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<html><head><meta charset="utf-8"><title>Function.Indexed.Relation.Binary.Equality</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">-- Function setoids and related constructions</a>
<a id="152" class="Comment">------------------------------------------------------------------------</a>
<a id="226" class="Symbol">{-#</a> <a id="230" class="Keyword">OPTIONS</a> <a id="238" class="Pragma">--cubical-compatible</a> <a id="259" class="Pragma">--safe</a> <a id="266" class="Symbol">#-}</a>
<a id="271" class="Keyword">module</a> <a id="278" href="Function.Indexed.Relation.Binary.Equality.html" class="Module">Function.Indexed.Relation.Binary.Equality</a> <a id="320" class="Keyword">where</a>
<a id="327" class="Keyword">open</a> <a id="332" class="Keyword">import</a> <a id="339" href="Relation.Binary.html" class="Module">Relation.Binary</a> <a id="355" class="Keyword">using</a> <a id="361" class="Symbol">(</a><a id="362" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a><a id="368" class="Symbol">)</a>
<a id="370" class="Keyword">open</a> <a id="375" class="Keyword">import</a> <a id="382" href="Relation.Binary.Indexed.Heterogeneous.html" class="Module">Relation.Binary.Indexed.Heterogeneous</a> <a id="420" class="Keyword">using</a> <a id="426" class="Symbol">(</a><a id="427" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#803" class="Record">IndexedSetoid</a><a id="440" class="Symbol">)</a>
<a id="443" class="Comment">-- A variant of setoid which uses the propositional equality setoid</a>
<a id="511" class="Comment">-- for the domain, and a more convenient definition of _≈_.</a>
<a id="≡-setoid"></a><a id="572" href="Function.Indexed.Relation.Binary.Equality.html#572" class="Function">≡-setoid</a> <a id="581" class="Symbol">:</a> <a id="583" class="Symbol">∀</a> <a id="585" class="Symbol">{</a><a id="586" href="Function.Indexed.Relation.Binary.Equality.html#586" class="Bound">f</a> <a id="588" href="Function.Indexed.Relation.Binary.Equality.html#588" class="Bound">t₁</a> <a id="591" href="Function.Indexed.Relation.Binary.Equality.html#591" class="Bound">t₂</a><a id="593" class="Symbol">}</a> <a id="595" class="Symbol">(</a><a id="596" href="Function.Indexed.Relation.Binary.Equality.html#596" class="Bound">From</a> <a id="601" class="Symbol">:</a> <a id="603" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="607" href="Function.Indexed.Relation.Binary.Equality.html#586" class="Bound">f</a><a id="608" class="Symbol">)</a> <a id="610" class="Symbol">→</a> <a id="612" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#803" class="Record">IndexedSetoid</a> <a id="626" href="Function.Indexed.Relation.Binary.Equality.html#596" class="Bound">From</a> <a id="631" href="Function.Indexed.Relation.Binary.Equality.html#588" class="Bound">t₁</a> <a id="634" href="Function.Indexed.Relation.Binary.Equality.html#591" class="Bound">t₂</a> <a id="637" class="Symbol">→</a> <a id="639" href="Relation.Binary.Bundles.html#1080" class="Record">Setoid</a> <a id="646" class="Symbol">_</a> <a id="648" class="Symbol">_</a>
<a id="650" href="Function.Indexed.Relation.Binary.Equality.html#572" class="Function">≡-setoid</a> <a id="659" href="Function.Indexed.Relation.Binary.Equality.html#659" class="Bound">From</a> <a id="664" href="Function.Indexed.Relation.Binary.Equality.html#664" class="Bound">To</a> <a id="667" class="Symbol">=</a> <a id="669" class="Keyword">record</a>
<a id="678" class="Symbol">{</a> <a id="680" href="Relation.Binary.Bundles.html#1143" class="Field">Carrier</a> <a id="694" class="Symbol">=</a> <a id="696" class="Symbol">(</a><a id="697" href="Function.Indexed.Relation.Binary.Equality.html#697" class="Bound">x</a> <a id="699" class="Symbol">:</a> <a id="701" href="Function.Indexed.Relation.Binary.Equality.html#659" class="Bound">From</a><a id="705" class="Symbol">)</a> <a id="707" class="Symbol">→</a> <a id="709" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#893" class="Field">Carrier</a> <a id="717" href="Function.Indexed.Relation.Binary.Equality.html#697" class="Bound">x</a>
<a id="721" class="Symbol">;</a> <a id="723" href="Relation.Binary.Bundles.html#1169" class="Field Operator">_≈_</a> <a id="737" class="Symbol">=</a> <a id="739" class="Symbol">λ</a> <a id="741" href="Function.Indexed.Relation.Binary.Equality.html#741" class="Bound">f</a> <a id="743" href="Function.Indexed.Relation.Binary.Equality.html#743" class="Bound">g</a> <a id="745" class="Symbol">→</a> <a id="747" class="Symbol">∀</a> <a id="749" href="Function.Indexed.Relation.Binary.Equality.html#749" class="Bound">x</a> <a id="751" class="Symbol">→</a> <a id="753" href="Function.Indexed.Relation.Binary.Equality.html#741" class="Bound">f</a> <a id="755" href="Function.Indexed.Relation.Binary.Equality.html#749" class="Bound">x</a> <a id="757" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#923" class="Field Operator">≈</a> <a id="759" href="Function.Indexed.Relation.Binary.Equality.html#743" class="Bound">g</a> <a id="761" href="Function.Indexed.Relation.Binary.Equality.html#749" class="Bound">x</a>
<a id="765" class="Symbol">;</a> <a id="767" href="Relation.Binary.Bundles.html#1203" class="Field">isEquivalence</a> <a id="781" class="Symbol">=</a> <a id="783" class="Keyword">record</a>
<a id="794" class="Symbol">{</a> <a id="796" href="Relation.Binary.Structures.html#1596" class="Field">refl</a> <a id="802" class="Symbol">=</a> <a id="804" class="Symbol">λ</a> <a id="806" class="Symbol">{</a><a id="807" href="Function.Indexed.Relation.Binary.Equality.html#807" class="Bound">f</a><a id="808" class="Symbol">}</a> <a id="810" href="Function.Indexed.Relation.Binary.Equality.html#810" class="Bound">x</a> <a id="812" class="Symbol">→</a> <a id="814" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#923" class="Function">refl</a>
<a id="823" class="Symbol">;</a> <a id="825" href="Relation.Binary.Structures.html#1622" class="Field">sym</a> <a id="831" class="Symbol">=</a> <a id="833" class="Symbol">λ</a> <a id="835" href="Function.Indexed.Relation.Binary.Equality.html#835" class="Bound">f∼g</a> <a id="839" href="Function.Indexed.Relation.Binary.Equality.html#839" class="Bound">x</a> <a id="841" class="Symbol">→</a> <a id="843" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#952" class="Function">sym</a> <a id="847" class="Symbol">(</a><a id="848" href="Function.Indexed.Relation.Binary.Equality.html#835" class="Bound">f∼g</a> <a id="852" href="Function.Indexed.Relation.Binary.Equality.html#839" class="Bound">x</a><a id="853" class="Symbol">)</a>
<a id="859" class="Symbol">;</a> <a id="861" href="Relation.Binary.Structures.html#1648" class="Field">trans</a> <a id="867" class="Symbol">=</a> <a id="869" class="Symbol">λ</a> <a id="871" href="Function.Indexed.Relation.Binary.Equality.html#871" class="Bound">f∼g</a> <a id="875" href="Function.Indexed.Relation.Binary.Equality.html#875" class="Bound">g∼h</a> <a id="879" href="Function.Indexed.Relation.Binary.Equality.html#879" class="Bound">x</a> <a id="881" class="Symbol">→</a> <a id="883" href="Relation.Binary.Indexed.Heterogeneous.Structures.html#981" class="Function">trans</a> <a id="889" class="Symbol">(</a><a id="890" href="Function.Indexed.Relation.Binary.Equality.html#871" class="Bound">f∼g</a> <a id="894" href="Function.Indexed.Relation.Binary.Equality.html#879" class="Bound">x</a><a id="895" class="Symbol">)</a> <a id="897" class="Symbol">(</a><a id="898" href="Function.Indexed.Relation.Binary.Equality.html#875" class="Bound">g∼h</a> <a id="902" href="Function.Indexed.Relation.Binary.Equality.html#879" class="Bound">x</a><a id="903" class="Symbol">)</a>
<a id="909" class="Symbol">}</a>
<a id="913" class="Symbol">}</a> <a id="915" class="Keyword">where</a> <a id="921" class="Keyword">open</a> <a id="926" href="Relation.Binary.Indexed.Heterogeneous.Bundles.html#803" class="Module">IndexedSetoid</a> <a id="940" href="Function.Indexed.Relation.Binary.Equality.html#664" class="Bound">To</a>
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