{"id":1742,"date":"2023-10-31T06:00:56","date_gmt":"2023-10-31T04:00:56","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1742"},"modified":"2023-10-30T17:47:50","modified_gmt":"2023-10-30T15:47:50","slug":"31-oct-23","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/31-oct-23\/","title":{"rendered":"La suma de dos funciones acotadas inferiormente tambi\u00e9n lo est\u00e1"},"content":{"rendered":"<p>Demostrar con Lean4 que la suma de dos funciones acotadas inferiormente tambi\u00e9n lo est\u00e1.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n<pre lang=\"lean\">\nimport src.Suma_de_cotas_inferiores\nvariable {f g : \u211d \u2192 \u211d}\n\n-- (acotadaInf f) afirma que f tiene cota inferior.\ndef acotadaInf (f : \u211d \u2192 \u211d) :=\n  \u2203 a, CotaInferior f a\n\nexample\n  (hf : acotadaInf f)\n  (hg : acotadaInf g)\n  : acotadaInf (f + g) :=\nby sorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n<p><br \/>\nDel ejercicio \u00abLa suma de una cota inferior de &#92;(f&#92;) y una cota inferior de &#92;(g&#92;) es una cota inferior de &#92;(f+g&#92;)\u00bb usaremos la definici\u00f3n de cota inferior (<code>CotaInferior<\/code>) y el lema <code>sumaCotaInf<\/code>.<\/p>\n<p>Puesto que &#92;(f&#92;) est\u00e1 acotada inferiormente, tiene una cota inferior.  &#92;(a&#92;) una de dichas cotas. An\u00e1logamentte, puesto que &#92;(g&#92;) est\u00e1 acotada inferiormente, tiene una cota inferior. Sea &#92;(b&#92;) una de dichas cotas. Por el lema <code>sumaCotaInf<\/code>, a+b es una cota inferior de &#92;(f+g&#92;). Por consiguiente, &#92;(f+g&#92;) est\u00e1 acotada inferiormente.<\/p>\n<p><b>Demostraciones con Lean4<\/b><\/p>\n<pre lang=\"lean\">\nimport src.Suma_de_cotas_inferiores\nvariable {f g : \u211d \u2192 \u211d}\n\n-- (acotadaInf f) afirma que f tiene cota inferior.\ndef acotadaInf (f : \u211d \u2192 \u211d) :=\n  \u2203 a, CotaInferior f a\n\n-- 1\u00aa demostraci\u00f3n\nexample\n  (hf : acotadaInf f)\n  (hg : acotadaInf g)\n  : acotadaInf (f + g) :=\nby\n  cases' hf with a ha\n  -- a : \u211d\n  -- ha : CotaInferior f a\n  cases' hg with b hb\n  -- b : \u211d\n  -- hb : CotaInferior g b\n  have h1 : CotaInferior (f + g) (a + b) := sumaCotaInf ha hb\n  have h2 : \u2203 z, CotaInferior (f + g) z :=\n    Exists.intro (a + b) h1\n  show acotadaInf (f + g)\n  exact h2\n\n-- 2\u00aa demostraci\u00f3n\nexample\n  (hf : acotadaInf f)\n  (hg : acotadaInf g)\n  : acotadaInf (f + g) :=\nby\n  cases' hf with a ha\n  -- a : \u211d\n  -- ha : FnLb f a\n  cases' hg with b hgb\n  -- b : \u211d\n  -- hgb : FnLb g b\n  use a + b\n  -- \u22a2 FnLb (f + g) (a + b)\n  apply sumaCotaInf ha hgb\n\n-- 3\u00aa demostraci\u00f3n\nexample\n  (hf : acotadaInf f)\n  (hg : acotadaInf g)\n  : acotadaInf (f + g) :=\nby\n  rcases hf with \u27e8a, ha\u27e9\n  -- a : \u211d\n  -- ha : FnLb f a\n  rcases hg with \u27e8b, hb\u27e9\n  -- b : \u211d\n  -- hb : FnLb g b\n  exact \u27e8a + b, sumaCotaInf ha hb\u27e9\n\n-- 4\u00aa demostraci\u00f3n\nexample :\n  acotadaInf f \u2192 acotadaInf g \u2192 acotadaInf (f + g) :=\nby\n  rintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n  -- a : \u211d\n  -- ha : FnLb f a\n  -- b : \u211d\n  -- hb : FnLb g b\n  exact \u27e8a + b, sumaCotaInf ha hb\u27e9\n\n-- 5\u00aa demostraci\u00f3n\nexample :\n  acotadaInf f \u2192 acotadaInf g \u2192 acotadaInf (f + g) :=\nfun \u27e8a, ha\u27e9 \u27e8b, hb\u27e9 \u21a6 \u27e8a + b, sumaCotaInf ha hb\u27e9\n\n-- Lemas usados\n-- ============\n\n-- #check (sumaCotaInf : FnLb f a \u2192 FnLb g b \u2192 FnLb (f + g) (a + b))\n<\/pre>\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. 29.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar con Lean4 que la suma de dos funciones acotadas inferiormente tambi\u00e9n lo est\u00e1. Para ello, completar la siguiente teor\u00eda de Lean4: import src.Suma_de_cotas_inferiores variable {f g : \u211d \u2192 \u211d} &#8212; (acotadaInf f) afirma que f tiene cota inferior. def acotadaInf (f : \u211d \u2192 \u211d) := \u2203 a, CotaInferior f a example (hf : acotadaInf f) (hg : acotadaInf g) : acotadaInf (f + g) := by sorry<\/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":[302,297],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1742"}],"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=1742"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1742\/revisions"}],"predecessor-version":[{"id":1744,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1742\/revisions\/1744"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1742"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1742"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}