{"id":549,"date":"2021-07-11T06:00:19","date_gmt":"2021-07-11T04:00:19","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=549"},"modified":"2021-07-04T17:07:44","modified_gmt":"2021-07-04T15:07:44","slug":"limite-de-sucesiones-constantes","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/limite-de-sucesiones-constantes\/","title":{"rendered":"L\u00edmite de sucesiones constantes"},"content":{"rendered":"<p>En Lean, una sucesi\u00f3n u\u2080, u\u2081, u\u2082, &#8230; se puede representar mediante una funci\u00f3n (u : \u2115 \u2192 \u211d) de forma que u(n) es u\u2099.<\/p>\n<p>Se define que a es el l\u00edmite de la sucesi\u00f3n u, por<\/p>\n<pre lang=\"text\">\n   def limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=\n   \u03bb u a, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - a| < \u03b5\n<\/pre>\n<p>donde se usa la notaci\u00f3n |x| para el valor absoluto de x<\/p>\n<pre lang=\"text\">\n   notation `|`x`|` := abs x\n<\/pre>\n<p>Demostrar que el l\u00edmite de la sucesi\u00f3n constante c es c.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariable (u : \u2115 \u2192 \u211d)\nvariable (c : \u211d)\n\nnotation `|`x`|` := abs x\n\ndef limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=\n\u03bb u a, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - a| < \u03b5\n\nexample :\n  limite (\u03bb n, c) c :=\nsorry\n<\/pre>\n<p>[expand title=\"Soluciones con Lean\"]<\/p>\n<pre lang=\"lean\">\r\nimport data.real.basic\r\n\r\nvariable (u : \u2115 \u2192 \u211d)\r\nvariable (c : \u211d)\r\n\r\nnotation `|`x`|` := abs x\r\n\r\ndef limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=\r\n\u03bb u a, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - a| < \u03b5\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\nbegin\r\n  unfold limite,\r\n  intros \u03b5 h\u03b5,\r\n  use 0,\r\n  intros n hn,\r\n  dsimp,\r\n  simp,\r\n  exact h\u03b5,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  use 0,\r\n  rintro n -,\r\n  norm_num,\r\n  assumption,\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  use 0,\r\n  intros n hn,\r\n  calc |(\u03bb n, c) n - c|\r\n       = |c - c|  : rfl\r\n   ... = 0        : by simp\r\n   ... < \u03b5        : h\u03b5\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  by finish,\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\n\u03bb \u03b5 h\u03b5, by finish\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample :\r\n  limite (\u03bb n, c) c :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nexists.intro 0\r\n  ( assume n,\r\n    assume hn : n \u2265 0,\r\n    show |(\u03bb n, c) n - c| < \u03b5, from\r\n      calc |(\u03bb n, c) n - c|\r\n           = |c - c|  : rfl\r\n       ... = 0        : by simp\r\n       ... < \u03b5        : h\u03b5)\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\/Limite_de_sucesiones_constantes.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=\"Soluciones con Isabelle\/HOL\"]<\/p>\n<pre lang=\"isar\">\r\n(* ---------------------------------------------------------------------\r\n-- En Isabelle\/HOL, una sucesi\u00f3n u\u2080, u\u2081, u\u2082, ... se puede representar \r\n-- mediante una funci\u00f3n (u : \u2115 \u2192 \u211d) de forma que u(n) es u\u2099.\r\n--\r\n-- Se define que a es el l\u00edmite de la sucesi\u00f3n u, por\r\n--    definition limite :: \"(nat \u21d2 real) \u21d2 real \u21d2 bool\"\r\n--      where \"limite u c \u27f7 (\u2200\u03b5>0. \u2203k::nat. \u2200n\u2265k. \u00a6u n - c\u00a6 < \u03b5)\"\r\n--\r\n-- Demostrar que el l\u00edmite de la sucesi\u00f3n constante c es c.\r\n-- ------------------------------------------------------------------ *)\r\n\r\ntheory Limite_de_sucesiones_constantes\r\nimports Main HOL.Real\r\nbegin\r\n\r\ndefinition limite :: \"(nat \u21d2 real) \u21d2 real \u21d2 bool\"\r\n  where \"limite u c \u27f7 (\u2200\u03b5>0. \u2203k::nat. \u2200n\u2265k. \u00a6u n - c\u00a6 < \u03b5)\"\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"limite (\u03bb n. c) c\"\r\nproof (unfold limite_def)\r\n  show \"\u2200\u03b5>0. \u2203k::nat. \u2200n\u2265k. \u00a6c - c\u00a6 < \u03b5\" \r\n  proof (intro allI impI)\r\n    fix \u03b5 :: real\r\n    assume \"0 < \u03b5\"\r\n    have \"\u2200n\u22650::nat. \u00a6c - c\u00a6 < \u03b5\" \r\n    proof (intro allI impI)\r\n      fix n :: nat\r\n      assume \"0 \u2264 n\"\r\n      have \"c - c = 0\" \r\n        by (simp only: diff_self)\r\n      then have \"\u00a6c - c\u00a6 = 0\" \r\n        by (simp only: abs_eq_0_iff)\r\n      also have \"\u2026 < \u03b5\"\r\n        by (simp only: \u20390 < \u03b5\u203a) \r\n      finally show \"\u00a6c - c\u00a6 < \u03b5\" \r\n        by this\r\n    qed\r\n    then show \"\u2203k::nat. \u2200n\u2265k. \u00a6c - c\u00a6 < \u03b5\" \r\n      by (rule exI)\r\n  qed\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"limite (\u03bb n. c) c\"\r\nproof (unfold limite_def)\r\n  show \"\u2200\u03b5>0. \u2203k::nat. \u2200n\u2265k. \u00a6c - c\u00a6 < \u03b5\" \r\n  proof (intro allI impI)\r\n    fix \u03b5 :: real\r\n    assume \"0 < \u03b5\"\r\n    have \"\u2200n\u22650::nat. \u00a6c - c\u00a6 < \u03b5\"          by (simp add: \u20390 < \u03b5\u203a)   \r\n    then show \"\u2203k::nat. \u2200n\u2265k. \u00a6c - c\u00a6 < \u03b5\" by (rule exI)\r\n  qed\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"limite (\u03bb n. c) c\" \r\n  unfolding limite_def\r\n  by simp\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"limite (\u03bb n. c) c\" \r\n  by (simp add: limite_def)\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>En Lean, una sucesi\u00f3n u\u2080, u\u2081, u\u2082, &#8230; se puede representar mediante una funci\u00f3n (u : \u2115 \u2192 \u211d) de forma que u(n) es u\u2099. Se define que a es el l\u00edmite de la sucesi\u00f3n u, por def limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop := \u03bb u a, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n &#8211; a| < \u03b5 donde se usa la notaci\u00f3n |x| para el valor absoluto de x notation `|`x`|` := abs x Demostrar que el l\u00edmite de la sucesi\u00f3n constante c es c. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variable (u : \u2115 \u2192 \u211d) variable (c : \u211d) notation `|`x`|` := abs x def limite : (\u2115 \u2192...\n<\/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":[13,14],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/549"}],"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=549"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/549\/revisions"}],"predecessor-version":[{"id":562,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/549\/revisions\/562"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=549"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=549"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=549"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}