{"id":1654,"date":"2023-10-11T06:00:14","date_gmt":"2023-10-11T04:00:14","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1654"},"modified":"2023-10-10T12:44:28","modified_gmt":"2023-10-10T10:44:28","slug":"11-oct-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/11-oct-23\/","title":{"rendered":"Si c es no negativo y f es mon\u00f3tona, entonces cf es mon\u00f3tona"},"content":{"rendered":"<p>Demostrar con Lean4 que si &#92;(c&#92;) es no negativo y &#92;(f&#92;) es mon\u00f3tona, entonces &#92;(cf&#92;) es mon\u00f3tona.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nvariable (f : \u211d \u2192 \u211d)\nvariable {c : \u211d}\n\nexample\n  (mf : Monotone f)\n  (nnc : 0 \u2264 c)\n  : Monotone (fun x \u21a6 c * f x) :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nSe usar\u00e1 el Lema<br \/>\n&#92;[ &#92;{b \u2264 c, 0 \u2264 a&#92;} \u22a2 ab \u2264 ac &#92;tag{L1} &#92;]<\/p>\n<p>Tenemos que demostrar que<br \/>\n&#92;[ (\u2200 a, b \u2208 \u211d) [a \u2264 b \u2192 (cf)(a) \u2264 (cf)(b)] &#92;]<br \/>\nSean &#92;(a, b \u2208 \u211d&#92;) tales que &#92;(a \u2264 b&#92;). Puesto que &#92;(f&#92;) es mon\u00f3tona, se tiene<br \/>\n&#92;[ f(a) \u2264 f(b) &#92;]<br \/>\ny, junto con la hip\u00f3tesis de que &#92;(c&#92;) es no negativo, usando el lema L1, se tiene que<br \/>\n&#92;[ cf(a) \u2264 cf(b) &#92;]<br \/>\nque es lo que hab\u00eda que demostrar.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nvariable (f : \u211d \u2192 \u211d)\nvariable {c : \u211d}\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (mf : Monotone f)\n  (nnc : 0 \u2264 c)\n  : Monotone (fun x \u21a6 c * f x) :=\nby\n  have h1 : \u2200 a b, a \u2264 b \u2192 (fun x \u21a6 c * f x) a \u2264 (fun x \u21a6 c * f x) b\n  { intros a b hab\n    have h2 : f a \u2264 f b := mf hab\n    show (fun x \u21a6 c * f x) a \u2264 (fun x \u21a6 c * f x) b\n    exact mul_le_mul_of_nonneg_left h2 nnc }\n  show Monotone (fun x \u21a6 c * f x)\n  exact h1\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (mf : Monotone f)\n  (nnc : 0 \u2264 c)\n  : Monotone (fun x \u21a6 c * f x) :=\nby\n  -- a b : \u211d\n  -- hab : a \u2264 b\n  intros a b hab\n  -- (fun x => c * f x) a \u2264 (fun x => c * f x) b\n  apply mul_le_mul_of_nonneg_left\n  . -- f a \u2264 f b\n    apply mf hab\n  . -- 0 \u2264 c\n    apply nnc\n\n-- 3\u00aa demostraci\u00f3n\nexample (mf : Monotone f) (nnc : 0 \u2264 c) :\n  Monotone (fun x \u21a6 c * f x) :=\n\u03bb _ _ hab \u21a6 mul_le_mul_of_nonneg_left (mf hab) nnc\n\n-- Lemas usados\n-- ============\n\n-- variable (a b : \u211d)\n-- #check (mul_le_mul_of_nonneg_left : b \u2264 c \u2192 0 \u2264 a \u2192 a * b \u2264 a * c)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Producto_de_un_positivo_por_una_funcion_monotona.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 26.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si &#92;(c&#92;) es no negativo y &#92;(f&#92;) es mon\u00f3tona, entonces &#92;(cf&#92;) es mon\u00f3tona. Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic variable (f : \u211d \u2192 \u211d) variable {c : \u211d} example (mf : Monotone f) (nnc : 0 \u2264 c) : Monotone (fun x \u21a6 c * f x) := by 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":[301,297],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1654"}],"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=1654"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1654\/revisions"}],"predecessor-version":[{"id":1707,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1654\/revisions\/1707"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}