{"id":2446,"date":"2024-05-03T12:54:38","date_gmt":"2024-05-03T10:54:38","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2446"},"modified":"2024-05-03T12:55:46","modified_gmt":"2024-05-03T10:55:46","slug":"03-may-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/03-may-24\/","title":{"rendered":"En los monoides, los inversos a la izquierda y a la derecha son iguales"},"content":{"rendered":"\n

Un monoide<\/a> es un conjunto junto con una operaci\u00f3n binaria que es asociativa y tiene elemento neutro.<\/p>\n

En Lean4, est\u00e1 definida la clase de los monoides (como Monoid<\/code>) y sus propiedades caracter\u00edsticas son<\/p>\n

\n   mul_assoc : (a * b) * c = a * (b * c)\n   one_mul :   1 * a = a\n   mul_one :   a * 1 = a\n<\/pre>\n
Demostrar que si \(M\) es un monoide, \(a \u2208 M\), \(b\) es un inverso de \(a\) por la izquierda y \(c\) es un inverso de \(a\) por la derecha, entonces \(b = c\).<\/p>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\nimport Mathlib.Algebra.Group.Defs\n\nvariable {M : Type} [Monoid M]\nvariable {a b c : M}\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Por la siguiente cadena de igualdades
\n\begin{align}
\n   b &= b * 1          &&\text{[por mul_one]} \\
\n     &= b * (a * c)    &&\text{[por hip\u00f3tesis]} \\
\n     &= (b * a) * c    &&\text{[por mul_assoc]} \\
\n     &= 1 * c          &&\text{[por hip\u00f3tesis]} \\
\n     &= c              &&\text{[por one_mul]} \\
\n\end{align}<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Algebra.Group.Defs\n\nvariable {M : Type} [Monoid M]\nvariable {a b c : M}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\ncalc b = b * 1       := (mul_one b).symm\n     _ = b * (a * c) := congrArg (b * .) hac.symm\n     _ = (b * a) * c := (mul_assoc b a c).symm\n     _ = 1 * c       := congrArg (. * c) hba\n     _ = c           := one_mul c\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\ncalc b  = b * 1       := by aesop\n      _ = b * (a * c) := by aesop\n      _ = (b * a) * c := (mul_assoc b a c).symm\n      _ = 1 * c       := by aesop\n      _ = c           := by aesop\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\nby\n  rw [\u2190one_mul c]\n  -- \u22a2 b = 1 * c\n  rw [\u2190hba]\n  -- \u22a2 b = (b * a) * c\n  rw [mul_assoc]\n  -- \u22a2 b = b * (a * c)\n  rw [hac]\n  -- \u22a2 b = b * 1\n  rw [mul_one b]\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\nby rw [\u2190one_mul c, \u2190hba, mul_assoc, hac, mul_one b]\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (hba : b * a = 1)\n  (hac : a * c = 1)\n  : b = c :=\nleft_inv_eq_right_inv hba hac\n\n-- Lemas usados\n-- ============\n\n-- #check (left_inv_eq_right_inv : b * a = 1 \u2192 a * c = 1 \u2192 b = c)\n-- #check (mul_assoc a b c : (a * b) * c = a * (b * c))\n-- #check (mul_one a : a * 1 = a)\n-- #check (one_mul a : 1 * a = a)\n<\/pre>\n
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\n
3. Demostraciones con Isabelle\/HOL<\/h2>\n
\ntheory En_los_monoides_los_inversos_a_la_izquierda_y_a_la_derecha_son_iguales\nimports Main\nbegin\n\ncontext monoid\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"b * a = 1\"\n          \"a * c = 1\"\n  shows   \"b = c\"\nproof -\n  have      \"b  = b * 1\"      by (simp only: right_neutral)\n  also have \"\u2026 = b * (a * c)\" by (simp only: \u2039a * c = 1\u203a)\n  also have \"\u2026 = (b * a) * c\" by (simp only: assoc)\n  also have \"\u2026 = 1 * c\"       by (simp only: \u2039b * a = 1\u203a)\n  also have \"\u2026 = c\"           by (simp only: left_neutral)\n  finally show \"b = c\"        by this\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"b * a = 1\"\n          \"a * c = 1\"\n  shows   \"b = c\"\nproof -\n  have      \"b  = b * 1\"      by simp\n  also have \"\u2026 = b * (a * c)\" using \u2039a * c = 1\u203a by simp\n  also have \"\u2026 = (b * a) * c\" by (simp add: assoc)\n  also have \"\u2026 = 1 * c\"       using \u2039b * a = 1\u203a by simp\n  also have \"\u2026 = c\"           by simp\n  finally show \"b = c\"        by this\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma\n  assumes \"b * a = 1\"\n          \"a * c = 1\"\n  shows   \"b = c\"\n  using assms\n  by (metis assoc left_neutral right_neutral)\n\nend\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Un monoide es un conjunto junto con una operaci\u00f3n binaria que es asociativa y tiene elemento neutro. En Lean4, 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 monoide, \(a \u2208 M\), \(b\) es un inverso de \(a\) por la izquierda y \(c\) es un inverso de \(a\) por la derecha, entonces \(b = c\). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Algebra.Group.Defs variable {M : Type} [Monoid M] variable {a b c : M} example (hba : b * a = 1) (hac :…<\/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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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\/2446"}],"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=2446"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2446\/revisions"}],"predecessor-version":[{"id":2448,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2446\/revisions\/2448"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2446"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2446"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2446"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}