{"id":1028,"date":"2022-09-13T06:00:00","date_gmt":"2022-09-13T04:00:00","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1028"},"modified":"2022-09-11T16:25:20","modified_gmt":"2022-09-11T14:25:20","slug":"si-r-es-un-anillo-y-a-%e2%88%88-r-entonces-2-a-a-a","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-r-es-un-anillo-y-a-%e2%88%88-r-entonces-2-a-a-a\/","title":{"rendered":"Si R es un anillo y a \u2208 R, entonces 2 * a = a + a"},"content":{"rendered":"<p>Demostrar que si R es un anillo y a \u2208 R, entonces<\/p>\n<pre lang=\"text\">\n2 * a = a + a.\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariables a : R\n\nexample : 2 * a = a + a :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.ring\n\nvariables {R : Type*} [ring R]\nvariables a : R\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 2 * a = a + a :=\ncalc\n  2 * a = (1 + 1) * a   : congr_fun (congr_arg has_mul.mul one_add_one_eq_two.symm) a\n  ...   = 1 * a + 1 * a : add_mul 1 1 a\n  ...   = a + 1 * a     : congr_arg (\u03bb x, x + 1 * a) (one_mul a)\n  ...   = a + a         : congr_arg (\u03bb x, a + x) (one_mul a)\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 2 * a = a + a :=\ncalc\n  2 * a = (1 + 1) * a   : by rw one_add_one_eq_two\n  ...   = 1 * a + 1 * a : by rw add_mul\n  ...   = a + a         : by rw one_mul\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 2 * a = a + a :=\nby rw [one_add_one_eq_two.symm, add_mul, one_mul]\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 2 * a = a + a :=\ncalc\n  2 * a = (1 + 1)  * a  : rfl\n  ...   = 1 * a + 1 * a : by simp [add_mul]\n  ...   = a + a         : by simp\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : 2 * a = a + a :=\n-- by library_search\ntwo_mul a\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\/Dos_por_a_igual_a_mas_a.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. 13.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si R es un anillo y a \u2208 R, entonces 2 * a = a + a. Para ello, completar la siguiente teor\u00eda de Lean: import algebra.ring variables {R : Type*} [ring R] variables a : R example : 2 * a = a + a := 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":[284],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1028"}],"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=1028"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1028\/revisions"}],"predecessor-version":[{"id":1095,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1028\/revisions\/1095"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1028"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1028"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1028"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}