{"id":976,"date":"2022-08-26T06:00:48","date_gmt":"2022-08-26T04:00:48","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=976"},"modified":"2022-09-11T13:42:23","modified_gmt":"2022-09-11T11:42:23","slug":"suma-por-diferencia","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/suma-por-diferencia\/","title":{"rendered":"Si a y b son n\u00fameros reales, entonces (a + b) * (a &#8211; b) = a^2 &#8211; b^2"},"content":{"rendered":"<p>Demostrar que si a y b son n\u00fameros reales, entonces<\/p>\n<pre lang=\"text\">\n(a + b) * (a - b) = a^2 - b^2\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\n\nvariables a b c d : \u211d\n\nexample : (a + b) * (a - b) = 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\n\nvariables a b c d : \u211d\n\n-- 1\u00aa demostraci\u00f3n\nexample : (a + b) * (a - b) = a^2 - b^2 :=\ncalc\n  (a + b) * (a - b)\n      = a * (a - b) + b * (a - b)         : by rw add_mul\n  ... = (a * a - a * b) + b * (a - b)     : by rw mul_sub\n  ... = (a^2 - a * b) + b * (a - b)       : by rw \u2190 pow_two\n  ... = (a^2 - a * b) + (b * a - b * b)   : by rw mul_sub\n  ... = (a^2 - a * b) + (b * a - b^2)     : by rw \u2190 pow_two\n  ... = (a^2 + -(a * b)) + (b * a - b^2)  : by ring\n  ... = a^2 + (-(a * b) + (b * a - b^2))  : by rw add_assoc\n  ... = a^2 + (-(a * b) + (b * a + -b^2)) : by ring\n  ... = a^2 + ((-(a * b) + b * a) + -b^2) : by rw \u2190 add_assoc\n                                               (-(a * b)) (b * a) (-b^2)\n  ... = a^2 + ((-(a * b) + a * b) + -b^2) : by rw mul_comm\n  ... = a^2 + (0 + -b^2)                  : by rw neg_add_self (a * b)\n  ... = (a^2 + 0) + -b^2                  : by rw \u2190 add_assoc\n  ... = a^2 + -b^2                        : by rw add_zero\n  ... = a^2 - b^2                         : by linarith\n\n\n-- 2\u00aa demostraci\u00f3n\nexample : (a + b) * (a - b) = a^2 - b^2 :=\ncalc\n  (a + b) * (a - b)\n      = a * (a - b) + b * (a - b)         : by ring\n  ... = (a * a - a * b) + b * (a - b)     : by ring\n  ... = (a^2 - a * b) + b * (a - b)       : by ring\n  ... = (a^2 - a * b) + (b * a - b * b)   : by ring\n  ... = (a^2 - a * b) + (b * a - b^2)     : by ring\n  ... = (a^2 + -(a * b)) + (b * a - b^2)  : by ring\n  ... = a^2 + (-(a * b) + (b * a - b^2))  : by ring\n  ... = a^2 + (-(a * b) + (b * a + -b^2)) : by ring\n  ... = a^2 + ((-(a * b) + b * a) + -b^2) : by ring\n  ... = a^2 + ((-(a * b) + a * b) + -b^2) : by ring\n  ... = a^2 + (0 + -b^2)                  : by ring\n  ... = (a^2 + 0) + -b^2                  : by ring\n  ... = a^2 + -b^2                        : by ring\n  ... = a^2 - b^2                         : by ring\n\n-- 3\u00aa demostraci\u00f3n\nexample : (a + b) * (a - b) = a^2 - b^2 :=\nby ring\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\/Suma_por_diferencia.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. 8.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que si a y b son n\u00fameros reales, entonces (a + b) * (a &#8211; b) = a^2 &#8211; b^2 Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables a b c d : \u211d example : (a + b) * (a &#8211; b) = a^2 &#8211; 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\/976"}],"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=976"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/976\/revisions"}],"predecessor-version":[{"id":1073,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/976\/revisions\/1073"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=976"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=976"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}