142" class="Pragma">∧-∨-distˡ</a>
|
|||
|
|
<a id="16152" class="String">"Warning: ∧-∨-distˡ was deprecated in v0.15.
|
|||
|
|
Please use ∧-distribˡ-∨ instead."</a>
|
|||
|
|
<a id="16231" class="Symbol">#-}</a>
|
|||
|
|
<a id="∧-∨-distʳ"></a><a id="16235" href="Data.Bool.Properties.html#16235" class="Function">∧-∨-distʳ</a> <a id="16247" class="Symbol">=</a> <a id="16249" href="Data.Bool.Properties.html#8447" class="Function">∧-distribʳ-∨</a>
|
|||
|
|
<a id="16262" class="Symbol">{-#</a> <a id="16266" class="Keyword">WARNING_ON_USAGE</a> <a id="16283" class="Pragma">∧-∨-distʳ</a>
|
|||
|
|
<a id="16293" class="String">"Warning: ∧-∨-distʳ was deprecated in v0.15.
|
|||
|
|
Please use ∧-distribʳ-∨ instead."</a>
|
|||
|
|
<a id="16372" class="Symbol">#-}</a>
|
|||
|
|
<a id="distrib-∧-∨"></a><a id="16376" href="Data.Bool.Properties.html#16376" class="Function">distrib-∧-∨</a> <a id="16388" class="Symbol">=</a> <a id="16390" href="Data.Bool.Properties.html#8676" class="Function">∧-distrib-∨</a>
|
|||
|
|
<a id="16402" class="Symbol">{-#</a> <a id="16406" class="Keyword">WARNING_ON_USAGE</a> <a id="16423" class="Pragma">distrib-∧-∨</a>
|
|||
|
|
<a id="16435" class="String">"Warning: distrib-∧-∨ was deprecated in v0.15.
|
|||
|
|
Please use ∧-distrib-∨ instead."</a>
|
|||
|
|
<a id="16515" class="Symbol">#-}</a>
|
|||
|
|
<a id="∨-∧-distˡ"></a><a id="16519" href="Data.Bool.Properties.html#16519" class="Function">∨-∧-distˡ</a> <a id="16531" class="Symbol">=</a> <a id="16533" href="Data.Bool.Properties.html#8757" class="Function">∨-distribˡ-∧</a>
|
|||
|
|
<a id="16546" class="Symbol">{-#</a> <a id="16550" class="Keyword">WARNING_ON_USAGE</a> <a id="16567" class="Pragma">∨-∧-distˡ</a>
|
|||
|
|
<a id="16577" class="String">"Warning: ∨-∧-distˡ was deprecated in v0.15.
|
|||
|
|
Please use ∨-distribˡ-∧ instead."</a>
|
|||
|
|
<a id="16656" class="Symbol">#-}</a>
|
|||
|
|
<a id="∨-∧-distʳ"></a><a id="16660" href="Data.Bool.Properties.html#16660" class="Function">∨-∧-distʳ</a> <a id="16672" class="Symbol">=</a> <a id="16674" href="Data.Bool.Properties.html#8858" class="Function">∨-distribʳ-∧</a>
|
|||
|
|
<a id="16687" class="Symbol">{-#</a> <a id="16691" class="Keyword">WARNING_ON_USAGE</a> <a id="16708" class="Pragma">∨-∧-distʳ</a>
|
|||
|
|
<a id="16718" class="String">"Warning: ∨-∧-distʳ was deprecated in v0.15.
|
|||
|
|
Please use ∨-distribʳ-∧ instead."</a>
|
|||
|
|
<a id="16797" class="Symbol">#-}</a>
|
|||
|
|
<a id="∨-∧-distrib"></a><a id="16801" href="Data.Bool.Properties.html#16801" class="Function">∨-∧-distrib</a> <a id="16813" class="Symbol">=</a> <a id="16815" href="Data.Bool.Properties.html#9099" class="Function">∨-distrib-∧</a>
|
|||
|
|
<a id="16827" class="Symbol">{-#</a> <a id="16831" class="Keyword">WARNING_ON_USAGE</a> <a id="16848" class="Pragma">∨-∧-distrib</a>
|
|||
|
|
<a id="16860" class="String">"Warning: ∨-∧-distrib was deprecated in v0.15.
|
|||
|
|
Please use ∨-distrib-∧ instead."</a>
|
|||
|
|
<a id="16940" class="Symbol">#-}</a>
|
|||
|
|
<a id="∨-∧-abs"></a><a id="16944" href="Data.Bool.Properties.html#16944" class="Function">∨-∧-abs</a> <a id="16955" class="Symbol">=</a> <a id="16957" href="Data.Bool.Properties.html#9253" class="Function">∨-abs-∧</a>
|
|||
|
|
<a id="16965" class="Symbol">{-#</a> <a id="16969" class="Keyword">WARNING_ON_USAGE</a> <a id="16986" class="Pragma">∨-∧-abs</a>
|
|||
|
|
<a id="16994" class="String">"Warning: ∨-∧-abs was deprecated in v0.15.
|
|||
|
|
Please use ∨-abs-∧ instead."</a>
|
|||
|
|
<a id="17066" class="Symbol">#-}</a>
|
|||
|
|
<a id="∧-∨-abs"></a><a id="17070" href="Data.Bool.Properties.html#17070" class="Function">∧-∨-abs</a> <a id="17081" class="Symbol">=</a> <a id="17083" href="Data.Bool.Properties.html#9180" class="Function">∧-abs-∨</a>
|
|||
|
|
<a id="17091" class="Symbol">{-#</a> <a id="17095" class="Keyword">WARNING_ON_USAGE</a> <a id="17112" class="Pragma">∧-∨-abs</a>
|
|||
|
|
<a id="17120" class="String">"Warning: ∧-∨-abs was deprecated in v0.15.
|
|||
|
|
Please use ∧-abs-∨ instead."</a>
|
|||
|
|
<a id="17192" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∧-inverseˡ"></a><a id="17196" href="Data.Bool.Properties.html#17196" class="Function">not-∧-inverseˡ</a> <a id="17211" class="Symbol">=</a> <a id="17213" href="Data.Bool.Properties.html#7952" class="Function">∧-inverseˡ</a>
|
|||
|
|
<a id="17224" class="Symbol">{-#</a> <a id="17228" class="Keyword">WARNING_ON_USAGE</a> <a id="17245" class="Pragma">not-∧-inverseˡ</a>
|
|||
|
|
<a id="17260" class="String">"Warning: not-∧-inverseˡ was deprecated in v0.15.
|
|||
|
|
Please use ∧-inverseˡ instead."</a>
|
|||
|
|
<a id="17342" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∧-inverseʳ"></a><a id="17346" href="Data.Bool.Properties.html#17346" class="Function">not-∧-inverseʳ</a> <a id="17361" class="Symbol">=</a> <a id="17363" href="Data.Bool.Properties.html#8039" class="Function">∧-inverseʳ</a>
|
|||
|
|
<a id="17374" class="Symbol">{-#</a> <a id="17378" class="Keyword">WARNING_ON_USAGE</a> <a id="17395" class="Pragma">not-∧-inverseʳ</a>
|
|||
|
|
<a id="17410" class="String">"Warning: not-∧-inverseʳ was deprecated in v0.15.
|
|||
|
|
Please use ∧-inverseʳ instead."</a>
|
|||
|
|
<a id="17492" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∧-inverse"></a><a id="17496" href="Data.Bool.Properties.html#17496" class="Function">not-∧-inverse</a> <a id="17510" class="Symbol">=</a> <a id="17512" href="Data.Bool.Properties.html#8135" class="Function">∧-inverse</a>
|
|||
|
|
<a id="17522" class="Symbol">{-#</a> <a id="17526" class="Keyword">WARNING_ON_USAGE</a> <a id="17543" class="Pragma">not-∧-inverse</a>
|
|||
|
|
<a id="17557" class="String">"Warning: not-∧-inverse was deprecated in v0.15.
|
|||
|
|
Please use ∧-inverse instead."</a>
|
|||
|
|
<a id="17637" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∨-inverseˡ"></a><a id="17641" href="Data.Bool.Properties.html#17641" class="Function">not-∨-inverseˡ</a> <a id="17656" class="Symbol">=</a> <a id="17658" href="Data.Bool.Properties.html#5366" class="Function">∨-inverseˡ</a>
|
|||
|
|
<a id="17669" class="Symbol">{-#</a> <a id="17673" class="Keyword">WARNING_ON_USAGE</a> <a id="17690" class="Pragma">not-∨-inverseˡ</a>
|
|||
|
|
<a id="17705" class="String">"Warning: not-∨-inverseˡ was deprecated in v0.15.
|
|||
|
|
Please use ∨-inverseˡ instead."</a>
|
|||
|
|
<a id="17787" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∨-inverseʳ"></a><a id="17791" href="Data.Bool.Properties.html#17791" class="Function">not-∨-inverseʳ</a> <a id="17806" class="Symbol">=</a> <a id="17808" href="Data.Bool.Properties.html#5453" class="Function">∨-inverseʳ</a>
|
|||
|
|
<a id="17819" class="Symbol">{-#</a> <a id="17823" class="Keyword">WARNING_ON_USAGE</a> <a id="17840" class="Pragma">not-∨-inverseʳ</a>
|
|||
|
|
<a id="17855" class="String">"Warning: not-∨-inverseʳ was deprecated in v0.15.
|
|||
|
|
Please use ∨-inverseʳ instead."</a>
|
|||
|
|
<a id="17937" class="Symbol">#-}</a>
|
|||
|
|
<a id="not-∨-inverse"></a><a id="17941" href="Data.Bool.Properties.html#17941" class="Function">not-∨-inverse</a> <a id="17955" class="Symbol">=</a> <a id="17957" href="Data.Bool.Properties.html#5548" class="Function">∨-inverse</a>
|
|||
|
|
<a id="17967" class="Symbol">{-#</a> <a id="17971" class="Keyword">WARNING_ON_USAGE</a> <a id="17988" class="Pragma">not-∨-inverse</a>
|
|||
|
|
<a id="18002" class="String">"Warning: not-∨-inverse was deprecated in v0.15.
|
|||
|
|
Please use ∨-inverse instead."</a>
|
|||
|
|
<a id="18082" class="Symbol">#-}</a>
|
|||
|
|
<a id="isCommutativeSemiring-∨-∧"></a><a id="18086" href="Data.Bool.Properties.html#18086" class="Function">isCommutativeSemiring-∨-∧</a> <a id="18112" class="Symbol">=</a> <a id="18114" href="Data.Bool.Properties.html#11140" class="Function">∨-∧-isCommutativeSemiring</a>
|
|||
|
|
<a id="18140" class="Symbol">{-#</a> <a id="18144" class="Keyword">WARNING_ON_USAGE</a> <a id="18161" class="Pragma">isCommutativeSemiring-∨-∧</a>
|
|||
|
|
<a id="18187" class="String">"Warning: isCommutativeSemiring-∨-∧ was deprecated in v0.15.
|
|||
|
|
Please use ∨-∧-isCommutativeSemiring instead."</a>
|
|||
|
|
<a id="18295" class="Symbol">#-}</a>
|
|||
|
|
<a id="commutativeSemiring-∨-∧"></a><a id="18299" href="Data.Bool.Properties.html#18299" class="Function">commutativeSemiring-∨-∧</a> <a id="18325" class="Symbol">=</a> <a id="18328" href="Data.Bool.Properties.html#11303" class="Function">∨-∧-commutativeSemiring</a>
|
|||
|
|
<a id="18352" class="Symbol">{-#</a> <a id="18356" class="Keyword">WARNING_ON_USAGE</a> <a id="18373" class="Pragma">commutativeSemiring-∨-∧</a>
|
|||
|
|
<a id="18397" class="String">"Warning: commutativeSemiring-∨-∧ was deprecated in v0.15.
|
|||
|
|
Please use ∨-∧-commutativeSemiring instead."</a>
|
|||
|
|
<a id="18501" class="Symbol">#-}</a>
|
|||
|
|
<a id="isCommutativeSemiring-∧-∨"></a><a id="18505" href="Data.Bool.Properties.html#18505" class="Function">isCommutativeSemiring-∧-∨</a> <a id="18531" class="Symbol">=</a> <a id="18533" href="Data.Bool.Properties.html#11835" class="Function">∧-∨-isCommutativeSemiring</a>
|
|||
|
|
<a id="18559" class="Symbol">{-#</a> <a id="18563" class="Keyword">WARNING_ON_USAGE</a> <a id="18580" class="Pragma">isCommutativeSemiring-∧-∨</a>
|
|||
|
|
<a id="18606" class="String">"Warning: isCommutativeSemiring-∧-∨ was deprecated in v0.15.
|
|||
|
|
Please use ∧-∨-isCommutativeSemiring instead."</a>
|
|||
|
|
<a id="18714" class="Symbol">#-}</a>
|
|||
|
|
<a id="commutativeSemiring-∧-∨"></a><a id="18718" href="Data.Bool.Properties.html#18718" class="Function">commutativeSemiring-∧-∨</a> <a id="18744" class="Symbol">=</a> <a id="18746" href="Data.Bool.Properties.html#11998" class="Function">∧-∨-commutativeSemiring</a>
|
|||
|
|
<a id="18770" class="Symbol">{-#</a> <a id="18774" class="Keyword">WARNING_ON_USAGE</a> <a id="18791" class="Pragma">commutativeSemiring-∧-∨</a>
|
|||
|
|
<a id="18815" class="String">"Warning: commutativeSemiring-∧-∨ was deprecated in v0.15.
|
|||
|
|
Please use ∧-∨-commutativeSemiring instead."</a>
|
|||
|
|
<a id="18919" class="Symbol">#-}</a>
|
|||
|
|
<a id="isBooleanAlgebra"></a><a id="18923" href="Data.Bool.Properties.html#18923" class="Function">isBooleanAlgebra</a> <a id="18949" class="Symbol">=</a> <a id="18951" href="Data.Bool.Properties.html#12965" class="Function">∨-∧-isBooleanAlgebra</a>
|
|||
|
|
<a id="18972" class="Symbol">{-#</a> <a id="18976" class="Keyword">WARNING_ON_USAGE</a> <a id="18993" class="Pragma">isBooleanAlgebra</a>
|
|||
|
|
<a id="19010" class="String">"Warning: isBooleanAlgebra was deprecated in v0.15.
|
|||
|
|
Please use ∨-∧-isBooleanAlgebra instead."</a>
|
|||
|
|
<a id="19104" class="Symbol">#-}</a>
|
|||
|
|
<a id="booleanAlgebra"></a><a id="19108" href="Data.Bool.Properties.html#19108" class="Function">booleanAlgebra</a> <a id="19134" class="Symbol">=</a> <a id="19136" href="Data.Bool.Properties.html#13208" class="Function">∨-∧-booleanAlgebra</a>
|
|||
|
|
<a id="19155" class="Symbol">{-#</a> <a id="19159" class="Keyword">WARNING_ON_USAGE</a> <a id="19176" class="Pragma">booleanAlgebra</a>
|
|||
|
|
<a id="19191" class="String">"Warning: booleanAlgebra was deprecated in v0.15.
|
|||
|
|
Please use ∨-∧-booleanAlgebra instead."</a>
|
|||
|
|
<a id="19281" class="Symbol">#-}</a>
|
|||
|
|
<a id="commutativeRing-xor-∧"></a><a id="19285" href="Data.Bool.Properties.html#19285" class="Function">commutativeRing-xor-∧</a> <a id="19311" class="Symbol">=</a> <a id="19313" href="Data.Bool.Properties.html#13542" class="Function">xor-∧-commutativeRing</a>
|
|||
|
|
<a id="19335" class="Symbol">{-#</a> <a id="19339" class="Keyword">WARNING_ON_USAGE</a> <a id="19356" class="Pragma">commutativeRing-xor-∧</a>
|
|||
|
|
<a id="19378" class="String">"Warning: commutativeRing-xor-∧ was deprecated in v0.15.
|
|||
|
|
Please use xor-∧-commutativeRing instead."</a>
|
|||
|
|
<a id="19478" class="Symbol">#-}</a>
|
|||
|
|
<a id="proof-irrelevance"></a><a id="19482" href="Data.Bool.Properties.html#19482" class="Function">proof-irrelevance</a> <a id="19500" class="Symbol">=</a> <a id="19502" href="Data.Bool.Properties.html#15414" class="Function">T-irrelevant</a>
|
|||
|
|
<a id="19515" class="Symbol">{-#</a> <a id="19519" class="Keyword">WARNING_ON_USAGE</a> <a id="19536" class="Pragma">proof-irrelevance</a>
|
|||
|
|
<a id="19554" class="String">"Warning: proof-irrelevance was deprecated in v0.15.
|
|||
|
|
Please use T-irrelevant instead."</a>
|
|||
|
|
<a id="19641" class="Symbol">#-}</a>
|
|||
|
|
|
|||
|
|
<a id="19646" class="Comment">-- Version 1.0</a>
|
|||
|
|
|
|||
|
|
<a id="T-irrelevance"></a><a id="19662" href="Data.Bool.Properties.html#19662" class="Function">T-irrelevance</a> <a id="19676" class="Symbol">=</a> <a id="19678" href="Data.Bool.Properties.html#15414" class="Function">T-irrelevant</a>
|
|||
|
|
<a id="19691" class="Symbol">{-#</a> <a id="19695" class="Keyword">WARNING_ON_USAGE</a> <a id="19712" class="Pragma">T-irrelevance</a>
|
|||
|
|
<a id="19726" class="String">"Warning: T-irrelevance was deprecated in v1.0.
|
|||
|
|
Please use T-irrelevant instead."</a>
|
|||
|
|
<a id="19808" class="Symbol">#-}</a>
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</pre></body></html>
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