{"id":2017,"date":"2024-01-30T06:00:37","date_gmt":"2024-01-30T04:00:37","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2017"},"modified":"2024-02-17T09:31:16","modified_gmt":"2024-02-17T07:31:16","slug":"30-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/30-ene-24\/","title":{"rendered":"Las funciones f(x,y) = (x + y)\u00b2 y g(x,y) = x\u00b2 + 2xy + y\u00b2 son iguales"},"content":{"rendered":"\n<p>Demostrar con Lean4 que las funciones &#92;(f(x,y) = (x + y)\u00b2&#92;) y &#92;(g(x,y) = x\u00b2 + 2xy + y&#92;)\u00b2 son iguales.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport  Mathlib.Data.Real.Basic\n\nexample : (fun x y : \u211d \u21a6 (x + y)^2) = (fun x y : \u211d \u21a6 x^2 + 2*x*y + y^2) :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<pre lang=\"lean\">\nimport  Mathlib.Data.Real.Basic\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (fun x y : \u211d \u21a6 (x + y)^2) = (fun x y : \u211d \u21a6 x^2 + 2*x*y + y^2) :=\nby\n  ext u v\n  -- u v : \u211d\n  -- \u22a2 (u + v) ^ 2 = u ^ 2 + 2 * u * v + v ^ 2\n  ring\n\n-- Comentario: La t\u00e1ctica ext transforma las conclusiones de la forma\n-- (fun x \u21a6 f x) = (fun x \u21a6 g x) en f x = g x.\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (fun x y : \u211d \u21a6 (x + y)^2) = (fun x y : \u211d \u21a6 x^2 + 2*x*y + y^2) :=\nby { ext ; ring }\n<\/pre>\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\/Demostracion_por_extensionalidad.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<h3>Referencias<\/h3>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 41.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que las funciones f(x,y) = (x + y)\u00b2 y g(x,y) = x\u00b2 + 2xy + y\u00b2 son iguales.<\/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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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":[17],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2017"}],"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=2017"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2017\/revisions"}],"predecessor-version":[{"id":2073,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2017\/revisions\/2073"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}