https://jaalonso.github.io/calculemus/esContents © 2026 <a href="mailto:">José A. Alonso</a> <a rel="license" href="https://creativecommons.org/licenses/by-nc-sa/4.0/"> <img alt="Creative Commons License BY-NC-SA" style="border-width:0; margin-bottom:12px;" src="https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png"></a>Wed, 14 Oct 2026 18:49:42 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rsshttps://jaalonso.github.io/calculemus/es/posts/2024/10/04-if_ff_is_biyective_then_f_is_biyective/José A. Alonso<div><p>Prove that if \(f ∘ f\) is bijective, then \(f\) is bijective.</p> <p>To do this, complete the following Lean4 theory:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</span> <span class="kn">open</span><span class="w"> </span><span class="n">Function</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">X</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="n">Z</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">Type</span><span class="o">}</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">f</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">X</span><span class="w"> </span><span class="bp">→</span><span class="w"> </span><span class="n">Y</span><span class="o">}</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">g</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">Y</span><span class="w"> </span><span class="bp">→</span><span class="w"> </span><span class="n">Z</span><span class="o">}</span> <span class="kd">example</span> <span class="w"> </span><span class="o">(</span><span class="n">f</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">X</span><span class="w"> </span><span class="bp">→</span><span class="w"> </span><span class="n">X</span><span class="o">)</span> <span class="w"> </span><span class="o">(</span><span class="n">h</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">Bijective</span><span class="w"> </span><span class="o">(</span><span class="n">f</span><span class="w"> </span><span class="bp">∘</span><span class="w"> </span><span class="n">f</span><span class="o">))</span> <span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">Bijective</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="o">:=</span> <span class="kd">by</span><span class="w"> </span><span class="gr">sorry</span> </pre></div> <p><a href="https://jaalonso.github.io/calculemus/es/posts/2024/10/04-if_ff_is_biyective_then_f_is_biyective/">Leer más…</a> (quedan 2 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/10/04-if_ff_is_biyective_then_f_is_biyective/Fri, 04 Oct 2024 04:00:00 GMT