{"id":1036,"date":"2022-09-19T06:00:41","date_gmt":"2022-09-19T04:00:41","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1036"},"modified":"2022-09-11T16:31:29","modified_gmt":"2022-09-11T14:31:29","slug":"si-g-es-un-grupo-y-a-b-%e2%88%88-g-entonces-a-b%e2%81%bb%c2%b9-b%e2%81%bb%c2%b9-a%e2%81%bb%c2%b9","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-g-es-un-grupo-y-a-b-%e2%88%88-g-entonces-a-b%e2%81%bb%c2%b9-b%e2%81%bb%c2%b9-a%e2%81%bb%c2%b9\/","title":{"rendered":"Si G es un grupo y a, b \u2208 G, entonces (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9"},"content":{"rendered":"<p>Demostrar que si G es un grupo y a, b \u2208 G, entonces<\/p>\n<pre lang=\"text\">\n(a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.group\nvariables {G : Type*} [group G]\nvariables a b : G\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport algebra.group\nvariables {G : Type*} [group G]\nvariables a b : G\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nbegin\n  apply mul_eq_one_iff_inv_eq.mp,\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : (mul_assoc _ _ _).symm\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : congr_arg (* a\u207b\u00b9) (mul_assoc a _ _)\n   ... = (a * 1) * a\u207b\u00b9         : congr_arg2 _ (congr_arg _ (mul_inv_self b)) rfl\n   ... = a * a\u207b\u00b9               : congr_arg (* a\u207b\u00b9) (mul_one a)\n   ... = 1                     : mul_inv_self a\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nbegin\n  apply mul_eq_one_iff_inv_eq.mp,\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : by simp only [mul_assoc]\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : by simp only [mul_assoc]\n   ... = (a * 1) * a\u207b\u00b9         : by simp only [mul_inv_self]\n   ... = a * a\u207b\u00b9               : by simp only [mul_one]\n   ... = 1                     : by simp only [mul_inv_self]\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nbegin\n  apply mul_eq_one_iff_inv_eq.mp,\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : by simp [mul_assoc]\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : by simp\n   ... = (a * 1) * a\u207b\u00b9         : by simp\n   ... = a * a\u207b\u00b9               : by simp\n   ... = 1                     : by simp,\nend\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\n-- by library_search\nmul_inv_rev a b\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nby simp\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\/Inverso_del_producto.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. 14.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si G es un grupo y a, b \u2208 G, entonces (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group variables {G : Type*} [group G] variables a b : G example : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 := 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":[285],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1036"}],"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=1036"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1036\/revisions"}],"predecessor-version":[{"id":1100,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1036\/revisions\/1100"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}