{"id":507,"date":"2021-06-29T06:00:28","date_gmt":"2021-06-29T04:00:28","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=507"},"modified":"2021-06-21T16:38:47","modified_gmt":"2021-06-21T14:38:47","slug":"en-los-monoides-los-inversos-a-la-izquierda-y-a-la-derecha-son-iguales","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/en-los-monoides-los-inversos-a-la-izquierda-y-a-la-derecha-son-iguales\/","title":{"rendered":"En los monoides, los inversos a la izquierda y a la derecha son iguales"},"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>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 si M es un monide, a \u2208 M, b es un inverso de a por la izquierda y c es un inverso de a por la derecha, entonce b = c.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport algebra.group.defs\n\nvariables {M : Type} [monoid M]\nvariables {a b c : M}\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\nsorry\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport algebra.group.defs\r\n\r\nvariables {M : Type} [monoid M]\r\nvariables {a b c : M}\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (hba : b * a = 1)\r\n  (hac : a * c = 1)\r\n  : b = c :=\r\nbegin\r\n rw \u2190one_mul c,\r\n rw \u2190hba,\r\n rw mul_assoc,\r\n rw hac,\r\n rw mul_one b,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (hba : b * a = 1)\r\n  (hac : a * c = 1)\r\n  : b = c :=\r\nby rw [\u2190one_mul c, \u2190hba, mul_assoc, hac, mul_one b]\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (hba : b * a = 1)\r\n  (hac : a * c = 1)\r\n  : b = c :=\r\ncalc b   = b * 1       : (mul_one b).symm\r\n     ... = b * (a * c) : congr_arg (\u03bb x, b * x) hac.symm\r\n     ... = (b * a) * c : (mul_assoc b a c).symm\r\n     ... = 1 * c       : congr_arg (\u03bb x, x * c) hba\r\n     ... = c           : one_mul c\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (hba : b * a = 1)\r\n  (hac : a * c = 1)\r\n  : b = c :=\r\ncalc b   = b * 1       : by finish\r\n     ... = b * (a * c) : by finish\r\n     ... = (b * a) * c : (mul_assoc b a c).symm\r\n     ... = 1 * c       : by finish\r\n     ... = c           : by finish\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (hba : b * a = 1)\r\n  (hac : a * c = 1)\r\n  : b = c :=\r\nleft_inv_eq_right_inv hba hac\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\/En_los_monoides_los_inversos_a_la_izquierda_y_a_la_derecha_son_iguales.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 En_los_monoides_los_inversos_a_la_izquierda_y_a_la_derecha_son_iguales\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 \"b \u2759* a = \u27591\"\r\n          \"a \u2759* c = \u27591\"\r\n  shows   \"b = c\"\r\nproof -\r\n  have      \"b  = b \u2759* \u27591\"      by (simp only: right_neutral)\r\n  also have \"\u2026 = b \u2759* (a \u2759* c)\" by (simp only: \u2039a \u2759* c = \u27591\u203a)\r\n  also have \"\u2026 = (b \u2759* a) \u2759* c\" by (simp only: assoc)\r\n  also have \"\u2026 = \u27591 \u2759* c\"       by (simp only: \u2039b \u2759* a = \u27591\u203a)\r\n  also have \"\u2026 = c\"             by (simp only: left_neutral)\r\n  finally show \"b = c\"          by this\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"b \u2759* a = \u27591\"\r\n          \"a \u2759* c = \u27591\"\r\n  shows   \"b = c\"\r\nproof -\r\n  have      \"b  = b \u2759* \u27591\"      by simp\r\n  also have \"\u2026 = b \u2759* (a \u2759* c)\" using \u2039a \u2759* c = \u27591\u203a by simp\r\n  also have \"\u2026 = (b \u2759* a) \u2759* c\" by (simp add: assoc)\r\n  also have \"\u2026 = \u27591 \u2759* c\"       using \u2039b \u2759* a = \u27591\u203a by simp\r\n  also have \"\u2026 = c\"             by simp\r\n  finally show \"b = c\"          by this\r\nqed\r\n\r\n(* 3\u00aa demostraci\u00f3n *)\r\n\r\nlemma\r\n  assumes \"b \u2759* a = \u27591\"\r\n          \"a \u2759* c = \u27591\"\r\n  shows   \"b = c\"\r\n  using assms\r\n  by (metis assoc 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>Un monoide es un conjunto junto con una operaci\u00f3n binaria que es asociativa y tiene elemento neutro. 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 si M es un monide, a \u2208 M, b es un inverso de a por la izquierda y c es un inverso de a por la derecha, entonce b = c. Para ello, completar la siguiente teor\u00eda de Lean: import algebra.group.defs variables {M : Type} [monoid M] variables {a b c : M} example (hba : b * a = 1) (hac :&#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\/507"}],"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=507"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/507\/revisions"}],"predecessor-version":[{"id":509,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/507\/revisions\/509"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=507"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}