{"id":1058,"date":"2022-09-27T06:00:36","date_gmt":"2022-09-27T04:00:36","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1058"},"modified":"2022-09-11T16:42:33","modified_gmt":"2022-09-11T14:42:33","slug":"si-a-b-%e2%88%88-%e2%84%9d-entonces-2ab-%e2%89%a4-a%c2%b2-b%c2%b2","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-b-%e2%88%88-%e2%84%9d-entonces-2ab-%e2%89%a4-a%c2%b2-b%c2%b2\/","title":{"rendered":"Si a, b \u2208 \u211d, entonces 2ab \u2264 a\u00b2 + b\u00b2"},"content":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d, entonces 2ab \u2264 a\u00b2 + b\u00b2<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nimport tactic\n\nvariables a b : \u211d\n\nexample : 2*a*b \u2264 a^2 + b^2 :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nimport tactic\n\nvariables a b : \u211d\n\n-- 1\u00aa demostraci\u00f3n\nexample : 2*a*b \u2264 a^2 + b^2 :=\nbegin\n  have : 0 \u2264 (a - b)^2 := sq_nonneg (a - b),\n  have : 0 \u2264 a^2 - 2*a*b + b^2, by linarith,\n  show 2*a*b \u2264 a^2 + b^2, by linarith,\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : 2*a*b \u2264 a^2 + b^2 :=\nbegin\n  have h : 0 \u2264 a^2 - 2*a*b + b^2,\n  { calc a^2 - 2*a*b + b^2\n        = (a - b)^2                   : by ring\n    ... \u2265 0                           : by apply pow_two_nonneg },\n  calc 2*a*b\n       = 2*a*b + 0                   : by ring\n   ... \u2264 2*a*b + (a^2 - 2*a*b + b^2) : add_le_add (le_refl _) h\n   ... = a^2 + b^2                   : by ring\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : 2*a*b \u2264 a^2 + b^2 :=\nbegin\n  have : 0 \u2264 a^2 - 2*a*b + b^2,\n  { calc a^2 - 2*a*b + b^2\n         = (a - b)^2       : by ring\n     ... \u2265 0               : by apply pow_two_nonneg },\n  linarith,\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample : 2*a*b \u2264 a^2 + b^2 :=\n-- by library_search\ntwo_mul_le_add_sq a b\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\/Doble_del_producto_menor_que_suma_de_cuadrados.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p><b>Referencias<\/b><\/p>\n<ul>\n<li>J. Avigad, K. Buzzard, R.Y. Lewis y P. Massot. <a href=\"https:\/\/bit.ly\/3U4UjBk\">Mathematics in Lean<\/a>, p. 17.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a, b \u2208 \u211d, entonces 2ab \u2264 a\u00b2 + b\u00b2 Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic import tactic variables a b : \u211d example : 2*a*b \u2264 a^2 + b^2 := 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":[286],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1058"}],"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=1058"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1058\/revisions"}],"predecessor-version":[{"id":1106,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1058\/revisions\/1106"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}