{"id":1905,"date":"2024-01-04T06:00:46","date_gmt":"2024-01-04T04:00:46","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1905"},"modified":"2023-12-31T19:14:34","modified_gmt":"2023-12-31T17:14:34","slug":"04-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/04-ene-24\/","title":{"rendered":"Si \u2264 es un preorden, entonces < es irreflexiva"},"content":{"rendered":"\n<p>Demostrar con Lean4 que si &#92;(\u2264&#92;) es un preorden, entonces &#92;(&lt;&#92;) es irreflexiva.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\nvariable {\u03b1 : Type _} [Preorder \u03b1]\nvariable (a : \u03b1)\n\nexample : \u00aca < a :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p>Se usar\u00e1 la siguiente propiedad de lo pre\u00f3rdenes<br \/>\n&#92;[ (\u2200 a, b)[a &lt; b \u2194 a \u2264 b \u2227 b \u2270 a] &#92;]<br \/>\nCon dicha propiedad, lo que tenemos que demostrar se transforma en<br \/>\n&#92;[ \u00ac(a \u2264 a \u2227 a \u2270 a) &#92;]<br \/>\nPara demostrarla, supongamos que<br \/>\n&#92;[ a \u2264 a \u2227 a \u2270 a &#92;]<br \/>\nlo que es una contradicci\u00f3n.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Tactic\nvariable {\u03b1 : Type _} [Preorder \u03b1]\nvariable (a : \u03b1)\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u00aca < a :=\nby\n  rw [lt_iff_le_not_le]\n  -- \u22a2 \u00ac(a \u2264 a \u2227 \u00aca \u2264 a)\n  rintro \u27e8h1, h2\u27e9\n  -- h1 : a \u2264 a\n  -- h2 : \u00aca \u2264 a\n  -- \u22a2 False\n  exact h2 h1\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : \u00aca < a :=\n  irrefl a\n\n-- Lemas usados\n-- ============\n\n-- variable (b : \u03b1)\n-- #check (lt_iff_le_not_le : a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a)\n-- #check (irrefl a : \u00aca < a)\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\/Preorden_es_irreflexivo.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. 38.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que si \u2264 es un preorden, entonces < es irreflexiva.\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\/1905"}],"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=1905"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1905\/revisions"}],"predecessor-version":[{"id":1909,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1905\/revisions\/1909"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}