{"id":1890,"date":"2023-12-29T06:00:44","date_gmt":"2023-12-29T04:00:44","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1890"},"modified":"2023-12-29T09:11:36","modified_gmt":"2023-12-29T07:11:36","slug":"29-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/29-dic-23\/","title":{"rendered":"3 divide al m\u00e1ximo com\u00fan divisor de 6 y 15"},"content":{"rendered":"\n<p>Demostrar con Lean4 que 3 divide al m\u00e1ximo com\u00fan divisor de 6 y 15.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nimport Mathlib.Data.Nat.GCD.Basic\n\nopen Nat\n\nexample : 3 \u2223 gcd 6 15 :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se usar\u00e1 el siguiente lema<br \/>\n&#92;[ (\u2200 k, m, n \u2208 \u2115)[k \u2223 \\gcd(m, n) \u2194 k \u2223 m \u2227 k \u2223 n] &#92;]<\/p>\n<p>Por el lema,<br \/>\n&#92;[ 3 \u2223 \\gcd(6, 15) &#92;]<br \/>\nse reduce a<br \/>\n&#92;[ 3 \u2223 6 \u2227 3 \u2223 15 &#92;]<br \/>\nque se verifican f\u00e1cilmente.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\nimport Mathlib.Data.Nat.GCD.Basic\n\nopen Nat\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 3 \u2223 gcd 6 15 :=\nby\n  rw [dvd_gcd_iff]\n  -- \u22a2 3 \u2223 6 \u2227 3 \u2223 15\n  constructor\n  . -- 3 \u2223 6\n    norm_num\n  . -- \u22a2 3 \u2223 15\n    norm_num\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 3 \u2223 gcd 6 15 :=\nby\n  rw [dvd_gcd_iff]\n  -- \u22a2 3 \u2223 6 \u2227 3 \u2223 15\n  constructor <;> norm_num\n\n-- Lemas usados\n-- ============\n\n-- variable (k m n : \u2115)\n-- #check (dvd_gcd_iff : k \u2223 gcd m n \u2194 k \u2223 m \u2227 k \u2223 n)\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\/Divisor_del_mcd.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. 37.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que 3 divide al m\u00e1ximo com\u00fan divisor de 6 y 15. Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.GCD.Basic open Nat example : 3 \u2223 gcd 6 15 := 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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1890"}],"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=1890"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1890\/revisions"}],"predecessor-version":[{"id":1893,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1890\/revisions\/1893"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1890"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1890"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1890"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}