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bsc-leon-vatthauer/agda/bsc-thesis/Relation.Binary.Lattice.Definitions.html
Leon Vatthauer 2bf4655eae
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<html><head><meta charset="utf-8"><title>Relation.Binary.Lattice.Definitions</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">-- Definitions for order-theoretic lattices</a>
<a id="150" class="Comment">------------------------------------------------------------------------</a>
<a id="224" class="Comment">-- The contents of this module should be accessed via</a>
<a id="278" class="Comment">-- `Relation.Binary.Lattice`.</a>
<a id="309" class="Symbol">{-#</a> <a id="313" class="Keyword">OPTIONS</a> <a id="321" class="Pragma">--cubical-compatible</a> <a id="342" class="Pragma">--safe</a> <a id="349" class="Symbol">#-}</a>
<a id="354" class="Keyword">module</a> <a id="361" href="Relation.Binary.Lattice.Definitions.html" class="Module">Relation.Binary.Lattice.Definitions</a> <a id="397" class="Keyword">where</a>
<a id="404" class="Keyword">open</a> <a id="409" class="Keyword">import</a> <a id="416" href="Algebra.Core.html" class="Module">Algebra.Core</a>
<a id="429" class="Keyword">open</a> <a id="434" class="Keyword">import</a> <a id="441" href="Data.Product.Base.html" class="Module">Data.Product.Base</a> <a id="459" class="Keyword">using</a> <a id="465" class="Symbol">(</a><a id="466" href="Data.Product.Base.html#1618" class="Function Operator">_×_</a><a id="469" class="Symbol">;</a> <a id="471" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">_,_</a><a id="474" class="Symbol">)</a>
<a id="476" class="Keyword">open</a> <a id="481" class="Keyword">import</a> <a id="488" href="Function.Base.html" class="Module">Function.Base</a> <a id="502" class="Keyword">using</a> <a id="508" class="Symbol">(</a><a id="509" href="Function.Base.html#1638" class="Function">flip</a><a id="513" class="Symbol">)</a>
<a id="515" class="Keyword">open</a> <a id="520" class="Keyword">import</a> <a id="527" href="Relation.Binary.Core.html" class="Module">Relation.Binary.Core</a> <a id="548" class="Keyword">using</a> <a id="554" class="Symbol">(</a><a id="555" href="Relation.Binary.Core.html#896" class="Function">Rel</a><a id="558" class="Symbol">)</a>
<a id="560" class="Keyword">open</a> <a id="565" class="Keyword">import</a> <a id="572" href="Level.html" class="Module">Level</a> <a id="578" class="Keyword">using</a> <a id="584" class="Symbol">(</a><a id="585" href="Agda.Primitive.html#742" class="Postulate">Level</a><a id="590" class="Symbol">)</a>
<a id="593" class="Keyword">private</a>
<a id="603" class="Keyword">variable</a>
<a id="616" href="Relation.Binary.Lattice.Definitions.html#616" class="Generalizable">a</a> <a id="618" href="Relation.Binary.Lattice.Definitions.html#618" class="Generalizable">ℓ</a> <a id="620" class="Symbol">:</a> <a id="622" href="Agda.Primitive.html#742" class="Postulate">Level</a>
<a id="632" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="634" class="Symbol">:</a> <a id="636" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="640" href="Relation.Binary.Lattice.Definitions.html#616" class="Generalizable">a</a>
<a id="643" class="Comment">------------------------------------------------------------------------</a>
<a id="716" class="Comment">-- Relationships between orders and operators</a>
<a id="Supremum"></a><a id="763" href="Relation.Binary.Lattice.Definitions.html#763" class="Function">Supremum</a> <a id="772" class="Symbol">:</a> <a id="774" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="778" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="780" href="Relation.Binary.Lattice.Definitions.html#618" class="Generalizable">ℓ</a> <a id="782" class="Symbol">→</a> <a id="784" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="788" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="790" class="Symbol">→</a> <a id="792" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="796" class="Symbol">_</a>
<a id="798" href="Relation.Binary.Lattice.Definitions.html#763" class="Function">Supremum</a> <a id="807" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">_≤_</a> <a id="811" href="Relation.Binary.Lattice.Definitions.html#811" class="Bound Operator">_∨_</a> <a id="815" class="Symbol">=</a>
<a id="819" class="Symbol">∀</a> <a id="821" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="823" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a> <a id="825" class="Symbol">→</a> <a id="827" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="829" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">≤</a> <a id="831" class="Symbol">(</a><a id="832" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="834" href="Relation.Binary.Lattice.Definitions.html#811" class="Bound Operator">∨</a> <a id="836" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a><a id="837" class="Symbol">)</a> <a id="839" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="841" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a> <a id="843" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">≤</a> <a id="845" class="Symbol">(</a><a id="846" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="848" href="Relation.Binary.Lattice.Definitions.html#811" class="Bound Operator">∨</a> <a id="850" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a><a id="851" class="Symbol">)</a> <a id="853" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="855" class="Symbol">∀</a> <a id="857" href="Relation.Binary.Lattice.Definitions.html#857" class="Bound">z</a> <a id="859" class="Symbol">→</a> <a id="861" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="863" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">≤</a> <a id="865" href="Relation.Binary.Lattice.Definitions.html#857" class="Bound">z</a> <a id="867" class="Symbol">→</a> <a id="869" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a> <a id="871" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">≤</a> <a id="873" href="Relation.Binary.Lattice.Definitions.html#857" class="Bound">z</a> <a id="875" class="Symbol">→</a> <a id="877" class="Symbol">(</a><a id="878" href="Relation.Binary.Lattice.Definitions.html#821" class="Bound">x</a> <a id="880" href="Relation.Binary.Lattice.Definitions.html#811" class="Bound Operator">∨</a> <a id="882" href="Relation.Binary.Lattice.Definitions.html#823" class="Bound">y</a><a id="883" class="Symbol">)</a> <a id="885" href="Relation.Binary.Lattice.Definitions.html#807" class="Bound Operator">≤</a> <a id="887" href="Relation.Binary.Lattice.Definitions.html#857" class="Bound">z</a>
<a id="Infimum"></a><a id="890" href="Relation.Binary.Lattice.Definitions.html#890" class="Function">Infimum</a> <a id="898" class="Symbol">:</a> <a id="900" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="904" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="906" href="Relation.Binary.Lattice.Definitions.html#618" class="Generalizable">ℓ</a> <a id="908" class="Symbol">→</a> <a id="910" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="914" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="916" class="Symbol">→</a> <a id="918" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="922" class="Symbol">_</a>
<a id="924" href="Relation.Binary.Lattice.Definitions.html#890" class="Function">Infimum</a> <a id="932" href="Relation.Binary.Lattice.Definitions.html#932" class="Bound Operator">_≤_</a> <a id="936" class="Symbol">=</a> <a id="938" href="Relation.Binary.Lattice.Definitions.html#763" class="Function">Supremum</a> <a id="947" class="Symbol">(</a><a id="948" href="Function.Base.html#1638" class="Function">flip</a> <a id="953" href="Relation.Binary.Lattice.Definitions.html#932" class="Bound Operator">_≤_</a><a id="956" class="Symbol">)</a>
<a id="Exponential"></a><a id="959" href="Relation.Binary.Lattice.Definitions.html#959" class="Function">Exponential</a> <a id="971" class="Symbol">:</a> <a id="973" href="Relation.Binary.Core.html#896" class="Function">Rel</a> <a id="977" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="979" href="Relation.Binary.Lattice.Definitions.html#618" class="Generalizable">ℓ</a> <a id="981" class="Symbol">→</a> <a id="983" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="987" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="989" class="Symbol">→</a> <a id="991" href="Algebra.Core.html#527" class="Function">Op₂</a> <a id="995" href="Relation.Binary.Lattice.Definitions.html#632" class="Generalizable">A</a> <a id="997" class="Symbol">→</a> <a id="999" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="1003" class="Symbol">_</a>
<a id="1005" href="Relation.Binary.Lattice.Definitions.html#959" class="Function">Exponential</a> <a id="1017" href="Relation.Binary.Lattice.Definitions.html#1017" class="Bound Operator">_≤_</a> <a id="1021" href="Relation.Binary.Lattice.Definitions.html#1021" class="Bound Operator">_∧_</a> <a id="1025" href="Relation.Binary.Lattice.Definitions.html#1025" class="Bound Operator">_⇨_</a> <a id="1029" class="Symbol">=</a>
<a id="1033" class="Symbol">∀</a> <a id="1035" href="Relation.Binary.Lattice.Definitions.html#1035" class="Bound">w</a> <a id="1037" href="Relation.Binary.Lattice.Definitions.html#1037" class="Bound">x</a> <a id="1039" href="Relation.Binary.Lattice.Definitions.html#1039" class="Bound">y</a> <a id="1041" class="Symbol">→</a> <a id="1043" class="Symbol">((</a><a id="1045" href="Relation.Binary.Lattice.Definitions.html#1035" class="Bound">w</a> <a id="1047" href="Relation.Binary.Lattice.Definitions.html#1021" class="Bound Operator">∧</a> <a id="1049" href="Relation.Binary.Lattice.Definitions.html#1037" class="Bound">x</a><a id="1050" class="Symbol">)</a> <a id="1052" href="Relation.Binary.Lattice.Definitions.html#1017" class="Bound Operator">≤</a> <a id="1054" href="Relation.Binary.Lattice.Definitions.html#1039" class="Bound">y</a> <a id="1056" class="Symbol">→</a> <a id="1058" href="Relation.Binary.Lattice.Definitions.html#1035" class="Bound">w</a> <a id="1060" href="Relation.Binary.Lattice.Definitions.html#1017" class="Bound Operator">≤</a> <a id="1062" class="Symbol">(</a><a id="1063" href="Relation.Binary.Lattice.Definitions.html#1037" class="Bound">x</a> <a id="1065" href="Relation.Binary.Lattice.Definitions.html#1025" class="Bound Operator">⇨</a> <a id="1067" href="Relation.Binary.Lattice.Definitions.html#1039" class="Bound">y</a><a id="1068" class="Symbol">))</a> <a id="1071" href="Data.Product.Base.html#1618" class="Function Operator">×</a> <a id="1073" class="Symbol">(</a><a id="1074" href="Relation.Binary.Lattice.Definitions.html#1035" class="Bound">w</a> <a id="1076" href="Relation.Binary.Lattice.Definitions.html#1017" class="Bound Operator">≤</a> <a id="1078" class="Symbol">(</a><a id="1079" href="Relation.Binary.Lattice.Definitions.html#1037" class="Bound">x</a> <a id="1081" href="Relation.Binary.Lattice.Definitions.html#1025" class="Bound Operator">⇨</a> <a id="1083" href="Relation.Binary.Lattice.Definitions.html#1039" class="Bound">y</a><a id="1084" class="Symbol">)</a> <a id="1086" class="Symbol">→</a> <a id="1088" class="Symbol">(</a><a id="1089" href="Relation.Binary.Lattice.Definitions.html#1035" class="Bound">w</a> <a id="1091" href="Relation.Binary.Lattice.Definitions.html#1021" class="Bound Operator">∧</a> <a id="1093" href="Relation.Binary.Lattice.Definitions.html#1037" class="Bound">x</a><a id="1094" class="Symbol">)</a> <a id="1096" href="Relation.Binary.Lattice.Definitions.html#1017" class="Bound Operator">≤</a> <a id="1098" href="Relation.Binary.Lattice.Definitions.html#1039" class="Bound">y</a><a id="1099" class="Symbol">)</a>
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