{"id":1772,"date":"2023-11-15T06:00:23","date_gmt":"2023-11-15T04:00:23","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1772"},"modified":"2023-11-05T12:51:04","modified_gmt":"2023-11-05T10:51:04","slug":"15-nov-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/15-nov-23\/","title":{"rendered":"En \u211d, a < b \u2192 \u00ac(b < a)"},"content":{"rendered":"<p>Demostrar con Lean4 que en &#92;(\u211d&#92;), &#92;(a &lt; b \u2192 \u00ac(b &lt; a)&#92;).<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nvariable (a b : \u211d)\n\nexample\n  (h : a < b)\n  : \u00ac b < a :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nPor hip\u00f3tesis &#92;(a &lt; b&#92;) y tenemos que demostrar que &#92;(\u00ac(b &lt; a)&#92;). Supongamos que &#92;(b &lt; a&#92;). Entonces, por la propiedad transiva &#92;(a &lt; a&#92;) que es una contradicci\u00f3n con la propiedad irreflexiva.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport Mathlib.Data.Real.Basic\n\nvariable (a b : \u211d)\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (h : a < b)\n  : \u00ac b < a :=\nby\n  intro h1\n  -- h1 : b < a\n  -- \u22a2 False\n  have : a < a := lt_trans h h1\n  apply lt_irrefl a this\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (h : a < b)\n  : \u00ac b < a :=\nby\n  intro h1\n  -- h1 : b < a\n  -- \u22a2 False\n  exact lt_irrefl a (lt_trans h h1)\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (h : a < b)\n  : \u00ac b < a :=\nfun h1 \u21a6 lt_irrefl a (lt_trans h h1)\n\n-- 4\u00aa demostraci\u00f3n\nexample\n  (h : a < b)\n  : \u00ac b < a :=\nlt_asymm h\n\n-- Lemas usados\n-- ============\n\n-- variable (c : \u211d)\n-- #check (lt_asymm : a < b \u2192 \u00acb < a)\n-- #check (lt_irrefl a : \u00aca < a)\n-- #check (lt_trans : a < b \u2192 b < c \u2192 a < c)\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\/Asimetria_de_menor.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. 32.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que en &#92;(\u211d&#92;), &#92;(a &lt; b \u2192 \u00ac(b &lt; a)&#92;). Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Real.Basic variable (a b : \u211d) example (h : a < b) : \u00ac b < a := by sorry\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\/1772"}],"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=1772"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1772\/revisions"}],"predecessor-version":[{"id":1773,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1772\/revisions\/1773"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}