{"id":728,"date":"2021-09-03T06:00:13","date_gmt":"2021-09-03T04:00:13","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=728"},"modified":"2021-08-29T12:49:51","modified_gmt":"2021-08-29T10:49:51","slug":"la-suma-de-los-n-primeros-impares-es-n2","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/la-suma-de-los-n-primeros-impares-es-n2\/","title":{"rendered":"La suma de los n primeros impares es n^2"},"content":{"rendered":"<p>En Lean, se puede definir el n-\u00e9simo n\u00famero primo por<\/p>\n<pre lang=\"text\">\r\n   def impar (n : \u2115) := 2 * n + 1\r\n<\/pre>\n<p>Adem\u00e1s, en la librer\u00eda finset est\u00e1n definidas las funciones<\/p>\n<pre lang=\"text\">\r\n   range :: \u2115 \u2192 finset \u2115\r\n   sum :: finset \u03b1 \u2192 (\u03b1 \u2192 \u03b2) \u2192 \u03b2\r\n<\/pre>\n<p>tales que<\/p>\n<p>+ (range n) es el conjunto de los n primeros n\u00fameros naturales. Por ejemplo, el valor de (range 3) es {0, 1, 2}.<br \/>\n+ (sum A f) es la suma del conjunto obtenido aplicando la funci\u00f3n f a los elementos del conjunto finito A. Por ejemplo, el valor de (sum (range 3) impar) es 9.<\/p>\n<p>Demostrar que la suma de los n primeros n\u00fameros impares es n\u00b2.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\r\nimport data.finset\r\nimport tactic.ring\r\nopen nat\r\n\r\nvariable (n : \u2115)\r\n\r\ndef impar (n : \u2115) := 2 * n + 1\r\n\r\nexample :\r\n  finset.sum (finset.range n) impar = n ^ 2 :=\r\nsorry\r\n<\/pre>\n<p>[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n<pre lang=\"lean\">\r\nimport data.finset\r\nimport tactic.ring\r\nopen nat\r\n\r\nset_option pp.structure_projections false\r\n\r\nvariable (n : \u2115)\r\n\r\ndef impar (n : \u2115) := 2 * n + 1\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample :\r\n  finset.sum (finset.range n) impar = n ^ 2 :=\r\nbegin\r\n  induction n with m HI,\r\n  { calc finset.sum (finset.range 0) impar\r\n          = 0\r\n            : by simp\r\n     ...  = 0 ^ 2\r\n            : rfl, },\r\n  { calc finset.sum (finset.range (succ m)) impar\r\n         = finset.sum (finset.range m) impar + impar m\r\n           : finset.sum_range_succ impar m\r\n     ... = m ^ 2 + impar m\r\n           : congr_arg2 (+) HI rfl\r\n     ... = m ^ 2 + 2 * m + 1\r\n           : rfl\r\n     ... = (m + 1) ^ 2\r\n           : by ring_nf\r\n     ... = succ m ^ 2\r\n           : rfl },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample :\r\n  finset.sum (finset.range n) impar = n ^ 2 :=\r\nbegin\r\n  induction n with d hd,\r\n  { refl, },\r\n  { rw finset.sum_range_succ,\r\n    rw hd,\r\n    change d ^ 2 + (2 * d + 1) = (d + 1) ^ 2,\r\n    ring_nf, },\r\nend\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/La_suma_de_los_n_primeros_impares_es_n^2.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\u00bbSoluciones con Isabelle\/HOL\u00bb]<\/p>\n<pre lang=\"isar\">\r\ntheory \"La_suma_de_los_n_primeros_impares_es_n^2\"\r\nimports Main\r\nbegin\r\n\r\ndefinition impar :: \"nat \u21d2 nat\" where\r\n  \"impar n \u2261 2 * n + 1\"\r\n\r\nlemma \"sum impar {i::nat. i < n} = n\u21e72\"\r\nproof (induct n)\r\n  show \"sum impar {i. i < 0} = 0\u21e72\"\r\n    by simp\r\nnext\r\n  fix n\r\n  assume HI : \"sum impar {i. i < n} = n\u21e72\"\r\n  have \"{i. i < Suc n} = {i. i < n} \u222a {n}\"\r\n    by auto\r\n  then have \"sum impar {i. i < Suc n} =\r\n             sum impar {i. i < n} + impar n\"\r\n    by simp\r\n  also have \"\u2026 = n\u21e72 + (2 * n + 1)\"\r\n    using HI impar_def by simp\r\n  also have \"\u2026 = (n + 1)\u21e72\"\r\n    by algebra\r\n  also have \"\u2026 = (Suc n)\u21e72\"\r\n    by simp\r\n  finally show \"sum impar {i. i < Suc n} = (Suc n)\u21e72\" .\r\nqed\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>En Lean, se puede definir el n-\u00e9simo n\u00famero primo por def impar (n : \u2115) := 2 * n + 1 Adem\u00e1s, en la librer\u00eda finset est\u00e1n definidas las funciones range :: \u2115 \u2192 finset \u2115 sum :: finset \u03b1 \u2192 (\u03b1 \u2192 \u03b2) \u2192 \u03b2 tales que + (range n) es el conjunto de los n primeros n\u00fameros naturales. Por ejemplo, el valor de (range 3) es {0, 1, 2}. + (sum A f) es la suma del conjunto obtenido aplicando la funci\u00f3n f a los elementos del conjunto finito A. Por ejemplo, el valor de (sum (range 3) impar) es 9. Demostrar que la suma de los n primeros n\u00fameros impares es n\u00b2. Para ello, completar la siguiente teor\u00eda de Lean: import data.finset&#8230;<\/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":[105,25],"tags":[201,77,177,100,198,205,84,203,202,204,169,154,147,82,111,99],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/728"}],"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=728"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/728\/revisions"}],"predecessor-version":[{"id":733,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/728\/revisions\/733"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=728"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=728"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=728"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}