{"id":1190,"date":"2022-11-10T18:36:41","date_gmt":"2022-11-10T16:36:41","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1190"},"modified":"2022-11-10T18:37:23","modified_gmt":"2022-11-10T16:37:23","slug":"10-nov-22","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/10-nov-22\/","title":{"rendered":"La composici\u00f3n de dos funciones mon\u00f3tonas es mon\u00f3tona"},"content":{"rendered":"<p>Demostrar que la composici\u00f3n de dos funciones mon\u00f3tonas es mon\u00f3tona.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables (f g : \u211d \u2192 \u211d)\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\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 g : \u211d \u2192 \u211d)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\nbegin\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      have h2 : f (g a) \u2264 f (g b) := mf h1,\n      show (f \u2218 g) a \u2264 (f \u2218 g) b,\n        by exact h2, },\n  show monotone (f \u2218 g),\n    by exact h1,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\nbegin\n  intros a b hab,\n  apply mf,\n  apply mg,\n  apply hab\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\n\u03bb a b hab, mf (mg hab)\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\n-- by library_search\nmonotone.comp mf mg\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (mf : monotone f)\n  (mg : monotone g)\n  : monotone (f \u2218 g) :=\n-- by hint\nby tauto\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\/Composicion_de_funciones_monotonas.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. 28.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que la composici\u00f3n de dos funciones mon\u00f3tonas es mon\u00f3tona. Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables (f g : \u211d \u2192 \u211d) example (mf : monotone f) (mg : monotone g) : monotone (f \u2218 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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1190"}],"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=1190"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1190\/revisions"}],"predecessor-version":[{"id":1192,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1190\/revisions\/1192"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1190"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1190"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}