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bsc-leon-vatthauer/agda/bsc-thesis/Algebra.Lattice.Properties.BooleanAlgebra.html
Leon Vatthauer 2bf4655eae
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<html><head><meta charset="utf-8"><title>Algebra.Lattice.Properties.BooleanAlgebra</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">-- Some derivable properties of Boolean algebras</a>
<a id="155" class="Comment">------------------------------------------------------------------------</a>
<a id="229" class="Symbol">{-#</a> <a id="233" class="Keyword">OPTIONS</a> <a id="241" class="Pragma">--cubical-compatible</a> <a id="262" class="Pragma">--safe</a> <a id="269" class="Symbol">#-}</a>
<a id="274" class="Keyword">open</a> <a id="279" class="Keyword">import</a> <a id="286" href="Algebra.Lattice.Bundles.html" class="Module">Algebra.Lattice.Bundles</a>
<a id="311" class="Keyword">module</a> <a id="318" href="Algebra.Lattice.Properties.BooleanAlgebra.html" class="Module">Algebra.Lattice.Properties.BooleanAlgebra</a>
<a id="362" class="Symbol">{</a><a id="363" href="Algebra.Lattice.Properties.BooleanAlgebra.html#363" class="Bound">b₁</a> <a id="366" href="Algebra.Lattice.Properties.BooleanAlgebra.html#366" class="Bound">b₂</a><a id="368" class="Symbol">}</a> <a id="370" class="Symbol">(</a><a id="371" href="Algebra.Lattice.Properties.BooleanAlgebra.html#371" class="Bound">B</a> <a id="373" class="Symbol">:</a> <a id="375" href="Algebra.Lattice.Bundles.html#5554" class="Record">BooleanAlgebra</a> <a id="390" href="Algebra.Lattice.Properties.BooleanAlgebra.html#363" class="Bound">b₁</a> <a id="393" href="Algebra.Lattice.Properties.BooleanAlgebra.html#366" class="Bound">b₂</a><a id="395" class="Symbol">)</a>
<a id="399" class="Keyword">where</a>
<a id="406" class="Keyword">open</a> <a id="411" href="Algebra.Lattice.Bundles.html#5554" class="Module">BooleanAlgebra</a> <a id="426" href="Algebra.Lattice.Properties.BooleanAlgebra.html#371" class="Bound">B</a>
<a id="429" class="Keyword">import</a> <a id="436" href="Algebra.Lattice.Properties.DistributiveLattice.html" class="Module">Algebra.Lattice.Properties.DistributiveLattice</a> <a id="483" class="Symbol">as</a> <a id="486" class="Module">DistribLatticeProperties</a>
<a id="511" class="Keyword">open</a> <a id="516" class="Keyword">import</a> <a id="523" href="Algebra.Core.html" class="Module">Algebra.Core</a>
<a id="536" class="Keyword">open</a> <a id="541" class="Keyword">import</a> <a id="548" href="Algebra.Structures.html" class="Module">Algebra.Structures</a> <a id="567" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">_≈_</a>
<a id="571" class="Keyword">open</a> <a id="576" class="Keyword">import</a> <a id="583" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="603" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">_≈_</a>
<a id="607" class="Keyword">open</a> <a id="612" class="Keyword">import</a> <a id="619" href="Algebra.Consequences.Setoid.html" class="Module">Algebra.Consequences.Setoid</a> <a id="647" href="Algebra.Lattice.Bundles.html#4863" class="Function">setoid</a>
<a id="654" class="Keyword">open</a> <a id="659" class="Keyword">import</a> <a id="666" href="Algebra.Bundles.html" class="Module">Algebra.Bundles</a>
<a id="682" class="Keyword">open</a> <a id="687" class="Keyword">import</a> <a id="694" href="Algebra.Lattice.Structures.html" class="Module">Algebra.Lattice.Structures</a> <a id="721" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">_≈_</a>
<a id="725" class="Keyword">open</a> <a id="730" class="Keyword">import</a> <a id="737" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="770" href="Algebra.Lattice.Bundles.html#4863" class="Function">setoid</a>
<a id="777" class="Keyword">open</a> <a id="782" class="Keyword">import</a> <a id="789" href="Function.Base.html" class="Module">Function.Base</a> <a id="803" class="Keyword">using</a> <a id="809" class="Symbol">(</a><a id="810" href="Function.Base.html#704" class="Function">id</a><a id="812" class="Symbol">;</a> <a id="814" href="Function.Base.html#1974" class="Function Operator">_$_</a><a id="817" class="Symbol">;</a> <a id="819" href="Function.Base.html#4322" class="Function Operator">_⟨_⟩_</a><a id="824" class="Symbol">)</a>
<a id="826" class="Keyword">open</a> <a id="831" class="Keyword">import</a> <a id="838" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="855" class="Keyword">using</a> <a id="861" class="Symbol">(</a><a id="862" href="Function.Bundles.html#12039" class="Function Operator">_⇔_</a><a id="865" class="Symbol">;</a> <a id="867" class="Keyword">module</a> <a id="874" href="Function.Bundles.html#4752" class="Module">Equivalence</a><a id="885" class="Symbol">)</a>
<a id="887" class="Keyword">open</a> <a id="892" class="Keyword">import</a> <a id="899" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="917" class="Keyword">using</a> <a id="923" class="Symbol">(</a><a id="924" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="927" class="Symbol">)</a>
<a id="930" class="Comment">------------------------------------------------------------------------</a>
<a id="1003" class="Comment">-- Export properties from distributive lattices</a>
<a id="1052" class="Keyword">open</a> <a id="1057" href="Algebra.Lattice.Properties.DistributiveLattice.html" class="Module">DistribLatticeProperties</a> <a id="1082" href="Algebra.Lattice.Bundles.html#6010" class="Function">distributiveLattice</a> <a id="1102" class="Keyword">public</a>
<a id="1110" class="Comment">------------------------------------------------------------------------</a>
<a id="1183" class="Comment">-- The dual construction is also a boolean algebra</a>
<a id="∧-∨-isBooleanAlgebra"></a><a id="1235" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1235" class="Function">∧-∨-isBooleanAlgebra</a> <a id="1256" class="Symbol">:</a> <a id="1258" href="Algebra.Lattice.Structures.html#5413" class="Record">IsBooleanAlgebra</a> <a id="1275" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="1279" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a> <a id="1283" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬_</a> <a id="1286" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="1288" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a>
<a id="1290" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1235" class="Function">∧-∨-isBooleanAlgebra</a> <a id="1311" class="Symbol">=</a> <a id="1313" class="Keyword">record</a>
<a id="1322" class="Symbol">{</a> <a id="1324" href="Algebra.Lattice.Structures.html#5501" class="Field">isDistributiveLattice</a> <a id="1346" class="Symbol">=</a> <a id="1348" href="Algebra.Lattice.Properties.DistributiveLattice.html#901" class="Function">∧-∨-isDistributiveLattice</a>
<a id="1376" class="Symbol">;</a> <a id="1378" href="Algebra.Lattice.Structures.html#5555" class="Field">∨-complement</a> <a id="1400" class="Symbol">=</a> <a id="1402" href="Algebra.Lattice.Structures.html#5597" class="Function">∧-complement</a>
<a id="1417" class="Symbol">;</a> <a id="1419" href="Algebra.Lattice.Structures.html#5597" class="Field">∧-complement</a> <a id="1441" class="Symbol">=</a> <a id="1443" href="Algebra.Lattice.Structures.html#5555" class="Function">∨-complement</a>
<a id="1458" class="Symbol">;</a> <a id="1460" href="Algebra.Lattice.Structures.html#5639" class="Field">¬-cong</a> <a id="1482" class="Symbol">=</a> <a id="1484" href="Algebra.Lattice.Structures.html#5639" class="Function">¬-cong</a>
<a id="1493" class="Symbol">}</a>
<a id="∧-∨-booleanAlgebra"></a><a id="1496" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1496" class="Function">∧-∨-booleanAlgebra</a> <a id="1515" class="Symbol">:</a> <a id="1517" href="Algebra.Lattice.Bundles.html#5554" class="Record">BooleanAlgebra</a> <a id="1532" class="Symbol">_</a> <a id="1534" class="Symbol">_</a>
<a id="1536" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1496" class="Function">∧-∨-booleanAlgebra</a> <a id="1555" class="Symbol">=</a> <a id="1557" class="Keyword">record</a>
<a id="1566" class="Symbol">{</a> <a id="1568" href="Algebra.Lattice.Bundles.html#5903" class="Field">isBooleanAlgebra</a> <a id="1585" class="Symbol">=</a> <a id="1587" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1235" class="Function">∧-∨-isBooleanAlgebra</a>
<a id="1610" class="Symbol">}</a>
<a id="1613" class="Comment">------------------------------------------------------------------------</a>
<a id="1686" class="Comment">-- (∨, ∧, ⊥, ⊤) and (∧, ∨, ⊤, ⊥) are commutative semirings</a>
<a id="∧-identityʳ"></a><a id="1746" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a> <a id="1758" class="Symbol">:</a> <a id="1760" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="1774" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="1776" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a>
<a id="1780" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a> <a id="1792" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a> <a id="1794" class="Symbol">=</a> <a id="1796" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1804" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a> <a id="1806" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="1808" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="1819" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1822" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="1830" class="Symbol">(</a><a id="1831" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="1835" class="Symbol">(</a><a id="1836" href="Algebra.Lattice.Structures.html#5812" class="Function">∨-complementʳ</a> <a id="1850" class="Symbol">_))</a> <a id="1854" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1858" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a> <a id="1860" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="1862" class="Symbol">(</a><a id="1863" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a> <a id="1865" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="1867" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="1869" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a><a id="1870" class="Symbol">)</a> <a id="1873" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1876" href="Algebra.Lattice.Structures.html#4454" class="Function">∧-absorbs-∨</a> <a id="1888" class="Symbol">_</a> <a id="1890" class="Symbol">_</a> <a id="1892" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1896" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1792" class="Bound">x</a> <a id="1911" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∧-identityˡ"></a><a id="1914" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1914" class="Function">∧-identityˡ</a> <a id="1926" class="Symbol">:</a> <a id="1928" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="1941" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="1943" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a>
<a id="1947" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1914" class="Function">∧-identityˡ</a> <a id="1959" class="Symbol">=</a> <a id="1961" href="Algebra.Consequences.Setoid.html#4478" class="Function">comm∧idʳ⇒idˡ</a> <a id="1974" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="1981" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a>
<a id="∧-identity"></a><a id="1994" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1994" class="Function">∧-identity</a> <a id="2005" class="Symbol">:</a> <a id="2007" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="2016" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="2018" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a>
<a id="2022" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1994" class="Function">∧-identity</a> <a id="2033" class="Symbol">=</a> <a id="2035" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1914" class="Function">∧-identityˡ</a> <a id="2047" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2049" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a>
<a id="∨-identityʳ"></a><a id="2062" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a> <a id="2074" class="Symbol">:</a> <a id="2076" href="Algebra.Definitions.html#1781" class="Function">RightIdentity</a> <a id="2090" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2092" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a>
<a id="2096" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a> <a id="2108" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2110" class="Symbol">=</a> <a id="2112" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2120" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2122" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="2124" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2135" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2138" href="Algebra.Lattice.Structures.html#4639" class="Function">∨-congˡ</a> <a id="2146" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="2148" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="2152" class="Symbol">(</a><a id="2153" href="Algebra.Lattice.Structures.html#5961" class="Function">∧-complementʳ</a> <a id="2167" class="Symbol">_)</a> <a id="2170" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2174" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2176" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="2178" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2180" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="2182" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="2184" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2189" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2192" href="Algebra.Lattice.Structures.html#4392" class="Function">∨-absorbs-∧</a> <a id="2204" class="Symbol">_</a> <a id="2206" class="Symbol">_</a> <a id="2208" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2212" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2108" class="Bound">x</a> <a id="2227" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∨-identityˡ"></a><a id="2230" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2230" class="Function">∨-identityˡ</a> <a id="2242" class="Symbol">:</a> <a id="2244" href="Algebra.Definitions.html#1708" class="Function">LeftIdentity</a> <a id="2257" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2259" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a>
<a id="2263" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2230" class="Function">∨-identityˡ</a> <a id="2275" class="Symbol">=</a> <a id="2277" href="Algebra.Consequences.Setoid.html#4478" class="Function">comm∧idʳ⇒idˡ</a> <a id="2290" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a> <a id="2297" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a>
<a id="∨-identity"></a><a id="2310" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2310" class="Function">∨-identity</a> <a id="2321" class="Symbol">:</a> <a id="2323" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="2332" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2334" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a>
<a id="2338" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2310" class="Function">∨-identity</a> <a id="2349" class="Symbol">=</a> <a id="2351" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2230" class="Function">∨-identityˡ</a> <a id="2363" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="2365" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a>
<a id="∧-zeroʳ"></a><a id="2378" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="2386" class="Symbol">:</a> <a id="2388" href="Algebra.Definitions.html#2007" class="Function">RightZero</a> <a id="2398" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2400" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a>
<a id="2404" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="2412" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2414" class="Symbol">=</a> <a id="2416" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2424" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2426" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="2428" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2439" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="2442" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="2450" class="Symbol">(</a><a id="2451" href="Algebra.Lattice.Structures.html#5961" class="Function">∧-complementʳ</a> <a id="2465" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a><a id="2466" class="Symbol">)</a> <a id="2468" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="2472" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2474" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="2477" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2480" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="2482" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="2484" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2487" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="2490" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="2498" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2500" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a> <a id="2502" class="Symbol">(</a><a id="2503" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="2505" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2412" class="Bound">x</a><a id="2506" class="Symbol">)</a> <a id="2508" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
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<a id="2593" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="2608" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="3920" href="Algebra.Lattice.Properties.BooleanAlgebra.html#3880" class="Function">∧-∨-isSemiring</a> <a id="3935" class="Symbol">=</a> <a id="3937" class="Keyword">record</a>
<a id="3946" class="Symbol">{</a> <a id="3948" href="Algebra.Structures.html#12800" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="3982" class="Symbol">=</a> <a id="3984" class="Keyword">record</a>
<a id="3995" class="Symbol">{</a> <a id="3997" href="Algebra.Structures.html#11137" class="Field">+-isCommutativeMonoid</a> <a id="4019" class="Symbol">=</a> <a id="4021" href="Algebra.Lattice.Properties.BooleanAlgebra.html#3442" class="Function">∧-⊤-isCommutativeMonoid</a>
<a id="4049" class="Symbol">;</a> <a id="4051" href="Algebra.Structures.html#11190" class="Field">*-cong</a> <a id="4058" class="Symbol">=</a> <a id="4060" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a>
<a id="4071" class="Symbol">;</a> <a id="4073" href="Algebra.Structures.html#11231" class="Field">*-assoc</a> <a id="4081" class="Symbol">=</a> <a id="4083" href="Algebra.Lattice.Structures.html#4147" class="Function">∨-assoc</a>
<a id="4095" class="Symbol">;</a> <a id="4097" href="Algebra.Structures.html#11273" class="Field">*-identity</a> <a id="4108" class="Symbol">=</a> <a id="4110" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2310" class="Function">∨-identity</a>
<a id="4125" class="Symbol">;</a> <a id="4127" href="Algebra.Structures.html#11315" class="Field">distrib</a> <a id="4135" class="Symbol">=</a> <a id="4137" href="Algebra.Lattice.Structures.html#4868" class="Function">∨-distrib-∧</a>
<a id="4153" class="Symbol">}</a>
<a id="4157" class="Symbol">;</a> <a id="4159" href="Algebra.Structures.html#12890" class="Field">zero</a> <a id="4164" class="Symbol">=</a> <a id="4166" href="Algebra.Lattice.Properties.BooleanAlgebra.html#3018" class="Function">∨-zero</a>
<a id="4175" class="Symbol">}</a>
<a id="∨-∧-isCommutativeSemiring"></a><a id="4178" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4178" class="Function">∨-∧-isCommutativeSemiring</a> <a id="4204" class="Symbol">:</a> <a id="4206" href="Algebra.Structures.html#13418" class="Record">IsCommutativeSemiring</a> <a id="4228" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a> <a id="4232" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="4236" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="4238" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a>
<a id="4240" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4178" class="Function">∨-∧-isCommutativeSemiring</a> <a id="4266" class="Symbol">=</a> <a id="4268" class="Keyword">record</a>
<a id="4277" class="Symbol">{</a> <a id="4279" href="Algebra.Structures.html#13498" class="Field">isSemiring</a> <a id="4290" class="Symbol">=</a> <a id="4292" href="Algebra.Lattice.Properties.BooleanAlgebra.html#3582" class="Function">∨-∧-isSemiring</a>
<a id="4309" class="Symbol">;</a> <a id="4311" href="Algebra.Structures.html#13536" class="Field">*-comm</a> <a id="4318" class="Symbol">=</a> <a id="4320" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a>
<a id="4329" class="Symbol">}</a>
<a id="∧-∨-isCommutativeSemiring"></a><a id="4332" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4332" class="Function">∧-∨-isCommutativeSemiring</a> <a id="4358" class="Symbol">:</a> <a id="4360" href="Algebra.Structures.html#13418" class="Record">IsCommutativeSemiring</a> <a id="4382" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="4386" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">_∨_</a> <a id="4390" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="4392" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a>
<a id="4394" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4332" class="Function">∧-∨-isCommutativeSemiring</a> <a id="4420" class="Symbol">=</a> <a id="4422" class="Keyword">record</a>
<a id="4431" class="Symbol">{</a> <a id="4433" href="Algebra.Structures.html#13498" class="Field">isSemiring</a> <a id="4444" class="Symbol">=</a> <a id="4446" href="Algebra.Lattice.Properties.BooleanAlgebra.html#3880" class="Function">∧-∨-isSemiring</a>
<a id="4463" class="Symbol">;</a> <a id="4465" href="Algebra.Structures.html#13536" class="Field">*-comm</a> <a id="4472" class="Symbol">=</a> <a id="4474" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a>
<a id="4483" class="Symbol">}</a>
<a id="∨-∧-commutativeSemiring"></a><a id="4486" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4486" class="Function">∨-∧-commutativeSemiring</a> <a id="4510" class="Symbol">:</a> <a id="4512" href="Algebra.Bundles.html#17471" class="Record">CommutativeSemiring</a> <a id="4532" class="Symbol">_</a> <a id="4534" class="Symbol">_</a>
<a id="4536" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4486" class="Function">∨-∧-commutativeSemiring</a> <a id="4560" class="Symbol">=</a> <a id="4562" class="Keyword">record</a>
<a id="4571" class="Symbol">{</a> <a id="4573" href="Algebra.Bundles.html#17806" class="Field">isCommutativeSemiring</a> <a id="4595" class="Symbol">=</a> <a id="4597" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4178" class="Function">∨-∧-isCommutativeSemiring</a>
<a id="4625" class="Symbol">}</a>
<a id="∧-∨-commutativeSemiring"></a><a id="4628" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4628" class="Function">∧-∨-commutativeSemiring</a> <a id="4652" class="Symbol">:</a> <a id="4654" href="Algebra.Bundles.html#17471" class="Record">CommutativeSemiring</a> <a id="4674" class="Symbol">_</a> <a id="4676" class="Symbol">_</a>
<a id="4678" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4628" class="Function">∧-∨-commutativeSemiring</a> <a id="4702" class="Symbol">=</a> <a id="4704" class="Keyword">record</a>
<a id="4713" class="Symbol">{</a> <a id="4715" href="Algebra.Bundles.html#17806" class="Field">isCommutativeSemiring</a> <a id="4737" class="Symbol">=</a> <a id="4739" href="Algebra.Lattice.Properties.BooleanAlgebra.html#4332" class="Function">∧-∨-isCommutativeSemiring</a>
<a id="4767" class="Symbol">}</a>
<a id="4770" class="Comment">------------------------------------------------------------------------</a>
<a id="4843" class="Comment">-- Some other properties</a>
<a id="4869" class="Comment">-- I took the statement of this lemma (called Uniqueness of</a>
<a id="4929" class="Comment">-- Complements) from some course notes, &quot;Boolean Algebra&quot;, written</a>
<a id="4996" class="Comment">-- by Gert Smolka.</a>
<a id="5016" class="Keyword">private</a>
<a id="lemma"></a><a id="5026" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5032" class="Symbol">:</a> <a id="5034" class="Symbol">∀</a> <a id="5036" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5036" class="Bound">x</a> <a id="5038" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5038" class="Bound">y</a> <a id="5040" class="Symbol">→</a> <a id="5042" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5036" class="Bound">x</a> <a id="5044" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5046" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5038" class="Bound">y</a> <a id="5048" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5050" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="5052" class="Symbol">→</a> <a id="5054" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5036" class="Bound">x</a> <a id="5056" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5058" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5038" class="Bound">y</a> <a id="5060" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5062" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5064" class="Symbol">→</a> <a id="5066" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5068" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5036" class="Bound">x</a> <a id="5070" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5072" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5038" class="Bound">y</a>
<a id="5076" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5082" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5084" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5086" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5086" class="Bound">x∧y=⊥</a> <a id="5092" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5092" class="Bound">x∨y=⊤</a> <a id="5098" class="Symbol">=</a> <a id="5100" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5110" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5112" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5129" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="5132" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a> <a id="5144" class="Symbol">_</a> <a id="5146" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="5152" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5154" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5156" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5158" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5171" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="5174" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="5182" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5092" class="Bound">x∨y=⊤</a> <a id="5188" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="5194" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5196" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5198" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5200" class="Symbol">(</a><a id="5201" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5203" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5205" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a><a id="5206" class="Symbol">)</a> <a id="5213" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5217" href="Algebra.Lattice.Structures.html#5126" class="Function">∧-distribˡ-∨</a> <a id="5230" class="Symbol">_</a> <a id="5232" class="Symbol">_</a> <a id="5234" class="Symbol">_</a> <a id="5236" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5242" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5244" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5246" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5248" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5250" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5252" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5254" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5256" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5258" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5261" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5265" href="Algebra.Lattice.Structures.html#4700" class="Function">∨-congʳ</a> <a id="5273" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5275" href="Algebra.Lattice.Structures.html#5887" class="Function">∧-complementˡ</a> <a id="5289" class="Symbol">_</a> <a id="5291" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5297" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="5299" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5301" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5303" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5305" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5307" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5316" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="5319" href="Algebra.Lattice.Structures.html#4700" class="Function">∨-congʳ</a> <a id="5327" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5086" class="Bound">x∧y=⊥</a> <a id="5333" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="5339" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5341" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5343" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5345" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5347" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5349" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5351" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5353" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5358" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="5361" href="Algebra.Lattice.Structures.html#5200" class="Function">∧-distribʳ-∨</a> <a id="5374" class="Symbol">_</a> <a id="5376" class="Symbol">_</a> <a id="5378" class="Symbol">_</a> <a id="5380" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="5386" class="Symbol">(</a><a id="5387" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a> <a id="5389" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5391" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5393" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5082" class="Bound">x</a><a id="5394" class="Symbol">)</a> <a id="5396" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5398" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5405" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5409" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="5417" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5419" href="Algebra.Lattice.Structures.html#5812" class="Function">∨-complementʳ</a> <a id="5433" class="Symbol">_</a> <a id="5435" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5441" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5443" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5445" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5460" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5464" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1914" class="Function">∧-identityˡ</a> <a id="5476" class="Symbol">_</a> <a id="5478" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5484" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5084" class="Bound">y</a> <a id="5503" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="⊥≉⊤"></a><a id="5506" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5506" class="Function">⊥≉⊤</a> <a id="5510" class="Symbol">:</a> <a id="5512" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5514" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="5516" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5518" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a>
<a id="5520" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5506" class="Function">⊥≉⊤</a> <a id="5524" class="Symbol">=</a> <a id="5526" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5532" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="5534" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5536" class="Symbol">(</a><a id="5537" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1746" class="Function">∧-identityʳ</a> <a id="5549" class="Symbol">_)</a> <a id="5552" class="Symbol">(</a><a id="5553" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2723" class="Function">∨-zeroʳ</a> <a id="5561" class="Symbol">_)</a>
<a id="⊤≉⊥"></a><a id="5565" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5565" class="Function">⊤≉⊥</a> <a id="5569" class="Symbol">:</a> <a id="5571" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5573" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5575" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5577" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a>
<a id="5579" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5565" class="Function">⊤≉⊥</a> <a id="5583" class="Symbol">=</a> <a id="5585" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5591" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="5593" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="5595" class="Symbol">(</a><a id="5596" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="5604" class="Symbol">_)</a> <a id="5607" class="Symbol">(</a><a id="5608" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a> <a id="5620" class="Symbol">_)</a>
<a id="¬-involutive"></a><a id="5624" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5624" class="Function">¬-involutive</a> <a id="5637" class="Symbol">:</a> <a id="5639" href="Algebra.Definitions.html#4153" class="Function">Involutive</a> <a id="5650" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬_</a>
<a id="5653" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5624" class="Function">¬-involutive</a> <a id="5666" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5666" class="Bound">x</a> <a id="5668" class="Symbol">=</a> <a id="5670" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5676" class="Symbol">(</a><a id="5677" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5679" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5666" class="Bound">x</a><a id="5680" class="Symbol">)</a> <a id="5682" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5666" class="Bound">x</a> <a id="5684" class="Symbol">(</a><a id="5685" href="Algebra.Lattice.Structures.html#5887" class="Function">∧-complementˡ</a> <a id="5699" class="Symbol">_)</a> <a id="5702" class="Symbol">(</a><a id="5703" href="Algebra.Lattice.Structures.html#5738" class="Function">∨-complementˡ</a> <a id="5717" class="Symbol">_)</a>
<a id="deMorgan₁"></a><a id="5721" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5721" class="Function">deMorgan₁</a> <a id="5731" class="Symbol">:</a> <a id="5733" class="Symbol">∀</a> <a id="5735" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5735" class="Bound">x</a> <a id="5737" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5737" class="Bound">y</a> <a id="5739" class="Symbol">→</a> <a id="5741" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5743" class="Symbol">(</a><a id="5744" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5735" class="Bound">x</a> <a id="5746" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5748" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5737" class="Bound">y</a><a id="5749" class="Symbol">)</a> <a id="5751" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="5753" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5755" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5735" class="Bound">x</a> <a id="5757" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5759" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5761" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5737" class="Bound">y</a>
<a id="5763" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5721" class="Function">deMorgan₁</a> <a id="5773" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5775" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="5777" class="Symbol">=</a> <a id="5779" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5026" class="Function">lemma</a> <a id="5785" class="Symbol">(</a><a id="5786" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5788" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5790" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5791" class="Symbol">)</a> <a id="5793" class="Symbol">(</a><a id="5794" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5796" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5798" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5800" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5802" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5803" class="Symbol">)</a> <a id="5805" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5825" class="Function">lem₁</a> <a id="5810" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6600" class="Function">lem₂</a>
<a id="5817" class="Keyword">where</a>
<a id="5825" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5825" class="Function">lem₁</a> <a id="5830" class="Symbol">=</a> <a id="5832" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="5842" class="Symbol">(</a><a id="5843" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5845" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5847" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5848" class="Symbol">)</a> <a id="5850" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5852" class="Symbol">(</a><a id="5853" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5855" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5857" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5859" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5861" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5862" class="Symbol">)</a> <a id="5873" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5876" href="Algebra.Lattice.Structures.html#5126" class="Function">∧-distribˡ-∨</a> <a id="5889" class="Symbol">_</a> <a id="5891" class="Symbol">_</a> <a id="5893" class="Symbol">_</a> <a id="5895" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5901" class="Symbol">(</a><a id="5902" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5904" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5906" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5907" class="Symbol">)</a> <a id="5909" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5911" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5913" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5915" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5917" class="Symbol">(</a><a id="5918" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5920" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5922" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5923" class="Symbol">)</a> <a id="5925" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5927" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5929" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="5932" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5935" href="Algebra.Lattice.Structures.html#4700" class="Function">∨-congʳ</a> <a id="5943" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5945" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="5953" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="5955" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="5962" class="Symbol">_</a> <a id="5964" class="Symbol">_</a> <a id="5966" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5972" class="Symbol">(</a><a id="5973" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="5975" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5977" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="5978" class="Symbol">)</a> <a id="5980" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5982" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="5984" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5986" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="5988" class="Symbol">(</a><a id="5989" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="5991" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5993" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="5994" class="Symbol">)</a> <a id="5996" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="5998" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6000" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6003" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6006" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="6014" class="Symbol">_</a> <a id="6016" class="Symbol">_</a> <a id="6018" class="Symbol">_</a> <a id="6020" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="6022" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a> <a id="6029" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="6031" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="6039" class="Symbol">_</a> <a id="6041" class="Symbol">_</a> <a id="6043" class="Symbol">_</a> <a id="6045" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6051" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6053" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6055" class="Symbol">(</a><a id="6056" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6058" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6060" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6062" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="6063" class="Symbol">)</a> <a id="6065" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6067" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6069" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6071" class="Symbol">(</a><a id="6072" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6074" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6076" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6078" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6079" class="Symbol">)</a> <a id="6082" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6085" class="Symbol">(</a><a id="6086" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="6094" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6096" href="Algebra.Lattice.Structures.html#5961" class="Function">∧-complementʳ</a> <a id="6110" class="Symbol">_)</a> <a id="6113" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="6115" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a> <a id="6122" href="Function.Base.html#4322" class="Function Operator">⟩</a>
<a id="6162" class="Symbol">(</a><a id="6163" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="6171" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6173" href="Algebra.Lattice.Structures.html#5961" class="Function">∧-complementʳ</a> <a id="6187" class="Symbol">_)</a> <a id="6190" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6196" class="Symbol">(</a><a id="6197" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6199" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6201" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a><a id="6202" class="Symbol">)</a> <a id="6204" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6206" class="Symbol">(</a><a id="6207" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6209" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6211" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a><a id="6212" class="Symbol">)</a> <a id="6227" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6230" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="6238" class="Symbol">_</a> <a id="6240" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="6242" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a> <a id="6249" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="6251" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="6259" class="Symbol">_</a> <a id="6261" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6267" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="6269" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6271" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="6298" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6301" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2062" class="Function">∨-identityʳ</a> <a id="6313" class="Symbol">_</a> <a id="6315" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6321" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="6352" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6357" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6357" class="Function">lem₃</a> <a id="6362" class="Symbol">=</a> <a id="6364" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6374" class="Symbol">(</a><a id="6375" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6377" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6379" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6380" class="Symbol">)</a> <a id="6382" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6384" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6386" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6397" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6400" href="Algebra.Lattice.Structures.html#5052" class="Function">∨-distribʳ-∧</a> <a id="6413" class="Symbol">_</a> <a id="6415" class="Symbol">_</a> <a id="6417" class="Symbol">_</a> <a id="6419" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6425" class="Symbol">(</a><a id="6426" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6428" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6430" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6432" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="6433" class="Symbol">)</a> <a id="6435" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6437" class="Symbol">(</a><a id="6438" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6440" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6442" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6444" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="6445" class="Symbol">)</a> <a id="6448" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6451" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="6459" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6461" href="Algebra.Lattice.Structures.html#5812" class="Function">∨-complementʳ</a> <a id="6475" class="Symbol">_</a> <a id="6477" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6483" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="6485" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6487" class="Symbol">(</a><a id="6488" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6490" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6492" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6494" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="6495" class="Symbol">)</a> <a id="6506" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6509" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1914" class="Function">∧-identityˡ</a> <a id="6521" class="Symbol">_</a> <a id="6523" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6529" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6531" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6533" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6535" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6552" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6555" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a> <a id="6562" class="Symbol">_</a> <a id="6564" class="Symbol">_</a> <a id="6566" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6572" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6574" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6576" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6578" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6595" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="6600" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6600" class="Function">lem₂</a> <a id="6605" class="Symbol">=</a> <a id="6607" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6617" class="Symbol">(</a><a id="6618" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6620" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6622" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6623" class="Symbol">)</a> <a id="6625" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6627" class="Symbol">(</a><a id="6628" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6630" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6632" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6634" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6636" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6637" class="Symbol">)</a> <a id="6640" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="6643" href="Algebra.Lattice.Structures.html#4147" class="Function">∨-assoc</a> <a id="6651" class="Symbol">_</a> <a id="6653" class="Symbol">_</a> <a id="6655" class="Symbol">_</a> <a id="6657" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="6663" class="Symbol">((</a><a id="6665" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6667" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6669" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6670" class="Symbol">)</a> <a id="6672" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6674" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6676" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a><a id="6677" class="Symbol">)</a> <a id="6679" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6681" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6683" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6686" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6689" href="Algebra.Lattice.Structures.html#4700" class="Function">∨-congʳ</a> <a id="6697" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6357" class="Function">lem₃</a> <a id="6702" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6708" class="Symbol">(</a><a id="6709" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6711" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6713" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6715" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6716" class="Symbol">)</a> <a id="6718" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6720" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6722" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6731" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6734" href="Algebra.Lattice.Structures.html#4147" class="Function">∨-assoc</a> <a id="6742" class="Symbol">_</a> <a id="6744" class="Symbol">_</a> <a id="6746" class="Symbol">_</a> <a id="6748" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6754" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6756" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6758" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6760" class="Symbol">(</a><a id="6761" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a> <a id="6763" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6765" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6767" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5775" class="Bound">y</a><a id="6768" class="Symbol">)</a> <a id="6777" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6780" href="Algebra.Lattice.Structures.html#4639" class="Function">∨-congˡ</a> <a id="6788" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6790" href="Algebra.Lattice.Structures.html#5812" class="Function">∨-complementʳ</a> <a id="6804" class="Symbol">_</a> <a id="6806" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6812" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6814" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5773" class="Bound">x</a> <a id="6816" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6818" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="6835" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="6838" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2723" class="Function">∨-zeroʳ</a> <a id="6846" class="Symbol">_</a> <a id="6848" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="6854" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a> <a id="6877" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="deMorgan₂"></a><a id="6880" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6880" class="Function">deMorgan₂</a> <a id="6890" class="Symbol">:</a> <a id="6892" class="Symbol">∀</a> <a id="6894" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6894" class="Bound">x</a> <a id="6896" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6896" class="Bound">y</a> <a id="6898" class="Symbol">→</a> <a id="6900" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6902" class="Symbol">(</a><a id="6903" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6894" class="Bound">x</a> <a id="6905" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6907" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6896" class="Bound">y</a><a id="6908" class="Symbol">)</a> <a id="6910" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="6912" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6914" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6894" class="Bound">x</a> <a id="6916" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="6918" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6920" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6896" class="Bound">y</a>
<a id="6922" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6880" class="Function">deMorgan₂</a> <a id="6932" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6932" class="Bound">x</a> <a id="6934" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6934" class="Bound">y</a> <a id="6936" class="Symbol">=</a> <a id="6938" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="6946" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="6948" class="Symbol">(</a><a id="6949" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6932" class="Bound">x</a> <a id="6951" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="6953" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6934" class="Bound">y</a><a id="6954" class="Symbol">)</a> <a id="6965" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="6968" href="Algebra.Lattice.Structures.html#5639" class="Function">¬-cong</a> <a id="6975" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="6977" class="Symbol">((</a><a id="6979" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5624" class="Function">¬-involutive</a> <a id="6992" class="Symbol">_)</a> <a id="6995" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="6997" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a> <a id="7004" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="7006" class="Symbol">(</a><a id="7007" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5624" class="Function">¬-involutive</a> <a id="7020" class="Symbol">_))</a> <a id="7024" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="7028" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7030" class="Symbol">(</a><a id="7031" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7033" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7035" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6932" class="Bound">x</a> <a id="7037" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="7039" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7041" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7043" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6934" class="Bound">y</a><a id="7044" class="Symbol">)</a> <a id="7047" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="7050" href="Algebra.Lattice.Structures.html#5639" class="Function">¬-cong</a> <a id="7057" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="7059" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5721" class="Function">deMorgan₁</a> <a id="7069" class="Symbol">_</a> <a id="7071" class="Symbol">_</a> <a id="7073" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="7077" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7079" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7081" class="Symbol">(</a><a id="7082" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7084" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6932" class="Bound">x</a> <a id="7086" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7088" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7090" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6934" class="Bound">y</a><a id="7091" class="Symbol">)</a> <a id="7096" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7099" href="Algebra.Lattice.Properties.BooleanAlgebra.html#5624" class="Function">¬-involutive</a> <a id="7112" class="Symbol">_</a> <a id="7114" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="7118" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7120" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6932" class="Bound">x</a> <a id="7122" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7124" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7126" href="Algebra.Lattice.Properties.BooleanAlgebra.html#6934" class="Bound">y</a> <a id="7137" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="7140" class="Comment">------------------------------------------------------------------------</a>
<a id="7213" class="Comment">-- (⊕, ∧, id, ⊥, ⊤) is a commutative ring</a>
<a id="7256" class="Comment">-- This construction is parameterised over the definition of xor.</a>
<a id="7323" class="Keyword">module</a> <a id="XorRing"></a><a id="7330" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7330" class="Module">XorRing</a>
<a id="7340" class="Symbol">(</a><a id="7341" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7341" class="Bound">xor</a> <a id="7345" class="Symbol">:</a> <a id="7347" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7351" href="Algebra.Lattice.Bundles.html#5670" class="Field">Carrier</a><a id="7358" class="Symbol">)</a>
<a id="7362" class="Symbol">(</a><a id="7363" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7363" class="Bound">⊕-def</a> <a id="7369" class="Symbol">:</a> <a id="7371" class="Symbol">∀</a> <a id="7373" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7373" class="Bound">x</a> <a id="7375" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7375" class="Bound">y</a> <a id="7377" class="Symbol">→</a> <a id="7379" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7341" class="Bound">xor</a> <a id="7383" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7373" class="Bound">x</a> <a id="7385" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7375" class="Bound">y</a> <a id="7387" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="7389" class="Symbol">(</a><a id="7390" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7373" class="Bound">x</a> <a id="7392" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="7394" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7375" class="Bound">y</a><a id="7395" class="Symbol">)</a> <a id="7397" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7399" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7401" class="Symbol">(</a><a id="7402" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7373" class="Bound">x</a> <a id="7404" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7406" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7375" class="Bound">y</a><a id="7407" class="Symbol">))</a>
<a id="7412" class="Keyword">where</a>
<a id="7421" class="Keyword">private</a>
<a id="7433" class="Keyword">infixl</a> <a id="7440" class="Number">6</a> <a id="7442" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a>
<a id="XorRing._⊕_"></a><a id="7451" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a> <a id="7455" class="Symbol">:</a> <a id="7457" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7461" href="Algebra.Lattice.Bundles.html#5670" class="Field">Carrier</a>
<a id="7473" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a> <a id="7477" class="Symbol">=</a> <a id="7479" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7341" class="Bound">xor</a>
<a id="XorRing.helper"></a><a id="7488" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7488" class="Function">helper</a> <a id="7495" class="Symbol">:</a> <a id="7497" class="Symbol">∀</a> <a id="7499" class="Symbol">{</a><a id="7500" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7500" class="Bound">x</a> <a id="7502" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7502" class="Bound">y</a> <a id="7504" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7504" class="Bound">u</a> <a id="7506" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7506" class="Bound">v</a><a id="7507" class="Symbol">}</a> <a id="7509" class="Symbol">→</a> <a id="7511" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7500" class="Bound">x</a> <a id="7513" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="7515" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7502" class="Bound">y</a> <a id="7517" class="Symbol">→</a> <a id="7519" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7504" class="Bound">u</a> <a id="7521" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="7523" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7506" class="Bound">v</a> <a id="7525" class="Symbol">→</a> <a id="7527" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7500" class="Bound">x</a> <a id="7529" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7531" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7533" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7504" class="Bound">u</a> <a id="7535" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a> <a id="7537" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7502" class="Bound">y</a> <a id="7539" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="7541" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="7543" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7506" class="Bound">v</a>
<a id="7549" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7488" class="Function">helper</a> <a id="7556" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7556" class="Bound">x≈y</a> <a id="7560" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7560" class="Bound">u≈v</a> <a id="7564" class="Symbol">=</a> <a id="7566" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7556" class="Bound">x≈y</a> <a id="7570" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="7572" href="Algebra.Lattice.Structures.html#4282" class="Function">∧-cong</a> <a id="7579" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="7581" href="Algebra.Lattice.Structures.html#5639" class="Function">¬-cong</a> <a id="7588" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7560" class="Bound">u≈v</a>
<a id="XorRing.⊕-cong"></a><a id="7595" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7595" class="Function">⊕-cong</a> <a id="7602" class="Symbol">:</a> <a id="7604" href="Algebra.Definitions.html#1302" class="Function">Congruent₂</a> <a id="7615" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a>
<a id="7621" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7595" class="Function">⊕-cong</a> <a id="7628" class="Symbol">{</a><a id="7629" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7629" class="Bound">x</a><a id="7630" class="Symbol">}</a> <a id="7632" class="Symbol">{</a><a id="7633" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7633" class="Bound">y</a><a id="7634" class="Symbol">}</a> <a id="7636" class="Symbol">{</a><a id="7637" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7637" class="Bound">u</a><a id="7638" class="Symbol">}</a> <a id="7640" class="Symbol">{</a><a id="7641" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7641" class="Bound">v</a><a id="7642" class="Symbol">}</a> <a id="7644" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7644" class="Bound">x≈y</a> <a id="7648" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7648" class="Bound">u≈v</a> <a id="7652" class="Symbol">=</a> <a id="7654" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="7664" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7629" class="Bound">x</a> <a id="7666" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">⊕</a> <a id="7668" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7637" class="Bound">u</a> <a id="7685" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="7689" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7363" class="Bound">⊕-def</a> <a id="7695" class="Symbol">_</a> <a id="7697" class="Symbol">_</a> <a id="7699" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
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<a id="11332" class="Keyword">where</a>
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<a id="11365" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11367" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11369" class="Symbol">(</a><a id="11370" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11372" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11374" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11375" class="Symbol">)</a> <a id="11378" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11381" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="11389" class="Symbol">_</a> <a id="11391" class="Symbol">_</a> <a id="11393" class="Symbol">_</a> <a id="11395" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11405" class="Symbol">(</a><a id="11406" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11408" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11410" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a><a id="11411" class="Symbol">)</a> <a id="11413" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11415" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a> <a id="11418" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11421" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="11429" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="11431" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="11438" class="Symbol">_</a> <a id="11440" class="Symbol">_</a> <a id="11442" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11452" class="Symbol">(</a><a id="11453" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11455" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11457" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a><a id="11458" class="Symbol">)</a> <a id="11460" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11462" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a> <a id="11465" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11468" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="11476" class="Symbol">_</a> <a id="11478" class="Symbol">_</a> <a id="11480" class="Symbol">_</a> <a id="11482" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11492" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11494" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11496" class="Symbol">(</a><a id="11497" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11499" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11501" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11502" class="Symbol">)</a> <a id="11505" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="11514" href="Algebra.Lattice.Properties.BooleanAlgebra.html#11514" class="Function">lem₁</a> <a id="11519" class="Symbol">=</a> <a id="11521" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="11535" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11537" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11539" class="Symbol">(</a><a id="11540" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11542" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11544" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11545" class="Symbol">)</a> <a id="11554" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11557" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="11565" class="Symbol">(</a><a id="11566" href="Algebra.Lattice.Properties.Lattice.html#893" class="Function">∧-idem</a> <a id="11573" class="Symbol">_)</a> <a id="11576" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11586" class="Symbol">(</a><a id="11587" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11589" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11591" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a><a id="11592" class="Symbol">)</a> <a id="11594" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11596" class="Symbol">(</a><a id="11597" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11599" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11601" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11602" class="Symbol">)</a> <a id="11605" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11608" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="11616" class="Symbol">_</a> <a id="11618" class="Symbol">_</a> <a id="11620" class="Symbol">_</a> <a id="11622" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11632" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11634" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11636" class="Symbol">(</a><a id="11637" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11639" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11641" class="Symbol">(</a><a id="11642" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11644" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11646" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11647" class="Symbol">))</a> <a id="11651" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11654" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="11662" href="Algebra.Lattice.Properties.BooleanAlgebra.html#11344" class="Function">lem₂</a> <a id="11667" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11677" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11679" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11681" class="Symbol">(</a><a id="11682" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11684" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11686" class="Symbol">(</a><a id="11687" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11689" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11691" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11692" class="Symbol">))</a> <a id="11696" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11699" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="11707" class="Symbol">_</a> <a id="11709" class="Symbol">_</a> <a id="11711" class="Symbol">_</a> <a id="11713" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11723" class="Symbol">(</a><a id="11724" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11726" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11728" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a><a id="11729" class="Symbol">)</a> <a id="11731" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11733" class="Symbol">(</a><a id="11734" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11736" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11738" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11739" class="Symbol">)</a> <a id="11742" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="11751" href="Algebra.Lattice.Properties.BooleanAlgebra.html#11751" class="Function">lem₃</a> <a id="11756" class="Symbol">=</a> <a id="11758" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="11772" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="11795" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11798" href="Algebra.Lattice.Properties.BooleanAlgebra.html#2378" class="Function">∧-zeroʳ</a> <a id="11806" class="Symbol">_</a> <a id="11808" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11818" class="Symbol">(</a><a id="11819" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11821" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="11823" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11824" class="Symbol">)</a> <a id="11826" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11828" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="11841" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11844" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="11852" class="Symbol">(</a><a id="11853" href="Algebra.Lattice.Structures.html#5961" class="Function">∧-complementʳ</a> <a id="11867" class="Symbol">_)</a> <a id="11870" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11880" class="Symbol">(</a><a id="11881" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11883" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="11885" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11886" class="Symbol">)</a> <a id="11888" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11890" class="Symbol">(</a><a id="11891" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11893" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11895" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="11897" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a><a id="11898" class="Symbol">)</a> <a id="11903" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="11906" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a> <a id="11914" class="Symbol">_</a> <a id="11916" class="Symbol">_</a> <a id="11918" class="Symbol">_</a> <a id="11920" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="11930" class="Symbol">((</a><a id="11932" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11934" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="11936" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11937" class="Symbol">)</a> <a id="11939" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11941" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a><a id="11942" class="Symbol">)</a> <a id="11944" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11946" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="11948" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11953" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="11956" href="Algebra.Lattice.Structures.html#4577" class="Function">∧-congʳ</a> <a id="11964" class="Symbol">(</a><a id="11965" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="11972" class="Symbol">_</a> <a id="11974" class="Symbol">_)</a> <a id="11977" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="11987" class="Symbol">(</a><a id="11988" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="11990" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="11992" class="Symbol">(</a><a id="11993" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10463" class="Bound">y</a> <a id="11995" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="11997" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10465" class="Bound">z</a><a id="11998" class="Symbol">))</a> <a id="12001" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="12003" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="12005" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10461" class="Bound">x</a> <a id="12010" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="XorRing.∧-distribʳ-⊕"></a><a id="12015" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12015" class="Function">∧-distribʳ-⊕</a> <a id="12028" class="Symbol">:</a> <a id="12030" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="12034" href="Algebra.Definitions.html#3274" class="Function Operator">DistributesOverʳ</a> <a id="12051" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a>
<a id="12057" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12015" class="Function">∧-distribʳ-⊕</a> <a id="12070" class="Symbol">=</a> <a id="12072" href="Algebra.Consequences.Setoid.html#7938" class="Function">comm∧distrˡ⇒distrʳ</a> <a id="12091" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7595" class="Function">⊕-cong</a> <a id="12098" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="12105" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10406" class="Function">∧-distribˡ-⊕</a>
<a id="XorRing.∧-distrib-⊕"></a><a id="12121" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12121" class="Function">∧-distrib-⊕</a> <a id="12133" class="Symbol">:</a> <a id="12135" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="12139" href="Algebra.Definitions.html#3393" class="Function Operator">DistributesOver</a> <a id="12155" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a>
<a id="12161" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12121" class="Function">∧-distrib-⊕</a> <a id="12173" class="Symbol">=</a> <a id="12175" href="Algebra.Lattice.Properties.BooleanAlgebra.html#10406" class="Function">∧-distribˡ-⊕</a> <a id="12188" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="12190" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12015" class="Function">∧-distribʳ-⊕</a>
<a id="12206" class="Keyword">private</a>
<a id="XorRing.lemma₂"></a><a id="12219" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12219" class="Function">lemma₂</a> <a id="12226" class="Symbol">:</a> <a id="12228" class="Symbol">∀</a> <a id="12230" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12230" class="Bound">x</a> <a id="12232" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12232" class="Bound">y</a> <a id="12234" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12234" class="Bound">u</a> <a id="12236" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12236" class="Bound">v</a> <a id="12238" class="Symbol">→</a>
<a id="12253" class="Symbol">(</a><a id="12254" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12230" class="Bound">x</a> <a id="12256" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="12258" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12232" class="Bound">y</a><a id="12259" class="Symbol">)</a> <a id="12261" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="12263" class="Symbol">(</a><a id="12264" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12234" class="Bound">u</a> <a id="12266" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="12268" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12236" class="Bound">v</a><a id="12269" class="Symbol">)</a> <a id="12271" href="Algebra.Lattice.Bundles.html#5699" class="Field Operator">≈</a>
<a id="12286" class="Symbol">((</a><a id="12288" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12230" class="Bound">x</a> <a id="12290" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="12292" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12234" class="Bound">u</a><a id="12293" class="Symbol">)</a> <a id="12295" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="12297" class="Symbol">(</a><a id="12298" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12232" class="Bound">y</a> <a id="12300" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="12302" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12234" class="Bound">u</a><a id="12303" class="Symbol">))</a> <a id="12306" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a>
<a id="12321" class="Symbol">((</a><a id="12323" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12230" class="Bound">x</a> <a id="12325" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="12327" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12236" class="Bound">v</a><a id="12328" class="Symbol">)</a> <a id="12330" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="12332" class="Symbol">(</a><a id="12333" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12232" class="Bound">y</a> <a id="12335" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="12337" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12236" class="Bound">v</a><a id="12338" class="Symbol">))</a>
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<a id="13650" class="Keyword">where</a>
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<a id="16932" class="Symbol">;</a> <a id="16934" href="Algebra.Structures.html#6329" class="Field">⁻¹-cong</a> <a id="16943" class="Symbol">=</a> <a id="16945" href="Function.Base.html#704" class="Function">id</a>
<a id="16952" class="Symbol">}</a>
<a id="XorRing.⊕-⊥-isAbelianGroup"></a><a id="16957" href="Algebra.Lattice.Properties.BooleanAlgebra.html#16957" class="Function">⊕-⊥-isAbelianGroup</a> <a id="16976" class="Symbol">:</a> <a id="16978" href="Algebra.Structures.html#7222" class="Record">IsAbelianGroup</a> <a id="16993" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a> <a id="16997" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="16999" href="Function.Base.html#704" class="Function">id</a>
<a id="17004" href="Algebra.Lattice.Properties.BooleanAlgebra.html#16957" class="Function">⊕-⊥-isAbelianGroup</a> <a id="17023" class="Symbol">=</a> <a id="17025" class="Keyword">record</a>
<a id="17036" class="Symbol">{</a> <a id="17038" href="Algebra.Structures.html#7324" class="Field">isGroup</a> <a id="17046" class="Symbol">=</a> <a id="17048" href="Algebra.Lattice.Properties.BooleanAlgebra.html#16817" class="Function">⊕-⊥-isGroup</a>
<a id="17064" class="Symbol">;</a> <a id="17066" href="Algebra.Structures.html#7353" class="Field">comm</a> <a id="17074" class="Symbol">=</a> <a id="17076" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7887" class="Function">⊕-comm</a>
<a id="17087" class="Symbol">}</a>
<a id="XorRing.⊕-∧-isRing"></a><a id="17092" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17092" class="Function">⊕-∧-isRing</a> <a id="17103" class="Symbol">:</a> <a id="17105" href="Algebra.Structures.html#21839" class="Record">IsRing</a> <a id="17112" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a> <a id="17116" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="17120" href="Function.Base.html#704" class="Function">id</a> <a id="17123" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="17125" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a>
<a id="17129" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17092" class="Function">⊕-∧-isRing</a> <a id="17140" class="Symbol">=</a> <a id="17142" class="Keyword">record</a>
<a id="17153" class="Symbol">{</a> <a id="17155" href="Algebra.Structures.html#21917" class="Field">+-isAbelianGroup</a> <a id="17172" class="Symbol">=</a> <a id="17174" href="Algebra.Lattice.Properties.BooleanAlgebra.html#16957" class="Function">⊕-⊥-isAbelianGroup</a>
<a id="17197" class="Symbol">;</a> <a id="17199" href="Algebra.Structures.html#21963" class="Field">*-cong</a> <a id="17206" class="Symbol">=</a> <a id="17208" href="Algebra.Lattice.Structures.html#4282" class="Function">∧-cong</a>
<a id="17219" class="Symbol">;</a> <a id="17221" href="Algebra.Structures.html#21999" class="Field">*-assoc</a> <a id="17229" class="Symbol">=</a> <a id="17231" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a>
<a id="17243" class="Symbol">;</a> <a id="17245" href="Algebra.Structures.html#22036" class="Field">*-identity</a> <a id="17256" class="Symbol">=</a> <a id="17258" href="Algebra.Lattice.Properties.BooleanAlgebra.html#1994" class="Function">∧-identity</a>
<a id="17273" class="Symbol">;</a> <a id="17275" href="Algebra.Structures.html#22073" class="Field">distrib</a> <a id="17283" class="Symbol">=</a> <a id="17285" href="Algebra.Lattice.Properties.BooleanAlgebra.html#12121" class="Function">∧-distrib-⊕</a>
<a id="17301" class="Symbol">}</a>
<a id="XorRing.⊕-∧-isCommutativeRing"></a><a id="17306" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17306" class="Function">⊕-∧-isCommutativeRing</a> <a id="17328" class="Symbol">:</a> <a id="17330" href="Algebra.Structures.html#23275" class="Record">IsCommutativeRing</a> <a id="17348" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7451" class="Function Operator">_⊕_</a> <a id="17352" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">_∧_</a> <a id="17356" href="Function.Base.html#704" class="Function">id</a> <a id="17359" href="Algebra.Lattice.Bundles.html#5872" class="Field">⊥</a> <a id="17361" href="Algebra.Lattice.Bundles.html#5841" class="Field">⊤</a>
<a id="17365" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17306" class="Function">⊕-∧-isCommutativeRing</a> <a id="17387" class="Symbol">=</a> <a id="17389" class="Keyword">record</a>
<a id="17400" class="Symbol">{</a> <a id="17402" href="Algebra.Structures.html#23372" class="Field">isRing</a> <a id="17409" class="Symbol">=</a> <a id="17411" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17092" class="Function">⊕-∧-isRing</a>
<a id="17426" class="Symbol">;</a> <a id="17428" href="Algebra.Structures.html#23404" class="Field">*-comm</a> <a id="17435" class="Symbol">=</a> <a id="17437" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a>
<a id="17448" class="Symbol">}</a>
<a id="XorRing.⊕-∧-commutativeRing"></a><a id="17453" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17453" class="Function">⊕-∧-commutativeRing</a> <a id="17473" class="Symbol">:</a> <a id="17475" href="Algebra.Bundles.html#27248" class="Record">CommutativeRing</a> <a id="17491" class="Symbol">_</a> <a id="17493" class="Symbol">_</a>
<a id="17497" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17453" class="Function">⊕-∧-commutativeRing</a> <a id="17517" class="Symbol">=</a> <a id="17519" class="Keyword">record</a>
<a id="17530" class="Symbol">{</a> <a id="17532" href="Algebra.Bundles.html#27605" class="Field">isCommutativeRing</a> <a id="17550" class="Symbol">=</a> <a id="17552" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17306" class="Function">⊕-∧-isCommutativeRing</a>
<a id="17578" class="Symbol">}</a>
<a id="17582" class="Keyword">infixl</a> <a id="17589" class="Number">6</a> <a id="17591" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17596" class="Function Operator">_⊕_</a>
<a id="_⊕_"></a><a id="17596" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17596" class="Function Operator">_⊕_</a> <a id="17600" class="Symbol">:</a> <a id="17602" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="17606" href="Algebra.Lattice.Bundles.html#5670" class="Field">Carrier</a>
<a id="17614" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17614" class="Bound">x</a> <a id="17616" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17596" class="Function Operator">⊕</a> <a id="17618" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17618" class="Bound">y</a> <a id="17620" class="Symbol">=</a> <a id="17622" class="Symbol">(</a><a id="17623" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17614" class="Bound">x</a> <a id="17625" href="Algebra.Lattice.Bundles.html#5736" class="Field Operator">∨</a> <a id="17627" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17618" class="Bound">y</a><a id="17628" class="Symbol">)</a> <a id="17630" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="17632" href="Algebra.Lattice.Bundles.html#5806" class="Field Operator">¬</a> <a id="17634" class="Symbol">(</a><a id="17635" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17614" class="Bound">x</a> <a id="17637" href="Algebra.Lattice.Bundles.html#5771" class="Field Operator">∧</a> <a id="17639" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17618" class="Bound">y</a><a id="17640" class="Symbol">)</a>
<a id="17643" class="Keyword">module</a> <a id="DefaultXorRing"></a><a id="17650" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17650" class="Module">DefaultXorRing</a> <a id="17665" class="Symbol">=</a> <a id="17667" href="Algebra.Lattice.Properties.BooleanAlgebra.html#7330" class="Module">XorRing</a> <a id="17675" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17596" class="Function Operator">_⊕_</a> <a id="17679" class="Symbol">(λ</a> <a id="17682" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17682" class="Bound">_</a> <a id="17684" href="Algebra.Lattice.Properties.BooleanAlgebra.html#17684" class="Bound">_</a> <a id="17686" class="Symbol">→</a> <a id="17688" href="Relation.Binary.Structures.html#1596" class="Function">refl</a><a id="17692" class="Symbol">)</a>
</pre></body></html>
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