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<html><head><meta charset="utf-8"><title>Data.Sum.Properties</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">-- Properties of sums (disjoint unions)</a>
<a id="146" class="Comment">------------------------------------------------------------------------</a>
<a id="220" class="Symbol">{-#</a> <a id="224" class="Keyword">OPTIONS</a> <a id="232" class="Pragma">--cubical-compatible</a> <a id="253" class="Pragma">--safe</a> <a id="260" class="Symbol">#-}</a>
<a id="265" class="Keyword">module</a> <a id="272" href="Data.Sum.Properties.html" class="Module">Data.Sum.Properties</a> <a id="292" class="Keyword">where</a>
<a id="299" class="Keyword">open</a> <a id="304" class="Keyword">import</a> <a id="311" href="Level.html" class="Module">Level</a>
<a id="317" class="Keyword">open</a> <a id="322" class="Keyword">import</a> <a id="329" href="Data.Sum.Base.html" class="Module">Data.Sum.Base</a>
<a id="343" class="Keyword">open</a> <a id="348" class="Keyword">import</a> <a id="355" href="Function.Base.html" class="Module">Function.Base</a> <a id="369" class="Keyword">using</a> <a id="375" class="Symbol">(</a><a id="376" href="Function.Base.html#4486" class="Function Operator">_∋_</a><a id="379" class="Symbol">;</a> <a id="381" href="Function.Base.html#1115" class="Function Operator">_∘_</a><a id="384" class="Symbol">;</a> <a id="386" href="Function.Base.html#704" class="Function">id</a><a id="388" class="Symbol">)</a>
<a id="390" class="Keyword">open</a> <a id="395" class="Keyword">import</a> <a id="402" href="Function.Bundles.html" class="Module">Function.Bundles</a> <a id="419" class="Keyword">using</a> <a id="425" class="Symbol">(</a><a id="426" href="Function.Bundles.html#14563" class="Function">mk↔ₛ′</a><a id="431" class="Symbol">;</a> <a id="433" href="Function.Bundles.html#12340" class="Function Operator">_↔_</a><a id="436" class="Symbol">)</a>
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<a id="655" class="Keyword">private</a>
<a id="665" class="Keyword">variable</a>
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<a id="716" href="Data.Sum.Properties.html#716" class="Generalizable">B</a> <a id="718" class="Symbol">:</a> <a id="720" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="724" href="Data.Sum.Properties.html#680" class="Generalizable">b</a>
<a id="730" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="732" class="Symbol">:</a> <a id="734" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="738" href="Data.Sum.Properties.html#682" class="Generalizable">c</a>
<a id="744" href="Data.Sum.Properties.html#744" class="Generalizable">D</a> <a id="746" class="Symbol">:</a> <a id="748" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="752" href="Data.Sum.Properties.html#684" class="Generalizable">d</a>
<a id="758" href="Data.Sum.Properties.html#758" class="Generalizable">E</a> <a id="760" class="Symbol">:</a> <a id="762" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="766" href="Data.Sum.Properties.html#686" class="Generalizable">e</a>
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<a id="844" href="Data.Sum.Properties.html#783" class="Function">inj₁-injective</a> <a id="859" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="864" class="Symbol">=</a> <a id="866" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="inj₂-injective"></a><a id="872" href="Data.Sum.Properties.html#872" class="Function">inj₂-injective</a> <a id="887" class="Symbol">:</a> <a id="889" class="Symbol">∀</a> <a id="891" class="Symbol">{</a><a id="892" href="Data.Sum.Properties.html#892" class="Bound">x</a> <a id="894" href="Data.Sum.Properties.html#894" class="Bound">y</a><a id="895" class="Symbol">}</a> <a id="897" class="Symbol">→</a> <a id="899" class="Symbol">(</a><a id="900" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="902" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="904" href="Data.Sum.Properties.html#716" class="Generalizable">B</a> <a id="906" href="Function.Base.html#4486" class="Function Operator">∋</a> <a id="908" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="913" href="Data.Sum.Properties.html#892" class="Bound">x</a><a id="914" class="Symbol">)</a> <a id="916" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="918" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="923" href="Data.Sum.Properties.html#894" class="Bound">y</a> <a id="925" class="Symbol">→</a> <a id="927" href="Data.Sum.Properties.html#892" class="Bound">x</a> <a id="929" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">≡</a> <a id="931" href="Data.Sum.Properties.html#894" class="Bound">y</a>
<a id="933" href="Data.Sum.Properties.html#872" class="Function">inj₂-injective</a> <a id="948" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a> <a id="953" class="Symbol">=</a> <a id="955" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="961" class="Keyword">module</a> <a id="968" href="Data.Sum.Properties.html#968" class="Module">_</a> <a id="970" class="Symbol">(</a><a id="971" href="Data.Sum.Properties.html#971" class="Bound">dec₁</a> <a id="976" class="Symbol">:</a> <a id="978" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="988" class="Symbol">{</a><a id="989" class="Argument">A</a> <a id="991" class="Symbol">=</a> <a id="993" href="Data.Sum.Properties.html#702" class="Generalizable">A</a><a id="994" class="Symbol">}</a> <a id="996" class="Symbol">{</a><a id="997" class="Argument">B</a> <a id="999" class="Symbol">=</a> <a id="1001" href="Data.Sum.Properties.html#702" class="Generalizable">A</a><a id="1002" class="Symbol">}</a> <a id="1004" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="1007" class="Symbol">)</a>
<a id="1018" class="Symbol">(</a><a id="1019" href="Data.Sum.Properties.html#1019" class="Bound">dec₂</a> <a id="1024" class="Symbol">:</a> <a id="1026" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="1036" class="Symbol">{</a><a id="1037" class="Argument">A</a> <a id="1039" class="Symbol">=</a> <a id="1041" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="1042" class="Symbol">}</a> <a id="1044" class="Symbol">{</a><a id="1045" class="Argument">B</a> <a id="1047" class="Symbol">=</a> <a id="1049" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="1050" class="Symbol">}</a> <a id="1052" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a><a id="1055" class="Symbol">)</a> <a id="1057" class="Keyword">where</a>
<a id="1066" href="Data.Sum.Properties.html#1066" class="Function">≡-dec</a> <a id="1072" class="Symbol">:</a> <a id="1074" href="Relation.Binary.Definitions.html#6161" class="Function">Decidable</a> <a id="1084" class="Symbol">{</a><a id="1085" class="Argument">A</a> <a id="1087" class="Symbol">=</a> <a id="1089" href="Data.Sum.Properties.html#993" class="Bound">A</a> <a id="1091" href="Data.Sum.Base.html#625" class="Datatype Operator">⊎</a> <a id="1093" href="Data.Sum.Properties.html#1041" class="Bound">B</a><a id="1094" class="Symbol">}</a> <a id="1096" href="Agda.Builtin.Equality.html#150" class="Datatype Operator">_≡_</a>
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<a id="1173" href="Data.Sum.Properties.html#1066" class="Function">≡-dec</a> <a id="1179" class="Symbol">(</a><a id="1180" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1185" href="Data.Sum.Properties.html#1185" class="Bound">x</a><a id="1186" class="Symbol">)</a> <a id="1188" class="Symbol">(</a><a id="1189" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1194" href="Data.Sum.Properties.html#1194" class="Bound">y</a><a id="1195" class="Symbol">)</a> <a id="1197" class="Symbol">=</a> <a id="1199" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="1202" class="Symbol">λ()</a>
<a id="1208" href="Data.Sum.Properties.html#1066" class="Function">≡-dec</a> <a id="1214" class="Symbol">(</a><a id="1215" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1220" href="Data.Sum.Properties.html#1220" class="Bound">x</a><a id="1221" class="Symbol">)</a> <a id="1223" class="Symbol">(</a><a id="1224" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1229" href="Data.Sum.Properties.html#1229" class="Bound">y</a><a id="1230" class="Symbol">)</a> <a id="1232" class="Symbol">=</a> <a id="1234" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="1237" class="Symbol">λ()</a>
<a id="1243" href="Data.Sum.Properties.html#1066" class="Function">≡-dec</a> <a id="1249" class="Symbol">(</a><a id="1250" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1255" href="Data.Sum.Properties.html#1255" class="Bound">x</a><a id="1256" class="Symbol">)</a> <a id="1258" class="Symbol">(</a><a id="1259" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1264" href="Data.Sum.Properties.html#1264" class="Bound">y</a><a id="1265" class="Symbol">)</a> <a id="1267" class="Symbol">=</a> <a id="1269" href="Relation.Nullary.Decidable.Core.html#4901" class="Function">map′</a> <a id="1274" class="Symbol">(</a><a id="1275" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="1280" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="1284" class="Symbol">)</a> <a id="1286" href="Data.Sum.Properties.html#872" class="Function">inj₂-injective</a> <a id="1301" class="Symbol">(</a><a id="1302" href="Data.Sum.Properties.html#1019" class="Bound">dec₂</a> <a id="1307" href="Data.Sum.Properties.html#1255" class="Bound">x</a> <a id="1309" href="Data.Sum.Properties.html#1264" class="Bound">y</a><a id="1310" class="Symbol">)</a>
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<a id="1540" href="Data.Sum.Properties.html#1500" class="Function">map-id</a> <a id="1547" class="Symbol">(</a><a id="1548" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="1553" class="Symbol">_)</a> <a id="1556" class="Symbol">=</a> <a id="1558" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="1563" href="Data.Sum.Properties.html#1500" class="Function">map-id</a> <a id="1570" class="Symbol">(</a><a id="1571" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="1576" class="Symbol">_)</a> <a id="1579" class="Symbol">=</a> <a id="1581" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="[,]-∘"></a><a id="1587" href="Data.Sum.Properties.html#1587" class="Function">[,]-∘</a> <a id="1593" class="Symbol">:</a> <a id="1595" class="Symbol">(</a><a id="1596" href="Data.Sum.Properties.html#1596" class="Bound">f</a> <a id="1598" class="Symbol">:</a> <a id="1600" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="1602" class="Symbol">→</a> <a id="1604" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="1605" class="Symbol">)</a>
<a id="1621" class="Symbol">{</a><a id="1622" href="Data.Sum.Properties.html#1622" class="Bound">g</a> <a id="1624" class="Symbol">:</a> <a id="1626" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="1628" class="Symbol">→</a> <a id="1630" href="Data.Sum.Properties.html#702" class="Generalizable">A</a><a id="1631" class="Symbol">}</a> <a id="1633" class="Symbol">{</a><a id="1634" href="Data.Sum.Properties.html#1634" class="Bound">h</a> <a id="1636" class="Symbol">:</a> <a id="1638" href="Data.Sum.Properties.html#744" class="Generalizable">D</a> <a id="1640" class="Symbol">→</a> <a id="1642" href="Data.Sum.Properties.html#702" class="Generalizable">A</a><a id="1643" class="Symbol">}</a> <a id="1645" class="Symbol">→</a>
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<a id="2079" href="Data.Sum.Properties.html#1920" class="Function">map-map</a> <a id="2087" class="Symbol">(</a><a id="2088" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2093" class="Symbol">_)</a> <a id="2096" class="Symbol">=</a> <a id="2098" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="map₁₂-map₂₁"></a><a id="2104" href="Data.Sum.Properties.html#2104" class="Function">map₁₂-map₂₁</a> <a id="2116" class="Symbol">:</a> <a id="2118" class="Symbol">{</a><a id="2119" href="Data.Sum.Properties.html#2119" class="Bound">f</a> <a id="2121" class="Symbol">:</a> <a id="2123" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="2125" class="Symbol">→</a> <a id="2127" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2128" class="Symbol">}</a> <a id="2130" class="Symbol">{</a><a id="2131" href="Data.Sum.Properties.html#2131" class="Bound">g</a> <a id="2133" class="Symbol">:</a> <a id="2135" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="2137" class="Symbol">→</a> <a id="2139" href="Data.Sum.Properties.html#744" class="Generalizable">D</a><a id="2140" class="Symbol">}</a> <a id="2142" class="Symbol">→</a>
<a id="2160" href="Data.Sum.Base.html#1331" class="Function">map₁</a> <a id="2165" href="Data.Sum.Properties.html#2119" class="Bound">f</a> <a id="2167" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2169" href="Data.Sum.Base.html#1383" class="Function">map₂</a> <a id="2174" href="Data.Sum.Properties.html#2131" class="Bound">g</a> <a id="2176" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2178" href="Data.Sum.Base.html#1383" class="Function">map₂</a> <a id="2183" href="Data.Sum.Properties.html#2131" class="Bound">g</a> <a id="2185" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2187" href="Data.Sum.Base.html#1331" class="Function">map₁</a> <a id="2192" href="Data.Sum.Properties.html#2119" class="Bound">f</a>
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<a id="2222" href="Data.Sum.Properties.html#2104" class="Function">map₁₂-map₂₁</a> <a id="2234" class="Symbol">(</a><a id="2235" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2240" class="Symbol">_)</a> <a id="2243" class="Symbol">=</a> <a id="2245" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
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<a id="2304" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2308" class="Symbol">(</a><a id="2309" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2313" href="Data.Sum.Properties.html#2265" class="Bound">f</a> <a id="2315" href="Data.Sum.Properties.html#2277" class="Bound">g</a><a id="2316" class="Symbol">)</a> <a id="2318" href="Data.Sum.Properties.html#2289" class="Bound">h</a> <a id="2320" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2322" href="Data.Sum.Base.html#1504" class="Function">assocˡ</a> <a id="2329" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2331" href="Data.Sum.Base.html#1504" class="Function">assocˡ</a> <a id="2338" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2340" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2344" href="Data.Sum.Properties.html#2265" class="Bound">f</a> <a id="2346" class="Symbol">(</a><a id="2347" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2351" href="Data.Sum.Properties.html#2277" class="Bound">g</a> <a id="2353" href="Data.Sum.Properties.html#2289" class="Bound">h</a><a id="2354" class="Symbol">)</a>
<a id="2356" href="Data.Sum.Properties.html#2251" class="Function">map-assocˡ</a> <a id="2367" class="Symbol">_</a> <a id="2369" class="Symbol">_</a> <a id="2371" class="Symbol">_</a> <a id="2373" class="Symbol">(</a><a id="2374" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2385" href="Data.Sum.Properties.html#2385" class="Bound">x</a> <a id="2387" class="Symbol">)</a> <a id="2389" class="Symbol">=</a> <a id="2391" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2396" href="Data.Sum.Properties.html#2251" class="Function">map-assocˡ</a> <a id="2407" class="Symbol">_</a> <a id="2409" class="Symbol">_</a> <a id="2411" class="Symbol">_</a> <a id="2413" class="Symbol">(</a><a id="2414" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2419" class="Symbol">(</a><a id="2420" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2425" href="Data.Sum.Properties.html#2425" class="Bound">y</a><a id="2426" class="Symbol">))</a> <a id="2429" class="Symbol">=</a> <a id="2431" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2436" href="Data.Sum.Properties.html#2251" class="Function">map-assocˡ</a> <a id="2447" class="Symbol">_</a> <a id="2449" class="Symbol">_</a> <a id="2451" class="Symbol">_</a> <a id="2453" class="Symbol">(</a><a id="2454" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2459" class="Symbol">(</a><a id="2460" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2465" href="Data.Sum.Properties.html#2465" class="Bound">z</a><a id="2466" class="Symbol">))</a> <a id="2469" class="Symbol">=</a> <a id="2471" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="map-assocʳ"></a><a id="2477" href="Data.Sum.Properties.html#2477" class="Function">map-assocʳ</a> <a id="2488" class="Symbol">:</a> <a id="2490" class="Symbol">(</a><a id="2491" href="Data.Sum.Properties.html#2491" class="Bound">f</a> <a id="2493" class="Symbol">:</a> <a id="2495" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="2497" class="Symbol">→</a> <a id="2499" href="Data.Sum.Properties.html#730" class="Generalizable">C</a><a id="2500" class="Symbol">)</a> <a id="2502" class="Symbol">(</a><a id="2503" href="Data.Sum.Properties.html#2503" class="Bound">g</a> <a id="2505" class="Symbol">:</a> <a id="2507" href="Data.Sum.Properties.html#716" class="Generalizable">B</a> <a id="2509" class="Symbol">→</a> <a id="2511" href="Data.Sum.Properties.html#744" class="Generalizable">D</a><a id="2512" class="Symbol">)</a> <a id="2514" class="Symbol">(</a><a id="2515" href="Data.Sum.Properties.html#2515" class="Bound">h</a> <a id="2517" class="Symbol">:</a> <a id="2519" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="2521" class="Symbol">→</a> <a id="2523" href="Data.Sum.Properties.html#772" class="Generalizable">F</a><a id="2524" class="Symbol">)</a> <a id="2526" class="Symbol">→</a>
<a id="2530" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2534" href="Data.Sum.Properties.html#2491" class="Bound">f</a> <a id="2536" class="Symbol">(</a><a id="2537" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2541" href="Data.Sum.Properties.html#2503" class="Bound">g</a> <a id="2543" href="Data.Sum.Properties.html#2515" class="Bound">h</a><a id="2544" class="Symbol">)</a> <a id="2546" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2548" href="Data.Sum.Base.html#1431" class="Function">assocʳ</a> <a id="2555" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2557" href="Data.Sum.Base.html#1431" class="Function">assocʳ</a> <a id="2564" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="2566" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2570" class="Symbol">(</a><a id="2571" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="2575" href="Data.Sum.Properties.html#2491" class="Bound">f</a> <a id="2577" href="Data.Sum.Properties.html#2503" class="Bound">g</a><a id="2578" class="Symbol">)</a> <a id="2580" href="Data.Sum.Properties.html#2515" class="Bound">h</a>
<a id="2582" href="Data.Sum.Properties.html#2477" class="Function">map-assocʳ</a> <a id="2593" class="Symbol">_</a> <a id="2595" class="Symbol">_</a> <a id="2597" class="Symbol">_</a> <a id="2599" class="Symbol">(</a><a id="2600" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2605" class="Symbol">(</a><a id="2606" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2611" href="Data.Sum.Properties.html#2611" class="Bound">x</a><a id="2612" class="Symbol">))</a> <a id="2615" class="Symbol">=</a> <a id="2617" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2622" href="Data.Sum.Properties.html#2477" class="Function">map-assocʳ</a> <a id="2633" class="Symbol">_</a> <a id="2635" class="Symbol">_</a> <a id="2637" class="Symbol">_</a> <a id="2639" class="Symbol">(</a><a id="2640" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="2645" class="Symbol">(</a><a id="2646" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2651" href="Data.Sum.Properties.html#2651" class="Bound">y</a><a id="2652" class="Symbol">))</a> <a id="2655" class="Symbol">=</a> <a id="2657" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="2662" href="Data.Sum.Properties.html#2477" class="Function">map-assocʳ</a> <a id="2673" class="Symbol">_</a> <a id="2675" class="Symbol">_</a> <a id="2677" class="Symbol">_</a> <a id="2679" class="Symbol">(</a><a id="2680" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="2691" href="Data.Sum.Properties.html#2691" class="Bound">z</a> <a id="2693" class="Symbol">)</a> <a id="2695" class="Symbol">=</a> <a id="2697" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a>
<a id="[,]-cong"></a><a id="2703" href="Data.Sum.Properties.html#2703" class="Function">[,]-cong</a> <a id="2712" class="Symbol">:</a> <a id="2714" class="Symbol">{</a><a id="2715" href="Data.Sum.Properties.html#2715" class="Bound">f</a> <a id="2717" href="Data.Sum.Properties.html#2717" class="Bound">f′</a> <a id="2720" class="Symbol">:</a> <a id="2722" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="2724" class="Symbol">→</a> <a id="2726" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2727" class="Symbol">}</a> <a id="2729" class="Symbol">{</a><a id="2730" href="Data.Sum.Properties.html#2730" class="Bound">g</a> <a id="2732" href="Data.Sum.Properties.html#2732" class="Bound">g′</a> <a id="2735" class="Symbol">:</a> <a id="2737" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="2739" class="Symbol">→</a> <a id="2741" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2742" class="Symbol">}</a> <a id="2744" class="Symbol">→</a>
<a id="2757" href="Data.Sum.Properties.html#2715" class="Bound">f</a> <a id="2759" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2761" href="Data.Sum.Properties.html#2717" class="Bound">f′</a> <a id="2764" class="Symbol">→</a> <a id="2766" href="Data.Sum.Properties.html#2730" class="Bound">g</a> <a id="2768" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2770" href="Data.Sum.Properties.html#2732" class="Bound">g′</a> <a id="2773" class="Symbol">→</a>
<a id="2786" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="2788" href="Data.Sum.Properties.html#2715" class="Bound">f</a> <a id="2790" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="2792" href="Data.Sum.Properties.html#2730" class="Bound">g</a> <a id="2794" href="Data.Sum.Base.html#811" class="Function Operator">]</a> <a id="2796" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2798" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="2800" href="Data.Sum.Properties.html#2717" class="Bound">f′</a> <a id="2803" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="2805" href="Data.Sum.Properties.html#2732" class="Bound">g′</a> <a id="2808" href="Data.Sum.Base.html#811" class="Function Operator">]</a>
<a id="2810" href="Data.Sum.Properties.html#2703" class="Function">[,]-cong</a> <a id="2819" class="Symbol">=</a> <a id="2821" href="Data.Sum.Base.html#811" class="Function Operator">[_,_]</a>
<a id="[-,]-cong"></a><a id="2828" href="Data.Sum.Properties.html#2828" class="Function">[-,]-cong</a> <a id="2838" class="Symbol">:</a> <a id="2840" class="Symbol">{</a><a id="2841" href="Data.Sum.Properties.html#2841" class="Bound">f</a> <a id="2843" href="Data.Sum.Properties.html#2843" class="Bound">f′</a> <a id="2846" class="Symbol">:</a> <a id="2848" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="2850" class="Symbol">→</a> <a id="2852" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2853" class="Symbol">}</a> <a id="2855" class="Symbol">{</a><a id="2856" href="Data.Sum.Properties.html#2856" class="Bound">g</a> <a id="2858" class="Symbol">:</a> <a id="2860" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="2862" class="Symbol">→</a> <a id="2864" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2865" class="Symbol">}</a> <a id="2867" class="Symbol">→</a>
<a id="2881" href="Data.Sum.Properties.html#2841" class="Bound">f</a> <a id="2883" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2885" href="Data.Sum.Properties.html#2843" class="Bound">f′</a> <a id="2888" class="Symbol">→</a>
<a id="2902" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="2904" href="Data.Sum.Properties.html#2841" class="Bound">f</a> <a id="2906" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="2908" href="Data.Sum.Properties.html#2856" class="Bound">g</a> <a id="2910" href="Data.Sum.Base.html#811" class="Function Operator">]</a> <a id="2912" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="2914" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="2916" href="Data.Sum.Properties.html#2843" class="Bound">f′</a> <a id="2919" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="2921" href="Data.Sum.Properties.html#2856" class="Bound">g</a> <a id="2923" href="Data.Sum.Base.html#811" class="Function Operator">]</a>
<a id="2925" href="Data.Sum.Properties.html#2828" class="Function">[-,]-cong</a> <a id="2935" class="Symbol">=</a> <a id="2937" href="Data.Sum.Base.html#811" class="Function Operator">[_,</a> <a id="2941" class="Symbol">(λ</a> <a id="2944" href="Data.Sum.Properties.html#2944" class="Bound">_</a> <a id="2946" class="Symbol">→</a> <a id="2948" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="2952" class="Symbol">)</a> <a id="2954" href="Data.Sum.Base.html#811" class="Function Operator">]</a>
<a id="[,-]-cong"></a><a id="2957" href="Data.Sum.Properties.html#2957" class="Function">[,-]-cong</a> <a id="2967" class="Symbol">:</a> <a id="2969" class="Symbol">{</a><a id="2970" href="Data.Sum.Properties.html#2970" class="Bound">f</a> <a id="2972" class="Symbol">:</a> <a id="2974" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="2976" class="Symbol">→</a> <a id="2978" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2979" class="Symbol">}</a> <a id="2981" class="Symbol">{</a><a id="2982" href="Data.Sum.Properties.html#2982" class="Bound">g</a> <a id="2984" href="Data.Sum.Properties.html#2984" class="Bound">g′</a> <a id="2987" class="Symbol">:</a> <a id="2989" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="2991" class="Symbol">→</a> <a id="2993" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="2994" class="Symbol">}</a> <a id="2996" class="Symbol">→</a>
<a id="3010" href="Data.Sum.Properties.html#2982" class="Bound">g</a> <a id="3012" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3014" href="Data.Sum.Properties.html#2984" class="Bound">g′</a> <a id="3017" class="Symbol">→</a>
<a id="3031" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="3033" href="Data.Sum.Properties.html#2970" class="Bound">f</a> <a id="3035" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="3037" href="Data.Sum.Properties.html#2982" class="Bound">g</a> <a id="3039" href="Data.Sum.Base.html#811" class="Function Operator">]</a> <a id="3041" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3043" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="3045" href="Data.Sum.Properties.html#2970" class="Bound">f</a> <a id="3047" href="Data.Sum.Base.html#811" class="Function Operator">,</a> <a id="3049" href="Data.Sum.Properties.html#2984" class="Bound">g′</a> <a id="3052" href="Data.Sum.Base.html#811" class="Function Operator">]</a>
<a id="3054" href="Data.Sum.Properties.html#2957" class="Function">[,-]-cong</a> <a id="3064" class="Symbol">=</a> <a id="3066" href="Data.Sum.Base.html#811" class="Function Operator">[</a> <a id="3068" class="Symbol">(λ</a> <a id="3071" href="Data.Sum.Properties.html#3071" class="Bound">_</a> <a id="3073" class="Symbol">→</a> <a id="3075" href="Agda.Builtin.Equality.html#207" class="InductiveConstructor">refl</a><a id="3079" class="Symbol">)</a> <a id="3081" href="Data.Sum.Base.html#811" class="Function Operator">,_]</a>
<a id="map-cong"></a><a id="3086" href="Data.Sum.Properties.html#3086" class="Function">map-cong</a> <a id="3095" class="Symbol">:</a> <a id="3097" class="Symbol">{</a><a id="3098" href="Data.Sum.Properties.html#3098" class="Bound">f</a> <a id="3100" href="Data.Sum.Properties.html#3100" class="Bound">f′</a> <a id="3103" class="Symbol">:</a> <a id="3105" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="3107" class="Symbol">→</a> <a id="3109" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="3110" class="Symbol">}</a> <a id="3112" class="Symbol">{</a><a id="3113" href="Data.Sum.Properties.html#3113" class="Bound">g</a> <a id="3115" href="Data.Sum.Properties.html#3115" class="Bound">g′</a> <a id="3118" class="Symbol">:</a> <a id="3120" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="3122" class="Symbol">→</a> <a id="3124" href="Data.Sum.Properties.html#744" class="Generalizable">D</a><a id="3125" class="Symbol">}</a> <a id="3127" class="Symbol">→</a>
<a id="3140" href="Data.Sum.Properties.html#3098" class="Bound">f</a> <a id="3142" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3144" href="Data.Sum.Properties.html#3100" class="Bound">f′</a> <a id="3147" class="Symbol">→</a> <a id="3149" href="Data.Sum.Properties.html#3113" class="Bound">g</a> <a id="3151" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3153" href="Data.Sum.Properties.html#3115" class="Bound">g′</a> <a id="3156" class="Symbol">→</a>
<a id="3169" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="3173" href="Data.Sum.Properties.html#3098" class="Bound">f</a> <a id="3175" href="Data.Sum.Properties.html#3113" class="Bound">g</a> <a id="3177" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3179" href="Data.Sum.Base.html#1253" class="Function">map</a> <a id="3183" href="Data.Sum.Properties.html#3100" class="Bound">f′</a> <a id="3186" href="Data.Sum.Properties.html#3115" class="Bound">g′</a>
<a id="3189" href="Data.Sum.Properties.html#3086" class="Function">map-cong</a> <a id="3198" href="Data.Sum.Properties.html#3198" class="Bound">f≗f′</a> <a id="3203" href="Data.Sum.Properties.html#3203" class="Bound">g≗g′</a> <a id="3208" class="Symbol">(</a><a id="3209" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3214" href="Data.Sum.Properties.html#3214" class="Bound">x</a><a id="3215" class="Symbol">)</a> <a id="3217" class="Symbol">=</a> <a id="3219" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3224" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a> <a id="3229" class="Symbol">(</a><a id="3230" href="Data.Sum.Properties.html#3198" class="Bound">f≗f′</a> <a id="3235" href="Data.Sum.Properties.html#3214" class="Bound">x</a><a id="3236" class="Symbol">)</a>
<a id="3238" href="Data.Sum.Properties.html#3086" class="Function">map-cong</a> <a id="3247" href="Data.Sum.Properties.html#3247" class="Bound">f≗f′</a> <a id="3252" href="Data.Sum.Properties.html#3252" class="Bound">g≗g′</a> <a id="3257" class="Symbol">(</a><a id="3258" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3263" href="Data.Sum.Properties.html#3263" class="Bound">x</a><a id="3264" class="Symbol">)</a> <a id="3266" class="Symbol">=</a> <a id="3268" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3273" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a> <a id="3278" class="Symbol">(</a><a id="3279" href="Data.Sum.Properties.html#3252" class="Bound">g≗g′</a> <a id="3284" href="Data.Sum.Properties.html#3263" class="Bound">x</a><a id="3285" class="Symbol">)</a>
<a id="map₁-cong"></a><a id="3288" href="Data.Sum.Properties.html#3288" class="Function">map₁-cong</a> <a id="3298" class="Symbol">:</a> <a id="3300" class="Symbol">{</a><a id="3301" href="Data.Sum.Properties.html#3301" class="Bound">f</a> <a id="3303" href="Data.Sum.Properties.html#3303" class="Bound">f′</a> <a id="3306" class="Symbol">:</a> <a id="3308" href="Data.Sum.Properties.html#702" class="Generalizable">A</a> <a id="3310" class="Symbol">→</a> <a id="3312" href="Data.Sum.Properties.html#716" class="Generalizable">B</a><a id="3313" class="Symbol">}</a> <a id="3315" class="Symbol">→</a>
<a id="3329" href="Data.Sum.Properties.html#3301" class="Bound">f</a> <a id="3331" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3333" href="Data.Sum.Properties.html#3303" class="Bound">f′</a> <a id="3336" class="Symbol">→</a>
<a id="3350" href="Data.Sum.Base.html#1331" class="Function">map₁</a> <a id="3355" class="Symbol">{</a><a id="3356" class="Argument">B</a> <a id="3358" class="Symbol">=</a> <a id="3360" href="Data.Sum.Properties.html#730" class="Generalizable">C</a><a id="3361" class="Symbol">}</a> <a id="3363" href="Data.Sum.Properties.html#3301" class="Bound">f</a> <a id="3365" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3367" href="Data.Sum.Base.html#1331" class="Function">map₁</a> <a id="3372" href="Data.Sum.Properties.html#3303" class="Bound">f′</a>
<a id="3375" href="Data.Sum.Properties.html#3288" class="Function">map₁-cong</a> <a id="3385" href="Data.Sum.Properties.html#3385" class="Bound">f≗f′</a> <a id="3390" class="Symbol">=</a> <a id="3392" href="Data.Sum.Properties.html#2828" class="Function">[-,]-cong</a> <a id="3402" class="Symbol">((</a><a id="3404" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3409" href="Data.Sum.Base.html#675" class="InductiveConstructor">inj₁</a><a id="3413" class="Symbol">)</a> <a id="3415" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="3417" href="Data.Sum.Properties.html#3385" class="Bound">f≗f′</a><a id="3421" class="Symbol">)</a>
<a id="map₂-cong"></a><a id="3424" href="Data.Sum.Properties.html#3424" class="Function">map₂-cong</a> <a id="3434" class="Symbol">:</a> <a id="3436" class="Symbol">{</a><a id="3437" href="Data.Sum.Properties.html#3437" class="Bound">g</a> <a id="3439" href="Data.Sum.Properties.html#3439" class="Bound">g′</a> <a id="3442" class="Symbol">:</a> <a id="3444" href="Data.Sum.Properties.html#730" class="Generalizable">C</a> <a id="3446" class="Symbol">→</a> <a id="3448" href="Data.Sum.Properties.html#744" class="Generalizable">D</a><a id="3449" class="Symbol">}</a> <a id="3451" class="Symbol">→</a>
<a id="3465" href="Data.Sum.Properties.html#3437" class="Bound">g</a> <a id="3467" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3469" href="Data.Sum.Properties.html#3439" class="Bound">g′</a> <a id="3472" class="Symbol">→</a>
<a id="3486" href="Data.Sum.Base.html#1383" class="Function">map₂</a> <a id="3491" class="Symbol">{</a><a id="3492" class="Argument">A</a> <a id="3494" class="Symbol">=</a> <a id="3496" href="Data.Sum.Properties.html#702" class="Generalizable">A</a><a id="3497" class="Symbol">}</a> <a id="3499" href="Data.Sum.Properties.html#3437" class="Bound">g</a> <a id="3501" href="Relation.Binary.PropositionalEquality.html#1557" class="Function Operator">≗</a> <a id="3503" href="Data.Sum.Base.html#1383" class="Function">map₂</a> <a id="3508" href="Data.Sum.Properties.html#3439" class="Bound">g′</a>
<a id="3511" href="Data.Sum.Properties.html#3424" class="Function">map₂-cong</a> <a id="3521" href="Data.Sum.Properties.html#3521" class="Bound">g≗g′</a> <a id="3526" class="Symbol">=</a> <a id="3528" href="Data.Sum.Properties.html#2957" class="Function">[,-]-cong</a> <a id="3538" class="Symbol">((</a><a id="3540" href="Relation.Binary.PropositionalEquality.Core.html#1158" class="Function">cong</a> <a id="3545" href="Data.Sum.Base.html#700" class="InductiveConstructor">inj₂</a><a id="3549" class="Symbol">)</a> <a id="3551" href="Function.Base.html#1115" class="Function Operator">∘</a> <a id="3553" href="Data.Sum.Properties.html#3521" class="Bound">g≗g′</a><a id="3557" class="Symbol">)</a>
<a id="3560" class="Comment">------------------------------------------------------------------------</a>
<a id="3633" class="Comment">-- DEPRECATED NAMES</a>
<a id="3653" class="Comment">------------------------------------------------------------------------</a>
<a id="3726" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="3797" class="Comment">-- not guaranteed.</a>
<a id="3817" class="Comment">-- Version 2.0</a>
<a id="[,]-∘-distr"></a><a id="3833" href="Data.Sum.Properties.html#3833" class="Function">[,]-∘-distr</a> <a id="3845" class="Symbol">=</a> <a id="3847" href="Data.Sum.Properties.html#1587" class="Function">[,]-∘</a>
<a id="3853" class="Symbol">{-#</a> <a id="3857" class="Keyword">WARNING_ON_USAGE</a> <a id="3874" class="Pragma">[,]-∘-distr</a>
<a id="3886" class="String">&quot;Warning: [,]-∘-distr was deprecated in v2.0.
Please use [,]-∘ instead.&quot;</a>
<a id="3959" class="Symbol">#-}</a>
<a id="[,]-map-commute"></a><a id="3964" href="Data.Sum.Properties.html#3964" class="Function">[,]-map-commute</a> <a id="3980" class="Symbol">=</a> <a id="3982" href="Data.Sum.Properties.html#1744" class="Function">[,]-map</a>
<a id="3990" class="Symbol">{-#</a> <a id="3994" class="Keyword">WARNING_ON_USAGE</a> <a id="4011" class="Pragma">[,]-map-commute</a>
<a id="4027" class="String">&quot;Warning: [,]-map-commute was deprecated in v2.0.
Please use [,]-map instead.&quot;</a>
<a id="4106" class="Symbol">#-}</a>
<a id="map-commute"></a><a id="4111" href="Data.Sum.Properties.html#4111" class="Function">map-commute</a> <a id="4123" class="Symbol">=</a> <a id="4125" href="Data.Sum.Properties.html#1920" class="Function">map-map</a>
<a id="4133" class="Symbol">{-#</a> <a id="4137" class="Keyword">WARNING_ON_USAGE</a> <a id="4154" class="Pragma">map-commute</a>
<a id="4166" class="String">&quot;Warning: map-commute was deprecated in v2.0.
Please use map-map instead.&quot;</a>
<a id="4241" class="Symbol">#-}</a>
<a id="map₁₂-commute"></a><a id="4246" href="Data.Sum.Properties.html#4246" class="Function">map₁₂-commute</a> <a id="4260" class="Symbol">=</a> <a id="4262" href="Data.Sum.Properties.html#2104" class="Function">map₁₂-map₂₁</a>
<a id="4274" class="Symbol">{-#</a> <a id="4278" class="Keyword">WARNING_ON_USAGE</a> <a id="4295" class="Pragma">map₁₂-commute</a>
<a id="4309" class="String">&quot;Warning: map₁₂-commute was deprecated in v2.0.
Please use map₁₂-map₂₁ instead.&quot;</a>
<a id="4390" class="Symbol">#-}</a>
</pre></body></html>
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