{"id":1179,"date":"2022-11-07T06:00:40","date_gmt":"2022-11-07T04:00:40","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1179"},"modified":"2022-11-06T17:01:49","modified_gmt":"2022-11-06T15:01:49","slug":"si-a-es-una-cota-superior-no-negativa-de-f-y-b-es-es-una-cota-superior-de-la-funcion-no-negativa-g-entonces-a%c2%b7b-es-una-cota-superior-de-f%c2%b7g","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-es-una-cota-superior-no-negativa-de-f-y-b-es-es-una-cota-superior-de-la-funcion-no-negativa-g-entonces-a%c2%b7b-es-una-cota-superior-de-f%c2%b7g\/","title":{"rendered":"Si a es una cota superior no negativa de f y b es es una cota superior de la funci\u00f3n no negativa g, entonces a\u00b7b es una cota superior de f\u00b7g"},"content":{"rendered":"<p>Demostrar que si a es una cota superior no negativa de f y b es es una cota superior de la funci\u00f3n no negativa g, entonces a\u00b7b es una cota superior de f\u00b7g.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\ndef cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop :=\n\u2200 x, f x \u2264 a\n\ndef no_negativa (f : \u211d \u2192 \u211d) : Prop :=\n\u2200 x, 0 \u2264 f x\n\nvariables (f g : \u211d \u2192 \u211d)\nvariables (a b : \u211d)\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\ndef cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop :=\n\u2200 x, f x \u2264 a\n\ndef no_negativa (f : \u211d \u2192 \u211d) : Prop :=\n\u2200 x, 0 \u2264 f x\n\nvariables (f g : \u211d \u2192 \u211d)\nvariables (a b : \u211d)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\nbegin\n  have h1 : \u2200 x, f x * g x \u2264 a * b,\n    { intro x,\n      have h2 : f x \u2264 a := hfa x,\n      have h3 : g x \u2264 b := hgb x,\n      have h4 : 0 \u2264 g x := nng x,\n      show f x * g x \u2264 a * b,\n        by exact mul_le_mul h2 h3 h4 nna, },\n  show cota_superior (f * g) (a * b),\n    by exact h1,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\nbegin\n  intro x,\n  change f x * g x \u2264 a * b,\n  apply mul_le_mul,\n  apply hfa,\n  apply hgb,\n  apply nng,\n  apply nna\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\nbegin\n  dunfold cota_superior no_negativa at *,\n  intro x,\n  have h1:= hfa x,\n  have h2:= hgb x,\n  have h3:= nng x,\n  exact mul_le_mul h1 h2 h3 nna,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\nbegin\n  dunfold cota_superior no_negativa at *,\n  intro x,\n  specialize hfa x,\n  specialize hgb x,\n  specialize nng x,\n  exact mul_le_mul hfa hgb nng nna,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hfa : cota_superior f a)\n  (hgb : cota_superior g b)\n  (nng : no_negativa g)\n  (nna : 0 \u2264 a)\n  : cota_superior (f * g) (a * b) :=\n\u03bb x, mul_le_mul (hfa x) (hgb x) (nng x) nna\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\/Cota_sup_del_producto_de_funciones.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. 27.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a es una cota superior no negativa de f y b es es una cota superior de la funci\u00f3n no negativa g, entonces a\u00b7b es una cota superior de f\u00b7g. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic def cota_superior (f : \u211d \u2192 \u211d) (a : \u211d) : Prop := \u2200 x, f x \u2264 a def no_negativa (f : \u211d \u2192 \u211d) : Prop := \u2200 x, 0 \u2264 f x variables (f g : \u211d \u2192 \u211d) variables (a b : \u211d) example (hfa : cota_superior f a) (hgb : cota_superior g b) (nng : no_negativa g) (nna : 0 \u2264 a) : cota_superior (f * g) (a * 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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1179"}],"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=1179"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1179\/revisions"}],"predecessor-version":[{"id":1180,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1179\/revisions\/1180"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}