{"id":2461,"date":"2024-05-09T06:00:45","date_gmt":"2024-05-09T04:00:45","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2461"},"modified":"2024-05-08T12:05:36","modified_gmt":"2024-05-08T10:05:36","slug":"09-may-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/09-may-24\/","title":{"rendered":"Caracterizaci\u00f3n de producto igual al primer factor"},"content":{"rendered":"\n

Un monoide cancelativo por la izquierda es un monoide \(M\) que cumple la propiedad cancelativa por la izquierda; es decir, para todo \(a, b \u2208 M\)
\n\[ a\u00b7b = a\u00b7c \u2194 b = c \]<\/p>\n

En Lean4 la clase de los monoides cancelativos por la izquierda es LeftCancelMonoid<\/code> y la propiedad cancelativa por la izquierda es<\/p>\n

\n   mul_left_cancel : a * b = a * c \u2192 b = c\n<\/pre>\n
Demostrar con Lean4 que si \(M\) es un monoide cancelativo por la izquierda y \(a, b \u2208 M\), entonces
\n\[ a\u00b7b = a \u2194 b = 1 \]<\/p>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\nimport Mathlib.Algebra.Group.Basic\n\nvariable {M : Type} [LeftCancelMonoid M]\nvariable {a b : M}\n\nexample : a * b = a \u2194 b = 1 :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Demostraremos las dos implicaciones.<\/p>\n
(\u27f9) Supongamos que
\n\[ a\u00b7b = a  \]
\nPor la propiedad del neutro por la derecha tenemos
\n\[ a\u00b71 = a \]
\nPor tanto,
\n\[ a\u00b7b = a\u00b71 \]
\ny, por la propiedad cancelativa,
\n\[ b = 1 \]<\/p>\n
(\u27f8) Supongamos que \(b = 1\). Entonces,
\n\begin{align}
\n   a\u00b7b &= a\u00b71    \\
\n       &= a      &&\text{[por el neutro por la derecha]}
\n\end{align}<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Algebra.Group.Basic\n\nvariable {M : Type} [LeftCancelMonoid M]\nvariable {a b : M}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * b = a \u2194 b = 1 :=\nby\n  constructor\n  . -- \u22a2 a * b = a \u2192 b = 1\n    intro h\n    -- h : a * b = a\n    -- \u22a2 b = 1\n    have h1 : a * b = a * 1 := by\n      calc a * b = a     := by exact h\n               _ = a * 1 := (mul_one a).symm\n    exact mul_left_cancel h1\n  . -- \u22a2 b = 1 \u2192 a * b = a\n    intro h\n    -- h : b = 1\n    -- \u22a2 a * b = a\n    rw [h]\n    -- \u22a2 a * 1 = a\n    exact mul_one a\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * b = a \u2194 b = 1 :=\ncalc a * b = a\n     \u2194 a * b = a * 1 := by rw [mul_one]\n   _ \u2194 b = 1         := mul_left_cancel_iff\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * b = a \u2194 b = 1 :=\nmul_right_eq_self\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : a * b = a \u2194 b = 1 :=\nby aesop\n\n-- Lemas usados\n-- ============\n\n-- variable (c : M)\n-- #check (mul_left_cancel : a * b = a * c \u2192 b = c)\n-- #check (mul_one a : a * 1 = a)\n-- #check (mul_right_eq_self : a * b = a \u2194 b = 1)\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 Caracterizacion_de_producto_igual_al_primer_factor\nimports Main\nbegin\n\ncontext cancel_comm_monoid_add\nbegin\n\n(* 1\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\nproof (rule iffI)\n  assume \"a + b = a\"\n  then have \"a + b = a + 0\"     by (simp only: add_0_right)\n  then show \"b = 0\"             by (simp only: add_left_cancel)\nnext\n  assume \"b = 0\"\n  have \"a + 0 = a\"              by (simp only: add_0_right)\n  with \u2039b = 0\u203a show \"a + b = a\" by (rule ssubst)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\nproof\n  assume \"a + b = a\"\n  then have \"a + b = a + 0\" by simp\n  then show \"b = 0\"         by simp\nnext\n  assume \"b = 0\"\n  have \"a + 0 = a\"          by simp\n  then show \"a + b = a\"     using \u2039b = 0\u203a by simp\nqed\n\n(* 3\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\nproof -\n  have \"(a + b = a) \u27f7 (a + b = a + 0)\" by (simp only: add_0_right)\n  also have \"\u2026 \u27f7 (b = 0)\"             by (simp only: add_left_cancel)\n  finally show \"a + b = a \u27f7 b = 0\"     by this\nqed\n\n(* 4\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\nproof -\n  have \"(a + b = a) \u27f7 (a + b = a + 0)\" by simp\n  also have \"\u2026 \u27f7 (b = 0)\"             by simp\n  finally show \"a + b = a \u27f7 b = 0\"     .\nqed\n\n(* 5\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\n  by (simp only: add_cancel_left_right)\n\n(* 6\u00aa demostraci\u00f3n *)\n\nlemma \"a + b = a \u27f7 b = 0\"\n  by auto\n\nend\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Un monoide cancelativo por la izquierda es un monoide \(M\) que cumple la propiedad cancelativa por la izquierda; es decir, para todo \(a, b \u2208 M\) \[ a\u00b7b = a\u00b7c \u2194 b = c \] En Lean4 la clase de los monoides cancelativos por la izquierda es LeftCancelMonoid y la propiedad cancelativa por la izquierda es mul_left_cancel : a * b = a * c \u2192 b = c Demostrar con Lean4 que si \(M\) es un monoide cancelativo por la izquierda y \(a, b \u2208 M\), entonces \[ a\u00b7b = a \u2194 b = 1 \] Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Algebra.Group.Basic variable {M : Type} [LeftCancelMonoid M] variable {a b : M} example : a * b = a…<\/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\/2461"}],"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=2461"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2461\/revisions"}],"predecessor-version":[{"id":2462,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2461\/revisions\/2462"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2461"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2461"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}