{"id":7552,"date":"2021-01-09T19:44:34","date_gmt":"2021-01-09T18:44:34","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7552"},"modified":"2021-01-09T19:45:14","modified_gmt":"2021-01-09T18:45:14","slug":"formatus-pruebas-en-lean-de-las-funciones-de-extraccion-no-estan-acotadas","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-las-funciones-de-extraccion-no-estan-acotadas\/","title":{"rendered":"ForMatUS: Pruebas en Lean de &#8220;Las funciones de extracci\u00f3n no est\u00e1n acotadas&#8221;"},"content":{"rendered":"<p>He a\u00f1adido a la lista <a href=\"https:\/\/bit.ly\/2QwnT30\">DAO (Demostraci\u00f3n Asistida por Ordenador) con Lean<\/a> el <a href=\"https:\/\/youtu.be\/zy8cWOQnJXo\">v\u00eddeo<\/a> en el que se comentan 11 pruebas en Lean de la propiedad<\/p>\n<blockquote><p>\n  Las funciones de extracci\u00f3n no est\u00e1n acotadas; es decir, que si \u03c6 es una funci\u00f3n de extracci\u00f3n, entonces<\/p>\n<blockquote><p>\n    \u2200 N N&#8217;, \u2203 n \u2265 N&#8217;, \u03c6 n \u2265 N\n  <\/p><\/blockquote>\n<\/blockquote>\n<p>usando los estilos declarativo, aplicativo y funcional.<\/p>\n<p>A continuaci\u00f3n, se muestra el v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/zy8cWOQnJXo\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>y el <a href=\"https:\/\/bit.ly\/3bq0R9M\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nopen nat\n\nvariable {\u03c6 : \u2115 \u2192 \u2115}\n\n-- ----------------------------------------------------\n-- Nota. Se usar\u00e1 la siguiente definici\u00f3n y lema\n-- estudiado anteriormente.\n-- ----------------------------------------------------\n\ndef extraccion : (\u2115 \u2192 \u2115) \u2192 Prop\n| \u03c6 := \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\n\nlemma id_mne_extraccion\n  (h : extraccion \u03c6)\n  : \u2200 n, n \u2264 \u03c6 n :=\nbegin\n  intros n,\n  induction n with m HI,\n  { linarith },\n  { exact nat.succ_le_of_lt (by linarith [h m (m+1) (by linarith)]) },\nend\n\n-- ----------------------------------------------------\n-- Ejercicio. Demostrar que las funciones de extracci\u00f3n\n-- no est\u00e1 acotadas; es decir, que si \u03c6 es una funci\u00f3n\n-- de extracci\u00f3n, entonces\n--     \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nbegin\n  intros N N',\n  let n := max N N',\n  use n,\n  split,\n  { exact le_max_right N N', },\n  { calc N \u2264 n   : le_max_left N N'\n       ... \u2264 \u03c6 n : id_mne_extraccion h n, },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nbegin\n  intros N N',\n  let n := max N N',\n  use n,\n  split,\n  { exact le_max_right N N', },\n  { exact le_trans (le_max_left N N')\n                   (id_mne_extraccion h n), },\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nbegin\n  intros N N',\n  use max N N',\n  split,\n  { exact le_max_right N N', },\n  { exact le_trans (le_max_left N N')\n                   (id_mne_extraccion h (max N N')), },\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nbegin\n  intros N N',\n  use max N N',\n  exact \u27e8le_max_right N N',\n         le_trans (le_max_left N N')\n                  (id_mne_extraccion h (max N N'))\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\n\u03bb N N',\n  \u27e8max N N', \u27e8le_max_right N N',\n              le_trans (le_max_left N N')\n                       (id_mne_extraccion h (max N N'))\u27e9\u27e9\n\n-- 6\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nassume N N',\nlet n := max N N' in\nhave h1 : n \u2265 N',\n  from le_max_right N N',\nshow \u2203 n \u2265 N', \u03c6 n \u2265 N, from\nexists.intro n\n  (exists.intro h1\n    (show \u03c6 n \u2265 N, from\n       calc N \u2264 n   : le_max_left N N'\n          ... \u2264 \u03c6 n : id_mne_extraccion h n))\n\n-- 7\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nassume N N',\nlet n := max N N' in\nhave h1 : n \u2265 N',\n  from le_max_right N N',\nshow \u2203 n \u2265 N', \u03c6 n \u2265 N, from\n\u27e8n, h1, calc N \u2264 n   : le_max_left N N'\n          ...  \u2264 \u03c6 n : id_mne_extraccion h n\u27e9\n\n-- 8\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nassume N N',\nlet n := max N N' in\nhave h1 : n \u2265 N',\n  from le_max_right N N',\nshow \u2203 n \u2265 N', \u03c6 n \u2265 N, from\n\u27e8n, h1, le_trans (le_max_left N N')\n                 (id_mne_extraccion h (max N N'))\u27e9\n\n-- 9\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nassume N N',\nlet n := max N N' in\nhave h1 : n \u2265 N',\n  from le_max_right N N',\n\u27e8n, h1, le_trans (le_max_left N N')\n                 (id_mne_extraccion h n)\u27e9\n\n-- 10\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\nassume N N',\n\u27e8max N N', le_max_right N N',\n           le_trans (le_max_left N N')\n                    (id_mne_extraccion h (max N N'))\u27e9\n\n-- 11\u00aa demostraci\u00f3n\nexample\n  (h : extraccion \u03c6)\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\n\u03bb N N',\n  \u27e8max N N', le_max_right N N',\n             le_trans (le_max_left N N')\n             (id_mne_extraccion h (max N N'))\u27e9\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>He a\u00f1adido a la lista DAO (Demostraci\u00f3n Asistida por Ordenador) con Lean el v\u00eddeo en el que se comentan 11 pruebas en Lean de la propiedad Las funciones de extracci\u00f3n no est\u00e1n acotadas; es decir, que si \u03c6 es una funci\u00f3n de extracci\u00f3n, entonces \u2200 N N&#8217;, \u2203 n \u2265 N&#8217;, \u03c6 n \u2265 N&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","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,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[336],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7552"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=7552"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7552\/revisions"}],"predecessor-version":[{"id":7555,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7552\/revisions\/7555"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7552"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7552"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7552"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}