{"id":1858,"date":"2023-12-13T06:00:32","date_gmt":"2023-12-13T04:00:32","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1858"},"modified":"2023-12-08T12:14:59","modified_gmt":"2023-12-08T10:14:59","slug":"13-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/13-dic-23\/","title":{"rendered":"Si (m \u2223 n \u2227 m \u2260 n), entonces (m \u2223 n \u2227 \u00ac(n \u2223 m))"},"content":{"rendered":"\n<p>Demostrar con Lean4 que si &#40;m \u2223 n \u2227 m \u2260 n&#41;, entonces &#40;m \u2223 n \u2227 \u00ac(n \u2223 m)&#41;.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.GCD.Basic\n\nvariable {m n : \u2115}\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>La primera parte de la conclusi\u00f3n coincide con la primera de la hip\u00f3tesis. Nos queda demostrar la segunda parte; es decir, que &#40;\u00ac(n | m)&#41;. Para ello, supongamos que &#40;n | m&#41;. Entonces, por la propiedad antisim\u00e9trica de la divisibilidad y la primera parte de la hip\u00f3tesis, se tiene que &#40;m = n&#41; en contradicci\u00f3n con la segunda parte de la hip\u00f3tesis.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Nat.GCD.Basic\n\nvariable {m n : \u2115}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\nby\n  constructor\n  . show m \u2223 n\n    exact h.left\n  . show \u00acn \u2223 m\n    { intro (h1 : n \u2223 m)\n      have h2 : m = n := dvd_antisymm h.left h1\n      show False\n      exact h.right h2 }\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\nby\n  constructor\n  . exact h.left\n  . intro (h1 : n \u2223 m)\n    exact h.right (dvd_antisymm h.left h1)\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\n\u27e8h.left, fun h1 \u21a6 h.right (dvd_antisymm h.left h1)\u27e9\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\nby\n  cases' h with h1 h2\n  -- h1 : m \u2223 n\n  -- h2 : m \u2260 n\n  constructor\n  . -- \u22a2 m \u2223 n\n    exact h1\n  . -- \u22a2 \u00acn \u2223 m\n    contrapose! h2\n    -- h2 : n \u2223 m\n    -- \u22a2 m = n\n    apply dvd_antisymm h1 h2\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : m \u2223 n \u2227 m \u2260 n)\n  : m \u2223 n \u2227 \u00ac n \u2223 m :=\nby\n  rcases h with \u27e8h1 : m \u2223 n, h2 : m \u2260 n\u27e9\n  constructor\n  . -- \u22a2 m \u2223 n\n    exact h1\n  . -- \u22a2 \u00acn \u2223 m\n    contrapose! h2\n    -- h2 : n \u2223 m\n    -- \u22a2 m = n\n    apply dvd_antisymm h1 h2\n\n-- Lemas usados\n-- ============\n\n-- #check (dvd_antisymm : m \u2223 n \u2192 n \u2223 m \u2192 m = n)\n<\/pre>\n<p><b>Demostraciones interactivas<\/b><\/p>\n<p>Se puede interactuar con las demostraciones anteriores en <a href=\"https:\/\/live.lean-lang.org\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus2\/main\/src\/Uso_de_conjuncion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">Lean 4 Web<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li> J. Avigad y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 36.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si m \u2223 n \u2227 m \u2260 n, entonces m \u2223 n \u2227 \u00ac(n \u2223 m).<\/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":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1858"}],"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=1858"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1858\/revisions"}],"predecessor-version":[{"id":1859,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1858\/revisions\/1859"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1858"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}