leonv/bsc-leon-vatthauer
1
0
Fork 0
mirror of https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git synced 2024-05-31 07:28:34 +02:00
Code Issues Projects Releases Packages Wiki Activity
bsc-leon-vatthauer/agda/bsc-thesis/Data.Bool.Properties.html

790 lines
201 KiB
HTML
Raw Normal View History

update index
2024-02-09 17:53:52 +01:00
<!DOCTYPE HTML>
<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>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- A bunch of properties</a>
<a id="131" class="Comment">------------------------------------------------------------------------</a>
<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>
<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>
<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>
<a id="313" class="Keyword">open</a> <a id="318" class="Keyword">import</a> <a id="325" href="Algebra.Lattice.Bundles.html" class="Module">Algebra.Lattice.Bundles</a>
<a id="349" class="Keyword">import</a> <a id="356" href="Algebra.Lattice.Properties.BooleanAlgebra.html" class="Module">Algebra.Lattice.Properties.BooleanAlgebra</a> <a id="398" class="Symbol">as</a> <a id="401" class="Module">BooleanAlgebraProperties</a>
<a id="426" class="Keyword">open</a> <a id="431" class="Keyword">import</a> <a id="438" href="Data.Bool.Base.html" class="Module">Data.Bool.Base</a>
<a id="453" class="Keyword">open</a> <a id="458" class="Keyword">import</a> <a id="465" href="Data.Empty.html" class="Module">Data.Empty</a>
<a id="476" class="Keyword">open</a> <a id="481" class="Keyword">import</a> <a id="488" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="506" class="Keyword">using</a> <a id="512" class="Symbol">(</a><a id="513" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="516" class="Symbol">;</a> <a id="518" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="521" class="Symbol">;</a> <a id="523" href="Data.Product.Base.html#636" class="Field">proj₁</a><a id="528" class="Symbol">;</a> <a id="530" href="Data.Product.Base.html#650" class="Field">proj₂</a><a id="535" class="Symbol">)</a>
<a id="537" class="Keyword">open</a> <a id="542" class="Keyword">import</a> <a id="549" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a> <a id="563" class="Keyword">using</a> <a id="569" class="Symbol">(</a><a id="570" href="Data.Sum.Base.html#625" class="Datatype Operator">_⊎_</a><a id="573" class="Symbol">;</a> <a id="575" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="579" class="Symbol">;</a> <a id="581" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="585" class="Symbol">;</a> <a id="587" href="Data.Sum.Base.html#811" class="Function Operator">[_,_]</a><a id="592" class="Symbol">)</a>
<a id="594" class="Keyword">open</a> <a id="599" class="Keyword">import</a> <a id="606" href="Function.Base.html" class="Module">Function.Base</a> <a id="620" class="Keyword">using</a> <a id="626" class="Symbol">(</a><a id="627" href="Function.Base.html#4322" class="Function Operator">_⟨_⟩_</a><a id="632" class="Symbol">;</a> <a id="634" href="Function.Base.html#725" class="Function">const</a><a id="639" class="Symbol">;</a> <a id="641" href="Function.Base.html#704" class="Function">id</a><a id="643" class="Symbol">)</a>
<a id="645" class="Keyword">open</a> <a id="650" class="Keyword">import</a> <a id="657" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="674" class="Keyword">hiding</a> <a id="681" class="Symbol">(</a><a id="682" href="Function.Bundles.html#5375" class="Record">LeftInverse</a><a id="693" class="Symbol">;</a> <a id="695" href="Function.Bundles.html#6509" class="Record">RightInverse</a><a id="707" class="Symbol">;</a> <a id="709" href="Function.Bundles.html#7340" class="Record">Inverse</a><a id="716" class="Symbol">)</a>
<a id="718" class="Keyword">open</a> <a id="723" class="Keyword">import</a> <a id="730" href="Induction.WellFounded.html" class="Module">Induction.WellFounded</a> <a id="752" class="Keyword">using</a> <a id="758" class="Symbol">(</a><a id="759" href="Induction.WellFounded.html#1604" class="Function">WellFounded</a><a id="770" class="Symbol">;</a> <a id="772" href="Induction.WellFounded.html#1356" class="Datatype">Acc</a><a id="775" class="Symbol">;</a> <a id="777" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a><a id="780" class="Symbol">)</a>
<a id="782" class="Keyword">open</a> <a id="787" class="Keyword">import</a> <a id="794" href="Level.html" class="Module">Level</a> <a id="800" class="Keyword">using</a> <a id="806" class="Symbol">(</a><a id="807" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="812" class="Symbol">;</a> <a id="814" href="Level.html#521" class="Function">0ℓ</a><a id="816" class="Symbol">)</a>
<a id="818" class="Keyword">open</a> <a id="823" class="Keyword">import</a> <a id="830" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="851" class="Keyword">using</a> <a id="857" class="Symbol">(</a><a id="858" href="Relation.Binary.Core.html#1268" class="Function Operator">_⇒_</a><a id="861" class="Symbol">)</a>
<a id="863" class="Keyword">open</a> <a id="868" class="Keyword">import</a> <a id="875" href="Relation.Binary.Structures.html" class="Module">Relation.Binary.Structures</a>
<a id="904" class="Keyword">using</a> <a id="910" class="Symbol">(</a><a id="911" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a><a id="921" class="Symbol">;</a> <a id="923" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a><a id="937" class="Symbol">;</a> <a id="939" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a><a id="951" class="Symbol">;</a> <a id="953" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a><a id="968" class="Symbol">;</a> <a id="970" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a><a id="990" class="Symbol">;</a> <a id="992" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a><a id="1010" class="Symbol">)</a>
<a id="1012" class="Keyword">open</a> <a id="1017" class="Keyword">import</a> <a id="1024" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a>
<a id="1050" class="Keyword">using</a> <a id="1056" class="Symbol">(</a><a id="1057" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="1063" class="Symbol">;</a> <a id="1065" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a><a id="1074" class="Symbol">;</a> <a id="1076" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="1081" class="Symbol">;</a> <a id="1083" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a><a id="1091" class="Symbol">;</a> <a id="1093" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a><a id="1103" class="Symbol">;</a> <a id="1105" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a><a id="1118" class="Symbol">;</a> <a id="1120" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a><a id="1138" class="Symbol">;</a> <a id="1140" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a><a id="1156" class="Symbol">)</a>
<a id="1158" class="Keyword">open</a> <a id="1163" class="Keyword">import</a> <a id="1170" href="Relation.Binary.Definitions.html" class="Module">Relation.Binary.Definitions</a>
<a id="1200" class="Keyword">using</a> <a id="1206" class="Symbol">(</a><a id="1207" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a><a id="1216" class="Symbol">;</a> <a id="1218" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a><a id="1227" class="Symbol">;</a> <a id="1229" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a><a id="1239" class="Symbol">;</a> <a id="1241" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a><a id="1254" class="Symbol">;</a> <a id="1256" href="Relation.Binary.Definitions.html#3663" class="Function">Minimum</a><a id="1263" class="Symbol">;</a> <a id="1265" href="Relation.Binary.Definitions.html#3514" class="Function">Maximum</a><a id="1272" class="Symbol">;</a> <a id="1274" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a><a id="1279" class="Symbol">;</a> <a id="1281" href="Relation.Binary.Definitions.html#6651" class="Function">Irrelevant</a><a id="1291" class="Symbol">;</a> <a id="1293" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a><a id="1304" class="Symbol">;</a> <a id="1306" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a><a id="1316" class="Symbol">;</a> <a id="1318" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a><a id="1323" class="Symbol">;</a> <a id="1325" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a><a id="1337" class="Symbol">;</a> <a id="1339" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a><a id="1343" class="Symbol">;</a> <a id="1345" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri&lt;</a><a id="1349" class="Symbol">;</a> <a id="1351" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri&gt;</a><a id="1355" class="Symbol">;</a> <a id="1357" href="Relation.Binary.Definitions.html#5663" class="Function Operator">_Respects₂_</a><a id="1368" class="Symbol">)</a>
<a id="1370" class="Keyword">open</a> <a id="1375" class="Keyword">import</a> <a id="1382" href="Relation.Binary.PropositionalEquality.Core.html" class="Module">Relation.Binary.PropositionalEquality.Core</a>
<a id="1425" class="Keyword">open</a> <a id="1430" class="Keyword">import</a> <a id="1437" href="Relation.Binary.PropositionalEquality.Properties.html" class="Module">Relation.Binary.PropositionalEquality.Properties</a>
<a id="1486" class="Keyword">open</a> <a id="1491" class="Keyword">import</a> <a id="1498" href="Relation.Nullary.Decidable.Core.html" class="Module">Relation.Nullary.Decidable.Core</a> <a id="1530" class="Keyword">using</a> <a id="1536" class="Symbol">(</a><a id="1537" href="Relation.Nullary.Decidable.Core.html#3468" class="Function">True</a><a id="1541" class="Symbol">;</a> <a id="1543" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a><a id="1546" class="Symbol">;</a> <a id="1548" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a><a id="1550" class="Symbol">;</a> <a id="1552" href="Relation.Nullary.Decidable.Core.html#3930" class="Function">fromWitness</a><a id="1563" class="Symbol">)</a>
<a id="1565" class="Keyword">import</a> <a id="1572" href="Relation.Unary.html" class="Module">Relation.Unary</a> <a id="1587" class="Symbol">as</a> <a id="1590" class="Module">U</a>
<a id="1593" class="Keyword">open</a> <a id="1598" class="Keyword">import</a> <a id="1605" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="1625" class="Symbol">{</a><a id="1626" class="Argument">A</a> <a id="1628" class="Symbol">=</a> <a id="1630" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a><a id="1634" class="Symbol">}</a> <a id="1636" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1640" class="Keyword">open</a> <a id="1645" class="Keyword">import</a> <a id="1652" href="Algebra.Structures.html" class="Module">Algebra.Structures</a> <a id="1671" class="Symbol">{</a><a id="1672" class="Argument">A</a> <a id="1674" class="Symbol">=</a> <a id="1676" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a><a id="1680" class="Symbol">}</a> <a id="1682" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1686" class="Keyword">open</a> <a id="1691" class="Keyword">import</a> <a id="1698" href="Algebra.Lattice.Structures.html" class="Module">Algebra.Lattice.Structures</a> <a id="1725" class="Symbol">{</a><a id="1726" class="Argument">A</a> <a id="1728" class="Symbol">=</a> <a id="1730" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a><a id="1734" class="Symbol">}</a> <a id="1736" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1741" class="Keyword">open</a> <a id="1746" href="Relation.Binary.PropositionalEquality.Properties.html#6744" class="Module">≡-Reasoning</a>
<a id="1759" class="Keyword">private</a>
<a id="1769" class="Keyword">variable</a>
<a id="1782" href="Data.Bool.Properties.html#1782" class="Generalizable">a</a> <a id="1784" href="Data.Bool.Properties.html#1784" class="Generalizable">b</a> <a id="1786" class="Symbol">:</a> <a id="1788" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="1798" href="Data.Bool.Properties.html#1798" class="Generalizable">A</a> <a id="1800" class="Symbol">:</a> <a id="1802" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1806" href="Data.Bool.Properties.html#1782" class="Generalizable">a</a>
<a id="1812" href="Data.Bool.Properties.html#1812" class="Generalizable">B</a> <a id="1814" class="Symbol">:</a> <a id="1816" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1820" href="Data.Bool.Properties.html#1784" class="Generalizable">b</a>
<a id="1823" class="Comment">------------------------------------------------------------------------</a>
<a id="1896" class="Comment">-- Properties of _≡_</a>
<a id="1918" class="Keyword">infix</a> <a id="1924" class="Number">4</a> <a id="1926" href="Data.Bool.Properties.html#1931" class="Function Operator">_≟_</a>
<a id="_≟_"></a><a id="1931" href="Data.Bool.Properties.html#1931" class="Function Operator">_≟_</a> <a id="1935" class="Symbol">:</a> <a id="1937" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="1947" class="Symbol">{</a><a id="1948" class="Argument">A</a> <a id="1950" class="Symbol">=</a> <a id="1952" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a><a id="1956" class="Symbol">}</a> <a id="1958" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="1962" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="1968" href="Data.Bool.Properties.html#1931" class="Function Operator">≟</a> <a id="1970" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="1976" class="Symbol">=</a> <a id="1978" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="1982" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="1987" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="1993" href="Data.Bool.Properties.html#1931" class="Function Operator">≟</a> <a id="1995" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2001" class="Symbol">=</a> <a id="2003" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="2007" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2012" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2018" href="Data.Bool.Properties.html#1931" class="Function Operator">≟</a> <a id="2020" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2026" class="Symbol">=</a> <a id="2028" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="2031" class="Symbol">λ()</a>
<a id="2035" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2041" href="Data.Bool.Properties.html#1931" class="Function Operator">≟</a> <a id="2043" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2049" class="Symbol">=</a> <a id="2051" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="2054" class="Symbol">λ()</a>
<a id="≡-setoid"></a><a id="2059" href="Data.Bool.Properties.html#2059" class="Function">≡-setoid</a> <a id="2068" class="Symbol">:</a> <a id="2070" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="2077" href="Level.html#521" class="Function">0ℓ</a> <a id="2080" href="Level.html#521" class="Function">0ℓ</a>
<a id="2083" href="Data.Bool.Properties.html#2059" class="Function">≡-setoid</a> <a id="2092" class="Symbol">=</a> <a id="2094" href="Relation.Binary.PropositionalEquality.Properties.html#5700" class="Function">setoid</a> <a id="2101" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a>
<a id="≡-decSetoid"></a><a id="2107" href="Data.Bool.Properties.html#2107" class="Function">≡-decSetoid</a> <a id="2119" class="Symbol">:</a> <a id="2121" href="Relation.Binary.Bundles.html#1548" class="Record">DecSetoid</a> <a id="2131" href="Level.html#521" class="Function">0ℓ</a> <a id="2134" href="Level.html#521" class="Function">0ℓ</a>
<a id="2137" href="Data.Bool.Properties.html#2107" class="Function">≡-decSetoid</a> <a id="2149" class="Symbol">=</a> <a id="2151" href="Relation.Binary.PropositionalEquality.Properties.html#5831" class="Function">decSetoid</a> <a id="2161" href="Data.Bool.Properties.html#1931" class="Function Operator">_≟_</a>
<a id="2166" class="Comment">------------------------------------------------------------------------</a>
<a id="2239" class="Comment">-- Properties of _≤_</a>
<a id="2261" class="Comment">-- Relational properties</a>
<a id="≤-reflexive"></a><a id="2287" href="Data.Bool.Properties.html#2287" class="Function">≤-reflexive</a> <a id="2299" class="Symbol">:</a> <a id="2301" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="2305" href="Relation.Binary.Core.html#1268" class="Function Operator">⇒</a> <a id="2307" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2311" href="Data.Bool.Properties.html#2287" class="Function">≤-reflexive</a> <a id="2323" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="2328" class="Symbol">=</a> <a id="2330" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
<a id="≤-refl"></a><a id="2335" href="Data.Bool.Properties.html#2335" class="Function">≤-refl</a> <a id="2342" class="Symbol">:</a> <a id="2344" href="Relation.Binary.Definitions.html#1428" class="Function">Reflexive</a> <a id="2354" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2358" href="Data.Bool.Properties.html#2335" class="Function">≤-refl</a> <a id="2365" class="Symbol">=</a> <a id="2367" href="Data.Bool.Properties.html#2287" class="Function">≤-reflexive</a> <a id="2379" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="≤-trans"></a><a id="2385" href="Data.Bool.Properties.html#2385" class="Function">≤-trans</a> <a id="2393" class="Symbol">:</a> <a id="2395" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="2406" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2410" href="Data.Bool.Properties.html#2385" class="Function">≤-trans</a> <a id="2418" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2422" href="Data.Bool.Properties.html#2422" class="Bound">p</a> <a id="2426" class="Symbol">=</a> <a id="2428" href="Data.Bool.Properties.html#2422" class="Bound">p</a>
<a id="2430" href="Data.Bool.Properties.html#2385" class="Function">≤-trans</a> <a id="2438" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="2442" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2446" class="Symbol">=</a> <a id="2448" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
<a id="≤-antisym"></a><a id="2453" href="Data.Bool.Properties.html#2453" class="Function">≤-antisym</a> <a id="2463" class="Symbol">:</a> <a id="2465" href="Relation.Binary.Definitions.html#2319" class="Function">Antisymmetric</a> <a id="2479" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="2483" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2487" href="Data.Bool.Properties.html#2453" class="Function">≤-antisym</a> <a id="2497" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2501" class="Symbol">_</a> <a id="2503" class="Symbol">=</a> <a id="2505" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="≤-minimum"></a><a id="2511" href="Data.Bool.Properties.html#2511" class="Function">≤-minimum</a> <a id="2521" class="Symbol">:</a> <a id="2523" href="Relation.Binary.Definitions.html#3663" class="Function">Minimum</a> <a id="2531" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="2535" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="2541" href="Data.Bool.Properties.html#2511" class="Function">≤-minimum</a> <a id="2551" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2557" class="Symbol">=</a> <a id="2559" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
<a id="2563" href="Data.Bool.Properties.html#2511" class="Function">≤-minimum</a> <a id="2573" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2579" class="Symbol">=</a> <a id="2581" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
<a id="≤-maximum"></a><a id="2586" href="Data.Bool.Properties.html#2586" class="Function">≤-maximum</a> <a id="2596" class="Symbol">:</a> <a id="2598" href="Relation.Binary.Definitions.html#3514" class="Function">Maximum</a> <a id="2606" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="2610" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="2615" href="Data.Bool.Properties.html#2586" class="Function">≤-maximum</a> <a id="2625" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2631" class="Symbol">=</a> <a id="2633" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a>
<a id="2637" href="Data.Bool.Properties.html#2586" class="Function">≤-maximum</a> <a id="2647" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2653" class="Symbol">=</a> <a id="2655" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
<a id="≤-total"></a><a id="2660" href="Data.Bool.Properties.html#2660" class="Function">≤-total</a> <a id="2668" class="Symbol">:</a> <a id="2670" href="Relation.Binary.Definitions.html#2910" class="Function">Total</a> <a id="2676" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2680" href="Data.Bool.Properties.html#2660" class="Function">≤-total</a> <a id="2688" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2694" href="Data.Bool.Properties.html#2694" class="Bound">b</a> <a id="2696" class="Symbol">=</a> <a id="2698" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2703" class="Symbol">(</a><a id="2704" href="Data.Bool.Properties.html#2511" class="Function">≤-minimum</a> <a id="2714" href="Data.Bool.Properties.html#2694" class="Bound">b</a><a id="2715" class="Symbol">)</a>
<a id="2717" href="Data.Bool.Properties.html#2660" class="Function">≤-total</a> <a id="2725" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2731" href="Data.Bool.Properties.html#2731" class="Bound">b</a> <a id="2733" class="Symbol">=</a> <a id="2735" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2740" class="Symbol">(</a><a id="2741" href="Data.Bool.Properties.html#2586" class="Function">≤-maximum</a> <a id="2751" href="Data.Bool.Properties.html#2731" class="Bound">b</a><a id="2752" class="Symbol">)</a>
<a id="2755" class="Keyword">infix</a> <a id="2761" class="Number">4</a> <a id="2763" href="Data.Bool.Properties.html#2769" class="Function Operator">_≤?_</a>
<a id="_≤?_"></a><a id="2769" href="Data.Bool.Properties.html#2769" class="Function Operator">_≤?_</a> <a id="2774" class="Symbol">:</a> <a id="2776" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="2786" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2790" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2796" href="Data.Bool.Properties.html#2769" class="Function Operator">≤?</a> <a id="2799" href="Data.Bool.Properties.html#2799" class="Bound">b</a> <a id="2805" class="Symbol">=</a> <a id="2807" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="2811" class="Symbol">(</a><a id="2812" href="Data.Bool.Properties.html#2511" class="Function">≤-minimum</a> <a id="2822" href="Data.Bool.Properties.html#2799" class="Bound">b</a><a id="2823" class="Symbol">)</a>
<a id="2825" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2831" href="Data.Bool.Properties.html#2769" class="Function Operator">≤?</a> <a id="2834" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="2840" class="Symbol">=</a> <a id="2842" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="2845" class="Symbol">λ</a> <a id="2847" class="Symbol">()</a>
<a id="2850" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2856" href="Data.Bool.Properties.html#2769" class="Function Operator">≤?</a> <a id="2859" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="2865" class="Symbol">=</a> <a id="2867" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="2871" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a>
<a id="≤-irrelevant"></a><a id="2876" href="Data.Bool.Properties.html#2876" class="Function">≤-irrelevant</a> <a id="2889" class="Symbol">:</a> <a id="2891" href="Relation.Binary.Definitions.html#6651" class="Function">Irrelevant</a> <a id="2902" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="2906" href="Data.Bool.Properties.html#2876" class="Function">≤-irrelevant</a> <a id="2919" class="Symbol">{_}</a> <a id="2927" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="2931" href="Data.Bool.Base.html#713" class="InductiveConstructor">f≤t</a> <a id="2935" class="Symbol">=</a> <a id="2937" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2942" href="Data.Bool.Properties.html#2876" class="Function">≤-irrelevant</a> <a id="2955" class="Symbol">{</a><a id="2956" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="2961" class="Symbol">}</a> <a id="2963" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2967" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="2971" class="Symbol">=</a> <a id="2973" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2978" href="Data.Bool.Properties.html#2876" class="Function">≤-irrelevant</a> <a id="2991" class="Symbol">{</a><a id="2992" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="2996" class="Symbol">}</a> <a id="2999" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="3003" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="3007" class="Symbol">=</a> <a id="3009" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="3015" class="Comment">-- Structures</a>
<a id="≤-isPreorder"></a><a id="3030" href="Data.Bool.Properties.html#3030" class="Function">≤-isPreorder</a> <a id="3043" class="Symbol">:</a> <a id="3045" href="Relation.Binary.Structures.html#2191" class="Record">IsPreorder</a> <a id="3056" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="3060" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="3064" href="Data.Bool.Properties.html#3030" class="Function">≤-isPreorder</a> <a id="3077" class="Symbol">=</a> <a id="3079" class="Keyword">record</a>
<a id="3088" class="Symbol">{</a> <a id="3090" href="Relation.Binary.Structures.html#2256" class="Field">isEquivalence</a> <a id="3104" class="Symbol">=</a> <a id="3106" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
<a id="3122" class="Symbol">;</a> <a id="3124" href="Relation.Binary.Structures.html#2359" class="Field">reflexive</a> <a id="3138" class="Symbol">=</a> <a id="3140" href="Data.Bool.Properties.html#2287" class="Function">≤-reflexive</a>
<a id="3154" class="Symbol">;</a> <a id="3156" href="Relation.Binary.Structures.html#2389" class="Field">trans</a> <a id="3170" class="Symbol">=</a> <a id="3172" href="Data.Bool.Properties.html#2385" class="Function">≤-trans</a>
<a id="3182" class="Symbol">}</a>
<a id="≤-isPartialOrder"></a><a id="3185" href="Data.Bool.Properties.html#3185" class="Function">≤-isPartialOrder</a> <a id="3202" class="Symbol">:</a> <a id="3204" href="Relation.Binary.Structures.html#3522" class="Record">IsPartialOrder</a> <a id="3219" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="3223" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="3227" href="Data.Bool.Properties.html#3185" class="Function">≤-isPartialOrder</a> <a id="3244" class="Symbol">=</a> <a id="3246" class="Keyword">record</a>
<a id="3255" class="Symbol">{</a> <a id="3257" href="Relation.Binary.Structures.html#3591" class="Field">isPreorder</a> <a id="3268" class="Symbol">=</a> <a id="3270" href="Data.Bool.Properties.html#3030" class="Function">≤-isPreorder</a>
<a id="3285" class="Symbol">;</a> <a id="3287" href="Relation.Binary.Structures.html#3623" class="Field">antisym</a> <a id="3298" class="Symbol">=</a> <a id="3300" href="Data.Bool.Properties.html#2453" class="Function">≤-antisym</a>
<a id="3312" class="Symbol">}</a>
<a id="≤-isTotalOrder"></a><a id="3315" href="Data.Bool.Properties.html#3315" class="Function">≤-isTotalOrder</a> <a id="3330" class="Symbol">:</a> <a id="3332" href="Relation.Binary.Structures.html#5508" class="Record">IsTotalOrder</a> <a id="3345" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="3349" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="3353" href="Data.Bool.Properties.html#3315" class="Function">≤-isTotalOrder</a> <a id="3368" class="Symbol">=</a> <a id="3370" class="Keyword">record</a>
<a id="3379" class="Symbol">{</a> <a id="3381" href="Relation.Binary.Structures.html#5575" class="Field">isPartialOrder</a> <a id="3396" class="Symbol">=</a> <a id="3398" href="Data.Bool.Properties.html#3185" class="Function">≤-isPartialOrder</a>
<a id="3417" class="Symbol">;</a> <a id="3419" href="Relation.Binary.Structures.html#5615" class="Field">total</a> <a id="3434" class="Symbol">=</a> <a id="3436" href="Data.Bool.Properties.html#2660" class="Function">≤-total</a>
<a id="3446" class="Symbol">}</a>
<a id="≤-isDecTotalOrder"></a><a id="3449" href="Data.Bool.Properties.html#3449" class="Function">≤-isDecTotalOrder</a> <a id="3467" class="Symbol">:</a> <a id="3469" href="Relation.Binary.Structures.html#5825" class="Record">IsDecTotalOrder</a> <a id="3485" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="3489" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a>
<a id="3493" href="Data.Bool.Properties.html#3449" class="Function">≤-isDecTotalOrder</a> <a id="3511" class="Symbol">=</a> <a id="3513" class="Keyword">record</a>
<a id="3522" class="Symbol">{</a> <a id="3524" href="Relation.Binary.Structures.html#5914" class="Field">isTotalOrder</a> <a id="3537" class="Symbol">=</a> <a id="3539" href="Data.Bool.Properties.html#3315" class="Function">≤-isTotalOrder</a>
<a id="3556" class="Symbol">;</a> <a id="3558" href="Relation.Binary.Structures.html#5950" class="Field Operator">_≟_</a> <a id="3571" class="Symbol">=</a> <a id="3573" href="Data.Bool.Properties.html#1931" class="Function Operator">_≟_</a>
<a id="3579" class="Symbol">;</a> <a id="3581" href="Relation.Binary.Structures.html#5983" class="Field Operator">_≤?_</a> <a id="3594" class="Symbol">=</a> <a id="3596" href="Data.Bool.Properties.html#2769" class="Function Operator">_≤?_</a>
<a id="3603" class="Symbol">}</a>
<a id="3606" class="Comment">-- Bundles</a>
<a id="≤-poset"></a><a id="3618" href="Data.Bool.Properties.html#3618" class="Function">≤-poset</a> <a id="3626" class="Symbol">:</a> <a id="3628" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a> <a id="3634" href="Level.html#521" class="Function">0ℓ</a> <a id="3637" href="Level.html#521" class="Function">0ℓ</a> <a id="3640" href="Level.html#521" class="Function">0ℓ</a>
<a id="3643" href="Data.Bool.Properties.html#3618" class="Function">≤-poset</a> <a id="3651" class="Symbol">=</a> <a id="3653" class="Keyword">record</a>
<a id="3662" class="Symbol">{</a> <a id="3664" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a> <a id="3679" class="Symbol">=</a> <a id="3681" href="Data.Bool.Properties.html#3185" class="Function">≤-isPartialOrder</a>
<a id="3700" class="Symbol">}</a>
<a id="≤-preorder"></a><a id="3703" href="Data.Bool.Properties.html#3703" class="Function">≤-preorder</a> <a id="3714" class="Symbol">:</a> <a id="3716" href="Relation.Binary.Bundles.html#2121" class="Record">Preorder</a> <a id="3725" href="Level.html#521" class="Function">0ℓ</a> <a id="3728" href="Level.html#521" class="Function">0ℓ</a> <a id="3731" href="Level.html#521" class="Function">0ℓ</a>
<a id="3734" href="Data.Bool.Properties.html#3703" class="Function">≤-preorder</a> <a id="3745" class="Symbol">=</a> <a id="3747" class="Keyword">record</a>
<a id="3756" class="Symbol">{</a> <a id="3758" href="Relation.Binary.Bundles.html#2334" class="Field">isPreorder</a> <a id="3769" class="Symbol">=</a> <a id="3771" href="Data.Bool.Properties.html#3030" class="Function">≤-isPreorder</a>
<a id="3786" class="Symbol">}</a>
<a id="≤-totalOrder"></a><a id="3789" href="Data.Bool.Properties.html#3789" class="Function">≤-totalOrder</a> <a id="3802" class="Symbol">:</a> <a id="3804" href="Relation.Binary.Bundles.html#6458" class="Record">TotalOrder</a> <a id="3815" href="Level.html#521" class="Function">0ℓ</a> <a id="3818" href="Level.html#521" class="Function">0ℓ</a> <a id="3821" href="Level.html#521" class="Function">0ℓ</a>
<a id="3824" href="Data.Bool.Properties.html#3789" class="Function">≤-totalOrder</a> <a id="3837" class="Symbol">=</a> <a id="3839" class="Keyword">record</a>
<a id="3848" class="Symbol">{</a> <a id="3850" href="Relation.Binary.Bundles.html#6632" class="Field">isTotalOrder</a> <a id="3863" class="Symbol">=</a> <a id="3865" href="Data.Bool.Properties.html#3315" class="Function">≤-isTotalOrder</a>
<a id="3882" class="Symbol">}</a>
<a id="≤-decTotalOrder"></a><a id="3885" href="Data.Bool.Properties.html#3885" class="Function">≤-decTotalOrder</a> <a id="3901" class="Symbol">:</a> <a id="3903" href="Relation.Binary.Bundles.html#7041" class="Record">DecTotalOrder</a> <a id="3917" href="Level.html#521" class="Function">0ℓ</a> <a id="3920" href="Level.html#521" class="Function">0ℓ</a> <a id="3923" href="Level.html#521" class="Function">0ℓ</a>
<a id="3926" href="Data.Bool.Properties.html#3885" class="Function">≤-decTotalOrder</a> <a id="3942" class="Symbol">=</a> <a id="3944" class="Keyword">record</a>
<a id="3953" class="Symbol">{</a> <a id="3955" href="Relation.Binary.Bundles.html#7227" class="Field">isDecTotalOrder</a> <a id="3971" class="Symbol">=</a> <a id="3973" href="Data.Bool.Properties.html#3449" class="Function">≤-isDecTotalOrder</a>
<a id="3993" class="Symbol">}</a>
<a id="3996" class="Comment">------------------------------------------------------------------------</a>
<a id="4069" class="Comment">-- Properties of _&lt;_</a>
<a id="4091" class="Comment">-- Relational properties</a>
<a id="&lt;-irrefl"></a><a id="4117" href="Data.Bool.Properties.html#4117" class="Function">&lt;-irrefl</a> <a id="4126" class="Symbol">:</a> <a id="4128" href="Relation.Binary.Definitions.html#2478" class="Function">Irreflexive</a> <a id="4140" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="4144" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4148" href="Data.Bool.Properties.html#4117" class="Function">&lt;-irrefl</a> <a id="4157" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4162" class="Symbol">()</a>
<a id="&lt;-asym"></a><a id="4166" href="Data.Bool.Properties.html#4166" class="Function">&lt;-asym</a> <a id="4173" class="Symbol">:</a> <a id="4175" href="Relation.Binary.Definitions.html#2590" class="Function">Asymmetric</a> <a id="4186" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4190" href="Data.Bool.Properties.html#4166" class="Function">&lt;-asym</a> <a id="4197" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4201" class="Symbol">()</a>
<a id="&lt;-trans"></a><a id="4205" href="Data.Bool.Properties.html#4205" class="Function">&lt;-trans</a> <a id="4213" class="Symbol">:</a> <a id="4215" href="Relation.Binary.Definitions.html#2103" class="Function">Transitive</a> <a id="4226" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4230" href="Data.Bool.Properties.html#4205" class="Function">&lt;-trans</a> <a id="4238" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4242" class="Symbol">()</a>
<a id="&lt;-transʳ"></a><a id="4246" href="Data.Bool.Properties.html#4246" class="Function">&lt;-transʳ</a> <a id="4255" class="Symbol">:</a> <a id="4257" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a> <a id="4263" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="4267" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a> <a id="4271" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4275" href="Data.Bool.Properties.html#4246" class="Function">&lt;-transʳ</a> <a id="4284" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="4288" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4292" class="Symbol">=</a> <a id="4294" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a>
<a id="&lt;-transˡ"></a><a id="4299" href="Data.Bool.Properties.html#4299" class="Function">&lt;-transˡ</a> <a id="4308" class="Symbol">:</a> <a id="4310" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a> <a id="4316" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a> <a id="4320" href="Data.Bool.Base.html#681" class="Datatype Operator">_≤_</a> <a id="4324" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4328" href="Data.Bool.Properties.html#4299" class="Function">&lt;-transˡ</a> <a id="4337" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4341" href="Data.Bool.Base.html#734" class="InductiveConstructor">b≤b</a> <a id="4345" class="Symbol">=</a> <a id="4347" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a>
<a id="&lt;-cmp"></a><a id="4352" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a> <a id="4358" class="Symbol">:</a> <a id="4360" href="Relation.Binary.Definitions.html#3281" class="Function">Trichotomous</a> <a id="4373" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="4377" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4381" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a> <a id="4387" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4393" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4399" class="Symbol">=</a> <a id="4401" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a> <a id="4406" class="Symbol">(λ())</a> <a id="4412" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4418" class="Symbol">(λ())</a>
<a id="4424" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a> <a id="4430" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4436" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4442" class="Symbol">=</a> <a id="4444" href="Relation.Binary.Definitions.html#3104" class="InductiveConstructor">tri&lt;</a> <a id="4449" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4455" class="Symbol">(λ())</a> <a id="4461" class="Symbol">(λ())</a>
<a id="4467" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a> <a id="4473" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4479" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4485" class="Symbol">=</a> <a id="4487" href="Relation.Binary.Definitions.html#3212" class="InductiveConstructor">tri&gt;</a> <a id="4492" class="Symbol">(λ())</a> <a id="4498" class="Symbol">(λ())</a> <a id="4504" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a>
<a id="4508" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a> <a id="4514" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4520" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4526" class="Symbol">=</a> <a id="4528" href="Relation.Binary.Definitions.html#3158" class="InductiveConstructor">tri≈</a> <a id="4533" class="Symbol">(λ())</a> <a id="4539" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="4545" class="Symbol">(λ())</a>
<a id="4552" class="Keyword">infix</a> <a id="4558" class="Number">4</a> <a id="4560" href="Data.Bool.Properties.html#4566" class="Function Operator">_&lt;?_</a>
<a id="_&lt;?_"></a><a id="4566" href="Data.Bool.Properties.html#4566" class="Function Operator">_&lt;?_</a> <a id="4571" class="Symbol">:</a> <a id="4573" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="4583" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4587" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4593" href="Data.Bool.Properties.html#4566" class="Function Operator">&lt;?</a> <a id="4596" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4602" class="Symbol">=</a> <a id="4604" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="4608" class="Symbol">(λ())</a>
<a id="4614" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="4620" href="Data.Bool.Properties.html#4566" class="Function Operator">&lt;?</a> <a id="4623" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4629" class="Symbol">=</a> <a id="4631" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="4635" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a>
<a id="4639" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="4645" href="Data.Bool.Properties.html#4566" class="Function Operator">&lt;?</a> <a id="4648" class="Symbol">_</a> <a id="4654" class="Symbol">=</a> <a id="4656" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="4660" class="Symbol">(λ())</a>
<a id="&lt;-resp₂-≡"></a><a id="4667" href="Data.Bool.Properties.html#4667" class="Function">&lt;-resp₂-≡</a> <a id="4677" class="Symbol">:</a> <a id="4679" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a> <a id="4683" href="Relation.Binary.Definitions.html#5663" class="Function Operator">Respects₂</a> <a id="4693" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
<a id="4697" href="Data.Bool.Properties.html#4667" class="Function">&lt;-resp₂-≡</a> <a id="4707" class="Symbol">=</a> <a id="4709" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4715" class="Symbol">(_</a> <a id="4718" href="Data.Bool.Base.html#760" class="Datatype Operator">&lt;_</a><a id="4720" class="Symbol">)</a> <a id="4722" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="4724" href="Relation.Binary.PropositionalEquality.Core.html#1808" class="Function">subst</a> <a id="4730" class="Symbol">(</a><a id="4731" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;</a> <a id="4734" class="Symbol">_)</a>
<a id="&lt;-irrelevant"></a><a id="4738" href="Data.Bool.Properties.html#4738" class="Function">&lt;-irrelevant</a> <a id="4751" class="Symbol">:</a> <a id="4753" href="Relation.Binary.Definitions.html#6651" class="Function">Irrelevant</a> <a id="4764" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4768" href="Data.Bool.Properties.html#4738" class="Function">&lt;-irrelevant</a> <a id="4781" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4785" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4789" class="Symbol">=</a> <a id="4791" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="&lt;-wellFounded"></a><a id="4797" href="Data.Bool.Properties.html#4797" class="Function">&lt;-wellFounded</a> <a id="4811" class="Symbol">:</a> <a id="4813" href="Induction.WellFounded.html#1604" class="Function">WellFounded</a> <a id="4825" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="4829" href="Data.Bool.Properties.html#4797" class="Function">&lt;-wellFounded</a> <a id="4843" class="Symbol">_</a> <a id="4845" class="Symbol">=</a> <a id="4847" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a> <a id="4851" href="Data.Bool.Properties.html#4869" class="Function">&lt;-acc</a>
<a id="4859" class="Keyword">where</a>
<a id="4869" href="Data.Bool.Properties.html#4869" class="Function">&lt;-acc</a> <a id="4875" class="Symbol">:</a> <a id="4877" class="Symbol">∀</a> <a id="4879" class="Symbol">{</a><a id="4880" href="Data.Bool.Properties.html#4880" class="Bound">x</a> <a id="4882" href="Data.Bool.Properties.html#4882" class="Bound">y</a><a id="4883" class="Symbol">}</a> <a id="4885" class="Symbol">→</a> <a id="4887" href="Data.Bool.Properties.html#4882" class="Bound">y</a> <a id="4889" href="Data.Bool.Base.html#760" class="Datatype Operator">&lt;</a> <a id="4891" href="Data.Bool.Properties.html#4880" class="Bound">x</a> <a id="4893" class="Symbol">→</a> <a id="4895" href="Induction.WellFounded.html#1356" class="Datatype">Acc</a> <a id="4899" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a> <a id="4903" href="Data.Bool.Properties.html#4882" class="Bound">y</a>
<a id="4909" href="Data.Bool.Properties.html#4869" class="Function">&lt;-acc</a> <a id="4915" href="Data.Bool.Base.html#792" class="InductiveConstructor">f&lt;t</a> <a id="4919" class="Symbol">=</a> <a id="4921" href="Induction.WellFounded.html#1418" class="InductiveConstructor">acc</a> <a id="4925" class="Symbol">λ</a> <a id="4927" class="Symbol">()</a>
<a id="4931" class="Comment">-- Structures</a>
<a id="&lt;-isStrictPartialOrder"></a><a id="4946" href="Data.Bool.Properties.html#4946" class="Function">&lt;-isStrictPartialOrder</a> <a id="4969" class="Symbol">:</a> <a id="4971" href="Relation.Binary.Structures.html#4298" class="Record">IsStrictPartialOrder</a> <a id="4992" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="4996" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="5000" href="Data.Bool.Properties.html#4946" class="Function">&lt;-isStrictPartialOrder</a> <a id="5023" class="Symbol">=</a> <a id="5025" class="Keyword">record</a>
<a id="5034" class="Symbol">{</a> <a id="5036" href="Relation.Binary.Structures.html#4373" class="Field">isEquivalence</a> <a id="5050" class="Symbol">=</a> <a id="5052" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
<a id="5068" class="Symbol">;</a> <a id="5070" href="Relation.Binary.Structures.html#4407" class="Field">irrefl</a> <a id="5084" class="Symbol">=</a> <a id="5086" href="Data.Bool.Properties.html#4117" class="Function">&lt;-irrefl</a>
<a id="5097" class="Symbol">;</a> <a id="5099" href="Relation.Binary.Structures.html#4447" class="Field">trans</a> <a id="5113" class="Symbol">=</a> <a id="5115" href="Data.Bool.Properties.html#4205" class="Function">&lt;-trans</a>
<a id="5125" class="Symbol">;</a> <a id="5127" href="Relation.Binary.Structures.html#4482" class="Field">&lt;-resp-≈</a> <a id="5141" class="Symbol">=</a> <a id="5143" href="Data.Bool.Properties.html#4667" class="Function">&lt;-resp₂-≡</a>
<a id="5155" class="Symbol">}</a>
<a id="&lt;-isStrictTotalOrder"></a><a id="5158" href="Data.Bool.Properties.html#5158" class="Function">&lt;-isStrictTotalOrder</a> <a id="5179" class="Symbol">:</a> <a id="5181" href="Relation.Binary.Structures.html#6732" class="Record">IsStrictTotalOrder</a> <a id="5200" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a> <a id="5204" href="Data.Bool.Base.html#760" class="Datatype Operator">_&lt;_</a>
<a id="5208" href="Data.Bool.Properties.html#5158" class="Function">&lt;-isStrictTotalOrder</a> <a id="5229" class="Symbol">=</a> <a id="5231" class="Keyword">record</a>
<a id="5240" class="Symbol">{</a> <a id="5242" href="Relation.Binary.Structures.html#6805" class="Field">isStrictPartialOrder</a> <a id="5263" class="Symbol">=</a> <a id="5265" href="Data.Bool.Properties.html#4946" class="Function">&lt;-isStrictPartialOrder</a>
<a id="5290" class="Symbol">;</a> <a id="5292" href="Relation.Binary.Structures.html#6857" class="Field">compare</a> <a id="5313" class="Symbol">=</a> <a id="5315" href="Data.Bool.Properties.html#4352" class="Function">&lt;-cmp</a>
<a id="5323" class="Symbol">}</a>
<a id="5326" class="Comment">-- Bundles</a>
<a id="&lt;-strictPartialOrder"></a><a id="5338" href="Data.Bool.Properties.html#5338" class="Function">&lt;-strictPartialOrder</a> <a id="5359" class="Symbol">:</a> <a id="5361" href="Relation.Binary.Bundles.html#4839" class="Record">StrictPartialOrder</a> <a id="5380" href="Level.html#521" class="Function">0ℓ</a> <a id="5383" href="Level.html#521" class="Function">0ℓ</a> <a id="5386" href="Level.html#521" class="Function">0ℓ</a>
<a id="5389" href="Data.Bool.Properties.html#5338" class="Function">&lt;-strictPartialOrder</a> <a id="5410" class="Symbol">=</a> <a id="5412" class="Keyword">record</a>
<a id="5421" class="Symbol">{</a> <a id="5423" href="Relation.Binary.Bundles.html#5045" class="Field">isStrictPartialOrder</a> <a id="5444" class="Symbol">=</a> <a id="5446" href="Data.Bool.Properties.html#4946" class="Function">&lt;-isStrictPartialOrder</a>
<a id="5471" class="Symbol">}</a>
<a id="&lt;-strictTotalOrder"></a><a id="5474" href="Data.Bool.Properties.html#5474" class="Function">&lt;-strictTotalOrder</a> <a id="5493" class="Symbol">:</a> <a id="5495" href="Relation.Binary.Bundles.html#8031" class="Record">StrictTotalOrder</a> <a id="5512" href="Level.html#521" class="Function">0ℓ</a> <a id="5515" href="Level.html#521" class="Function">0ℓ</a> <a id="5518" href="Level.html#521" class="Function">0ℓ</a>
<a id="5521" href="Data.Bool.Properties.html#5474" class="Function">&lt;-strictTotalOrder</a> <a id="5540" class="Symbol">=</a> <a id="5542" class="Keyword">record</a>
<a id="5551" class="Symbol">{</a> <a id="5553" href="Relation.Binary.Bundles.html#8229" class="Field">isStrictTotalOrder</a> <a id="5572" class="Symbol">=</a> <a id="5574" href="Data.Bool.Properties.html#5158" class="Function">&lt;-isStrictTotalOrder</a>
<a id="5597" class="Symbol">}</a>
<a id="5600" class="Comment">------------------------------------------------------------------------</a>
<a id="5673" class="Comment">-- Properties of _∨_</a>
<a id="∨-assoc"></a><a id="5695" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a> <a id="5703" class="Symbol">:</a> <a id="5705" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="5717" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="5721" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a> <a id="5729" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="5735" href="Data.Bool.Properties.html#5735" class="Bound">y</a> <a id="5737" href="Data.Bool.Properties.html#5737" class="Bound">z</a> <a id="5739" class="Symbol">=</a> <a id="5741" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="5746" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a> <a id="5754" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5760" href="Data.Bool.Properties.html#5760" class="Bound">y</a> <a id="5762" href="Data.Bool.Properties.html#5762" class="Bound">z</a> <a id="5764" class="Symbol">=</a> <a id="5766" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-comm"></a><a id="5772" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="5779" class="Symbol">:</a> <a id="5781" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="5793" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="5797" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="5804" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="5810" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="5816" class="Symbol">=</a> <a id="5818" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="5823" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="5830" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="5836" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5842" class="Symbol">=</a> <a id="5844" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="5849" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="5856" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5862" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="5868" class="Symbol">=</a> <a id="5870" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="5875" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="5882" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5888" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5894" class="Symbol">=</a> <a id="5896" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-identityˡ"></a><a id="5902" href="Data.Bool.Properties.html#5902" class="Function">∨-identityˡ</a> <a id="5914" class="Symbol">:</a> <a id="5916" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="5929" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5935" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="5939" href="Data.Bool.Properties.html#5902" class="Function">∨-identityˡ</a> <a id="5951" class="Symbol">_</a> <a id="5953" class="Symbol">=</a> <a id="5955" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-identityʳ"></a><a id="5961" href="Data.Bool.Properties.html#5961" class="Function">∨-identityʳ</a> <a id="5973" class="Symbol">:</a> <a id="5975" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="5989" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="5995" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="5999" href="Data.Bool.Properties.html#5961" class="Function">∨-identityʳ</a> <a id="6011" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6017" class="Symbol">=</a> <a id="6019" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="6024" href="Data.Bool.Properties.html#5961" class="Function">∨-identityʳ</a> <a id="6036" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6042" class="Symbol">=</a> <a id="6044" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-identity"></a><a id="6050" href="Data.Bool.Properties.html#6050" class="Function">∨-identity</a> <a id="6061" class="Symbol">:</a> <a id="6063" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="6072" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6078" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6082" href="Data.Bool.Properties.html#6050" class="Function">∨-identity</a> <a id="6093" class="Symbol">=</a> <a id="6095" href="Data.Bool.Properties.html#5902" class="Function">∨-identityˡ</a> <a id="6107" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6109" href="Data.Bool.Properties.html#5961" class="Function">∨-identityʳ</a>
<a id="∨-zeroˡ"></a><a id="6122" href="Data.Bool.Properties.html#6122" class="Function">∨-zeroˡ</a> <a id="6130" class="Symbol">:</a> <a id="6132" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="6141" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6146" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6150" href="Data.Bool.Properties.html#6122" class="Function">∨-zeroˡ</a> <a id="6158" class="Symbol">_</a> <a id="6160" class="Symbol">=</a> <a id="6162" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-zeroʳ"></a><a id="6168" href="Data.Bool.Properties.html#6168" class="Function">∨-zeroʳ</a> <a id="6176" class="Symbol">:</a> <a id="6178" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="6188" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6193" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6197" href="Data.Bool.Properties.html#6168" class="Function">∨-zeroʳ</a> <a id="6205" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6211" class="Symbol">=</a> <a id="6213" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="6218" href="Data.Bool.Properties.html#6168" class="Function">∨-zeroʳ</a> <a id="6226" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6232" class="Symbol">=</a> <a id="6234" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-zero"></a><a id="6240" href="Data.Bool.Properties.html#6240" class="Function">∨-zero</a> <a id="6247" class="Symbol">:</a> <a id="6249" href="Algebra.Definitions.html#2074" class="Function">Zero</a> <a id="6254" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6259" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6263" href="Data.Bool.Properties.html#6240" class="Function">∨-zero</a> <a id="6270" class="Symbol">=</a> <a id="6272" href="Data.Bool.Properties.html#6122" class="Function">∨-zeroˡ</a> <a id="6280" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6282" href="Data.Bool.Properties.html#6168" class="Function">∨-zeroʳ</a>
<a id="∨-inverseˡ"></a><a id="6291" href="Data.Bool.Properties.html#6291" class="Function">∨-inverseˡ</a> <a id="6302" class="Symbol">:</a> <a id="6304" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="6316" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6321" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="6325" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6329" href="Data.Bool.Properties.html#6291" class="Function">∨-inverseˡ</a> <a id="6340" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6346" class="Symbol">=</a> <a id="6348" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="6353" href="Data.Bool.Properties.html#6291" class="Function">∨-inverseˡ</a> <a id="6364" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6370" class="Symbol">=</a> <a id="6372" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-inverseʳ"></a><a id="6378" href="Data.Bool.Properties.html#6378" class="Function">∨-inverseʳ</a> <a id="6389" class="Symbol">:</a> <a id="6391" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="6404" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6409" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="6413" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6417" href="Data.Bool.Properties.html#6378" class="Function">∨-inverseʳ</a> <a id="6428" href="Data.Bool.Properties.html#6428" class="Bound">x</a> <a id="6430" class="Symbol">=</a> <a id="6432" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="6439" href="Data.Bool.Properties.html#6428" class="Bound">x</a> <a id="6441" class="Symbol">(</a><a id="6442" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="6446" href="Data.Bool.Properties.html#6428" class="Bound">x</a><a id="6447" class="Symbol">)</a> <a id="6449" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="6451" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="6457" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="6459" href="Data.Bool.Properties.html#6291" class="Function">∨-inverseˡ</a> <a id="6470" href="Data.Bool.Properties.html#6428" class="Bound">x</a>
<a id="∨-inverse"></a><a id="6473" href="Data.Bool.Properties.html#6473" class="Function">∨-inverse</a> <a id="6483" class="Symbol">:</a> <a id="6485" href="Algebra.Definitions.html#2322" class="Function">Inverse</a> <a id="6493" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6498" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="6502" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6506" href="Data.Bool.Properties.html#6473" class="Function">∨-inverse</a> <a id="6516" class="Symbol">=</a> <a id="6518" href="Data.Bool.Properties.html#6291" class="Function">∨-inverseˡ</a> <a id="6529" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6531" href="Data.Bool.Properties.html#6378" class="Function">∨-inverseʳ</a>
<a id="∨-idem"></a><a id="6543" href="Data.Bool.Properties.html#6543" class="Function">∨-idem</a> <a id="6550" class="Symbol">:</a> <a id="6552" href="Algebra.Definitions.html#3706" class="Function">Idempotent</a> <a id="6563" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6567" href="Data.Bool.Properties.html#6543" class="Function">∨-idem</a> <a id="6574" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6580" class="Symbol">=</a> <a id="6582" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="6587" href="Data.Bool.Properties.html#6543" class="Function">∨-idem</a> <a id="6594" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6600" class="Symbol">=</a> <a id="6602" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-sel"></a><a id="6608" href="Data.Bool.Properties.html#6608" class="Function">∨-sel</a> <a id="6614" class="Symbol">:</a> <a id="6616" href="Algebra.Definitions.html#3841" class="Function">Selective</a> <a id="6626" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6630" href="Data.Bool.Properties.html#6608" class="Function">∨-sel</a> <a id="6636" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6642" href="Data.Bool.Properties.html#6642" class="Bound">y</a> <a id="6644" class="Symbol">=</a> <a id="6646" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="6651" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="6656" href="Data.Bool.Properties.html#6608" class="Function">∨-sel</a> <a id="6662" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="6667" href="Data.Bool.Properties.html#6667" class="Bound">y</a> <a id="6670" class="Symbol">=</a> <a id="6672" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="6677" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-conicalˡ"></a><a id="6683" href="Data.Bool.Properties.html#6683" class="Function">∨-conicalˡ</a> <a id="6694" class="Symbol">:</a> <a id="6696" href="Algebra.Definitions.html#2909" class="Function">LeftConical</a> <a id="6708" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6714" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6718" href="Data.Bool.Properties.html#6683" class="Function">∨-conicalˡ</a> <a id="6729" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6735" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6741" class="Symbol">_</a> <a id="6743" class="Symbol">=</a> <a id="6745" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-conicalʳ"></a><a id="6751" href="Data.Bool.Properties.html#6751" class="Function">∨-conicalʳ</a> <a id="6762" class="Symbol">:</a> <a id="6764" href="Algebra.Definitions.html#2990" class="Function">RightConical</a> <a id="6777" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6783" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6787" href="Data.Bool.Properties.html#6751" class="Function">∨-conicalʳ</a> <a id="6798" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6804" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6810" class="Symbol">_</a> <a id="6812" class="Symbol">=</a> <a id="6814" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-conical"></a><a id="6820" href="Data.Bool.Properties.html#6820" class="Function">∨-conical</a> <a id="6830" class="Symbol">:</a> <a id="6832" href="Algebra.Definitions.html#3073" class="Function">Conical</a> <a id="6840" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="6846" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6850" href="Data.Bool.Properties.html#6820" class="Function">∨-conical</a> <a id="6860" class="Symbol">=</a> <a id="6862" href="Data.Bool.Properties.html#6683" class="Function">∨-conicalˡ</a> <a id="6873" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="6875" href="Data.Bool.Properties.html#6751" class="Function">∨-conicalʳ</a>
<a id="∨-isMagma"></a><a id="6887" href="Data.Bool.Properties.html#6887" class="Function">∨-isMagma</a> <a id="6897" class="Symbol">:</a> <a id="6899" href="Algebra.Structures.html#1225" class="Record">IsMagma</a> <a id="6907" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6911" href="Data.Bool.Properties.html#6887" class="Function">∨-isMagma</a> <a id="6921" class="Symbol">=</a> <a id="6923" class="Keyword">record</a>
<a id="6932" class="Symbol">{</a> <a id="6934" href="Algebra.Structures.html#1277" class="Field">isEquivalence</a> <a id="6948" class="Symbol">=</a> <a id="6950" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
<a id="6966" class="Symbol">;</a> <a id="6968" href="Algebra.Structures.html#1315" class="Field">∙-cong</a> <a id="6982" class="Symbol">=</a> <a id="6984" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="6990" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="6996" class="Symbol">}</a>
<a id="∨-magma"></a><a id="6999" href="Data.Bool.Properties.html#6999" class="Function">∨-magma</a> <a id="7007" class="Symbol">:</a> <a id="7009" href="Algebra.Bundles.html#1141" class="Record">Magma</a> <a id="7015" href="Level.html#521" class="Function">0ℓ</a> <a id="7018" href="Level.html#521" class="Function">0ℓ</a>
<a id="7021" href="Data.Bool.Properties.html#6999" class="Function">∨-magma</a> <a id="7029" class="Symbol">=</a> <a id="7031" class="Keyword">record</a>
<a id="7040" class="Symbol">{</a> <a id="7042" href="Algebra.Bundles.html#1293" class="Field">isMagma</a> <a id="7050" class="Symbol">=</a> <a id="7052" href="Data.Bool.Properties.html#6887" class="Function">∨-isMagma</a>
<a id="7064" class="Symbol">}</a>
<a id="∨-isSemigroup"></a><a id="7067" href="Data.Bool.Properties.html#7067" class="Function">∨-isSemigroup</a> <a id="7081" class="Symbol">:</a> <a id="7083" href="Algebra.Structures.html#2897" class="Record">IsSemigroup</a> <a id="7095" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="7099" href="Data.Bool.Properties.html#7067" class="Function">∨-isSemigroup</a> <a id="7113" class="Symbol">=</a> <a id="7115" class="Keyword">record</a>
<a id="7124" class="Symbol">{</a> <a id="7126" href="Algebra.Structures.html#2953" class="Field">isMagma</a> <a id="7134" class="Symbol">=</a> <a id="7136" href="Data.Bool.Properties.html#6887" class="Function">∨-isMagma</a>
<a id="7148" class="Symbol">;</a> <a id="7150" href="Algebra.Structures.html#2977" class="Field">assoc</a> <a id="7158" class="Symbol">=</a> <a id="7160" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a>
<a id="7170" class="Symbol">}</a>
<a id="∨-semigroup"></a><a id="7173" href="Data.Bool.Properties.html#7173" class="Function">∨-semigroup</a> <a id="7185" class="Symbol">:</a> <a id="7187" href="Algebra.Bundles.html#4139" class="Record">Semigroup</a> <a id="7197" href="Level.html#521" class="Function">0ℓ</a> <a id="7200" href="Level.html#521" class="Function">0ℓ</a>
<a id="7203" href="Data.Bool.Properties.html#7173" class="Function">∨-semigroup</a> <a id="7215" class="Symbol">=</a> <a id="7217" class="Keyword">record</a>
<a id="7226" class="Symbol">{</a> <a id="7228" href="Algebra.Bundles.html#4307" class="Field">isSemigroup</a> <a id="7240" class="Symbol">=</a> <a id="7242" href="Data.Bool.Properties.html#7067" class="Function">∨-isSemigroup</a>
<a id="7258" class="Symbol">}</a>
<a id="∨-isBand"></a><a id="7261" href="Data.Bool.Properties.html#7261" class="Function">∨-isBand</a> <a id="7270" class="Symbol">:</a> <a id="7272" href="Algebra.Structures.html#3041" class="Record">IsBand</a> <a id="7279" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="7283" href="Data.Bool.Properties.html#7261" class="Function">∨-isBand</a> <a id="7292" class="Symbol">=</a> <a id="7294" class="Keyword">record</a>
<a id="7303" class="Symbol">{</a> <a id="7305" href="Algebra.Structures.html#3092" class="Field">isSemigroup</a> <a id="7317" class="Symbol">=</a> <a id="7319" href="Data.Bool.Properties.html#7067" class="Function">∨-isSemigroup</a>
<a id="7335" class="Symbol">;</a> <a id="7337" href="Algebra.Structures.html#3124" class="Field">idem</a> <a id="7349" class="Symbol">=</a> <a id="7351" href="Data.Bool.Properties.html#6543" class="Function">∨-idem</a>
<a id="7360" class="Symbol">}</a>
<a id="∨-band"></a><a id="7363" href="Data.Bool.Properties.html#7363" class="Function">∨-band</a> <a id="7370" class="Symbol">:</a> <a id="7372" href="Algebra.Bundles.html#4502" class="Record">Band</a> <a id="7377" href="Level.html#521" class="Function">0ℓ</a> <a id="7380" href="Level.html#521" class="Function">0ℓ</a>
<a id="7383" href="Data.Bool.Properties.html#7363" class="Function">∨-band</a> <a id="7390" class="Symbol">=</a> <a id="7392" class="Keyword">record</a>
<a id="7401" class="Symbol">{</a> <a id="7403" href="Algebra.Bundles.html#4653" class="Field">isBand</a> <a id="7410" class="Symbol">=</a> <a id="7412" href="Data.Bool.Properties.html#7261" class="Function">∨-isBand</a>
<a id="7423" class="Symbol">}</a>
<a id="∨-isSemilattice"></a><a id="7426" href="Data.Bool.Properties.html#7426" class="Function">∨-isSemilattice</a> <a id="7442" class="Symbol">:</a> <a id="7444" href="Algebra.Lattice.Structures.html#1160" class="Record">IsSemilattice</a> <a id="7458" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="7462" href="Data.Bool.Properties.html#7426" class="Function">∨-isSemilattice</a> <a id="7478" class="Symbol">=</a> <a id="7480" class="Keyword">record</a>
<a id="7489" class="Symbol">{</a> <a id="7491" href="Algebra.Lattice.Structures.html#1218" class="Field">isBand</a> <a id="7498" class="Symbol">=</a> <a id="7500" href="Data.Bool.Properties.html#7261" class="Function">∨-isBand</a>
<a id="7511" class="Symbol">;</a> <a id="7513" href="Algebra.Lattice.Structures.html#1240" class="Field">comm</a> <a id="7520" class="Symbol">=</a> <a id="7522" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a>
<a id="7531" class="Symbol">}</a>
<a id="∨-semilattice"></a><a id="7534" href="Data.Bool.Properties.html#7534" class="Function">∨-semilattice</a> <a id="7548" class="Symbol">:</a> <a id="7550" href="Algebra.Lattice.Bundles.html#1223" class="Record">Semilattice</a> <a id="7562" href="Level.html#521" class="Function">0ℓ</a> <a id="7565" href="Level.html#521" class="Function">0ℓ</a>
<a id="7568" href="Data.Bool.Properties.html#7534" class="Function">∨-semilattice</a> <a id="7582" class="Symbol">=</a> <a id="7584" class="Keyword">record</a>
<a id="7593" class="Symbol">{</a> <a id="7595" href="Algebra.Lattice.Bundles.html#1399" class="Field">isSemilattice</a> <a id="7609" class="Symbol">=</a> <a id="7611" href="Data.Bool.Properties.html#7426" class="Function">∨-isSemilattice</a>
<a id="7629" class="Symbol">}</a>
<a id="∨-isMonoid"></a><a id="7632" href="Data.Bool.Properties.html#7632" class="Function">∨-isMonoid</a> <a id="7643" class="Symbol">:</a> <a id="7645" href="Algebra.Structures.html#3974" class="Record">IsMonoid</a> <a id="7654" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="7658" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="7664" href="Data.Bool.Properties.html#7632" class="Function">∨-isMonoid</a> <a id="7675" class="Symbol">=</a> <a id="7677" class="Keyword">record</a>
<a id="7686" class="Symbol">{</a> <a id="7688" href="Algebra.Structures.html#4035" class="Field">isSemigroup</a> <a id="7700" class="Symbol">=</a> <a id="7702" href="Data.Bool.Properties.html#7067" class="Function">∨-isSemigroup</a>
<a id="7718" class="Symbol">;</a> <a id="7720" href="Algebra.Structures.html#4067" class="Field">identity</a> <a id="7729" class="Symbol">=</a> <a id="7731" href="Data.Bool.Properties.html#6050" class="Function">∨-identity</a>
<a id="7744" class="Symbol">}</a>
<a id="∨-isCommutativeMonoid"></a><a id="7747" href="Data.Bool.Properties.html#7747" class="Function">∨-isCommutativeMonoid</a> <a id="7769" class="Symbol">:</a> <a id="7771" href="Algebra.Structures.html#4384" class="Record">IsCommutativeMonoid</a> <a id="7791" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="7795" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="7801" href="Data.Bool.Properties.html#7747" class="Function">∨-isCommutativeMonoid</a> <a id="7823" class="Symbol">=</a> <a id="7825" class="Keyword">record</a>
<a id="7834" class="Symbol">{</a> <a id="7836" href="Algebra.Structures.html#4456" class="Field">isMonoid</a> <a id="7845" class="Symbol">=</a> <a id="7847" href="Data.Bool.Properties.html#7632" class="Function">∨-isMonoid</a>
<a id="7860" class="Symbol">;</a> <a id="7862" href="Algebra.Structures.html#4484" class="Field">comm</a> <a id="7867" class="Symbol">=</a> <a id="7869" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a>
<a id="7878" class="Symbol">}</a>
<a id="∨-commutativeMonoid"></a><a id="7881" href="Data.Bool.Properties.html#7881" class="Function">∨-commutativeMonoid</a> <a id="7901" class="Symbol">:</a> <a id="7903" href="Algebra.Bundles.html#6622" class="Record">CommutativeMonoid</a> <a id="7921" href="Level.html#521" class="Function">0ℓ</a> <a id="7924" href="Level.html#521" class="Function">0ℓ</a>
<a id="7927" href="Data.Bool.Properties.html#7881" class="Function">∨-commutativeMonoid</a> <a id="7947" class="Symbol">=</a> <a id="7949" class="Keyword">record</a>
<a id="7958" class="Symbol">{</a> <a id="7960" href="Algebra.Bundles.html#6856" class="Field">isCommutativeMonoid</a> <a id="7980" class="Symbol">=</a> <a id="7982" href="Data.Bool.Properties.html#7747" class="Function">∨-isCommutativeMonoid</a>
<a id="8006" class="Symbol">}</a>
<a id="∨-isIdempotentCommutativeMonoid"></a><a id="8009" href="Data.Bool.Properties.html#8009" class="Function">∨-isIdempotentCommutativeMonoid</a> <a id="8041" class="Symbol">:</a>
<a id="8045" href="Algebra.Structures.html#4793" class="Record">IsIdempotentCommutativeMonoid</a> <a id="8075" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="8079" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="8085" href="Data.Bool.Properties.html#8009" class="Function">∨-isIdempotentCommutativeMonoid</a> <a id="8117" class="Symbol">=</a> <a id="8119" class="Keyword">record</a>
<a id="8128" class="Symbol">{</a> <a id="8130" href="Algebra.Structures.html#4912" class="Field">isCommutativeMonoid</a> <a id="8150" class="Symbol">=</a> <a id="8152" href="Data.Bool.Properties.html#7747" class="Function">∨-isCommutativeMonoid</a>
<a id="8176" class="Symbol">;</a> <a id="8178" href="Algebra.Structures.html#4962" class="Field">idem</a> <a id="8198" class="Symbol">=</a> <a id="8200" href="Data.Bool.Properties.html#6543" class="Function">∨-idem</a>
<a id="8209" class="Symbol">}</a>
<a id="∨-idempotentCommutativeMonoid"></a><a id="8212" href="Data.Bool.Properties.html#8212" class="Function">∨-idempotentCommutativeMonoid</a> <a id="8242" class="Symbol">:</a> <a id="8244" href="Algebra.Bundles.html#7355" class="Record">IdempotentCommutativeMonoid</a> <a id="8272" href="Level.html#521" class="Function">0ℓ</a> <a id="8275" href="Level.html#521" class="Function">0ℓ</a>
<a id="8278" href="Data.Bool.Properties.html#8212" class="Function">∨-idempotentCommutativeMonoid</a> <a id="8308" class="Symbol">=</a> <a id="8310" class="Keyword">record</a>
<a id="8319" class="Symbol">{</a> <a id="8321" href="Algebra.Bundles.html#7639" class="Field">isIdempotentCommutativeMonoid</a> <a id="8351" class="Symbol">=</a> <a id="8353" href="Data.Bool.Properties.html#8009" class="Function">∨-isIdempotentCommutativeMonoid</a>
<a id="8387" class="Symbol">}</a>
<a id="8390" class="Comment">------------------------------------------------------------------------</a>
<a id="8463" class="Comment">-- Properties of _∧_</a>
<a id="∧-assoc"></a><a id="8485" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a> <a id="8493" class="Symbol">:</a> <a id="8495" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="8507" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8511" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a> <a id="8519" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8525" href="Data.Bool.Properties.html#8525" class="Bound">y</a> <a id="8527" href="Data.Bool.Properties.html#8527" class="Bound">z</a> <a id="8529" class="Symbol">=</a> <a id="8531" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="8536" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a> <a id="8544" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8550" href="Data.Bool.Properties.html#8550" class="Bound">y</a> <a id="8552" href="Data.Bool.Properties.html#8552" class="Bound">z</a> <a id="8554" class="Symbol">=</a> <a id="8556" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-comm"></a><a id="8562" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="8569" class="Symbol">:</a> <a id="8571" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="8583" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8587" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="8594" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8600" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8606" class="Symbol">=</a> <a id="8608" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="8613" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="8620" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8626" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8632" class="Symbol">=</a> <a id="8634" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="8639" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="8646" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8652" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8658" class="Symbol">=</a> <a id="8660" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="8665" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="8672" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8678" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8684" class="Symbol">=</a> <a id="8686" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-identityˡ"></a><a id="8692" href="Data.Bool.Properties.html#8692" class="Function">∧-identityˡ</a> <a id="8704" class="Symbol">:</a> <a id="8706" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="8719" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8724" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8728" href="Data.Bool.Properties.html#8692" class="Function">∧-identityˡ</a> <a id="8740" class="Symbol">_</a> <a id="8742" class="Symbol">=</a> <a id="8744" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-identityʳ"></a><a id="8750" href="Data.Bool.Properties.html#8750" class="Function">∧-identityʳ</a> <a id="8762" class="Symbol">:</a> <a id="8764" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="8778" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8783" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8787" href="Data.Bool.Properties.html#8750" class="Function">∧-identityʳ</a> <a id="8799" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8805" class="Symbol">=</a> <a id="8807" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="8812" href="Data.Bool.Properties.html#8750" class="Function">∧-identityʳ</a> <a id="8824" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8830" class="Symbol">=</a> <a id="8832" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-identity"></a><a id="8838" href="Data.Bool.Properties.html#8838" class="Function">∧-identity</a> <a id="8849" class="Symbol">:</a> <a id="8851" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="8860" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="8865" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8869" href="Data.Bool.Properties.html#8838" class="Function">∧-identity</a> <a id="8880" class="Symbol">=</a> <a id="8882" href="Data.Bool.Properties.html#8692" class="Function">∧-identityˡ</a> <a id="8894" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="8896" href="Data.Bool.Properties.html#8750" class="Function">∧-identityʳ</a>
<a id="∧-zeroˡ"></a><a id="8909" href="Data.Bool.Properties.html#8909" class="Function">∧-zeroˡ</a> <a id="8917" class="Symbol">:</a> <a id="8919" href="Algebra.Definitions.html#1942" class="Function">LeftZero</a> <a id="8928" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8934" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8938" href="Data.Bool.Properties.html#8909" class="Function">∧-zeroˡ</a> <a id="8946" class="Symbol">_</a> <a id="8948" class="Symbol">=</a> <a id="8950" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-zeroʳ"></a><a id="8956" href="Data.Bool.Properties.html#8956" class="Function">∧-zeroʳ</a> <a id="8964" class="Symbol">:</a> <a id="8966" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="8976" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="8982" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="8986" href="Data.Bool.Properties.html#8956" class="Function">∧-zeroʳ</a> <a id="8994" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9000" class="Symbol">=</a> <a id="9002" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="9007" href="Data.Bool.Properties.html#8956" class="Function">∧-zeroʳ</a> <a id="9015" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9021" class="Symbol">=</a> <a id="9023" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-zero"></a><a id="9029" href="Data.Bool.Properties.html#9029" class="Function">∧-zero</a> <a id="9036" class="Symbol">:</a> <a id="9038" href="Algebra.Definitions.html#2074" class="Function">Zero</a> <a id="9043" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9049" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9053" href="Data.Bool.Properties.html#9029" class="Function">∧-zero</a> <a id="9060" class="Symbol">=</a> <a id="9062" href="Data.Bool.Properties.html#8909" class="Function">∧-zeroˡ</a> <a id="9070" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="9072" href="Data.Bool.Properties.html#8956" class="Function">∧-zeroʳ</a>
<a id="∧-inverseˡ"></a><a id="9081" href="Data.Bool.Properties.html#9081" class="Function">∧-inverseˡ</a> <a id="9092" class="Symbol">:</a> <a id="9094" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="9106" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9112" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="9116" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9120" href="Data.Bool.Properties.html#9081" class="Function">∧-inverseˡ</a> <a id="9131" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9137" class="Symbol">=</a> <a id="9139" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="9144" href="Data.Bool.Properties.html#9081" class="Function">∧-inverseˡ</a> <a id="9155" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9160" class="Symbol">=</a> <a id="9162" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-inverseʳ"></a><a id="9168" href="Data.Bool.Properties.html#9168" class="Function">∧-inverseʳ</a> <a id="9179" class="Symbol">:</a> <a id="9181" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="9194" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9200" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="9204" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9208" href="Data.Bool.Properties.html#9168" class="Function">∧-inverseʳ</a> <a id="9219" href="Data.Bool.Properties.html#9219" class="Bound">x</a> <a id="9221" class="Symbol">=</a> <a id="9223" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="9230" href="Data.Bool.Properties.html#9219" class="Bound">x</a> <a id="9232" class="Symbol">(</a><a id="9233" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="9237" href="Data.Bool.Properties.html#9219" class="Bound">x</a><a id="9238" class="Symbol">)</a> <a id="9240" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="9242" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="9248" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="9250" href="Data.Bool.Properties.html#9081" class="Function">∧-inverseˡ</a> <a id="9261" href="Data.Bool.Properties.html#9219" class="Bound">x</a>
<a id="∧-inverse"></a><a id="9264" href="Data.Bool.Properties.html#9264" class="Function">∧-inverse</a> <a id="9274" class="Symbol">:</a> <a id="9276" href="Algebra.Definitions.html#2322" class="Function">Inverse</a> <a id="9284" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9290" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="9294" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9298" href="Data.Bool.Properties.html#9264" class="Function">∧-inverse</a> <a id="9308" class="Symbol">=</a> <a id="9310" href="Data.Bool.Properties.html#9081" class="Function">∧-inverseˡ</a> <a id="9321" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="9323" href="Data.Bool.Properties.html#9168" class="Function">∧-inverseʳ</a>
<a id="∧-idem"></a><a id="9335" href="Data.Bool.Properties.html#9335" class="Function">∧-idem</a> <a id="9342" class="Symbol">:</a> <a id="9344" href="Algebra.Definitions.html#3706" class="Function">Idempotent</a> <a id="9355" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9359" href="Data.Bool.Properties.html#9335" class="Function">∧-idem</a> <a id="9366" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9372" class="Symbol">=</a> <a id="9374" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="9379" href="Data.Bool.Properties.html#9335" class="Function">∧-idem</a> <a id="9386" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9392" class="Symbol">=</a> <a id="9394" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-sel"></a><a id="9400" href="Data.Bool.Properties.html#9400" class="Function">∧-sel</a> <a id="9406" class="Symbol">:</a> <a id="9408" href="Algebra.Definitions.html#3841" class="Function">Selective</a> <a id="9418" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9422" href="Data.Bool.Properties.html#9400" class="Function">∧-sel</a> <a id="9428" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9434" href="Data.Bool.Properties.html#9434" class="Bound">y</a> <a id="9436" class="Symbol">=</a> <a id="9438" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="9443" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="9448" href="Data.Bool.Properties.html#9400" class="Function">∧-sel</a> <a id="9454" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9459" href="Data.Bool.Properties.html#9459" class="Bound">y</a> <a id="9462" class="Symbol">=</a> <a id="9464" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="9469" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-conicalˡ"></a><a id="9475" href="Data.Bool.Properties.html#9475" class="Function">∧-conicalˡ</a> <a id="9486" class="Symbol">:</a> <a id="9488" href="Algebra.Definitions.html#2909" class="Function">LeftConical</a> <a id="9500" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9505" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9509" href="Data.Bool.Properties.html#9475" class="Function">∧-conicalˡ</a> <a id="9520" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9525" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9530" class="Symbol">_</a> <a id="9532" class="Symbol">=</a> <a id="9534" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-conicalʳ"></a><a id="9540" href="Data.Bool.Properties.html#9540" class="Function">∧-conicalʳ</a> <a id="9551" class="Symbol">:</a> <a id="9553" href="Algebra.Definitions.html#2990" class="Function">RightConical</a> <a id="9566" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9571" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9575" href="Data.Bool.Properties.html#9540" class="Function">∧-conicalʳ</a> <a id="9586" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9591" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9596" class="Symbol">_</a> <a id="9598" class="Symbol">=</a> <a id="9600" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-conical"></a><a id="9606" href="Data.Bool.Properties.html#9606" class="Function">∧-conical</a> <a id="9616" class="Symbol">:</a> <a id="9618" href="Algebra.Definitions.html#3073" class="Function">Conical</a> <a id="9626" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9631" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="9635" href="Data.Bool.Properties.html#9606" class="Function">∧-conical</a> <a id="9645" class="Symbol">=</a> <a id="9647" href="Data.Bool.Properties.html#9475" class="Function">∧-conicalˡ</a> <a id="9658" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="9660" href="Data.Bool.Properties.html#9540" class="Function">∧-conicalʳ</a>
<a id="∧-distribˡ-∨"></a><a id="9672" href="Data.Bool.Properties.html#9672" class="Function">∧-distribˡ-∨</a> <a id="9685" class="Symbol">:</a> <a id="9687" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="9691" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="9708" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="9712" href="Data.Bool.Properties.html#9672" class="Function">∧-distribˡ-∨</a> <a id="9725" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="9731" href="Data.Bool.Properties.html#9731" class="Bound">y</a> <a id="9733" href="Data.Bool.Properties.html#9733" class="Bound">z</a> <a id="9735" class="Symbol">=</a> <a id="9737" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="9742" href="Data.Bool.Properties.html#9672" class="Function">∧-distribˡ-∨</a> <a id="9755" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="9761" href="Data.Bool.Properties.html#9761" class="Bound">y</a> <a id="9763" href="Data.Bool.Properties.html#9763" class="Bound">z</a> <a id="9765" class="Symbol">=</a> <a id="9767" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-distribʳ-∨"></a><a id="9773" href="Data.Bool.Properties.html#9773" class="Function">∧-distribʳ-∨</a> <a id="9786" class="Symbol">:</a> <a id="9788" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="9792" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="9809" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="9813" href="Data.Bool.Properties.html#9773" class="Function">∧-distribʳ-∨</a> <a id="9826" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9828" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9830" href="Data.Bool.Properties.html#9830" class="Bound">z</a> <a id="9832" class="Symbol">=</a> <a id="9834" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="9842" class="Symbol">(</a><a id="9843" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9845" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9847" href="Data.Bool.Properties.html#9830" class="Bound">z</a><a id="9848" class="Symbol">)</a> <a id="9850" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9852" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9858" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="9861" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="9868" class="Symbol">(</a><a id="9869" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9871" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9873" href="Data.Bool.Properties.html#9830" class="Bound">z</a><a id="9874" class="Symbol">)</a> <a id="9876" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9878" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="9882" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9884" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9886" class="Symbol">(</a><a id="9887" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9889" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9891" href="Data.Bool.Properties.html#9830" class="Bound">z</a><a id="9892" class="Symbol">)</a> <a id="9898" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="9901" href="Data.Bool.Properties.html#9672" class="Function">∧-distribˡ-∨</a> <a id="9914" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9916" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9918" href="Data.Bool.Properties.html#9830" class="Bound">z</a> <a id="9920" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="9924" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9926" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9928" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9930" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9932" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9934" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9936" href="Data.Bool.Properties.html#9830" class="Bound">z</a> <a id="9940" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="9943" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="9949" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="9953" class="Symbol">(</a><a id="9954" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="9961" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9963" href="Data.Bool.Properties.html#9828" class="Bound">y</a><a id="9964" class="Symbol">)</a> <a id="9966" class="Symbol">(</a><a id="9967" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a> <a id="9974" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9976" href="Data.Bool.Properties.html#9830" class="Bound">z</a><a id="9977" class="Symbol">)</a> <a id="9979" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="9983" href="Data.Bool.Properties.html#9828" class="Bound">y</a> <a id="9985" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9987" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9989" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="9991" href="Data.Bool.Properties.html#9830" class="Bound">z</a> <a id="9993" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="9995" href="Data.Bool.Properties.html#9826" class="Bound">x</a> <a id="9999" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∧-distrib-∨"></a><a id="10002" href="Data.Bool.Properties.html#10002" class="Function">∧-distrib-∨</a> <a id="10014" class="Symbol">:</a> <a id="10016" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10020" href="Algebra.Definitions.html#3393" class="Function Operator">DistributesOver</a> <a id="10036" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="10040" href="Data.Bool.Properties.html#10002" class="Function">∧-distrib-∨</a> <a id="10052" class="Symbol">=</a> <a id="10054" href="Data.Bool.Properties.html#9672" class="Function">∧-distribˡ-∨</a> <a id="10067" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10069" href="Data.Bool.Properties.html#9773" class="Function">∧-distribʳ-∨</a>
<a id="∨-distribˡ-∧"></a><a id="10083" href="Data.Bool.Properties.html#10083" class="Function">∨-distribˡ-∧</a> <a id="10096" class="Symbol">:</a> <a id="10098" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10102" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="10119" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10123" href="Data.Bool.Properties.html#10083" class="Function">∨-distribˡ-∧</a> <a id="10136" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="10142" href="Data.Bool.Properties.html#10142" class="Bound">y</a> <a id="10144" href="Data.Bool.Properties.html#10144" class="Bound">z</a> <a id="10146" class="Symbol">=</a> <a id="10148" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="10153" href="Data.Bool.Properties.html#10083" class="Function">∨-distribˡ-∧</a> <a id="10166" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="10172" href="Data.Bool.Properties.html#10172" class="Bound">y</a> <a id="10174" href="Data.Bool.Properties.html#10174" class="Bound">z</a> <a id="10176" class="Symbol">=</a> <a id="10178" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-distribʳ-∧"></a><a id="10184" href="Data.Bool.Properties.html#10184" class="Function">∨-distribʳ-∧</a> <a id="10197" class="Symbol">:</a> <a id="10199" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10203" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="10220" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10224" href="Data.Bool.Properties.html#10184" class="Function">∨-distribʳ-∧</a> <a id="10237" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10239" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10241" href="Data.Bool.Properties.html#10241" class="Bound">z</a> <a id="10243" class="Symbol">=</a> <a id="10245" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="10253" class="Symbol">(</a><a id="10254" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10256" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="10258" href="Data.Bool.Properties.html#10241" class="Bound">z</a><a id="10259" class="Symbol">)</a> <a id="10261" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10263" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10272" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="10275" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="10282" class="Symbol">(</a><a id="10283" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10285" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="10287" href="Data.Bool.Properties.html#10241" class="Bound">z</a><a id="10288" class="Symbol">)</a> <a id="10290" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10292" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="10296" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10298" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10300" class="Symbol">(</a><a id="10301" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10303" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="10305" href="Data.Bool.Properties.html#10241" class="Bound">z</a><a id="10306" class="Symbol">)</a> <a id="10315" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="10318" href="Data.Bool.Properties.html#10083" class="Function">∨-distribˡ-∧</a> <a id="10331" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10333" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10335" href="Data.Bool.Properties.html#10241" class="Bound">z</a> <a id="10337" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="10341" class="Symbol">(</a><a id="10342" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10344" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10346" href="Data.Bool.Properties.html#10239" class="Bound">y</a><a id="10347" class="Symbol">)</a> <a id="10349" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="10351" class="Symbol">(</a><a id="10352" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10354" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10356" href="Data.Bool.Properties.html#10241" class="Bound">z</a><a id="10357" class="Symbol">)</a> <a id="10360" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">≡⟨</a> <a id="10363" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="10369" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10373" class="Symbol">(</a><a id="10374" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="10381" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10383" href="Data.Bool.Properties.html#10239" class="Bound">y</a><a id="10384" class="Symbol">)</a> <a id="10386" class="Symbol">(</a><a id="10387" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a> <a id="10394" href="Data.Bool.Properties.html#10237" class="Bound">x</a> <a id="10396" href="Data.Bool.Properties.html#10241" class="Bound">z</a><a id="10397" class="Symbol">)</a> <a id="10399" href="Relation.Binary.Reasoning.Syntax.html#10986" class="Function">⟩</a>
<a id="10403" class="Symbol">(</a><a id="10404" href="Data.Bool.Properties.html#10239" class="Bound">y</a> <a id="10406" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10408" href="Data.Bool.Properties.html#10237" class="Bound">x</a><a id="10409" class="Symbol">)</a> <a id="10411" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="10413" class="Symbol">(</a><a id="10414" href="Data.Bool.Properties.html#10241" class="Bound">z</a> <a id="10416" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="10418" href="Data.Bool.Properties.html#10237" class="Bound">x</a><a id="10419" class="Symbol">)</a> <a id="10422" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∨-distrib-∧"></a><a id="10425" href="Data.Bool.Properties.html#10425" class="Function">∨-distrib-∧</a> <a id="10437" class="Symbol">:</a> <a id="10439" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10443" href="Algebra.Definitions.html#3393" class="Function Operator">DistributesOver</a> <a id="10459" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10463" href="Data.Bool.Properties.html#10425" class="Function">∨-distrib-∧</a> <a id="10475" class="Symbol">=</a> <a id="10477" href="Data.Bool.Properties.html#10083" class="Function">∨-distribˡ-∧</a> <a id="10490" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10492" href="Data.Bool.Properties.html#10184" class="Function">∨-distribʳ-∧</a>
<a id="∧-abs-∨"></a><a id="10506" href="Data.Bool.Properties.html#10506" class="Function">∧-abs-∨</a> <a id="10514" class="Symbol">:</a> <a id="10516" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="10520" href="Algebra.Definitions.html#3918" class="Function Operator">Absorbs</a> <a id="10528" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="10532" href="Data.Bool.Properties.html#10506" class="Function">∧-abs-∨</a> <a id="10540" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="10546" href="Data.Bool.Properties.html#10546" class="Bound">y</a> <a id="10548" class="Symbol">=</a> <a id="10550" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="10555" href="Data.Bool.Properties.html#10506" class="Function">∧-abs-∨</a> <a id="10563" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="10569" href="Data.Bool.Properties.html#10569" class="Bound">y</a> <a id="10571" class="Symbol">=</a> <a id="10573" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-abs-∧"></a><a id="10579" href="Data.Bool.Properties.html#10579" class="Function">∨-abs-∧</a> <a id="10587" class="Symbol">:</a> <a id="10589" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10593" href="Algebra.Definitions.html#3918" class="Function Operator">Absorbs</a> <a id="10601" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10605" href="Data.Bool.Properties.html#10579" class="Function">∨-abs-∧</a> <a id="10613" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="10619" href="Data.Bool.Properties.html#10619" class="Bound">y</a> <a id="10621" class="Symbol">=</a> <a id="10623" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="10628" href="Data.Bool.Properties.html#10579" class="Function">∨-abs-∧</a> <a id="10636" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="10642" href="Data.Bool.Properties.html#10642" class="Bound">y</a> <a id="10644" class="Symbol">=</a> <a id="10646" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∨-∧-absorptive"></a><a id="10652" href="Data.Bool.Properties.html#10652" class="Function">∨-∧-absorptive</a> <a id="10667" class="Symbol">:</a> <a id="10669" href="Algebra.Definitions.html#3997" class="Function">Absorptive</a> <a id="10680" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="10684" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10688" href="Data.Bool.Properties.html#10652" class="Function">∨-∧-absorptive</a> <a id="10703" class="Symbol">=</a> <a id="10705" href="Data.Bool.Properties.html#10579" class="Function">∨-abs-∧</a> <a id="10713" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="10715" href="Data.Bool.Properties.html#10506" class="Function">∧-abs-∨</a>
<a id="∧-isMagma"></a><a id="10724" href="Data.Bool.Properties.html#10724" class="Function">∧-isMagma</a> <a id="10734" class="Symbol">:</a> <a id="10736" href="Algebra.Structures.html#1225" class="Record">IsMagma</a> <a id="10744" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10748" href="Data.Bool.Properties.html#10724" class="Function">∧-isMagma</a> <a id="10758" class="Symbol">=</a> <a id="10760" class="Keyword">record</a>
<a id="10769" class="Symbol">{</a> <a id="10771" href="Algebra.Structures.html#1277" class="Field">isEquivalence</a> <a id="10785" class="Symbol">=</a> <a id="10787" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
<a id="10803" class="Symbol">;</a> <a id="10805" href="Algebra.Structures.html#1315" class="Field">∙-cong</a> <a id="10819" class="Symbol">=</a> <a id="10821" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="10827" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10833" class="Symbol">}</a>
<a id="∧-magma"></a><a id="10836" href="Data.Bool.Properties.html#10836" class="Function">∧-magma</a> <a id="10844" class="Symbol">:</a> <a id="10846" href="Algebra.Bundles.html#1141" class="Record">Magma</a> <a id="10852" href="Level.html#521" class="Function">0ℓ</a> <a id="10855" href="Level.html#521" class="Function">0ℓ</a>
<a id="10858" href="Data.Bool.Properties.html#10836" class="Function">∧-magma</a> <a id="10866" class="Symbol">=</a> <a id="10868" class="Keyword">record</a>
<a id="10877" class="Symbol">{</a> <a id="10879" href="Algebra.Bundles.html#1293" class="Field">isMagma</a> <a id="10887" class="Symbol">=</a> <a id="10889" href="Data.Bool.Properties.html#10724" class="Function">∧-isMagma</a>
<a id="10901" class="Symbol">}</a>
<a id="∧-isSemigroup"></a><a id="10904" href="Data.Bool.Properties.html#10904" class="Function">∧-isSemigroup</a> <a id="10918" class="Symbol">:</a> <a id="10920" href="Algebra.Structures.html#2897" class="Record">IsSemigroup</a> <a id="10932" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="10936" href="Data.Bool.Properties.html#10904" class="Function">∧-isSemigroup</a> <a id="10950" class="Symbol">=</a> <a id="10952" class="Keyword">record</a>
<a id="10961" class="Symbol">{</a> <a id="10963" href="Algebra.Structures.html#2953" class="Field">isMagma</a> <a id="10971" class="Symbol">=</a> <a id="10973" href="Data.Bool.Properties.html#10724" class="Function">∧-isMagma</a>
<a id="10985" class="Symbol">;</a> <a id="10987" href="Algebra.Structures.html#2977" class="Field">assoc</a> <a id="10995" class="Symbol">=</a> <a id="10997" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a>
<a id="11007" class="Symbol">}</a>
<a id="∧-semigroup"></a><a id="11010" href="Data.Bool.Properties.html#11010" class="Function">∧-semigroup</a> <a id="11022" class="Symbol">:</a> <a id="11024" href="Algebra.Bundles.html#4139" class="Record">Semigroup</a> <a id="11034" href="Level.html#521" class="Function">0ℓ</a> <a id="11037" href="Level.html#521" class="Function">0ℓ</a>
<a id="11040" href="Data.Bool.Properties.html#11010" class="Function">∧-semigroup</a> <a id="11052" class="Symbol">=</a> <a id="11054" class="Keyword">record</a>
<a id="11063" class="Symbol">{</a> <a id="11065" href="Algebra.Bundles.html#4307" class="Field">isSemigroup</a> <a id="11077" class="Symbol">=</a> <a id="11079" href="Data.Bool.Properties.html#10904" class="Function">∧-isSemigroup</a>
<a id="11095" class="Symbol">}</a>
<a id="∧-isBand"></a><a id="11098" href="Data.Bool.Properties.html#11098" class="Function">∧-isBand</a> <a id="11107" class="Symbol">:</a> <a id="11109" href="Algebra.Structures.html#3041" class="Record">IsBand</a> <a id="11116" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="11120" href="Data.Bool.Properties.html#11098" class="Function">∧-isBand</a> <a id="11129" class="Symbol">=</a> <a id="11131" class="Keyword">record</a>
<a id="11140" class="Symbol">{</a> <a id="11142" href="Algebra.Structures.html#3092" class="Field">isSemigroup</a> <a id="11154" class="Symbol">=</a> <a id="11156" href="Data.Bool.Properties.html#10904" class="Function">∧-isSemigroup</a>
<a id="11172" class="Symbol">;</a> <a id="11174" href="Algebra.Structures.html#3124" class="Field">idem</a> <a id="11186" class="Symbol">=</a> <a id="11188" href="Data.Bool.Properties.html#9335" class="Function">∧-idem</a>
<a id="11197" class="Symbol">}</a>
<a id="∧-band"></a><a id="11200" href="Data.Bool.Properties.html#11200" class="Function">∧-band</a> <a id="11207" class="Symbol">:</a> <a id="11209" href="Algebra.Bundles.html#4502" class="Record">Band</a> <a id="11214" href="Level.html#521" class="Function">0ℓ</a> <a id="11217" href="Level.html#521" class="Function">0ℓ</a>
<a id="11220" href="Data.Bool.Properties.html#11200" class="Function">∧-band</a> <a id="11227" class="Symbol">=</a> <a id="11229" class="Keyword">record</a>
<a id="11238" class="Symbol">{</a> <a id="11240" href="Algebra.Bundles.html#4653" class="Field">isBand</a> <a id="11247" class="Symbol">=</a> <a id="11249" href="Data.Bool.Properties.html#11098" class="Function">∧-isBand</a>
<a id="11260" class="Symbol">}</a>
<a id="∧-isSemilattice"></a><a id="11263" href="Data.Bool.Properties.html#11263" class="Function">∧-isSemilattice</a> <a id="11279" class="Symbol">:</a> <a id="11281" href="Algebra.Lattice.Structures.html#1160" class="Record">IsSemilattice</a> <a id="11295" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="11299" href="Data.Bool.Properties.html#11263" class="Function">∧-isSemilattice</a> <a id="11315" class="Symbol">=</a> <a id="11317" class="Keyword">record</a>
<a id="11326" class="Symbol">{</a> <a id="11328" href="Algebra.Lattice.Structures.html#1218" class="Field">isBand</a> <a id="11335" class="Symbol">=</a> <a id="11337" href="Data.Bool.Properties.html#11098" class="Function">∧-isBand</a>
<a id="11348" class="Symbol">;</a> <a id="11350" href="Algebra.Lattice.Structures.html#1240" class="Field">comm</a> <a id="11357" class="Symbol">=</a> <a id="11359" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a>
<a id="11368" class="Symbol">}</a>
<a id="∧-semilattice"></a><a id="11371" href="Data.Bool.Properties.html#11371" class="Function">∧-semilattice</a> <a id="11385" class="Symbol">:</a> <a id="11387" href="Algebra.Lattice.Bundles.html#1223" class="Record">Semilattice</a> <a id="11399" href="Level.html#521" class="Function">0ℓ</a> <a id="11402" href="Level.html#521" class="Function">0ℓ</a>
<a id="11405" href="Data.Bool.Properties.html#11371" class="Function">∧-semilattice</a> <a id="11419" class="Symbol">=</a> <a id="11421" class="Keyword">record</a>
<a id="11430" class="Symbol">{</a> <a id="11432" href="Algebra.Lattice.Bundles.html#1399" class="Field">isSemilattice</a> <a id="11446" class="Symbol">=</a> <a id="11448" href="Data.Bool.Properties.html#11263" class="Function">∧-isSemilattice</a>
<a id="11466" class="Symbol">}</a>
<a id="∧-isMonoid"></a><a id="11469" href="Data.Bool.Properties.html#11469" class="Function">∧-isMonoid</a> <a id="11480" class="Symbol">:</a> <a id="11482" href="Algebra.Structures.html#3974" class="Record">IsMonoid</a> <a id="11491" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11495" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="11500" href="Data.Bool.Properties.html#11469" class="Function">∧-isMonoid</a> <a id="11511" class="Symbol">=</a> <a id="11513" class="Keyword">record</a>
<a id="11522" class="Symbol">{</a> <a id="11524" href="Algebra.Structures.html#4035" class="Field">isSemigroup</a> <a id="11536" class="Symbol">=</a> <a id="11538" href="Data.Bool.Properties.html#10904" class="Function">∧-isSemigroup</a>
<a id="11554" class="Symbol">;</a> <a id="11556" href="Algebra.Structures.html#4067" class="Field">identity</a> <a id="11565" class="Symbol">=</a> <a id="11567" href="Data.Bool.Properties.html#8838" class="Function">∧-identity</a>
<a id="11580" class="Symbol">}</a>
<a id="∧-isCommutativeMonoid"></a><a id="11583" href="Data.Bool.Properties.html#11583" class="Function">∧-isCommutativeMonoid</a> <a id="11605" class="Symbol">:</a> <a id="11607" href="Algebra.Structures.html#4384" class="Record">IsCommutativeMonoid</a> <a id="11627" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11631" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="11636" href="Data.Bool.Properties.html#11583" class="Function">∧-isCommutativeMonoid</a> <a id="11658" class="Symbol">=</a> <a id="11660" class="Keyword">record</a>
<a id="11669" class="Symbol">{</a> <a id="11671" href="Algebra.Structures.html#4456" class="Field">isMonoid</a> <a id="11680" class="Symbol">=</a> <a id="11682" href="Data.Bool.Properties.html#11469" class="Function">∧-isMonoid</a>
<a id="11695" class="Symbol">;</a> <a id="11697" href="Algebra.Structures.html#4484" class="Field">comm</a> <a id="11702" class="Symbol">=</a> <a id="11704" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a>
<a id="11713" class="Symbol">}</a>
<a id="∧-commutativeMonoid"></a><a id="11716" href="Data.Bool.Properties.html#11716" class="Function">∧-commutativeMonoid</a> <a id="11736" class="Symbol">:</a> <a id="11738" href="Algebra.Bundles.html#6622" class="Record">CommutativeMonoid</a> <a id="11756" href="Level.html#521" class="Function">0ℓ</a> <a id="11759" href="Level.html#521" class="Function">0ℓ</a>
<a id="11762" href="Data.Bool.Properties.html#11716" class="Function">∧-commutativeMonoid</a> <a id="11782" class="Symbol">=</a> <a id="11784" class="Keyword">record</a>
<a id="11793" class="Symbol">{</a> <a id="11795" href="Algebra.Bundles.html#6856" class="Field">isCommutativeMonoid</a> <a id="11815" class="Symbol">=</a> <a id="11817" href="Data.Bool.Properties.html#11583" class="Function">∧-isCommutativeMonoid</a>
<a id="11841" class="Symbol">}</a>
<a id="∧-isIdempotentCommutativeMonoid"></a><a id="11844" href="Data.Bool.Properties.html#11844" class="Function">∧-isIdempotentCommutativeMonoid</a> <a id="11876" class="Symbol">:</a>
<a id="11880" href="Algebra.Structures.html#4793" class="Record">IsIdempotentCommutativeMonoid</a> <a id="11910" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="11914" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="11919" href="Data.Bool.Properties.html#11844" class="Function">∧-isIdempotentCommutativeMonoid</a> <a id="11951" class="Symbol">=</a> <a id="11953" class="Keyword">record</a>
<a id="11962" class="Symbol">{</a> <a id="11964" href="Algebra.Structures.html#4912" class="Field">isCommutativeMonoid</a> <a id="11984" class="Symbol">=</a> <a id="11986" href="Data.Bool.Properties.html#11583" class="Function">∧-isCommutativeMonoid</a>
<a id="12010" class="Symbol">;</a> <a id="12012" href="Algebra.Structures.html#4962" class="Field">idem</a> <a id="12017" class="Symbol">=</a> <a id="12019" href="Data.Bool.Properties.html#9335" class="Function">∧-idem</a>
<a id="12028" class="Symbol">}</a>
<a id="∧-idempotentCommutativeMonoid"></a><a id="12031" href="Data.Bool.Properties.html#12031" class="Function">∧-idempotentCommutativeMonoid</a> <a id="12061" class="Symbol">:</a> <a id="12063" href="Algebra.Bundles.html#7355" class="Record">IdempotentCommutativeMonoid</a> <a id="12091" href="Level.html#521" class="Function">0ℓ</a> <a id="12094" href="Level.html#521" class="Function">0ℓ</a>
<a id="12097" href="Data.Bool.Properties.html#12031" class="Function">∧-idempotentCommutativeMonoid</a> <a id="12127" class="Symbol">=</a> <a id="12129" class="Keyword">record</a>
<a id="12138" class="Symbol">{</a> <a id="12140" href="Algebra.Bundles.html#7639" class="Field">isIdempotentCommutativeMonoid</a> <a id="12170" class="Symbol">=</a> <a id="12172" href="Data.Bool.Properties.html#11844" class="Function">∧-isIdempotentCommutativeMonoid</a>
<a id="12206" class="Symbol">}</a>
<a id="∨-∧-isSemiring"></a><a id="12209" href="Data.Bool.Properties.html#12209" class="Function">∨-∧-isSemiring</a> <a id="12224" class="Symbol">:</a> <a id="12226" href="Algebra.Structures.html#12731" class="Record">IsSemiring</a> <a id="12237" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="12241" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="12245" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="12251" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="12256" href="Data.Bool.Properties.html#12209" class="Function">∨-∧-isSemiring</a> <a id="12271" class="Symbol">=</a> <a id="12273" class="Keyword">record</a>
<a id="12282" class="Symbol">{</a> <a id="12284" href="Algebra.Structures.html#12800" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="12318" class="Symbol">=</a> <a id="12320" class="Keyword">record</a>
<a id="12331" class="Symbol">{</a> <a id="12333" href="Algebra.Structures.html#11137" class="Field">+-isCommutativeMonoid</a> <a id="12355" class="Symbol">=</a> <a id="12357" href="Data.Bool.Properties.html#7747" class="Function">∨-isCommutativeMonoid</a>
<a id="12383" class="Symbol">;</a> <a id="12385" href="Algebra.Structures.html#11190" class="Field">*-cong</a> <a id="12392" class="Symbol">=</a> <a id="12394" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="12400" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="12408" class="Symbol">;</a> <a id="12410" href="Algebra.Structures.html#11231" class="Field">*-assoc</a> <a id="12418" class="Symbol">=</a> <a id="12420" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a>
<a id="12432" class="Symbol">;</a> <a id="12434" href="Algebra.Structures.html#11273" class="Field">*-identity</a> <a id="12445" class="Symbol">=</a> <a id="12447" href="Data.Bool.Properties.html#8838" class="Function">∧-identity</a>
<a id="12462" class="Symbol">;</a> <a id="12464" href="Algebra.Structures.html#11315" class="Field">distrib</a> <a id="12472" class="Symbol">=</a> <a id="12474" href="Data.Bool.Properties.html#10002" class="Function">∧-distrib-∨</a>
<a id="12490" class="Symbol">}</a>
<a id="12494" class="Symbol">;</a> <a id="12496" href="Algebra.Structures.html#12890" class="Field">zero</a> <a id="12501" class="Symbol">=</a> <a id="12503" href="Data.Bool.Properties.html#9029" class="Function">∧-zero</a>
<a id="12512" class="Symbol">}</a>
<a id="∨-∧-isCommutativeSemiring"></a><a id="12515" href="Data.Bool.Properties.html#12515" class="Function">∨-∧-isCommutativeSemiring</a>
<a id="12543" class="Symbol">:</a> <a id="12545" href="Algebra.Structures.html#13418" class="Record">IsCommutativeSemiring</a> <a id="12567" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="12571" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="12575" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="12581" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="12586" href="Data.Bool.Properties.html#12515" class="Function">∨-∧-isCommutativeSemiring</a> <a id="12612" class="Symbol">=</a> <a id="12614" class="Keyword">record</a>
<a id="12623" class="Symbol">{</a> <a id="12625" href="Algebra.Structures.html#13498" class="Field">isSemiring</a> <a id="12636" class="Symbol">=</a> <a id="12638" href="Data.Bool.Properties.html#12209" class="Function">∨-∧-isSemiring</a>
<a id="12655" class="Symbol">;</a> <a id="12657" href="Algebra.Structures.html#13536" class="Field">*-comm</a> <a id="12664" class="Symbol">=</a> <a id="12666" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a>
<a id="12675" class="Symbol">}</a>
<a id="∨-∧-commutativeSemiring"></a><a id="12678" href="Data.Bool.Properties.html#12678" class="Function">∨-∧-commutativeSemiring</a> <a id="12702" class="Symbol">:</a> <a id="12704" href="Algebra.Bundles.html#17471" class="Record">CommutativeSemiring</a> <a id="12724" href="Level.html#521" class="Function">0ℓ</a> <a id="12727" href="Level.html#521" class="Function">0ℓ</a>
<a id="12730" href="Data.Bool.Properties.html#12678" class="Function">∨-∧-commutativeSemiring</a> <a id="12754" class="Symbol">=</a> <a id="12756" class="Keyword">record</a>
<a id="12765" class="Symbol">{</a> <a id="12767" href="Algebra.Bundles.html#17654" class="Field Operator">_+_</a> <a id="12789" class="Symbol">=</a> <a id="12791" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="12797" class="Symbol">;</a> <a id="12799" href="Algebra.Bundles.html#17694" class="Field Operator">_*_</a> <a id="12821" class="Symbol">=</a> <a id="12823" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="12829" class="Symbol">;</a> <a id="12831" href="Algebra.Bundles.html#17734" class="Field">0#</a> <a id="12853" class="Symbol">=</a> <a id="12855" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="12863" class="Symbol">;</a> <a id="12865" href="Algebra.Bundles.html#17770" class="Field">1#</a> <a id="12887" class="Symbol">=</a> <a id="12889" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="12896" class="Symbol">;</a> <a id="12898" href="Algebra.Bundles.html#17806" class="Field">isCommutativeSemiring</a> <a id="12920" class="Symbol">=</a> <a id="12922" href="Data.Bool.Properties.html#12515" class="Function">∨-∧-isCommutativeSemiring</a>
<a id="12950" class="Symbol">}</a>
<a id="∧-∨-isSemiring"></a><a id="12953" href="Data.Bool.Properties.html#12953" class="Function">∧-∨-isSemiring</a> <a id="12968" class="Symbol">:</a> <a id="12970" href="Algebra.Structures.html#12731" class="Record">IsSemiring</a> <a id="12981" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="12985" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="12989" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="12994" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="13000" href="Data.Bool.Properties.html#12953" class="Function">∧-∨-isSemiring</a> <a id="13015" class="Symbol">=</a> <a id="13017" class="Keyword">record</a>
<a id="13026" class="Symbol">{</a> <a id="13028" href="Algebra.Structures.html#12800" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="13062" class="Symbol">=</a> <a id="13064" class="Keyword">record</a>
<a id="13075" class="Symbol">{</a> <a id="13077" href="Algebra.Structures.html#11137" class="Field">+-isCommutativeMonoid</a> <a id="13099" class="Symbol">=</a> <a id="13101" href="Data.Bool.Properties.html#11583" class="Function">∧-isCommutativeMonoid</a>
<a id="13127" class="Symbol">;</a> <a id="13129" href="Algebra.Structures.html#11190" class="Field">*-cong</a> <a id="13136" class="Symbol">=</a> <a id="13138" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="13144" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="13152" class="Symbol">;</a> <a id="13154" href="Algebra.Structures.html#11231" class="Field">*-assoc</a> <a id="13162" class="Symbol">=</a> <a id="13164" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a>
<a id="13176" class="Symbol">;</a> <a id="13178" href="Algebra.Structures.html#11273" class="Field">*-identity</a> <a id="13189" class="Symbol">=</a> <a id="13191" href="Data.Bool.Properties.html#6050" class="Function">∨-identity</a>
<a id="13206" class="Symbol">;</a> <a id="13208" href="Algebra.Structures.html#11315" class="Field">distrib</a> <a id="13216" class="Symbol">=</a> <a id="13218" href="Data.Bool.Properties.html#10425" class="Function">∨-distrib-∧</a>
<a id="13234" class="Symbol">}</a>
<a id="13238" class="Symbol">;</a> <a id="13240" href="Algebra.Structures.html#12890" class="Field">zero</a> <a id="13245" class="Symbol">=</a> <a id="13247" href="Data.Bool.Properties.html#6240" class="Function">∨-zero</a>
<a id="13256" class="Symbol">}</a>
<a id="∧-∨-isCommutativeSemiring"></a><a id="13259" href="Data.Bool.Properties.html#13259" class="Function">∧-∨-isCommutativeSemiring</a>
<a id="13287" class="Symbol">:</a> <a id="13289" href="Algebra.Structures.html#13418" class="Record">IsCommutativeSemiring</a> <a id="13311" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="13315" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="13319" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="13324" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="13330" href="Data.Bool.Properties.html#13259" class="Function">∧-∨-isCommutativeSemiring</a> <a id="13356" class="Symbol">=</a> <a id="13358" class="Keyword">record</a>
<a id="13367" class="Symbol">{</a> <a id="13369" href="Algebra.Structures.html#13498" class="Field">isSemiring</a> <a id="13380" class="Symbol">=</a> <a id="13382" href="Data.Bool.Properties.html#12953" class="Function">∧-∨-isSemiring</a>
<a id="13399" class="Symbol">;</a> <a id="13401" href="Algebra.Structures.html#13536" class="Field">*-comm</a> <a id="13408" class="Symbol">=</a> <a id="13410" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a>
<a id="13419" class="Symbol">}</a>
<a id="∧-∨-commutativeSemiring"></a><a id="13422" href="Data.Bool.Properties.html#13422" class="Function">∧-∨-commutativeSemiring</a> <a id="13446" class="Symbol">:</a> <a id="13448" href="Algebra.Bundles.html#17471" class="Record">CommutativeSemiring</a> <a id="13468" href="Level.html#521" class="Function">0ℓ</a> <a id="13471" href="Level.html#521" class="Function">0ℓ</a>
<a id="13474" href="Data.Bool.Properties.html#13422" class="Function">∧-∨-commutativeSemiring</a> <a id="13498" class="Symbol">=</a> <a id="13500" class="Keyword">record</a>
<a id="13509" class="Symbol">{</a> <a id="13511" href="Algebra.Bundles.html#17654" class="Field Operator">_+_</a> <a id="13533" class="Symbol">=</a> <a id="13535" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="13541" class="Symbol">;</a> <a id="13543" href="Algebra.Bundles.html#17694" class="Field Operator">_*_</a> <a id="13565" class="Symbol">=</a> <a id="13567" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="13573" class="Symbol">;</a> <a id="13575" href="Algebra.Bundles.html#17734" class="Field">0#</a> <a id="13597" class="Symbol">=</a> <a id="13599" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="13606" class="Symbol">;</a> <a id="13608" href="Algebra.Bundles.html#17770" class="Field">1#</a> <a id="13630" class="Symbol">=</a> <a id="13632" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="13640" class="Symbol">;</a> <a id="13642" href="Algebra.Bundles.html#17806" class="Field">isCommutativeSemiring</a> <a id="13664" class="Symbol">=</a> <a id="13666" href="Data.Bool.Properties.html#13259" class="Function">∧-∨-isCommutativeSemiring</a>
<a id="13694" class="Symbol">}</a>
<a id="∨-∧-isLattice"></a><a id="13697" href="Data.Bool.Properties.html#13697" class="Function">∨-∧-isLattice</a> <a id="13711" class="Symbol">:</a> <a id="13713" href="Algebra.Lattice.Structures.html#4019" class="Record">IsLattice</a> <a id="13723" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="13727" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="13731" href="Data.Bool.Properties.html#13697" class="Function">∨-∧-isLattice</a> <a id="13745" class="Symbol">=</a> <a id="13747" class="Keyword">record</a>
<a id="13756" class="Symbol">{</a> <a id="13758" href="Algebra.Lattice.Structures.html#4075" class="Field">isEquivalence</a> <a id="13772" class="Symbol">=</a> <a id="13774" href="Relation.Binary.PropositionalEquality.Properties.html#5422" class="Function">isEquivalence</a>
<a id="13790" class="Symbol">;</a> <a id="13792" href="Algebra.Lattice.Structures.html#4113" class="Field">∨-comm</a> <a id="13806" class="Symbol">=</a> <a id="13808" href="Data.Bool.Properties.html#5772" class="Function">∨-comm</a>
<a id="13817" class="Symbol">;</a> <a id="13819" href="Algebra.Lattice.Structures.html#4147" class="Field">∨-assoc</a> <a id="13833" class="Symbol">=</a> <a id="13835" href="Data.Bool.Properties.html#5695" class="Function">∨-assoc</a>
<a id="13845" class="Symbol">;</a> <a id="13847" href="Algebra.Lattice.Structures.html#4181" class="Field">∨-cong</a> <a id="13861" class="Symbol">=</a> <a id="13863" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="13869" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a>
<a id="13875" class="Symbol">;</a> <a id="13877" href="Algebra.Lattice.Structures.html#4214" class="Field">∧-comm</a> <a id="13891" class="Symbol">=</a> <a id="13893" href="Data.Bool.Properties.html#8562" class="Function">∧-comm</a>
<a id="13902" class="Symbol">;</a> <a id="13904" href="Algebra.Lattice.Structures.html#4248" class="Field">∧-assoc</a> <a id="13918" class="Symbol">=</a> <a id="13920" href="Data.Bool.Properties.html#8485" class="Function">∧-assoc</a>
<a id="13930" class="Symbol">;</a> <a id="13932" href="Algebra.Lattice.Structures.html#4282" class="Field">∧-cong</a> <a id="13946" class="Symbol">=</a> <a id="13948" href="Relation.Binary.PropositionalEquality.Core.html#1395" class="Function">cong₂</a> <a id="13954" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="13960" class="Symbol">;</a> <a id="13962" href="Algebra.Lattice.Structures.html#4315" class="Field">absorptive</a> <a id="13976" class="Symbol">=</a> <a id="13978" href="Data.Bool.Properties.html#10652" class="Function">∨-∧-absorptive</a>
<a id="13995" class="Symbol">}</a>
<a id="∨-∧-lattice"></a><a id="13998" href="Data.Bool.Properties.html#13998" class="Function">∨-∧-lattice</a> <a id="14010" class="Symbol">:</a> <a id="14012" href="Algebra.Lattice.Bundles.html#4406" class="Record">Lattice</a> <a id="14020" href="Level.html#521" class="Function">0ℓ</a> <a id="14023" href="Level.html#521" class="Function">0ℓ</a>
<a id="14026" href="Data.Bool.Properties.html#13998" class="Function">∨-∧-lattice</a> <a id="14038" class="Symbol">=</a> <a id="14040" class="Keyword">record</a>
<a id="14049" class="Symbol">{</a> <a id="14051" href="Algebra.Lattice.Bundles.html#4609" class="Field">isLattice</a> <a id="14061" class="Symbol">=</a> <a id="14063" href="Data.Bool.Properties.html#13697" class="Function">∨-∧-isLattice</a>
<a id="14079" class="Symbol">}</a>
<a id="∨-∧-isDistributiveLattice"></a><a id="14082" href="Data.Bool.Properties.html#14082" class="Function">∨-∧-isDistributiveLattice</a> <a id="14108" class="Symbol">:</a> <a id="14110" href="Algebra.Lattice.Structures.html#4768" class="Record">IsDistributiveLattice</a> <a id="14132" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="14136" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a>
<a id="14140" href="Data.Bool.Properties.html#14082" class="Function">∨-∧-isDistributiveLattice</a> <a id="14166" class="Symbol">=</a> <a id="14168" class="Keyword">record</a>
<a id="14177" class="Symbol">{</a> <a id="14179" href="Algebra.Lattice.Structures.html#4836" class="Field">isLattice</a> <a id="14191" class="Symbol">=</a> <a id="14193" href="Data.Bool.Properties.html#13697" class="Function">∨-∧-isLattice</a>
<a id="14209" class="Symbol">;</a> <a id="14211" href="Algebra.Lattice.Structures.html#4868" class="Field">∨-distrib-∧</a> <a id="14223" class="Symbol">=</a> <a id="14225" href="Data.Bool.Properties.html#10425" class="Function">∨-distrib-∧</a>
<a id="14239" class="Symbol">;</a> <a id="14241" href="Algebra.Lattice.Structures.html#4906" class="Field">∧-distrib-∨</a> <a id="14253" class="Symbol">=</a> <a id="14255" href="Data.Bool.Properties.html#10002" class="Function">∧-distrib-∨</a>
<a id="14269" class="Symbol">}</a>
<a id="∨-∧-distributiveLattice"></a><a id="14272" href="Data.Bool.Properties.html#14272" class="Function">∨-∧-distributiveLattice</a> <a id="14296" class="Symbol">:</a> <a id="14298" href="Algebra.Lattice.Bundles.html#4989" class="Record">DistributiveLattice</a> <a id="14318" href="Level.html#521" class="Function">0ℓ</a> <a id="14321" href="Level.html#521" class="Function">0ℓ</a>
<a id="14324" href="Data.Bool.Properties.html#14272" class="Function">∨-∧-distributiveLattice</a> <a id="14348" class="Symbol">=</a> <a id="14350" class="Keyword">record</a>
<a id="14359" class="Symbol">{</a> <a id="14361" href="Algebra.Lattice.Bundles.html#5252" class="Field">isDistributiveLattice</a> <a id="14383" class="Symbol">=</a> <a id="14385" href="Data.Bool.Properties.html#14082" class="Function">∨-∧-isDistributiveLattice</a>
<a id="14413" class="Symbol">}</a>
<a id="∨-∧-isBooleanAlgebra"></a><a id="14416" href="Data.Bool.Properties.html#14416" class="Function">∨-∧-isBooleanAlgebra</a> <a id="14437" class="Symbol">:</a> <a id="14439" href="Algebra.Lattice.Structures.html#5413" class="Record">IsBooleanAlgebra</a> <a id="14456" href="Data.Bool.Base.html#1053" class="Function Operator">_∨_</a> <a id="14460" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="14464" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="14468" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="14473" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="14479" href="Data.Bool.Properties.html#14416" class="Function">∨-∧-isBooleanAlgebra</a> <a id="14500" class="Symbol">=</a> <a id="14502" class="Keyword">record</a>
<a id="14511" class="Symbol">{</a> <a id="14513" href="Algebra.Lattice.Structures.html#5501" class="Field">isDistributiveLattice</a> <a id="14535" class="Symbol">=</a> <a id="14537" href="Data.Bool.Properties.html#14082" class="Function">∨-∧-isDistributiveLattice</a>
<a id="14565" class="Symbol">;</a> <a id="14567" href="Algebra.Lattice.Structures.html#5555" class="Field">∨-complement</a> <a id="14589" class="Symbol">=</a> <a id="14591" href="Data.Bool.Properties.html#6473" class="Function">∨-inverse</a>
<a id="14603" class="Symbol">;</a> <a id="14605" href="Algebra.Lattice.Structures.html#5597" class="Field">∧-complement</a> <a id="14627" class="Symbol">=</a> <a id="14629" href="Data.Bool.Properties.html#9264" class="Function">∧-inverse</a>
<a id="14641" class="Symbol">;</a> <a id="14643" href="Algebra.Lattice.Structures.html#5639" class="Field">¬-cong</a> <a id="14665" class="Symbol">=</a> <a id="14667" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="14672" href="Data.Bool.Base.html#941" class="Function">not</a>
<a id="14678" class="Symbol">}</a>
<a id="∨-∧-booleanAlgebra"></a><a id="14681" href="Data.Bool.Properties.html#14681" class="Function">∨-∧-booleanAlgebra</a> <a id="14700" class="Symbol">:</a> <a id="14702" href="Algebra.Lattice.Bundles.html#5554" class="Record">BooleanAlgebra</a> <a id="14717" href="Level.html#521" class="Function">0ℓ</a> <a id="14720" href="Level.html#521" class="Function">0ℓ</a>
<a id="14723" href="Data.Bool.Properties.html#14681" class="Function">∨-∧-booleanAlgebra</a> <a id="14742" class="Symbol">=</a> <a id="14744" class="Keyword">record</a>
<a id="14753" class="Symbol">{</a> <a id="14755" href="Algebra.Lattice.Bundles.html#5903" class="Field">isBooleanAlgebra</a> <a id="14772" class="Symbol">=</a> <a id="14774" href="Data.Bool.Properties.html#14416" class="Function">∨-∧-isBooleanAlgebra</a>
<a id="14797" class="Symbol">}</a>
<a id="14800" class="Comment">------------------------------------------------------------------------</a>
<a id="14873" class="Comment">-- Properties of not</a>
<a id="not-involutive"></a><a id="14895" href="Data.Bool.Properties.html#14895" class="Function">not-involutive</a> <a id="14910" class="Symbol">:</a> <a id="14912" href="Algebra.Definitions.html#4153" class="Function">Involutive</a> <a id="14923" href="Data.Bool.Base.html#941" class="Function">not</a>
<a id="14927" href="Data.Bool.Properties.html#14895" class="Function">not-involutive</a> <a id="14942" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="14948" class="Symbol">=</a> <a id="14950" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="14955" href="Data.Bool.Properties.html#14895" class="Function">not-involutive</a> <a id="14970" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="14976" class="Symbol">=</a> <a id="14978" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="not-injective"></a><a id="14984" href="Data.Bool.Properties.html#14984" class="Function">not-injective</a> <a id="14998" class="Symbol">:</a> <a id="15000" class="Symbol">∀</a> <a id="15002" class="Symbol">{</a><a id="15003" href="Data.Bool.Properties.html#15003" class="Bound">x</a> <a id="15005" href="Data.Bool.Properties.html#15005" class="Bound">y</a><a id="15006" class="Symbol">}</a> <a id="15008" class="Symbol">→</a> <a id="15010" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15014" href="Data.Bool.Properties.html#15003" class="Bound">x</a> <a id="15016" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15018" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15022" href="Data.Bool.Properties.html#15005" class="Bound">y</a> <a id="15024" class="Symbol">→</a> <a id="15026" href="Data.Bool.Properties.html#15003" class="Bound">x</a> <a id="15028" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15030" href="Data.Bool.Properties.html#15005" class="Bound">y</a>
<a id="15032" href="Data.Bool.Properties.html#14984" class="Function">not-injective</a> <a id="15046" class="Symbol">{</a><a id="15047" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15052" class="Symbol">}</a> <a id="15054" class="Symbol">{</a><a id="15055" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15060" class="Symbol">}</a> <a id="15062" href="Data.Bool.Properties.html#15062" class="Bound">nx≢ny</a> <a id="15068" class="Symbol">=</a> <a id="15070" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15075" href="Data.Bool.Properties.html#14984" class="Function">not-injective</a> <a id="15089" class="Symbol">{</a><a id="15090" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15094" class="Symbol">}</a> <a id="15097" class="Symbol">{</a><a id="15098" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15102" class="Symbol">}</a> <a id="15105" href="Data.Bool.Properties.html#15105" class="Bound">nx≢ny</a> <a id="15111" class="Symbol">=</a> <a id="15113" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="not-¬"></a><a id="15119" href="Data.Bool.Properties.html#15119" class="Function">not-¬</a> <a id="15125" class="Symbol">:</a> <a id="15127" class="Symbol">∀</a> <a id="15129" class="Symbol">{</a><a id="15130" href="Data.Bool.Properties.html#15130" class="Bound">x</a> <a id="15132" href="Data.Bool.Properties.html#15132" class="Bound">y</a><a id="15133" class="Symbol">}</a> <a id="15135" class="Symbol">→</a> <a id="15137" href="Data.Bool.Properties.html#15130" class="Bound">x</a> <a id="15139" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15141" href="Data.Bool.Properties.html#15132" class="Bound">y</a> <a id="15143" class="Symbol">→</a> <a id="15145" href="Data.Bool.Properties.html#15130" class="Bound">x</a> <a id="15147" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">≢</a> <a id="15149" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15153" href="Data.Bool.Properties.html#15132" class="Bound">y</a>
<a id="15155" href="Data.Bool.Properties.html#15119" class="Function">not-¬</a> <a id="15161" class="Symbol">{</a><a id="15162" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15166" class="Symbol">}</a> <a id="15169" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="15174" class="Symbol">()</a>
<a id="15177" href="Data.Bool.Properties.html#15119" class="Function">not-¬</a> <a id="15183" class="Symbol">{</a><a id="15184" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15189" class="Symbol">}</a> <a id="15191" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="15196" class="Symbol">()</a>
<a id="¬-not"></a><a id="15200" href="Data.Bool.Properties.html#15200" class="Function">¬-not</a> <a id="15206" class="Symbol">:</a> <a id="15208" class="Symbol">∀</a> <a id="15210" class="Symbol">{</a><a id="15211" href="Data.Bool.Properties.html#15211" class="Bound">x</a> <a id="15213" href="Data.Bool.Properties.html#15213" class="Bound">y</a><a id="15214" class="Symbol">}</a> <a id="15216" class="Symbol">→</a> <a id="15218" href="Data.Bool.Properties.html#15211" class="Bound">x</a> <a id="15220" href="Relation.Binary.PropositionalEquality.Core.html#858" class="Function Operator">≢</a> <a id="15222" href="Data.Bool.Properties.html#15213" class="Bound">y</a> <a id="15224" class="Symbol">→</a> <a id="15226" href="Data.Bool.Properties.html#15211" class="Bound">x</a> <a id="15228" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15230" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15234" href="Data.Bool.Properties.html#15213" class="Bound">y</a>
<a id="15236" href="Data.Bool.Properties.html#15200" class="Function">¬-not</a> <a id="15242" class="Symbol">{</a><a id="15243" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15247" class="Symbol">}</a> <a id="15250" class="Symbol">{</a><a id="15251" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15255" class="Symbol">}</a> <a id="15258" href="Data.Bool.Properties.html#15258" class="Bound">x≢y</a> <a id="15262" class="Symbol">=</a> <a id="15264" href="Data.Empty.html#1050" class="Function">⊥-elim</a> <a id="15271" class="Symbol">(</a><a id="15272" href="Data.Bool.Properties.html#15258" class="Bound">x≢y</a> <a id="15276" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="15280" class="Symbol">)</a>
<a id="15282" href="Data.Bool.Properties.html#15200" class="Function">¬-not</a> <a id="15288" class="Symbol">{</a><a id="15289" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15293" class="Symbol">}</a> <a id="15296" class="Symbol">{</a><a id="15297" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15302" class="Symbol">}</a> <a id="15304" class="Symbol">_</a> <a id="15308" class="Symbol">=</a> <a id="15310" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15315" href="Data.Bool.Properties.html#15200" class="Function">¬-not</a> <a id="15321" class="Symbol">{</a><a id="15322" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15327" class="Symbol">}</a> <a id="15329" class="Symbol">{</a><a id="15330" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="15334" class="Symbol">}</a> <a id="15337" class="Symbol">_</a> <a id="15341" class="Symbol">=</a> <a id="15343" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15348" href="Data.Bool.Properties.html#15200" class="Function">¬-not</a> <a id="15354" class="Symbol">{</a><a id="15355" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15360" class="Symbol">}</a> <a id="15362" class="Symbol">{</a><a id="15363" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="15368" class="Symbol">}</a> <a id="15370" href="Data.Bool.Properties.html#15370" class="Bound">x≢y</a> <a id="15374" class="Symbol">=</a> <a id="15376" href="Data.Empty.html#1050" class="Function">⊥-elim</a> <a id="15383" class="Symbol">(</a><a id="15384" href="Data.Bool.Properties.html#15370" class="Bound">x≢y</a> <a id="15388" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="15392" class="Symbol">)</a>
<a id="15395" class="Comment">------------------------------------------------------------------------</a>
<a id="15468" class="Comment">-- Properties of _xor_</a>
<a id="xor-is-ok"></a><a id="15492" href="Data.Bool.Properties.html#15492" class="Function">xor-is-ok</a> <a id="15502" class="Symbol">:</a> <a id="15504" class="Symbol">∀</a> <a id="15506" href="Data.Bool.Properties.html#15506" class="Bound">x</a> <a id="15508" href="Data.Bool.Properties.html#15508" class="Bound">y</a> <a id="15510" class="Symbol">→</a> <a id="15512" href="Data.Bool.Properties.html#15506" class="Bound">x</a> <a id="15514" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15518" href="Data.Bool.Properties.html#15508" class="Bound">y</a> <a id="15520" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15522" class="Symbol">(</a><a id="15523" href="Data.Bool.Properties.html#15506" class="Bound">x</a> <a id="15525" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="15527" href="Data.Bool.Properties.html#15508" class="Bound">y</a><a id="15528" class="Symbol">)</a> <a id="15530" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="15532" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15536" class="Symbol">(</a><a id="15537" href="Data.Bool.Properties.html#15506" class="Bound">x</a> <a id="15539" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="15541" href="Data.Bool.Properties.html#15508" class="Bound">y</a><a id="15542" class="Symbol">)</a>
<a id="15544" href="Data.Bool.Properties.html#15492" class="Function">xor-is-ok</a> <a id="15554" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="15560" href="Data.Bool.Properties.html#15560" class="Bound">y</a> <a id="15562" class="Symbol">=</a> <a id="15564" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15569" href="Data.Bool.Properties.html#15492" class="Function">xor-is-ok</a> <a id="15579" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="15585" href="Data.Bool.Properties.html#15585" class="Bound">y</a> <a id="15587" class="Symbol">=</a> <a id="15589" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="15593" class="Symbol">(</a><a id="15594" href="Data.Bool.Properties.html#8750" class="Function">∧-identityʳ</a> <a id="15606" class="Symbol">_)</a>
<a id="true-xor"></a><a id="15610" href="Data.Bool.Properties.html#15610" class="Function">true-xor</a> <a id="15619" class="Symbol">:</a> <a id="15621" class="Symbol">∀</a> <a id="15623" href="Data.Bool.Properties.html#15623" class="Bound">x</a> <a id="15625" class="Symbol">→</a> <a id="15627" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="15632" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15636" href="Data.Bool.Properties.html#15623" class="Bound">x</a> <a id="15638" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15640" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15644" href="Data.Bool.Properties.html#15623" class="Bound">x</a>
<a id="15646" href="Data.Bool.Properties.html#15610" class="Function">true-xor</a> <a id="15655" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="15661" class="Symbol">=</a> <a id="15663" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15668" href="Data.Bool.Properties.html#15610" class="Function">true-xor</a> <a id="15677" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="15683" class="Symbol">=</a> <a id="15685" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-same"></a><a id="15691" href="Data.Bool.Properties.html#15691" class="Function">xor-same</a> <a id="15700" class="Symbol">:</a> <a id="15702" class="Symbol">∀</a> <a id="15704" href="Data.Bool.Properties.html#15704" class="Bound">x</a> <a id="15706" class="Symbol">→</a> <a id="15708" href="Data.Bool.Properties.html#15704" class="Bound">x</a> <a id="15710" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15714" href="Data.Bool.Properties.html#15704" class="Bound">x</a> <a id="15716" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15718" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="15724" href="Data.Bool.Properties.html#15691" class="Function">xor-same</a> <a id="15733" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="15739" class="Symbol">=</a> <a id="15741" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15746" href="Data.Bool.Properties.html#15691" class="Function">xor-same</a> <a id="15755" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="15761" class="Symbol">=</a> <a id="15763" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="not-distribˡ-xor"></a><a id="15769" href="Data.Bool.Properties.html#15769" class="Function">not-distribˡ-xor</a> <a id="15786" class="Symbol">:</a> <a id="15788" class="Symbol">∀</a> <a id="15790" href="Data.Bool.Properties.html#15790" class="Bound">x</a> <a id="15792" href="Data.Bool.Properties.html#15792" class="Bound">y</a> <a id="15794" class="Symbol">→</a> <a id="15796" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15800" class="Symbol">(</a><a id="15801" href="Data.Bool.Properties.html#15790" class="Bound">x</a> <a id="15803" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15807" href="Data.Bool.Properties.html#15792" class="Bound">y</a><a id="15808" class="Symbol">)</a> <a id="15810" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15812" class="Symbol">(</a><a id="15813" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15817" href="Data.Bool.Properties.html#15790" class="Bound">x</a><a id="15818" class="Symbol">)</a> <a id="15820" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15824" href="Data.Bool.Properties.html#15792" class="Bound">y</a>
<a id="15826" href="Data.Bool.Properties.html#15769" class="Function">not-distribˡ-xor</a> <a id="15843" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="15849" href="Data.Bool.Properties.html#15849" class="Bound">y</a> <a id="15851" class="Symbol">=</a> <a id="15853" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15858" href="Data.Bool.Properties.html#15769" class="Function">not-distribˡ-xor</a> <a id="15875" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="15881" href="Data.Bool.Properties.html#15881" class="Bound">y</a> <a id="15883" class="Symbol">=</a> <a id="15885" href="Data.Bool.Properties.html#14895" class="Function">not-involutive</a> <a id="15900" class="Symbol">_</a>
<a id="not-distribʳ-xor"></a><a id="15903" href="Data.Bool.Properties.html#15903" class="Function">not-distribʳ-xor</a> <a id="15920" class="Symbol">:</a> <a id="15922" class="Symbol">∀</a> <a id="15924" href="Data.Bool.Properties.html#15924" class="Bound">x</a> <a id="15926" href="Data.Bool.Properties.html#15926" class="Bound">y</a> <a id="15928" class="Symbol">→</a> <a id="15930" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15934" class="Symbol">(</a><a id="15935" href="Data.Bool.Properties.html#15924" class="Bound">x</a> <a id="15937" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15941" href="Data.Bool.Properties.html#15926" class="Bound">y</a><a id="15942" class="Symbol">)</a> <a id="15944" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="15946" href="Data.Bool.Properties.html#15924" class="Bound">x</a> <a id="15948" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="15952" class="Symbol">(</a><a id="15953" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="15957" href="Data.Bool.Properties.html#15926" class="Bound">y</a><a id="15958" class="Symbol">)</a>
<a id="15960" href="Data.Bool.Properties.html#15903" class="Function">not-distribʳ-xor</a> <a id="15977" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="15983" href="Data.Bool.Properties.html#15983" class="Bound">y</a> <a id="15985" class="Symbol">=</a> <a id="15987" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="15992" href="Data.Bool.Properties.html#15903" class="Function">not-distribʳ-xor</a> <a id="16009" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16015" href="Data.Bool.Properties.html#16015" class="Bound">y</a> <a id="16017" class="Symbol">=</a> <a id="16019" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-assoc"></a><a id="16025" href="Data.Bool.Properties.html#16025" class="Function">xor-assoc</a> <a id="16035" class="Symbol">:</a> <a id="16037" href="Algebra.Definitions.html#1548" class="Function">Associative</a> <a id="16049" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16055" href="Data.Bool.Properties.html#16025" class="Function">xor-assoc</a> <a id="16065" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16071" href="Data.Bool.Properties.html#16071" class="Bound">y</a> <a id="16073" href="Data.Bool.Properties.html#16073" class="Bound">z</a> <a id="16075" class="Symbol">=</a> <a id="16077" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="16081" class="Symbol">(</a><a id="16082" href="Data.Bool.Properties.html#15769" class="Function">not-distribˡ-xor</a> <a id="16099" href="Data.Bool.Properties.html#16071" class="Bound">y</a> <a id="16101" href="Data.Bool.Properties.html#16073" class="Bound">z</a><a id="16102" class="Symbol">)</a>
<a id="16104" href="Data.Bool.Properties.html#16025" class="Function">xor-assoc</a> <a id="16114" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16120" href="Data.Bool.Properties.html#16120" class="Bound">y</a> <a id="16122" href="Data.Bool.Properties.html#16122" class="Bound">z</a> <a id="16124" class="Symbol">=</a> <a id="16126" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-comm"></a><a id="16132" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16141" class="Symbol">:</a> <a id="16143" href="Algebra.Definitions.html#1635" class="Function">Commutative</a> <a id="16155" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16161" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16170" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16176" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16182" class="Symbol">=</a> <a id="16184" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16189" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16198" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16204" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16210" class="Symbol">=</a> <a id="16212" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16217" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16226" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16232" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16238" class="Symbol">=</a> <a id="16240" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16245" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16254" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16260" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16266" class="Symbol">=</a> <a id="16268" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-identityˡ"></a><a id="16274" href="Data.Bool.Properties.html#16274" class="Function">xor-identityˡ</a> <a id="16288" class="Symbol">:</a> <a id="16290" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="16303" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16309" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16315" href="Data.Bool.Properties.html#16274" class="Function">xor-identityˡ</a> <a id="16329" class="Symbol">_</a> <a id="16331" class="Symbol">=</a> <a id="16333" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-identityʳ"></a><a id="16339" href="Data.Bool.Properties.html#16339" class="Function">xor-identityʳ</a> <a id="16353" class="Symbol">:</a> <a id="16355" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="16369" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16375" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16381" href="Data.Bool.Properties.html#16339" class="Function">xor-identityʳ</a> <a id="16395" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16401" class="Symbol">=</a> <a id="16403" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16408" href="Data.Bool.Properties.html#16339" class="Function">xor-identityʳ</a> <a id="16422" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16428" class="Symbol">=</a> <a id="16430" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-identity"></a><a id="16436" href="Data.Bool.Properties.html#16436" class="Function">xor-identity</a> <a id="16449" class="Symbol">:</a> <a id="16451" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="16460" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16466" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16472" href="Data.Bool.Properties.html#16436" class="Function">xor-identity</a> <a id="16485" class="Symbol">=</a> <a id="16487" href="Data.Bool.Properties.html#16274" class="Function">xor-identityˡ</a> <a id="16501" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="16503" href="Data.Bool.Properties.html#16339" class="Function">xor-identityʳ</a>
<a id="xor-inverseˡ"></a><a id="16518" href="Data.Bool.Properties.html#16518" class="Function">xor-inverseˡ</a> <a id="16531" class="Symbol">:</a> <a id="16533" href="Algebra.Definitions.html#2144" class="Function">LeftInverse</a> <a id="16545" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16550" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="16554" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16560" href="Data.Bool.Properties.html#16518" class="Function">xor-inverseˡ</a> <a id="16573" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16579" class="Symbol">=</a> <a id="16581" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16586" href="Data.Bool.Properties.html#16518" class="Function">xor-inverseˡ</a> <a id="16599" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16604" class="Symbol">=</a> <a id="16606" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-inverseʳ"></a><a id="16612" href="Data.Bool.Properties.html#16612" class="Function">xor-inverseʳ</a> <a id="16625" class="Symbol">:</a> <a id="16627" href="Algebra.Definitions.html#2232" class="Function">RightInverse</a> <a id="16640" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16645" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="16649" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16655" href="Data.Bool.Properties.html#16612" class="Function">xor-inverseʳ</a> <a id="16668" href="Data.Bool.Properties.html#16668" class="Bound">x</a> <a id="16670" class="Symbol">=</a> <a id="16672" href="Data.Bool.Properties.html#16132" class="Function">xor-comm</a> <a id="16681" href="Data.Bool.Properties.html#16668" class="Bound">x</a> <a id="16683" class="Symbol">(</a><a id="16684" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="16688" href="Data.Bool.Properties.html#16668" class="Bound">x</a><a id="16689" class="Symbol">)</a> <a id="16691" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="16693" href="Relation.Binary.PropositionalEquality.Core.html#1757" class="Function">trans</a> <a id="16699" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="16701" href="Data.Bool.Properties.html#16518" class="Function">xor-inverseˡ</a> <a id="16714" href="Data.Bool.Properties.html#16668" class="Bound">x</a>
<a id="xor-inverse"></a><a id="16717" href="Data.Bool.Properties.html#16717" class="Function">xor-inverse</a> <a id="16729" class="Symbol">:</a> <a id="16731" href="Algebra.Definitions.html#2322" class="Function">Inverse</a> <a id="16739" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16744" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="16748" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16754" href="Data.Bool.Properties.html#16717" class="Function">xor-inverse</a> <a id="16766" class="Symbol">=</a> <a id="16768" href="Data.Bool.Properties.html#16518" class="Function">xor-inverseˡ</a> <a id="16781" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="16783" href="Data.Bool.Properties.html#16612" class="Function">xor-inverseʳ</a>
<a id="∧-distribˡ-xor"></a><a id="16797" href="Data.Bool.Properties.html#16797" class="Function">∧-distribˡ-xor</a> <a id="16812" class="Symbol">:</a> <a id="16814" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="16818" href="Algebra.Definitions.html#3155" class="Function Operator">DistributesOverˡ</a> <a id="16835" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16841" href="Data.Bool.Properties.html#16797" class="Function">∧-distribˡ-xor</a> <a id="16856" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16862" href="Data.Bool.Properties.html#16862" class="Bound">y</a> <a id="16864" href="Data.Bool.Properties.html#16864" class="Bound">z</a> <a id="16866" class="Symbol">=</a> <a id="16868" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16873" href="Data.Bool.Properties.html#16797" class="Function">∧-distribˡ-xor</a> <a id="16888" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="16894" href="Data.Bool.Properties.html#16894" class="Bound">y</a> <a id="16896" href="Data.Bool.Properties.html#16896" class="Bound">z</a> <a id="16898" class="Symbol">=</a> <a id="16900" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="∧-distribʳ-xor"></a><a id="16906" href="Data.Bool.Properties.html#16906" class="Function">∧-distribʳ-xor</a> <a id="16921" class="Symbol">:</a> <a id="16923" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="16927" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="16944" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="16950" href="Data.Bool.Properties.html#16906" class="Function">∧-distribʳ-xor</a> <a id="16965" href="Data.Bool.Properties.html#16965" class="Bound">x</a> <a id="16967" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="16973" href="Data.Bool.Properties.html#16973" class="Bound">z</a> <a id="16978" class="Symbol">=</a> <a id="16980" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="16985" href="Data.Bool.Properties.html#16906" class="Function">∧-distribʳ-xor</a> <a id="17000" href="Data.Bool.Properties.html#17000" class="Bound">x</a> <a id="17002" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="17007" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="17013" class="Symbol">=</a> <a id="17015" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="17019" class="Symbol">(</a><a id="17020" href="Data.Bool.Properties.html#16339" class="Function">xor-identityʳ</a> <a id="17034" href="Data.Bool.Properties.html#17000" class="Bound">x</a><a id="17035" class="Symbol">)</a>
<a id="17037" href="Data.Bool.Properties.html#16906" class="Function">∧-distribʳ-xor</a> <a id="17052" href="Data.Bool.Properties.html#17052" class="Bound">x</a> <a id="17054" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="17059" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="17065" class="Symbol">=</a> <a id="17067" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="17071" class="Symbol">(</a><a id="17072" href="Data.Bool.Properties.html#15691" class="Function">xor-same</a> <a id="17081" href="Data.Bool.Properties.html#17052" class="Bound">x</a><a id="17082" class="Symbol">)</a>
<a id="∧-distrib-xor"></a><a id="17085" href="Data.Bool.Properties.html#17085" class="Function">∧-distrib-xor</a> <a id="17099" class="Symbol">:</a> <a id="17101" href="Data.Bool.Base.html#995" class="Function Operator">_∧_</a> <a id="17105" href="Algebra.Definitions.html#3393" class="Function Operator">DistributesOver</a> <a id="17121" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a>
<a id="17127" href="Data.Bool.Properties.html#17085" class="Function">∧-distrib-xor</a> <a id="17141" class="Symbol">=</a> <a id="17143" href="Data.Bool.Properties.html#16797" class="Function">∧-distribˡ-xor</a> <a id="17158" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="17160" href="Data.Bool.Properties.html#16906" class="Function">∧-distribʳ-xor</a>
<a id="xor-annihilates-not"></a><a id="17176" href="Data.Bool.Properties.html#17176" class="Function">xor-annihilates-not</a> <a id="17196" class="Symbol">:</a> <a id="17198" class="Symbol">∀</a> <a id="17200" href="Data.Bool.Properties.html#17200" class="Bound">x</a> <a id="17202" href="Data.Bool.Properties.html#17202" class="Bound">y</a> <a id="17204" class="Symbol">→</a> <a id="17206" class="Symbol">(</a><a id="17207" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="17211" href="Data.Bool.Properties.html#17200" class="Bound">x</a><a id="17212" class="Symbol">)</a> <a id="17214" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="17218" class="Symbol">(</a><a id="17219" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="17223" href="Data.Bool.Properties.html#17202" class="Bound">y</a><a id="17224" class="Symbol">)</a> <a id="17226" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="17228" href="Data.Bool.Properties.html#17200" class="Bound">x</a> <a id="17230" href="Data.Bool.Base.html#1110" class="Function Operator">xor</a> <a id="17234" href="Data.Bool.Properties.html#17202" class="Bound">y</a>
<a id="17236" href="Data.Bool.Properties.html#17176" class="Function">xor-annihilates-not</a> <a id="17256" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="17262" href="Data.Bool.Properties.html#17262" class="Bound">y</a> <a id="17264" class="Symbol">=</a> <a id="17266" href="Data.Bool.Properties.html#14895" class="Function">not-involutive</a> <a id="17281" class="Symbol">_</a>
<a id="17283" href="Data.Bool.Properties.html#17176" class="Function">xor-annihilates-not</a> <a id="17303" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="17309" href="Data.Bool.Properties.html#17309" class="Bound">y</a> <a id="17311" class="Symbol">=</a> <a id="17313" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="xor-∧-commutativeRing"></a><a id="17319" href="Data.Bool.Properties.html#17319" class="Function">xor-∧-commutativeRing</a> <a id="17341" class="Symbol">:</a> <a id="17343" href="Algebra.Bundles.html#27248" class="Record">CommutativeRing</a> <a id="17359" href="Level.html#521" class="Function">0ℓ</a> <a id="17362" href="Level.html#521" class="Function">0ℓ</a>
<a id="17365" href="Data.Bool.Properties.html#17319" class="Function">xor-∧-commutativeRing</a> <a id="17387" class="Symbol">=</a> <a id="17389" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17453" class="Function">⊕-∧-commutativeRing</a>
<a id="17411" class="Keyword">where</a>
<a id="17419" class="Keyword">open</a> <a id="17424" href="Algebra.Lattice.Properties.BooleanAlgebra.html" class="Module">BooleanAlgebraProperties</a> <a id="17449" href="Data.Bool.Properties.html#14681" class="Function">∨-∧-booleanAlgebra</a>
<a id="17470" class="Keyword">open</a> <a id="17475" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7330" class="Module">XorRing</a> <a id="17483" href="Data.Bool.Base.html#1110" class="Function Operator">_xor_</a> <a id="17489" href="Data.Bool.Properties.html#15492" class="Function">xor-is-ok</a>
<a id="17500" class="Comment">------------------------------------------------------------------------</a>
<a id="17573" class="Comment">-- Properties of if_then_else_</a>
<a id="if-float"></a><a id="17605" href="Data.Bool.Properties.html#17605" class="Function">if-float</a> <a id="17614" class="Symbol">:</a> <a id="17616" class="Symbol">∀</a> <a id="17618" class="Symbol">(</a><a id="17619" href="Data.Bool.Properties.html#17619" class="Bound">f</a> <a id="17621" class="Symbol">:</a> <a id="17623" href="Data.Bool.Properties.html#1798" class="Generalizable">A</a> <a id="17625" class="Symbol">→</a> <a id="17627" href="Data.Bool.Properties.html#1812" class="Generalizable">B</a><a id="17628" class="Symbol">)</a> <a id="17630" href="Data.Bool.Properties.html#17630" class="Bound">b</a> <a id="17632" class="Symbol">{</a><a id="17633" href="Data.Bool.Properties.html#17633" class="Bound">x</a> <a id="17635" href="Data.Bool.Properties.html#17635" class="Bound">y</a><a id="17636" class="Symbol">}</a> <a id="17638" class="Symbol">→</a>
<a id="17651" href="Data.Bool.Properties.html#17619" class="Bound">f</a> <a id="17653" class="Symbol">(</a><a id="17654" href="Data.Bool.Base.html#1505" class="Function Operator">if</a> <a id="17657" href="Data.Bool.Properties.html#17630" class="Bound">b</a> <a id="17659" href="Data.Bool.Base.html#1505" class="Function Operator">then</a> <a id="17664" href="Data.Bool.Properties.html#17633" class="Bound">x</a> <a id="17666" href="Data.Bool.Base.html#1505" class="Function Operator">else</a> <a id="17671" href="Data.Bool.Properties.html#17635" class="Bound">y</a><a id="17672" class="Symbol">)</a> <a id="17674" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="17676" class="Symbol">(</a><a id="17677" href="Data.Bool.Base.html#1505" class="Function Operator">if</a> <a id="17680" href="Data.Bool.Properties.html#17630" class="Bound">b</a> <a id="17682" href="Data.Bool.Base.html#1505" class="Function Operator">then</a> <a id="17687" href="Data.Bool.Properties.html#17619" class="Bound">f</a> <a id="17689" href="Data.Bool.Properties.html#17633" class="Bound">x</a> <a id="17691" href="Data.Bool.Base.html#1505" class="Function Operator">else</a> <a id="17696" href="Data.Bool.Properties.html#17619" class="Bound">f</a> <a id="17698" href="Data.Bool.Properties.html#17635" class="Bound">y</a><a id="17699" class="Symbol">)</a>
<a id="17701" href="Data.Bool.Properties.html#17605" class="Function">if-float</a> <a id="17710" class="Symbol">_</a> <a id="17712" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="17718" class="Symbol">=</a> <a id="17720" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="17725" href="Data.Bool.Properties.html#17605" class="Function">if-float</a> <a id="17734" class="Symbol">_</a> <a id="17736" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a> <a id="17742" class="Symbol">=</a> <a id="17744" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="17750" class="Comment">------------------------------------------------------------------------</a>
<a id="17823" class="Comment">-- Properties of T</a>
<a id="17843" class="Keyword">open</a> <a id="17848" href="Relation.Nullary.Decidable.Core.html" class="Module">Relation.Nullary.Decidable.Core</a> <a id="17880" class="Keyword">public</a> <a id="17887" class="Keyword">using</a> <a id="17893" class="Symbol">(</a><a id="17894" href="Relation.Nullary.Decidable.Core.html#2465" class="Function">T?</a><a id="17896" class="Symbol">)</a>
<a id="T-≡"></a><a id="17899" href="Data.Bool.Properties.html#17899" class="Function">T-≡</a> <a id="17903" class="Symbol">:</a> <a id="17905" class="Symbol">∀</a> <a id="17907" class="Symbol">{</a><a id="17908" href="Data.Bool.Properties.html#17908" class="Bound">x</a><a id="17909" class="Symbol">}</a> <a id="17911" class="Symbol">→</a> <a id="17913" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="17915" href="Data.Bool.Properties.html#17908" class="Bound">x</a> <a id="17917" href="Function.Bundles.html#12039" class="Function Operator">⇔</a> <a id="17919" href="Data.Bool.Properties.html#17908" class="Bound">x</a> <a id="17921" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="17923" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a>
<a id="17928" href="Data.Bool.Properties.html#17899" class="Function">T-≡</a> <a id="17932" class="Symbol">{</a><a id="17933" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="17938" class="Symbol">}</a> <a id="17940" class="Symbol">=</a> <a id="17942" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="17946" class="Symbol">(λ</a> <a id="17949" class="Symbol">())</a> <a id="17959" class="Symbol">(λ</a> <a id="17962" class="Symbol">())</a>
<a id="17966" href="Data.Bool.Properties.html#17899" class="Function">T-≡</a> <a id="17970" class="Symbol">{</a><a id="17971" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="17975" class="Symbol">}</a> <a id="17978" class="Symbol">=</a> <a id="17980" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="17984" class="Symbol">(</a><a id="17985" href="Function.Base.html#725" class="Function">const</a> <a id="17991" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="17995" class="Symbol">)</a> <a id="17997" class="Symbol">(</a><a id="17998" href="Function.Base.html#725" class="Function">const</a> <a id="18004" class="Symbol">_)</a>
<a id="T-not-≡"></a><a id="18008" href="Data.Bool.Properties.html#18008" class="Function">T-not-≡</a> <a id="18016" class="Symbol">:</a> <a id="18018" class="Symbol">∀</a> <a id="18020" class="Symbol">{</a><a id="18021" href="Data.Bool.Properties.html#18021" class="Bound">x</a><a id="18022" class="Symbol">}</a> <a id="18024" class="Symbol">→</a> <a id="18026" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18028" class="Symbol">(</a><a id="18029" href="Data.Bool.Base.html#941" class="Function">not</a> <a id="18033" href="Data.Bool.Properties.html#18021" class="Bound">x</a><a id="18034" class="Symbol">)</a> <a id="18036" href="Function.Bundles.html#12039" class="Function Operator">⇔</a> <a id="18038" href="Data.Bool.Properties.html#18021" class="Bound">x</a> <a id="18040" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="18042" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a>
<a id="18048" href="Data.Bool.Properties.html#18008" class="Function">T-not-≡</a> <a id="18056" class="Symbol">{</a><a id="18057" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18062" class="Symbol">}</a> <a id="18064" class="Symbol">=</a> <a id="18066" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18070" class="Symbol">(</a><a id="18071" href="Function.Base.html#725" class="Function">const</a> <a id="18077" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="18081" class="Symbol">)</a> <a id="18083" class="Symbol">(</a><a id="18084" href="Function.Base.html#725" class="Function">const</a> <a id="18090" class="Symbol">_)</a>
<a id="18093" href="Data.Bool.Properties.html#18008" class="Function">T-not-≡</a> <a id="18101" class="Symbol">{</a><a id="18102" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18106" class="Symbol">}</a> <a id="18109" class="Symbol">=</a> <a id="18111" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18115" class="Symbol">(λ</a> <a id="18118" class="Symbol">())</a> <a id="18128" class="Symbol">(λ</a> <a id="18131" class="Symbol">())</a>
<a id="T-∧"></a><a id="18136" href="Data.Bool.Properties.html#18136" class="Function">T-∧</a> <a id="18140" class="Symbol">:</a> <a id="18142" class="Symbol">∀</a> <a id="18144" class="Symbol">{</a><a id="18145" href="Data.Bool.Properties.html#18145" class="Bound">x</a> <a id="18147" href="Data.Bool.Properties.html#18147" class="Bound">y</a><a id="18148" class="Symbol">}</a> <a id="18150" class="Symbol">→</a> <a id="18152" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18154" class="Symbol">(</a><a id="18155" href="Data.Bool.Properties.html#18145" class="Bound">x</a> <a id="18157" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="18159" href="Data.Bool.Properties.html#18147" class="Bound">y</a><a id="18160" class="Symbol">)</a> <a id="18162" href="Function.Bundles.html#12039" class="Function Operator">⇔</a> <a id="18164" class="Symbol">(</a><a id="18165" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18167" href="Data.Bool.Properties.html#18145" class="Bound">x</a> <a id="18169" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="18171" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18173" href="Data.Bool.Properties.html#18147" class="Bound">y</a><a id="18174" class="Symbol">)</a>
<a id="18176" href="Data.Bool.Properties.html#18136" class="Function">T-∧</a> <a id="18180" class="Symbol">{</a><a id="18181" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18185" class="Symbol">}</a> <a id="18188" class="Symbol">{</a><a id="18189" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18193" class="Symbol">}</a> <a id="18196" class="Symbol">=</a> <a id="18198" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18202" class="Symbol">(</a><a id="18203" href="Function.Base.html#725" class="Function">const</a> <a id="18209" class="Symbol">(_</a> <a id="18212" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="18214" class="Symbol">_))</a> <a id="18218" class="Symbol">(</a><a id="18219" href="Function.Base.html#725" class="Function">const</a> <a id="18225" class="Symbol">_)</a>
<a id="18228" href="Data.Bool.Properties.html#18136" class="Function">T-∧</a> <a id="18232" class="Symbol">{</a><a id="18233" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18237" class="Symbol">}</a> <a id="18240" class="Symbol">{</a><a id="18241" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18246" class="Symbol">}</a> <a id="18248" class="Symbol">=</a> <a id="18250" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18254" class="Symbol">(λ</a> <a id="18257" class="Symbol">())</a> <a id="18270" href="Data.Product.Base.html#650" class="Field">proj₂</a>
<a id="18276" href="Data.Bool.Properties.html#18136" class="Function">T-∧</a> <a id="18280" class="Symbol">{</a><a id="18281" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18286" class="Symbol">}</a> <a id="18288" class="Symbol">{_}</a> <a id="18296" class="Symbol">=</a> <a id="18298" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18302" class="Symbol">(λ</a> <a id="18305" class="Symbol">())</a> <a id="18318" href="Data.Product.Base.html#636" class="Field">proj₁</a>
<a id="T-∨"></a><a id="18325" href="Data.Bool.Properties.html#18325" class="Function">T-∨</a> <a id="18329" class="Symbol">:</a> <a id="18331" class="Symbol">∀</a> <a id="18333" class="Symbol">{</a><a id="18334" href="Data.Bool.Properties.html#18334" class="Bound">x</a> <a id="18336" href="Data.Bool.Properties.html#18336" class="Bound">y</a><a id="18337" class="Symbol">}</a> <a id="18339" class="Symbol">→</a> <a id="18341" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18343" class="Symbol">(</a><a id="18344" href="Data.Bool.Properties.html#18334" class="Bound">x</a> <a id="18346" href="Data.Bool.Base.html#1053" class="Function Operator">∨</a> <a id="18348" href="Data.Bool.Properties.html#18336" class="Bound">y</a><a id="18349" class="Symbol">)</a> <a id="18351" href="Function.Bundles.html#12039" class="Function Operator">⇔</a> <a id="18353" class="Symbol">(</a><a id="18354" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18356" href="Data.Bool.Properties.html#18334" class="Bound">x</a> <a id="18358" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="18360" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18362" href="Data.Bool.Properties.html#18336" class="Bound">y</a><a id="18363" class="Symbol">)</a>
<a id="18365" href="Data.Bool.Properties.html#18325" class="Function">T-∨</a> <a id="18369" class="Symbol">{</a><a id="18370" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18374" class="Symbol">}</a> <a id="18377" class="Symbol">{_}</a> <a id="18385" class="Symbol">=</a> <a id="18387" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18391" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="18396" class="Symbol">(</a><a id="18397" href="Function.Base.html#725" class="Function">const</a> <a id="18403" class="Symbol">_)</a>
<a id="18406" href="Data.Bool.Properties.html#18325" class="Function">T-∨</a> <a id="18410" class="Symbol">{</a><a id="18411" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18416" class="Symbol">}</a> <a id="18418" class="Symbol">{</a><a id="18419" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18423" class="Symbol">}</a> <a id="18426" class="Symbol">=</a> <a id="18428" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18432" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="18437" class="Symbol">(</a><a id="18438" href="Function.Base.html#725" class="Function">const</a> <a id="18444" class="Symbol">_)</a>
<a id="18447" href="Data.Bool.Properties.html#18325" class="Function">T-∨</a> <a id="18451" class="Symbol">{</a><a id="18452" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18457" class="Symbol">}</a> <a id="18459" class="Symbol">{</a><a id="18460" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18465" class="Symbol">}</a> <a id="18467" class="Symbol">=</a> <a id="18469" href="Function.Bundles.html#13138" class="Function">mk⇔</a> <a id="18473" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="18478" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="18480" href="Function.Base.html#704" class="Function">id</a> <a id="18483" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="18485" href="Function.Base.html#704" class="Function">id</a> <a id="18488" href="Data.Sum.Base.html#811" class="Function Operator">]</a>
<a id="T-irrelevant"></a><a id="18491" href="Data.Bool.Properties.html#18491" class="Function">T-irrelevant</a> <a id="18504" class="Symbol">:</a> <a id="18506" href="Relation.Unary.html#4125" class="Function">U.Irrelevant</a> <a id="18519" href="Data.Bool.Base.html#1348" class="Function">T</a>
<a id="18521" href="Data.Bool.Properties.html#18491" class="Function">T-irrelevant</a> <a id="18534" class="Symbol">{</a><a id="18535" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a><a id="18539" class="Symbol">}</a> <a id="18542" class="Symbol">_</a> <a id="18545" class="Symbol">_</a> <a id="18548" class="Symbol">=</a> <a id="18550" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="T?-diag"></a><a id="18556" href="Data.Bool.Properties.html#18556" class="Function">T?-diag</a> <a id="18564" class="Symbol">:</a> <a id="18566" class="Symbol">∀</a> <a id="18568" href="Data.Bool.Properties.html#18568" class="Bound">b</a> <a id="18570" class="Symbol">→</a> <a id="18572" href="Data.Bool.Base.html#1348" class="Function">T</a> <a id="18574" href="Data.Bool.Properties.html#18568" class="Bound">b</a> <a id="18576" class="Symbol">→</a> <a id="18578" href="Relation.Nullary.Decidable.Core.html#3468" class="Function">True</a> <a id="18583" class="Symbol">(</a><a id="18584" href="Relation.Nullary.Decidable.Core.html#2465" class="Function">T?</a> <a id="18587" href="Data.Bool.Properties.html#18568" class="Bound">b</a><a id="18588" class="Symbol">)</a>
<a id="18590" href="Data.Bool.Properties.html#18556" class="Function">T?-diag</a> <a id="18598" href="Data.Bool.Properties.html#18598" class="Bound">b</a> <a id="18600" class="Symbol">=</a> <a id="18602" href="Relation.Nullary.Decidable.Core.html#3930" class="Function">fromWitness</a>
<a id="18615" class="Comment">------------------------------------------------------------------------</a>
<a id="18688" class="Comment">-- Miscellaneous other properties</a>
<a id="⇔→≡"></a><a id="18723" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18727" class="Symbol">:</a> <a id="18729" class="Symbol">{</a><a id="18730" href="Data.Bool.Properties.html#18730" class="Bound">x</a> <a id="18732" href="Data.Bool.Properties.html#18732" class="Bound">y</a> <a id="18734" href="Data.Bool.Properties.html#18734" class="Bound">z</a> <a id="18736" class="Symbol">:</a> <a id="18738" href="Agda.Builtin.Bool.html#173" class="Datatype">Bool</a><a id="18742" class="Symbol">}</a> <a id="18744" class="Symbol">→</a> <a id="18746" href="Data.Bool.Properties.html#18730" class="Bound">x</a> <a id="18748" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="18750" href="Data.Bool.Properties.html#18734" class="Bound">z</a> <a id="18752" href="Function.Bundles.html#12039" class="Function Operator">⇔</a> <a id="18754" href="Data.Bool.Properties.html#18732" class="Bound">y</a> <a id="18756" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="18758" href="Data.Bool.Properties.html#18734" class="Bound">z</a> <a id="18760" class="Symbol">→</a> <a id="18762" href="Data.Bool.Properties.html#18730" class="Bound">x</a> <a id="18764" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="18766" href="Data.Bool.Properties.html#18732" class="Bound">y</a>
<a id="18768" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18772" class="Symbol">{</a><a id="18773" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18778" class="Symbol">}</a> <a id="18780" class="Symbol">{</a><a id="18781" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18786" class="Symbol">}</a> <a id="18796" href="Data.Bool.Properties.html#18796" class="Bound">hyp</a> <a id="18800" class="Symbol">=</a> <a id="18802" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="18807" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18811" class="Symbol">{</a><a id="18812" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18817" class="Symbol">}</a> <a id="18819" class="Symbol">{</a><a id="18820" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18825" class="Symbol">}</a> <a id="18827" class="Symbol">{</a><a id="18828" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18833" class="Symbol">}</a> <a id="18835" href="Data.Bool.Properties.html#18835" class="Bound">hyp</a> <a id="18839" class="Symbol">=</a> <a id="18841" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="18845" class="Symbol">(</a><a id="18846" href="Function.Bundles.html#4810" class="Field">Equivalence.to</a> <a id="18861" href="Data.Bool.Properties.html#18835" class="Bound">hyp</a> <a id="18865" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="18869" class="Symbol">)</a>
<a id="18871" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18875" class="Symbol">{</a><a id="18876" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18881" class="Symbol">}</a> <a id="18883" class="Symbol">{</a><a id="18884" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18889" class="Symbol">}</a> <a id="18891" class="Symbol">{</a><a id="18892" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18897" class="Symbol">}</a> <a id="18899" href="Data.Bool.Properties.html#18899" class="Bound">hyp</a> <a id="18903" class="Symbol">=</a> <a id="18905" href="Function.Bundles.html#4834" class="Field">Equivalence.from</a> <a id="18922" href="Data.Bool.Properties.html#18899" class="Bound">hyp</a> <a id="18926" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="18931" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18935" class="Symbol">{</a><a id="18936" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="18941" class="Symbol">}</a> <a id="18943" class="Symbol">{</a><a id="18944" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18949" class="Symbol">}</a> <a id="18951" class="Symbol">{</a><a id="18952" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="18957" class="Symbol">}</a> <a id="18959" href="Data.Bool.Properties.html#18959" class="Bound">hyp</a> <a id="18963" class="Symbol">=</a> <a id="18965" href="Function.Bundles.html#4834" class="Field">Equivalence.from</a> <a id="18982" href="Data.Bool.Properties.html#18959" class="Bound">hyp</a> <a id="18986" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="18991" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="18995" class="Symbol">{</a><a id="18996" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="19001" class="Symbol">}</a> <a id="19003" class="Symbol">{</a><a id="19004" href="Agda.Builtin.Bool.html#198" class="InductiveConstructor">true</a> <a id="19009" class="Symbol">}</a> <a id="19011" class="Symbol">{</a><a id="19012" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="19017" class="Symbol">}</a> <a id="19019" href="Data.Bool.Properties.html#19019" class="Bound">hyp</a> <a id="19023" class="Symbol">=</a> <a id="19025" href="Relation.Binary.PropositionalEquality.Core.html#1712" class="Function">sym</a> <a id="19029" class="Symbol">(</a><a id="19030" href="Function.Bundles.html#4810" class="Field">Equivalence.to</a> <a id="19045" href="Data.Bool.Properties.html#19019" class="Bound">hyp</a> <a id="19049" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="19053" class="Symbol">)</a>
<a id="19055" href="Data.Bool.Properties.html#18723" class="Function">⇔→≡</a> <a id="19059" class="Symbol">{</a><a id="19060" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="19065" class="Symbol">}</a> <a id="19067" class="Symbol">{</a><a id="19068" href="Agda.Builtin.Bool.html#192" class="InductiveConstructor">false</a><a id="19073" class="Symbol">}</a> <a id="19083" href="Data.Bool.Properties.html#19083" class="Bound">hyp</a> <a id="19087" class="Symbol">=</a> <a id="19089" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="19096" class="Comment">------------------------------------------------------------------------</a>
<a id="19169" class="Comment">-- DEPRECATED NAMES</a>
<a id="19189" class="Comment">------------------------------------------------------------------------</a>
<a id="19262" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="19333" class="Comment">-- not guaranteed.</a>
<a id="19353" class="Comment">-- Version 2.0</a>
<a id="push-function-into-if"></a><a id="19369" href="Data.Bool.Properties.html#19369" class="Function">push-function-into-if</a> <a id="19391" class="Symbol">=</a> <a id="19393" href="Data.Bool.Properties.html#17605" class="Function">if-float</a>
<a id="19402" class="Symbol">{-#</a> <a id="19406" class="Keyword">WARNING_ON_USAGE</a> <a id="19423" class="Pragma">push-function-into-if</a>
<a id="19445" class="String">&quot;Warning: push-function-into-if was deprecated in v2.0.
Please use if-float instead.&quot;</a>
<a id="19531" class="Symbol">#-}</a>
</pre></body></html>
Reference in a new issue Copy permalink