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bsc-leon-vatthauer/agda/bsc-thesis/Algebra.Properties.Group.html
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<html><head><meta charset="utf-8"><title>Algebra.Properties.Group</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">-- Some derivable properties</a>
<a id="135" class="Comment">------------------------------------------------------------------------</a>
<a id="209" class="Symbol">{-#</a> <a id="213" class="Keyword">OPTIONS</a> <a id="221" class="Pragma">--cubical-compatible</a> <a id="242" class="Pragma">--safe</a> <a id="249" class="Symbol">#-}</a>
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<a id="1526" href="Algebra.Properties.Group.html#1498" class="Function">∙-cancel</a> <a id="1535" class="Symbol">=</a> <a id="1537" href="Algebra.Properties.Group.html#1125" class="Function">∙-cancelˡ</a> <a id="1547" href="Agda.Builtin.Sigma.html#235" class="InductiveConstructor Operator">,</a> <a id="1549" href="Algebra.Properties.Group.html#1316" class="Function">∙-cancelʳ</a>
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<a id="1759" href="Algebra.Properties.Group.html#1608" class="Bound">x</a> <a id="1780" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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<a id="1907" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="1916" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="1919" href="Algebra.Structures.html#6506" class="Function">inverseʳ</a> <a id="1928" href="Algebra.Properties.Group.html#1846" class="Bound">y</a> <a id="1930" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
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<a id="1961" href="Algebra.Properties.Group.html#1846" class="Bound">y</a> <a id="1963" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="1965" href="Algebra.Properties.Group.html#1842" class="Bound">x</a> <a id="1967" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="1970" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a> <a id="1972" class="Symbol">)</a>
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<a id="2112" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="2134" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">≈⟨</a> <a id="2137" href="Algebra.Structures.html#6506" class="Function">inverseʳ</a> <a id="2146" class="Symbol">_</a> <a id="2148" href="Relation.Binary.Reasoning.Syntax.html#7074" class="Function">⟨</a>
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<a id="2370" href="Algebra.Properties.Group.html#2353" class="Bound">x</a> <a id="2385" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2388" href="Algebra.Properties.Group.html#669" class="Function">left-helper</a> <a id="2400" href="Algebra.Properties.Group.html#2353" class="Bound">x</a> <a id="2402" href="Algebra.Properties.Group.html#2355" class="Bound">y</a> <a id="2404" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2408" class="Symbol">(</a><a id="2409" href="Algebra.Properties.Group.html#2353" class="Bound">x</a> <a id="2411" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2413" href="Algebra.Properties.Group.html#2355" class="Bound">y</a><a id="2414" class="Symbol">)</a> <a id="2416" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2418" href="Algebra.Properties.Group.html#2355" class="Bound">y</a> <a id="2420" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2423" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2426" href="Algebra.Structures.html#1526" class="Function">∙-congʳ</a> <a id="2434" href="Algebra.Properties.Group.html#2357" class="Bound">eq</a> <a id="2437" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2446" href="Algebra.Properties.Group.html#2355" class="Bound">y</a> <a id="2449" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2451" href="Algebra.Properties.Group.html#2355" class="Bound">y</a> <a id="2453" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2456" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2459" href="Algebra.Structures.html#6506" class="Function">inverseʳ</a> <a id="2468" href="Algebra.Properties.Group.html#2355" class="Bound">y</a> <a id="2470" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2474" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="2489" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="identityʳ-unique"></a><a id="2492" href="Algebra.Properties.Group.html#2492" class="Function">identityʳ-unique</a> <a id="2509" class="Symbol">:</a> <a id="2511" class="Symbol">∀</a> <a id="2513" href="Algebra.Properties.Group.html#2513" class="Bound">x</a> <a id="2515" href="Algebra.Properties.Group.html#2515" class="Bound">y</a> <a id="2517" class="Symbol">→</a> <a id="2519" href="Algebra.Properties.Group.html#2513" class="Bound">x</a> <a id="2521" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2523" href="Algebra.Properties.Group.html#2515" class="Bound">y</a> <a id="2525" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="2527" href="Algebra.Properties.Group.html#2513" class="Bound">x</a> <a id="2529" class="Symbol">→</a> <a id="2531" href="Algebra.Properties.Group.html#2515" class="Bound">y</a> <a id="2533" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="2535" href="Algebra.Bundles.html#9934" class="Field">ε</a>
<a id="2537" href="Algebra.Properties.Group.html#2492" class="Function">identityʳ-unique</a> <a id="2554" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2556" href="Algebra.Properties.Group.html#2556" class="Bound">y</a> <a id="2558" href="Algebra.Properties.Group.html#2558" class="Bound">eq</a> <a id="2561" class="Symbol">=</a> <a id="2563" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2571" href="Algebra.Properties.Group.html#2556" class="Bound">y</a> <a id="2586" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2589" href="Algebra.Properties.Group.html#900" class="Function">right-helper</a> <a id="2602" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2604" href="Algebra.Properties.Group.html#2556" class="Bound">y</a> <a id="2606" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2610" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2612" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2615" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2617" class="Symbol">(</a><a id="2618" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2620" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2622" href="Algebra.Properties.Group.html#2556" class="Bound">y</a><a id="2623" class="Symbol">)</a> <a id="2625" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2628" href="Relation.Binary.Structures.html#1596" class="Function">refl</a> <a id="2633" href="Function.Base.html#4322" class="Function Operator">⟨</a> <a id="2635" href="Algebra.Structures.html#1315" class="Function">∙-cong</a> <a id="2642" href="Function.Base.html#4322" class="Function Operator">⟩</a> <a id="2644" href="Algebra.Properties.Group.html#2558" class="Bound">eq</a> <a id="2647" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2651" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2653" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2656" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2659" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2666" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2669" href="Algebra.Structures.html#6443" class="Function">inverseˡ</a> <a id="2678" href="Algebra.Properties.Group.html#2554" class="Bound">x</a> <a id="2680" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2684" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="2699" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="identity-unique"></a><a id="2702" href="Algebra.Properties.Group.html#2702" class="Function">identity-unique</a> <a id="2718" class="Symbol">:</a> <a id="2720" class="Symbol">∀</a> <a id="2722" class="Symbol">{</a><a id="2723" href="Algebra.Properties.Group.html#2723" class="Bound">x</a><a id="2724" class="Symbol">}</a> <a id="2726" class="Symbol">→</a> <a id="2728" href="Algebra.Definitions.html#1856" class="Function">Identity</a> <a id="2737" href="Algebra.Properties.Group.html#2723" class="Bound">x</a> <a id="2739" href="Algebra.Bundles.html#9908" class="Field Operator">_∙_</a> <a id="2743" class="Symbol">→</a> <a id="2745" href="Algebra.Properties.Group.html#2723" class="Bound">x</a> <a id="2747" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="2749" href="Algebra.Bundles.html#9934" class="Field">ε</a>
<a id="2751" href="Algebra.Properties.Group.html#2702" class="Function">identity-unique</a> <a id="2767" class="Symbol">{</a><a id="2768" href="Algebra.Properties.Group.html#2768" class="Bound">x</a><a id="2769" class="Symbol">}</a> <a id="2771" href="Algebra.Properties.Group.html#2771" class="Bound">id</a> <a id="2774" class="Symbol">=</a> <a id="2776" href="Algebra.Properties.Group.html#2291" class="Function">identityˡ-unique</a> <a id="2793" href="Algebra.Properties.Group.html#2768" class="Bound">x</a> <a id="2795" href="Algebra.Properties.Group.html#2768" class="Bound">x</a> <a id="2797" class="Symbol">(</a><a id="2798" href="Data.Product.Base.html#650" class="Field">proj₂</a> <a id="2804" href="Algebra.Properties.Group.html#2771" class="Bound">id</a> <a id="2807" href="Algebra.Properties.Group.html#2768" class="Bound">x</a><a id="2808" class="Symbol">)</a>
<a id="inverseˡ-unique"></a><a id="2811" href="Algebra.Properties.Group.html#2811" class="Function">inverseˡ-unique</a> <a id="2827" class="Symbol">:</a> <a id="2829" class="Symbol">∀</a> <a id="2831" href="Algebra.Properties.Group.html#2831" class="Bound">x</a> <a id="2833" href="Algebra.Properties.Group.html#2833" class="Bound">y</a> <a id="2835" class="Symbol">→</a> <a id="2837" href="Algebra.Properties.Group.html#2831" class="Bound">x</a> <a id="2839" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2841" href="Algebra.Properties.Group.html#2833" class="Bound">y</a> <a id="2843" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="2845" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="2847" class="Symbol">→</a> <a id="2849" href="Algebra.Properties.Group.html#2831" class="Bound">x</a> <a id="2851" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="2853" href="Algebra.Properties.Group.html#2833" class="Bound">y</a> <a id="2855" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a>
<a id="2858" href="Algebra.Properties.Group.html#2811" class="Function">inverseˡ-unique</a> <a id="2874" href="Algebra.Properties.Group.html#2874" class="Bound">x</a> <a id="2876" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="2878" href="Algebra.Properties.Group.html#2878" class="Bound">eq</a> <a id="2881" class="Symbol">=</a> <a id="2883" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="2891" href="Algebra.Properties.Group.html#2874" class="Bound">x</a> <a id="2906" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2909" href="Algebra.Properties.Group.html#669" class="Function">left-helper</a> <a id="2921" href="Algebra.Properties.Group.html#2874" class="Bound">x</a> <a id="2923" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="2925" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2929" class="Symbol">(</a><a id="2930" href="Algebra.Properties.Group.html#2874" class="Bound">x</a> <a id="2932" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2934" href="Algebra.Properties.Group.html#2876" class="Bound">y</a><a id="2935" class="Symbol">)</a> <a id="2937" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2939" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="2941" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2944" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2947" href="Algebra.Structures.html#1526" class="Function">∙-congʳ</a> <a id="2955" href="Algebra.Properties.Group.html#2878" class="Bound">eq</a> <a id="2958" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="2967" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="2970" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="2972" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="2974" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="2977" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="2980" href="Algebra.Structures.html#4136" class="Function">identityˡ</a> <a id="2990" class="Symbol">(</a><a id="2991" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="2993" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a><a id="2995" class="Symbol">)</a> <a id="2997" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3011" href="Algebra.Properties.Group.html#2876" class="Bound">y</a> <a id="3013" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="3016" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
<a id="inverseʳ-unique"></a><a id="3019" href="Algebra.Properties.Group.html#3019" class="Function">inverseʳ-unique</a> <a id="3035" class="Symbol">:</a> <a id="3037" class="Symbol">∀</a> <a id="3039" href="Algebra.Properties.Group.html#3039" class="Bound">x</a> <a id="3041" href="Algebra.Properties.Group.html#3041" class="Bound">y</a> <a id="3043" class="Symbol">→</a> <a id="3045" href="Algebra.Properties.Group.html#3039" class="Bound">x</a> <a id="3047" href="Algebra.Bundles.html#9908" class="Field Operator">∙</a> <a id="3049" href="Algebra.Properties.Group.html#3041" class="Bound">y</a> <a id="3051" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="3053" href="Algebra.Bundles.html#9934" class="Field">ε</a> <a id="3055" class="Symbol">→</a> <a id="3057" href="Algebra.Properties.Group.html#3041" class="Bound">y</a> <a id="3059" href="Algebra.Bundles.html#9880" class="Field Operator">≈</a> <a id="3061" href="Algebra.Properties.Group.html#3039" class="Bound">x</a> <a id="3063" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a>
<a id="3066" href="Algebra.Properties.Group.html#3019" class="Function">inverseʳ-unique</a> <a id="3082" href="Algebra.Properties.Group.html#3082" class="Bound">x</a> <a id="3084" href="Algebra.Properties.Group.html#3084" class="Bound">y</a> <a id="3086" href="Algebra.Properties.Group.html#3086" class="Bound">eq</a> <a id="3089" class="Symbol">=</a> <a id="3091" href="Relation.Binary.Reasoning.Syntax.html#1510" class="Function Operator">begin</a>
<a id="3099" href="Algebra.Properties.Group.html#3084" class="Bound">y</a> <a id="3107" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3110" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="3114" class="Symbol">(</a><a id="3115" href="Algebra.Properties.Group.html#1560" class="Function">⁻¹-involutive</a> <a id="3129" href="Algebra.Properties.Group.html#3084" class="Bound">y</a><a id="3130" class="Symbol">)</a> <a id="3132" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3136" href="Algebra.Properties.Group.html#3084" class="Bound">y</a> <a id="3138" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="3141" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="3144" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">≈⟨</a> <a id="3147" href="Algebra.Structures.html#6329" class="Function">⁻¹-cong</a> <a id="3155" class="Symbol">(</a><a id="3156" href="Relation.Binary.Structures.html#1622" class="Function">sym</a> <a id="3160" class="Symbol">(</a><a id="3161" href="Algebra.Properties.Group.html#2811" class="Function">inverseˡ-unique</a> <a id="3177" href="Algebra.Properties.Group.html#3082" class="Bound">x</a> <a id="3179" href="Algebra.Properties.Group.html#3084" class="Bound">y</a> <a id="3181" href="Algebra.Properties.Group.html#3086" class="Bound">eq</a><a id="3183" class="Symbol">))</a> <a id="3186" href="Relation.Binary.Reasoning.Syntax.html#7049" class="Function">⟩</a>
<a id="3190" href="Algebra.Properties.Group.html#3082" class="Bound">x</a> <a id="3192" href="Algebra.Bundles.html#9956" class="Field Operator">⁻¹</a> <a id="3198" href="Relation.Binary.Reasoning.Syntax.html#12283" class="Function Operator">∎</a>
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