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614 lines
290 KiB
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614 lines
290 KiB
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<html><head><meta charset="utf-8"><title>Algebra.Properties.BooleanAlgebra</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
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<a id="74" class="Comment">-- The Agda standard library</a>
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<a id="103" class="Comment">--</a>
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<a id="106" class="Comment">-- Some derivable properties</a>
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<a id="135" class="Comment">------------------------------------------------------------------------</a>
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<a id="209" class="Symbol">{-#</a> <a id="213" class="Keyword">OPTIONS</a> <a id="221" class="Pragma">--cubical-compatible</a> <a id="242" class="Pragma">--safe</a> <a id="249" class="Symbol">#-}</a>
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<a id="254" class="Comment">-- Disabled to prevent warnings from deprecated names</a>
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<a id="308" class="Symbol">{-#</a> <a id="312" class="Keyword">OPTIONS</a> <a id="320" class="Pragma">--warn=noUserWarning</a> <a id="341" class="Symbol">#-}</a>
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<a id="346" class="Keyword">open</a> <a id="351" class="Keyword">import</a> <a id="358" href="Algebra.Bundles.html" class="Module">Algebra.Bundles</a>
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<a id="375" class="Keyword">module</a> <a id="382" href="Algebra.Properties.BooleanAlgebra.html" class="Module">Algebra.Properties.BooleanAlgebra</a>
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<a id="418" class="Symbol">{</a><a id="419" href="Algebra.Properties.BooleanAlgebra.html#419" class="Bound">b₁</a> <a id="422" href="Algebra.Properties.BooleanAlgebra.html#422" class="Bound">b₂</a><a id="424" class="Symbol">}</a> <a id="426" class="Symbol">(</a><a id="427" href="Algebra.Properties.BooleanAlgebra.html#427" class="Bound">B</a> <a id="429" class="Symbol">:</a> <a id="431" href="Algebra.Bundles.html#23152" class="Record">BooleanAlgebra</a> <a id="446" href="Algebra.Properties.BooleanAlgebra.html#419" class="Bound">b₁</a> <a id="449" href="Algebra.Properties.BooleanAlgebra.html#422" class="Bound">b₂</a><a id="451" class="Symbol">)</a>
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<a id="455" class="Keyword">where</a>
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<a id="462" class="Keyword">open</a> <a id="467" href="Algebra.Bundles.html#23152" class="Module">BooleanAlgebra</a> <a id="482" href="Algebra.Properties.BooleanAlgebra.html#427" class="Bound">B</a>
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<a id="485" class="Keyword">import</a> <a id="492" href="Algebra.Properties.DistributiveLattice.html" class="Module">Algebra.Properties.DistributiveLattice</a> <a id="531" class="Symbol">as</a> <a id="534" class="Module">DistribLatticeProperties</a>
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<a id="559" class="Keyword">open</a> <a id="564" class="Keyword">import</a> <a id="571" href="Algebra.Core.html" class="Module">Algebra.Core</a>
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<a id="584" class="Keyword">open</a> <a id="589" class="Keyword">import</a> <a id="596" href="Algebra.Structures.html" class="Module">Algebra.Structures</a> <a id="615" href="Algebra.Bundles.html#23297" class="Field Operator">_≈_</a>
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<a id="619" class="Keyword">open</a> <a id="624" class="Keyword">import</a> <a id="631" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="651" href="Algebra.Bundles.html#23297" class="Field Operator">_≈_</a>
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<a id="655" class="Keyword">open</a> <a id="660" class="Keyword">import</a> <a id="667" href="Algebra.Consequences.Setoid.html" class="Module">Algebra.Consequences.Setoid</a> <a id="695" href="Algebra.Bundles.html#9620" class="Function">setoid</a>
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<a id="702" class="Keyword">open</a> <a id="707" class="Keyword">import</a> <a id="714" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="747" href="Algebra.Bundles.html#9620" class="Function">setoid</a>
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<a id="754" class="Keyword">open</a> <a id="759" class="Keyword">import</a> <a id="766" href="Relation.Binary.html" class="Module">Relation.Binary</a>
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<a id="782" class="Keyword">open</a> <a id="787" class="Keyword">import</a> <a id="794" href="Function.Base.html" class="Module">Function.Base</a>
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<a id="808" class="Keyword">open</a> <a id="813" class="Keyword">import</a> <a id="820" href="Function.Equality.html" class="Module">Function.Equality</a> <a id="838" class="Keyword">using</a> <a id="844" class="Symbol">(</a><a id="845" href="Function.Equality.html#1073" class="Field Operator">_⟨$⟩_</a><a id="850" class="Symbol">)</a>
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<a id="852" class="Keyword">open</a> <a id="857" class="Keyword">import</a> <a id="864" href="Function.Equivalence.html" class="Module">Function.Equivalence</a> <a id="885" class="Keyword">using</a> <a id="891" class="Symbol">(</a><a id="892" href="Function.Equivalence.html#1213" class="Function Operator">_⇔_</a><a id="895" class="Symbol">;</a> <a id="897" class="Keyword">module</a> <a id="904" href="Function.Equivalence.html#838" class="Module">Equivalence</a><a id="915" class="Symbol">)</a>
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<a id="917" class="Keyword">open</a> <a id="922" class="Keyword">import</a> <a id="929" href="Data.Product.html" class="Module">Data.Product</a> <a id="942" class="Keyword">using</a> <a id="948" class="Symbol">(</a><a id="949" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">_,_</a><a id="952" class="Symbol">)</a>
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<a id="955" class="Comment">------------------------------------------------------------------------</a>
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<a id="1028" class="Comment">-- Export properties from distributive lattices</a>
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<a id="1077" class="Keyword">open</a> <a id="1082" href="Algebra.Properties.DistributiveLattice.html" class="Module">DistribLatticeProperties</a> <a id="1107" href="Algebra.Bundles.html#23608" class="Function">distributiveLattice</a> <a id="1127" class="Keyword">public</a>
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<a id="1136" class="Keyword">hiding</a> <a id="1143" class="Symbol">(</a><a id="1144" href="Algebra.Properties.DistributiveLattice.html#3612" class="Function">replace-equality</a><a id="1160" class="Symbol">)</a>
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<a id="1163" class="Comment">------------------------------------------------------------------------</a>
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<a id="1236" class="Comment">-- Some simple consequences</a>
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<a id="∨-complementˡ"></a><a id="1265" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="1279" class="Symbol">:</a> <a id="1281" href="Algebra.Definitions.html#1713" class="Function">LeftInverse</a> <a id="1293" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="1295" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a> <a id="1298" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
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<a id="1302" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="1316" class="Symbol">=</a> <a id="1318" href="Algebra.Consequences.Setoid.html#3361" class="Function">comm+invʳ⇒invˡ</a> <a id="1333" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="1340" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a>
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<a id="∨-complement"></a><a id="1355" href="Algebra.Properties.BooleanAlgebra.html#1355" class="Function">∨-complement</a> <a id="1368" class="Symbol">:</a> <a id="1370" href="Algebra.Definitions.html#1891" class="Function">Inverse</a> <a id="1378" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="1380" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a> <a id="1383" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
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<a id="1387" href="Algebra.Properties.BooleanAlgebra.html#1355" class="Function">∨-complement</a> <a id="1400" class="Symbol">=</a> <a id="1402" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="1416" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1418" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a>
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<a id="∧-complementˡ"></a><a id="1433" href="Algebra.Properties.BooleanAlgebra.html#1433" class="Function">∧-complementˡ</a> <a id="1447" class="Symbol">:</a> <a id="1449" href="Algebra.Definitions.html#1713" class="Function">LeftInverse</a> <a id="1461" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="1463" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a> <a id="1466" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
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<a id="1470" href="Algebra.Properties.BooleanAlgebra.html#1433" class="Function">∧-complementˡ</a> <a id="1484" class="Symbol">=</a> <a id="1486" href="Algebra.Consequences.Setoid.html#3361" class="Function">comm+invʳ⇒invˡ</a> <a id="1501" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="1508" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a>
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<a id="∧-complement"></a><a id="1523" href="Algebra.Properties.BooleanAlgebra.html#1523" class="Function">∧-complement</a> <a id="1536" class="Symbol">:</a> <a id="1538" href="Algebra.Definitions.html#1891" class="Function">Inverse</a> <a id="1546" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="1548" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a> <a id="1551" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
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<a id="1555" href="Algebra.Properties.BooleanAlgebra.html#1523" class="Function">∧-complement</a> <a id="1568" class="Symbol">=</a> <a id="1570" href="Algebra.Properties.BooleanAlgebra.html#1433" class="Function">∧-complementˡ</a> <a id="1584" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="1586" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a>
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<a id="1601" class="Comment">------------------------------------------------------------------------</a>
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<a id="1674" class="Comment">-- The dual construction is also a boolean algebra</a>
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<a id="∧-∨-isBooleanAlgebra"></a><a id="1726" href="Algebra.Properties.BooleanAlgebra.html#1726" class="Function">∧-∨-isBooleanAlgebra</a> <a id="1747" class="Symbol">:</a> <a id="1749" href="Algebra.Structures.html#15866" class="Record">IsBooleanAlgebra</a> <a id="1766" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="1770" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="1774" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a> <a id="1777" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="1779" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
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<a id="1781" href="Algebra.Properties.BooleanAlgebra.html#1726" class="Function">∧-∨-isBooleanAlgebra</a> <a id="1802" class="Symbol">=</a> <a id="1804" class="Keyword">record</a>
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<a id="1813" class="Symbol">{</a> <a id="1815" href="Algebra.Structures.html#15960" class="Field">isDistributiveLattice</a> <a id="1837" class="Symbol">=</a> <a id="1839" href="Algebra.Properties.DistributiveLattice.html#2288" class="Function">∧-∨-isDistributiveLattice</a>
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<a id="1867" class="Symbol">;</a> <a id="1869" href="Algebra.Structures.html#16014" class="Field">∨-complementʳ</a> <a id="1891" class="Symbol">=</a> <a id="1893" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a>
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<a id="1909" class="Symbol">;</a> <a id="1911" href="Algebra.Structures.html#16061" class="Field">∧-complementʳ</a> <a id="1933" class="Symbol">=</a> <a id="1935" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a>
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<a id="1951" class="Symbol">;</a> <a id="1953" href="Algebra.Structures.html#16108" class="Field">¬-cong</a> <a id="1975" class="Symbol">=</a> <a id="1977" href="Algebra.Structures.html#16108" class="Function">¬-cong</a>
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<a id="1986" class="Symbol">}</a>
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<a id="∧-∨-booleanAlgebra"></a><a id="1989" href="Algebra.Properties.BooleanAlgebra.html#1989" class="Function">∧-∨-booleanAlgebra</a> <a id="2008" class="Symbol">:</a> <a id="2010" href="Algebra.Bundles.html#23152" class="Record">BooleanAlgebra</a> <a id="2025" class="Symbol">_</a> <a id="2027" class="Symbol">_</a>
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<a id="2029" href="Algebra.Properties.BooleanAlgebra.html#1989" class="Function">∧-∨-booleanAlgebra</a> <a id="2048" class="Symbol">=</a> <a id="2050" class="Keyword">record</a>
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<a id="2059" class="Symbol">{</a> <a id="2061" href="Algebra.Bundles.html#23501" class="Field">isBooleanAlgebra</a> <a id="2078" class="Symbol">=</a> <a id="2080" href="Algebra.Properties.BooleanAlgebra.html#1726" class="Function">∧-∨-isBooleanAlgebra</a>
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<a id="2103" class="Symbol">}</a>
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<a id="2106" class="Comment">------------------------------------------------------------------------</a>
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<a id="2179" class="Comment">-- (∨, ∧, ⊥, ⊤) and (∧, ∨, ⊤, ⊥) are commutative semirings</a>
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<a id="∧-identityʳ"></a><a id="2239" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="2251" class="Symbol">:</a> <a id="2253" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="2267" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="2269" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
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<a id="2273" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="2285" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a> <a id="2287" class="Symbol">=</a> <a id="2289" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="2297" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a> <a id="2299" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2301" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="2312" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2315" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="2323" class="Symbol">(</a><a id="2324" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="2328" class="Symbol">(</a><a id="2329" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="2343" class="Symbol">_))</a> <a id="2347" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="2351" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a> <a id="2353" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2355" class="Symbol">(</a><a id="2356" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a> <a id="2358" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="2360" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="2362" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a><a id="2363" class="Symbol">)</a> <a id="2366" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2369" href="Algebra.Structures.html#6565" class="Function">∧-absorbs-∨</a> <a id="2381" class="Symbol">_</a> <a id="2383" class="Symbol">_</a> <a id="2385" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="2389" href="Algebra.Properties.BooleanAlgebra.html#2285" class="Bound">x</a> <a id="2404" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="∧-identityˡ"></a><a id="2407" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="2419" class="Symbol">:</a> <a id="2421" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="2434" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="2436" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
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<a id="2440" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="2452" class="Symbol">=</a> <a id="2454" href="Algebra.Consequences.Setoid.html#1929" class="Function">comm+idʳ⇒idˡ</a> <a id="2467" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="2474" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a>
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<a id="∧-identity"></a><a id="2487" href="Algebra.Properties.BooleanAlgebra.html#2487" class="Function">∧-identity</a> <a id="2498" class="Symbol">:</a> <a id="2500" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="2509" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="2511" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
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<a id="2515" href="Algebra.Properties.BooleanAlgebra.html#2487" class="Function">∧-identity</a> <a id="2526" class="Symbol">=</a> <a id="2528" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="2540" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2542" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a>
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<a id="∨-identityʳ"></a><a id="2555" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a> <a id="2567" class="Symbol">:</a> <a id="2569" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="2583" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2585" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
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<a id="2589" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a> <a id="2601" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2603" class="Symbol">=</a> <a id="2605" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="2613" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2615" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="2617" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2628" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2631" href="Algebra.Structures.html#6750" class="Function">∨-congˡ</a> <a id="2639" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="2641" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="2645" class="Symbol">(</a><a id="2646" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="2660" class="Symbol">_)</a> <a id="2663" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="2667" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2669" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="2671" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2673" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2675" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="2677" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2682" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="2685" href="Algebra.Structures.html#6503" class="Function">∨-absorbs-∧</a> <a id="2697" class="Symbol">_</a> <a id="2699" class="Symbol">_</a> <a id="2701" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="2705" href="Algebra.Properties.BooleanAlgebra.html#2601" class="Bound">x</a> <a id="2720" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="∨-identityˡ"></a><a id="2723" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="2735" class="Symbol">:</a> <a id="2737" href="Algebra.Definitions.html#1277" class="Function">LeftIdentity</a> <a id="2750" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2752" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
|
|||
|
|
<a id="2756" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="2768" class="Symbol">=</a> <a id="2770" href="Algebra.Consequences.Setoid.html#1929" class="Function">comm+idʳ⇒idˡ</a> <a id="2783" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="2790" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a>
|
|||
|
|
|
|||
|
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<a id="∨-identity"></a><a id="2803" href="Algebra.Properties.BooleanAlgebra.html#2803" class="Function">∨-identity</a> <a id="2814" class="Symbol">:</a> <a id="2816" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="2825" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2827" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
|
|||
|
|
<a id="2831" href="Algebra.Properties.BooleanAlgebra.html#2803" class="Function">∨-identity</a> <a id="2842" class="Symbol">=</a> <a id="2844" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="2856" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="2858" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zeroʳ"></a><a id="2871" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="2879" class="Symbol">:</a> <a id="2881" href="Algebra.Definitions.html#1576" class="Function">RightZero</a> <a id="2891" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2893" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
|
|||
|
|
<a id="2897" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="2905" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2907" class="Symbol">=</a> <a id="2909" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="2917" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2919" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2921" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="2932" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2936" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="2944" class="Symbol">(</a><a id="2945" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="2959" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a><a id="2960" class="Symbol">)</a> <a id="2962" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="2966" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2968" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2971" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2974" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="2976" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="2978" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2981" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="2985" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="2993" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2995" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="2997" class="Symbol">(</a><a id="2998" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3000" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a><a id="3001" class="Symbol">)</a> <a id="3003" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="3007" class="Symbol">(</a><a id="3008" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="3010" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="3012" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a><a id="3013" class="Symbol">)</a> <a id="3015" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="3017" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3019" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="3022" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3026" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="3034" class="Symbol">(</a><a id="3035" href="Algebra.Properties.Lattice.html#5107" class="Function">∧-idempotent</a> <a id="3048" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a><a id="3049" class="Symbol">)</a> <a id="3051" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="3055" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="3063" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="3065" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3067" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="3070" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3074" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="3088" href="Algebra.Properties.BooleanAlgebra.html#2905" class="Bound">x</a> <a id="3090" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="3094" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="3109" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zeroˡ"></a><a id="3112" href="Algebra.Properties.BooleanAlgebra.html#3112" class="Function">∧-zeroˡ</a> <a id="3120" class="Symbol">:</a> <a id="3122" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="3131" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="3133" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
|
|||
|
|
<a id="3137" href="Algebra.Properties.BooleanAlgebra.html#3112" class="Function">∧-zeroˡ</a> <a id="3145" class="Symbol">=</a> <a id="3147" href="Algebra.Consequences.Setoid.html#2211" class="Function">comm+zeʳ⇒zeˡ</a> <a id="3160" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="3167" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∧-zero"></a><a id="3176" href="Algebra.Properties.BooleanAlgebra.html#3176" class="Function">∧-zero</a> <a id="3183" class="Symbol">:</a> <a id="3185" href="Algebra.Definitions.html#1643" class="Function">Zero</a> <a id="3190" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="3192" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a>
|
|||
|
|
<a id="3196" href="Algebra.Properties.BooleanAlgebra.html#3176" class="Function">∧-zero</a> <a id="3203" class="Symbol">=</a> <a id="3205" href="Algebra.Properties.BooleanAlgebra.html#3112" class="Function">∧-zeroˡ</a> <a id="3213" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3215" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zeroʳ"></a><a id="3224" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a> <a id="3232" class="Symbol">:</a> <a id="3234" class="Symbol">∀</a> <a id="3236" href="Algebra.Properties.BooleanAlgebra.html#3236" class="Bound">x</a> <a id="3238" class="Symbol">→</a> <a id="3240" href="Algebra.Properties.BooleanAlgebra.html#3236" class="Bound">x</a> <a id="3242" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3244" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="3246" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="3248" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="3250" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a> <a id="3258" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3260" class="Symbol">=</a> <a id="3262" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="3270" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3272" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3274" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="3285" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="3289" href="Algebra.Structures.html#6750" class="Function">∨-congˡ</a> <a id="3297" class="Symbol">(</a><a id="3298" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="3312" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a><a id="3313" class="Symbol">)</a> <a id="3315" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="3319" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3321" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3324" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3327" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3329" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3331" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3334" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="3338" href="Algebra.Structures.html#6258" class="Function">∨-assoc</a> <a id="3346" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3348" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3350" class="Symbol">(</a><a id="3351" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3353" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a><a id="3354" class="Symbol">)</a> <a id="3356" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="3360" class="Symbol">(</a><a id="3361" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3363" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3365" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a><a id="3366" class="Symbol">)</a> <a id="3368" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3370" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3372" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3375" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3378" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="3386" class="Symbol">(</a><a id="3387" href="Algebra.Properties.Lattice.html#5242" class="Function">∨-idempotent</a> <a id="3400" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a><a id="3401" class="Symbol">)</a> <a id="3403" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="3407" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3415" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="3417" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="3419" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3422" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="3425" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="3439" href="Algebra.Properties.BooleanAlgebra.html#3258" class="Bound">x</a> <a id="3441" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="3445" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="3460" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zeroˡ"></a><a id="3463" href="Algebra.Properties.BooleanAlgebra.html#3463" class="Function">∨-zeroˡ</a> <a id="3471" class="Symbol">:</a> <a id="3473" href="Algebra.Definitions.html#1511" class="Function">LeftZero</a> <a id="3482" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="3484" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
|
|||
|
|
<a id="3488" href="Algebra.Properties.BooleanAlgebra.html#3463" class="Function">∨-zeroˡ</a> <a id="3496" class="Symbol">=</a> <a id="3498" href="Algebra.Consequences.Setoid.html#2211" class="Function">comm+zeʳ⇒zeˡ</a> <a id="3511" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="3518" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-zero"></a><a id="3527" href="Algebra.Properties.BooleanAlgebra.html#3527" class="Function">∨-zero</a> <a id="3534" class="Symbol">:</a> <a id="3536" href="Algebra.Definitions.html#1643" class="Function">Zero</a> <a id="3541" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="3543" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a>
|
|||
|
|
<a id="3547" href="Algebra.Properties.BooleanAlgebra.html#3527" class="Function">∨-zero</a> <a id="3554" class="Symbol">=</a> <a id="3556" href="Algebra.Properties.BooleanAlgebra.html#3463" class="Function">∨-zeroˡ</a> <a id="3564" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="3566" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a>
|
|||
|
|
|
|||
|
|
<a id="∨-⊥-isMonoid"></a><a id="3575" href="Algebra.Properties.BooleanAlgebra.html#3575" class="Function">∨-⊥-isMonoid</a> <a id="3588" class="Symbol">:</a> <a id="3590" href="Algebra.Structures.html#2810" class="Record">IsMonoid</a> <a id="3599" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="3603" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
|
|
<a id="3605" href="Algebra.Properties.BooleanAlgebra.html#3575" class="Function">∨-⊥-isMonoid</a> <a id="3618" class="Symbol">=</a> <a id="3620" class="Keyword">record</a>
|
|||
|
|
<a id="3629" class="Symbol">{</a> <a id="3631" href="Algebra.Structures.html#2871" class="Field">isSemigroup</a> <a id="3643" class="Symbol">=</a> <a id="3645" href="Algebra.Properties.Lattice.html#2127" class="Function">∨-isSemigroup</a>
|
|||
|
|
<a id="3661" class="Symbol">;</a> <a id="3663" href="Algebra.Structures.html#2903" class="Field">identity</a> <a id="3675" class="Symbol">=</a> <a id="3677" href="Algebra.Properties.BooleanAlgebra.html#2803" class="Function">∨-identity</a>
|
|||
|
|
<a id="3690" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-⊤-isMonoid"></a><a id="3693" href="Algebra.Properties.BooleanAlgebra.html#3693" class="Function">∧-⊤-isMonoid</a> <a id="3706" class="Symbol">:</a> <a id="3708" href="Algebra.Structures.html#2810" class="Record">IsMonoid</a> <a id="3717" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="3721" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="3723" href="Algebra.Properties.BooleanAlgebra.html#3693" class="Function">∧-⊤-isMonoid</a> <a id="3736" class="Symbol">=</a> <a id="3738" class="Keyword">record</a>
|
|||
|
|
<a id="3747" class="Symbol">{</a> <a id="3749" href="Algebra.Structures.html#2871" class="Field">isSemigroup</a> <a id="3761" class="Symbol">=</a> <a id="3763" href="Algebra.Properties.Lattice.html#1163" class="Function">∧-isSemigroup</a>
|
|||
|
|
<a id="3779" class="Symbol">;</a> <a id="3781" href="Algebra.Structures.html#2903" class="Field">identity</a> <a id="3793" class="Symbol">=</a> <a id="3795" href="Algebra.Properties.BooleanAlgebra.html#2487" class="Function">∧-identity</a>
|
|||
|
|
<a id="3808" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-⊥-isCommutativeMonoid"></a><a id="3811" href="Algebra.Properties.BooleanAlgebra.html#3811" class="Function">∨-⊥-isCommutativeMonoid</a> <a id="3835" class="Symbol">:</a> <a id="3837" href="Algebra.Structures.html#3101" class="Record">IsCommutativeMonoid</a> <a id="3857" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="3861" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
|
|
<a id="3863" href="Algebra.Properties.BooleanAlgebra.html#3811" class="Function">∨-⊥-isCommutativeMonoid</a> <a id="3887" class="Symbol">=</a> <a id="3889" class="Keyword">record</a>
|
|||
|
|
<a id="3898" class="Symbol">{</a> <a id="3900" href="Algebra.Structures.html#3173" class="Field">isMonoid</a> <a id="3909" class="Symbol">=</a> <a id="3911" href="Algebra.Properties.BooleanAlgebra.html#3575" class="Function">∨-⊥-isMonoid</a>
|
|||
|
|
<a id="3926" class="Symbol">;</a> <a id="3928" href="Algebra.Structures.html#3201" class="Field">comm</a> <a id="3937" class="Symbol">=</a> <a id="3939" href="Algebra.Structures.html#6224" class="Function">∨-comm</a>
|
|||
|
|
<a id="3948" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-⊤-isCommutativeMonoid"></a><a id="3951" href="Algebra.Properties.BooleanAlgebra.html#3951" class="Function">∧-⊤-isCommutativeMonoid</a> <a id="3975" class="Symbol">:</a> <a id="3977" href="Algebra.Structures.html#3101" class="Record">IsCommutativeMonoid</a> <a id="3997" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="4001" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="4003" href="Algebra.Properties.BooleanAlgebra.html#3951" class="Function">∧-⊤-isCommutativeMonoid</a> <a id="4027" class="Symbol">=</a> <a id="4029" class="Keyword">record</a>
|
|||
|
|
<a id="4038" class="Symbol">{</a> <a id="4040" href="Algebra.Structures.html#3173" class="Field">isMonoid</a> <a id="4049" class="Symbol">=</a> <a id="4051" href="Algebra.Properties.BooleanAlgebra.html#3693" class="Function">∧-⊤-isMonoid</a>
|
|||
|
|
<a id="4066" class="Symbol">;</a> <a id="4068" href="Algebra.Structures.html#3201" class="Field">comm</a> <a id="4077" class="Symbol">=</a> <a id="4079" href="Algebra.Structures.html#6325" class="Function">∧-comm</a>
|
|||
|
|
<a id="4088" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isSemiring"></a><a id="4091" href="Algebra.Properties.BooleanAlgebra.html#4091" class="Function">∨-∧-isSemiring</a> <a id="4106" class="Symbol">:</a> <a id="4108" href="Algebra.Structures.html#11370" class="Record">IsSemiring</a> <a id="4119" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="4123" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="4127" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="4129" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="4131" href="Algebra.Properties.BooleanAlgebra.html#4091" class="Function">∨-∧-isSemiring</a> <a id="4146" class="Symbol">=</a> <a id="4148" class="Keyword">record</a>
|
|||
|
|
<a id="4157" class="Symbol">{</a> <a id="4159" href="Algebra.Structures.html#11439" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="4193" class="Symbol">=</a> <a id="4195" class="Keyword">record</a>
|
|||
|
|
<a id="4206" class="Symbol">{</a> <a id="4208" href="Algebra.Structures.html#10106" class="Field">+-isCommutativeMonoid</a> <a id="4230" class="Symbol">=</a> <a id="4232" href="Algebra.Properties.BooleanAlgebra.html#3811" class="Function">∨-⊥-isCommutativeMonoid</a>
|
|||
|
|
<a id="4260" class="Symbol">;</a> <a id="4262" href="Algebra.Structures.html#10159" class="Field">*-isMonoid</a> <a id="4273" class="Symbol">=</a> <a id="4275" href="Algebra.Properties.BooleanAlgebra.html#3693" class="Function">∧-⊤-isMonoid</a>
|
|||
|
|
<a id="4292" class="Symbol">;</a> <a id="4294" href="Algebra.Structures.html#10201" class="Field">distrib</a> <a id="4302" class="Symbol">=</a> <a id="4304" href="Algebra.Properties.DistributiveLattice.html#3453" class="Function">∧-∨-distrib</a>
|
|||
|
|
<a id="4320" class="Symbol">}</a>
|
|||
|
|
<a id="4324" class="Symbol">;</a> <a id="4326" href="Algebra.Structures.html#11529" class="Field">zero</a> <a id="4331" class="Symbol">=</a> <a id="4333" href="Algebra.Properties.BooleanAlgebra.html#3176" class="Function">∧-zero</a>
|
|||
|
|
<a id="4342" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-isSemiring"></a><a id="4345" href="Algebra.Properties.BooleanAlgebra.html#4345" class="Function">∧-∨-isSemiring</a> <a id="4360" class="Symbol">:</a> <a id="4362" href="Algebra.Structures.html#11370" class="Record">IsSemiring</a> <a id="4373" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="4377" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="4381" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="4383" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
|
|
<a id="4385" href="Algebra.Properties.BooleanAlgebra.html#4345" class="Function">∧-∨-isSemiring</a> <a id="4400" class="Symbol">=</a> <a id="4402" class="Keyword">record</a>
|
|||
|
|
<a id="4411" class="Symbol">{</a> <a id="4413" href="Algebra.Structures.html#11439" class="Field">isSemiringWithoutAnnihilatingZero</a> <a id="4447" class="Symbol">=</a> <a id="4449" class="Keyword">record</a>
|
|||
|
|
<a id="4460" class="Symbol">{</a> <a id="4462" href="Algebra.Structures.html#10106" class="Field">+-isCommutativeMonoid</a> <a id="4484" class="Symbol">=</a> <a id="4486" href="Algebra.Properties.BooleanAlgebra.html#3951" class="Function">∧-⊤-isCommutativeMonoid</a>
|
|||
|
|
<a id="4514" class="Symbol">;</a> <a id="4516" href="Algebra.Structures.html#10159" class="Field">*-isMonoid</a> <a id="4527" class="Symbol">=</a> <a id="4529" href="Algebra.Properties.BooleanAlgebra.html#3575" class="Function">∨-⊥-isMonoid</a>
|
|||
|
|
<a id="4546" class="Symbol">;</a> <a id="4548" href="Algebra.Structures.html#10201" class="Field">distrib</a> <a id="4556" class="Symbol">=</a> <a id="4558" href="Algebra.Properties.DistributiveLattice.html#3017" class="Function">∨-∧-distrib</a>
|
|||
|
|
<a id="4574" class="Symbol">}</a>
|
|||
|
|
<a id="4578" class="Symbol">;</a> <a id="4580" href="Algebra.Structures.html#11529" class="Field">zero</a> <a id="4585" class="Symbol">=</a> <a id="4587" href="Algebra.Properties.BooleanAlgebra.html#3527" class="Function">∨-zero</a>
|
|||
|
|
<a id="4596" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-isCommutativeSemiring"></a><a id="4599" href="Algebra.Properties.BooleanAlgebra.html#4599" class="Function">∨-∧-isCommutativeSemiring</a> <a id="4625" class="Symbol">:</a> <a id="4627" href="Algebra.Structures.html#12026" class="Record">IsCommutativeSemiring</a> <a id="4649" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="4653" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="4657" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="4659" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="4661" href="Algebra.Properties.BooleanAlgebra.html#4599" class="Function">∨-∧-isCommutativeSemiring</a> <a id="4687" class="Symbol">=</a> <a id="4689" class="Keyword">record</a>
|
|||
|
|
<a id="4698" class="Symbol">{</a> <a id="4700" href="Algebra.Structures.html#12106" class="Field">isSemiring</a> <a id="4711" class="Symbol">=</a> <a id="4713" href="Algebra.Properties.BooleanAlgebra.html#4091" class="Function">∨-∧-isSemiring</a>
|
|||
|
|
<a id="4730" class="Symbol">;</a> <a id="4732" href="Algebra.Structures.html#12144" class="Field">*-comm</a> <a id="4739" class="Symbol">=</a> <a id="4741" href="Algebra.Structures.html#6325" class="Function">∧-comm</a>
|
|||
|
|
<a id="4750" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-isCommutativeSemiring"></a><a id="4753" href="Algebra.Properties.BooleanAlgebra.html#4753" class="Function">∧-∨-isCommutativeSemiring</a> <a id="4779" class="Symbol">:</a> <a id="4781" href="Algebra.Structures.html#12026" class="Record">IsCommutativeSemiring</a> <a id="4803" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="4807" href="Algebra.Bundles.html#23334" class="Field Operator">_∨_</a> <a id="4811" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="4813" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
|
|
<a id="4815" href="Algebra.Properties.BooleanAlgebra.html#4753" class="Function">∧-∨-isCommutativeSemiring</a> <a id="4841" class="Symbol">=</a> <a id="4843" class="Keyword">record</a>
|
|||
|
|
<a id="4852" class="Symbol">{</a> <a id="4854" href="Algebra.Structures.html#12106" class="Field">isSemiring</a> <a id="4865" class="Symbol">=</a> <a id="4867" href="Algebra.Properties.BooleanAlgebra.html#4345" class="Function">∧-∨-isSemiring</a>
|
|||
|
|
<a id="4884" class="Symbol">;</a> <a id="4886" href="Algebra.Structures.html#12144" class="Field">*-comm</a> <a id="4893" class="Symbol">=</a> <a id="4895" href="Algebra.Structures.html#6224" class="Function">∨-comm</a>
|
|||
|
|
<a id="4904" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∨-∧-commutativeSemiring"></a><a id="4907" href="Algebra.Properties.BooleanAlgebra.html#4907" class="Function">∨-∧-commutativeSemiring</a> <a id="4931" class="Symbol">:</a> <a id="4933" href="Algebra.Bundles.html#17360" class="Record">CommutativeSemiring</a> <a id="4953" class="Symbol">_</a> <a id="4955" class="Symbol">_</a>
|
|||
|
|
<a id="4957" href="Algebra.Properties.BooleanAlgebra.html#4907" class="Function">∨-∧-commutativeSemiring</a> <a id="4981" class="Symbol">=</a> <a id="4983" class="Keyword">record</a>
|
|||
|
|
<a id="4992" class="Symbol">{</a> <a id="4994" href="Algebra.Bundles.html#17695" class="Field">isCommutativeSemiring</a> <a id="5016" class="Symbol">=</a> <a id="5018" href="Algebra.Properties.BooleanAlgebra.html#4599" class="Function">∨-∧-isCommutativeSemiring</a>
|
|||
|
|
<a id="5046" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="∧-∨-commutativeSemiring"></a><a id="5049" href="Algebra.Properties.BooleanAlgebra.html#5049" class="Function">∧-∨-commutativeSemiring</a> <a id="5073" class="Symbol">:</a> <a id="5075" href="Algebra.Bundles.html#17360" class="Record">CommutativeSemiring</a> <a id="5095" class="Symbol">_</a> <a id="5097" class="Symbol">_</a>
|
|||
|
|
<a id="5099" href="Algebra.Properties.BooleanAlgebra.html#5049" class="Function">∧-∨-commutativeSemiring</a> <a id="5123" class="Symbol">=</a> <a id="5125" class="Keyword">record</a>
|
|||
|
|
<a id="5134" class="Symbol">{</a> <a id="5136" href="Algebra.Bundles.html#17695" class="Field">isCommutativeSemiring</a> <a id="5158" class="Symbol">=</a> <a id="5160" href="Algebra.Properties.BooleanAlgebra.html#4753" class="Function">∧-∨-isCommutativeSemiring</a>
|
|||
|
|
<a id="5188" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="5191" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="5264" class="Comment">-- Some other properties</a>
|
|||
|
|
|
|||
|
|
<a id="5290" class="Comment">-- I took the statement of this lemma (called Uniqueness of</a>
|
|||
|
|
<a id="5350" class="Comment">-- Complements) from some course notes, "Boolean Algebra", written</a>
|
|||
|
|
<a id="5417" class="Comment">-- by Gert Smolka.</a>
|
|||
|
|
|
|||
|
|
<a id="5437" class="Keyword">private</a>
|
|||
|
|
<a id="lemma"></a><a id="5447" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="5453" class="Symbol">:</a> <a id="5455" class="Symbol">∀</a> <a id="5457" href="Algebra.Properties.BooleanAlgebra.html#5457" class="Bound">x</a> <a id="5459" href="Algebra.Properties.BooleanAlgebra.html#5459" class="Bound">y</a> <a id="5461" class="Symbol">→</a> <a id="5463" href="Algebra.Properties.BooleanAlgebra.html#5457" class="Bound">x</a> <a id="5465" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5467" href="Algebra.Properties.BooleanAlgebra.html#5459" class="Bound">y</a> <a id="5469" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="5471" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="5473" class="Symbol">→</a> <a id="5475" href="Algebra.Properties.BooleanAlgebra.html#5457" class="Bound">x</a> <a id="5477" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5479" href="Algebra.Properties.BooleanAlgebra.html#5459" class="Bound">y</a> <a id="5481" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="5483" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="5485" class="Symbol">→</a> <a id="5487" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5489" href="Algebra.Properties.BooleanAlgebra.html#5457" class="Bound">x</a> <a id="5491" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="5493" href="Algebra.Properties.BooleanAlgebra.html#5459" class="Bound">y</a>
|
|||
|
|
<a id="5497" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="5503" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5505" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5507" href="Algebra.Properties.BooleanAlgebra.html#5507" class="Bound">x∧y=⊥</a> <a id="5513" href="Algebra.Properties.BooleanAlgebra.html#5513" class="Bound">x∨y=⊤</a> <a id="5519" class="Symbol">=</a> <a id="5521" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="5531" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5533" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5550" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5554" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="5566" class="Symbol">_</a> <a id="5568" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="5574" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5576" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5578" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5580" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="5593" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5597" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="5605" href="Algebra.Properties.BooleanAlgebra.html#5513" class="Bound">x∨y=⊤</a> <a id="5611" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="5617" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5619" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5621" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5623" class="Symbol">(</a><a id="5624" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5626" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5628" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a><a id="5629" class="Symbol">)</a> <a id="5636" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5640" href="Algebra.Properties.DistributiveLattice.html#3159" class="Function">∧-∨-distribˡ</a> <a id="5653" class="Symbol">_</a> <a id="5655" class="Symbol">_</a> <a id="5657" class="Symbol">_</a> <a id="5659" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="5665" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5667" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5669" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5671" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5673" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5675" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5677" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5679" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5681" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5684" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5688" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="5696" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="5698" href="Algebra.Properties.BooleanAlgebra.html#1433" class="Function">∧-complementˡ</a> <a id="5712" class="Symbol">_</a> <a id="5714" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="5720" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="5722" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5724" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5726" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5728" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5730" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5739" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5743" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="5751" href="Algebra.Properties.BooleanAlgebra.html#5507" class="Bound">x∧y=⊥</a> <a id="5757" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="5763" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5765" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5767" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5769" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5771" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5773" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5775" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5777" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5782" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="5786" href="Algebra.Properties.DistributiveLattice.html#3306" class="Function">∧-∨-distribʳ</a> <a id="5799" class="Symbol">_</a> <a id="5801" class="Symbol">_</a> <a id="5803" class="Symbol">_</a> <a id="5805" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="5811" class="Symbol">(</a><a id="5812" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a> <a id="5814" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="5816" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5818" href="Algebra.Properties.BooleanAlgebra.html#5503" class="Bound">x</a><a id="5819" class="Symbol">)</a> <a id="5821" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5823" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5830" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5834" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="5842" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="5844" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="5858" class="Symbol">_</a> <a id="5860" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="5866" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="5868" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="5870" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5885" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="5889" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="5901" class="Symbol">_</a> <a id="5903" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="5909" href="Algebra.Properties.BooleanAlgebra.html#5505" class="Bound">y</a> <a id="5928" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="⊥≉⊤"></a><a id="5931" href="Algebra.Properties.BooleanAlgebra.html#5931" class="Function">⊥≉⊤</a> <a id="5935" class="Symbol">:</a> <a id="5937" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5939" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="5941" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="5943" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
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<a id="5945" href="Algebra.Properties.BooleanAlgebra.html#5931" class="Function">⊥≉⊤</a> <a id="5949" class="Symbol">=</a> <a id="5951" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="5957" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="5959" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="5961" class="Symbol">(</a><a id="5962" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="5974" class="Symbol">_)</a> <a id="5977" class="Symbol">(</a><a id="5978" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a> <a id="5986" class="Symbol">_)</a>
|
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|||
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<a id="⊤≉⊥"></a><a id="5990" href="Algebra.Properties.BooleanAlgebra.html#5990" class="Function">⊤≉⊥</a> <a id="5994" class="Symbol">:</a> <a id="5996" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="5998" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="6000" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="6002" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
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|
<a id="6004" href="Algebra.Properties.BooleanAlgebra.html#5990" class="Function">⊤≉⊥</a> <a id="6008" class="Symbol">=</a> <a id="6010" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="6016" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="6018" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="6020" class="Symbol">(</a><a id="6021" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="6029" class="Symbol">_)</a> <a id="6032" class="Symbol">(</a><a id="6033" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a> <a id="6045" class="Symbol">_)</a>
|
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<a id="¬-involutive"></a><a id="6049" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="6062" class="Symbol">:</a> <a id="6064" href="Algebra.Definitions.html#3030" class="Function">Involutive</a> <a id="6075" href="Algebra.Bundles.html#23404" class="Field Operator">¬_</a>
|
|||
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|
<a id="6078" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="6091" href="Algebra.Properties.BooleanAlgebra.html#6091" class="Bound">x</a> <a id="6093" class="Symbol">=</a> <a id="6095" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="6101" class="Symbol">(</a><a id="6102" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6104" href="Algebra.Properties.BooleanAlgebra.html#6091" class="Bound">x</a><a id="6105" class="Symbol">)</a> <a id="6107" href="Algebra.Properties.BooleanAlgebra.html#6091" class="Bound">x</a> <a id="6109" class="Symbol">(</a><a id="6110" href="Algebra.Properties.BooleanAlgebra.html#1433" class="Function">∧-complementˡ</a> <a id="6124" class="Symbol">_)</a> <a id="6127" class="Symbol">(</a><a id="6128" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="6142" class="Symbol">_)</a>
|
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<a id="deMorgan₁"></a><a id="6146" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="6156" class="Symbol">:</a> <a id="6158" class="Symbol">∀</a> <a id="6160" href="Algebra.Properties.BooleanAlgebra.html#6160" class="Bound">x</a> <a id="6162" href="Algebra.Properties.BooleanAlgebra.html#6162" class="Bound">y</a> <a id="6164" class="Symbol">→</a> <a id="6166" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6168" class="Symbol">(</a><a id="6169" href="Algebra.Properties.BooleanAlgebra.html#6160" class="Bound">x</a> <a id="6171" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6173" href="Algebra.Properties.BooleanAlgebra.html#6162" class="Bound">y</a><a id="6174" class="Symbol">)</a> <a id="6176" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="6178" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6180" href="Algebra.Properties.BooleanAlgebra.html#6160" class="Bound">x</a> <a id="6182" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6184" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6186" href="Algebra.Properties.BooleanAlgebra.html#6162" class="Bound">y</a>
|
|||
|
|
<a id="6188" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="6198" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6200" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6202" class="Symbol">=</a> <a id="6204" href="Algebra.Properties.BooleanAlgebra.html#5447" class="Function">lemma</a> <a id="6210" class="Symbol">(</a><a id="6211" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6213" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6215" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6216" class="Symbol">)</a> <a id="6218" class="Symbol">(</a><a id="6219" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6221" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6223" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6225" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6227" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6228" class="Symbol">)</a> <a id="6230" href="Algebra.Properties.BooleanAlgebra.html#6250" class="Function">lem₁</a> <a id="6235" href="Algebra.Properties.BooleanAlgebra.html#7025" class="Function">lem₂</a>
|
|||
|
|
<a id="6242" class="Keyword">where</a>
|
|||
|
|
<a id="6250" href="Algebra.Properties.BooleanAlgebra.html#6250" class="Function">lem₁</a> <a id="6255" class="Symbol">=</a> <a id="6257" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="6267" class="Symbol">(</a><a id="6268" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6270" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6272" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6273" class="Symbol">)</a> <a id="6275" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6277" class="Symbol">(</a><a id="6278" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6280" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6282" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6284" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6286" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6287" class="Symbol">)</a> <a id="6298" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6301" href="Algebra.Properties.DistributiveLattice.html#3159" class="Function">∧-∨-distribˡ</a> <a id="6314" class="Symbol">_</a> <a id="6316" class="Symbol">_</a> <a id="6318" class="Symbol">_</a> <a id="6320" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6326" class="Symbol">(</a><a id="6327" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6329" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6331" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6332" class="Symbol">)</a> <a id="6334" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6336" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6338" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6340" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6342" class="Symbol">(</a><a id="6343" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6345" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6347" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6348" class="Symbol">)</a> <a id="6350" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6352" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6354" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6357" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6360" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="6368" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6370" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="6378" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6380" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="6387" class="Symbol">_</a> <a id="6389" class="Symbol">_</a> <a id="6391" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6397" class="Symbol">(</a><a id="6398" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6400" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6402" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="6403" class="Symbol">)</a> <a id="6405" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6407" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6409" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6411" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6413" class="Symbol">(</a><a id="6414" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6416" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6418" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6419" class="Symbol">)</a> <a id="6421" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6423" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6425" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6428" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6431" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="6439" class="Symbol">_</a> <a id="6441" class="Symbol">_</a> <a id="6443" class="Symbol">_</a> <a id="6445" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="6447" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="6454" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="6456" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="6464" class="Symbol">_</a> <a id="6466" class="Symbol">_</a> <a id="6468" class="Symbol">_</a> <a id="6470" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6476" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6478" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6480" class="Symbol">(</a><a id="6481" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6483" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6485" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6487" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="6488" class="Symbol">)</a> <a id="6490" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6492" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6494" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6496" class="Symbol">(</a><a id="6497" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6499" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6501" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6503" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6504" class="Symbol">)</a> <a id="6507" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6510" class="Symbol">(</a><a id="6511" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="6519" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6521" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="6535" class="Symbol">_)</a> <a id="6538" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="6540" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="6547" href="Function.Base.html#4407" class="Function Operator">⟩</a>
|
|||
|
|
<a id="6587" class="Symbol">(</a><a id="6588" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="6596" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6598" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="6612" class="Symbol">_)</a> <a id="6615" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6621" class="Symbol">(</a><a id="6622" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6624" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6626" href="Algebra.Bundles.html#23470" class="Field">⊥</a><a id="6627" class="Symbol">)</a> <a id="6629" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6631" class="Symbol">(</a><a id="6632" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6634" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6636" href="Algebra.Bundles.html#23470" class="Field">⊥</a><a id="6637" class="Symbol">)</a> <a id="6652" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6655" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="6663" class="Symbol">_</a> <a id="6665" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="6667" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="6674" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="6676" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="6684" class="Symbol">_</a> <a id="6686" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6692" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="6694" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6696" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="6723" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6726" href="Algebra.Properties.BooleanAlgebra.html#2555" class="Function">∨-identityʳ</a> <a id="6738" class="Symbol">_</a> <a id="6740" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6746" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="6777" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="6782" href="Algebra.Properties.BooleanAlgebra.html#6782" class="Function">lem₃</a> <a id="6787" class="Symbol">=</a> <a id="6789" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="6799" class="Symbol">(</a><a id="6800" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6802" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6804" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="6805" class="Symbol">)</a> <a id="6807" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6809" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6811" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6822" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6825" href="Algebra.Structures.html#7054" class="Function">∨-∧-distribʳ</a> <a id="6838" class="Symbol">_</a> <a id="6840" class="Symbol">_</a> <a id="6842" class="Symbol">_</a> <a id="6844" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6850" class="Symbol">(</a><a id="6851" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6853" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6855" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6857" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="6858" class="Symbol">)</a> <a id="6860" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6862" class="Symbol">(</a><a id="6863" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6865" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6867" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6869" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="6870" class="Symbol">)</a> <a id="6873" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6876" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="6884" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="6886" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="6900" class="Symbol">_</a> <a id="6902" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6908" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="6910" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="6912" class="Symbol">(</a><a id="6913" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6915" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6917" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6919" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="6920" class="Symbol">)</a> <a id="6931" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6934" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="6946" class="Symbol">_</a> <a id="6948" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6954" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="6956" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="6958" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6960" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="6977" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="6980" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="6987" class="Symbol">_</a> <a id="6989" class="Symbol">_</a> <a id="6991" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="6997" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="6999" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7001" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7003" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="7020" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="7025" href="Algebra.Properties.BooleanAlgebra.html#7025" class="Function">lem₂</a> <a id="7030" class="Symbol">=</a> <a id="7032" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="7042" class="Symbol">(</a><a id="7043" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7045" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7047" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="7048" class="Symbol">)</a> <a id="7050" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7052" class="Symbol">(</a><a id="7053" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7055" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7057" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7059" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7061" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="7062" class="Symbol">)</a> <a id="7065" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="7069" href="Algebra.Structures.html#6258" class="Function">∨-assoc</a> <a id="7077" class="Symbol">_</a> <a id="7079" class="Symbol">_</a> <a id="7081" class="Symbol">_</a> <a id="7083" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="7089" class="Symbol">((</a><a id="7091" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7093" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7095" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="7096" class="Symbol">)</a> <a id="7098" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7100" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7102" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a><a id="7103" class="Symbol">)</a> <a id="7105" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7107" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7109" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="7112" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7115" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="7123" href="Algebra.Properties.BooleanAlgebra.html#6782" class="Function">lem₃</a> <a id="7128" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="7134" class="Symbol">(</a><a id="7135" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7137" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7139" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7141" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="7142" class="Symbol">)</a> <a id="7144" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7146" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7148" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="7157" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7160" href="Algebra.Structures.html#6258" class="Function">∨-assoc</a> <a id="7168" class="Symbol">_</a> <a id="7170" class="Symbol">_</a> <a id="7172" class="Symbol">_</a> <a id="7174" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="7180" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7182" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7184" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7186" class="Symbol">(</a><a id="7187" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a> <a id="7189" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7191" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7193" href="Algebra.Properties.BooleanAlgebra.html#6200" class="Bound">y</a><a id="7194" class="Symbol">)</a> <a id="7203" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7206" href="Algebra.Structures.html#6750" class="Function">∨-congˡ</a> <a id="7214" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="7216" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="7230" class="Symbol">_</a> <a id="7232" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="7238" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7240" href="Algebra.Properties.BooleanAlgebra.html#6198" class="Bound">x</a> <a id="7242" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7244" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="7261" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7264" href="Algebra.Properties.BooleanAlgebra.html#3224" class="Function">∨-zeroʳ</a> <a id="7272" class="Symbol">_</a> <a id="7274" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="7280" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="7303" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="deMorgan₂"></a><a id="7306" href="Algebra.Properties.BooleanAlgebra.html#7306" class="Function">deMorgan₂</a> <a id="7316" class="Symbol">:</a> <a id="7318" class="Symbol">∀</a> <a id="7320" href="Algebra.Properties.BooleanAlgebra.html#7320" class="Bound">x</a> <a id="7322" href="Algebra.Properties.BooleanAlgebra.html#7322" class="Bound">y</a> <a id="7324" class="Symbol">→</a> <a id="7326" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7328" class="Symbol">(</a><a id="7329" href="Algebra.Properties.BooleanAlgebra.html#7320" class="Bound">x</a> <a id="7331" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7333" href="Algebra.Properties.BooleanAlgebra.html#7322" class="Bound">y</a><a id="7334" class="Symbol">)</a> <a id="7336" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="7338" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7340" href="Algebra.Properties.BooleanAlgebra.html#7320" class="Bound">x</a> <a id="7342" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7344" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7346" href="Algebra.Properties.BooleanAlgebra.html#7322" class="Bound">y</a>
|
|||
|
|
<a id="7348" href="Algebra.Properties.BooleanAlgebra.html#7306" class="Function">deMorgan₂</a> <a id="7358" href="Algebra.Properties.BooleanAlgebra.html#7358" class="Bound">x</a> <a id="7360" href="Algebra.Properties.BooleanAlgebra.html#7360" class="Bound">y</a> <a id="7362" class="Symbol">=</a> <a id="7364" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="7372" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7374" class="Symbol">(</a><a id="7375" href="Algebra.Properties.BooleanAlgebra.html#7358" class="Bound">x</a> <a id="7377" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7379" href="Algebra.Properties.BooleanAlgebra.html#7360" class="Bound">y</a><a id="7380" class="Symbol">)</a> <a id="7391" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="7395" href="Algebra.Structures.html#16108" class="Function">¬-cong</a> <a id="7402" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="7404" class="Symbol">((</a><a id="7406" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="7419" class="Symbol">_)</a> <a id="7422" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="7424" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="7431" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="7433" class="Symbol">(</a><a id="7434" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="7447" class="Symbol">_))</a> <a id="7451" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="7455" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7457" class="Symbol">(</a><a id="7458" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7460" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7462" href="Algebra.Properties.BooleanAlgebra.html#7358" class="Bound">x</a> <a id="7464" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7466" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7468" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7470" href="Algebra.Properties.BooleanAlgebra.html#7360" class="Bound">y</a><a id="7471" class="Symbol">)</a> <a id="7474" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="7478" href="Algebra.Structures.html#16108" class="Function">¬-cong</a> <a id="7485" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="7487" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="7497" class="Symbol">_</a> <a id="7499" class="Symbol">_</a> <a id="7501" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="7505" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7507" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7509" class="Symbol">(</a><a id="7510" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7512" href="Algebra.Properties.BooleanAlgebra.html#7358" class="Bound">x</a> <a id="7514" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7516" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7518" href="Algebra.Properties.BooleanAlgebra.html#7360" class="Bound">y</a><a id="7519" class="Symbol">)</a> <a id="7524" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="7527" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="7540" class="Symbol">_</a> <a id="7542" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="7546" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7548" href="Algebra.Properties.BooleanAlgebra.html#7358" class="Bound">x</a> <a id="7550" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7552" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7554" href="Algebra.Properties.BooleanAlgebra.html#7360" class="Bound">y</a> <a id="7565" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="7568" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="7641" class="Comment">-- (⊕, ∧, id, ⊥, ⊤) is a commutative ring</a>
|
|||
|
|
|
|||
|
|
<a id="7684" class="Comment">-- This construction is parameterised over the definition of xor.</a>
|
|||
|
|
|
|||
|
|
<a id="7751" class="Keyword">module</a> <a id="XorRing"></a><a id="7758" href="Algebra.Properties.BooleanAlgebra.html#7758" class="Module">XorRing</a>
|
|||
|
|
<a id="7768" class="Symbol">(</a><a id="7769" href="Algebra.Properties.BooleanAlgebra.html#7769" class="Bound">xor</a> <a id="7773" class="Symbol">:</a> <a id="7775" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7779" href="Algebra.Bundles.html#23268" class="Field">Carrier</a><a id="7786" class="Symbol">)</a>
|
|||
|
|
<a id="7790" class="Symbol">(</a><a id="7791" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="7797" class="Symbol">:</a> <a id="7799" class="Symbol">∀</a> <a id="7801" href="Algebra.Properties.BooleanAlgebra.html#7801" class="Bound">x</a> <a id="7803" href="Algebra.Properties.BooleanAlgebra.html#7803" class="Bound">y</a> <a id="7805" class="Symbol">→</a> <a id="7807" href="Algebra.Properties.BooleanAlgebra.html#7769" class="Bound">xor</a> <a id="7811" href="Algebra.Properties.BooleanAlgebra.html#7801" class="Bound">x</a> <a id="7813" href="Algebra.Properties.BooleanAlgebra.html#7803" class="Bound">y</a> <a id="7815" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="7817" class="Symbol">(</a><a id="7818" href="Algebra.Properties.BooleanAlgebra.html#7801" class="Bound">x</a> <a id="7820" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="7822" href="Algebra.Properties.BooleanAlgebra.html#7803" class="Bound">y</a><a id="7823" class="Symbol">)</a> <a id="7825" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7827" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7829" class="Symbol">(</a><a id="7830" href="Algebra.Properties.BooleanAlgebra.html#7801" class="Bound">x</a> <a id="7832" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7834" href="Algebra.Properties.BooleanAlgebra.html#7803" class="Bound">y</a><a id="7835" class="Symbol">))</a>
|
|||
|
|
<a id="7840" class="Keyword">where</a>
|
|||
|
|
|
|||
|
|
<a id="7849" class="Keyword">private</a>
|
|||
|
|
<a id="7861" class="Keyword">infixl</a> <a id="7868" class="Number">6</a> <a id="7870" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing._⊕_"></a><a id="7879" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="7883" class="Symbol">:</a> <a id="7885" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="7889" href="Algebra.Bundles.html#23268" class="Field">Carrier</a>
|
|||
|
|
<a id="7901" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="7905" class="Symbol">=</a> <a id="7907" href="Algebra.Properties.BooleanAlgebra.html#7769" class="Bound">xor</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.helper"></a><a id="7916" href="Algebra.Properties.BooleanAlgebra.html#7916" class="Function">helper</a> <a id="7923" class="Symbol">:</a> <a id="7925" class="Symbol">∀</a> <a id="7927" class="Symbol">{</a><a id="7928" href="Algebra.Properties.BooleanAlgebra.html#7928" class="Bound">x</a> <a id="7930" href="Algebra.Properties.BooleanAlgebra.html#7930" class="Bound">y</a> <a id="7932" href="Algebra.Properties.BooleanAlgebra.html#7932" class="Bound">u</a> <a id="7934" href="Algebra.Properties.BooleanAlgebra.html#7934" class="Bound">v</a><a id="7935" class="Symbol">}</a> <a id="7937" class="Symbol">→</a> <a id="7939" href="Algebra.Properties.BooleanAlgebra.html#7928" class="Bound">x</a> <a id="7941" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="7943" href="Algebra.Properties.BooleanAlgebra.html#7930" class="Bound">y</a> <a id="7945" class="Symbol">→</a> <a id="7947" href="Algebra.Properties.BooleanAlgebra.html#7932" class="Bound">u</a> <a id="7949" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="7951" href="Algebra.Properties.BooleanAlgebra.html#7934" class="Bound">v</a> <a id="7953" class="Symbol">→</a> <a id="7955" href="Algebra.Properties.BooleanAlgebra.html#7928" class="Bound">x</a> <a id="7957" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7959" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7961" href="Algebra.Properties.BooleanAlgebra.html#7932" class="Bound">u</a> <a id="7963" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="7965" href="Algebra.Properties.BooleanAlgebra.html#7930" class="Bound">y</a> <a id="7967" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="7969" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="7971" href="Algebra.Properties.BooleanAlgebra.html#7934" class="Bound">v</a>
|
|||
|
|
<a id="7977" href="Algebra.Properties.BooleanAlgebra.html#7916" class="Function">helper</a> <a id="7984" href="Algebra.Properties.BooleanAlgebra.html#7984" class="Bound">x≈y</a> <a id="7988" href="Algebra.Properties.BooleanAlgebra.html#7988" class="Bound">u≈v</a> <a id="7992" class="Symbol">=</a> <a id="7994" href="Algebra.Properties.BooleanAlgebra.html#7984" class="Bound">x≈y</a> <a id="7998" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="8000" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="8007" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="8009" href="Algebra.Structures.html#16108" class="Function">¬-cong</a> <a id="8016" href="Algebra.Properties.BooleanAlgebra.html#7988" class="Bound">u≈v</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-cong"></a><a id="8023" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a> <a id="8030" class="Symbol">:</a> <a id="8032" href="Algebra.Definitions.html#871" class="Function">Congruent₂</a> <a id="8043" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
|
|
<a id="8049" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a> <a id="8056" class="Symbol">{</a><a id="8057" href="Algebra.Properties.BooleanAlgebra.html#8057" class="Bound">x</a><a id="8058" class="Symbol">}</a> <a id="8060" class="Symbol">{</a><a id="8061" href="Algebra.Properties.BooleanAlgebra.html#8061" class="Bound">y</a><a id="8062" class="Symbol">}</a> <a id="8064" class="Symbol">{</a><a id="8065" href="Algebra.Properties.BooleanAlgebra.html#8065" class="Bound">u</a><a id="8066" class="Symbol">}</a> <a id="8068" class="Symbol">{</a><a id="8069" href="Algebra.Properties.BooleanAlgebra.html#8069" class="Bound">v</a><a id="8070" class="Symbol">}</a> <a id="8072" href="Algebra.Properties.BooleanAlgebra.html#8072" class="Bound">x≈y</a> <a id="8076" href="Algebra.Properties.BooleanAlgebra.html#8076" class="Bound">u≈v</a> <a id="8080" class="Symbol">=</a> <a id="8082" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="8092" href="Algebra.Properties.BooleanAlgebra.html#8057" class="Bound">x</a> <a id="8094" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="8096" href="Algebra.Properties.BooleanAlgebra.html#8065" class="Bound">u</a> <a id="8113" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="8117" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="8123" class="Symbol">_</a> <a id="8125" class="Symbol">_</a> <a id="8127" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="9269" class="Keyword">where</a>
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<a id="9343" href="Algebra.Properties.BooleanAlgebra.html#9325" class="Bound">x</a> <a id="9345" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="9347" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9349" class="Symbol">(</a><a id="9350" href="Algebra.Properties.BooleanAlgebra.html#9325" class="Bound">x</a> <a id="9352" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="9354" href="Algebra.Properties.BooleanAlgebra.html#9327" class="Bound">y</a><a id="9355" class="Symbol">)</a> <a id="9366" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="9369" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="9377" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="9379" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="9389" class="Symbol">_</a> <a id="9391" class="Symbol">_</a> <a id="9393" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="9514" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="9516" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="9518" class="Symbol">(</a><a id="9519" href="Algebra.Properties.BooleanAlgebra.html#9325" class="Bound">x</a> <a id="9521" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="9523" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9525" href="Algebra.Properties.BooleanAlgebra.html#9327" class="Bound">y</a><a id="9526" class="Symbol">)</a> <a id="9537" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="9540" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="9552" class="Symbol">_</a> <a id="9554" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="9562" href="Algebra.Properties.BooleanAlgebra.html#9325" class="Bound">x</a> <a id="9564" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="9566" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9568" href="Algebra.Properties.BooleanAlgebra.html#9327" class="Bound">y</a> <a id="9585" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="XorRing.¬-distribʳ-⊕"></a><a id="9590" href="Algebra.Properties.BooleanAlgebra.html#9590" class="Function">¬-distribʳ-⊕</a> <a id="9603" class="Symbol">:</a> <a id="9605" class="Symbol">∀</a> <a id="9607" href="Algebra.Properties.BooleanAlgebra.html#9607" class="Bound">x</a> <a id="9609" href="Algebra.Properties.BooleanAlgebra.html#9609" class="Bound">y</a> <a id="9611" class="Symbol">→</a> <a id="9613" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9615" class="Symbol">(</a><a id="9616" href="Algebra.Properties.BooleanAlgebra.html#9607" class="Bound">x</a> <a id="9618" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9620" href="Algebra.Properties.BooleanAlgebra.html#9609" class="Bound">y</a><a id="9621" class="Symbol">)</a> <a id="9623" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="9625" href="Algebra.Properties.BooleanAlgebra.html#9607" class="Bound">x</a> <a id="9627" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9629" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9631" href="Algebra.Properties.BooleanAlgebra.html#9609" class="Bound">y</a>
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<a id="9635" href="Algebra.Properties.BooleanAlgebra.html#9590" class="Function">¬-distribʳ-⊕</a> <a id="9648" href="Algebra.Properties.BooleanAlgebra.html#9648" class="Bound">x</a> <a id="9650" href="Algebra.Properties.BooleanAlgebra.html#9650" class="Bound">y</a> <a id="9652" class="Symbol">=</a> <a id="9654" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="9704" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9706" class="Symbol">(</a><a id="9707" href="Algebra.Properties.BooleanAlgebra.html#9650" class="Bound">y</a> <a id="9709" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9711" href="Algebra.Properties.BooleanAlgebra.html#9648" class="Bound">x</a><a id="9712" class="Symbol">)</a> <a id="9715" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="9718" href="Algebra.Properties.BooleanAlgebra.html#8538" class="Function">¬-distribˡ-⊕</a> <a id="9731" class="Symbol">_</a> <a id="9733" class="Symbol">_</a> <a id="9735" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="9772" href="Algebra.Properties.BooleanAlgebra.html#9648" class="Bound">x</a> <a id="9774" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9776" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9778" href="Algebra.Properties.BooleanAlgebra.html#9650" class="Bound">y</a> <a id="9783" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="XorRing.⊕-annihilates-¬"></a><a id="9788" href="Algebra.Properties.BooleanAlgebra.html#9788" class="Function">⊕-annihilates-¬</a> <a id="9804" class="Symbol">:</a> <a id="9806" class="Symbol">∀</a> <a id="9808" href="Algebra.Properties.BooleanAlgebra.html#9808" class="Bound">x</a> <a id="9810" href="Algebra.Properties.BooleanAlgebra.html#9810" class="Bound">y</a> <a id="9812" class="Symbol">→</a> <a id="9814" href="Algebra.Properties.BooleanAlgebra.html#9808" class="Bound">x</a> <a id="9816" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9818" href="Algebra.Properties.BooleanAlgebra.html#9810" class="Bound">y</a> <a id="9820" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="9822" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9824" href="Algebra.Properties.BooleanAlgebra.html#9808" class="Bound">x</a> <a id="9826" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9828" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9830" href="Algebra.Properties.BooleanAlgebra.html#9810" class="Bound">y</a>
|
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<a id="9834" href="Algebra.Properties.BooleanAlgebra.html#9788" class="Function">⊕-annihilates-¬</a> <a id="9850" href="Algebra.Properties.BooleanAlgebra.html#9850" class="Bound">x</a> <a id="9852" href="Algebra.Properties.BooleanAlgebra.html#9852" class="Bound">y</a> <a id="9854" class="Symbol">=</a> <a id="9856" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="9866" href="Algebra.Properties.BooleanAlgebra.html#9850" class="Bound">x</a> <a id="9868" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9870" href="Algebra.Properties.BooleanAlgebra.html#9852" class="Bound">y</a> <a id="9879" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="9883" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="9896" class="Symbol">_</a> <a id="9898" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="9904" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9906" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9908" class="Symbol">(</a><a id="9909" href="Algebra.Properties.BooleanAlgebra.html#9850" class="Bound">x</a> <a id="9911" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9913" href="Algebra.Properties.BooleanAlgebra.html#9852" class="Bound">y</a><a id="9914" class="Symbol">)</a> <a id="9917" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="9921" href="Algebra.Structures.html#16108" class="Function">¬-cong</a> <a id="9928" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="9930" href="Algebra.Properties.BooleanAlgebra.html#8538" class="Function">¬-distribˡ-⊕</a> <a id="9943" class="Symbol">_</a> <a id="9945" class="Symbol">_</a> <a id="9947" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="9953" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9955" class="Symbol">(</a><a id="9956" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="9958" href="Algebra.Properties.BooleanAlgebra.html#9850" class="Bound">x</a> <a id="9960" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="9962" href="Algebra.Properties.BooleanAlgebra.html#9852" class="Bound">y</a><a id="9963" class="Symbol">)</a> <a id="9966" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="9970" href="Algebra.Properties.BooleanAlgebra.html#9590" class="Function">¬-distribʳ-⊕</a> <a id="9983" class="Symbol">_</a> <a id="9985" class="Symbol">_</a> <a id="9987" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="10046" href="Algebra.Properties.BooleanAlgebra.html#10011" class="Function">⊕-identityˡ</a> <a id="10058" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a> <a id="10060" class="Symbol">=</a> <a id="10062" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="10072" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10074" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="10076" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a> <a id="10093" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10096" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="10102" class="Symbol">_</a> <a id="10104" class="Symbol">_</a> <a id="10106" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="10112" class="Symbol">(</a><a id="10113" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10115" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="10117" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a><a id="10118" class="Symbol">)</a> <a id="10120" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10122" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="10124" class="Symbol">(</a><a id="10125" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10127" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10129" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a><a id="10130" class="Symbol">)</a> <a id="10133" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10136" href="Algebra.Properties.BooleanAlgebra.html#7916" class="Function">helper</a> <a id="10143" class="Symbol">(</a><a id="10144" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="10156" class="Symbol">_)</a> <a id="10159" class="Symbol">(</a><a id="10160" href="Algebra.Properties.BooleanAlgebra.html#3112" class="Function">∧-zeroˡ</a> <a id="10168" class="Symbol">_)</a> <a id="10171" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="10177" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a> <a id="10179" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10181" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="10183" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10198" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10201" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="10209" href="Algebra.Properties.BooleanAlgebra.html#5931" class="Function">⊥≉⊤</a> <a id="10213" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="10219" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a> <a id="10221" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10223" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="10240" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10243" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="10255" class="Symbol">_</a> <a id="10257" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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<a id="10263" href="Algebra.Properties.BooleanAlgebra.html#10058" class="Bound">x</a> <a id="10284" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
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<a id="XorRing.⊕-identityʳ"></a><a id="10289" href="Algebra.Properties.BooleanAlgebra.html#10289" class="Function">⊕-identityʳ</a> <a id="10301" class="Symbol">:</a> <a id="10303" href="Algebra.Definitions.html#1350" class="Function">RightIdentity</a> <a id="10317" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10319" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
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<a id="10325" href="Algebra.Properties.BooleanAlgebra.html#10289" class="Function">⊕-identityʳ</a> <a id="10337" class="Symbol">_</a> <a id="10339" class="Symbol">=</a> <a id="10341" href="Algebra.Properties.BooleanAlgebra.html#8316" class="Function">⊕-comm</a> <a id="10348" class="Symbol">_</a> <a id="10350" class="Symbol">_</a> <a id="10352" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="10354" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="10360" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="10362" href="Algebra.Properties.BooleanAlgebra.html#10011" class="Function">⊕-identityˡ</a> <a id="10374" class="Symbol">_</a>
|
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<a id="XorRing.⊕-identity"></a><a id="10379" href="Algebra.Properties.BooleanAlgebra.html#10379" class="Function">⊕-identity</a> <a id="10390" class="Symbol">:</a> <a id="10392" href="Algebra.Definitions.html#1425" class="Function">Identity</a> <a id="10401" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10403" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
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<a id="10409" href="Algebra.Properties.BooleanAlgebra.html#10379" class="Function">⊕-identity</a> <a id="10420" class="Symbol">=</a> <a id="10422" href="Algebra.Properties.BooleanAlgebra.html#10011" class="Function">⊕-identityˡ</a> <a id="10434" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10436" href="Algebra.Properties.BooleanAlgebra.html#10289" class="Function">⊕-identityʳ</a>
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<a id="XorRing.⊕-inverseˡ"></a><a id="10451" href="Algebra.Properties.BooleanAlgebra.html#10451" class="Function">⊕-inverseˡ</a> <a id="10462" class="Symbol">:</a> <a id="10464" href="Algebra.Definitions.html#1713" class="Function">LeftInverse</a> <a id="10476" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10478" href="Function.Base.html#624" class="Function">id</a> <a id="10481" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
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<a id="10487" href="Algebra.Properties.BooleanAlgebra.html#10451" class="Function">⊕-inverseˡ</a> <a id="10498" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10500" class="Symbol">=</a> <a id="10502" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
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<a id="10512" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10514" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="10516" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10532" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10535" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="10541" class="Symbol">_</a> <a id="10543" class="Symbol">_</a> <a id="10545" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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<a id="10551" class="Symbol">(</a><a id="10552" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10554" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="10556" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a><a id="10557" class="Symbol">)</a> <a id="10559" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10561" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="10563" class="Symbol">(</a><a id="10564" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10566" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10568" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a><a id="10569" class="Symbol">)</a> <a id="10571" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10574" href="Algebra.Properties.BooleanAlgebra.html#7916" class="Function">helper</a> <a id="10581" class="Symbol">(</a><a id="10582" href="Algebra.Properties.Lattice.html#5242" class="Function">∨-idempotent</a> <a id="10595" class="Symbol">_)</a> <a id="10598" class="Symbol">(</a><a id="10599" href="Algebra.Properties.Lattice.html#5107" class="Function">∧-idempotent</a> <a id="10612" class="Symbol">_)</a> <a id="10615" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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<a id="10621" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10623" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10625" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="10627" href="Algebra.Properties.BooleanAlgebra.html#10498" class="Bound">x</a> <a id="10641" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10644" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="10658" class="Symbol">_</a> <a id="10660" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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<a id="10666" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10686" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
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<a id="XorRing.⊕-inverseʳ"></a><a id="10691" href="Algebra.Properties.BooleanAlgebra.html#10691" class="Function">⊕-inverseʳ</a> <a id="10702" class="Symbol">:</a> <a id="10704" href="Algebra.Definitions.html#1801" class="Function">RightInverse</a> <a id="10717" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10719" href="Function.Base.html#624" class="Function">id</a> <a id="10722" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
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<a id="10728" href="Algebra.Properties.BooleanAlgebra.html#10691" class="Function">⊕-inverseʳ</a> <a id="10739" class="Symbol">_</a> <a id="10741" class="Symbol">=</a> <a id="10743" href="Algebra.Properties.BooleanAlgebra.html#8316" class="Function">⊕-comm</a> <a id="10750" class="Symbol">_</a> <a id="10752" class="Symbol">_</a> <a id="10754" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="10756" href="Relation.Binary.Structures.html#1629" class="Function">trans</a> <a id="10762" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="10764" href="Algebra.Properties.BooleanAlgebra.html#10451" class="Function">⊕-inverseˡ</a> <a id="10775" class="Symbol">_</a>
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<a id="XorRing.⊕-inverse"></a><a id="10780" href="Algebra.Properties.BooleanAlgebra.html#10780" class="Function">⊕-inverse</a> <a id="10790" class="Symbol">:</a> <a id="10792" href="Algebra.Definitions.html#1891" class="Function">Inverse</a> <a id="10800" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="10802" href="Function.Base.html#624" class="Function">id</a> <a id="10805" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
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<a id="10811" href="Algebra.Properties.BooleanAlgebra.html#10780" class="Function">⊕-inverse</a> <a id="10821" class="Symbol">=</a> <a id="10823" href="Algebra.Properties.BooleanAlgebra.html#10451" class="Function">⊕-inverseˡ</a> <a id="10834" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="10836" href="Algebra.Properties.BooleanAlgebra.html#10691" class="Function">⊕-inverseʳ</a>
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<a id="XorRing.∧-distribˡ-⊕"></a><a id="10850" href="Algebra.Properties.BooleanAlgebra.html#10850" class="Function">∧-distribˡ-⊕</a> <a id="10863" class="Symbol">:</a> <a id="10865" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="10869" href="Algebra.Definitions.html#2236" class="Function Operator">DistributesOverˡ</a> <a id="10886" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
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<a id="10892" href="Algebra.Properties.BooleanAlgebra.html#10850" class="Function">∧-distribˡ-⊕</a> <a id="10905" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="10907" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="10909" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a> <a id="10911" class="Symbol">=</a> <a id="10913" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
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<a id="10923" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="10925" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10927" class="Symbol">(</a><a id="10928" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="10930" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="10932" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="10933" class="Symbol">)</a> <a id="10950" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="10953" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="10961" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="10963" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="10969" class="Symbol">_</a> <a id="10971" class="Symbol">_</a> <a id="10973" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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<a id="10979" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="10981" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10983" class="Symbol">((</a><a id="10985" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="10987" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="10989" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="10990" class="Symbol">)</a> <a id="10992" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="10994" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="10996" class="Symbol">(</a><a id="10997" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="10999" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11001" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11002" class="Symbol">))</a> <a id="11006" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11010" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="11018" class="Symbol">_</a> <a id="11020" class="Symbol">_</a> <a id="11022" class="Symbol">_</a> <a id="11024" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="11030" class="Symbol">(</a><a id="11031" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11033" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11035" class="Symbol">(</a><a id="11036" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11038" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11040" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11041" class="Symbol">))</a> <a id="11044" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11046" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11048" class="Symbol">(</a><a id="11049" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11051" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11053" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11054" class="Symbol">)</a> <a id="11057" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11060" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="11068" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="11070" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="11080" class="Symbol">_</a> <a id="11082" class="Symbol">_</a> <a id="11084" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="11090" class="Symbol">(</a><a id="11091" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11093" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11095" class="Symbol">(</a><a id="11096" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11098" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11100" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11101" class="Symbol">))</a> <a id="11104" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11110" class="Symbol">(</a><a id="11111" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11113" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11115" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11117" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11119" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11120" class="Symbol">)</a> <a id="11137" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11141" href="Algebra.Properties.BooleanAlgebra.html#2723" class="Function">∨-identityˡ</a> <a id="11153" class="Symbol">_</a> <a id="11155" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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<a id="11161" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="11163" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a>
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<a id="11169" class="Symbol">((</a><a id="11171" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11173" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11175" class="Symbol">(</a><a id="11176" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11178" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11180" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11181" class="Symbol">))</a> <a id="11184" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11190" class="Symbol">(</a><a id="11191" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11193" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11195" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11197" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11199" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11200" class="Symbol">))</a> <a id="11217" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11220" href="Algebra.Structures.html#6811" class="Function">∨-congʳ</a> <a id="11228" href="Algebra.Properties.BooleanAlgebra.html#12210" class="Function">lem₃</a> <a id="11233" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="11239" class="Symbol">((</a><a id="11241" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11243" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11245" class="Symbol">(</a><a id="11246" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11248" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11250" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11251" class="Symbol">))</a> <a id="11254" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11256" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11258" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a><a id="11259" class="Symbol">)</a> <a id="11261" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a>
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<a id="11267" class="Symbol">((</a><a id="11269" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11271" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11273" class="Symbol">(</a><a id="11274" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11276" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11278" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11279" class="Symbol">))</a> <a id="11282" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11288" class="Symbol">(</a><a id="11289" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11291" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11293" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11295" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11297" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11298" class="Symbol">))</a> <a id="11315" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11319" href="Algebra.Properties.DistributiveLattice.html#3159" class="Function">∧-∨-distribˡ</a> <a id="11332" class="Symbol">_</a> <a id="11334" class="Symbol">_</a> <a id="11336" class="Symbol">_</a> <a id="11338" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="11344" class="Symbol">(</a><a id="11345" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11347" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11349" class="Symbol">(</a><a id="11350" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11352" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11354" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11355" class="Symbol">))</a> <a id="11358" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11364" class="Symbol">(</a><a id="11365" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11367" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11369" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11371" class="Symbol">(</a><a id="11372" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11374" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11376" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11378" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11380" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11381" class="Symbol">))</a> <a id="11391" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11395" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="11403" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="11405" href="Algebra.Structures.html#6750" class="Function">∨-congˡ</a> <a id="11413" class="Symbol">(</a><a id="11414" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="11424" class="Symbol">_</a> <a id="11426" class="Symbol">_)</a> <a id="11429" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="11435" class="Symbol">(</a><a id="11436" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11438" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11440" class="Symbol">(</a><a id="11441" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11443" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11445" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11446" class="Symbol">))</a> <a id="11449" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11455" class="Symbol">(</a><a id="11456" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11458" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11460" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11462" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11464" class="Symbol">(</a><a id="11465" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11467" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11469" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11470" class="Symbol">))</a> <a id="11482" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11486" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="11494" class="Symbol">(</a><a id="11495" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="11505" class="Symbol">_</a> <a id="11507" class="Symbol">_)</a> <a id="11510" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="11516" class="Symbol">(</a><a id="11517" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11519" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11521" class="Symbol">(</a><a id="11522" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11524" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11526" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11527" class="Symbol">))</a> <a id="11530" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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|
<a id="11536" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11538" class="Symbol">(</a><a id="11539" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11541" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11543" class="Symbol">(</a><a id="11544" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11546" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11548" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11549" class="Symbol">))</a> <a id="11563" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11566" href="Algebra.Properties.BooleanAlgebra.html#7916" class="Function">helper</a> <a id="11573" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="11578" href="Algebra.Properties.BooleanAlgebra.html#11965" class="Function">lem₁</a> <a id="11583" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="11589" class="Symbol">(</a><a id="11590" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11592" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11594" class="Symbol">(</a><a id="11595" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11597" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11599" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11600" class="Symbol">))</a> <a id="11603" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11609" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11611" class="Symbol">((</a><a id="11613" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11615" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11617" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="11618" class="Symbol">)</a> <a id="11620" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11622" class="Symbol">(</a><a id="11623" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11625" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11627" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11628" class="Symbol">))</a> <a id="11636" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11639" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="11647" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="11649" href="Algebra.Properties.DistributiveLattice.html#3159" class="Function">∧-∨-distribˡ</a> <a id="11662" class="Symbol">_</a> <a id="11664" class="Symbol">_</a> <a id="11666" class="Symbol">_</a> <a id="11668" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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|
<a id="11674" class="Symbol">((</a><a id="11676" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11678" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11680" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="11681" class="Symbol">)</a> <a id="11683" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="11685" class="Symbol">(</a><a id="11686" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11688" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11690" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11691" class="Symbol">))</a> <a id="11694" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="11700" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="11702" class="Symbol">((</a><a id="11704" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11706" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11708" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="11709" class="Symbol">)</a> <a id="11711" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11713" class="Symbol">(</a><a id="11714" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11716" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11718" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11719" class="Symbol">))</a> <a id="11727" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11731" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="11737" class="Symbol">_</a> <a id="11739" class="Symbol">_</a> <a id="11741" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="11747" class="Symbol">(</a><a id="11748" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11750" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11752" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="11753" class="Symbol">)</a> <a id="11755" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="11757" class="Symbol">(</a><a id="11758" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11760" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11762" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11763" class="Symbol">)</a> <a id="11774" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
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|
<a id="11782" class="Keyword">where</a>
|
|||
|
|
<a id="11794" href="Algebra.Properties.BooleanAlgebra.html#11794" class="Function">lem₂</a> <a id="11799" class="Symbol">=</a> <a id="11801" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
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<a id="11815" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11817" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11819" class="Symbol">(</a><a id="11820" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11822" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11824" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11825" class="Symbol">)</a> <a id="11828" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="11832" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="11840" class="Symbol">_</a> <a id="11842" class="Symbol">_</a> <a id="11844" class="Symbol">_</a> <a id="11846" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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|
<a id="11856" class="Symbol">(</a><a id="11857" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11859" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11861" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="11862" class="Symbol">)</a> <a id="11864" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11866" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a> <a id="11869" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11872" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="11880" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="11882" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="11889" class="Symbol">_</a> <a id="11891" class="Symbol">_</a> <a id="11893" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="11903" class="Symbol">(</a><a id="11904" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11906" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11908" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a><a id="11909" class="Symbol">)</a> <a id="11911" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11913" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a> <a id="11916" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="11919" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="11927" class="Symbol">_</a> <a id="11929" class="Symbol">_</a> <a id="11931" class="Symbol">_</a> <a id="11933" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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|
|
<a id="11943" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11945" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11947" class="Symbol">(</a><a id="11948" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11950" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11952" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11953" class="Symbol">)</a> <a id="11956" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
|
|
|||
|
|
<a id="11965" href="Algebra.Properties.BooleanAlgebra.html#11965" class="Function">lem₁</a> <a id="11970" class="Symbol">=</a> <a id="11972" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
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|
<a id="11986" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="11988" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11990" class="Symbol">(</a><a id="11991" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="11993" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="11995" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="11996" class="Symbol">)</a> <a id="12005" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="12009" href="Algebra.Structures.html#6688" class="Function">∧-congʳ</a> <a id="12017" class="Symbol">(</a><a id="12018" href="Algebra.Properties.Lattice.html#5107" class="Function">∧-idempotent</a> <a id="12031" class="Symbol">_)</a> <a id="12034" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
|
|
<a id="12044" class="Symbol">(</a><a id="12045" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12047" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12049" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a><a id="12050" class="Symbol">)</a> <a id="12052" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12054" class="Symbol">(</a><a id="12055" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12057" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12059" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12060" class="Symbol">)</a> <a id="12063" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="12066" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="12074" class="Symbol">_</a> <a id="12076" class="Symbol">_</a> <a id="12078" class="Symbol">_</a> <a id="12080" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
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|
<a id="12090" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12092" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12094" class="Symbol">(</a><a id="12095" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12097" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12099" class="Symbol">(</a><a id="12100" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12102" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12104" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12105" class="Symbol">))</a> <a id="12109" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="12112" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="12120" href="Algebra.Properties.BooleanAlgebra.html#11794" class="Function">lem₂</a> <a id="12125" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="12135" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12137" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12139" class="Symbol">(</a><a id="12140" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12142" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12144" class="Symbol">(</a><a id="12145" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12147" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12149" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12150" class="Symbol">))</a> <a id="12154" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="12158" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="12166" class="Symbol">_</a> <a id="12168" class="Symbol">_</a> <a id="12170" class="Symbol">_</a> <a id="12172" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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<a id="12182" class="Symbol">(</a><a id="12183" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12185" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12187" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a><a id="12188" class="Symbol">)</a> <a id="12190" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12192" class="Symbol">(</a><a id="12193" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12195" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12197" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12198" class="Symbol">)</a> <a id="12201" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="12210" href="Algebra.Properties.BooleanAlgebra.html#12210" class="Function">lem₃</a> <a id="12215" class="Symbol">=</a> <a id="12217" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
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<a id="12231" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="12254" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="12258" href="Algebra.Properties.BooleanAlgebra.html#2871" class="Function">∧-zeroʳ</a> <a id="12266" class="Symbol">_</a> <a id="12268" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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<a id="12278" class="Symbol">(</a><a id="12279" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12281" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12283" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12284" class="Symbol">)</a> <a id="12286" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12288" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="12301" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="12305" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="12313" class="Symbol">(</a><a id="12314" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="12328" class="Symbol">_)</a> <a id="12331" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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<a id="12341" class="Symbol">(</a><a id="12342" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12344" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12346" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12347" class="Symbol">)</a> <a id="12349" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12351" class="Symbol">(</a><a id="12352" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12354" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12356" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="12358" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a><a id="12359" class="Symbol">)</a> <a id="12364" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="12368" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="12376" class="Symbol">_</a> <a id="12378" class="Symbol">_</a> <a id="12380" class="Symbol">_</a> <a id="12382" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
|||
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<a id="12392" class="Symbol">((</a><a id="12394" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12396" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12398" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12399" class="Symbol">)</a> <a id="12401" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12403" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a><a id="12404" class="Symbol">)</a> <a id="12406" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12408" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="12410" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12415" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="12418" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="12425" class="Symbol">_</a> <a id="12427" class="Symbol">_</a> <a id="12429" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="12431" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="12438" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="12440" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="12446" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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|
<a id="12456" class="Symbol">(</a><a id="12457" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12459" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12461" class="Symbol">(</a><a id="12462" href="Algebra.Properties.BooleanAlgebra.html#10907" class="Bound">y</a> <a id="12464" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12466" href="Algebra.Properties.BooleanAlgebra.html#10909" class="Bound">z</a><a id="12467" class="Symbol">))</a> <a id="12470" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12472" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="12474" href="Algebra.Properties.BooleanAlgebra.html#10905" class="Bound">x</a> <a id="12479" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
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|
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<a id="XorRing.∧-distribʳ-⊕"></a><a id="12484" href="Algebra.Properties.BooleanAlgebra.html#12484" class="Function">∧-distribʳ-⊕</a> <a id="12497" class="Symbol">:</a> <a id="12499" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="12503" href="Algebra.Definitions.html#2355" class="Function Operator">DistributesOverʳ</a> <a id="12520" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
|
|
<a id="12526" href="Algebra.Properties.BooleanAlgebra.html#12484" class="Function">∧-distribʳ-⊕</a> <a id="12539" class="Symbol">=</a> <a id="12541" href="Algebra.Consequences.Setoid.html#4794" class="Function">comm+distrˡ⇒distrʳ</a> <a id="12560" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a> <a id="12567" href="Algebra.Structures.html#6325" class="Function">∧-comm</a> <a id="12574" href="Algebra.Properties.BooleanAlgebra.html#10850" class="Function">∧-distribˡ-⊕</a>
|
|||
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<a id="XorRing.∧-distrib-⊕"></a><a id="12590" href="Algebra.Properties.BooleanAlgebra.html#12590" class="Function">∧-distrib-⊕</a> <a id="12602" class="Symbol">:</a> <a id="12604" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="12608" href="Algebra.Definitions.html#2474" class="Function Operator">DistributesOver</a> <a id="12624" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
|
|
<a id="12630" href="Algebra.Properties.BooleanAlgebra.html#12590" class="Function">∧-distrib-⊕</a> <a id="12642" class="Symbol">=</a> <a id="12644" href="Algebra.Properties.BooleanAlgebra.html#10850" class="Function">∧-distribˡ-⊕</a> <a id="12657" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="12659" href="Algebra.Properties.BooleanAlgebra.html#12484" class="Function">∧-distribʳ-⊕</a>
|
|||
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|||
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<a id="12675" class="Keyword">private</a>
|
|||
|
|
|
|||
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<a id="XorRing.lemma₂"></a><a id="12688" href="Algebra.Properties.BooleanAlgebra.html#12688" class="Function">lemma₂</a> <a id="12695" class="Symbol">:</a> <a id="12697" class="Symbol">∀</a> <a id="12699" href="Algebra.Properties.BooleanAlgebra.html#12699" class="Bound">x</a> <a id="12701" href="Algebra.Properties.BooleanAlgebra.html#12701" class="Bound">y</a> <a id="12703" href="Algebra.Properties.BooleanAlgebra.html#12703" class="Bound">u</a> <a id="12705" href="Algebra.Properties.BooleanAlgebra.html#12705" class="Bound">v</a> <a id="12707" class="Symbol">→</a>
|
|||
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<a id="12722" class="Symbol">(</a><a id="12723" href="Algebra.Properties.BooleanAlgebra.html#12699" class="Bound">x</a> <a id="12725" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12727" href="Algebra.Properties.BooleanAlgebra.html#12701" class="Bound">y</a><a id="12728" class="Symbol">)</a> <a id="12730" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12732" class="Symbol">(</a><a id="12733" href="Algebra.Properties.BooleanAlgebra.html#12703" class="Bound">u</a> <a id="12735" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12737" href="Algebra.Properties.BooleanAlgebra.html#12705" class="Bound">v</a><a id="12738" class="Symbol">)</a> <a id="12740" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a>
|
|||
|
|
<a id="12755" class="Symbol">((</a><a id="12757" href="Algebra.Properties.BooleanAlgebra.html#12699" class="Bound">x</a> <a id="12759" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12761" href="Algebra.Properties.BooleanAlgebra.html#12703" class="Bound">u</a><a id="12762" class="Symbol">)</a> <a id="12764" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12766" class="Symbol">(</a><a id="12767" href="Algebra.Properties.BooleanAlgebra.html#12701" class="Bound">y</a> <a id="12769" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12771" href="Algebra.Properties.BooleanAlgebra.html#12703" class="Bound">u</a><a id="12772" class="Symbol">))</a> <a id="12775" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
|
|||
|
|
<a id="12790" class="Symbol">((</a><a id="12792" href="Algebra.Properties.BooleanAlgebra.html#12699" class="Bound">x</a> <a id="12794" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12796" href="Algebra.Properties.BooleanAlgebra.html#12705" class="Bound">v</a><a id="12797" class="Symbol">)</a> <a id="12799" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12801" class="Symbol">(</a><a id="12802" href="Algebra.Properties.BooleanAlgebra.html#12701" class="Bound">y</a> <a id="12804" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12806" href="Algebra.Properties.BooleanAlgebra.html#12705" class="Bound">v</a><a id="12807" class="Symbol">))</a>
|
|||
|
|
<a id="12814" href="Algebra.Properties.BooleanAlgebra.html#12688" class="Function">lemma₂</a> <a id="12821" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="12823" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a> <a id="12825" href="Algebra.Properties.BooleanAlgebra.html#12825" class="Bound">u</a> <a id="12827" href="Algebra.Properties.BooleanAlgebra.html#12827" class="Bound">v</a> <a id="12829" class="Symbol">=</a> <a id="12831" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
|
|
<a id="12845" class="Symbol">(</a><a id="12846" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="12848" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12850" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a><a id="12851" class="Symbol">)</a> <a id="12853" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12855" class="Symbol">(</a><a id="12856" href="Algebra.Properties.BooleanAlgebra.html#12825" class="Bound">u</a> <a id="12858" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12860" href="Algebra.Properties.BooleanAlgebra.html#12827" class="Bound">v</a><a id="12861" class="Symbol">)</a> <a id="12876" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="12879" href="Algebra.Properties.DistributiveLattice.html#2870" class="Function">∨-∧-distribˡ</a> <a id="12892" class="Symbol">_</a> <a id="12894" class="Symbol">_</a> <a id="12896" class="Symbol">_</a> <a id="12898" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="12908" class="Symbol">((</a><a id="12910" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="12912" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12914" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a><a id="12915" class="Symbol">)</a> <a id="12917" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12919" href="Algebra.Properties.BooleanAlgebra.html#12825" class="Bound">u</a><a id="12920" class="Symbol">)</a> <a id="12922" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12924" class="Symbol">((</a><a id="12926" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="12928" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="12930" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a><a id="12931" class="Symbol">)</a> <a id="12933" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="12935" href="Algebra.Properties.BooleanAlgebra.html#12827" class="Bound">v</a><a id="12936" class="Symbol">)</a> <a id="12939" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="12942" href="Algebra.Structures.html#7054" class="Function">∨-∧-distribʳ</a> <a id="12955" class="Symbol">_</a> <a id="12957" class="Symbol">_</a> <a id="12959" class="Symbol">_</a>
|
|||
|
|
<a id="13005" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="13007" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="13014" href="Function.Base.html#4407" class="Function Operator">⟩</a>
|
|||
|
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<a id="13058" href="Algebra.Structures.html#7054" class="Function">∨-∧-distribʳ</a> <a id="13071" class="Symbol">_</a> <a id="13073" class="Symbol">_</a> <a id="13075" class="Symbol">_</a> <a id="13077" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
|
|
<a id="13087" class="Symbol">((</a><a id="13089" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="13091" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13093" href="Algebra.Properties.BooleanAlgebra.html#12825" class="Bound">u</a><a id="13094" class="Symbol">)</a> <a id="13096" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13098" class="Symbol">(</a><a id="13099" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a> <a id="13101" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13103" href="Algebra.Properties.BooleanAlgebra.html#12825" class="Bound">u</a><a id="13104" class="Symbol">))</a> <a id="13107" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
|
|||
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<a id="13117" class="Symbol">((</a><a id="13119" href="Algebra.Properties.BooleanAlgebra.html#12821" class="Bound">x</a> <a id="13121" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13123" href="Algebra.Properties.BooleanAlgebra.html#12827" class="Bound">v</a><a id="13124" class="Symbol">)</a> <a id="13126" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13128" class="Symbol">(</a><a id="13129" href="Algebra.Properties.BooleanAlgebra.html#12823" class="Bound">y</a> <a id="13131" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13133" href="Algebra.Properties.BooleanAlgebra.html#12827" class="Bound">v</a><a id="13134" class="Symbol">))</a> <a id="13148" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
|||
|
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|
|||
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<a id="XorRing.⊕-assoc"></a><a id="13153" href="Algebra.Properties.BooleanAlgebra.html#13153" class="Function">⊕-assoc</a> <a id="13161" class="Symbol">:</a> <a id="13163" href="Algebra.Definitions.html#1117" class="Function">Associative</a> <a id="13175" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
|
|||
|
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<a id="13181" href="Algebra.Properties.BooleanAlgebra.html#13153" class="Function">⊕-assoc</a> <a id="13189" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13191" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13193" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a> <a id="13195" class="Symbol">=</a> <a id="13197" href="Relation.Binary.Structures.html#1603" class="Function">sym</a> <a id="13201" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="13203" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
|
|||
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<a id="13213" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13215" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="13217" class="Symbol">(</a><a id="13218" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13220" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="13222" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13223" class="Symbol">)</a> <a id="13256" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="13259" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="13264" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="13266" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a> <a id="13273" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="13275" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="13281" class="Symbol">_</a> <a id="13283" class="Symbol">_</a> <a id="13285" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="13291" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13293" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="13295" class="Symbol">((</a><a id="13297" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13299" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13301" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13302" class="Symbol">)</a> <a id="13304" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13306" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13308" class="Symbol">(</a><a id="13309" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13311" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13313" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13314" class="Symbol">))</a> <a id="13334" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="13337" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="13343" class="Symbol">_</a> <a id="13345" class="Symbol">_</a> <a id="13347" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
|
|||
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<a id="13355" class="Symbol">(</a><a id="13356" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13358" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13360" class="Symbol">((</a><a id="13362" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13364" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13366" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13367" class="Symbol">)</a> <a id="13369" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13371" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13373" class="Symbol">(</a><a id="13374" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="13376" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13378" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13379" class="Symbol">)))</a> <a id="13383" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
|
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<a id="13462" class="Symbol">(((</a><a id="13465" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13467" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13469" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13470" class="Symbol">)</a> <a id="13472" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13474" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13475" class="Symbol">)</a> <a id="13477" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13479" class="Symbol">((</a><a id="13481" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13483" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13485" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13487" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13488" class="Symbol">)</a> <a id="13490" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13492" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13494" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13495" class="Symbol">))</a> <a id="13498" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="13504" class="Symbol">(((</a><a id="13507" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13509" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13511" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13513" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13515" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13516" class="Symbol">)</a> <a id="13518" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13520" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13521" class="Symbol">)</a> <a id="13523" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13525" class="Symbol">((</a><a id="13527" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13529" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13531" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13533" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13534" class="Symbol">)</a> <a id="13536" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13538" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13540" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13541" class="Symbol">))</a> <a id="13547" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="13550" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="13558" class="Symbol">_</a> <a id="13560" class="Symbol">_</a> <a id="13562" class="Symbol">_</a> <a id="13564" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="13570" class="Symbol">((</a><a id="13572" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13574" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13576" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13577" class="Symbol">)</a> <a id="13579" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13581" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13582" class="Symbol">)</a> <a id="13584" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="13590" class="Symbol">(((</a><a id="13593" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13595" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13597" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13599" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13600" class="Symbol">)</a> <a id="13602" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13604" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13606" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13607" class="Symbol">)</a> <a id="13609" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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|||
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<a id="13616" class="Symbol">(((</a><a id="13619" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13621" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13623" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13625" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13627" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13628" class="Symbol">)</a> <a id="13630" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13632" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13633" class="Symbol">)</a> <a id="13635" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13637" class="Symbol">((</a><a id="13639" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13641" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13643" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13645" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13646" class="Symbol">)</a> <a id="13648" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13650" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13652" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13653" class="Symbol">)))</a> <a id="13658" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="13661" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="13669" href="Algebra.Properties.BooleanAlgebra.html#16530" class="Function">lem₅</a> <a id="13674" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="13680" class="Symbol">((</a><a id="13682" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13684" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13686" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13687" class="Symbol">)</a> <a id="13689" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13691" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13692" class="Symbol">)</a> <a id="13694" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
|
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<a id="13700" class="Symbol">(((</a><a id="13703" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13705" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13707" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13709" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13711" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13712" class="Symbol">)</a> <a id="13714" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13716" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13717" class="Symbol">)</a> <a id="13719" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="13726" class="Symbol">(((</a><a id="13729" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13731" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13733" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13735" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13736" class="Symbol">)</a> <a id="13738" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13740" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13742" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13743" class="Symbol">)</a> <a id="13745" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13747" class="Symbol">((</a><a id="13749" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13751" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13753" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13755" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13756" class="Symbol">)</a> <a id="13758" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13760" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13762" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13763" class="Symbol">)))</a> <a id="13768" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="13772" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="13780" class="Symbol">_</a> <a id="13782" class="Symbol">_</a> <a id="13784" class="Symbol">_</a> <a id="13786" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="13792" class="Symbol">(((</a><a id="13795" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13797" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13799" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13800" class="Symbol">)</a> <a id="13802" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13804" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13805" class="Symbol">)</a> <a id="13807" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13809" class="Symbol">((</a><a id="13811" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13813" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13815" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13817" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13819" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13820" class="Symbol">)</a> <a id="13822" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13824" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13825" class="Symbol">))</a> <a id="13828" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="13834" class="Symbol">(((</a><a id="13837" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13839" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13841" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13843" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13844" class="Symbol">)</a> <a id="13846" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13848" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13850" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13851" class="Symbol">)</a> <a id="13853" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13855" class="Symbol">((</a><a id="13857" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13859" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13861" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13863" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13864" class="Symbol">)</a> <a id="13866" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13868" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13870" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13871" class="Symbol">))</a> <a id="13877" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="13880" href="Algebra.Properties.BooleanAlgebra.html#14144" class="Function">lem₁</a> <a id="13885" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="13887" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="13894" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="13896" href="Algebra.Properties.BooleanAlgebra.html#14963" class="Function">lem₂</a> <a id="13901" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="13909" class="Symbol">(((</a><a id="13912" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13914" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13916" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13917" class="Symbol">)</a> <a id="13919" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13921" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13923" class="Symbol">(</a><a id="13924" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13926" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13928" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13929" class="Symbol">))</a> <a id="13932" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13934" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13935" class="Symbol">)</a> <a id="13937" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="13943" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13945" class="Symbol">(((</a><a id="13948" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13950" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="13952" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13953" class="Symbol">)</a> <a id="13955" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13957" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="13959" class="Symbol">(</a><a id="13960" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="13962" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13964" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="13965" class="Symbol">))</a> <a id="13968" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="13970" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="13971" class="Symbol">)</a> <a id="13986" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="13990" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="13996" class="Symbol">_</a> <a id="13998" class="Symbol">_</a> <a id="14000" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
|
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<a id="14006" class="Symbol">((</a><a id="14008" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14010" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14012" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14013" class="Symbol">)</a> <a id="14015" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14017" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14019" class="Symbol">(</a><a id="14020" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14022" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14024" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14025" class="Symbol">))</a> <a id="14028" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="14030" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a> <a id="14049" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="14053" href="Algebra.Properties.BooleanAlgebra.html#7791" class="Bound">⊕-def</a> <a id="14059" class="Symbol">_</a> <a id="14061" class="Symbol">_</a> <a id="14063" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="14065" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a> <a id="14072" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="14074" href="Relation.Binary.Structures.html#1577" class="Function">refl</a> <a id="14079" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="14085" class="Symbol">(</a><a id="14086" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14088" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="14090" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14091" class="Symbol">)</a> <a id="14093" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">⊕</a> <a id="14095" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a> <a id="14128" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
|
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<a id="14134" class="Keyword">where</a>
|
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<a id="14144" href="Algebra.Properties.BooleanAlgebra.html#14144" class="Function">lem₁</a> <a id="14149" class="Symbol">=</a> <a id="14151" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="14352" href="Algebra.Properties.BooleanAlgebra.html#14352" class="Function">lem₂′</a> <a id="14358" class="Symbol">=</a> <a id="14360" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="14372" class="Symbol">(</a><a id="14373" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14375" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14377" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14379" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14380" class="Symbol">)</a> <a id="14382" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14384" class="Symbol">(</a><a id="14385" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14387" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14389" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14391" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14392" class="Symbol">)</a> <a id="14407" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="14411" href="Algebra.Properties.BooleanAlgebra.html#2407" class="Function">∧-identityˡ</a> <a id="14423" class="Symbol">_</a> <a id="14425" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="14427" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="14434" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="14436" href="Algebra.Properties.BooleanAlgebra.html#2239" class="Function">∧-identityʳ</a> <a id="14448" class="Symbol">_</a> <a id="14450" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="14458" class="Symbol">(</a><a id="14459" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="14461" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14463" class="Symbol">(</a><a id="14464" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14466" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14468" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14470" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14471" class="Symbol">))</a> <a id="14474" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14476" class="Symbol">((</a><a id="14478" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14480" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14482" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14484" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14485" class="Symbol">)</a> <a id="14487" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14489" href="Algebra.Bundles.html#23439" class="Field">⊤</a><a id="14490" class="Symbol">)</a> <a id="14493" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="14498" class="Symbol">(</a><a id="14499" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="14513" class="Symbol">_</a> <a id="14515" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="14517" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="14524" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="14526" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="14533" class="Symbol">_</a> <a id="14535" class="Symbol">_)</a>
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<a id="14586" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="14588" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="14595" href="Function.Base.html#4407" class="Function Operator">⟩</a>
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<a id="14643" class="Symbol">(</a><a id="14644" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="14652" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="14654" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="14668" class="Symbol">_)</a> <a id="14671" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="14773" class="Symbol">(</a><a id="14774" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14776" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14778" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14780" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="14782" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14783" class="Symbol">)</a> <a id="14785" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="14787" class="Symbol">(</a><a id="14788" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="14790" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="14792" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="14793" class="Symbol">)</a> <a id="14808" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="14812" href="Algebra.Properties.BooleanAlgebra.html#7306" class="Function">deMorgan₂</a> <a id="14822" class="Symbol">_</a> <a id="14824" class="Symbol">_</a> <a id="14826" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="14828" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="14835" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="14837" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="14850" class="Symbol">_</a> <a id="14852" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="15227" href="Algebra.Properties.BooleanAlgebra.html#15227" class="Function">lem₃</a> <a id="15232" class="Symbol">=</a> <a id="15234" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="15246" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15248" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15250" class="Symbol">((</a><a id="15252" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15254" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15256" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15257" class="Symbol">)</a> <a id="15259" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15261" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15263" class="Symbol">(</a><a id="15264" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15266" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15268" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15269" class="Symbol">))</a> <a id="15281" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15284" href="Algebra.Structures.html#6750" class="Function">∨-congˡ</a> <a id="15292" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="15294" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="15302" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="15304" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="15314" class="Symbol">_</a> <a id="15316" class="Symbol">_</a> <a id="15318" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15326" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15328" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15330" class="Symbol">((</a><a id="15332" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15334" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15336" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15337" class="Symbol">)</a> <a id="15339" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15341" class="Symbol">(</a><a id="15342" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15344" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15346" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15348" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15350" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15351" class="Symbol">))</a> <a id="15361" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15364" href="Algebra.Properties.DistributiveLattice.html#2870" class="Function">∨-∧-distribˡ</a> <a id="15377" class="Symbol">_</a> <a id="15379" class="Symbol">_</a> <a id="15381" class="Symbol">_</a> <a id="15383" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15391" class="Symbol">(</a><a id="15392" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15394" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15396" class="Symbol">(</a><a id="15397" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15399" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15401" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15402" class="Symbol">))</a> <a id="15405" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15407" class="Symbol">(</a><a id="15408" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15410" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15412" class="Symbol">(</a><a id="15413" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15415" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15417" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15419" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15421" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15422" class="Symbol">))</a> <a id="15426" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">≈˘⟨</a> <a id="15430" href="Algebra.Structures.html#6258" class="Function">∨-assoc</a> <a id="15438" class="Symbol">_</a> <a id="15440" class="Symbol">_</a> <a id="15442" class="Symbol">_</a> <a id="15444" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="15446" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="15453" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="15455" href="Algebra.Structures.html#6258" class="Function">∨-assoc</a> <a id="15463" class="Symbol">_</a> <a id="15465" class="Symbol">_</a> <a id="15467" class="Symbol">_</a> <a id="15469" href="Relation.Binary.Reasoning.Setoid.html#1162" class="Function">⟩</a>
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<a id="15477" class="Symbol">((</a><a id="15479" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15481" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15483" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="15484" class="Symbol">)</a> <a id="15486" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15488" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15489" class="Symbol">)</a> <a id="15491" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15493" class="Symbol">((</a><a id="15495" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="15497" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15499" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15501" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="15502" class="Symbol">)</a> <a id="15504" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15506" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15508" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15509" class="Symbol">)</a> <a id="15512" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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|||
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|
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<a id="15519" href="Algebra.Properties.BooleanAlgebra.html#15519" class="Function">lem₄′</a> <a id="15525" class="Symbol">=</a> <a id="15527" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="15539" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15541" class="Symbol">((</a><a id="15543" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15545" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15547" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15548" class="Symbol">)</a> <a id="15550" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15552" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15554" class="Symbol">(</a><a id="15555" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15557" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15559" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15560" class="Symbol">))</a> <a id="15566" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15569" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="15579" class="Symbol">_</a> <a id="15581" class="Symbol">_</a> <a id="15583" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15591" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15593" class="Symbol">(</a><a id="15594" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15596" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15598" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15599" class="Symbol">)</a> <a id="15601" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15603" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15605" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15607" class="Symbol">(</a><a id="15608" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15610" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15612" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15613" class="Symbol">)</a> <a id="15618" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15621" href="Algebra.Properties.BooleanAlgebra.html#7306" class="Function">deMorgan₂</a> <a id="15631" class="Symbol">_</a> <a id="15633" class="Symbol">_</a> <a id="15635" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="15637" href="Algebra.Structures.html#6292" class="Function">∨-cong</a> <a id="15644" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="15646" href="Algebra.Properties.BooleanAlgebra.html#6049" class="Function">¬-involutive</a> <a id="15659" class="Symbol">_</a> <a id="15661" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15669" class="Symbol">(</a><a id="15670" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15672" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15674" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15676" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15678" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15679" class="Symbol">)</a> <a id="15681" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15683" class="Symbol">(</a><a id="15684" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15686" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15688" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15689" class="Symbol">)</a> <a id="15696" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15699" href="Algebra.Properties.BooleanAlgebra.html#12688" class="Function">lemma₂</a> <a id="15706" class="Symbol">_</a> <a id="15708" class="Symbol">_</a> <a id="15710" class="Symbol">_</a> <a id="15712" class="Symbol">_</a> <a id="15714" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15722" class="Symbol">((</a><a id="15724" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15726" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15728" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15730" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="15731" class="Symbol">)</a> <a id="15733" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15735" class="Symbol">(</a><a id="15736" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15738" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a> <a id="15740" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15742" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="15743" class="Symbol">))</a> <a id="15746" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="15754" class="Symbol">((</a><a id="15756" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15758" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15760" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15762" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15763" class="Symbol">)</a> <a id="15765" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15767" class="Symbol">(</a><a id="15768" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15770" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a> <a id="15772" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15774" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15775" class="Symbol">))</a> <a id="15781" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="15784" class="Symbol">(</a><a id="15785" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="15799" class="Symbol">_</a> <a id="15801" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="15803" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="15810" href="Function.Base.html#4407" class="Function Operator">⟩</a> <a id="15812" href="Algebra.Structures.html#6224" class="Function">∨-comm</a> <a id="15819" class="Symbol">_</a> <a id="15821" class="Symbol">_)</a>
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<a id="15862" href="Function.Base.html#4407" class="Function Operator">⟨</a> <a id="15864" href="Algebra.Structures.html#6393" class="Function">∧-cong</a> <a id="15871" href="Function.Base.html#4407" class="Function Operator">⟩</a>
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<a id="15908" class="Symbol">(</a><a id="15909" href="Algebra.Structures.html#6627" class="Function">∧-congˡ</a> <a id="15917" href="Function.Base.html#1928" class="Function Operator">$</a> <a id="15919" href="Algebra.Properties.BooleanAlgebra.html#1265" class="Function">∨-complementˡ</a> <a id="15933" class="Symbol">_)</a> <a id="15936" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="15944" class="Symbol">(</a><a id="15945" href="Algebra.Bundles.html#23439" class="Field">⊤</a> <a id="15947" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="15949" class="Symbol">(</a><a id="15950" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="15952" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="15954" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="15956" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="15957" class="Symbol">))</a> <a id="15960" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="16045" class="Symbol">(</a><a id="16046" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="16048" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16050" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16052" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16053" class="Symbol">)</a> <a id="16055" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16057" class="Symbol">(</a><a id="16058" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16060" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="16062" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16064" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16065" class="Symbol">)</a> <a id="16072" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="16079" href="Algebra.Properties.BooleanAlgebra.html#16079" class="Function">lem₄</a> <a id="16084" class="Symbol">=</a> <a id="16086" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="16098" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16100" class="Symbol">(</a><a id="16101" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16103" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16105" class="Symbol">((</a><a id="16107" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="16109" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16111" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16112" class="Symbol">)</a> <a id="16114" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16116" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16118" class="Symbol">(</a><a id="16119" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a> <a id="16121" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16123" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16124" class="Symbol">)))</a> <a id="16129" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="16132" href="Algebra.Properties.BooleanAlgebra.html#6146" class="Function">deMorgan₁</a> <a id="16142" class="Symbol">_</a> <a id="16144" class="Symbol">_</a> <a id="16146" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="16530" href="Algebra.Properties.BooleanAlgebra.html#16530" class="Function">lem₅</a> <a id="16535" class="Symbol">=</a> <a id="16537" href="Relation.Binary.Reasoning.Base.Single.html#1925" class="Function Operator">begin</a>
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<a id="16767" class="Symbol">(((</a><a id="16770" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16772" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16774" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16776" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16778" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16779" class="Symbol">)</a> <a id="16781" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16783" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16784" class="Symbol">)</a> <a id="16786" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16788" class="Symbol">((</a><a id="16790" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16792" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16794" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16796" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16797" class="Symbol">)</a> <a id="16799" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16801" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16803" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16804" class="Symbol">))</a> <a id="16807" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="16815" class="Symbol">((</a><a id="16817" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16819" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16821" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16823" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16824" class="Symbol">)</a> <a id="16826" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16828" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16830" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16831" class="Symbol">)</a> <a id="16858" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">≈⟨</a> <a id="16861" href="Algebra.Structures.html#6359" class="Function">∧-assoc</a> <a id="16869" class="Symbol">_</a> <a id="16871" class="Symbol">_</a> <a id="16873" class="Symbol">_</a> <a id="16875" href="Relation.Binary.Reasoning.Setoid.html#1061" class="Function">⟩</a>
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<a id="16883" class="Symbol">((</a><a id="16885" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16887" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16889" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16891" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16893" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16894" class="Symbol">)</a> <a id="16896" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16898" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16899" class="Symbol">)</a> <a id="16901" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a>
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<a id="16909" class="Symbol">(((</a><a id="16912" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16914" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16916" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16918" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16919" class="Symbol">)</a> <a id="16921" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16923" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16925" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16926" class="Symbol">)</a> <a id="16928" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="16930" class="Symbol">((</a><a id="16932" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16934" href="Algebra.Properties.BooleanAlgebra.html#13189" class="Bound">x</a> <a id="16936" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16938" href="Algebra.Properties.BooleanAlgebra.html#13191" class="Bound">y</a><a id="16939" class="Symbol">)</a> <a id="16941" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="16943" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="16945" href="Algebra.Properties.BooleanAlgebra.html#13193" class="Bound">z</a><a id="16946" class="Symbol">))</a> <a id="16952" href="Relation.Binary.Reasoning.Base.Single.html#2564" class="Function Operator">∎</a>
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<a id="XorRing.⊕-isMagma"></a><a id="16957" href="Algebra.Properties.BooleanAlgebra.html#16957" class="Function">⊕-isMagma</a> <a id="16967" class="Symbol">:</a> <a id="16969" href="Algebra.Structures.html#1129" class="Record">IsMagma</a> <a id="16977" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
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<a id="16983" href="Algebra.Properties.BooleanAlgebra.html#16957" class="Function">⊕-isMagma</a> <a id="16993" class="Symbol">=</a> <a id="16995" class="Keyword">record</a>
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<a id="17006" class="Symbol">{</a> <a id="17008" href="Algebra.Structures.html#1181" class="Field">isEquivalence</a> <a id="17022" class="Symbol">=</a> <a id="17024" href="Algebra.Structures.html#6186" class="Function">isEquivalence</a>
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<a id="17042" class="Symbol">;</a> <a id="17044" href="Algebra.Structures.html#1219" class="Field">∙-cong</a> <a id="17058" class="Symbol">=</a> <a id="17060" href="Algebra.Properties.BooleanAlgebra.html#8023" class="Function">⊕-cong</a>
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<a id="17071" class="Symbol">}</a>
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<a id="XorRing.⊕-isSemigroup"></a><a id="17076" href="Algebra.Properties.BooleanAlgebra.html#17076" class="Function">⊕-isSemigroup</a> <a id="17090" class="Symbol">:</a> <a id="17092" href="Algebra.Structures.html#1796" class="Record">IsSemigroup</a> <a id="17104" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a>
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<a id="17110" href="Algebra.Properties.BooleanAlgebra.html#17076" class="Function">⊕-isSemigroup</a> <a id="17124" class="Symbol">=</a> <a id="17126" class="Keyword">record</a>
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|||
|
|
<a id="17137" class="Symbol">{</a> <a id="17139" href="Algebra.Structures.html#1852" class="Field">isMagma</a> <a id="17147" class="Symbol">=</a> <a id="17149" href="Algebra.Properties.BooleanAlgebra.html#16957" class="Function">⊕-isMagma</a>
|
|||
|
|
<a id="17163" class="Symbol">;</a> <a id="17165" href="Algebra.Structures.html#1876" class="Field">assoc</a> <a id="17173" class="Symbol">=</a> <a id="17175" href="Algebra.Properties.BooleanAlgebra.html#13153" class="Function">⊕-assoc</a>
|
|||
|
|
<a id="17187" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-⊥-isMonoid"></a><a id="17192" href="Algebra.Properties.BooleanAlgebra.html#17192" class="Function">⊕-⊥-isMonoid</a> <a id="17205" class="Symbol">:</a> <a id="17207" href="Algebra.Structures.html#2810" class="Record">IsMonoid</a> <a id="17216" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="17220" href="Algebra.Bundles.html#23470" class="Field">⊥</a>
|
|||
|
|
<a id="17224" href="Algebra.Properties.BooleanAlgebra.html#17192" class="Function">⊕-⊥-isMonoid</a> <a id="17237" class="Symbol">=</a> <a id="17239" class="Keyword">record</a>
|
|||
|
|
<a id="17250" class="Symbol">{</a> <a id="17252" href="Algebra.Structures.html#2871" class="Field">isSemigroup</a> <a id="17264" class="Symbol">=</a> <a id="17266" href="Algebra.Properties.BooleanAlgebra.html#17076" class="Function">⊕-isSemigroup</a>
|
|||
|
|
<a id="17284" class="Symbol">;</a> <a id="17286" href="Algebra.Structures.html#2903" class="Field">identity</a> <a id="17298" class="Symbol">=</a> <a id="17300" href="Algebra.Properties.BooleanAlgebra.html#10379" class="Function">⊕-identity</a>
|
|||
|
|
<a id="17315" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-⊥-isGroup"></a><a id="17320" href="Algebra.Properties.BooleanAlgebra.html#17320" class="Function">⊕-⊥-isGroup</a> <a id="17332" class="Symbol">:</a> <a id="17334" href="Algebra.Structures.html#4430" class="Record">IsGroup</a> <a id="17342" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="17346" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="17348" href="Function.Base.html#624" class="Function">id</a>
|
|||
|
|
<a id="17353" href="Algebra.Properties.BooleanAlgebra.html#17320" class="Function">⊕-⊥-isGroup</a> <a id="17365" class="Symbol">=</a> <a id="17367" class="Keyword">record</a>
|
|||
|
|
<a id="17378" class="Symbol">{</a> <a id="17380" href="Algebra.Structures.html#4506" class="Field">isMonoid</a> <a id="17389" class="Symbol">=</a> <a id="17391" href="Algebra.Properties.BooleanAlgebra.html#17192" class="Function">⊕-⊥-isMonoid</a>
|
|||
|
|
<a id="17408" class="Symbol">;</a> <a id="17410" href="Algebra.Structures.html#4537" class="Field">inverse</a> <a id="17419" class="Symbol">=</a> <a id="17421" href="Algebra.Properties.BooleanAlgebra.html#10780" class="Function">⊕-inverse</a>
|
|||
|
|
<a id="17435" class="Symbol">;</a> <a id="17437" href="Algebra.Structures.html#4571" class="Field">⁻¹-cong</a> <a id="17446" class="Symbol">=</a> <a id="17448" href="Function.Base.html#624" class="Function">id</a>
|
|||
|
|
<a id="17455" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-⊥-isAbelianGroup"></a><a id="17460" href="Algebra.Properties.BooleanAlgebra.html#17460" class="Function">⊕-⊥-isAbelianGroup</a> <a id="17479" class="Symbol">:</a> <a id="17481" href="Algebra.Structures.html#5132" class="Record">IsAbelianGroup</a> <a id="17496" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="17500" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="17502" href="Function.Base.html#624" class="Function">id</a>
|
|||
|
|
<a id="17507" href="Algebra.Properties.BooleanAlgebra.html#17460" class="Function">⊕-⊥-isAbelianGroup</a> <a id="17526" class="Symbol">=</a> <a id="17528" class="Keyword">record</a>
|
|||
|
|
<a id="17539" class="Symbol">{</a> <a id="17541" href="Algebra.Structures.html#5234" class="Field">isGroup</a> <a id="17549" class="Symbol">=</a> <a id="17551" href="Algebra.Properties.BooleanAlgebra.html#17320" class="Function">⊕-⊥-isGroup</a>
|
|||
|
|
<a id="17567" class="Symbol">;</a> <a id="17569" href="Algebra.Structures.html#5263" class="Field">comm</a> <a id="17577" class="Symbol">=</a> <a id="17579" href="Algebra.Properties.BooleanAlgebra.html#8316" class="Function">⊕-comm</a>
|
|||
|
|
<a id="17590" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-∧-isRing"></a><a id="17595" href="Algebra.Properties.BooleanAlgebra.html#17595" class="Function">⊕-∧-isRing</a> <a id="17606" class="Symbol">:</a> <a id="17608" href="Algebra.Structures.html#13205" class="Record">IsRing</a> <a id="17615" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="17619" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="17623" href="Function.Base.html#624" class="Function">id</a> <a id="17626" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="17628" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="17632" href="Algebra.Properties.BooleanAlgebra.html#17595" class="Function">⊕-∧-isRing</a> <a id="17643" class="Symbol">=</a> <a id="17645" class="Keyword">record</a>
|
|||
|
|
<a id="17656" class="Symbol">{</a> <a id="17658" href="Algebra.Structures.html#13283" class="Field">+-isAbelianGroup</a> <a id="17675" class="Symbol">=</a> <a id="17677" href="Algebra.Properties.BooleanAlgebra.html#17460" class="Function">⊕-⊥-isAbelianGroup</a>
|
|||
|
|
<a id="17700" class="Symbol">;</a> <a id="17702" href="Algebra.Structures.html#13329" class="Field">*-isMonoid</a> <a id="17713" class="Symbol">=</a> <a id="17715" href="Algebra.Properties.BooleanAlgebra.html#3693" class="Function">∧-⊤-isMonoid</a>
|
|||
|
|
<a id="17732" class="Symbol">;</a> <a id="17734" href="Algebra.Structures.html#13366" class="Field">distrib</a> <a id="17742" class="Symbol">=</a> <a id="17744" href="Algebra.Properties.BooleanAlgebra.html#12590" class="Function">∧-distrib-⊕</a>
|
|||
|
|
<a id="17760" class="Symbol">;</a> <a id="17762" href="Algebra.Structures.html#13409" class="Field">zero</a> <a id="17767" class="Symbol">=</a> <a id="17769" href="Algebra.Properties.BooleanAlgebra.html#3176" class="Function">∧-zero</a>
|
|||
|
|
<a id="17780" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-∧-isCommutativeRing"></a><a id="17785" href="Algebra.Properties.BooleanAlgebra.html#17785" class="Function">⊕-∧-isCommutativeRing</a> <a id="17807" class="Symbol">:</a> <a id="17809" href="Algebra.Structures.html#15327" class="Record">IsCommutativeRing</a> <a id="17827" href="Algebra.Properties.BooleanAlgebra.html#7879" class="Function Operator">_⊕_</a> <a id="17831" href="Algebra.Bundles.html#23369" class="Field Operator">_∧_</a> <a id="17835" href="Function.Base.html#624" class="Function">id</a> <a id="17838" href="Algebra.Bundles.html#23470" class="Field">⊥</a> <a id="17840" href="Algebra.Bundles.html#23439" class="Field">⊤</a>
|
|||
|
|
<a id="17844" href="Algebra.Properties.BooleanAlgebra.html#17785" class="Function">⊕-∧-isCommutativeRing</a> <a id="17866" class="Symbol">=</a> <a id="17868" class="Keyword">record</a>
|
|||
|
|
<a id="17879" class="Symbol">{</a> <a id="17881" href="Algebra.Structures.html#15424" class="Field">isRing</a> <a id="17888" class="Symbol">=</a> <a id="17890" href="Algebra.Properties.BooleanAlgebra.html#17595" class="Function">⊕-∧-isRing</a>
|
|||
|
|
<a id="17905" class="Symbol">;</a> <a id="17907" href="Algebra.Structures.html#15456" class="Field">*-comm</a> <a id="17914" class="Symbol">=</a> <a id="17916" href="Algebra.Structures.html#6325" class="Function">∧-comm</a>
|
|||
|
|
<a id="17927" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-∧-commutativeRing"></a><a id="17932" href="Algebra.Properties.BooleanAlgebra.html#17932" class="Function">⊕-∧-commutativeRing</a> <a id="17952" class="Symbol">:</a> <a id="17954" href="Algebra.Bundles.html#21958" class="Record">CommutativeRing</a> <a id="17970" class="Symbol">_</a> <a id="17972" class="Symbol">_</a>
|
|||
|
|
<a id="17976" href="Algebra.Properties.BooleanAlgebra.html#17932" class="Function">⊕-∧-commutativeRing</a> <a id="17996" class="Symbol">=</a> <a id="17998" class="Keyword">record</a>
|
|||
|
|
<a id="18009" class="Symbol">{</a> <a id="18011" href="Algebra.Bundles.html#22315" class="Field">isCommutativeRing</a> <a id="18029" class="Symbol">=</a> <a id="18031" href="Algebra.Properties.BooleanAlgebra.html#17785" class="Function">⊕-∧-isCommutativeRing</a>
|
|||
|
|
<a id="18057" class="Symbol">}</a>
|
|||
|
|
|
|||
|
|
<a id="XorRing.⊕-¬-distribˡ"></a><a id="18062" href="Algebra.Properties.BooleanAlgebra.html#18062" class="Function">⊕-¬-distribˡ</a> <a id="18075" class="Symbol">=</a> <a id="18077" href="Algebra.Properties.BooleanAlgebra.html#8538" class="Function">¬-distribˡ-⊕</a>
|
|||
|
|
<a id="18092" class="Symbol">{-#</a> <a id="18096" class="Keyword">WARNING_ON_USAGE</a> <a id="18113" class="Pragma">⊕-¬-distribˡ</a>
|
|||
|
|
<a id="18128" class="String">"Warning: ⊕-¬-distribˡ was deprecated in v1.1.
|
|||
|
|
Please use ¬-distribˡ-⊕ instead."</a>
|
|||
|
|
<a id="18213" class="Symbol">#-}</a>
|
|||
|
|
<a id="XorRing.⊕-¬-distribʳ"></a><a id="18219" href="Algebra.Properties.BooleanAlgebra.html#18219" class="Function">⊕-¬-distribʳ</a> <a id="18232" class="Symbol">=</a> <a id="18234" href="Algebra.Properties.BooleanAlgebra.html#9590" class="Function">¬-distribʳ-⊕</a>
|
|||
|
|
<a id="18249" class="Symbol">{-#</a> <a id="18253" class="Keyword">WARNING_ON_USAGE</a> <a id="18270" class="Pragma">⊕-¬-distribʳ</a>
|
|||
|
|
<a id="18285" class="String">"Warning: ⊕-¬-distribʳ was deprecated in v1.1.
|
|||
|
|
Please use ¬-distribʳ-⊕ instead."</a>
|
|||
|
|
<a id="18370" class="Symbol">#-}</a>
|
|||
|
|
<a id="XorRing.isCommutativeRing"></a><a id="18376" href="Algebra.Properties.BooleanAlgebra.html#18376" class="Function">isCommutativeRing</a> <a id="18394" class="Symbol">=</a> <a id="18396" href="Algebra.Properties.BooleanAlgebra.html#17785" class="Function">⊕-∧-isCommutativeRing</a>
|
|||
|
|
<a id="18420" class="Symbol">{-#</a> <a id="18424" class="Keyword">WARNING_ON_USAGE</a> <a id="18441" class="Pragma">isCommutativeRing</a>
|
|||
|
|
<a id="18461" class="String">"Warning: isCommutativeRing was deprecated in v1.1.
|
|||
|
|
Please use ⊕-∧-isCommutativeRing instead."</a>
|
|||
|
|
<a id="18560" class="Symbol">#-}</a>
|
|||
|
|
<a id="XorRing.commutativeRing"></a><a id="18566" href="Algebra.Properties.BooleanAlgebra.html#18566" class="Function">commutativeRing</a> <a id="18582" class="Symbol">=</a> <a id="18584" href="Algebra.Properties.BooleanAlgebra.html#17932" class="Function">⊕-∧-commutativeRing</a>
|
|||
|
|
<a id="18606" class="Symbol">{-#</a> <a id="18610" class="Keyword">WARNING_ON_USAGE</a> <a id="18627" class="Pragma">commutativeRing</a>
|
|||
|
|
<a id="18645" class="String">"Warning: commutativeRing was deprecated in v1.1.
|
|||
|
|
Please use ⊕-∧-commutativeRing instead."</a>
|
|||
|
|
<a id="18740" class="Symbol">#-}</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
<a id="18746" class="Keyword">infixl</a> <a id="18753" class="Number">6</a> <a id="18755" href="Algebra.Properties.BooleanAlgebra.html#18760" class="Function Operator">_⊕_</a>
|
|||
|
|
|
|||
|
|
<a id="_⊕_"></a><a id="18760" href="Algebra.Properties.BooleanAlgebra.html#18760" class="Function Operator">_⊕_</a> <a id="18764" class="Symbol">:</a> <a id="18766" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="18770" href="Algebra.Bundles.html#23268" class="Field">Carrier</a>
|
|||
|
|
<a id="18778" href="Algebra.Properties.BooleanAlgebra.html#18778" class="Bound">x</a> <a id="18780" href="Algebra.Properties.BooleanAlgebra.html#18760" class="Function Operator">⊕</a> <a id="18782" href="Algebra.Properties.BooleanAlgebra.html#18782" class="Bound">y</a> <a id="18784" class="Symbol">=</a> <a id="18786" class="Symbol">(</a><a id="18787" href="Algebra.Properties.BooleanAlgebra.html#18778" class="Bound">x</a> <a id="18789" href="Algebra.Bundles.html#23334" class="Field Operator">∨</a> <a id="18791" href="Algebra.Properties.BooleanAlgebra.html#18782" class="Bound">y</a><a id="18792" class="Symbol">)</a> <a id="18794" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="18796" href="Algebra.Bundles.html#23404" class="Field Operator">¬</a> <a id="18798" class="Symbol">(</a><a id="18799" href="Algebra.Properties.BooleanAlgebra.html#18778" class="Bound">x</a> <a id="18801" href="Algebra.Bundles.html#23369" class="Field Operator">∧</a> <a id="18803" href="Algebra.Properties.BooleanAlgebra.html#18782" class="Bound">y</a><a id="18804" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
<a id="18807" class="Keyword">module</a> <a id="DefaultXorRing"></a><a id="18814" href="Algebra.Properties.BooleanAlgebra.html#18814" class="Module">DefaultXorRing</a> <a id="18829" class="Symbol">=</a> <a id="18831" href="Algebra.Properties.BooleanAlgebra.html#7758" class="Module">XorRing</a> <a id="18839" href="Algebra.Properties.BooleanAlgebra.html#18760" class="Function Operator">_⊕_</a> <a id="18843" class="Symbol">(λ</a> <a id="18846" href="Algebra.Properties.BooleanAlgebra.html#18846" class="Bound">_</a> <a id="18848" href="Algebra.Properties.BooleanAlgebra.html#18848" class="Bound">_</a> <a id="18850" class="Symbol">→</a> <a id="18852" href="Relation.Binary.Structures.html#1577" class="Function">refl</a><a id="18856" class="Symbol">)</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
<a id="18860" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="18933" class="Comment">-- DEPRECATED NAMES</a>
|
|||
|
|
<a id="18953" class="Comment">------------------------------------------------------------------------</a>
|
|||
|
|
<a id="19026" class="Comment">-- Please use the new names as continuing support for the old names is</a>
|
|||
|
|
<a id="19097" class="Comment">-- not guaranteed.</a>
|
|||
|
|
|
|||
|
|
<a id="19117" class="Comment">-- Version 1.1</a>
|
|||
|
|
|
|||
|
|
<a id="¬⊥=⊤"></a><a id="19133" href="Algebra.Properties.BooleanAlgebra.html#19133" class="Function">¬⊥=⊤</a> <a id="19138" class="Symbol">=</a> <a id="19140" href="Algebra.Properties.BooleanAlgebra.html#5931" class="Function">⊥≉⊤</a>
|
|||
|
|
<a id="19144" class="Symbol">{-#</a> <a id="19148" class="Keyword">WARNING_ON_USAGE</a> <a id="19165" class="Pragma">¬⊥=⊤</a>
|
|||
|
|
<a id="19170" class="String">"Warning: ¬⊥=⊤ was deprecated in v1.1.
|
|||
|
|
Please use ⊥≉⊤ instead."</a>
|
|||
|
|
<a id="19234" class="Symbol">#-}</a>
|
|||
|
|
<a id="¬⊤=⊥"></a><a id="19238" href="Algebra.Properties.BooleanAlgebra.html#19238" class="Function">¬⊤=⊥</a> <a id="19243" class="Symbol">=</a> <a id="19245" href="Algebra.Properties.BooleanAlgebra.html#5990" class="Function">⊤≉⊥</a>
|
|||
|
|
<a id="19249" class="Symbol">{-#</a> <a id="19253" class="Keyword">WARNING_ON_USAGE</a> <a id="19270" class="Pragma">¬⊤=⊥</a>
|
|||
|
|
<a id="19275" class="String">"Warning: ¬⊤=⊥ was deprecated in v1.1.
|
|||
|
|
Please use ⊤≉⊥ instead."</a>
|
|||
|
|
<a id="19339" class="Symbol">#-}</a>
|
|||
|
|
|
|||
|
|
<a id="19344" class="Comment">-- Version 1.4</a>
|
|||
|
|
|
|||
|
|
<a id="replace-equality"></a><a id="19360" href="Algebra.Properties.BooleanAlgebra.html#19360" class="Function">replace-equality</a> <a id="19377" class="Symbol">:</a> <a id="19379" class="Symbol">{</a><a id="19380" href="Algebra.Properties.BooleanAlgebra.html#19380" class="Bound Operator">_≈′_</a> <a id="19385" class="Symbol">:</a> <a id="19387" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="19391" href="Algebra.Bundles.html#23268" class="Field">Carrier</a> <a id="19399" href="Algebra.Properties.BooleanAlgebra.html#422" class="Bound">b₂</a><a id="19401" class="Symbol">}</a> <a id="19403" class="Symbol">→</a>
|
|||
|
|
<a id="19424" class="Symbol">(∀</a> <a id="19427" class="Symbol">{</a><a id="19428" href="Algebra.Properties.BooleanAlgebra.html#19428" class="Bound">x</a> <a id="19430" href="Algebra.Properties.BooleanAlgebra.html#19430" class="Bound">y</a><a id="19431" class="Symbol">}</a> <a id="19433" class="Symbol">→</a> <a id="19435" href="Algebra.Properties.BooleanAlgebra.html#19428" class="Bound">x</a> <a id="19437" href="Algebra.Bundles.html#23297" class="Field Operator">≈</a> <a id="19439" href="Algebra.Properties.BooleanAlgebra.html#19430" class="Bound">y</a> <a id="19441" href="Function.Equivalence.html#1213" class="Function Operator">⇔</a> <a id="19443" class="Symbol">(</a><a id="19444" href="Algebra.Properties.BooleanAlgebra.html#19428" class="Bound">x</a> <a id="19446" href="Algebra.Properties.BooleanAlgebra.html#19380" class="Bound Operator">≈′</a> <a id="19449" href="Algebra.Properties.BooleanAlgebra.html#19430" class="Bound">y</a><a id="19450" class="Symbol">))</a> <a id="19453" class="Symbol">→</a>
|
|||
|
|
<a id="19474" href="Algebra.Bundles.html#23152" class="Record">BooleanAlgebra</a> <a id="19489" class="Symbol">_</a> <a id="19491" class="Symbol">_</a>
|
|||
|
|
<a id="19493" href="Algebra.Properties.BooleanAlgebra.html#19360" class="Function">replace-equality</a> <a id="19510" class="Symbol">{</a><a id="19511" href="Algebra.Properties.BooleanAlgebra.html#19511" class="Bound Operator">_≈′_</a><a id="19515" class="Symbol">}</a> <a id="19517" href="Algebra.Properties.BooleanAlgebra.html#19517" class="Bound">≈⇔≈′</a> <a id="19522" class="Symbol">=</a> <a id="19524" class="Keyword">record</a>
|
|||
|
|
<a id="19533" class="Symbol">{</a> <a id="19535" href="Algebra.Bundles.html#23501" class="Field">isBooleanAlgebra</a> <a id="19552" class="Symbol">=</a> <a id="19555" class="Keyword">record</a>
|
|||
|
|
<a id="19566" class="Symbol">{</a> <a id="19568" href="Algebra.Structures.html#15960" class="Field">isDistributiveLattice</a> <a id="19590" class="Symbol">=</a> <a id="19592" href="Algebra.Bundles.html#10009" class="Field">DistributiveLattice.isDistributiveLattice</a>
|
|||
|
|
<a id="19642" class="Symbol">(</a><a id="19643" href="Algebra.Properties.DistributiveLattice.html#3612" class="Function">DistribLatticeProperties.replace-equality</a> <a id="19685" href="Algebra.Bundles.html#23608" class="Function">distributiveLattice</a> <a id="19705" href="Algebra.Properties.BooleanAlgebra.html#19517" class="Bound">≈⇔≈′</a><a id="19709" class="Symbol">)</a>
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<a id="19715" class="Symbol">;</a> <a id="19717" href="Algebra.Structures.html#16014" class="Field">∨-complementʳ</a> <a id="19739" class="Symbol">=</a> <a id="19741" class="Symbol">λ</a> <a id="19743" href="Algebra.Properties.BooleanAlgebra.html#19743" class="Bound">x</a> <a id="19745" class="Symbol">→</a> <a id="19747" href="Function.Equivalence.html#988" class="Function">to</a> <a id="19750" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="19754" href="Algebra.Structures.html#16014" class="Function">∨-complementʳ</a> <a id="19768" href="Algebra.Properties.BooleanAlgebra.html#19743" class="Bound">x</a>
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<a id="19774" class="Symbol">;</a> <a id="19776" href="Algebra.Structures.html#16061" class="Field">∧-complementʳ</a> <a id="19798" class="Symbol">=</a> <a id="19800" class="Symbol">λ</a> <a id="19802" href="Algebra.Properties.BooleanAlgebra.html#19802" class="Bound">x</a> <a id="19804" class="Symbol">→</a> <a id="19806" href="Function.Equivalence.html#988" class="Function">to</a> <a id="19809" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="19813" href="Algebra.Structures.html#16061" class="Function">∧-complementʳ</a> <a id="19827" href="Algebra.Properties.BooleanAlgebra.html#19802" class="Bound">x</a>
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<a id="19833" class="Symbol">;</a> <a id="19835" href="Algebra.Structures.html#16108" class="Field">¬-cong</a> <a id="19857" class="Symbol">=</a> <a id="19859" class="Symbol">λ</a> <a id="19861" href="Algebra.Properties.BooleanAlgebra.html#19861" class="Bound">i≈j</a> <a id="19865" class="Symbol">→</a> <a id="19867" href="Function.Equivalence.html#988" class="Function">to</a> <a id="19870" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="19874" href="Algebra.Structures.html#16108" class="Function">¬-cong</a> <a id="19881" class="Symbol">(</a><a id="19882" href="Function.Equivalence.html#1009" class="Function">from</a> <a id="19887" href="Function.Equality.html#1073" class="Field Operator">⟨$⟩</a> <a id="19891" href="Algebra.Properties.BooleanAlgebra.html#19861" class="Bound">i≈j</a><a id="19894" class="Symbol">)</a>
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<a id="19900" class="Symbol">}</a>
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<a id="19904" class="Symbol">}</a> <a id="19906" class="Keyword">where</a> <a id="19912" class="Keyword">open</a> <a id="19917" class="Keyword">module</a> <a id="19924" href="Algebra.Properties.BooleanAlgebra.html#19924" class="Module">E</a> <a id="19926" class="Symbol">{</a><a id="19927" href="Algebra.Properties.BooleanAlgebra.html#19927" class="Bound">x</a> <a id="19929" href="Algebra.Properties.BooleanAlgebra.html#19929" class="Bound">y</a><a id="19930" class="Symbol">}</a> <a id="19932" class="Symbol">=</a> <a id="19934" href="Function.Equivalence.html#838" class="Module">Equivalence</a> <a id="19946" class="Symbol">(</a><a id="19947" href="Algebra.Properties.BooleanAlgebra.html#19517" class="Bound">≈⇔≈′</a> <a id="19952" class="Symbol">{</a><a id="19953" href="Algebra.Properties.BooleanAlgebra.html#19927" class="Bound">x</a><a id="19954" class="Symbol">}</a> <a id="19956" class="Symbol">{</a><a id="19957" href="Algebra.Properties.BooleanAlgebra.html#19929" class="Bound">y</a><a id="19958" class="Symbol">})</a>
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<a id="19961" class="Symbol">{-#</a> <a id="19965" class="Keyword">WARNING_ON_USAGE</a> <a id="19982" class="Pragma">replace-equality</a>
|
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|
<a id="19999" class="String">"Warning: replace-equality was deprecated in v1.4.
|
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|
Please use isBooleanAlgebra from `Algebra.Construct.Subst.Equality` instead."</a>
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<a id="20128" class="Symbol">#-}</a>
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</pre></body></html>
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