teorema del emparedado<\/a>; es decir, que si para todo n, u(n) \u2264 v(n) \u2264 w(n) y u(n) tiene el mismo l\u00edmite que w(n), entonces v(n) tambi\u00e9n tiene dicho l\u00edmite.<\/p>\nPara ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport data.real.basic\n\nvariables (u v w : \u2115 \u2192 \u211d)\nvariable (a : \u211d)\n\nnotation `|`x`|` := abs x\n\ndef limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=\n\u03bb u c, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - c| \u2264 \u03b5\n\nexample\n (hu : limite u a)\n (hw : limite w a)\n (h : \u2200 n, u n \u2264 v n)\n (h' : \u2200 n, v n \u2264 w n) :\n limite v a :=\nsorry\n<\/pre>\n[expand title=\"Soluciones con Lean\"]<\/p>\n
\r\nimport data.real.basic\r\n\r\nvariables (u v w : \u2115 \u2192 \u211d)\r\nvariable (a : \u211d)\r\n\r\nnotation `|`x`|` := abs x\r\n\r\ndef limite : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=\r\n\u03bb u c, \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, |u n - c| \u2264 \u03b5\r\n\r\n-- Nota. En la demostraci\u00f3n se usar\u00e1 el siguiente lema:\r\nlemma max_ge_iff\r\n {p q r : \u2115}\r\n : r \u2265 max p q \u2194 r \u2265 p \u2227 r \u2265 q :=\r\nmax_le_iff\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n (hu : limite u a)\r\n (hw : limite w a)\r\n (h : \u2200 n, u n \u2264 v n)\r\n (h' : \u2200 n, v n \u2264 w n) :\r\n limite v a :=\r\nbegin\r\n intros \u03b5 h\u03b5,\r\n cases hu \u03b5 h\u03b5 with N hN, clear hu,\r\n cases hw \u03b5 h\u03b5 with N' hN', clear hw h\u03b5,\r\n use max N N',\r\n intros n hn,\r\n rw max_ge_iff at hn,\r\n specialize hN n hn.1,\r\n specialize hN' n hn.2,\r\n specialize h n,\r\n specialize h' n,\r\n clear hn,\r\n rw abs_le at *,\r\n split,\r\n { calc -\u03b5\r\n \u2264 u n - a : hN.1\r\n ... \u2264 v n - a : by linarith, },\r\n { calc v n - a\r\n \u2264 w n - a : by linarith\r\n ... \u2264 \u03b5 : hN'.2, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n (hu : limite u a)\r\n (hw : limite w a)\r\n (h : \u2200 n, u n \u2264 v n)\r\n (h' : \u2200 n, v n \u2264 w n) :\r\n limite v a :=\r\nbegin\r\n intros \u03b5 h\u03b5,\r\n cases hu \u03b5 h\u03b5 with N hN, clear hu,\r\n cases hw \u03b5 h\u03b5 with N' hN', clear hw h\u03b5,\r\n use max N N',\r\n intros n hn,\r\n rw max_ge_iff at hn,\r\n specialize hN n (by linarith),\r\n specialize hN' n (by linarith),\r\n specialize h n,\r\n specialize h' n,\r\n rw abs_le at *,\r\n split,\r\n { linarith, },\r\n { linarith, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n (hu : limite u a)\r\n (hw : limite w a)\r\n (h : \u2200 n, u n \u2264 v n)\r\n (h' : \u2200 n, v n \u2264 w n) :\r\n limite v a :=\r\nbegin\r\n intros \u03b5 h\u03b5,\r\n cases hu \u03b5 h\u03b5 with N hN, clear hu,\r\n cases hw \u03b5 h\u03b5 with N' hN', clear hw h\u03b5,\r\n use max N N',\r\n intros n hn,\r\n rw max_ge_iff at hn,\r\n specialize hN n (by linarith),\r\n specialize hN' n (by linarith),\r\n specialize h n,\r\n specialize h' n,\r\n rw abs_le at *,\r\n split ; linarith,\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\nexample\r\n (hu : limite u a)\r\n (hw : limite w a)\r\n (h : \u2200 n, u n \u2264 v n)\r\n (h' : \u2200 n, v n \u2264 w n) :\r\n limite v a :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nexists.elim (hu \u03b5 h\u03b5)\r\n ( assume N,\r\n assume hN : \u2200 (n : \u2115), n \u2265 N \u2192 |u n - a| \u2264 \u03b5,\r\n exists.elim (hw \u03b5 h\u03b5)\r\n ( assume N',\r\n assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |w n - a| \u2264 \u03b5,\r\n show \u2203 N, \u2200 n, n \u2265 N \u2192 |v n - a| \u2264 \u03b5, from\r\n exists.intro (max N N')\r\n ( assume n,\r\n assume hn : n \u2265 max N N',\r\n have h1 : n \u2265 N \u2227 n \u2265 N',\r\n from max_ge_iff.mp hn,\r\n have h2 : -\u03b5 \u2264 v n - a,\r\n { have h2a : |u n - a| \u2264 \u03b5,\r\n from hN n h1.1,\r\n calc -\u03b5\r\n \u2264 u n - a : and.left (abs_le.mp h2a)\r\n ... \u2264 v n - a : by linarith [h n], },\r\n have h3 : v n - a \u2264 \u03b5,\r\n { have h3a : |w n - a| \u2264 \u03b5,\r\n from hN' n h1.2,\r\n calc v n - a\r\n \u2264 w n - a : by linarith [h' n]\r\n ... \u2264 \u03b5 : and.right (abs_le.mp h3a), },\r\n show |v n - a| \u2264 \u03b5,\r\n from abs_le.mpr (and.intro h2 h3))))\r\n<\/pre>\nSe puede interactuar con la prueba anterior en esta sesi\u00f3n con Lean<\/a>.<\/p>\nEn los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="lean"> y otra con <\/pre>
\n[\/expand]<\/p>\n
[expand title=\"Soluciones con Isabelle\/HOL\"]<\/p>\n
\r\ntheory Teorema_del_emparedado\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\nlemma\r\n assumes \"limite u a\"\r\n \"limite w a\"\r\n \"\u2200n. u n \u2264 v n\"\r\n \"\u2200n. v n \u2264 w n\"\r\n shows \"limite v a\"\r\nproof (unfold limite_def; intro allI impI)\r\n fix \u03b5 :: real\r\n assume h\u03b5 : \"0 < \u03b5\"\r\n obtain N where hN : \"\u2200n\u2265N. \u00a6u n - a\u00a6 < \u03b5\"\r\n using assms(1) h\u03b5 limite_def\r\n by auto\r\n obtain N' where hN' : \"\u2200n\u2265N'. \u00a6w n - a\u00a6 < \u03b5\"\r\n using assms(2) h\u03b5 limite_def\r\n by auto\r\n have \"\u2200n\u2265max N N'. \u00a6v n - a\u00a6 < \u03b5\"\r\n proof (intro allI impI)\r\n fix n\r\n assume hn : \"n\u2265max N N'\"\r\n have \"v n - a < \u03b5\"\r\n proof -\r\n have \"v n - a \u2264 w n - a\"\r\n using assms(4) by simp\r\n also have \"\u2026 \u2264 \u00a6w n - a\u00a6\"\r\n by simp\r\n also have \"\u2026 < \u03b5\"\r\n using hN' hn by auto\r\n finally show \"v n - a < \u03b5\" .\r\n qed\r\n moreover\r\n have \"-(v n - a) < \u03b5\"\r\n proof -\r\n have \"-(v n - a) \u2264 -(u n - a)\"\r\n using assms(3) by auto\r\n also have \"\u2026 \u2264 \u00a6u n - a\u00a6\"\r\n by simp\r\n also have \"\u2026 < \u03b5\"\r\n using hN hn by auto\r\n finally show \"-(v n - a) < \u03b5\" .\r\n qed\r\n ultimately show \"\u00a6v n - a\u00a6 < \u03b5\"\r\n by (simp only: abs_less_iff)\r\n qed\r\n then show \"\u2203k. \u2200n\u2265k. \u00a6v n - a\u00a6 < \u03b5\"\r\n by (rule exI)\r\nqed\r\n\r\nend\r\n<\/pre>\nEn los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="isar"> y otra con <\/pre>
\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"
En Lean, una sucesi\u00f3n u\u2080, u\u2081, u\u2082, … 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 – a| < \u03b5 donde se usa la notaci\u00f3n |x| para el valor absoluto de x notation `|`x`|` := abs x Demostrar el teorema del emparedado; es decir, que si para todo n, u(n) \u2264 v(n) \u2264 w(n) y u(n) tiene el mismo l\u00edmite que w(n), entonces v(n) tambi\u00e9n tiene dicho l\u00edmite. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic...\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":[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\/580"}],"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=580"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/580\/revisions"}],"predecessor-version":[{"id":582,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/580\/revisions\/582"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=580"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}