{"id":1656,"date":"2023-10-12T06:00:53","date_gmt":"2023-10-12T04:00:53","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1656"},"modified":"2023-10-10T12:46:06","modified_gmt":"2023-10-10T10:46:06","slug":"12-oct-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/12-oct-23\/","title":{"rendered":"La composici\u00f3n de dos funciones mon\u00f3tonas es mon\u00f3tona"},"content":{"rendered":"<p>Demostrar con Lean4 que la composici\u00f3n de dos funciones mon\u00f3tonas 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 g : \u211d \u2192 \u211d)\n\nexample\n  (mf : Monotone f)\n  (mg : Monotone g)\n  : Monotone (f \u2218 g) :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nSean &#92;(f&#92;) y &#92;(g&#92;) dos funciones mon\u00f3tonas de &#92;(\u211d&#92;) en &#92;(\u211d&#92;). Tenemos que demostrar que &#92;(f \u2218 g&#92;) es mon\u00f3tona; es decir, que<br \/>\n&#92;[ (\u2200 a, b \u2208 \u211d) [a \u2264 b \u2192 (f \u2218 g)(a) \u2264 (f \u2218 g)(b)] &#92;]<br \/>\nSean &#92;(a, b \u2208 \u211d&#92;) tales que &#92;(a \u2264 b&#92;). Por ser &#92;(g&#92;) mon\u00f3tona, se tiene<br \/>\n&#92;[ g(a) \u2264 g(b) &#92;]<br \/>\ny, por ser f mon\u00f3tona, se tiene<br \/>\n&#92;[ f(g(a)) \u2264 f(g(b)) &#92;]<br \/>\nFinalmente, por la definici\u00f3n de composici\u00f3n,<br \/>\n&#92;[ (f \u2218 g)(a) \u2264 (f \u2218 g)(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 g : \u211d \u2192 \u211d)\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (mf : Monotone f)\n  (mg : Monotone g)\n  : Monotone (f \u2218 g) :=\nby\n  have h1 : \u2200 a b, a \u2264 b \u2192 (f \u2218 g) a \u2264 (f \u2218 g) b\n  { intros a b hab\n    have h1 : g a \u2264 g b := mg hab\n    show (f \u2218 g) a \u2264 (f \u2218 g) b\n    exact mf h1 }\n  show Monotone (f \u2218 g)\n  exact h1\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (mf : Monotone f)\n  (mg : Monotone g)\n  : Monotone (f \u2218 g) :=\nby\n  have h1 : \u2200 a b, a \u2264 b \u2192 (f \u2218 g) a \u2264 (f \u2218 g) b\n  { intros a b hab\n    show (f \u2218 g) a \u2264 (f \u2218 g) b\n    exact mf (mg hab) }\n  show Monotone (f \u2218 g)\n  exact h1\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (mf : Monotone f)\n  (mg : Monotone g)\n  : Monotone (f \u2218 g) :=\nby\n  -- a b : \u211d\n  -- hab : a \u2264 b\n  intros a b hab\n  -- (f \u2218 g) a \u2264 (f \u2218 g) b\n  apply mf\n  -- g a \u2264 g b\n  apply mg\n  -- a \u2264 b\n  apply hab\n\n-- 4\u00aa demostraci\u00f3n\nexample (mf : Monotone f) (mg : Monotone g) :\n  Monotone (f \u2218 g) :=\n\u03bb _ _ hab \u21a6 mf (mg hab)\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\/Composicion_de_funciones_monotonas.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 la composici\u00f3n de dos funciones mon\u00f3tonas es mon\u00f3tona. Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic variable (f g : \u211d \u2192 \u211d) example (mf : Monotone f) (mg : Monotone g) : Monotone (f \u2218 g) := 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":[301,297,286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1656"}],"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=1656"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1656\/revisions"}],"predecessor-version":[{"id":1708,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1656\/revisions\/1708"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1656"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1656"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}