{"id":535,"date":"2021-07-07T06:00:03","date_gmt":"2021-07-07T04:00:03","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=535"},"modified":"2021-07-04T17:05:44","modified_gmt":"2021-07-04T15:05:44","slug":"inverso-del-inverso-en-grupos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/inverso-del-inverso-en-grupos\/","title":{"rendered":"Inverso del inverso en grupos"},"content":{"rendered":"

Sea G un grupo y a \u2208 G. Demostrar que<\/p>\n

\n    (a\u207b\u00b9)\u207b\u00b9 = a\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport algebra.group.basic\n\nuniverse  u\nvariables {G : Type u} [group G]\nvariables {a b : G}\n\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\nsorry\n<\/pre>\n
[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n
\r\nimport algebra.group.basic\r\n\r\nuniverse  u\r\nvariables {G : Type u} [group G]\r\nvariables {a b : G}\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\r\ncalc (a\u207b\u00b9)\u207b\u00b9\r\n     = (a\u207b\u00b9)\u207b\u00b9 * 1         : (mul_one (a\u207b\u00b9)\u207b\u00b9).symm\r\n ... = (a\u207b\u00b9)\u207b\u00b9 * (a\u207b\u00b9 * a) : congr_arg ((*) (a\u207b\u00b9)\u207b\u00b9) (inv_mul_self a).symm\r\n ... = ((a\u207b\u00b9)\u207b\u00b9 * a\u207b\u00b9) * a : (mul_assoc _ _ _).symm\r\n ... = 1 * a               : congr_arg (* a) (inv_mul_self a\u207b\u00b9)\r\n ... = a                   : one_mul a\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\r\ncalc (a\u207b\u00b9)\u207b\u00b9\r\n     = (a\u207b\u00b9)\u207b\u00b9 * 1         : by simp only [mul_one]\r\n ... = (a\u207b\u00b9)\u207b\u00b9 * (a\u207b\u00b9 * a) : by simp only [inv_mul_self]\r\n ... = ((a\u207b\u00b9)\u207b\u00b9 * a\u207b\u00b9) * a : by simp only [mul_assoc]\r\n ... = 1 * a               : by simp only [inv_mul_self]\r\n ... = a                   : by simp only [one_mul]\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\r\ncalc (a\u207b\u00b9)\u207b\u00b9\r\n     = (a\u207b\u00b9)\u207b\u00b9 * 1         : by simp\r\n ... = (a\u207b\u00b9)\u207b\u00b9 * (a\u207b\u00b9 * a) : by simp\r\n ... = ((a\u207b\u00b9)\u207b\u00b9 * a\u207b\u00b9) * a : by simp\r\n ... = 1 * a               : by simp\r\n ... = a                   : by simp\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\r\nbegin\r\n  apply inv_eq_of_mul_eq_one,\r\n  exact mul_left_inv a,\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a :=\r\ninv_eq_of_mul_eq_one (mul_left_inv a)\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a:=\r\ninv_inv a\r\n\r\n-- 7\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a\u207b\u00b9)\u207b\u00b9 = a:=\r\nby simp\r\n<\/pre>\n
Se puede interactuar con la prueba anterior en esta sesi\u00f3n con Lean<\/a>.<\/p>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="lean"> y otra con <\/pre>
\n[\/expand]<\/p>\n
[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n
\r\ntheory Inverso_del_inverso_en_grupos\r\nimports Main\r\nbegin\r\n\r\ncontext group\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\nproof -\r\n  have \"inverse (inverse a) =\r\n        (inverse (inverse a)) * 1\"\r\n    by (simp only: right_neutral)\r\n  also have \"\u2026 = inverse (inverse a) * (inverse a * a)\"\r\n    by (simp only: left_inverse)\r\n  also have \"\u2026 = (inverse (inverse a) * inverse a) * a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = 1 * a\"\r\n    by (simp only: left_inverse)\r\n  also have \"\u2026 = a\"\r\n    by (simp only: left_neutral)\r\n  finally show \"inverse (inverse a) = a\"\r\n    by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\nproof -\r\n  have \"inverse (inverse a) =\r\n        (inverse (inverse a)) * 1\"                      by simp\r\n  also have \"\u2026 = inverse (inverse a) * (inverse a * a)\" by simp\r\n  also have \"\u2026 = (inverse (inverse a) * inverse a) * a\" by simp\r\n  also have \"\u2026 = 1 * a\"                                 by simp\r\n  finally show \"inverse (inverse a) = a\"                by simp\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\nproof (rule inverse_unique)\r\n  show \"inverse a * a = 1\"\r\n    by (simp only: left_inverse)\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\nproof (rule inverse_unique)\r\n  show \"inverse a * a = 1\" by simp\r\nqed\r\n\r\n(* 5\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\n  by (rule inverse_unique) simp\r\n\r\n(* 6\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\n  by (simp only: inverse_inverse)\r\n\r\n(* 7\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (inverse a) = a\"\r\n  by simp\r\n\r\nend\r\n\r\nend\r\n<\/pre>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="isar"> y otra con <\/pre>
\n[\/expand]<\/p>\n
Referencia<\/h4>\n
Propiedad 3.20 del libro Abstract algebra: Theory and applications<\/a> de Thomas W. Judson.<\/p>\n","protected":false},"excerpt":{"rendered":"
Sea G un grupo y a \u2208 G. Demostrar que (a\u207b\u00b9)\u207b\u00b9 = a Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group.basic universe u variables {G : Type u} [group G] variables {a b : G} example : (a\u207b\u00b9)\u207b\u00b9 = a := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import algebra.group.basic universe u variables {G : Type u} [group G] variables {a b : G} — 1\u00aa demostraci\u00f3n — =============== example : (a\u207b\u00b9)\u207b\u00b9 = a := calc (a\u207b\u00b9)\u207b\u00b9 = (a\u207b\u00b9)\u207b\u00b9 * 1 : (mul_one (a\u207b\u00b9)\u207b\u00b9).symm … = (a\u207b\u00b9)\u207b\u00b9 * (a\u207b\u00b9 * a) : congr_arg ((*) (a\u207b\u00b9)\u207b\u00b9) (inv_mul_self a).symm … = ((a\u207b\u00b9)\u207b\u00b9 * a\u207b\u00b9) * a : (mul_assoc _ _ _).symm … = 1 * a : congr_arg (* a) (inv_mul_self a\u207b\u00b9) … = a : one_mul…<\/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":[11],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/535"}],"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=535"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/535\/revisions"}],"predecessor-version":[{"id":558,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/535\/revisions\/558"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=535"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=535"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}