{"id":7577,"date":"2021-01-22T11:46:53","date_gmt":"2021-01-22T10:46:53","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=7577"},"modified":"2021-01-22T11:46:53","modified_gmt":"2021-01-22T10:46:53","slug":"negacion-del-existencial-en-lean-caracterizacion-de-numeros-no-pares","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/negacion-del-existencial-en-lean-caracterizacion-de-numeros-no-pares\/","title":{"rendered":"Negaci\u00f3n del existencial en Lean: Caracterizaci\u00f3n de n\u00fameros no pares"},"content":{"rendered":"<p>He a\u00f1adido a la lista <a href=\"https:\/\/bit.ly\/2QwnT30\">DAO (Demostraci\u00f3n Asistida por Ordenador) con Lean<\/a> el <a href=\"https:\/\/youtu.be\/YNZRDMejdYA\">v\u00eddeo<\/a> en el que se comentan 15 pruebas en Lean de la propiedad<\/p>\n<blockquote><p>\n  \u00ac(\u2203 k, n = 2<em>k) \u2194 \u2200 k, n \u2260 2<\/em>k\n<\/p><\/blockquote>\n<p>usando los estilos declarativo, aplicativo y funcional.<\/p>\n<p>A continuaci\u00f3n, se muestra el v\u00eddeo<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/YNZRDMejdYA\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p>y el <a href=\"https:\/\/bit.ly\/3sNuKXO\">c\u00f3digo<\/a> de la teor\u00eda utilizada<\/p>\n<pre lang=\"lean\">\nimport tactic\n\nvariable (n : \u2124)\n\n-- ----------------------------------------------------\n-- Ejercicio. Demostrar que si n es un n\u00famero entero,\n-- entonces\n--    \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k\n-- ----------------------------------------------------\n\n-- 1\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nbegin\n  split,\n  { intro h,\n    intro k,\n    intro hk,\n    apply h,\n    use k,\n    exact hk, },\n  { intro h1,\n    intro h2,\n    cases h2 with k hk,\n    apply h1 k,\n    exact hk, },\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nbegin\n  split,\n  { intros h k hk,\n    exact h \u27e8k, hk\u27e9 },\n  { rintros h \u27e8k, rfl\u27e9,\n    apply h k,\n    refl, },\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nbegin\n  split,\n  { intros h k hk,\n    exact h \u27e8k, hk\u27e9 },\n  { rintros h \u27e8k, rfl\u27e9,\n    exact h k rfl },\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nbegin\n  push_neg,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nby push_neg\n\n-- 6\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  ( assume h1 : \u00ac(\u2203 k, n = 2*k),\n    show \u2200 k, n \u2260 2*k, from\n      assume k,\n      show n \u2260 2*k, from\n        assume h2 : n = 2*k,\n        have h3 : \u2203 k, n = 2*k,\n          from exists.intro k h2,\n        show false,\n          from h1 h3)\n  ( assume h1 : \u2200 k, n \u2260 2*k,\n    show \u00ac(\u2203 k, n = 2*k), from\n      assume h2 : \u2203 k, n = 2*k,\n      show false, from\n        exists.elim h2\n          ( assume k,\n            assume hk : n = 2*k,\n            have h3 : n \u2260 2*k,\n              from h1 k,\n            show false,\n              from h3 hk))\n\n-- 7\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  ( assume h1 : \u00ac(\u2203 k, n = 2*k),\n    show \u2200 k, n \u2260 2*k, from\n      assume k,\n      show n \u2260 2*k, from\n        assume h2 : n = 2*k,\n        have h3 : \u2203 k, n = 2*k,\n          from exists.intro k h2,\n        h1 h3)\n  ( assume h1 : \u2200 k, n \u2260 2*k,\n    show \u00ac(\u2203 k, n = 2*k), from\n      assume h2 : \u2203 k, n = 2*k,\n      show false, from\n        exists.elim h2\n          ( assume k,\n            assume hk : n = 2*k,\n            have h3 : n \u2260 2*k,\n              from h1 k,\n            h3 hk))\n\n-- 8\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  ( assume h1 : \u00ac(\u2203 k, n = 2*k),\n    show \u2200 k, n \u2260 2*k, from\n      assume k,\n      show n \u2260 2*k, from\n        assume h2 : n = 2*k,\n        h1 (exists.intro k h2))\n  ( assume h1 : \u2200 k, n \u2260 2*k,\n    show \u00ac(\u2203 k, n = 2*k), from\n      assume h2 : \u2203 k, n = 2*k,\n      show false, from\n        exists.elim h2\n          ( assume k,\n            assume hk : n = 2*k,\n            (h1 k) hk))\n\n-- 9\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  ( assume h1 : \u00ac(\u2203 k, n = 2*k),\n    show \u2200 k, n \u2260 2*k, from\n      assume k,\n      \u03bb h2, h1 (exists.intro k h2))\n  ( assume h1 : \u2200 k, n \u2260 2*k,\n    show \u00ac(\u2203 k, n = 2*k), from\n      assume h2 : \u2203 k, n = 2*k,\n      show false, from\n        exists.elim h2\n          (\u03bb k hk, (h1 k) hk))\n\n-- 10\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  ( assume h1 : \u00ac(\u2203 k, n = 2*k),\n    \u03bb k, \u03bb h2, h1 (exists.intro k h2))\n  ( assume h1 : \u2200 k, n \u2260 2*k,\n    show \u00ac(\u2203 k, n = 2*k), from\n      assume h2 : \u2203 k, n = 2*k,\n      exists.elim h2 (\u03bb k hk, (h1 k) hk))\n\n-- 11\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\niff.intro\n  (\u03bb h1 k h2, h1 (exists.intro k h2))\n  (\u03bb h1 h2, exists.elim h2 (\u03bb k hk, (h1 k) hk))\n\n-- 12\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\n\u27e8\u03bb h1 k h2, h1 \u27e8k, h2\u27e9,\n \u03bb h1 h2, exists.elim h2 (\u03bb k hk, (h1 k) hk)\u27e9\n\n-- 13\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\n-- by hint\nby simp\n\n-- 14\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nby finish\n\n-- 15\u00aa demostraci\u00f3n\nexample : \u00ac(\u2203 k, n = 2*k) \u2194 \u2200 k, n \u2260 2*k :=\nby norm_num\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>He a\u00f1adido a la lista DAO (Demostraci\u00f3n Asistida por Ordenador) con Lean el v\u00eddeo en el que se comentan 15 pruebas en Lean de la propiedad \u00ac(\u2203 k, n = 2k) \u2194 \u2200 k, n \u2260 2k usando los estilos declarativo, aplicativo y funcional. A continuaci\u00f3n, se muestra el v\u00eddeo y el c\u00f3digo de la&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","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,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[335],"tags":[336],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7577"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=7577"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7577\/revisions"}],"predecessor-version":[{"id":7578,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/7577\/revisions\/7578"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=7577"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=7577"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=7577"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}