{"id":1034,"date":"2022-09-16T06:00:35","date_gmt":"2022-09-16T04:00:35","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1034"},"modified":"2022-09-11T16:30:29","modified_gmt":"2022-09-11T14:30:29","slug":"si-g-es-un-grupo-y-a-b-%e2%88%88-g-tales-que-b-a-1-entonces-a%e2%81%bb%c2%b9-b","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-g-es-un-grupo-y-a-b-%e2%88%88-g-tales-que-b-a-1-entonces-a%e2%81%bb%c2%b9-b\/","title":{"rendered":"Si G es un grupo y a, b \u2208 G tales que b * a = 1, entonces a\u207b\u00b9 = b"},"content":{"rendered":"<p>Demostrar que si G es un grupo y a, b \u2208 G tales que<\/p>\n<pre lang=\"text\">\nb * a = 1\n<\/pre>\n<p>entonces<\/p>\n<pre lang=\"text\">\na\u207b\u00b9 = b\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\n  (h : b * a = 1)\n  : a\u207b\u00b9 = b :=\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\n  (h : b * a = 1)\n  : a\u207b\u00b9 = b :=\ncalc\n  a\u207b\u00b9 =  1 * a\u207b\u00b9       : (one_mul a\u207b\u00b9).symm\n  ... =  (b * a) * a\u207b\u00b9 : congr_arg (\u03bb x, x * a\u207b\u00b9) h.symm\n  ... =  b * (a * a\u207b\u00b9) : mul_assoc b a a\u207b\u00b9\n  ... =  b * 1         : congr_arg (\u03bb x, b * x) (mul_right_inv a)\n  ... =  b             : mul_one b\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : b * a = 1)\n  : a\u207b\u00b9 = b :=\ncalc\n  a\u207b\u00b9 =  1 * a\u207b\u00b9       : by rw one_mul\n  ... =  (b * a) * a\u207b\u00b9 : by rw h\n  ... =  b * (a * a\u207b\u00b9) : by rw mul_assoc\n  ... =  b * 1         : by rw mul_right_inv\n  ... =  b             : by rw mul_one\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : b * a = 1)\n  : a\u207b\u00b9 = b :=\ncalc\n  a\u207b\u00b9 =  1 * a\u207b\u00b9       : by simp\n  ... =  (b * a) * a\u207b\u00b9 : by simp [h]\n  ... =  b * (a * a\u207b\u00b9) : by simp\n  ... =  b * 1         : by simp\n  ... =  b             : by simp\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : b * a = 1)\n  : a\u207b\u00b9 = b :=\n-- by library_search\ninv_eq_of_mul_eq_one_left h\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\/CS_inverso_izquierda.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>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si G es un grupo y a, b \u2208 G tales que b * a = 1 entonces a\u207b\u00b9 = b Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group variables {G : Type*} [group G] variables a b : G example (h : b * a = 1) : a\u207b\u00b9 = b := 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\/1034"}],"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=1034"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1034\/revisions"}],"predecessor-version":[{"id":1099,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1034\/revisions\/1099"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1034"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1034"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1034"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}