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7.8 KiB
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31 lines
7.8 KiB
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<html><head><meta charset="utf-8"><title>Function.Definitions.Core2</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">-- Definitions for types of functions that only require an equality</a>
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<a id="174" class="Comment">-- relation over the co-domain.</a>
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<a id="206" class="Comment">------------------------------------------------------------------------</a>
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<a id="280" class="Comment">-- The contents of this file should usually be accessed from `Function`.</a>
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<a id="354" class="Symbol">{-#</a> <a id="358" class="Keyword">OPTIONS</a> <a id="366" class="Pragma">--cubical-compatible</a> <a id="387" class="Pragma">--safe</a> <a id="394" class="Symbol">#-}</a>
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<a id="399" class="Keyword">open</a> <a id="404" class="Keyword">import</a> <a id="411" href="Relation.Binary.html" class="Module">Relation.Binary</a>
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<a id="428" class="Keyword">module</a> <a id="435" href="Function.Definitions.Core2.html" class="Module">Function.Definitions.Core2</a>
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<a id="464" class="Symbol">{</a><a id="465" href="Function.Definitions.Core2.html#465" class="Bound">b</a> <a id="467" href="Function.Definitions.Core2.html#467" class="Bound">ℓ₂</a><a id="469" class="Symbol">}</a> <a id="471" class="Symbol">{</a><a id="472" href="Function.Definitions.Core2.html#472" class="Bound">B</a> <a id="474" class="Symbol">:</a> <a id="476" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="480" href="Function.Definitions.Core2.html#465" class="Bound">b</a><a id="481" class="Symbol">}</a> <a id="483" class="Symbol">(</a><a id="484" href="Function.Definitions.Core2.html#484" class="Bound Operator">_≈₂_</a> <a id="489" class="Symbol">:</a> <a id="491" href="Relation.Binary.Core.html#891" class="Function">Rel</a> <a id="495" href="Function.Definitions.Core2.html#472" class="Bound">B</a> <a id="497" href="Function.Definitions.Core2.html#467" class="Bound">ℓ₂</a><a id="499" class="Symbol">)</a>
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<a id="503" class="Keyword">where</a>
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<a id="510" class="Keyword">open</a> <a id="515" class="Keyword">import</a> <a id="522" href="Data.Product.html" class="Module">Data.Product</a> <a id="535" class="Keyword">using</a> <a id="541" class="Symbol">(</a><a id="542" href="Data.Product.html#1378" class="Function">∃</a><a id="543" class="Symbol">)</a>
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<a id="545" class="Keyword">open</a> <a id="550" class="Keyword">import</a> <a id="557" href="Level.html" class="Module">Level</a> <a id="563" class="Keyword">using</a> <a id="569" class="Symbol">(</a><a id="570" href="Agda.Primitive.html#591" class="Postulate">Level</a><a id="575" class="Symbol">;</a> <a id="577" href="Agda.Primitive.html#804" class="Primitive Operator">_⊔_</a><a id="580" class="Symbol">)</a>
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<a id="583" class="Comment">------------------------------------------------------------------------</a>
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<a id="656" class="Comment">-- Definitions</a>
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<a id="Surjective"></a><a id="672" href="Function.Definitions.Core2.html#672" class="Function">Surjective</a> <a id="683" class="Symbol">:</a> <a id="685" class="Symbol">∀</a> <a id="687" class="Symbol">{</a><a id="688" href="Function.Definitions.Core2.html#688" class="Bound">a</a><a id="689" class="Symbol">}</a> <a id="691" class="Symbol">{</a><a id="692" href="Function.Definitions.Core2.html#692" class="Bound">A</a> <a id="694" class="Symbol">:</a> <a id="696" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="700" href="Function.Definitions.Core2.html#688" class="Bound">a</a><a id="701" class="Symbol">}</a> <a id="703" class="Symbol">→</a> <a id="705" class="Symbol">(</a><a id="706" href="Function.Definitions.Core2.html#692" class="Bound">A</a> <a id="708" class="Symbol">→</a> <a id="710" href="Function.Definitions.Core2.html#472" class="Bound">B</a><a id="711" class="Symbol">)</a> <a id="713" class="Symbol">→</a> <a id="715" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="719" class="Symbol">(</a><a id="720" href="Function.Definitions.Core2.html#688" class="Bound">a</a> <a id="722" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="724" href="Function.Definitions.Core2.html#465" class="Bound">b</a> <a id="726" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="728" href="Function.Definitions.Core2.html#467" class="Bound">ℓ₂</a><a id="730" class="Symbol">)</a>
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<a id="732" href="Function.Definitions.Core2.html#672" class="Function">Surjective</a> <a id="743" href="Function.Definitions.Core2.html#743" class="Bound">f</a> <a id="745" class="Symbol">=</a> <a id="747" class="Symbol">∀</a> <a id="749" href="Function.Definitions.Core2.html#749" class="Bound">y</a> <a id="751" class="Symbol">→</a> <a id="753" href="Data.Product.html#1378" class="Function">∃</a> <a id="755" class="Symbol">λ</a> <a id="757" href="Function.Definitions.Core2.html#757" class="Bound">x</a> <a id="759" class="Symbol">→</a> <a id="761" href="Function.Definitions.Core2.html#743" class="Bound">f</a> <a id="763" href="Function.Definitions.Core2.html#757" class="Bound">x</a> <a id="765" href="Function.Definitions.Core2.html#484" class="Bound Operator">≈₂</a> <a id="768" href="Function.Definitions.Core2.html#749" class="Bound">y</a>
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<a id="771" class="Comment">-- (Note the name `LeftInverse` is used for the bundle)</a>
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<a id="Inverseˡ"></a><a id="827" href="Function.Definitions.Core2.html#827" class="Function">Inverseˡ</a> <a id="836" class="Symbol">:</a> <a id="838" class="Symbol">∀</a> <a id="840" class="Symbol">{</a><a id="841" href="Function.Definitions.Core2.html#841" class="Bound">a</a><a id="842" class="Symbol">}</a> <a id="844" class="Symbol">{</a><a id="845" href="Function.Definitions.Core2.html#845" class="Bound">A</a> <a id="847" class="Symbol">:</a> <a id="849" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="853" href="Function.Definitions.Core2.html#841" class="Bound">a</a><a id="854" class="Symbol">}</a> <a id="856" class="Symbol">→</a> <a id="858" class="Symbol">(</a><a id="859" href="Function.Definitions.Core2.html#845" class="Bound">A</a> <a id="861" class="Symbol">→</a> <a id="863" href="Function.Definitions.Core2.html#472" class="Bound">B</a><a id="864" class="Symbol">)</a> <a id="866" class="Symbol">→</a> <a id="868" class="Symbol">(</a><a id="869" href="Function.Definitions.Core2.html#472" class="Bound">B</a> <a id="871" class="Symbol">→</a> <a id="873" href="Function.Definitions.Core2.html#845" class="Bound">A</a><a id="874" class="Symbol">)</a> <a id="876" class="Symbol">→</a> <a id="878" href="Agda.Primitive.html#320" class="Primitive">Set</a> <a id="882" class="Symbol">(</a><a id="883" href="Function.Definitions.Core2.html#465" class="Bound">b</a> <a id="885" href="Agda.Primitive.html#804" class="Primitive Operator">⊔</a> <a id="887" href="Function.Definitions.Core2.html#467" class="Bound">ℓ₂</a><a id="889" class="Symbol">)</a>
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<a id="891" href="Function.Definitions.Core2.html#827" class="Function">Inverseˡ</a> <a id="900" href="Function.Definitions.Core2.html#900" class="Bound">f</a> <a id="902" href="Function.Definitions.Core2.html#902" class="Bound">g</a> <a id="904" class="Symbol">=</a> <a id="906" class="Symbol">∀</a> <a id="908" href="Function.Definitions.Core2.html#908" class="Bound">x</a> <a id="910" class="Symbol">→</a> <a id="912" href="Function.Definitions.Core2.html#900" class="Bound">f</a> <a id="914" class="Symbol">(</a><a id="915" href="Function.Definitions.Core2.html#902" class="Bound">g</a> <a id="917" href="Function.Definitions.Core2.html#908" class="Bound">x</a><a id="918" class="Symbol">)</a> <a id="920" href="Function.Definitions.Core2.html#484" class="Bound Operator">≈₂</a> <a id="923" href="Function.Definitions.Core2.html#908" class="Bound">x</a>
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</pre></body></html>
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