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 18:49:44 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rsshttps://jaalonso.github.io/calculemus/es/posts/2020/02/24-praeclarum_theoremaes/José A. Alonso<p>Demostrar con Isabelle/HOL el <a href="https://tinyurl.com/25yt3ef9">Praeclarum theorema</a> de Leibniz: \[ (p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s)) \]</p> <p>Para ello, completar la siguiente teoría</p> <div class="code"><pre class="code literal-block"><span class="k">theory</span><span class="w"> </span><span class="n">Praeclarum_theorema</span> <span class="kp">imports</span><span class="w"> </span><span class="n">Main</span> <span class="k">begin</span> <span class="kn">lemma</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s))"</span> <span class="w"> </span><span class="gr">oops</span> <span class="k">end</span> </pre></div> <h3>Soluciones con Isabelle/HOL</h3> <div class="code"><pre class="code literal-block"><span class="k">theory</span><span class="w"> </span><span class="n">Praeclarum_theorema</span> <span class="kp">imports</span><span class="w"> </span><span class="n">Main</span> <span class="k">begin</span> <span class="c">(* 1ª demostración: detallada *)</span> <span class="kn">lemma</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s))"</span> <span class="k">proof</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">impI</span><span class="o">)</span> <span class="w"> </span><span class="k">assume</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s)"</span> <span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"(p ∧ r) ⟶ (q ∧ s)"</span> <span class="w"> </span><span class="k">proof</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">impI</span><span class="o">)</span> <span class="w"> </span><span class="k">assume</span><span class="w"> </span><span class="s">"p ∧ r"</span> <span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"q ∧ s"</span> <span class="w"> </span><span class="k">proof</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjI</span><span class="o">)</span> <span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"p ⟶ q"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹(p ⟶ q) ∧ (r ⟶ s)›</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjunct1</span><span class="o">)</span> <span class="w"> </span><span class="k">moreover</span><span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"p"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹p ∧ r›</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjunct1</span><span class="o">)</span> <span class="w"> </span><span class="k">ultimately</span><span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"q"</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">mp</span><span class="o">)</span> <span class="w"> </span><span class="k">next</span> <span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"r ⟶ s"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹(p ⟶ q) ∧ (r ⟶ s)›</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjunct2</span><span class="o">)</span> <span class="w"> </span><span class="k">moreover</span><span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"r"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹p ∧ r›</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjunct2</span><span class="o">)</span> <span class="w"> </span><span class="k">ultimately</span><span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"s"</span><span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">mp</span><span class="o">)</span> <span class="w"> </span><span class="k">qed</span> <span class="w"> </span><span class="k">qed</span> <span class="k">qed</span> <span class="c">(* 2ª demostración: estructurada *)</span> <span class="kn">lemma</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s))"</span> <span class="k">proof</span> <span class="w"> </span><span class="k">assume</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s)"</span> <span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"(p ∧ r) ⟶ (q ∧ s)"</span> <span class="w"> </span><span class="k">proof</span> <span class="w"> </span><span class="k">assume</span><span class="w"> </span><span class="s">"p ∧ r"</span> <span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"q ∧ s"</span> <span class="w"> </span><span class="k">proof</span> <span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"p ⟶ q"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹(p ⟶ q) ∧ (r ⟶ s)›</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">moreover</span><span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"p"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹p ∧ r›</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">ultimately</span><span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"q"</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">next</span> <span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"r ⟶ s"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹(p ⟶ q) ∧ (r ⟶ s)›</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">moreover</span><span class="w"> </span><span class="k">have</span><span class="w"> </span><span class="s">"r"</span><span class="w"> </span><span class="k">using</span><span class="w"> </span><span class="s">‹p ∧ r›</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">ultimately</span><span class="w"> </span><span class="k">show</span><span class="w"> </span><span class="s">"s"</span><span class="w"> </span><span class="ow">..</span> <span class="w"> </span><span class="k">qed</span> <span class="w"> </span><span class="k">qed</span> <span class="k">qed</span> <span class="c">(* 3ª demostración: aplicativa *)</span> <span class="kn">lemma</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s))"</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">impI</span><span class="o">)</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">impI</span><span class="o">)</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">erule</span><span class="w"> </span><span class="n">conjE</span><span class="o">)+</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">rule</span><span class="w"> </span><span class="n">conjI</span><span class="o">)</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">erule</span><span class="w"> </span><span class="n">mp</span><span class="o">)</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="n">assumption</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="o">(</span><span class="n">erule</span><span class="w"> </span><span class="n">mp</span><span class="o">)</span> <span class="w"> </span><span class="kp">apply</span><span class="w"> </span><span class="n">assumption</span> <span class="w"> </span><span class="k">done</span> <span class="c">(* 4ª demostración: automática *)</span> <span class="kn">lemma</span><span class="w"> </span><span class="s">"(p ⟶ q) ∧ (r ⟶ s) ⟶ ((p ∧ r) ⟶ (q ∧ s))"</span> <span class="w"> </span><span class="k">by</span><span class="w"> </span><span class="n">simp</span> <span class="k">end</span> </pre></div>https://jaalonso.github.io/calculemus/es/posts/2020/02/24-praeclarum_theoremaes/Mon, 24 Feb 2020 04:00:00 GMT