{"id":1234,"date":"2022-12-05T06:00:03","date_gmt":"2022-12-05T04:00:03","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1234"},"modified":"2022-12-04T09:26:06","modified_gmt":"2022-12-04T07:26:06","slug":"05-dic-22","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/05-dic-22\/","title":{"rendered":"Si a divide a b y a c, entonces tambi\u00e9n divide a b + c"},"content":{"rendered":"<p>Demostrar que si a divide a b y a c, entonces tambi\u00e9n divide a b + c.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariables {a b c : \u2115}\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariables {a b c : \u2115}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nbegin\n  rcases divab with \u27e8d, beq : b = a * d\u27e9,\n  rcases divac with \u27e8e, ceq: c = a * e\u27e9,\n  have h1 : b + c = a * (d + e),\n    calc b + c\n         = (a * d) + c       : congr_arg (+ c) beq\n     ... = (a * d) + (a * e) : congr_arg ((+) (a * d)) ceq\n     ... = a * (d + e)       : by rw \u2190 mul_add,\n  show a \u2223 (b + c),\n    by exact dvd.intro (d + e) (eq.symm h1),\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nbegin\n  rcases divab with \u27e8d, beq : b = a * d\u27e9,\n  rcases divac with \u27e8e, ceq: c = a * e\u27e9,\n  have h1 : b + c = a * (d + e), by linarith,\n  show a \u2223 (b + c),\n    by exact dvd.intro (d + e) (eq.symm h1),\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nbegin\n  rcases divab with \u27e8d, beq : b = a * d\u27e9,\n  rcases divac with \u27e8e, ceq: c = a * e\u27e9,\n  show a \u2223 (b + c),\n    by exact dvd.intro (d + e) (by linarith),\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nbegin\n  cases divab with d beq,\n  cases divac with e ceq,\n  rw [ceq, beq],\n  use (d + e),\n  ring\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (divab : a \u2223 b)\n  (divac : a \u2223 c)\n  : a \u2223 (b + c) :=\nbegin\n  rcases divab with \u27e8d, rfl\u27e9,\n  rcases divac with \u27e8e, rfl\u27e9,\n  use (d + e),\n  ring,\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\/Suma_divisible.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. 33.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a divide a b y a c, entonces tambi\u00e9n divide a b + c. Para ello, completar la siguiente teor\u00eda de Lean: import tactic variables {a b c : \u2115} example (divab : a \u2223 b) (divac : a \u2223 c) : a \u2223 (b + c) := 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\/1234"}],"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=1234"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1234\/revisions"}],"predecessor-version":[{"id":1237,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1234\/revisions\/1237"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}