{"id":1050,"date":"2022-09-22T06:00:06","date_gmt":"2022-09-22T04:00:06","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1050"},"modified":"2022-09-11T16:38:54","modified_gmt":"2022-09-11T14:38:54","slug":"si-a-b-c-d-f-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-y-c-d-entonces-a-e%e1%b6%9c-f-b-e%e1%b5%88-f","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-c-d-f-%e2%88%88-%e2%84%9d-tales-que-a-%e2%89%a4-b-y-c-d-entonces-a-e%e1%b6%9c-f-b-e%e1%b5%88-f\/","title":{"rendered":"Si a, b, c, d, f \u2208 \u211d tales que a \u2264 b y c < d, entonces a + e\u1d9c + f < b + e\u1d48 + f"},"content":{"rendered":"<p>Demostrar que si a, b, c, d, f \u2208 \u211d tales que<\/p>\n<pre lang=\"text\">\na \u2264 b\nc < d\n<\/pre>\n<p>entonces<\/p>\n<pre lang=\"text\">\na + e\u1d9c + f < b + e\u1d48 + f\n<\/pre>\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 c d f : \u211d\n\nexample\n  (hab : a \u2264 b)\n  (hcd : c < d)\n  : a + exp c + f < b + exp d + f :=\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 c d f : \u211d\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hab : a \u2264 b)\n  (hcd : c < d)\n  : a + exp c + f < b + exp d + f :=\nbegin\n  apply add_lt_add_of_lt_of_le,\n  { apply add_lt_add_of_le_of_lt,\n    { exact hab, },\n    { apply exp_lt_exp.mpr,\n      exact hcd, }},\n  { apply le_refl, },\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hab : a \u2264 b)\n  (hcd : c < d)\n  : a + exp c + f < b + exp d + f :=\nbegin\n  apply add_lt_add_of_lt_of_le,\n  { apply add_lt_add_of_le_of_lt hab (exp_lt_exp.mpr hcd), },\n  { refl, },\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hab : a \u2264 b)\n  (hcd : c < d)\n  : a + exp c + f < b + exp d + f :=\nadd_lt_add_of_lt_of_le\n  (add_lt_add_of_le_of_lt hab (exp_lt_exp.mpr hcd))\n  (le_refl f)\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hab : a \u2264 b)\n  (hcd : c < d)\n  : a + exp c + f < b + exp d + f :=\nby linarith [exp_lt_exp.mpr hcd]\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_2.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, c, d, f \u2208 \u211d tales que a \u2264 b c < d entonces a + e\u1d9c + f < b + e\u1d48 + f Para ello, completar la siguiente teor\u00eda de Lean: import analysis.special_functions.log.basic open real variables a b c d f : \u211d example (hab : a \u2264 b) (hcd : c < d) : a + exp c + f < b + exp d + f := sorry\n<\/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\/1050"}],"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=1050"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1050\/revisions"}],"predecessor-version":[{"id":1103,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1050\/revisions\/1103"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1050"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1050"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1050"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}