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<html><head><meta charset="utf-8"><title>Algebra.Lattice.Properties.Lattice</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Comment">------------------------------------------------------------------------</a>
<a id="74" class="Comment">-- The Agda standard library</a>
<a id="103" class="Comment">--</a>
<a id="106" class="Comment">-- Some derivable properties of lattices</a>
<a id="147" class="Comment">------------------------------------------------------------------------</a>
<a id="221" class="Symbol">{-#</a> <a id="225" class="Keyword">OPTIONS</a> <a id="233" class="Pragma">--cubical-compatible</a> <a id="254" class="Pragma">--safe</a> <a id="261" class="Symbol">#-}</a>
<a id="266" class="Keyword">open</a> <a id="271" class="Keyword">import</a> <a id="278" href="Algebra.Lattice.Bundles.html" class="Module">Algebra.Lattice.Bundles</a>
<a id="302" class="Keyword">import</a> <a id="309" href="Algebra.Lattice.Properties.Semilattice.html" class="Module">Algebra.Lattice.Properties.Semilattice</a> <a id="348" class="Symbol">as</a> <a id="351" class="Module">SemilatticeProperties</a>
<a id="373" class="Keyword">open</a> <a id="378" class="Keyword">import</a> <a id="385" href="Relation.Binary.Bundles.html" class="Module">Relation.Binary.Bundles</a> <a id="409" class="Keyword">using</a> <a id="415" class="Symbol">(</a><a id="416" href="Relation.Binary.Bundles.html#3530" class="Record">Poset</a><a id="421" class="Symbol">)</a>
<a id="423" class="Keyword">import</a> <a id="430" href="Relation.Binary.Lattice.html" class="Module">Relation.Binary.Lattice</a> <a id="454" class="Symbol">as</a> <a id="457" class="Module">R</a>
<a id="459" class="Keyword">open</a> <a id="464" class="Keyword">import</a> <a id="471" href="Function.Base.html" class="Module">Function.Base</a>
<a id="485" class="Keyword">open</a> <a id="490" class="Keyword">import</a> <a id="497" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="515" class="Keyword">using</a> <a id="521" class="Symbol">(</a><a id="522" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="525" class="Symbol">;</a> <a id="527" href="Data.Product.Base.html#5054" class="Function">swap</a><a id="531" class="Symbol">)</a>
<a id="534" class="Keyword">module</a> <a id="541" href="Algebra.Lattice.Properties.Lattice.html" class="Module">Algebra.Lattice.Properties.Lattice</a>
<a id="578" class="Symbol">{</a><a id="579" href="Algebra.Lattice.Properties.Lattice.html#579" class="Bound">l₁</a> <a id="582" href="Algebra.Lattice.Properties.Lattice.html#582" class="Bound">l₂</a><a id="584" class="Symbol">}</a> <a id="586" class="Symbol">(</a><a id="587" href="Algebra.Lattice.Properties.Lattice.html#587" class="Bound">L</a> <a id="589" class="Symbol">:</a> <a id="591" href="Algebra.Lattice.Bundles.html#4406" class="Record">Lattice</a> <a id="599" href="Algebra.Lattice.Properties.Lattice.html#579" class="Bound">l₁</a> <a id="602" href="Algebra.Lattice.Properties.Lattice.html#582" class="Bound">l₂</a><a id="604" class="Symbol">)</a> <a id="606" class="Keyword">where</a>
<a id="613" class="Keyword">open</a> <a id="618" href="Algebra.Lattice.Bundles.html#4406" class="Module">Lattice</a> <a id="626" href="Algebra.Lattice.Properties.Lattice.html#587" class="Bound">L</a>
<a id="628" class="Keyword">open</a> <a id="633" class="Keyword">import</a> <a id="640" href="Algebra.Definitions.html" class="Module">Algebra.Definitions</a> <a id="660" href="Algebra.Lattice.Bundles.html#4523" class="Field Operator">_≈_</a>
<a id="664" class="Keyword">open</a> <a id="669" class="Keyword">import</a> <a id="676" href="Algebra.Structures.html" class="Module">Algebra.Structures</a> <a id="695" href="Algebra.Lattice.Bundles.html#4523" class="Field Operator">_≈_</a>
<a id="699" class="Keyword">open</a> <a id="704" class="Keyword">import</a> <a id="711" href="Algebra.Lattice.Structures.html" class="Module">Algebra.Lattice.Structures</a> <a id="738" href="Algebra.Lattice.Bundles.html#4523" class="Field Operator">_≈_</a>
<a id="742" class="Keyword">open</a> <a id="747" class="Keyword">import</a> <a id="754" href="Relation.Binary.Reasoning.Setoid.html" class="Module">Relation.Binary.Reasoning.Setoid</a> <a id="787" href="Algebra.Lattice.Bundles.html#4863" class="Function">setoid</a>
<a id="795" class="Comment">------------------------------------------------------------------------</a>
<a id="868" class="Comment">-- _∧_ is a semilattice</a>
<a id="∧-idem"></a><a id="893" href="Algebra.Lattice.Properties.Lattice.html#893" class="Function">∧-idem</a> <a id="900" class="Symbol">:</a> <a id="902" href="Algebra.Definitions.html#3706" class="Function">Idempotent</a> <a id="913" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="917" href="Algebra.Lattice.Properties.Lattice.html#893" class="Function">∧-idem</a> <a id="924" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="926" class="Symbol">=</a> <a id="928" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="936" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="938" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="940" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="953" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="956" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="964" class="Symbol">(</a><a id="965" href="Algebra.Lattice.Structures.html#4392" class="Function">∨-absorbs-∧</a> <a id="977" class="Symbol">_</a> <a id="979" class="Symbol">_)</a> <a id="982" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="986" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="988" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="990" class="Symbol">(</a><a id="991" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="993" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="995" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="997" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="999" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a><a id="1000" class="Symbol">)</a> <a id="1003" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1007" href="Algebra.Lattice.Structures.html#4454" class="Function">∧-absorbs-∨</a> <a id="1019" class="Symbol">_</a> <a id="1021" class="Symbol">_</a> <a id="1023" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1027" href="Algebra.Lattice.Properties.Lattice.html#924" class="Bound">x</a> <a id="1044" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∧-isMagma"></a><a id="1047" href="Algebra.Lattice.Properties.Lattice.html#1047" class="Function">∧-isMagma</a> <a id="1057" class="Symbol">:</a> <a id="1059" href="Algebra.Structures.html#1225" class="Record">IsMagma</a> <a id="1067" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="1071" href="Algebra.Lattice.Properties.Lattice.html#1047" class="Function">∧-isMagma</a> <a id="1081" class="Symbol">=</a> <a id="1083" class="Keyword">record</a>
<a id="1092" class="Symbol">{</a> <a id="1094" href="Algebra.Structures.html#1277" class="Field">isEquivalence</a> <a id="1108" class="Symbol">=</a> <a id="1110" href="Algebra.Lattice.Structures.html#4075" class="Function">isEquivalence</a>
<a id="1126" class="Symbol">;</a> <a id="1128" href="Algebra.Structures.html#1315" class="Field">∙-cong</a> <a id="1142" class="Symbol">=</a> <a id="1144" href="Algebra.Lattice.Structures.html#4282" class="Function">∧-cong</a>
<a id="1153" class="Symbol">}</a>
<a id="∧-isSemigroup"></a><a id="1156" href="Algebra.Lattice.Properties.Lattice.html#1156" class="Function">∧-isSemigroup</a> <a id="1170" class="Symbol">:</a> <a id="1172" href="Algebra.Structures.html#2897" class="Record">IsSemigroup</a> <a id="1184" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="1188" href="Algebra.Lattice.Properties.Lattice.html#1156" class="Function">∧-isSemigroup</a> <a id="1202" class="Symbol">=</a> <a id="1204" class="Keyword">record</a>
<a id="1213" class="Symbol">{</a> <a id="1215" href="Algebra.Structures.html#2953" class="Field">isMagma</a> <a id="1223" class="Symbol">=</a> <a id="1225" href="Algebra.Lattice.Properties.Lattice.html#1047" class="Function">∧-isMagma</a>
<a id="1237" class="Symbol">;</a> <a id="1239" href="Algebra.Structures.html#2977" class="Field">assoc</a> <a id="1247" class="Symbol">=</a> <a id="1249" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a>
<a id="1259" class="Symbol">}</a>
<a id="∧-isBand"></a><a id="1262" href="Algebra.Lattice.Properties.Lattice.html#1262" class="Function">∧-isBand</a> <a id="1271" class="Symbol">:</a> <a id="1273" href="Algebra.Structures.html#3041" class="Record">IsBand</a> <a id="1280" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="1284" href="Algebra.Lattice.Properties.Lattice.html#1262" class="Function">∧-isBand</a> <a id="1293" class="Symbol">=</a> <a id="1295" class="Keyword">record</a>
<a id="1304" class="Symbol">{</a> <a id="1306" href="Algebra.Structures.html#3092" class="Field">isSemigroup</a> <a id="1318" class="Symbol">=</a> <a id="1320" href="Algebra.Lattice.Properties.Lattice.html#1156" class="Function">∧-isSemigroup</a>
<a id="1336" class="Symbol">;</a> <a id="1338" href="Algebra.Structures.html#3124" class="Field">idem</a> <a id="1350" class="Symbol">=</a> <a id="1352" href="Algebra.Lattice.Properties.Lattice.html#893" class="Function">∧-idem</a>
<a id="1361" class="Symbol">}</a>
<a id="∧-isSemilattice"></a><a id="1364" href="Algebra.Lattice.Properties.Lattice.html#1364" class="Function">∧-isSemilattice</a> <a id="1380" class="Symbol">:</a> <a id="1382" href="Algebra.Lattice.Structures.html#1160" class="Record">IsSemilattice</a> <a id="1396" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="1400" href="Algebra.Lattice.Properties.Lattice.html#1364" class="Function">∧-isSemilattice</a> <a id="1416" class="Symbol">=</a> <a id="1418" class="Keyword">record</a>
<a id="1427" class="Symbol">{</a> <a id="1429" href="Algebra.Lattice.Structures.html#1218" class="Field">isBand</a> <a id="1436" class="Symbol">=</a> <a id="1438" href="Algebra.Lattice.Properties.Lattice.html#1262" class="Function">∧-isBand</a>
<a id="1449" class="Symbol">;</a> <a id="1451" href="Algebra.Lattice.Structures.html#1240" class="Field">comm</a> <a id="1458" class="Symbol">=</a> <a id="1460" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a>
<a id="1469" class="Symbol">}</a>
<a id="∧-semilattice"></a><a id="1472" href="Algebra.Lattice.Properties.Lattice.html#1472" class="Function">∧-semilattice</a> <a id="1486" class="Symbol">:</a> <a id="1488" href="Algebra.Lattice.Bundles.html#1223" class="Record">Semilattice</a> <a id="1500" href="Algebra.Lattice.Properties.Lattice.html#579" class="Bound">l₁</a> <a id="1503" href="Algebra.Lattice.Properties.Lattice.html#582" class="Bound">l₂</a>
<a id="1506" href="Algebra.Lattice.Properties.Lattice.html#1472" class="Function">∧-semilattice</a> <a id="1520" class="Symbol">=</a> <a id="1522" class="Keyword">record</a>
<a id="1531" class="Symbol">{</a> <a id="1533" href="Algebra.Lattice.Bundles.html#1399" class="Field">isSemilattice</a> <a id="1547" class="Symbol">=</a> <a id="1549" href="Algebra.Lattice.Properties.Lattice.html#1364" class="Function">∧-isSemilattice</a>
<a id="1567" class="Symbol">}</a>
<a id="1570" class="Keyword">open</a> <a id="1575" href="Algebra.Lattice.Properties.Semilattice.html" class="Module">SemilatticeProperties</a> <a id="1597" href="Algebra.Lattice.Properties.Lattice.html#1472" class="Function">∧-semilattice</a> <a id="1611" class="Keyword">public</a>
<a id="1620" class="Keyword">using</a>
<a id="1628" class="Symbol">(</a> <a id="1630" href="Algebra.Lattice.Properties.Semilattice.html#1201" class="Function">∧-isOrderTheoreticMeetSemilattice</a>
<a id="1666" class="Symbol">;</a> <a id="1668" href="Algebra.Lattice.Properties.Semilattice.html#1413" class="Function">∧-isOrderTheoreticJoinSemilattice</a>
<a id="1704" class="Symbol">;</a> <a id="1706" href="Algebra.Lattice.Properties.Semilattice.html#1673" class="Function">∧-orderTheoreticMeetSemilattice</a>
<a id="1740" class="Symbol">;</a> <a id="1742" href="Algebra.Lattice.Properties.Semilattice.html#1835" class="Function">∧-orderTheoreticJoinSemilattice</a>
<a id="1776" class="Symbol">)</a>
<a id="1779" class="Comment">------------------------------------------------------------------------</a>
<a id="1852" class="Comment">-- _∨_ is a semilattice</a>
<a id="∨-idem"></a><a id="1877" href="Algebra.Lattice.Properties.Lattice.html#1877" class="Function">∨-idem</a> <a id="1884" class="Symbol">:</a> <a id="1886" href="Algebra.Definitions.html#3706" class="Function">Idempotent</a> <a id="1897" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="1901" href="Algebra.Lattice.Properties.Lattice.html#1877" class="Function">∨-idem</a> <a id="1908" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1910" class="Symbol">=</a> <a id="1912" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="1920" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1922" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="1924" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1931" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="1934" href="Algebra.Lattice.Structures.html#4639" class="Function">∨-congˡ</a> <a id="1942" class="Symbol">(</a><a id="1943" href="Algebra.Lattice.Properties.Lattice.html#893" class="Function">∧-idem</a> <a id="1950" class="Symbol">_)</a> <a id="1953" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
<a id="1957" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1959" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="1961" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1963" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="1965" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="1968" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="1972" href="Algebra.Lattice.Structures.html#4392" class="Function">∨-absorbs-∧</a> <a id="1984" class="Symbol">_</a> <a id="1986" class="Symbol">_</a> <a id="1988" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="1992" href="Algebra.Lattice.Properties.Lattice.html#1908" class="Bound">x</a> <a id="2003" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="∨-isMagma"></a><a id="2006" href="Algebra.Lattice.Properties.Lattice.html#2006" class="Function">∨-isMagma</a> <a id="2016" class="Symbol">:</a> <a id="2018" href="Algebra.Structures.html#1225" class="Record">IsMagma</a> <a id="2026" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="2030" href="Algebra.Lattice.Properties.Lattice.html#2006" class="Function">∨-isMagma</a> <a id="2040" class="Symbol">=</a> <a id="2042" class="Keyword">record</a>
<a id="2051" class="Symbol">{</a> <a id="2053" href="Algebra.Structures.html#1277" class="Field">isEquivalence</a> <a id="2067" class="Symbol">=</a> <a id="2069" href="Algebra.Lattice.Structures.html#4075" class="Function">isEquivalence</a>
<a id="2085" class="Symbol">;</a> <a id="2087" href="Algebra.Structures.html#1315" class="Field">∙-cong</a> <a id="2101" class="Symbol">=</a> <a id="2103" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a>
<a id="2112" class="Symbol">}</a>
<a id="∨-isSemigroup"></a><a id="2115" href="Algebra.Lattice.Properties.Lattice.html#2115" class="Function">∨-isSemigroup</a> <a id="2129" class="Symbol">:</a> <a id="2131" href="Algebra.Structures.html#2897" class="Record">IsSemigroup</a> <a id="2143" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="2147" href="Algebra.Lattice.Properties.Lattice.html#2115" class="Function">∨-isSemigroup</a> <a id="2161" class="Symbol">=</a> <a id="2163" class="Keyword">record</a>
<a id="2172" class="Symbol">{</a> <a id="2174" href="Algebra.Structures.html#2953" class="Field">isMagma</a> <a id="2182" class="Symbol">=</a> <a id="2184" href="Algebra.Lattice.Properties.Lattice.html#2006" class="Function">∨-isMagma</a>
<a id="2196" class="Symbol">;</a> <a id="2198" href="Algebra.Structures.html#2977" class="Field">assoc</a> <a id="2206" class="Symbol">=</a> <a id="2208" href="Algebra.Lattice.Structures.html#4147" class="Function">∨-assoc</a>
<a id="2218" class="Symbol">}</a>
<a id="∨-isBand"></a><a id="2221" href="Algebra.Lattice.Properties.Lattice.html#2221" class="Function">∨-isBand</a> <a id="2230" class="Symbol">:</a> <a id="2232" href="Algebra.Structures.html#3041" class="Record">IsBand</a> <a id="2239" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="2243" href="Algebra.Lattice.Properties.Lattice.html#2221" class="Function">∨-isBand</a> <a id="2252" class="Symbol">=</a> <a id="2254" class="Keyword">record</a>
<a id="2263" class="Symbol">{</a> <a id="2265" href="Algebra.Structures.html#3092" class="Field">isSemigroup</a> <a id="2277" class="Symbol">=</a> <a id="2279" href="Algebra.Lattice.Properties.Lattice.html#2115" class="Function">∨-isSemigroup</a>
<a id="2295" class="Symbol">;</a> <a id="2297" href="Algebra.Structures.html#3124" class="Field">idem</a> <a id="2309" class="Symbol">=</a> <a id="2311" href="Algebra.Lattice.Properties.Lattice.html#1877" class="Function">∨-idem</a>
<a id="2320" class="Symbol">}</a>
<a id="∨-isSemilattice"></a><a id="2323" href="Algebra.Lattice.Properties.Lattice.html#2323" class="Function">∨-isSemilattice</a> <a id="2339" class="Symbol">:</a> <a id="2341" href="Algebra.Lattice.Structures.html#1160" class="Record">IsSemilattice</a> <a id="2355" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="2359" href="Algebra.Lattice.Properties.Lattice.html#2323" class="Function">∨-isSemilattice</a> <a id="2375" class="Symbol">=</a> <a id="2377" class="Keyword">record</a>
<a id="2386" class="Symbol">{</a> <a id="2388" href="Algebra.Lattice.Structures.html#1218" class="Field">isBand</a> <a id="2395" class="Symbol">=</a> <a id="2397" href="Algebra.Lattice.Properties.Lattice.html#2221" class="Function">∨-isBand</a>
<a id="2408" class="Symbol">;</a> <a id="2410" href="Algebra.Lattice.Structures.html#1240" class="Field">comm</a> <a id="2417" class="Symbol">=</a> <a id="2419" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a>
<a id="2428" class="Symbol">}</a>
<a id="∨-semilattice"></a><a id="2431" href="Algebra.Lattice.Properties.Lattice.html#2431" class="Function">∨-semilattice</a> <a id="2445" class="Symbol">:</a> <a id="2447" href="Algebra.Lattice.Bundles.html#1223" class="Record">Semilattice</a> <a id="2459" href="Algebra.Lattice.Properties.Lattice.html#579" class="Bound">l₁</a> <a id="2462" href="Algebra.Lattice.Properties.Lattice.html#582" class="Bound">l₂</a>
<a id="2465" href="Algebra.Lattice.Properties.Lattice.html#2431" class="Function">∨-semilattice</a> <a id="2479" class="Symbol">=</a> <a id="2481" class="Keyword">record</a>
<a id="2490" class="Symbol">{</a> <a id="2492" href="Algebra.Lattice.Bundles.html#1399" class="Field">isSemilattice</a> <a id="2506" class="Symbol">=</a> <a id="2508" href="Algebra.Lattice.Properties.Lattice.html#2323" class="Function">∨-isSemilattice</a>
<a id="2526" class="Symbol">}</a>
<a id="2529" class="Keyword">open</a> <a id="2534" href="Algebra.Lattice.Properties.Semilattice.html" class="Module">SemilatticeProperties</a> <a id="2556" href="Algebra.Lattice.Properties.Lattice.html#2431" class="Function">∨-semilattice</a> <a id="2570" class="Keyword">public</a>
<a id="2579" class="Keyword">using</a> <a id="2585" class="Symbol">()</a>
<a id="2590" class="Keyword">renaming</a>
<a id="2601" class="Symbol">(</a> <a id="2603" href="Algebra.Lattice.Properties.Semilattice.html#1201" class="Function">∧-isOrderTheoreticMeetSemilattice</a> <a id="2637" class="Symbol">to</a> <a id="2640" class="Function">∨-isOrderTheoreticMeetSemilattice</a>
<a id="2676" class="Symbol">;</a> <a id="2678" href="Algebra.Lattice.Properties.Semilattice.html#1413" class="Function">∧-isOrderTheoreticJoinSemilattice</a> <a id="2712" class="Symbol">to</a> <a id="2715" class="Function">∨-isOrderTheoreticJoinSemilattice</a>
<a id="2751" class="Symbol">;</a> <a id="2753" href="Algebra.Lattice.Properties.Semilattice.html#1673" class="Function">∧-orderTheoreticMeetSemilattice</a> <a id="2787" class="Symbol">to</a> <a id="2790" class="Function">∨-orderTheoreticMeetSemilattice</a>
<a id="2824" class="Symbol">;</a> <a id="2826" href="Algebra.Lattice.Properties.Semilattice.html#1835" class="Function">∧-orderTheoreticJoinSemilattice</a> <a id="2860" class="Symbol">to</a> <a id="2863" class="Function">∨-orderTheoreticJoinSemilattice</a>
<a id="2897" class="Symbol">)</a>
<a id="2900" class="Comment">------------------------------------------------------------------------</a>
<a id="2973" class="Comment">-- The dual construction is also a lattice.</a>
<a id="∧-∨-isLattice"></a><a id="3018" href="Algebra.Lattice.Properties.Lattice.html#3018" class="Function">∧-∨-isLattice</a> <a id="3032" class="Symbol">:</a> <a id="3034" href="Algebra.Lattice.Structures.html#4019" class="Record">IsLattice</a> <a id="3044" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a> <a id="3048" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="3052" href="Algebra.Lattice.Properties.Lattice.html#3018" class="Function">∧-∨-isLattice</a> <a id="3066" class="Symbol">=</a> <a id="3068" class="Keyword">record</a>
<a id="3077" class="Symbol">{</a> <a id="3079" href="Algebra.Lattice.Structures.html#4075" class="Field">isEquivalence</a> <a id="3093" class="Symbol">=</a> <a id="3095" href="Algebra.Lattice.Structures.html#4075" class="Function">isEquivalence</a>
<a id="3111" class="Symbol">;</a> <a id="3113" href="Algebra.Lattice.Structures.html#4113" class="Field">∨-comm</a> <a id="3127" class="Symbol">=</a> <a id="3129" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a>
<a id="3138" class="Symbol">;</a> <a id="3140" href="Algebra.Lattice.Structures.html#4147" class="Field">∨-assoc</a> <a id="3154" class="Symbol">=</a> <a id="3156" href="Algebra.Lattice.Structures.html#4248" class="Function">∧-assoc</a>
<a id="3166" class="Symbol">;</a> <a id="3168" href="Algebra.Lattice.Structures.html#4181" class="Field">∨-cong</a> <a id="3182" class="Symbol">=</a> <a id="3184" href="Algebra.Lattice.Structures.html#4282" class="Function">∧-cong</a>
<a id="3193" class="Symbol">;</a> <a id="3195" href="Algebra.Lattice.Structures.html#4214" class="Field">∧-comm</a> <a id="3209" class="Symbol">=</a> <a id="3211" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a>
<a id="3220" class="Symbol">;</a> <a id="3222" href="Algebra.Lattice.Structures.html#4248" class="Field">∧-assoc</a> <a id="3236" class="Symbol">=</a> <a id="3238" href="Algebra.Lattice.Structures.html#4147" class="Function">∨-assoc</a>
<a id="3248" class="Symbol">;</a> <a id="3250" href="Algebra.Lattice.Structures.html#4282" class="Field">∧-cong</a> <a id="3264" class="Symbol">=</a> <a id="3266" href="Algebra.Lattice.Structures.html#4181" class="Function">∨-cong</a>
<a id="3275" class="Symbol">;</a> <a id="3277" href="Algebra.Lattice.Structures.html#4315" class="Field">absorptive</a> <a id="3291" class="Symbol">=</a> <a id="3293" href="Data.Product.Base.html#5054" class="Function">swap</a> <a id="3298" href="Algebra.Lattice.Structures.html#4315" class="Function">absorptive</a>
<a id="3311" class="Symbol">}</a>
<a id="∧-∨-lattice"></a><a id="3314" href="Algebra.Lattice.Properties.Lattice.html#3314" class="Function">∧-∨-lattice</a> <a id="3326" class="Symbol">:</a> <a id="3328" href="Algebra.Lattice.Bundles.html#4406" class="Record">Lattice</a> <a id="3336" class="Symbol">_</a> <a id="3338" class="Symbol">_</a>
<a id="3340" href="Algebra.Lattice.Properties.Lattice.html#3314" class="Function">∧-∨-lattice</a> <a id="3352" class="Symbol">=</a> <a id="3354" class="Keyword">record</a>
<a id="3363" class="Symbol">{</a> <a id="3365" href="Algebra.Lattice.Bundles.html#4609" class="Field">isLattice</a> <a id="3375" class="Symbol">=</a> <a id="3377" href="Algebra.Lattice.Properties.Lattice.html#3018" class="Function">∧-∨-isLattice</a>
<a id="3393" class="Symbol">}</a>
<a id="3396" class="Comment">------------------------------------------------------------------------</a>
<a id="3469" class="Comment">-- Every algebraic lattice can be turned into an order-theoretic one.</a>
<a id="3540" class="Keyword">open</a> <a id="3545" href="Algebra.Lattice.Properties.Semilattice.html" class="Module">SemilatticeProperties</a> <a id="3567" href="Algebra.Lattice.Properties.Lattice.html#1472" class="Function">∧-semilattice</a> <a id="3581" class="Keyword">public</a> <a id="3588" class="Keyword">using</a> <a id="3594" class="Symbol">(</a><a id="3595" href="Algebra.Lattice.Properties.Semilattice.html#884" class="Function">poset</a><a id="3600" class="Symbol">)</a>
<a id="3602" class="Keyword">open</a> <a id="3607" href="Relation.Binary.Bundles.html#3530" class="Module">Poset</a> <a id="3613" href="Algebra.Lattice.Properties.Semilattice.html#884" class="Function">poset</a> <a id="3619" class="Keyword">using</a> <a id="3625" class="Symbol">(</a><a id="3626" href="Relation.Binary.Bundles.html#3669" class="Field Operator">_≤_</a><a id="3629" class="Symbol">;</a> <a id="3631" href="Relation.Binary.Bundles.html#3705" class="Field">isPartialOrder</a><a id="3645" class="Symbol">)</a>
<a id="∨-∧-isOrderTheoreticLattice"></a><a id="3648" href="Algebra.Lattice.Properties.Lattice.html#3648" class="Function">∨-∧-isOrderTheoreticLattice</a> <a id="3676" class="Symbol">:</a> <a id="3678" href="Relation.Binary.Lattice.Structures.html#2895" class="Record">R.IsLattice</a> <a id="3690" href="Algebra.Lattice.Bundles.html#4523" class="Field Operator">_≈_</a> <a id="3694" href="Relation.Binary.Bundles.html#3669" class="Function Operator">_≤_</a> <a id="3698" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a> <a id="3702" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">_∧_</a>
<a id="3706" href="Algebra.Lattice.Properties.Lattice.html#3648" class="Function">∨-∧-isOrderTheoreticLattice</a> <a id="3734" class="Symbol">=</a> <a id="3736" class="Keyword">record</a>
<a id="3745" class="Symbol">{</a> <a id="3747" href="Relation.Binary.Lattice.Structures.html#3057" class="Field">isPartialOrder</a> <a id="3762" class="Symbol">=</a> <a id="3764" href="Relation.Binary.Bundles.html#3705" class="Function">isPartialOrder</a>
<a id="3781" class="Symbol">;</a> <a id="3783" href="Relation.Binary.Lattice.Structures.html#3101" class="Field">supremum</a> <a id="3798" class="Symbol">=</a> <a id="3800" href="Algebra.Lattice.Properties.Lattice.html#4531" class="Function">supremum</a>
<a id="3811" class="Symbol">;</a> <a id="3813" href="Relation.Binary.Lattice.Structures.html#3139" class="Field">infimum</a> <a id="3828" class="Symbol">=</a> <a id="3830" href="Relation.Binary.Lattice.Structures.html#2104" class="Function">infimum</a>
<a id="3840" class="Symbol">}</a>
<a id="3844" class="Keyword">where</a>
<a id="3852" class="Keyword">open</a> <a id="3857" href="Relation.Binary.Lattice.Bundles.html#2108" class="Module">R.MeetSemilattice</a> <a id="3875" href="Algebra.Lattice.Properties.Semilattice.html#1673" class="Function">∧-orderTheoreticMeetSemilattice</a> <a id="3907" class="Keyword">using</a> <a id="3913" class="Symbol">(</a><a id="3914" href="Relation.Binary.Lattice.Structures.html#2104" class="Function">infimum</a><a id="3921" class="Symbol">)</a>
<a id="3925" class="Keyword">open</a> <a id="3930" href="Relation.Binary.Lattice.Bundles.html#709" class="Module">R.JoinSemilattice</a> <a id="3948" href="Algebra.Lattice.Properties.Lattice.html#2863" class="Function">∨-orderTheoreticJoinSemilattice</a> <a id="3980" class="Keyword">using</a> <a id="3986" class="Symbol">(</a><a id="3987" href="Relation.Binary.Lattice.Structures.html#1178" class="Function">x≤x∨y</a><a id="3992" class="Symbol">;</a> <a id="3994" href="Relation.Binary.Lattice.Structures.html#1259" class="Function">y≤x∨y</a><a id="3999" class="Symbol">;</a> <a id="4001" href="Relation.Binary.Lattice.Structures.html#1340" class="Function">∨-least</a><a id="4008" class="Symbol">)</a>
<a id="4014" class="Keyword">renaming</a> <a id="4023" class="Symbol">(</a><a id="4024" href="Relation.Binary.Lattice.Bundles.html#909" class="Field Operator">_≤_</a> <a id="4028" class="Symbol">to</a> <a id="4031" class="Field Operator">_≤′_</a><a id="4035" class="Symbol">)</a>
<a id="4040" class="Comment">-- An alternative but equivalent interpretation of the order _≤_.</a>
<a id="4109" href="Algebra.Lattice.Properties.Lattice.html#4109" class="Function">sound</a> <a id="4115" class="Symbol">:</a> <a id="4117" class="Symbol">∀</a> <a id="4119" class="Symbol">{</a><a id="4120" href="Algebra.Lattice.Properties.Lattice.html#4120" class="Bound">x</a> <a id="4122" href="Algebra.Lattice.Properties.Lattice.html#4122" class="Bound">y</a><a id="4123" class="Symbol">}</a> <a id="4125" class="Symbol">→</a> <a id="4127" href="Algebra.Lattice.Properties.Lattice.html#4120" class="Bound">x</a> <a id="4129" href="Algebra.Lattice.Properties.Lattice.html#4031" class="Function Operator">≤′</a> <a id="4132" href="Algebra.Lattice.Properties.Lattice.html#4122" class="Bound">y</a> <a id="4134" class="Symbol">→</a> <a id="4136" href="Algebra.Lattice.Properties.Lattice.html#4120" class="Bound">x</a> <a id="4138" href="Relation.Binary.Bundles.html#3669" class="Function Operator">≤</a> <a id="4140" href="Algebra.Lattice.Properties.Lattice.html#4122" class="Bound">y</a>
<a id="4144" href="Algebra.Lattice.Properties.Lattice.html#4109" class="Function">sound</a> <a id="4150" class="Symbol">{</a><a id="4151" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a><a id="4152" class="Symbol">}</a> <a id="4154" class="Symbol">{</a><a id="4155" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a><a id="4156" class="Symbol">}</a> <a id="4158" href="Algebra.Lattice.Properties.Lattice.html#4158" class="Bound">y≈y∨x</a> <a id="4164" class="Symbol">=</a> <a id="4166" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4170" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="4172" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4182" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4184" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="4186" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a> <a id="4195" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4198" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="4206" href="Algebra.Lattice.Properties.Lattice.html#4158" class="Bound">y≈y∨x</a> <a id="4212" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4218" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4220" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="4222" class="Symbol">(</a><a id="4223" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a> <a id="4225" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="4227" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a><a id="4228" class="Symbol">)</a> <a id="4231" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4234" href="Algebra.Lattice.Structures.html#4516" class="Function">∧-congˡ</a> <a id="4242" class="Symbol">(</a><a id="4243" href="Algebra.Lattice.Structures.html#4113" class="Function">∨-comm</a> <a id="4250" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a> <a id="4252" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a><a id="4253" class="Symbol">)</a> <a id="4255" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4261" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4263" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="4265" class="Symbol">(</a><a id="4266" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4268" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="4270" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a><a id="4271" class="Symbol">)</a> <a id="4274" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4277" href="Algebra.Lattice.Structures.html#4454" class="Function">∧-absorbs-∨</a> <a id="4289" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4291" href="Algebra.Lattice.Properties.Lattice.html#4155" class="Bound">y</a> <a id="4293" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4299" href="Algebra.Lattice.Properties.Lattice.html#4151" class="Bound">x</a> <a id="4312" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4317" href="Algebra.Lattice.Properties.Lattice.html#4317" class="Function">complete</a> <a id="4326" class="Symbol">:</a> <a id="4328" class="Symbol">∀</a> <a id="4330" class="Symbol">{</a><a id="4331" href="Algebra.Lattice.Properties.Lattice.html#4331" class="Bound">x</a> <a id="4333" href="Algebra.Lattice.Properties.Lattice.html#4333" class="Bound">y</a><a id="4334" class="Symbol">}</a> <a id="4336" class="Symbol">→</a> <a id="4338" href="Algebra.Lattice.Properties.Lattice.html#4331" class="Bound">x</a> <a id="4340" href="Relation.Binary.Bundles.html#3669" class="Function Operator">≤</a> <a id="4342" href="Algebra.Lattice.Properties.Lattice.html#4333" class="Bound">y</a> <a id="4344" class="Symbol">→</a> <a id="4346" href="Algebra.Lattice.Properties.Lattice.html#4331" class="Bound">x</a> <a id="4348" href="Algebra.Lattice.Properties.Lattice.html#4031" class="Function Operator">≤′</a> <a id="4351" href="Algebra.Lattice.Properties.Lattice.html#4333" class="Bound">y</a>
<a id="4355" href="Algebra.Lattice.Properties.Lattice.html#4317" class="Function">complete</a> <a id="4364" class="Symbol">{</a><a id="4365" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a><a id="4366" class="Symbol">}</a> <a id="4368" class="Symbol">{</a><a id="4369" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a><a id="4370" class="Symbol">}</a> <a id="4372" href="Algebra.Lattice.Properties.Lattice.html#4372" class="Bound">x≈x∧y</a> <a id="4378" class="Symbol">=</a> <a id="4380" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="4384" href="Function.Base.html#1974" class="Function Operator">$</a> <a id="4386" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4396" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4398" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="4400" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a> <a id="4409" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4412" href="Algebra.Lattice.Structures.html#4639" class="Function">∨-congˡ</a> <a id="4420" href="Algebra.Lattice.Properties.Lattice.html#4372" class="Bound">x≈x∧y</a> <a id="4426" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4432" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4434" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="4436" class="Symbol">(</a><a id="4437" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a> <a id="4439" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="4441" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a><a id="4442" class="Symbol">)</a> <a id="4445" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4448" href="Algebra.Lattice.Structures.html#4639" class="Function">∨-congˡ</a> <a id="4456" class="Symbol">(</a><a id="4457" href="Algebra.Lattice.Structures.html#4214" class="Function">∧-comm</a> <a id="4464" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a> <a id="4466" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a><a id="4467" class="Symbol">)</a> <a id="4469" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4475" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4477" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">∨</a> <a id="4479" class="Symbol">(</a><a id="4480" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4482" href="Algebra.Lattice.Bundles.html#4581" class="Field Operator">∧</a> <a id="4484" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a><a id="4485" class="Symbol">)</a> <a id="4488" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4491" href="Algebra.Lattice.Structures.html#4392" class="Function">∨-absorbs-∧</a> <a id="4503" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4505" href="Algebra.Lattice.Properties.Lattice.html#4365" class="Bound">x</a> <a id="4507" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4513" href="Algebra.Lattice.Properties.Lattice.html#4369" class="Bound">y</a> <a id="4526" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4531" href="Algebra.Lattice.Properties.Lattice.html#4531" class="Function">supremum</a> <a id="4540" class="Symbol">:</a> <a id="4542" href="Relation.Binary.Lattice.Definitions.html#763" class="Function">R.Supremum</a> <a id="4553" href="Relation.Binary.Bundles.html#3669" class="Function Operator">_≤_</a> <a id="4557" href="Algebra.Lattice.Bundles.html#4553" class="Field Operator">_∨_</a>
<a id="4563" href="Algebra.Lattice.Properties.Lattice.html#4531" class="Function">supremum</a> <a id="4572" href="Algebra.Lattice.Properties.Lattice.html#4572" class="Bound">x</a> <a id="4574" href="Algebra.Lattice.Properties.Lattice.html#4574" class="Bound">y</a> <a id="4576" class="Symbol">=</a>
<a id="4583" href="Algebra.Lattice.Properties.Lattice.html#4109" class="Function">sound</a> <a id="4589" class="Symbol">(</a><a id="4590" href="Relation.Binary.Lattice.Structures.html#1178" class="Function">x≤x∨y</a> <a id="4596" href="Algebra.Lattice.Properties.Lattice.html#4572" class="Bound">x</a> <a id="4598" href="Algebra.Lattice.Properties.Lattice.html#4574" class="Bound">y</a><a id="4599" class="Symbol">)</a> <a id="4601" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a>
<a id="4608" href="Algebra.Lattice.Properties.Lattice.html#4109" class="Function">sound</a> <a id="4614" class="Symbol">(</a><a id="4615" href="Relation.Binary.Lattice.Structures.html#1259" class="Function">y≤x∨y</a> <a id="4621" href="Algebra.Lattice.Properties.Lattice.html#4572" class="Bound">x</a> <a id="4623" href="Algebra.Lattice.Properties.Lattice.html#4574" class="Bound">y</a><a id="4624" class="Symbol">)</a> <a id="4626" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a>
<a id="4633" class="Symbol">λ</a> <a id="4635" href="Algebra.Lattice.Properties.Lattice.html#4635" class="Bound">z</a> <a id="4637" href="Algebra.Lattice.Properties.Lattice.html#4637" class="Bound">x≤z</a> <a id="4641" href="Algebra.Lattice.Properties.Lattice.html#4641" class="Bound">y≤z</a> <a id="4645" class="Symbol">→</a> <a id="4647" href="Algebra.Lattice.Properties.Lattice.html#4109" class="Function">sound</a> <a id="4653" class="Symbol">(</a><a id="4654" href="Relation.Binary.Lattice.Structures.html#1340" class="Function">∨-least</a> <a id="4662" class="Symbol">(</a><a id="4663" href="Algebra.Lattice.Properties.Lattice.html#4317" class="Function">complete</a> <a id="4672" href="Algebra.Lattice.Properties.Lattice.html#4637" class="Bound">x≤z</a><a id="4675" class="Symbol">)</a> <a id="4677" class="Symbol">(</a><a id="4678" href="Algebra.Lattice.Properties.Lattice.html#4317" class="Function">complete</a> <a id="4687" href="Algebra.Lattice.Properties.Lattice.html#4641" class="Bound">y≤z</a><a id="4690" class="Symbol">))</a>
<a id="∨-∧-orderTheoreticLattice"></a><a id="4694" href="Algebra.Lattice.Properties.Lattice.html#4694" class="Function">∨-∧-orderTheoreticLattice</a> <a id="4720" class="Symbol">:</a> <a id="4722" href="Relation.Binary.Lattice.Bundles.html#3497" class="Record">R.Lattice</a> <a id="4732" class="Symbol">_</a> <a id="4734" class="Symbol">_</a> <a id="4736" class="Symbol">_</a>
<a id="4738" href="Algebra.Lattice.Properties.Lattice.html#4694" class="Function">∨-∧-orderTheoreticLattice</a> <a id="4764" class="Symbol">=</a> <a id="4766" class="Keyword">record</a>
<a id="4775" class="Symbol">{</a> <a id="4777" href="Relation.Binary.Lattice.Bundles.html#3852" class="Field">isLattice</a> <a id="4787" class="Symbol">=</a> <a id="4789" href="Algebra.Lattice.Properties.Lattice.html#3648" class="Function">∨-∧-isOrderTheoreticLattice</a>
<a id="4819" class="Symbol">}</a>
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