{"id":613,"date":"2021-07-31T06:00:12","date_gmt":"2021-07-31T04:00:12","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=613"},"modified":"2021-07-29T16:53:58","modified_gmt":"2021-07-29T14:53:58","slug":"las-sucesiones-acotadas-por-cero-son-nulas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/las-sucesiones-acotadas-por-cero-son-nulas\/","title":{"rendered":"Las sucesiones acotadas por cero son nulas"},"content":{"rendered":"<p>Demostrar que las sucesiones acotadas por cero son nulas.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nimport tactic\n\nvariable (u : \u2115 \u2192 \u211d)\n\nnotation `|`x`|` := abs x\n\nexample\n  (h : \u2200 n, |u n| \u2264 0)\n  : \u2200 n, u n = 0 :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport data.real.basic\r\nimport tactic\r\n\r\nvariable (u : \u2115 \u2192 \u211d)\r\n\r\nnotation `|`x`|` := abs x\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\nbegin\r\n  intro n,\r\n  rw \u2190 abs_eq_zero,\r\n  specialize h n,\r\n  apply le_antisymm,\r\n  { exact h, },\r\n  { exact abs_nonneg (u n), },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\nbegin\r\n  intro n,\r\n  rw \u2190 abs_eq_zero,\r\n  specialize h n,\r\n  exact le_antisymm h (abs_nonneg (u n)),\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\nbegin\r\n  intro n,\r\n  rw \u2190 abs_eq_zero,\r\n  exact le_antisymm (h n) (abs_nonneg (u n)),\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\nbegin\r\n  intro n,\r\n  exact abs_eq_zero.mp (le_antisymm (h n) (abs_nonneg (u n))),\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\n\u03bb n, abs_eq_zero.mp (le_antisymm (h n) (abs_nonneg (u n)))\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\nexample\r\n  (h : \u2200 n, |u n| \u2264 0)\r\n  : \u2200 n, u n = 0 :=\r\nby finish\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Las_sucesiones_acotadas_por_cero_son_nulas.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory Las_sucesiones_acotadas_por_cero_son_nulas\r\nimports Main HOL.Real\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  fixes a :: \"nat \u21d2 real\"\r\n  assumes \"\u2200n. \u00a6a n\u00a6 \u2264 0\"\r\n  shows   \"\u2200n. a n = 0\"\r\nproof (rule allI)\r\n  fix n\r\n  have \"\u00a6a n\u00a6 = 0\"\r\n  proof (rule antisym)\r\n    show \"\u00a6a n\u00a6 \u2264 0\"\r\n      using assms by (rule allE)\r\n  next\r\n    show \" 0 \u2264 \u00a6a n\u00a6\"\r\n      by (rule abs_ge_zero)\r\n  qed\r\n  then show \"a n = 0\"\r\n    by (simp only: abs_eq_0_iff)\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  fixes a :: \"nat \u21d2 real\"\r\n  assumes \"\u2200n. \u00a6a n\u00a6 \u2264 0\"\r\n  shows   \"\u2200n. a n = 0\"\r\nproof (rule allI)\r\n  fix n\r\n  have \"\u00a6a n\u00a6 = 0\"\r\n  proof (rule antisym)\r\n    show \"\u00a6a n\u00a6 \u2264 0\" try\r\n      using assms by (rule allE)\r\n  next\r\n    show \" 0 \u2264 \u00a6a n\u00a6\"\r\n      by simp\r\n  qed\r\n  then show \"a n = 0\"\r\n    by simp\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\nlemma\r\n  fixes a :: \"nat \u21d2 real\"\r\n  assumes \"\u2200n. \u00a6a n\u00a6 \u2264 0\"\r\n  shows   \"\u2200n. a n = 0\"\r\nproof (rule allI)\r\n  fix n\r\n  have \"\u00a6a n\u00a6 = 0\"\r\n    using assms by auto\r\n  then show \"a n = 0\"\r\n    by simp\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\nlemma\r\n  fixes a :: \"nat \u21d2 real\"\r\n  assumes \"\u2200n. \u00a6a n\u00a6 \u2264 0\"\r\n  shows   \"\u2200n. a n = 0\"\r\nusing assms by auto\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que las sucesiones acotadas por cero son nulas. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic import tactic variable (u : \u2115 \u2192 \u211d) notation `|`x`|` := abs x example (h : \u2200 n, |u n| \u2264 0) : \u2200 n, u n = 0 := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import data.real.basic import tactic variable (u : \u2115 \u2192 \u211d) notation `|`x`|` := abs x &#8212; 1\u00aa demostraci\u00f3n example (h : \u2200 n, |u n| \u2264 0) : \u2200 n, u n = 0 := begin intro n, rw \u2190 abs_eq_zero, specialize h n, apply le_antisymm, { exact h, }, { exact abs_nonneg (u n), }, end &#8212; 2\u00aa demostraci\u00f3n example (h : \u2200 n, |u n| \u2264 0) : \u2200&#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":[13],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/613"}],"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=613"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/613\/revisions"}],"predecessor-version":[{"id":614,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/613\/revisions\/614"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=613"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=613"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=613"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}