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 18:49:41 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rsshttps://jaalonso.github.io/calculemus/es/posts/2026/01/10-propiedad_arquimedianaes/José A. Alonso<div><p>Demostrar la propiedad arquimediana de los números reales; es decir, que para cualquier \(ε ∈ ℝ\) con \(0 &lt; ε\), existe \(N ∈ ℕ\) tal que \(1/ε &lt; N\).</p> <p>Para ello, completar la siguiente teoría de Lean 4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">ε</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">}</span> <span class="kd">variable</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="mi">0</span><span class="w"> </span><span class="bp">&lt;</span><span class="w"> </span><span class="n">ε</span><span class="o">)</span> <span class="kd">example</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="bp">∃</span><span class="w"> </span><span class="o">(</span><span class="n">N</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℕ</span><span class="o">),</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="bp">/</span><span class="w"> </span><span class="n">ε</span><span class="w"> </span><span class="bp">&lt;</span><span class="w"> </span><span class="n">N</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> <hr> <p><a href="https://jaalonso.github.io/calculemus/es/posts/2026/01/10-propiedad_arquimedianaes/">Leer más…</a> (quedan 2 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2026/01/10-propiedad_arquimedianaes/Sat, 10 Jan 2026 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/05/30-propiedad_de_la_densidad_de_los_realeses/José A. Alonso<div><p>Demostrar con Lean4 que si \(x, y ∈ ℝ\) tales que \((∀ z)[y &lt; z → x ≤ z]\), entonces \(x ≤ y\).</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</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="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</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="bp">∀</span><span class="w"> </span><span class="n">z</span><span class="o">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="bp">&lt;</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="bp">→</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">z</span><span class="o">)</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="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/2024/05/30-propiedad_de_la_densidad_de_los_realeses/">Leer más…</a> (quedan 4 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/05/30-propiedad_de_la_densidad_de_los_realeses/Thu, 30 May 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/31-demostracion_por_congruenciaes/José A. Alonso<div><p>Demostrar con Lean4 que en \(ℝ\), \(|a| = |a - b + b|\)</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</span> <span class="kd">variable</span><span class="w"> </span><span class="o">(</span><span class="n">a</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</span> <span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="bp">|</span><span class="n">a</span><span class="bp">|</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="bp">|</span><span class="n">a</span><span class="w"> </span><span class="bp">-</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="n">b</span><span class="bp">|</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/2024/01/31-demostracion_por_congruenciaes/">Leer más…</a> (quedan 1 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/31-demostracion_por_congruenciaes/Wed, 31 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/30-demostracion_por_extensionalidades/José A. Alonso<div><p>Demostrar con Lean4 que las funciones \(f(x,y) = (x + y)²\) y \(g(x,y) = x² + 2xy + y\)² son iguales.</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</span> <span class="kd">example</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">(</span><span class="k">fun</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="w"> </span><span class="bp">↦</span><span class="w"> </span><span class="o">(</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="bp">^</span><span class="mi">2</span><span class="o">)</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="o">(</span><span class="k">fun</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="w"> </span><span class="bp">↦</span><span class="w"> </span><span class="n">x</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="mi">2</span><span class="bp">*</span><span class="n">x</span><span class="bp">*</span><span class="n">y</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="n">y</span><span class="bp">^</span><span class="mi">2</span><span class="o">)</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/2024/01/30-demostracion_por_extensionalidades/">Leer más…</a> (quedan 1 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/30-demostracion_por_extensionalidades/Tue, 30 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/29-irracionalidad_de_la_raiz_cuadrada_de_2es/José A. Alonso<div><p>Demostrar con Lean4 que la raíz cuadrada de 2 es irracional; es decir, que no existen \(m, n ∈ ℕ\) tales que \(m\) y \(n\) son coprimos (es decir, que no tienen factores comunes distintos de uno) y \(m² = 2n²\).</p> <p>Para ello, completar la siguiente teoría de Lean4:</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">import</span><span class="w"> </span><span class="n">Mathlib.Data.Nat.Prime.Defs</span> <span class="kn">open</span><span class="w"> </span><span class="n">Nat</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">m</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℕ</span><span class="o">}</span> <span class="kd">example</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="bp">¬∃</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">n</span><span class="o">,</span><span class="w"> </span><span class="n">coprime</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="bp">∧</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="bp">^</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="bp">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="bp">^</span><span class="w"> </span><span class="mi">2</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/2024/01/29-irracionalidad_de_la_raiz_cuadrada_de_2es/">Leer más…</a> (quedan 3 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/29-irracionalidad_de_la_raiz_cuadrada_de_2es/Mon, 29 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/22-cuadrado_igual_a_cuadradoes/José A. Alonso<div><p>Demostrar con Lean4 que en \(ℝ\), \[x² = y² → x = y ∨ x = -y\]</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</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="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</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">x</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="n">y</span><span class="bp">^</span><span class="mi">2</span><span class="o">)</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="w"> </span><span class="bp">∨</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="bp">-</span><span class="n">y</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/2024/01/22-cuadrado_igual_a_cuadradoes/">Leer más…</a> (quedan 3 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/22-cuadrado_igual_a_cuadradoes/Mon, 22 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/19-cuadrado_igual_a_unoes/José A. Alonso<div><p>Demostrar con Lean4 que en \(ℝ\), si \(x² = 1\) entonces \(x = 1\) ó \(x = -1\).</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</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="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</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">x</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="mi">1</span><span class="o">)</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="mi">1</span><span class="w"> </span><span class="bp">∨</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="bp">-</span><span class="mi">1</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/2024/01/19-cuadrado_igual_a_unoes/">Leer más…</a> (quedan 3 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/19-cuadrado_igual_a_unoes/Fri, 19 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/18-desigualdad_con_rcaseses/José A. Alonso<div><p>Demostrar con Lean4 que si \((∃ x, y ∈ ℝ)[z = x² + y² ∨ z = x² + y² + 1]\), entonces \(z ≥ 0\).</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</span> <span class="kd">variable</span><span class="w"> </span><span class="o">{</span><span class="n">z</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</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="bp">∃</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">y</span><span class="o">,</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="n">x</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="n">y</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">∨</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="bp">=</span><span class="w"> </span><span class="n">x</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="n">y</span><span class="bp">^</span><span class="mi">2</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="mi">1</span><span class="o">)</span> <span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="bp">≥</span><span class="w"> </span><span class="mi">0</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/2024/01/18-desigualdad_con_rcaseses/">Leer más…</a> (quedan 5 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/18-desigualdad_con_rcaseses/Thu, 18 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/16-eliminacion_de_la_disyuncion_con_rcaseses/José A. Alonso<div><p>Demostrar con Lean4 que en ℝ, si \(x ≠ 0\) entonces \(x &lt; 0\) ó \(x &gt; 0\).</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</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="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</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">x</span><span class="w"> </span><span class="bp">≠</span><span class="w"> </span><span class="mi">0</span><span class="o">)</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">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="bp">∨</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="bp">&gt;</span><span class="w"> </span><span class="mi">0</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/2024/01/16-eliminacion_de_la_disyuncion_con_rcaseses/">Leer más…</a> (quedan 1 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/16-eliminacion_de_la_disyuncion_con_rcaseses/Tue, 16 Jan 2024 04:00:00 GMThttps://jaalonso.github.io/calculemus/es/posts/2024/01/15-desigualdad_triangular_para_valor_absolutoes/José A. Alonso<div><p>Demostrar con Lean4 que en \(ℝ\), \[ |x + y| ≤ |x| + |y| \]</p> <p>Para ello, completar la siguiente teoría de Lean4:</p> <div class="code"><pre class="code literal-block"><span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Data.Real.Basic</span> <span class="kn">import</span><span class="w"> </span><span class="n">Mathlib.Tactic</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="o">:</span><span class="w"> </span><span class="n">ℝ</span><span class="o">}</span> <span class="kd">example</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="bp">|</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="bp">|</span><span class="w"> </span><span class="bp">≤</span><span class="w"> </span><span class="bp">|</span><span class="n">x</span><span class="bp">|</span><span class="w"> </span><span class="bp">+</span><span class="w"> </span><span class="bp">|</span><span class="n">y</span><span class="bp">|</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/2024/01/15-desigualdad_triangular_para_valor_absolutoes/">Leer más…</a> (quedan 3 minutos de lectura)</p></div>https://jaalonso.github.io/calculemus/es/posts/2024/01/15-desigualdad_triangular_para_valor_absolutoes/Mon, 15 Jan 2024 04:00:00 GMT