{"id":352,"date":"2021-05-26T06:00:22","date_gmt":"2021-05-26T04:00:22","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=352"},"modified":"2021-05-25T18:17:02","modified_gmt":"2021-05-25T16:17:02","slug":"union-con-su-interseccion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/union-con-su-interseccion\/","title":{"rendered":"Uni\u00f3n con su intersecci\u00f3n"},"content":{"rendered":"<p>Demostrar que<\/p>\n<blockquote><p>\n  s \u222a (s \u2229 t) = s\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 : set \u03b1\n\nexample : s \u222a (s \u2229 t) = s :=\nsorry\n<\/pre>\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 : set \u03b1\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nbegin\n  ext x,\n  split,\n  { intro hx,\n    cases hx with xs xst,\n    { exact xs, },\n    { exact xst.1, }},\n  { intro xs,\n    left,\n    exact xs, },\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nbegin\n  ext x,\n  exact \u27e8\u03bb hx, or.dcases_on hx id and.left,\n         \u03bb xs, or.inl xs\u27e9,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nbegin\n  ext x,\n  split,\n  { rintros (xs | \u27e8xs, xt\u27e9);\n    exact xs },\n  { intro xs,\n    left,\n    exact xs },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : s \u222a (s \u2229 t) = s :=\nsup_inf_self\n<\/pre>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Union_con_su_interseccion\nimports Main\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\nproof (rule equalityI)\n  show \"s \u222a (s \u2229 t) \u2286 s\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s \u222a (s \u2229 t)\"\n    then show \"x \u2208 s\"\n    proof\n      assume \"x \u2208 s\"\n      then show \"x \u2208 s\"\n        by this\n    next\n      assume \"x \u2208 s \u2229 t\"\n      then show \"x \u2208 s\"\n        by (simp only: IntD1)\n    qed\n  qed\nnext\n  show \"s \u2286 s \u222a (s \u2229 t)\"\n  proof (rule subsetI)\n    fix x\n    assume \"x \u2208 s\"\n    then show \"x \u2208 s \u222a (s \u2229 t)\"\n      by (simp only: UnI1)\n  qed\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\nproof\n  show \"s \u222a s \u2229 t \u2286 s\"\n  proof\n    fix x\n    assume \"x \u2208 s \u222a (s \u2229 t)\"\n    then show \"x \u2208 s\"\n    proof\n      assume \"x \u2208 s\"\n      then show \"x \u2208 s\"\n        by this\n    next\n      assume \"x \u2208 s \u2229 t\"\n      then show \"x \u2208 s\"\n        by simp\n    qed\n  qed\nnext\n  show \"s \u2286 s \u222a (s \u2229 t)\"\n  proof\n    fix x\n    assume \"x \u2208 s\"\n    then show \"x \u2208 s \u222a (s \u2229 t)\"\n      by simp\n  qed\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"s \u222a (s \u2229 t) = s\"\n  by auto\n\nend\n<\/pre>\n<p><strong>Nuevas soluciones<\/strong><\/p>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que s \u222a (s \u2229 t) = s Para ello, completar la siguiente teor\u00eda de Lean: import data.set.basic open set variable {\u03b1 : Type} variables s t : set \u03b1 example : s \u222a (s \u2229 t) = s := sorry<\/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\/352"}],"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=352"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/352\/revisions"}],"predecessor-version":[{"id":353,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/352\/revisions\/353"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}