{"id":291,"date":"2021-05-17T13:03:05","date_gmt":"2021-05-17T11:03:05","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=291"},"modified":"2022-04-18T11:37:04","modified_gmt":"2022-04-18T09:37:04","slug":"propiedad-de-monotonia-de-la-interseccion-2021","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/propiedad-de-monotonia-de-la-interseccion-2021\/","title":{"rendered":"Propiedad de monoton\u00eda de la intersecci\u00f3n"},"content":{"rendered":"<p>Demostrar que la intersecci\u00f3n es mon\u00f3tona por la izquierda; es decir, si<\/p>\n<blockquote><p>\n  s \u2286 t,\n<\/p><\/blockquote>\n<p>entonces<\/p>\n<blockquote><p>\n  s \u2229 u \u2286 t \u2229 u.\n<\/p><\/blockquote>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nsorry\n<\/pre>\n<p><strong>Notas<\/strong><\/p>\n<ul>\n<li>En <a href=\"https:\/\/bit.ly\/3frC0T0\">este enlace<\/a> se puede escribir las soluciones en Lean.<\/li>\n<li>A continuaci\u00f3n se muestran algunas soluciones (que se pueden probar en <a href=\"https:\/\/bit.ly\/3tZEP3i\">este enlace<\/a>).<\/li>\n<li>En los comentarios se pueden publicar otras soluciones, en Lean o en otros sistemas de razonamiento.\n<ul>\n<li>Para publicar las demostraciones en Lean se deben de escribir entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62;<\/li>\n<li>Para publicar las demostraciones en Isabelle\/HOL se deben de escribir entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.set.basic\nopen set\n\nvariable {\u03b1 : Type}\nvariables s t u : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  rw subset_def,\n  rw inter_def,\n  rw inter_def,\n  dsimp,\n  intros x h,\n  cases h with xs xu,\n  split,\n  { rw subset_def at h,\n    apply h,\n    assumption },\n  { assumption },\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  rw [subset_def, inter_def, inter_def],\n  dsimp,\n  rintros x \u27e8xs, xu\u27e9,\n  rw subset_def at h,\n  exact \u27e8h _ xs, xu\u27e9,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  simp only [subset_def, mem_inter_eq] at *,\n  rintros x \u27e8xs, xu\u27e9,\n  exact \u27e8h _ xs, xu\u27e9,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\nbegin\n  intros x xsu,\n  exact \u27e8h xsu.1, xsu.2\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : s \u2286 t)\n  : s \u2229 u \u2286 t \u2229 u :=\ninter_subset_inter_left u h\n<\/pre>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Propiedad_de_monotonia_de_la_interseccion\nimports Main\nbegin\n\n(* 1\u00aa soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nproof  (rule subsetI)\n  fix x\n  assume hx: \"x \u2208 s \u2229 u\"\n  have xs: \"x \u2208 s\"\n    using hx\n    by (simp only: IntD1)\n  then have xt: \"x \u2208 t\"\n    using assms\n    by (simp only: subset_eq)\n  have xu: \"x \u2208 u\"\n    using hx\n    by (simp only: IntD2)\n  show \"x \u2208 t \u2229 u\"\n    using xt xu\n    by (simp only: Int_iff)\nqed\n\n(* 2 soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nproof\n  fix x\n  assume hx: \"x \u2208 s \u2229 u\"\n  have xs: \"x \u2208 s\"\n    using hx\n    by simp\n  then have xt: \"x \u2208 t\"\n    using assms\n    by auto\n  have xu: \"x \u2208 u\"\n    using hx\n    by simp\n  show \"x \u2208 t \u2229 u\"\n    using xt xu\n    by simp\nqed\n\n(* 3\u00aa soluci\u00f3n *)\nlemma\n  assumes \"s \u2286 t\"\n  shows   \"s \u2229 u \u2286 t \u2229 u\"\nusing assms\nby auto\n\n(* 4\u00aa soluci\u00f3n *)\nlemma\n  \"s \u2286 t \u27f9 s \u2229 u \u2286 t \u2229 u\"\nby auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que la intersecci\u00f3n es mon\u00f3tona por la izquierda; es decir, si s \u2286 t, entonces s \u2229 u \u2286 t \u2229 u. Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t u : set \u03b1 example (h : s \u2286 t) : s \u2229 u \u2286 t \u2229 u := sorry Notas En este enlace se puede escribir las soluciones en Lean. A continuaci\u00f3n se muestran algunas soluciones (que se pueden probar en este enlace). En los comentarios se pueden publicar otras soluciones, en Lean o en otros sistemas de razonamiento. Para publicar las demostraciones en Lean se deben de escribir entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62; Para publicar las&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[7],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/291"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=291"}],"version-history":[{"count":14,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/291\/revisions"}],"predecessor-version":[{"id":329,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/291\/revisions\/329"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}