{"id":533,"date":"2021-07-06T06:00:34","date_gmt":"2021-07-06T04:00:34","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=533"},"modified":"2021-07-04T17:05:18","modified_gmt":"2021-07-04T15:05:18","slug":"inverso-del-producto","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/inverso-del-producto\/","title":{"rendered":"Inverso del producto"},"content":{"rendered":"

Sea G un grupo y a, b \u2208 G. Entonces,<\/p>\n

\n   (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\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 * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\nbegin\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 * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\r\nbegin\r\n  apply inv_eq_of_mul_eq_one,\r\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\r\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : (mul_assoc _ _ _).symm\r\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : congr_arg (* a\u207b\u00b9) (mul_assoc a _ _)\r\n   ... = (a * 1) * a\u207b\u00b9         : congr_arg2 _ (congr_arg _ (mul_inv_self b)) rfl\r\n   ... = a * a\u207b\u00b9               : congr_arg (* a\u207b\u00b9) (mul_one a)\r\n   ... = 1                     : mul_inv_self a\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\r\nbegin\r\n  apply inv_eq_of_mul_eq_one,\r\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\r\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : by simp only [mul_assoc]\r\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : by simp only [mul_assoc]\r\n   ... = (a * 1) * a\u207b\u00b9         : by simp only [mul_inv_self]\r\n   ... = a * a\u207b\u00b9               : by simp only [mul_one]\r\n   ... = 1                     : by simp only [mul_inv_self]\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\r\nbegin\r\n  apply inv_eq_of_mul_eq_one,\r\n  calc a * b * (b\u207b\u00b9 * a\u207b\u00b9)\r\n       = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : by simp [mul_assoc]\r\n   ... = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : by simp\r\n   ... = (a * 1) * a\u207b\u00b9         : by simp\r\n   ... = a * a\u207b\u00b9               : by simp\r\n   ... = 1                     : by simp,\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\r\nmul_inv_rev a b\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 :=\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_producto\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 (a * b) = inverse b * inverse a\"\r\nproof (rule inverse_unique)\r\n  have \"(a * b) * (inverse b * inverse a) =\r\n        ((a * b) * inverse b) * inverse a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (a * (b * inverse b)) * inverse a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (a * 1) * inverse a\"\r\n    by (simp only: right_inverse)\r\n  also have \"\u2026 = a * inverse a\"\r\n    by (simp only: right_neutral)\r\n  also have \"\u2026 = 1\"\r\n    by (simp only: right_inverse)\r\n  finally show \"a * b * (inverse b * inverse a) = 1\"\r\n    by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (a * b) = inverse b * inverse a\"\r\nproof (rule inverse_unique)\r\n  have \"(a * b) * (inverse b * inverse a) =\r\n        ((a * b) * inverse b) * inverse a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (a * (b * inverse b)) * inverse a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (a * 1) * inverse a\"\r\n    by simp\r\n  also have \"\u2026 = a * inverse a\"\r\n    by simp\r\n  also have \"\u2026 = 1\"\r\n    by simp\r\n  finally show \"a * b * (inverse b * inverse a) = 1\"\r\n    .\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (a * b) = inverse b * inverse a\"\r\nproof (rule inverse_unique)\r\n  have \"a * b * (inverse b * inverse a) =\r\n        a * (b * inverse b) * inverse a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = 1\"\r\n    by simp\r\n  finally show \"a * b * (inverse b * inverse a) = 1\" .\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma \"inverse (a * b) = inverse b * inverse a\"\r\n  by (simp only: inverse_distrib_swap)\r\n\r\n(* ?\u00aa demostraci\u00f3n *)\r\n\r\nend\r\n\r\nend\r\n\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 3.19 del libro Abstract algebra: Theory and applications<\/a> de Thomas W. Judson.<\/p>\n","protected":false},"excerpt":{"rendered":"
Sea G 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.basic universe u variables {G : Type u} [group G] variables {a b : G} example : (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 := begin 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 * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9 := begin apply inv_eq_of_mul_eq_one, calc a * b * (b\u207b\u00b9 * a\u207b\u00b9) = ((a * b) * b\u207b\u00b9) * a\u207b\u00b9 : (mul_assoc _ _ _).symm … = (a * (b * b\u207b\u00b9)) * a\u207b\u00b9 : congr_arg (* a\u207b\u00b9)…<\/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\/533"}],"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=533"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/533\/revisions"}],"predecessor-version":[{"id":557,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/533\/revisions\/557"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=533"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=533"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}