{"id":1898,"date":"2024-01-02T06:00:58","date_gmt":"2024-01-02T04:00:58","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1898"},"modified":"2023-12-29T14:27:04","modified_gmt":"2023-12-29T12:27:04","slug":"02-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/02-ene-24\/","title":{"rendered":"La funci\u00f3n x \u21a6 -x no es mon\u00f3tona creciente"},"content":{"rendered":"\n<p>Demostrar con Lean4 que la funci\u00f3n &#92;(x \u21a6 -x&#92;) no es mon\u00f3tona creciente.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nimport src.CNS_de_no_monotona\n\nexample : \u00acMonotone fun x : \u211d \u21a6 -x :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Usando el lema del ejercicio anterior que afirma que una funci\u00f3n f no es mon\u00f3tona syss existen x e y tales que x \u2264 y y f(x) > f(y), basta demostrar que<br \/>\n&#92;[ (\u2203 x, y)[x \u2264 y \u2227 -x > -y] &#92;]<br \/>\nBasta elegir 2 y 3 ya que<br \/>\n&#92;[ 2 \u2264 3 \u2227 -2 > -3 &#92;]<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nimport src.CNS_de_no_monotona\n\nexample : \u00acMonotone fun x : \u211d \u21a6 -x :=\nby\n  apply not_Monotone_iff.mpr\n  -- \u22a2 \u2203 x y, x \u2264 y \u2227 -x > -y\n  use 2, 3\n  -- \u22a2 2 \u2264 3 \u2227 -2 > -3\n  norm_num\n<\/pre>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 37.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que la funci\u00f3n x \u21a6 -x no es mon\u00f3tona creciente.<\/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":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1898"}],"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=1898"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1898\/revisions"}],"predecessor-version":[{"id":1900,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1898\/revisions\/1900"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}