{"id":1227,"date":"2022-11-30T17:46:36","date_gmt":"2022-11-30T15:46:36","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1227"},"modified":"2022-12-04T08:35:16","modified_gmt":"2022-12-04T06:35:16","slug":"30-nov-22","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/30-nov-22\/","title":{"rendered":"Si c \u2265 0 y f est\u00e1 acotada superiormente, entonces c * f tambi\u00e9n lo est\u00e1"},"content":{"rendered":"<p>Demostrar que si c \u2265 0 y f est\u00e1 acotada superiormente, entonces c * f tambi\u00e9n lo est\u00e1.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables {f : \u211d \u2192 \u211d}                    \nvariables {a c : \u211d}                      \n\n-- (cota_superior f a) se verifica si a es una cota superior de f.\ndef cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop := \u2200 x, f x \u2264 a\n\n-- (acotada_sup f) se verifica si f tiene cota superior.\ndef acotada_sup (f : \u211d \u2192 \u211d) := \u2203 a, cota_superior f a\n\nexample\n  (hf : acotada_sup f)\n  (h : c \u2265 0)\n  : acotada_sup (\u03bb x, c * f x) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables {f : \u211d \u2192 \u211d}\nvariables {a c : \u211d}\n\n-- (cota_superior f a) se verifica si a es una cota superior de f.\ndef cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop := \u2200 x, f x \u2264 a\n\n-- (acotada_sup f) se verifica si f tiene cota superior.\ndef acotada_sup (f : \u211d \u2192 \u211d) := \u2203 a, cota_superior f a\n\n-- Lema auxiliar\n-- ============\n\nlemma cota_superior_mul\n  (hfa : cota_superior f a)\n  (h : c \u2265 0)\n  : cota_superior (\u03bb x, c * f x) (c * a) :=\n\u03bb x, mul_le_mul_of_nonneg_left (hfa x) h\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hf : acotada_sup f)\n  (h : c \u2265 0)\n  : acotada_sup (\u03bb x, c * f x) :=\nbegin\n  cases hf with a ha,\n  have h1 : cota_superior (\u03bb x, c * f x) (c * a) := cota_superior_mul ha h,\n  have h2 : \u2203 z, \u2200 x, (\u03bb x, c * f x) x \u2264 z,\n    by exact Exists.intro (c * a) h1,\n  show acotada_sup (\u03bb x, c * f x),\n    by exact h2,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hf : acotada_sup f)\n  (h : c \u2265 0)\n  : acotada_sup (\u03bb x, c * f x) :=\nbegin\n  cases hf with a ha,\n  use c * a,\n  apply cota_superior_mul ha h,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hf : acotada_sup f)\n  (h : c \u2265 0)\n  : acotada_sup (\u03bb x, c * f x) :=\nbegin\n  rcases hf with \u27e8a, ha\u27e9,\n  exact \u27e8c * a, cota_superior_mul ha h\u27e9,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : c \u2265 0)\n  : acotada_sup f \u2192 acotada_sup (\u03bb x, c * f x) :=\nbegin\n  rintro \u27e8a, ha\u27e9,\n  exact \u27e8c * a, cota_superior_mul ha h\u27e9,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : c \u2265 0)\n  : acotada_sup f \u2192 acotada_sup (\u03bb x, c * f x) :=\n\u03bb \u27e8a, ha\u27e9, \u27e8c * a, cota_superior_mul ha h\u27e9\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\/Producto_por_funcion_acotada_superiormente.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. 31.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si c \u2265 0 y f est\u00e1 acotada superiormente, entonces c * f tambi\u00e9n lo est\u00e1. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables {f : \u211d \u2192 \u211d} variables {a c : \u211d} &#8212; (cota_superior f a) se verifica si a es una cota superior de f. def cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop := \u2200 x, f x \u2264 a &#8212; (acotada_sup f) se verifica si f tiene cota superior. def acotada_sup (f : \u211d \u2192 \u211d) := \u2203 a, cota_superior f a example (hf : acotada_sup f) (h : c \u2265 0) : acotada_sup (\u03bb x, c * f x) := 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\/1227"}],"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=1227"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1227\/revisions"}],"predecessor-version":[{"id":1233,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1227\/revisions\/1233"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1227"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1227"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}