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<html><head><meta charset="utf-8"><title>Relation.Binary.Reasoning.Base.Double</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">-- The basic code for equational reasoning with two relations:</a>
<a id="169" class="Comment">-- equality and some other ordering.</a>
<a id="206" class="Comment">------------------------------------------------------------------------</a>
<a id="279" class="Comment">--</a>
<a id="282" class="Comment">-- See `Data.Nat.Properties` or `Relation.Binary.Reasoning.PartialOrder`</a>
<a id="355" class="Comment">-- for examples of how to instantiate this module.</a>
<a id="407" class="Symbol">{-#</a> <a id="411" class="Keyword">OPTIONS</a> <a id="419" class="Pragma">--cubical-compatible</a> <a id="440" class="Pragma">--safe</a> <a id="447" class="Symbol">#-}</a>
<a id="452" class="Keyword">open</a> <a id="457" class="Keyword">import</a> <a id="464" href="Level.html" class="Module">Level</a> <a id="470" class="Keyword">using</a> <a id="476" class="Symbol">(</a><a id="477" href="Agda.Primitive.html#961" class="Primitive Operator">_⊔_</a><a id="480" class="Symbol">)</a>
<a id="482" class="Keyword">open</a> <a id="487" class="Keyword">import</a> <a id="494" href="Function.html" class="Module">Function</a> <a id="503" class="Keyword">using</a> <a id="509" class="Symbol">(</a><a id="510" href="Function.Base.html#4042" class="Function Operator">case_of_</a><a id="518" class="Symbol">)</a>
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<a id="830" class="Keyword">open</a> <a id="835" class="Keyword">import</a> <a id="842" href="Relation.Binary.Reasoning.Syntax.html" class="Module">Relation.Binary.Reasoning.Syntax</a>
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<a id="1016" class="Keyword">where</a>
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<a id="1051" class="Comment">------------------------------------------------------------------------</a>
<a id="1124" class="Comment">-- A datatype to hide the current relation type</a>
<a id="1173" class="Keyword">infix</a> <a id="1179" class="Number">4</a> <a id="1181" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a>
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<a id="_IsRelatedTo_.equals"></a><a id="1299" href="Relation.Binary.Reasoning.Base.Double.html#1299" class="InductiveConstructor">equals</a> <a id="1309" class="Symbol">:</a> <a id="1311" class="Symbol">(</a><a id="1312" href="Relation.Binary.Reasoning.Base.Double.html#1312" class="Bound">x≈y</a> <a id="1316" class="Symbol">:</a> <a id="1318" href="Relation.Binary.Reasoning.Base.Double.html#1216" class="Bound">x</a> <a id="1320" href="Relation.Binary.Reasoning.Base.Double.html#947" class="Bound Operator">≈</a> <a id="1322" href="Relation.Binary.Reasoning.Base.Double.html#1218" class="Bound">y</a><a id="1323" class="Symbol">)</a> <a id="1325" class="Symbol">→</a> <a id="1327" href="Relation.Binary.Reasoning.Base.Double.html#1216" class="Bound">x</a> <a id="1329" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">IsRelatedTo</a> <a id="1341" href="Relation.Binary.Reasoning.Base.Double.html#1218" class="Bound">y</a>
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<a id="1407" href="Relation.Binary.Reasoning.Base.Double.html#1344" class="Function">start</a> <a id="1413" class="Symbol">(</a><a id="1414" href="Relation.Binary.Reasoning.Base.Double.html#1253" class="InductiveConstructor">nonstrict</a> <a id="1424" href="Relation.Binary.Reasoning.Base.Double.html#1424" class="Bound">x≲y</a><a id="1427" class="Symbol">)</a> <a id="1429" class="Symbol">=</a> <a id="1431" href="Relation.Binary.Reasoning.Base.Double.html#1424" class="Bound">x≲y</a>
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<a id="≈-go"></a><a id="1773" href="Relation.Binary.Reasoning.Base.Double.html#1773" class="Function">≈-go</a> <a id="1778" class="Symbol">:</a> <a id="1780" href="Relation.Binary.Definitions.html#1674" class="Function">Trans</a> <a id="1786" href="Relation.Binary.Reasoning.Base.Double.html#947" class="Bound Operator">_≈_</a> <a id="1790" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="1804" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a>
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<a id="1989" class="Comment">------------------------------------------------------------------------</a>
<a id="2062" class="Comment">-- A record that is used to ensure that the final relation proved by the</a>
<a id="2135" class="Comment">-- chain of reasoning can be converted into the required relation.</a>
<a id="2203" class="Keyword">data</a> <a id="IsEquality"></a><a id="2208" href="Relation.Binary.Reasoning.Base.Double.html#2208" class="Datatype">IsEquality</a> <a id="2219" class="Symbol">{</a><a id="2220" href="Relation.Binary.Reasoning.Base.Double.html#2220" class="Bound">x</a> <a id="2222" href="Relation.Binary.Reasoning.Base.Double.html#2222" class="Bound">y</a><a id="2223" class="Symbol">}</a> <a id="2225" class="Symbol">:</a> <a id="2227" href="Relation.Binary.Reasoning.Base.Double.html#2220" class="Bound">x</a> <a id="2229" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">IsRelatedTo</a> <a id="2241" href="Relation.Binary.Reasoning.Base.Double.html#2222" class="Bound">y</a> <a id="2243" class="Symbol">→</a> <a id="2245" href="Agda.Primitive.html#388" class="Primitive">Set</a> <a id="2249" class="Symbol">(</a><a id="2250" href="Relation.Binary.Reasoning.Base.Double.html#923" class="Bound">a</a> <a id="2252" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="2254" href="Relation.Binary.Reasoning.Base.Double.html#925" class="Bound">ℓ₁</a> <a id="2257" href="Agda.Primitive.html#961" class="Primitive Operator">⊔</a> <a id="2259" href="Relation.Binary.Reasoning.Base.Double.html#928" class="Bound">ℓ₂</a><a id="2261" class="Symbol">)</a> <a id="2263" class="Keyword">where</a>
<a id="IsEquality.isEquality"></a><a id="2271" href="Relation.Binary.Reasoning.Base.Double.html#2271" class="InductiveConstructor">isEquality</a> <a id="2282" class="Symbol">:</a> <a id="2284" class="Symbol">∀</a> <a id="2286" href="Relation.Binary.Reasoning.Base.Double.html#2286" class="Bound">x≈y</a> <a id="2290" class="Symbol">→</a> <a id="2292" href="Relation.Binary.Reasoning.Base.Double.html#2208" class="Datatype">IsEquality</a> <a id="2303" class="Symbol">(</a><a id="2304" href="Relation.Binary.Reasoning.Base.Double.html#1299" class="InductiveConstructor">equals</a> <a id="2311" href="Relation.Binary.Reasoning.Base.Double.html#2286" class="Bound">x≈y</a><a id="2314" class="Symbol">)</a>
<a id="IsEquality?"></a><a id="2317" href="Relation.Binary.Reasoning.Base.Double.html#2317" class="Function">IsEquality?</a> <a id="2329" class="Symbol">:</a> <a id="2331" class="Symbol">∀</a> <a id="2333" class="Symbol">{</a><a id="2334" href="Relation.Binary.Reasoning.Base.Double.html#2334" class="Bound">x</a> <a id="2336" href="Relation.Binary.Reasoning.Base.Double.html#2336" class="Bound">y</a><a id="2337" class="Symbol">}</a> <a id="2339" class="Symbol">(</a><a id="2340" href="Relation.Binary.Reasoning.Base.Double.html#2340" class="Bound">x≲y</a> <a id="2344" class="Symbol">:</a> <a id="2346" href="Relation.Binary.Reasoning.Base.Double.html#2334" class="Bound">x</a> <a id="2348" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">IsRelatedTo</a> <a id="2360" href="Relation.Binary.Reasoning.Base.Double.html#2336" class="Bound">y</a><a id="2361" class="Symbol">)</a> <a id="2363" class="Symbol">→</a> <a id="2365" href="Relation.Nullary.Decidable.Core.html#1485" class="Record">Dec</a> <a id="2369" class="Symbol">(</a><a id="2370" href="Relation.Binary.Reasoning.Base.Double.html#2208" class="Datatype">IsEquality</a> <a id="2381" href="Relation.Binary.Reasoning.Base.Double.html#2340" class="Bound">x≲y</a><a id="2384" class="Symbol">)</a>
<a id="2386" href="Relation.Binary.Reasoning.Base.Double.html#2317" class="Function">IsEquality?</a> <a id="2398" class="Symbol">(</a><a id="2399" href="Relation.Binary.Reasoning.Base.Double.html#1253" class="InductiveConstructor">nonstrict</a> <a id="2409" class="Symbol">_)</a> <a id="2412" class="Symbol">=</a> <a id="2414" href="Relation.Nullary.Decidable.Core.html#1655" class="InductiveConstructor">no</a> <a id="2417" class="Symbol">λ()</a>
<a id="2421" href="Relation.Binary.Reasoning.Base.Double.html#2317" class="Function">IsEquality?</a> <a id="2433" class="Symbol">(</a><a id="2434" href="Relation.Binary.Reasoning.Base.Double.html#1299" class="InductiveConstructor">equals</a> <a id="2441" href="Relation.Binary.Reasoning.Base.Double.html#2441" class="Bound">x≈y</a><a id="2444" class="Symbol">)</a> <a id="2447" class="Symbol">=</a> <a id="2449" href="Relation.Nullary.Decidable.Core.html#1618" class="InductiveConstructor">yes</a> <a id="2453" class="Symbol">(</a><a id="2454" href="Relation.Binary.Reasoning.Base.Double.html#2271" class="InductiveConstructor">isEquality</a> <a id="2465" href="Relation.Binary.Reasoning.Base.Double.html#2441" class="Bound">x≈y</a><a id="2468" class="Symbol">)</a>
<a id="extractEquality"></a><a id="2471" href="Relation.Binary.Reasoning.Base.Double.html#2471" class="Function">extractEquality</a> <a id="2487" class="Symbol">:</a> <a id="2489" class="Symbol">∀</a> <a id="2491" class="Symbol">{</a><a id="2492" href="Relation.Binary.Reasoning.Base.Double.html#2492" class="Bound">x</a> <a id="2494" href="Relation.Binary.Reasoning.Base.Double.html#2494" class="Bound">y</a><a id="2495" class="Symbol">}</a> <a id="2497" class="Symbol">{</a><a id="2498" href="Relation.Binary.Reasoning.Base.Double.html#2498" class="Bound">x≲y</a> <a id="2502" class="Symbol">:</a> <a id="2504" href="Relation.Binary.Reasoning.Base.Double.html#2492" class="Bound">x</a> <a id="2506" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">IsRelatedTo</a> <a id="2518" href="Relation.Binary.Reasoning.Base.Double.html#2494" class="Bound">y</a><a id="2519" class="Symbol">}</a> <a id="2521" class="Symbol">→</a> <a id="2523" href="Relation.Binary.Reasoning.Base.Double.html#2208" class="Datatype">IsEquality</a> <a id="2534" href="Relation.Binary.Reasoning.Base.Double.html#2498" class="Bound">x≲y</a> <a id="2538" class="Symbol">→</a> <a id="2540" href="Relation.Binary.Reasoning.Base.Double.html#2492" class="Bound">x</a> <a id="2542" href="Relation.Binary.Reasoning.Base.Double.html#947" class="Bound Operator">≈</a> <a id="2544" href="Relation.Binary.Reasoning.Base.Double.html#2494" class="Bound">y</a>
<a id="2546" href="Relation.Binary.Reasoning.Base.Double.html#2471" class="Function">extractEquality</a> <a id="2562" class="Symbol">(</a><a id="2563" href="Relation.Binary.Reasoning.Base.Double.html#2271" class="InductiveConstructor">isEquality</a> <a id="2574" href="Relation.Binary.Reasoning.Base.Double.html#2574" class="Bound">x≈y</a><a id="2577" class="Symbol">)</a> <a id="2579" class="Symbol">=</a> <a id="2581" href="Relation.Binary.Reasoning.Base.Double.html#2574" class="Bound">x≈y</a>
<a id="equalitySubRelation"></a><a id="2586" href="Relation.Binary.Reasoning.Base.Double.html#2586" class="Function">equalitySubRelation</a> <a id="2606" class="Symbol">:</a> <a id="2608" href="Relation.Binary.Reasoning.Syntax.html#1920" class="Record">SubRelation</a> <a id="2621" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="2635" class="Symbol">_</a> <a id="2637" class="Symbol">_</a>
<a id="2639" href="Relation.Binary.Reasoning.Base.Double.html#2586" class="Function">equalitySubRelation</a> <a id="2659" class="Symbol">=</a> <a id="2661" class="Keyword">record</a>
<a id="2670" class="Symbol">{</a> <a id="2672" href="Relation.Binary.Reasoning.Syntax.html#2033" class="Field">IsS</a> <a id="2676" class="Symbol">=</a> <a id="2678" href="Relation.Binary.Reasoning.Base.Double.html#2208" class="Datatype">IsEquality</a>
<a id="2691" class="Symbol">;</a> <a id="2693" href="Relation.Binary.Reasoning.Syntax.html#2058" class="Field">IsS?</a> <a id="2698" class="Symbol">=</a> <a id="2700" href="Relation.Binary.Reasoning.Base.Double.html#2317" class="Function">IsEquality?</a>
<a id="2714" class="Symbol">;</a> <a id="2716" href="Relation.Binary.Reasoning.Syntax.html#2101" class="Field">extract</a> <a id="2724" class="Symbol">=</a> <a id="2726" href="Relation.Binary.Reasoning.Base.Double.html#2471" class="Function">extractEquality</a>
<a id="2744" class="Symbol">}</a>
<a id="2747" class="Comment">------------------------------------------------------------------------</a>
<a id="2820" class="Comment">-- Reasoning combinators</a>
<a id="2846" class="Keyword">open</a> <a id="2851" href="Relation.Binary.Reasoning.Syntax.html#1408" class="Module">begin-syntax</a> <a id="2865" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="2879" href="Relation.Binary.Reasoning.Base.Double.html#1344" class="Function">start</a> <a id="2885" class="Keyword">public</a>
<a id="2892" class="Keyword">open</a> <a id="2897" href="Relation.Binary.Reasoning.Syntax.html#2412" class="Module">begin-equality-syntax</a> <a id="2920" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="2934" href="Relation.Binary.Reasoning.Base.Double.html#2586" class="Function">equalitySubRelation</a> <a id="2954" class="Keyword">public</a>
<a id="2961" class="Keyword">open</a> <a id="2966" href="Relation.Binary.Reasoning.Syntax.html#10883" class="Module">≡-syntax</a> <a id="2975" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="2989" href="Relation.Binary.Reasoning.Base.Double.html#1436" class="Function">≡-go</a> <a id="2994" class="Keyword">public</a>
<a id="3001" class="Keyword">open</a> <a id="3006" href="Relation.Binary.Reasoning.Syntax.html#6995" class="Module">≈-syntax</a> <a id="3015" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3029" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3043" href="Relation.Binary.Reasoning.Base.Double.html#1773" class="Function">≈-go</a> <a id="3048" href="Relation.Binary.Structures.html#1622" class="Function">Eq.sym</a> <a id="3055" class="Keyword">public</a>
<a id="3062" class="Keyword">open</a> <a id="3067" href="Relation.Binary.Reasoning.Syntax.html#5305" class="Module">≲-syntax</a> <a id="3076" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3090" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3104" href="Relation.Binary.Reasoning.Base.Double.html#1620" class="Function">≲-go</a> <a id="3109" class="Keyword">public</a>
<a id="3116" class="Keyword">open</a> <a id="3121" href="Relation.Binary.Reasoning.Syntax.html#12202" class="Module">end-syntax</a> <a id="3132" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3146" href="Relation.Binary.Reasoning.Base.Double.html#1935" class="Function">stop</a> <a id="3151" class="Keyword">public</a>
<a id="3160" class="Comment">------------------------------------------------------------------------</a>
<a id="3233" class="Comment">-- DEPRECATED NAMES</a>
<a id="3253" class="Comment">------------------------------------------------------------------------</a>
<a id="3326" class="Comment">-- Please use the new names as continuing support for the old names is</a>
<a id="3397" class="Comment">-- not guaranteed.</a>
<a id="3417" class="Comment">-- Version 2.0</a>
<a id="3433" class="Keyword">open</a> <a id="3438" href="Relation.Binary.Reasoning.Syntax.html#5172" class="Module">∼-syntax</a> <a id="3447" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3461" href="Relation.Binary.Reasoning.Base.Double.html#1201" class="Datatype Operator">_IsRelatedTo_</a> <a id="3475" href="Relation.Binary.Reasoning.Base.Double.html#1620" class="Function">≲-go</a> <a id="3480" class="Keyword">public</a>
<a id="3487" class="Symbol">{-#</a> <a id="3491" class="Keyword">WARNING_ON_USAGE</a> <a id="3508" class="Pragma">step-∼</a>
<a id="3515" class="String">&quot;Warning: step-∼ and _∼⟨_⟩_ syntax was deprecated in v2.0.
Please use step-≲ and _≲⟨_⟩_ instead. &quot;</a>
<a id="3614" class="Symbol">#-}</a>
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