{"id":1780,"date":"2023-11-20T06:00:31","date_gmt":"2023-11-20T04:00:31","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1780"},"modified":"2023-11-07T19:24:34","modified_gmt":"2023-11-07T17:24:34","slug":"20-nov-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/20-nov-23\/","title":{"rendered":"La funci\u00f3n identidad no est\u00e1 acotada superiormente"},"content":{"rendered":"<p>Demostrar con Lean4 que la funci\u00f3n identidad no est\u00e1 acotada superiormente.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport src.Funcion_no_acotada_superiormente\n\nexample : \u00ac acotadaSup (fun x \u21a6 x) :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nUsando el lema de ejercicio anterior (que afirma que si para cada &#92;(a&#92;), existe un &#92;(x&#92;) tal que &#92;(f(x) > a&#92;), entonces &#92;(f&#92;) no tiene cota superior) basta demostrar que<br \/>\n&#92;[ (\u2200a \u2208 \u211d)(\u2203x \u2208 \u211d) [x > a] &#92;]<br \/>\nSea &#92;(a \u2208 \u211d&#92;). Entonces &#92;(a + 1 > a&#92;) y, por tanto, &#92;((\u2203x \u2208 \u211d) [x > a]&#92;).<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport src.Funcion_no_acotada_superiormente\n\n-- 1\u00aa demostraci\u00f3n\nexample : \u00ac acotadaSup (fun x \u21a6 x) :=\nby\n  apply sinCotaSup\n  -- \u22a2 \u2200 (a : \u211d), \u2203 x, x > a\n  intro a\n  -- a : \u211d\n  -- \u22a2 \u2203 x, x > a\n  use a + 1\n  -- \u22a2 a + 1 > a\n  linarith\n\n-- 2\u00aa demostraci\u00f3n\nexample : \u00ac acotadaSup (fun x \u21a6 x) :=\nby\n  apply sinCotaSup\n  -- \u22a2 \u2200 (a : \u211d), \u2203 x, x > a\n  intro a\n  -- a : \u211d\n  -- \u22a2 \u2203 x, x > a\n  exact \u27e8a + 1, by linarith\u27e9\n\n-- 3\u00aa demostraci\u00f3n\nexample : \u00ac acotadaSup (fun x \u21a6 x) :=\nby\n  apply sinCotaSup\n  -- \u22a2 \u2200 (a : \u211d), \u2203 x, x > a\n  exact fun a \u21a6 \u27e8a + 1, by linarith\u27e9\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\/La_identidad_no_esta_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. 32.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que la funci\u00f3n identidad no est\u00e1 acotada superiormente. Para ello, completar la siguiente teor\u00eda de Lean4: import src.Funcion_no_acotada_superiormente example : \u00ac acotadaSup (fun x \u21a6 x) := 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\/1780"}],"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=1780"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1780\/revisions"}],"predecessor-version":[{"id":1781,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1780\/revisions\/1781"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1780"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1780"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}