{"id":1126,"date":"2022-10-11T06:00:41","date_gmt":"2022-10-11T04:00:41","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1126"},"modified":"2022-10-05T12:05:07","modified_gmt":"2022-10-05T10:05:07","slug":"si-m-n-%e2%88%88-%e2%84%95-entonces-gcdmn-gcdnm","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-m-n-%e2%88%88-%e2%84%95-entonces-gcdmn-gcdnm\/","title":{"rendered":"Si m, n \u2208 \u2115, entonces gcd(m,n) = gcd(n,m)"},"content":{"rendered":"<p>Demostrar que si m, n \u2208 \u2115, entonces gcd(m,n) = gcd(n,m).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.nat.gcd\nopen nat\nvariables k m n : \u2115\n\nexample : gcd m n = gcd n m :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.nat.gcd\n\nopen nat\n\nvariables k m n : \u2115\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : gcd m n = gcd n m :=\nbegin\n  have h1 : gcd m n \u2223 gcd n m,\n    { have h1a : gcd m n \u2223 n,\n        by exact gcd_dvd_right m n,\n      have h1b : gcd m n \u2223 m,\n        by exact gcd_dvd_left m n,\n      show gcd m n \u2223 gcd n m,\n        by exact dvd_gcd h1a h1b, },\n  have h2 : gcd n m \u2223 gcd m n,\n    { have h2a : gcd n m \u2223 m,\n        by exact gcd_dvd_right n m,\n      have h2b : gcd n m \u2223 n,\n        by exact gcd_dvd_left n m,\n      show gcd n m \u2223 gcd m n,\n        by exact dvd_gcd h2a h2b, },\n  show gcd m n = gcd n m,\n    by exact dvd_antisymm h1 h2,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : gcd m n = gcd n m :=\nbegin\n  apply dvd_antisymm,\n  { apply dvd_gcd,\n    { exact gcd_dvd_right m n, },\n    { exact gcd_dvd_left m n, }},\n  { apply dvd_gcd,\n    { exact gcd_dvd_right n m, },\n    { exact gcd_dvd_left n m, }},\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : gcd m n = gcd n m :=\nbegin\n  apply dvd_antisymm,\n  { apply dvd_gcd (gcd_dvd_right m n) (gcd_dvd_left m n)},\n  { apply dvd_gcd (gcd_dvd_right n m) (gcd_dvd_left n m)},\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : gcd m n = gcd n m :=\ndvd_antisymm\n  (dvd_gcd (gcd_dvd_right m n) (gcd_dvd_left m n))\n  (dvd_gcd (gcd_dvd_right n m) (gcd_dvd_left n m))\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux : gcd m n \u2223 gcd n m :=\nbegin\n  have h1 : gcd m n \u2223 n,\n    by exact gcd_dvd_right m n,\n  have h2 : gcd m n \u2223 m,\n    by exact gcd_dvd_left m n,\n  show gcd m n \u2223 gcd n m,\n    by exact dvd_gcd h1 h2,\nend\n\nexample : gcd m n = gcd n m :=\nbegin\n  apply dvd_antisymm,\n  { exact aux m n, },\n  { exact aux n m, },\nend\n\n-- 6\u00aa demostraci\u00f3n\n-- ===============\n\nlemma aux2 : gcd m n \u2223 gcd n m :=\ndvd_gcd (gcd_dvd_right m n) (gcd_dvd_left m n)\n\nexample : gcd m n = gcd n m :=\ndvd_antisymm (aux2 m n) (aux2 n m)\n\n-- 7\u00aa demostraci\u00f3n\n-- ===============\n\nexample : gcd m n = gcd n m :=\n-- by library_search\ngcd_comm m n\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\/Conmutatividad_del_gcd.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. 21.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si m, n \u2208 \u2115, entonces gcd(m,n) = gcd(n,m). Para ello, completar la siguiente teor\u00eda de Lean: import data.nat.gcd open nat variables k m n : \u2115 example : gcd m n = gcd n m := 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":[292],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1126"}],"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=1126"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1126\/revisions"}],"predecessor-version":[{"id":1127,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1126\/revisions\/1127"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}