{"id":1120,"date":"2022-10-06T06:00:51","date_gmt":"2022-10-06T04:00:51","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1120"},"modified":"2022-09-26T17:55:45","modified_gmt":"2022-09-26T15:55:45","slug":"si-x-y-z-%e2%88%88-%e2%84%95-entonces-x-%e2%88%a3-y-x-z","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-x-y-z-%e2%88%88-%e2%84%95-entonces-x-%e2%88%a3-y-x-z\/","title":{"rendered":"Si x, y, z \u2208 \u2115, entonces x \u2223 y * x * z"},"content":{"rendered":"<p>Demostrar que si x, y, z \u2208 N, entonces<\/p>\n<pre lang=\"text\">\n   x \u2223 y * x * z\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.nat.basic\nvariables x y z : \u2115\n\nexample : x \u2223 y * x * z :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.nat.basic\nvariables x y z : \u2115\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2223 y * x * z :=\nbegin\n  have h1 : x \u2223 y * x,\n    { exact dvd_mul_left x y },\n  have h2 : (y * x) \u2223 (y * x * z),\n    { exact dvd.intro z rfl},\n  show x \u2223 y * x * z,\n    { exact dvd_trans h1 h2},\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2223 y * x * z :=\ndvd_trans (dvd_mul_left x y) (dvd.intro z rfl)\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : x \u2223 y * x * z :=\nbegin\n  apply dvd_mul_of_dvd_left,\n  apply dvd_mul_left\nend\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\/Divisivilidad_productos.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. 20.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si x, y, z \u2208 N, entonces x \u2223 y * x * z Para ello, completar la siguiente teor\u00eda de Lean: import data.nat.basic variables x y z : \u2115 example : x \u2223 y * x * z := 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":[291],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1120"}],"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=1120"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1120\/revisions"}],"predecessor-version":[{"id":1121,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1120\/revisions\/1121"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}