{"id":7546,"date":"2021-01-08T19:44:24","date_gmt":"2021-01-08T18:44:24","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7546"},"modified":"2021-01-08T19:48:54","modified_gmt":"2021-01-08T18:48:54","slug":"formatus-pruebas-en-lean-de-la-funcion-identidad-es-menor-o-igual-que-la-funcion-de-extraccion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/formatus-pruebas-en-lean-de-la-funcion-identidad-es-menor-o-igual-que-la-funcion-de-extraccion\/","title":{"rendered":"ForMatUS: Pruebas en Lean de &#8220;La funci\u00f3n identidad es menor o igual que la funci\u00f3n de extracci\u00f3n&#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\/xS_dJyL6OZA\">v\u00eddeo<\/a> en el que se comentan 5 pruebas en Lean de la propiedad<\/p>\n<blockquote><p>\n  Si \u03c6 es una funci\u00f3n de extracci\u00f3n (es decir, una funci\u00f3n creciente de \u2115 en \u2115), entonces n \u2264 \u03c6 n (para todo n).\n<\/p><\/blockquote>\n<p>usando los estilos declarativo y aplicativo.<\/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\/xS_dJyL6OZA\" 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\/38rQrVe\">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\nset_option pp.structure_projections false\n\n-- ----------------------------------------------------\n-- Ejercicio 1. Para extraer una sucesi\u00f3n se aplica una\n-- funci\u00f3n de extracci\u00f3n que conserva el orden; por\n-- ejemplo, la subsucesi\u00f3n\n--    u\u2092, u\u2082, u\u2084, u\u2086, ...\n-- se ha obtenido con la funci\u00f3n de extracci\u00f3n \u03c6 tal\n-- que \u03c6(n) = 2*n.\n--\n-- Definir la funci\u00f3n\n--    extraccion : (\u2115 \u2192 \u2115) \u2192 Prop\n-- tal que (extraccion \u03c6) expresa que \u03c6 es una funci\u00f3n\n-- de extracci\u00f3n\n-- ----------------------------------------------------\n\ndef extraccion : (\u2115 \u2192 \u2115) \u2192 Prop\n| \u03c6 := \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\n\n-- ----------------------------------------------------\n-- Ejercicio 2. Demostrar que si \u03c6 es una funci\u00f3n de\n-- extracci\u00f3n, entonces\n--    \u2200 n, n \u2264 \u03c6 n\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample :\n  extraccion \u03c6 \u2192 \u2200 n, n \u2264 \u03c6 n :=\nbegin\n  intros h n,\n  induction n with m HI,\n  { exact nat.zero_le (\u03c6 0), },\n  { apply nat.succ_le_of_lt,\n    have h1 : m < succ m,\n      from lt_add_one m,\n    calc m \u2264 \u03c6 m        : HI\n       ... < \u03c6 (succ m) : h m (succ m) h1, },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample :\n  extraccion \u03c6 \u2192 \u2200 n, n \u2264 \u03c6 n :=\nbegin\n  intros h n,\n  induction n with m HI,\n  { exact nat.zero_le _ },\n  { apply nat.succ_le_of_lt,\n    calc m \u2264 \u03c6 m        : HI\n       ... < \u03c6 (succ m) : by linarith [h m (m+1) (by linarith)] },\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample :\n  extraccion \u03c6 \u2192 \u2200 n, n \u2264 \u03c6 n :=\nbegin\n  intros h n,\n  induction n with m HI,\n  { linarith },\n  { apply nat.succ_le_of_lt,\n    linarith [h m (m+1) (by linarith)] },\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample :\n  extraccion \u03c6 \u2192 \u2200 n, n \u2264 \u03c6 n :=\nbegin\n  intros h 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-- 5\u00aa demostraci\u00f3n\nexample :\n  extraccion \u03c6 \u2192 \u2200 n, n \u2264 \u03c6 n :=\nassume h : extraccion \u03c6,\nassume n,\nnat.rec_on n\n  ( show 0 \u2264 \u03c6 0,\n      from nat.zero_le (\u03c6 0) )\n  ( assume m,\n    assume HI : m \u2264 \u03c6 m,\n    have h1 : m < succ m,\n      from lt_add_one m,\n    have h2 : m < \u03c6 (succ m), from\n      calc m \u2264 \u03c6 m        : HI\n         ... < \u03c6 (succ m) : h m (succ m) h1,\n    show succ m \u2264 \u03c6 (succ m),\n      from nat.succ_le_of_lt h2)\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 5 pruebas en Lean de la propiedad Si \u03c6 es una funci\u00f3n de extracci\u00f3n (es decir, una funci\u00f3n creciente de \u2115 en \u2115), entonces n \u2264 \u03c6 n (para todo n). usando los estilos declarativo y&#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\/7546"}],"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=7546"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7546\/revisions"}],"predecessor-version":[{"id":7549,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7546\/revisions\/7549"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7546"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7546"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7546"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}