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<html><head><meta charset="utf-8"><title>Categories.Morphism.Reasoning.Iso</title><link rel="stylesheet" href="Agda.css"></head><body><pre class="Agda"><a id="1" class="Symbol">{-#</a> <a id="5" class="Keyword">OPTIONS</a> <a id="13" class="Pragma">--without-K</a> <a id="25" class="Pragma">--safe</a> <a id="32" class="Symbol">#-}</a>
<a id="36" class="Keyword">open</a> <a id="41" class="Keyword">import</a> <a id="48" href="Categories.Category.html" class="Module">Categories.Category</a>
<a id="69" class="Comment">{-
Various combinators for working with Isomorphisms in the
context of morphism equalities
both for Category (Switch) and IsGroupoid (GroupoidR)
-}</a>
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<a id="595" class="Keyword">private</a>
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<a id="2544" href="Categories.Morphism.Reasoning.Iso.html#2506" class="Function">flip-fromʳ</a> <a id="2555" class="Symbol">{</a><a id="2556" href="Categories.Morphism.Reasoning.Iso.html#2556" class="Bound">h</a><a id="2557" class="Symbol">}</a> <a id="2559" href="Categories.Morphism.Reasoning.Iso.html#2559" class="Bound">eq</a> <a id="2562" class="Symbol">=</a> <a id="2564" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2574" href="Categories.Morphism.Reasoning.Iso.html#2556" class="Bound">h</a> <a id="2588" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2591" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="2598" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="2603" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2609" href="Categories.Morphism.Reasoning.Iso.html#2556" class="Bound">h</a> <a id="2611" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2613" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2618" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2620" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2623" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2626" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="2634" href="Categories.Morphism.Reasoning.Iso.html#2559" class="Bound">eq</a> <a id="2637" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2643" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2657" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="Switch.flip-toˡ"></a><a id="2662" href="Categories.Morphism.Reasoning.Iso.html#2662" class="Function">flip-toˡ</a> <a id="2671" class="Symbol">:</a> <a id="2673" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2676" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2678" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="2680" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2682" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2685" class="Symbol">→</a> <a id="2687" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="2689" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2691" href="Categories.Morphism.html#2006" class="Field">from</a>
<a id="2698" href="Categories.Morphism.Reasoning.Iso.html#2662" class="Function">flip-toˡ</a> <a id="2707" class="Symbol">{</a><a id="2708" href="Categories.Morphism.Reasoning.Iso.html#2708" class="Bound">h</a><a id="2709" class="Symbol">}</a> <a id="2711" href="Categories.Morphism.Reasoning.Iso.html#2711" class="Bound">eq</a> <a id="2714" class="Symbol">=</a> <a id="2716" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2726" href="Categories.Morphism.Reasoning.Iso.html#2708" class="Bound">h</a> <a id="2742" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2745" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="2752" href="Categories.Morphism.html#1612" class="Function">isoʳ</a> <a id="2757" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2763" class="Symbol">(</a><a id="2764" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2769" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2771" href="Categories.Morphism.html#2023" class="Field">to</a><a id="2773" class="Symbol">)</a> <a id="2775" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2777" href="Categories.Morphism.Reasoning.Iso.html#2708" class="Bound">h</a> <a id="2779" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2782" href="Categories.Morphism.Reasoning.Core.html#6665" class="Function">cancelʳ</a> <a id="2790" href="Categories.Morphism.Reasoning.Iso.html#2711" class="Bound">eq</a> <a id="2793" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2799" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2815" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="Switch.flip-toʳ"></a><a id="2820" href="Categories.Morphism.Reasoning.Iso.html#2820" class="Function">flip-toʳ</a> <a id="2829" class="Symbol">:</a> <a id="2831" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="2833" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2835" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2838" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2840" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="2843" class="Symbol">→</a> <a id="2845" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="2847" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="2849" href="Categories.Morphism.html#2006" class="Field">from</a>
<a id="2856" href="Categories.Morphism.Reasoning.Iso.html#2820" class="Function">flip-toʳ</a> <a id="2865" class="Symbol">{</a><a id="2866" href="Categories.Morphism.Reasoning.Iso.html#2866" class="Bound">h</a><a id="2867" class="Symbol">}</a> <a id="2869" href="Categories.Morphism.Reasoning.Iso.html#2869" class="Bound">eq</a> <a id="2872" class="Symbol">=</a> <a id="2874" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2884" href="Categories.Morphism.Reasoning.Iso.html#2866" class="Bound">h</a> <a id="2898" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2901" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="2908" href="Categories.Morphism.html#1586" class="Function">isoˡ</a> <a id="2913" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2919" href="Categories.Morphism.Reasoning.Iso.html#2866" class="Bound">h</a> <a id="2921" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2923" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="2926" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="2928" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2933" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2936" href="Categories.Morphism.Reasoning.Core.html#6851" class="Function">cancelˡ</a> <a id="2944" href="Categories.Morphism.Reasoning.Iso.html#2869" class="Bound">eq</a> <a id="2947" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2953" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="2967" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="2972" class="Comment">-- We can flip an iso i in a commuting triangle, like so:</a>
<a id="3032" class="Comment">--</a>
<a id="3037" class="Comment">-- i i⁻¹</a>
<a id="3079" class="Comment">-- X --------&gt; Y X &lt;-------- Y</a>
<a id="3126" class="Comment">-- \ ≃ / \ ≃ /</a>
<a id="3172" class="Comment">-- \ / \ /</a>
<a id="3217" class="Comment">-- g \ / h ===&gt; g \ / h</a>
<a id="3263" class="Comment">-- \ / \ /</a>
<a id="3306" class="Comment">-- V V V V</a>
<a id="3348" class="Comment">-- A A</a>
<a id="3389" class="Comment">--</a>
<a id="Switch.flip-iso"></a><a id="3394" href="Categories.Morphism.Reasoning.Iso.html#3394" class="Function">flip-iso</a> <a id="3403" class="Symbol">:</a> <a id="3405" class="Symbol">{</a><a id="3406" href="Categories.Morphism.Reasoning.Iso.html#3406" class="Bound">g</a> <a id="3408" class="Symbol">:</a> <a id="3410" href="Categories.Morphism.Reasoning.Iso.html#691" class="Bound">X</a> <a id="3412" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="3414" href="Categories.Morphism.Reasoning.Iso.html#618" class="Generalizable">A</a><a id="3415" class="Symbol">}</a> <a id="3417" class="Symbol">{</a><a id="3418" href="Categories.Morphism.Reasoning.Iso.html#3418" class="Bound">h</a> <a id="3420" class="Symbol">:</a> <a id="3422" href="Categories.Morphism.Reasoning.Iso.html#695" class="Bound">Y</a> <a id="3424" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="3426" href="Categories.Morphism.Reasoning.Iso.html#618" class="Generalizable">A</a><a id="3427" class="Symbol">}</a> <a id="3429" class="Symbol">→</a> <a id="3431" href="Categories.Morphism.Reasoning.Iso.html#3406" class="Bound">g</a> <a id="3433" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="3435" href="Categories.Morphism.Reasoning.Iso.html#3418" class="Bound">h</a> <a id="3437" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3439" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="3444" class="Symbol">→</a> <a id="3446" href="Categories.Morphism.Reasoning.Iso.html#3406" class="Bound">g</a> <a id="3448" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="3450" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="3453" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="3455" href="Categories.Morphism.Reasoning.Iso.html#3418" class="Bound">h</a>
<a id="3459" href="Categories.Morphism.Reasoning.Iso.html#3394" class="Function">flip-iso</a> <a id="3468" href="Categories.Morphism.Reasoning.Iso.html#3468" class="Bound">tr₁</a> <a id="3472" class="Symbol">=</a> <a id="3474" href="Relation.Binary.Structures.html#1622" class="Function">Equiv.sym</a> <a id="3484" class="Symbol">(</a><a id="3485" href="Categories.Morphism.Reasoning.Iso.html#1094" class="Function">switch-fromtoʳ</a> <a id="3500" class="Symbol">(</a><a id="3501" href="Relation.Binary.Structures.html#1622" class="Function">Equiv.sym</a> <a id="3511" href="Categories.Morphism.Reasoning.Iso.html#3468" class="Bound">tr₁</a><a id="3514" class="Symbol">))</a>
<a id="3520" class="Comment">-- Consider two commuting squares</a>
<a id="3556" class="Comment">--</a>
<a id="3561" class="Comment">-- f₁ f₂</a>
<a id="3601" class="Comment">-- X -------&gt; A X -------&gt; A</a>
<a id="3646" class="Comment">-- | | | |</a>
<a id="3691" class="Comment">-- | | | |</a>
<a id="3736" class="Comment">-- ≃ | i | h ≃ | i | h</a>
<a id="3783" class="Comment">-- | | | |</a>
<a id="3828" class="Comment">-- V V V V</a>
<a id="3873" class="Comment">-- Y -------&gt; B Y -------&gt; B</a>
<a id="3918" class="Comment">-- g₁ g₂</a>
<a id="3958" class="Comment">--</a>
<a id="3963" class="Comment">-- with i an isomorphism. Then g₁ ≈ g₂ if f₁ ≈ f₂.</a>
<a id="Switch.push-eq"></a><a id="4018" href="Categories.Morphism.Reasoning.Iso.html#4018" class="Function">push-eq</a> <a id="4026" class="Symbol">:</a> <a id="4028" class="Symbol">{</a><a id="4029" href="Categories.Morphism.Reasoning.Iso.html#4029" class="Bound">f₁</a> <a id="4032" href="Categories.Morphism.Reasoning.Iso.html#4032" class="Bound">f₂</a> <a id="4035" class="Symbol">:</a> <a id="4037" href="Categories.Morphism.Reasoning.Iso.html#691" class="Bound">X</a> <a id="4039" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="4041" href="Categories.Morphism.Reasoning.Iso.html#618" class="Generalizable">A</a><a id="4042" class="Symbol">}</a> <a id="4044" class="Symbol">{</a><a id="4045" href="Categories.Morphism.Reasoning.Iso.html#4045" class="Bound">g₁</a> <a id="4048" href="Categories.Morphism.Reasoning.Iso.html#4048" class="Bound">g₂</a> <a id="4051" class="Symbol">:</a> <a id="4053" href="Categories.Morphism.Reasoning.Iso.html#695" class="Bound">Y</a> <a id="4055" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="4057" href="Categories.Morphism.Reasoning.Iso.html#620" class="Generalizable">B</a><a id="4058" class="Symbol">}</a> <a id="4060" class="Symbol">{</a><a id="4061" href="Categories.Morphism.Reasoning.Iso.html#4061" class="Bound">h</a> <a id="4063" class="Symbol">:</a> <a id="4065" href="Categories.Morphism.Reasoning.Iso.html#618" class="Generalizable">A</a> <a id="4067" href="Categories.Category.Core.html#575" class="Field Operator">⇒</a> <a id="4069" href="Categories.Morphism.Reasoning.Iso.html#620" class="Generalizable">B</a><a id="4070" class="Symbol">}</a> <a id="4072" class="Symbol">→</a>
<a id="4086" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="4104" href="Categories.Morphism.Reasoning.Iso.html#4029" class="Bound">f₁</a> <a id="4107" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="4112" href="Categories.Morphism.Reasoning.Iso.html#4061" class="Bound">h</a> <a id="4114" href="Categories.Morphism.Reasoning.Iso.html#4045" class="Bound">g₁</a> <a id="4117" class="Symbol">→</a>
<a id="4131" href="Categories.Category.html#884" class="Function">CommutativeSquare</a> <a id="4149" href="Categories.Morphism.Reasoning.Iso.html#4032" class="Bound">f₂</a> <a id="4152" href="Categories.Morphism.html#2006" class="Field">from</a> <a id="4157" href="Categories.Morphism.Reasoning.Iso.html#4061" class="Bound">h</a> <a id="4159" href="Categories.Morphism.Reasoning.Iso.html#4048" class="Bound">g₂</a> <a id="4162" class="Symbol">→</a>
<a id="4176" href="Categories.Morphism.Reasoning.Iso.html#4029" class="Bound">f₁</a> <a id="4179" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4181" href="Categories.Morphism.Reasoning.Iso.html#4032" class="Bound">f₂</a> <a id="4184" class="Symbol">→</a> <a id="4186" href="Categories.Morphism.Reasoning.Iso.html#4045" class="Bound">g₁</a> <a id="4189" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4191" href="Categories.Morphism.Reasoning.Iso.html#4048" class="Bound">g₂</a>
<a id="4196" href="Categories.Morphism.Reasoning.Iso.html#4018" class="Function">push-eq</a> <a id="4204" class="Symbol">{</a><a id="4205" class="Argument">f₁</a> <a id="4208" class="Symbol">=</a> <a id="4210" href="Categories.Morphism.Reasoning.Iso.html#4210" class="Bound">f₁</a><a id="4212" class="Symbol">}</a> <a id="4214" class="Symbol">{</a><a id="4215" href="Categories.Morphism.Reasoning.Iso.html#4215" class="Bound">f₂</a><a id="4217" class="Symbol">}</a> <a id="4219" class="Symbol">{</a><a id="4220" href="Categories.Morphism.Reasoning.Iso.html#4220" class="Bound">g₁</a><a id="4222" class="Symbol">}</a> <a id="4224" class="Symbol">{</a><a id="4225" href="Categories.Morphism.Reasoning.Iso.html#4225" class="Bound">g₂</a><a id="4227" class="Symbol">}</a> <a id="4229" class="Symbol">{</a><a id="4230" href="Categories.Morphism.Reasoning.Iso.html#4230" class="Bound">h₂</a><a id="4232" class="Symbol">}</a> <a id="4234" href="Categories.Morphism.Reasoning.Iso.html#4234" class="Bound">sq₁</a> <a id="4238" href="Categories.Morphism.Reasoning.Iso.html#4238" class="Bound">sq₂</a> <a id="4242" href="Categories.Morphism.Reasoning.Iso.html#4242" class="Bound">hyp</a> <a id="4246" class="Symbol">=</a> <a id="4248" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4258" href="Categories.Morphism.Reasoning.Iso.html#4220" class="Bound">g₁</a> <a id="4275" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">≈˘⟨</a> <a id="4279" href="Categories.Morphism.Reasoning.Iso.html#3394" class="Function">flip-iso</a> <a id="4288" href="Categories.Morphism.Reasoning.Iso.html#4234" class="Bound">sq₁</a> <a id="4292" href="Relation.Binary.Reasoning.Syntax.html#7400" class="Function">⟩</a>
<a id="4298" class="Symbol">(</a><a id="4299" href="Categories.Morphism.Reasoning.Iso.html#4230" class="Bound">h₂</a> <a id="4302" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4304" href="Categories.Morphism.Reasoning.Iso.html#4210" class="Bound">f₁</a><a id="4306" class="Symbol">)</a> <a id="4308" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4310" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="4315" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4318" href="Categories.Category.Core.html#1597" class="Function">∘-resp-≈ˡ</a> <a id="4328" class="Symbol">(</a><a id="4329" href="Categories.Category.Core.html#1706" class="Function">∘-resp-≈ʳ</a> <a id="4339" href="Categories.Morphism.Reasoning.Iso.html#4242" class="Bound">hyp</a><a id="4342" class="Symbol">)</a> <a id="4344" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4350" class="Symbol">(</a><a id="4351" href="Categories.Morphism.Reasoning.Iso.html#4230" class="Bound">h₂</a> <a id="4354" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4356" href="Categories.Morphism.Reasoning.Iso.html#4215" class="Bound">f₂</a><a id="4358" class="Symbol">)</a> <a id="4360" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4362" href="Categories.Morphism.html#2023" class="Field">to</a> <a id="4367" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4370" href="Categories.Morphism.Reasoning.Iso.html#3394" class="Function">flip-iso</a> <a id="4379" href="Categories.Morphism.Reasoning.Iso.html#4238" class="Bound">sq₂</a> <a id="4383" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4389" href="Categories.Morphism.Reasoning.Iso.html#4225" class="Bound">g₂</a> <a id="4406" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4409" class="Keyword">open</a> <a id="4414" href="Categories.Morphism.Reasoning.Iso.html#679" class="Module">Switch</a> <a id="4421" class="Keyword">public</a>
<a id="4429" class="Comment">-- conjugates</a>
<a id="4443" class="Keyword">module</a> <a id="4450" href="Categories.Morphism.Reasoning.Iso.html#4450" class="Module">_</a> <a id="4452" class="Symbol">(</a><a id="4453" href="Categories.Morphism.Reasoning.Iso.html#4453" class="Bound">i</a> <a id="4455" class="Symbol">:</a> <a id="4457" href="Categories.Morphism.Reasoning.Iso.html#618" class="Generalizable">A</a> <a id="4459" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="4461" href="Categories.Morphism.Reasoning.Iso.html#620" class="Generalizable">B</a><a id="4462" class="Symbol">)</a> <a id="4464" class="Symbol">(</a><a id="4465" href="Categories.Morphism.Reasoning.Iso.html#4465" class="Bound">j</a> <a id="4467" class="Symbol">:</a> <a id="4469" href="Categories.Morphism.Reasoning.Iso.html#622" class="Generalizable">X</a> <a id="4471" href="Categories.Morphism.html#1958" class="Record Operator">≅</a> <a id="4473" href="Categories.Morphism.Reasoning.Iso.html#624" class="Generalizable">Y</a><a id="4474" class="Symbol">)</a> <a id="4476" class="Keyword">where</a>
<a id="4484" class="Keyword">private</a>
<a id="4496" class="Keyword">module</a> <a id="4503" href="Categories.Morphism.Reasoning.Iso.html#4503" class="Module">i</a> <a id="4505" class="Symbol">=</a> <a id="4507" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="4511" href="Categories.Morphism.Reasoning.Iso.html#4453" class="Bound">i</a>
<a id="4517" class="Keyword">module</a> <a id="4524" href="Categories.Morphism.Reasoning.Iso.html#4524" class="Module">j</a> <a id="4526" class="Symbol">=</a> <a id="4528" href="Categories.Morphism.html#1958" class="Module Operator">_≅_</a> <a id="4532" href="Categories.Morphism.Reasoning.Iso.html#4465" class="Bound">j</a>
<a id="4537" href="Categories.Morphism.Reasoning.Iso.html#4537" class="Function">conjugate-from</a> <a id="4552" class="Symbol">:</a> <a id="4554" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="4556" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4558" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="4565" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4567" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="4574" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4576" href="Categories.Morphism.Reasoning.Iso.html#638" class="Generalizable">g</a> <a id="4578" class="Symbol">→</a> <a id="4580" href="Categories.Morphism.html#2023" class="Field">j.to</a> <a id="4585" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4587" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="4589" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4591" href="Categories.Morphism.Reasoning.Iso.html#638" class="Generalizable">g</a> <a id="4593" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4595" href="Categories.Morphism.html#2023" class="Function">i.to</a>
<a id="4602" href="Categories.Morphism.Reasoning.Iso.html#4537" class="Function">conjugate-from</a> <a id="4617" class="Symbol">{</a><a id="4618" class="Argument">f</a> <a id="4620" class="Symbol">=</a> <a id="4622" href="Categories.Morphism.Reasoning.Iso.html#4622" class="Bound">f</a><a id="4623" class="Symbol">}</a> <a id="4625" class="Symbol">{</a><a id="4626" class="Argument">g</a> <a id="4628" class="Symbol">=</a> <a id="4630" href="Categories.Morphism.Reasoning.Iso.html#4630" class="Bound">g</a><a id="4631" class="Symbol">}</a> <a id="4633" href="Categories.Morphism.Reasoning.Iso.html#4633" class="Bound">eq</a> <a id="4636" class="Symbol">=</a> <a id="4638" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4648" href="Categories.Morphism.html#2023" class="Field">j.to</a> <a id="4653" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4655" href="Categories.Morphism.Reasoning.Iso.html#4622" class="Bound">f</a> <a id="4675" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4678" href="Categories.Morphism.Reasoning.Core.html#2899" class="Function">introʳ</a> <a id="4685" href="Categories.Morphism.html#1612" class="Function">i.isoʳ</a> <a id="4692" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4698" class="Symbol">(</a><a id="4699" href="Categories.Morphism.html#2023" class="Field">j.to</a> <a id="4704" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4706" href="Categories.Morphism.Reasoning.Iso.html#4622" class="Bound">f</a><a id="4707" class="Symbol">)</a> <a id="4709" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4711" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="4718" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4720" href="Categories.Morphism.html#2023" class="Function">i.to</a> <a id="4725" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4728" href="Categories.Morphism.Reasoning.Core.html#7259" class="Function">center</a> <a id="4735" href="Categories.Morphism.Reasoning.Iso.html#4633" class="Bound">eq</a> <a id="4738" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4744" href="Categories.Morphism.html#2023" class="Field">j.to</a> <a id="4749" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4751" class="Symbol">(</a><a id="4752" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="4759" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4761" href="Categories.Morphism.Reasoning.Iso.html#4630" class="Bound">g</a><a id="4762" class="Symbol">)</a> <a id="4764" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4766" href="Categories.Morphism.html#2023" class="Function">i.to</a> <a id="4771" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4774" href="Categories.Morphism.Reasoning.Core.html#7391" class="Function">center⁻¹</a> <a id="4783" href="Categories.Morphism.html#1586" class="Function">j.isoˡ</a> <a id="4790" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a> <a id="4801" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4807" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="4810" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4812" href="Categories.Morphism.Reasoning.Iso.html#4630" class="Bound">g</a> <a id="4814" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4816" href="Categories.Morphism.html#2023" class="Function">i.to</a> <a id="4834" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="4837" href="Categories.Category.Core.html#1096" class="Field">identityˡ</a> <a id="4847" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="4853" href="Categories.Morphism.Reasoning.Iso.html#4630" class="Bound">g</a> <a id="4855" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4857" href="Categories.Morphism.html#2023" class="Function">i.to</a> <a id="4880" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="4885" href="Categories.Morphism.Reasoning.Iso.html#4885" class="Function">conjugate-to</a> <a id="4898" class="Symbol">:</a> <a id="4900" href="Categories.Morphism.html#2023" class="Field">j.to</a> <a id="4905" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4907" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="4909" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4911" href="Categories.Morphism.Reasoning.Iso.html#638" class="Generalizable">g</a> <a id="4913" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4915" href="Categories.Morphism.html#2023" class="Function">i.to</a> <a id="4920" class="Symbol">→</a> <a id="4922" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="4924" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4926" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="4933" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="4935" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="4942" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4944" href="Categories.Morphism.Reasoning.Iso.html#638" class="Generalizable">g</a>
<a id="4948" href="Categories.Morphism.Reasoning.Iso.html#4885" class="Function">conjugate-to</a> <a id="4961" class="Symbol">{</a><a id="4962" class="Argument">f</a> <a id="4964" class="Symbol">=</a> <a id="4966" href="Categories.Morphism.Reasoning.Iso.html#4966" class="Bound">f</a><a id="4967" class="Symbol">}</a> <a id="4969" class="Symbol">{</a><a id="4970" class="Argument">g</a> <a id="4972" class="Symbol">=</a> <a id="4974" href="Categories.Morphism.Reasoning.Iso.html#4974" class="Bound">g</a><a id="4975" class="Symbol">}</a> <a id="4977" href="Categories.Morphism.Reasoning.Iso.html#4977" class="Bound">eq</a> <a id="4980" class="Symbol">=</a> <a id="4982" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="4992" href="Categories.Morphism.Reasoning.Iso.html#4966" class="Bound">f</a> <a id="4994" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="4996" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="5021" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5024" href="Categories.Morphism.Reasoning.Core.html#3063" class="Function">introˡ</a> <a id="5031" href="Categories.Morphism.html#1612" class="Function">j.isoʳ</a> <a id="5038" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5044" class="Symbol">(</a><a id="5045" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="5052" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5054" href="Categories.Morphism.html#2023" class="Field">j.to</a><a id="5058" class="Symbol">)</a> <a id="5060" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5062" href="Categories.Morphism.Reasoning.Iso.html#4966" class="Bound">f</a> <a id="5064" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5066" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="5073" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5076" href="Categories.Morphism.Reasoning.Core.html#7259" class="Function">center</a> <a id="5083" href="Categories.Morphism.Reasoning.Iso.html#4977" class="Bound">eq</a> <a id="5086" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5092" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="5099" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5101" class="Symbol">(</a><a id="5102" href="Categories.Morphism.Reasoning.Iso.html#4974" class="Bound">g</a> <a id="5104" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5106" href="Categories.Morphism.html#2023" class="Function">i.to</a><a id="5110" class="Symbol">)</a> <a id="5112" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5114" href="Categories.Morphism.html#2006" class="Function">i.from</a> <a id="5121" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5124" href="Categories.Morphism.Reasoning.Core.html#7391" class="Function">center⁻¹</a> <a id="5133" href="Relation.Binary.Structures.html#1596" class="Function">Equiv.refl</a> <a id="5144" href="Categories.Morphism.html#1586" class="Function">i.isoˡ</a> <a id="5151" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5157" class="Symbol">(</a><a id="5158" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="5165" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5167" href="Categories.Morphism.Reasoning.Iso.html#4974" class="Bound">g</a><a id="5168" class="Symbol">)</a> <a id="5170" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5172" href="Categories.Category.Core.html#630" class="Field">id</a> <a id="5186" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="5189" href="Categories.Category.Core.html#1145" class="Field">identityʳ</a> <a id="5199" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="5205" href="Categories.Morphism.html#2006" class="Field">j.from</a> <a id="5212" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5214" href="Categories.Morphism.Reasoning.Iso.html#4974" class="Bound">g</a> <a id="5234" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="5237" class="Keyword">module</a> <a id="GroupoidR"></a><a id="5244" href="Categories.Morphism.Reasoning.Iso.html#5244" class="Module">GroupoidR</a> <a id="5254" class="Symbol">(</a><a id="5255" href="Categories.Morphism.Reasoning.Iso.html#5255" class="Bound">G</a> <a id="5257" class="Symbol">:</a> <a id="5259" href="Categories.Category.Groupoid.html#189" class="Record">IsGroupoid</a> <a id="5270" href="Categories.Morphism.Reasoning.Iso.html#274" class="Bound">C</a><a id="5271" class="Symbol">)</a> <a id="5273" class="Keyword">where</a>
<a id="5281" class="Keyword">open</a> <a id="5286" href="Categories.Category.Groupoid.html#189" class="Module">IsGroupoid</a> <a id="5297" href="Categories.Morphism.Reasoning.Iso.html#5255" class="Bound">G</a> <a id="5299" class="Keyword">using</a> <a id="5305" class="Symbol">(</a><a id="5306" href="Categories.Category.Groupoid.html#365" class="Field Operator">_⁻¹</a><a id="5309" class="Symbol">;</a> <a id="5311" href="Categories.Category.Groupoid.html#399" class="Field">iso</a><a id="5314" class="Symbol">;</a> <a id="5316" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a><a id="5325" class="Symbol">)</a>
<a id="GroupoidR.switch-fromtoˡ′"></a><a id="5330" href="Categories.Morphism.Reasoning.Iso.html#5330" class="Function">switch-fromtoˡ′</a> <a id="5346" class="Symbol">:</a> <a id="5348" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5350" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5352" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5354" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5356" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5358" class="Symbol">→</a> <a id="5360" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5362" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5364" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5366" href="Categories.Category.Groupoid.html#365" class="Field Operator">⁻¹</a> <a id="5369" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5371" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a>
<a id="5375" href="Categories.Morphism.Reasoning.Iso.html#5330" class="Function">switch-fromtoˡ′</a> <a id="5391" class="Symbol">=</a> <a id="5393" href="Categories.Morphism.Reasoning.Iso.html#720" class="Function">switch-fromtoˡ</a> <a id="5408" class="Symbol">(</a><a id="5409" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="5419" class="Symbol">_)</a>
<a id="GroupoidR.switch-tofromˡ′"></a><a id="5425" href="Categories.Morphism.Reasoning.Iso.html#5425" class="Function">switch-tofromˡ′</a> <a id="5441" class="Symbol">:</a> <a id="5443" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5445" href="Categories.Category.Groupoid.html#365" class="Field Operator">⁻¹</a> <a id="5448" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5450" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5452" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5454" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5456" class="Symbol">→</a> <a id="5458" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5460" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5462" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5464" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5466" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a>
<a id="5470" href="Categories.Morphism.Reasoning.Iso.html#5425" class="Function">switch-tofromˡ′</a> <a id="5486" class="Symbol">=</a> <a id="5488" href="Categories.Morphism.Reasoning.Iso.html#907" class="Function">switch-tofromˡ</a> <a id="5503" class="Symbol">(</a><a id="5504" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="5514" class="Symbol">_)</a>
<a id="GroupoidR.switch-fromtoʳ′"></a><a id="5520" href="Categories.Morphism.Reasoning.Iso.html#5520" class="Function">switch-fromtoʳ′</a> <a id="5536" class="Symbol">:</a> <a id="5538" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5540" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5542" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5544" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5546" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5548" class="Symbol">→</a> <a id="5550" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5552" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5554" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5556" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5558" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5560" href="Categories.Category.Groupoid.html#365" class="Field Operator">⁻¹</a>
<a id="5565" href="Categories.Morphism.Reasoning.Iso.html#5520" class="Function">switch-fromtoʳ′</a> <a id="5581" class="Symbol">=</a> <a id="5583" href="Categories.Morphism.Reasoning.Iso.html#1094" class="Function">switch-fromtoʳ</a> <a id="5598" class="Symbol">(</a><a id="5599" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="5609" class="Symbol">_)</a>
<a id="GroupoidR.switch-tofromʳ′"></a><a id="5615" href="Categories.Morphism.Reasoning.Iso.html#5615" class="Function">switch-tofromʳ′</a> <a id="5631" class="Symbol">:</a> <a id="5633" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5635" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5637" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a> <a id="5639" href="Categories.Category.Groupoid.html#365" class="Field Operator">⁻¹</a> <a id="5642" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5644" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5646" class="Symbol">→</a> <a id="5648" href="Categories.Morphism.Reasoning.Iso.html#640" class="Generalizable">h</a> <a id="5650" href="Categories.Category.Core.html#595" class="Field Operator">≈</a> <a id="5652" href="Categories.Morphism.Reasoning.Iso.html#642" class="Generalizable">k</a> <a id="5654" href="Categories.Category.Core.html#656" class="Field Operator">∘</a> <a id="5656" href="Categories.Morphism.Reasoning.Iso.html#636" class="Generalizable">f</a>
<a id="5660" href="Categories.Morphism.Reasoning.Iso.html#5615" class="Function">switch-tofromʳ′</a> <a id="5676" class="Symbol">=</a> <a id="5678" href="Categories.Morphism.Reasoning.Iso.html#1281" class="Function">switch-tofromʳ</a> <a id="5693" class="Symbol">(</a><a id="5694" href="Categories.Category.Groupoid.html#489" class="Function">equiv-obj</a> <a id="5704" class="Symbol">_)</a>
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