{"id":1840,"date":"2023-12-08T06:00:46","date_gmt":"2023-12-08T04:00:46","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1840"},"modified":"2023-11-28T13:42:48","modified_gmt":"2023-11-28T11:42:48","slug":"08-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/08-dic-23\/","title":{"rendered":"Si 0 < 0, entonces a > 37 para cualquier n\u00famero a"},"content":{"rendered":"\n<p>Demostrar con Lean4 que si &#92;(0 &lt; 0&#92;), entonces &#92;(a > 37&#92;) para cualquier n\u00famero &#92;(a&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\n\nvariable (a : \u2115)\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Basta demostrar una contradicci\u00f3n, ya que de una contradicci\u00f3n se sigue cualquier cosa.<\/p>\n<p>La hip\u00f3tesis es una contradicci\u00f3n con la propiedad irreflexiva de la relaci\u00f3n &#92;(&lt;&#92;).<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\n\nvariable (a : \u2115)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nby\n  exfalso\n  -- \u22a2 False\n  show False\n  exact lt_irrefl 0 h\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nby\n  exfalso\n  -- \u22a2 False\n  apply lt_irrefl 0 h\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nabsurd h (lt_irrefl 0)\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nby\n  have : \u00ac 0 < 0 :=  lt_irrefl 0\n  contradiction\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  (h : 0 < 0)\n  : a > 37 :=\nby linarith\n\n-- Lemas usados\n-- ============\n\n-- variable (p q : Prop)\n-- #check (lt_irrefl a : \u00aca < a)\n-- #check (absurd : p \u2192 \u00acp \u2192 q)\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\/Principio_de_explosion.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. 34.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si 0 < 0, entonces a > 37 para cualquier n\u00famero 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":"","_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\/1840"}],"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=1840"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1840\/revisions"}],"predecessor-version":[{"id":1846,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1840\/revisions\/1846"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1840"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}