{"id":1173,"date":"2022-11-04T11:37:27","date_gmt":"2022-11-04T09:37:27","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1173"},"modified":"2022-11-04T11:37:27","modified_gmt":"2022-11-04T09:37:27","slug":"el-producto-de-dos-funciones-no-negativas-es-no-negativa","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/el-producto-de-dos-funciones-no-negativas-es-no-negativa\/","title":{"rendered":"El producto de dos funciones no negativas es no negativa"},"content":{"rendered":"<p>Demostrar que el producto de dos funciones no negativas es no negativa.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nvariables (f g : \u211d \u2192 \u211d)\n\ndef no_negativa (f : \u211d \u2192 \u211d) : Prop := \u2200 x, 0 \u2264 f x\n\nexample\n  (nnf : no_negativa f)\n  (nng : no_negativa g)\n  : no_negativa (f * g) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nvariables (f g : \u211d \u2192 \u211d)\n\ndef no_negativa (f : \u211d \u2192 \u211d) : Prop := \u2200 x, 0 \u2264 f x\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (nnf : no_negativa f)\n  (nng : no_negativa g)\n  : no_negativa (f * g) :=\nbegin\n  have h1 : \u2200x, 0 \u2264 f x * g x,\n  { intro x,\n    have h2: 0 \u2264 f x := nnf x,\n    have h3: 0 \u2264 g x := nng x,\n    show 0 \u2264 f x * g x,\n      by exact mul_nonneg (nnf x) (nng x), },\n  show no_negativa (f * g),\n    by exact h1,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (nnf : no_negativa f)\n  (nng : no_negativa g)\n  : no_negativa (f * g) :=\nbegin\n  intro x,\n  change 0 \u2264 f x * g x,\n  apply mul_nonneg,\n  apply nnf,\n  apply nng\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (nnf : no_negativa f)\n  (nng : no_negativa g)\n  : no_negativa (f * g) :=\n\u03bb x, mul_nonneg (nnf x) (nng x)\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_no_negativas.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 el producto de dos funciones no negativas es no negativa. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables (f g : \u211d \u2192 \u211d) def no_negativa (f : \u211d \u2192 \u211d) : Prop := \u2200 x, 0 \u2264 f x example (nnf : no_negativa f) (nng : no_negativa g) : no_negativa (f * g) := 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":[286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1173"}],"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=1173"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1173\/revisions"}],"predecessor-version":[{"id":1174,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1173\/revisions\/1174"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}