{"id":547,"date":"2021-07-10T06:00:56","date_gmt":"2021-07-10T04:00:56","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=547"},"modified":"2021-07-04T17:07:10","modified_gmt":"2021-07-04T15:07:10","slug":"los-monoides-booleanos-son-conmutativos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/los-monoides-booleanos-son-conmutativos\/","title":{"rendered":"Los monoides booleanos son conmutativos"},"content":{"rendered":"<p>Un <a href=\"https:\/\/en.wikipedia.org\/wiki\/Monoid\">monoide<\/a> es un conjunto junto con una operaci\u00f3n binaria que es asociativa y tiene elemento neutro.<\/p>\n<p>Un monoide M es booleano si<\/p>\n<pre lang=\"text\">\n   \u2200 x \u2208 M, x * x = 1\n<\/pre>\n<p>y es conmutativo si<\/p>\n<pre lang=\"text\">\n   \u2200 x y \u2208 M, x * y = y * x\n<\/pre>\n<p>En Lean, est\u00e1 definida la clase de los monoides (como <code>monoid<\/code>) y sus propiedades caracter\u00edsticas son<\/p>\n<pre lang=\"text\">\n   mul_assoc : (a * b) * c = a * (b * c)\n   one_mul :   1 * a = a\n   mul_one :   a * 1 = a\n<\/pre>\n<p>Demostrar que los monoides booleanos son conmutativos.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.group.basic\n\nuniverse  u\nvariables {M : Type u} [monoid M]\n\nexample\n  (h : \u2200 x : M, x * x = 1)\n  : \u2200 x y : M, x * y = y * x :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport algebra.group.basic\r\n\r\nuniverse  u\r\nvariables {M : Type u} [monoid M]\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : \u2200 x : M, x * x = 1)\r\n  : \u2200 x y : M, x * y = y * x :=\r\nbegin\r\n  intros a b,\r\n  calc a * b\r\n       = (a * b) * 1                   : (mul_one (a * b)).symm\r\n   ... = (a * b) * (a * a)             : congr_arg ((*) (a*b)) (h a).symm\r\n   ... = ((a * b) * a) * a             : (mul_assoc (a*b) a a).symm\r\n   ... = (a * (b * a)) * a             : congr_arg (* a) (mul_assoc a b a)\r\n   ... = (1 * (a * (b * a))) * a       : congr_arg (* a) (one_mul (a*(b*a))).symm\r\n   ... = ((b * b) * (a * (b * a))) * a : congr_arg (* a) (congr_arg (* (a*(b*a))) (h b).symm)\r\n   ... = (b * (b * (a * (b * a)))) * a : congr_arg (* a) (mul_assoc b b (a*(b*a)))\r\n   ... = (b * ((b * a) * (b * a))) * a : congr_arg (* a) (congr_arg ((*) b) (mul_assoc b a (b*a)).symm)\r\n   ... = (b * 1) * a                   : congr_arg (* a) (congr_arg ((*) b) (h (b*a)))\r\n   ... = b * a                         : congr_arg (* a) (mul_one b),\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : \u2200 x : M, x * x = 1)\r\n  : \u2200 x y : M, x * y = y * x :=\r\nbegin\r\n  intros a b,\r\n  calc a * b\r\n       = (a * b) * 1                   : by simp only [mul_one]\r\n   ... = (a * b) * (a * a)             : by simp only [h]\r\n   ... = ((a * b) * a) * a             : by simp only [mul_assoc]\r\n   ... = (a * (b * a)) * a             : by simp only [mul_assoc]\r\n   ... = (1 * (a * (b * a))) * a       : by simp only [one_mul]\r\n   ... = ((b * b) * (a * (b * a))) * a : by simp only [h]\r\n   ... = (b * (b * (a * (b * a)))) * a : by simp only [mul_assoc]\r\n   ... = (b * ((b * a) * (b * a))) * a : by simp only [mul_assoc]\r\n   ... = (b * 1) * a                   : by simp only [h]\r\n   ... = b * a                         : by simp only [mul_one]\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : \u2200 x : M, x * x = 1)\r\n  : \u2200 x y : M, x * y = y * x :=\r\nbegin\r\n  intros a b,\r\n  calc a * b\r\n       = (a * b) * 1                   : by simp\r\n   ... = (a * b) * (a * a)             : by simp [h]\r\n   ... = ((a * b) * a) * a             : by simp [mul_assoc]\r\n   ... = (a * (b * a)) * a             : by simp [mul_assoc]\r\n   ... = (1 * (a * (b * a))) * a       : by simp\r\n   ... = ((b * b) * (a * (b * a))) * a : by simp [h]\r\n   ... = (b * (b * (a * (b * a)))) * a : by simp [mul_assoc]\r\n   ... = (b * ((b * a) * (b * a))) * a : by simp [mul_assoc]\r\n   ... = (b * 1) * a                   : by simp [h]\r\n   ... = b * a                         : by simp,\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : \u2200 x : M, x * x = 1)\r\n  : \u2200 x y : M, x * y = y * x :=\r\nbegin\r\n  intros a b,\r\n  calc a * b\r\n       = (a * b) * (a * a)             : by simp [h]\r\n   ... = (1 * (a * (b * a))) * a       : by simp [mul_assoc]\r\n   ... = ((b * b) * (a * (b * a))) * a : by simp [h]\r\n   ... = (b * ((b * a) * (b * a))) * a : by simp [mul_assoc]\r\n   ... = b * a                         : by simp [h],\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : \u2200 x : M, x * x = 1)\r\n  : \u2200 x y : M, x * y = y * x :=\r\nbegin\r\n  intros a b,\r\n  calc a * b\r\n       = ((b * b) * (a * (b * a))) * a : by simp [h, mul_assoc]\r\n   ... = (b * ((b * a) * (b * a))) * a : by simp [mul_assoc]\r\n   ... = b * a                         : by simp [h],\r\nend\r\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\/Los_monoides_booleanos_son_conmutativos.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;lean&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 Los_monoides_booleanos_son_conmutativos\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 \"\u2200 x. x * x = 1\"\r\n  shows   \"\u2200 x y. x * y = y * x\"\r\nproof (rule allI)+\r\n  fix a b\r\n  have \"a * b = (a * b) * 1\"\r\n    by (simp only: right_neutral)\r\n  also have \"\u2026 = (a * b) * (a * a)\"\r\n    by (simp only: assms)\r\n  also have \"\u2026 = ((a * b) * a) * a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (a * (b * a)) * a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (1 * (a * (b * a))) * a\"\r\n    by (simp only: left_neutral)\r\n  also have \"\u2026 = ((b * b) * (a * (b * a))) * a\"\r\n    by (simp only: assms)\r\n  also have \"\u2026 = (b * (b * (a * (b * a)))) * a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (b * ((b * a) * (b * a))) * a\"\r\n    by (simp only: assoc)\r\n  also have \"\u2026 = (b * 1) * a\"\r\n    by (simp only: assms)\r\n  also have \"\u2026 = b * a\"\r\n    by (simp only: right_neutral)\r\n  finally show \"a * b = b * a\"\r\n    by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"\u2200 x. x * x = 1\"\r\n  shows   \"\u2200 x y. x * y = y * x\"\r\nproof (rule allI)+\r\n  fix a b\r\n  have \"a * b = (a * b) * 1\"                    by simp\r\n  also have \"\u2026 = (a * b) * (a * a)\"             by (simp add: assms)\r\n  also have \"\u2026 = ((a * b) * a) * a\"             by (simp add: assoc)\r\n  also have \"\u2026 = (a * (b * a)) * a\"             by (simp add: assoc)\r\n  also have \"\u2026 = (1 * (a * (b * a))) * a\"       by simp\r\n  also have \"\u2026 = ((b * b) * (a * (b * a))) * a\" by (simp add: assms)\r\n  also have \"\u2026 = (b * (b * (a * (b * a)))) * a\" by (simp add: assoc)\r\n  also have \"\u2026 = (b * ((b * a) * (b * a))) * a\" by (simp add: assoc)\r\n  also have \"\u2026 = (b * 1) * a\"                   by (simp add: assms)\r\n  also have \"\u2026 = b * a\"                         by simp\r\n  finally show \"a * b = b * a\"                  by this\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"\u2200 x. x * x = 1\"\r\n  shows   \"\u2200 x y. x * y = y * x\"\r\nproof (rule allI)+\r\n  fix a b\r\n  have \"a * b = (a * b) * (a * a)\"              by (simp add: assms)\r\n  also have \"\u2026 = (a * (b * a)) * a\"             by (simp add: assoc)\r\n  also have \"\u2026 = ((b * b) * (a * (b * a))) * a\" by (simp add: assms)\r\n  also have \"\u2026 = (b * ((b * a) * (b * a))) * a\" by (simp add: assoc)\r\n  also have \"\u2026 = (b * 1) * a\"                   by (simp add: assms)\r\n  finally show \"a * b = b * a\"                  by simp\r\nqed\r\n\r\n(* 4\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"\u2200 x. x * x = 1\"\r\n  shows   \"\u2200 x y. x * y = y * x\"\r\n  by (metis assms assoc 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>Un monoide es un conjunto junto con una operaci\u00f3n binaria que es asociativa y tiene elemento neutro. Un monoide M es booleano si \u2200 x \u2208 M, x * x = 1 y es conmutativo si \u2200 x y \u2208 M, x * y = y * x En Lean, est\u00e1 definida la clase de los monoides (como monoid) y sus propiedades caracter\u00edsticas son mul_assoc : (a * b) * c = a * (b * c) one_mul : 1 * a = a mul_one : a * 1 = a Demostrar que los monoides booleanos son conmutativos. Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group.basic universe u variables {M : Type u} [monoid M] example (h : \u2200 x : M, x&#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\/547"}],"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=547"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/547\/revisions"}],"predecessor-version":[{"id":561,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/547\/revisions\/561"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=547"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=547"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=547"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}