{"id":1802,"date":"2023-11-23T06:00:18","date_gmt":"2023-11-23T04:00:18","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1802"},"modified":"2023-11-12T11:47:41","modified_gmt":"2023-11-12T09:47:41","slug":"23-nov-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/23-nov-23\/","title":{"rendered":"No para toda f : \u211d \u2192 \u211d mon\u00f3tona, (\u2200a,b)[f(a) \u2264 f(b) \u2192 a \u2264 b]"},"content":{"rendered":"\n<p>Demostrar con Lean4 que no para toda &#92;(f : \u211d \u2192 \u211d&#92;) mon\u00f3tona, &#92;((\u2200a,b)[f(a) \u2264 f(b) \u2192 a \u2264 b]&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nexample :\n  \u00ac\u2200 {f : \u211d \u2192 \u211d}, Monotone f \u2192 \u2200 {a b}, f a \u2264 f b \u2192 a \u2264 b :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Supongamos que<br \/>\n&#92;[ (\u2200f)[f &#92;text{ es mon\u00f3tona } \u2192 (\u2200a, b)[f(a) \u2264 f(b) \u2192 a \u2264 b]] &#92;tag{1} &#92;]<br \/>\nSea &#92;(f : \u211d \u2192 \u211d&#92;) la funci\u00f3n constante igual a cero; es decir,<br \/>\n&#92;[ (\u2200x \u2208 \u211d)[f(x) = 0] &#92;]<br \/>\nEntonces, &#92;(f&#92;) es mon\u00f3tona y &#92;(f(1) \u2264 f(0)&#92;) (ya que &#92;(f(1) = 0 \u2264 0 = f(0)&#92;)). Luego, por (1), &#92;(1 \u2264 0&#92;) que es una contradicci\u00f3n.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample :\n  \u00ac\u2200 {f : \u211d \u2192 \u211d}, Monotone f \u2192 \u2200 {a b}, f a \u2264 f b \u2192 a \u2264 b :=\nby\n  intro h1\n  -- h1 : \u2200 {f : \u211d \u2192 \u211d}, Monotone f \u2192 \u2200 {a b : \u211d}, f a \u2264 f b \u2192 a \u2264 b\n  -- \u22a2 False\n  let f := fun _ : \u211d \u21a6 (0 : \u211d)\n  have h2 : Monotone f := monotone_const\n  have h3 : f 1 \u2264 f 0 := le_refl 0\n  have h4 : 1 \u2264 0 := h1 h2 h3\n  linarith\n\n-- Lemas usados\n-- ============\n\n-- variable (a c : \u211d)\n-- #check (le_refl a : a \u2264 a)\n-- #check (monotone_const : Monotone fun _ : \u211d \u21a6 c)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Propiedad_de_monotona.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\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. 32.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que no para toda &#92;(f : \u211d \u2192 \u211d&#92;) mon\u00f3tona, &#92;((\u2200a,b)[f(a) \u2264 f(b) \u2192 a \u2264 b]&#92;). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic example : \u00ac\u2200 {f : \u211d \u2192 \u211d}, Monotone f \u2192 \u2200 {a b}, f a \u2264 f b \u2192 a \u2264 b := by sorry<\/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\/1802"}],"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=1802"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1802\/revisions"}],"predecessor-version":[{"id":1805,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1802\/revisions\/1805"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}