{"id":7538,"date":"2021-01-05T06:00:24","date_gmt":"2021-01-05T05:00:24","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7538"},"modified":"2021-01-04T12:45:10","modified_gmt":"2021-01-04T11:45:10","slug":"pruebas-en-lean-de-si-x-y-%e2%89%a4-%ce%b5-para-todo-%ce%b5-0-entonces-x-y","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/pruebas-en-lean-de-si-x-y-%e2%89%a4-%ce%b5-para-todo-%ce%b5-0-entonces-x-y\/","title":{"rendered":"Pruebas en Lean de &#8220;Si |x &#8211; y| \u2264 \u03b5, para todo \u03b5 > 0, entonces x = y&#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\/_k7e173__6o\">v\u00eddeo<\/a> en el que se comentan pruebas en Lean de la propiedad:<\/p>\n<blockquote><p>\n  Si |x &#8211; y| \u2264 \u03b5, para todo \u03b5 > 0, entonces x = y\n<\/p><\/blockquote>\n<p>usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico.<\/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\/_k7e173__6o\" 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\/3b4PPGH\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables (x y : \u211d)\n\n-- ----------------------------------------------------\n-- Ejercicio 1. Definir la notaci\u00f3n |x| para el valor\n-- absoluto de x.\n-- ----------------------------------------------------\n\nnotation `|`x`|` := abs x\n\n-- ----------------------------------------------------\n-- Ejercicio 2. Demostrar que si |x - y| \u2264 \u03b5, para todo\n-- \u03b5 > 0, entonces x = y.\n-- ----------------------------------------------------\n\n-- Se usar\u00e1 el lema demostrado anteriormente\nlemma cero_de_abs_mne_todos\n  (h : \u2200 \u03b5 > 0, |x| \u2264 \u03b5)\n  : x = 0 :=\nabs_eq_zero.mp\n  (eq_of_le_of_forall_le_of_dense (abs_nonneg x) h)\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : \u2200 \u03b5 > 0, |x - y| \u2264 \u03b5)\n  : x = y :=\nbegin\n  rw \u2190 sub_eq_zero,\n  exact cero_de_abs_mne_todos (x - y) h,\nend\n\n-- 2\u00aa demostraci\u00f3n\nlemma ig_de_abs_sub_mne_todos\n  (h : \u2200 \u03b5 > 0, |x - y| \u2264 \u03b5)\n  : x = y :=\nsub_eq_zero.mp (cero_de_abs_mne_todos (x - y) h)\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 pruebas en Lean de la propiedad: Si |x &#8211; y| \u2264 \u03b5, para todo \u03b5 > 0, entonces x = y usando los estilos declarativos, aplicativos, funcional y autom\u00e1tico. A continuaci\u00f3n, se muestra el v\u00eddeo 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\/7538"}],"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=7538"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7538\/revisions"}],"predecessor-version":[{"id":7539,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7538\/revisions\/7539"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7538"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7538"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7538"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}