{"id":1043,"date":"2022-09-21T06:00:03","date_gmt":"2022-09-21T04:00:03","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1043"},"modified":"2022-09-11T16:37:49","modified_gmt":"2022-09-11T14:37:49","slug":"si-a-b-d-%e2%88%88-%e2%84%9d-tales-que-1-%e2%89%a4-a-y-b-%e2%89%a4-d-entonces-2-a-e%e1%b5%87-%e2%89%a4-3a-e%e1%b5%88","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-d-%e2%88%88-%e2%84%9d-tales-que-1-%e2%89%a4-a-y-b-%e2%89%a4-d-entonces-2-a-e%e1%b5%87-%e2%89%a4-3a-e%e1%b5%88\/","title":{"rendered":"Si a, b, d \u2208 \u211d tales que 1 \u2264 a y b \u2264 d, entonces 2 + a + e\u1d47 \u2264 3a + e\u1d48"},"content":{"rendered":"<p>Demostrar que si a, b, d \u2208 \u211d tales que <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=1+%5Cleq+a&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"1 &#92;leq a\" class=\"latex\" \/> y <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=b+%5Cleq+d&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"b &#92;leq d\" class=\"latex\" \/>, entonces <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=2+%2B+a+%2B+e%5Eb+%5Cleq+3a+%2B+e%5Ed&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"2 + a + e^b &#92;leq 3a + e^d\" class=\"latex\" \/>.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport analysis.special_functions.log.basic\nopen real\nvariables a b d : \u211d\n\nexample\n  (h  : 1 \u2264 a)\n  (h' : b \u2264 d)\n  : 2 + a + exp b \u2264 3 * a + exp d :=\nsorry\n<\/pre>\n<p><strong>Nota<\/strong>: Se pueden usar los lemas<\/p>\n<pre lang=\"text\">\nadd_le_add : a \u2264 b \u2192 c \u2264 d \u2192 a + c \u2264 b + d\nexp_le_exp : exp a \u2264 exp b \u2194 a \u2264 b\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport analysis.special_functions.log.basic\nopen real\nvariables a b d : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h  : 1 \u2264 a)\n  (h' : b \u2264 d)\n  : 2 + a + exp b \u2264 3 * a + exp d :=\nbegin\n  apply add_le_add,\n  { calc 2 + a\n         = (1 + 1) + a : by refl\n     ... \u2264 (1 + a) + a : by simp [h]\n     ... \u2264 (a + a) + a : by simp [h]\n     ... = 3 * a       : by ring },\n  { exact exp_le_exp.mpr h', },\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h  : 1 \u2264 a)\n  (h' : b \u2264 d)\n  : 2 + a + exp b \u2264 3 * a + exp d :=\nby linarith [exp_le_exp.mpr h']\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Desigualdad_con_exponencial.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li>J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 16.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a, b, d \u2208 \u211d tales que y , entonces . Para ello, completar la siguiente teor\u00eda de Lean: import analysis.special_functions.log.basic open real variables a b d : \u211d example (h : 1 \u2264 a) (h&#8217; : b \u2264 d) : 2 + a + exp b \u2264 3 * a + exp d := sorry Nota: Se pueden usar los lemas add_le_add : a \u2264 b \u2192 c \u2264 d \u2192 a + c \u2264 b + d exp_le_exp : exp a \u2264 exp b \u2194 a \u2264 b<\/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":[288,286,287],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1043"}],"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=1043"}],"version-history":[{"count":7,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1043\/revisions"}],"predecessor-version":[{"id":1102,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1043\/revisions\/1102"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1043"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1043"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1043"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}