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39 lines
9.6 KiB
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39 lines
9.6 KiB
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<html><head><meta charset="utf-8"><title>Relation.Nullary.Product</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">-- Products of nullary relations</a>
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<a id="139" class="Comment">------------------------------------------------------------------------</a>
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<a id="213" class="Symbol">{-#</a> <a id="217" class="Keyword">OPTIONS</a> <a id="225" class="Pragma">--cubical-compatible</a> <a id="246" class="Pragma">--safe</a> <a id="253" class="Symbol">#-}</a>
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<a id="258" class="Keyword">module</a> <a id="265" href="Relation.Nullary.Product.html" class="Module">Relation.Nullary.Product</a> <a id="290" class="Keyword">where</a>
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<a id="297" class="Keyword">open</a> <a id="302" class="Keyword">import</a> <a id="309" href="Data.Bool.Base.html" class="Module">Data.Bool.Base</a>
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<a id="324" class="Keyword">open</a> <a id="329" class="Keyword">import</a> <a id="336" href="Data.Product.html" class="Module">Data.Product</a>
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<a id="349" class="Keyword">open</a> <a id="354" class="Keyword">import</a> <a id="361" href="Function.Base.html" class="Module">Function.Base</a> <a id="375" class="Keyword">using</a> <a id="381" class="Symbol">(</a><a id="382" href="Function.Base.html#1040" class="Function Operator">_∘_</a><a id="385" class="Symbol">)</a>
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<a id="387" class="Keyword">open</a> <a id="392" class="Keyword">import</a> <a id="399" href="Level.html" class="Module">Level</a>
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<a id="405" class="Keyword">open</a> <a id="410" class="Keyword">import</a> <a id="417" href="Relation.Nullary.Reflects.html" class="Module">Relation.Nullary.Reflects</a>
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<a id="443" class="Keyword">open</a> <a id="448" class="Keyword">import</a> <a id="455" href="Relation.Nullary.html" class="Module">Relation.Nullary</a>
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<a id="473" class="Keyword">private</a>
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<a id="483" class="Keyword">variable</a>
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<a id="496" href="Relation.Nullary.Product.html#496" class="Generalizable">p</a> <a id="498" href="Relation.Nullary.Product.html#498" class="Generalizable">q</a> <a id="500" class="Symbol">:</a> <a id="502" href="Agda.Primitive.html#591" class="Postulate">Level</a>
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<a id="512" href="Relation.Nullary.Product.html#512" class="Generalizable">P</a> <a id="514" class="Symbol">:</a> <a id="516" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="520" href="Relation.Nullary.Product.html#496" class="Generalizable">p</a>
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<a id="526" href="Relation.Nullary.Product.html#526" class="Generalizable">Q</a> <a id="528" class="Symbol">:</a> <a id="530" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="534" href="Relation.Nullary.Product.html#498" class="Generalizable">q</a>
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<a id="537" class="Comment">------------------------------------------------------------------------</a>
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<a id="610" class="Comment">-- Some properties which are preserved by _×_.</a>
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<a id="658" class="Keyword">infixr</a> <a id="665" class="Number">2</a> <a id="667" href="Relation.Nullary.Product.html#689" class="Function Operator">_×-reflects_</a> <a id="680" href="Relation.Nullary.Product.html#930" class="Function Operator">_×-dec_</a>
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<a id="_×-reflects_"></a><a id="689" href="Relation.Nullary.Product.html#689" class="Function Operator">_×-reflects_</a> <a id="702" class="Symbol">:</a> <a id="704" class="Symbol">∀</a> <a id="706" class="Symbol">{</a><a id="707" href="Relation.Nullary.Product.html#707" class="Bound">bp</a> <a id="710" href="Relation.Nullary.Product.html#710" class="Bound">bq</a><a id="712" class="Symbol">}</a> <a id="714" class="Symbol">→</a> <a id="716" href="Relation.Nullary.html#863" class="Datatype">Reflects</a> <a id="725" href="Relation.Nullary.Product.html#512" class="Generalizable">P</a> <a id="727" href="Relation.Nullary.Product.html#707" class="Bound">bp</a> <a id="730" class="Symbol">→</a> <a id="732" href="Relation.Nullary.html#863" class="Datatype">Reflects</a> <a id="741" href="Relation.Nullary.Product.html#526" class="Generalizable">Q</a> <a id="743" href="Relation.Nullary.Product.html#710" class="Bound">bq</a> <a id="746" class="Symbol">→</a>
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<a id="775" href="Relation.Nullary.html#863" class="Datatype">Reflects</a> <a id="784" class="Symbol">(</a><a id="785" href="Relation.Nullary.Product.html#512" class="Generalizable">P</a> <a id="787" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="789" href="Relation.Nullary.Product.html#526" class="Generalizable">Q</a><a id="790" class="Symbol">)</a> <a id="792" class="Symbol">(</a><a id="793" href="Relation.Nullary.Product.html#707" class="Bound">bp</a> <a id="796" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="798" href="Relation.Nullary.Product.html#710" class="Bound">bq</a><a id="800" class="Symbol">)</a>
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<a id="802" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a> <a id="807" href="Relation.Nullary.Product.html#807" class="Bound">p</a> <a id="809" href="Relation.Nullary.Product.html#689" class="Function Operator">×-reflects</a> <a id="820" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a> <a id="825" href="Relation.Nullary.Product.html#825" class="Bound">q</a> <a id="827" class="Symbol">=</a> <a id="829" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a> <a id="833" class="Symbol">(</a><a id="834" href="Relation.Nullary.Product.html#807" class="Bound">p</a> <a id="836" href="Agda.Builtin.Sigma.html#218" class="InductiveConstructor Operator">,</a> <a id="838" href="Relation.Nullary.Product.html#825" class="Bound">q</a><a id="839" class="Symbol">)</a>
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<a id="841" href="Relation.Nullary.html#911" class="InductiveConstructor">ofʸ</a> <a id="846" href="Relation.Nullary.Product.html#846" class="Bound">p</a> <a id="848" href="Relation.Nullary.Product.html#689" class="Function Operator">×-reflects</a> <a id="859" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a> <a id="863" href="Relation.Nullary.Product.html#863" class="Bound">¬q</a> <a id="866" class="Symbol">=</a> <a id="868" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a> <a id="872" class="Symbol">(</a><a id="873" href="Relation.Nullary.Product.html#863" class="Bound">¬q</a> <a id="876" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="878" href="Agda.Builtin.Sigma.html#246" class="Field">proj₂</a><a id="883" class="Symbol">)</a>
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<a id="885" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a> <a id="889" href="Relation.Nullary.Product.html#889" class="Bound">¬p</a> <a id="892" href="Relation.Nullary.Product.html#689" class="Function Operator">×-reflects</a> <a id="903" class="Symbol">_</a> <a id="910" class="Symbol">=</a> <a id="912" href="Relation.Nullary.html#948" class="InductiveConstructor">ofⁿ</a> <a id="916" class="Symbol">(</a><a id="917" href="Relation.Nullary.Product.html#889" class="Bound">¬p</a> <a id="920" href="Function.Base.html#1040" class="Function Operator">∘</a> <a id="922" href="Agda.Builtin.Sigma.html#234" class="Field">proj₁</a><a id="927" class="Symbol">)</a>
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<a id="_×-dec_"></a><a id="930" href="Relation.Nullary.Product.html#930" class="Function Operator">_×-dec_</a> <a id="938" class="Symbol">:</a> <a id="940" href="Relation.Nullary.html#1520" class="Record">Dec</a> <a id="944" href="Relation.Nullary.Product.html#512" class="Generalizable">P</a> <a id="946" class="Symbol">→</a> <a id="948" href="Relation.Nullary.html#1520" class="Record">Dec</a> <a id="952" href="Relation.Nullary.Product.html#526" class="Generalizable">Q</a> <a id="954" class="Symbol">→</a> <a id="956" href="Relation.Nullary.html#1520" class="Record">Dec</a> <a id="960" class="Symbol">(</a><a id="961" href="Relation.Nullary.Product.html#512" class="Generalizable">P</a> <a id="963" href="Data.Product.html#1176" class="Function Operator">×</a> <a id="965" href="Relation.Nullary.Product.html#526" class="Generalizable">Q</a><a id="966" class="Symbol">)</a>
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<a id="968" href="Relation.Nullary.html#1590" class="Field">does</a> <a id="974" class="Symbol">(</a><a id="975" href="Relation.Nullary.Product.html#975" class="Bound">p?</a> <a id="978" href="Relation.Nullary.Product.html#930" class="Function Operator">×-dec</a> <a id="984" href="Relation.Nullary.Product.html#984" class="Bound">q?</a><a id="986" class="Symbol">)</a> <a id="988" class="Symbol">=</a> <a id="990" href="Relation.Nullary.html#1590" class="Field">does</a> <a id="995" href="Relation.Nullary.Product.html#975" class="Bound">p?</a> <a id="998" href="Data.Bool.Base.html#995" class="Function Operator">∧</a> <a id="1000" href="Relation.Nullary.html#1590" class="Field">does</a> <a id="1005" href="Relation.Nullary.Product.html#984" class="Bound">q?</a>
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<a id="1008" href="Relation.Nullary.html#1607" class="Field">proof</a> <a id="1014" class="Symbol">(</a><a id="1015" href="Relation.Nullary.Product.html#1015" class="Bound">p?</a> <a id="1018" href="Relation.Nullary.Product.html#930" class="Function Operator">×-dec</a> <a id="1024" href="Relation.Nullary.Product.html#1024" class="Bound">q?</a><a id="1026" class="Symbol">)</a> <a id="1028" class="Symbol">=</a> <a id="1030" href="Relation.Nullary.html#1607" class="Field">proof</a> <a id="1036" href="Relation.Nullary.Product.html#1015" class="Bound">p?</a> <a id="1039" href="Relation.Nullary.Product.html#689" class="Function Operator">×-reflects</a> <a id="1050" href="Relation.Nullary.html#1607" class="Field">proof</a> <a id="1056" href="Relation.Nullary.Product.html#1024" class="Bound">q?</a>
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</pre></body></html>
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