{"id":1860,"date":"2023-12-14T06:00:54","date_gmt":"2023-12-14T04:00:54","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1860"},"modified":"2023-12-08T14:10:46","modified_gmt":"2023-12-08T12:10:46","slug":"14-dic-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/14-dic-23\/","title":{"rendered":"(\u2203x \u2208 \u211d)[2 < x < 3]"},"content":{"rendered":"\n<p>Demostrar con Lean4 que &#92;((\u2203x \u2208 \u211d)[2 &lt; x &lt; 3]&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nexample : \u2203 x : \u211d, 2 < x \u2227 x < 3 :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Podemos usar el n\u00famero &#92;(&#92;dfrac{5}{2}&#92;) y comprobar que &#92;(2 &lt; &#92;dfrac{5}{2} &lt; 3&#92;).<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u2203 x : \u211d, 2 < x \u2227 x < 3 :=\nby\n  use 5 \/ 2\n  show 2 < 5 \/ 2 \u2227 5 \/ 2 < 3\n  constructor\n  . show 2 < 5 \/ 2\n    norm_num\n  . show 5 \/ 2 < 3\n    norm_num\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u2203 x : \u211d, 2 < x \u2227 x < 3 :=\nby\n  use 5 \/ 2\n  constructor\n  . norm_num\n  . norm_num\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u2203 x : \u211d, 2 < x \u2227 x < 3 :=\nby\n  use 5 \/ 2\n  constructor <;> norm_num\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u2203 x : \u211d, 2 < x \u2227 x < 3 :=\n\u27e85\/2, by norm_num\u27e9\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\/Entre_2_y_3.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 (\u2203x \u2208 \u211d)[2 < x < 3].\n<\/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\/1860"}],"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=1860"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1860\/revisions"}],"predecessor-version":[{"id":1863,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1860\/revisions\/1863"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}