{"id":1488,"date":"2023-08-25T06:00:16","date_gmt":"2023-08-25T04:00:16","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1488"},"modified":"2023-08-09T17:44:13","modified_gmt":"2023-08-09T15:44:13","slug":"25-ago-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/25-ago-23\/","title":{"rendered":"En \u211d, si 1 \u2264 a y b \u2264 d, entonces 2 + a + e\u1d47 \u2264 3a + e\u1d48"},"content":{"rendered":"<p>Demostrar con Lean4 que si \\(a\\), \\(b\\) y \\(d\\) n\u00fameros reales tales que \\(1 \\leq a\\) y \\(b \\leq d\\), entonces \\(2 + a + e^b \\leq 3a + e^d\\).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\r\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\r\n\r\nopen Real\r\n\r\nvariable (a b d : \u211d)\r\n\r\nexample\r\n  (h1 : 1 \u2264 a)\r\n  (h2 : b \u2264 d)\r\n  : 2 + a + exp b \u2264 3 * a + exp d :=\r\nby sorry\r\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nDe la primera hip\u00f3tesis (\\(1 \\leq a\\)), multiplicando por \\(2\\), se obtiene<br \/>\n\\[2 \\leq 2a\\]<br \/>\ny, sumando a ambos lados, se tiene<br \/>\n\\[2 + a \\leq 3a \\tag{1}\\]<\/p>\n<p>De la hip\u00f3tesis 2 (\\(b \\leq d\\)) y de la monoton\u00eda de la funci\u00f3n exponencial se tiene<br \/>\n\\[e^b \\leq e^d \\tag{2} \\]<\/p>\n<p>Finalmente, de (1) y (2) se tiene<br \/>\n\\[2 + a + e^b \\leq 3a + e^d\\]<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\r\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\r\n\r\nopen Real\r\n\r\nvariable (a b d : \u211d)\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (h1 : 1 \u2264 a)\r\n  (h2 : b \u2264 d)\r\n  : 2 + a + exp b \u2264 3 * a + exp d :=\r\nby\r\n  have h3 : 2 + a \u2264 3 * a := calc\r\n    2 + a = 2 * 1 + a := by linarith only []\r\n        _ \u2264 2 * a + a := by linarith only [h1]\r\n        _ \u2264 3 * a     := by linarith only []\r\n  have h4 : exp b \u2264 exp d := by\r\n    linarith only [exp_le_exp.mpr h2]\r\n  show 2 + a + exp b \u2264 3 * a + exp d\r\n  exact add_le_add h3 h4\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (h1 : 1 \u2264 a)\r\n  (h2 : b \u2264 d)\r\n  : 2 + a + exp b \u2264 3 * a + exp d :=\r\ncalc\r\n  2 + a + exp b\r\n    \u2264 3 * a + exp b := by linarith only [h1]\r\n  _ \u2264 3 * a + exp d := by linarith only [exp_le_exp.mpr h2]\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (h1 : 1 \u2264 a)\r\n  (h2 : b \u2264 d)\r\n  : 2 + a + exp b \u2264 3 * a + exp d :=\r\nby linarith [exp_le_exp.mpr h2]\r\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/lean.math.hhu.de\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Inecuaciones_con_exponenciales.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. 15.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si \\(a\\), \\(b\\) y \\(d\\) n\u00fameros reales tales que \\(1 \\leq a\\) y \\(b \\leq d\\), entonces \\(2 + a + e^b \\leq 3a + e^d\\). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Analysis.SpecialFunctions.Log.Basic open Real variable (a b d : \u211d) example (h1 : 1 \u2264 a) (h2 : b \u2264 d) : 2 + a + exp b \u2264 3 * a + exp d := 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":[297,286,287],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1488"}],"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=1488"}],"version-history":[{"count":9,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1488\/revisions"}],"predecessor-version":[{"id":1497,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1488\/revisions\/1497"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1488"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1488"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}