{"id":1054,"date":"2022-09-23T06:00:01","date_gmt":"2022-09-23T04:00:01","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1054"},"modified":"2022-09-11T16:40:12","modified_gmt":"2022-09-11T14:40:12","slug":"si-a-b-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-entonces-log1-e%e1%b5%83-%e2%89%a4-log1-e%e1%b5%87","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-entonces-log1-e%e1%b5%83-%e2%89%a4-log1-e%e1%b5%87\/","title":{"rendered":"Si a, b \u2208 \u211d tales que a \u2264 b, entonces log(1 + e\u1d43) \u2264 log(1 + e\u1d47)"},"content":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d tales que a \u2264 b, entonces log(1 + e\u1d43) \u2264 log(1 + e\u1d47).<\/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 : \u211d\n\nexample\n  (h : a \u2264 b)\n  : log (1 + exp a) \u2264 log (1 + exp b) :=\nsorry\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 : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a \u2264 b)\n  : log (1 + exp a) \u2264 log (1 + exp b) :=\nbegin\n  have h\u2080 : 0 < 1 + exp a,\n  { apply add_pos,\n    exact one_pos,\n    apply exp_pos, },\n  have h\u2081 : 0 < 1 + exp b,\n  { apply add_pos,\n    exact one_pos,\n    apply exp_pos },\n  apply (log_le_log h\u2080 h\u2081).mpr,\n  apply add_le_add,\n   apply le_refl,\n  apply exp_le_exp.mpr h,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a \u2264 b)\n  : log (1 + exp a) \u2264 log (1 + exp b) :=\nbegin\n  have h\u2080 : 0 < 1 + exp a := add_pos one_pos (exp_pos a),\n  have h\u2081 : 0 < 1 + exp b := add_pos one_pos (exp_pos b),\n  exact (log_le_log h\u2080 h\u2081).mpr (add_le_add rfl.ge (exp_le_exp.mpr h))\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux : 0 < 1 + exp a :=\nadd_pos one_pos (exp_pos a)\n\nexample\n  (h : a \u2264 b)\n  : log (1 + exp a) \u2264 log (1 + exp b) :=\nbegin\n  have h\u2080 : 0 < 1 + exp a := aux a,\n  have h\u2081 : 0 < 1 + exp b := aux b,\n  exact (log_le_log h\u2080 h\u2081).mpr (add_le_add rfl.ge (exp_le_exp.mpr h))\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : a \u2264 b)\n  : log (1 + exp a) \u2264 log (1 + exp b) :=\n(log_le_log (aux a) (aux b)).mpr (add_le_add rfl.ge (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_logaritmos.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. 17.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d tales que a \u2264 b, entonces log(1 + e\u1d43) \u2264 log(1 + e\u1d47). Para ello, completar la siguiente teor\u00eda de Lean: import analysis.special_functions.log.basic open real variables a b : \u211d example (h : a \u2264 b) : log (1 + exp a) \u2264 log (1 + exp b) := 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":[289,286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1054"}],"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=1054"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1054\/revisions"}],"predecessor-version":[{"id":1104,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1054\/revisions\/1104"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1054"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1054"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}