{"id":514,"date":"2021-07-01T06:00:58","date_gmt":"2021-07-01T04:00:58","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=514"},"modified":"2021-06-22T12:00:46","modified_gmt":"2021-06-22T10:00:46","slug":"equivalencia-de-inversos-iguales-al-neutro","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/equivalencia-de-inversos-iguales-al-neutro\/","title":{"rendered":"Equivalencia de inversos iguales al neutro"},"content":{"rendered":"<p>Sea M un monoide y a, b \u2208 M tales que a * b = 1. Demostrar que a = 1 si y s\u00f3lo si b = 1.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.group.basic\n\nvariables {M : Type} [monoid M]\nvariables {a b : M}\n\nexample\n  (h : a * b = 1)\n  : a = 1 \u2194 b = 1 :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport algebra.group.basic\r\n\r\nvariables {M : Type} [monoid M]\r\nvariables {a b : M}\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nbegin\r\n  split,\r\n  { intro a1,\r\n    rw a1 at h,\r\n    rw one_mul at h,\r\n    exact h, },\r\n  { intro b1,\r\n    rw b1 at h,\r\n    rw mul_one at h,\r\n    exact h, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nbegin\r\n  split,\r\n  { intro a1,\r\n    calc b = 1 * b : (one_mul b).symm\r\n       ... = a * b : congr_arg (* b) a1.symm\r\n       ... = 1     : h, },\r\n  { intro b1,\r\n    calc a = a * 1 : (mul_one a).symm\r\n       ... = a * b : congr_arg ((*) a) b1.symm\r\n       ... = 1     : h, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nbegin\r\n  split,\r\n  { rintro rfl,\r\n    simpa using h, },\r\n  { rintro rfl,\r\n    simpa using h, },\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nby split ; { rintro rfl, simpa using h }\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nby split ; finish\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\nby finish [iff_def]\r\n\r\n-- 7\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : a * b = 1)\r\n  : a = 1 \u2194 b = 1 :=\r\neq_one_iff_eq_one_of_mul_eq_one h\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/www.cs.us.es\/~jalonso\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Equivalencia_de_inversos_iguales_al_neutro.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>,<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory Equivalencia_de_inversos_iguales_al_neutro\r\nimports Main\r\nbegin\r\n\r\ncontext monoid\r\nbegin\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"a * b = 1\"\r\n  shows   \"a = 1 \u27f7 b = 1\"\r\nproof (rule iffI)\r\n  assume \"a = 1\"\r\n  have \"b = 1 * b\"      by (simp only: left_neutral)\r\n  also have \"\u2026 = a * b\" by (simp only: \u2039a = 1\u203a)\r\n  also have \"\u2026 = 1\"     by (simp only: \u2039a * b = 1\u203a)\r\n  finally show \"b = 1\"  by this\r\nnext\r\n  assume \"b = 1\"\r\n  have \"a = a * 1\"      by (simp only: right_neutral)\r\n  also have \"\u2026 = a * b\" by (simp only: \u2039b = 1\u203a)\r\n  also have \"\u2026 = 1\"     by (simp only: \u2039a * b = 1\u203a)\r\n  finally show \"a = 1\"  by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"a * b = 1\"\r\n  shows   \"a = 1 \u27f7 b = 1\"\r\nproof\r\n  assume \"a = 1\"\r\n  have \"b = 1 * b\"      by simp\r\n  also have \"\u2026 = a * b\" using \u2039a = 1\u203a by simp\r\n  also have \"\u2026 = 1\"     using \u2039a * b = 1\u203a by simp\r\n  finally show \"b = 1\"  .\r\nnext\r\n  assume \"b = 1\"\r\n  have \"a = a * 1\"      by simp\r\n  also have \"\u2026 = a * b\" using \u2039b = 1\u203a by simp\r\n  also have \"\u2026 = 1\"     using \u2039a * b = 1\u203a by simp\r\n  finally show \"a = 1\"  .\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"a * b = 1\"\r\n  shows   \"a = 1 \u27f7 b = 1\"\r\n  by (metis assms left_neutral right_neutral)\r\n\r\nend\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sea M un monoide y a, b \u2208 M tales que a * b = 1. Demostrar que a = 1 si y s\u00f3lo si b = 1. Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group.basic variables {M : Type} [monoid M] variables {a b : M} example (h : a * b = 1) : a = 1 \u2194 b = 1 := sorry [expand title=\u00bbSoluciones con Lean\u00bb] import algebra.group.basic variables {M : Type} [monoid M] variables {a b : M} &#8212; 1\u00aa demostraci\u00f3n &#8212; =============== example (h : a * b = 1) : a = 1 \u2194 b = 1 := begin split, { intro a1, rw a1 at h, rw one_mul at h, exact h, }, { intro b1,&#8230;<\/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":[9],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/514"}],"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=514"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/514\/revisions"}],"predecessor-version":[{"id":515,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/514\/revisions\/515"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}