{"id":1832,"date":"2023-12-06T06:00:22","date_gmt":"2023-12-06T04:00:22","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1832"},"modified":"2023-12-07T14:24:47","modified_gmt":"2023-12-07T12:24:47","slug":"06-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/06-dic-23\/","title":{"rendered":"Si \u00ac(\u2200a)(\u2203x)[f(x) > a]\u200b, entonces f est\u00e1 acotada superiormente"},"content":{"rendered":"\n<p>Demostrar con Lean4 que si &#92;(\u00ac(\u2200a)(\u2203x)[f(x) > a]&#92;)\u200b, entonces &#92;(f&#92;) est\u00e1 acotada superiormente.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\ndef CotaSuperior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop :=\n  \u2200 x, f x \u2264 a\n\ndef acotadaSup (f : \u211d \u2192 \u211d) :=\n  \u2203 a, CotaSuperior f a\n\nvariable (f : \u211d \u2192 \u211d)\n\nexample\n  (h : \u00ac\u2200 a, \u2203 x, f x > a)\n  : acotadaSup f :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Tenemos que demostrar que &#92;(f&#92;) es acotada superiormente; es decir, que<br \/>\n&#92;[ (\u2203a)(\u2200x)[f(x) \u2264 a] &#92;]<br \/>\nque es exactamente la f\u00f3rmula obtenida interiorizando la negaci\u00f3n en la hip\u00f3tesis.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\ndef CotaSuperior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop :=\n  \u2200 x, f x \u2264 a\n\ndef acotadaSup (f : \u211d \u2192 \u211d) :=\n  \u2203 a, CotaSuperior f a\n\nvariable (f : \u211d \u2192 \u211d)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u2200 a, \u2203 x, f x > a)\n  : acotadaSup f :=\nby\n  unfold acotadaSup\n  -- \u22a2 \u2203 a, CotaSuperior f a\n  unfold CotaSuperior\n  -- \u22a2 \u2203 a, \u2200 (x : \u211d), f x \u2264 a\n  push_neg at h\n  -- h : \u2203 a, \u2200 (x : \u211d), f x \u2264 a\n  exact h\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u2200 a, \u2203 x, f x > a)\n  : acotadaSup f :=\nby\n  unfold acotadaSup CotaSuperior\n  -- \u22a2 \u2203 a, \u2200 (x : \u211d), f x \u2264 a\n  push_neg at h\n  -- h : \u2203 a, \u2200 (x : \u211d), f x \u2264 a\n  exact h\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : \u00ac\u2200 a, \u2203 x, f x > a)\n  : acotadaSup f :=\nby\n  push_neg at h\n  -- h : \u2203 a, \u2200 (x : \u211d), f x \u2264 a\n  exact h\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\/CS_de_acotada_superiormente.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. 34.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si \u00ac(\u2200a)(\u2203x)[f(x) > a], entonces f est\u00e1 acotada superiormente.<\/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\/1832"}],"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=1832"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1832\/revisions"}],"predecessor-version":[{"id":1857,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1832\/revisions\/1857"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1832"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1832"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}