{"id":974,"date":"2022-08-25T06:00:51","date_gmt":"2022-08-25T04:00:51","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=974"},"modified":"2022-09-11T13:40:49","modified_gmt":"2022-09-11T11:40:49","slug":"producto-de-dos-binomios","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/producto-de-dos-binomios\/","title":{"rendered":"Si a, b, c y d son n\u00fameros reales, entonces (a + b) * (c + d) = a * c + a * d + b * c + b * d"},"content":{"rendered":"<p>Demostrar que si a, b, c y d son n\u00fameros reales, entonces<\/p>\n<pre lang=\"text\">\n(a + b) * (c + d) = a * c + a * d + b * c + b * d\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\n  : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\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\n-- ===============\n\nexample\n  : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\ncalc\n  (a + b) * (c + d)\n      = a * (c + d) + b * (c + d)       : by rw add_mul\n  ... = a * c + a * d + b * (c + d)     : by rw mul_add\n  ... = a * c + a * d + (b * c + b * d) : by rw mul_add\n  ... = a * c + a * d + b * c + b * d   : by rw \u2190add_assoc\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\ncalc\n  (a + b) * (c + d)\n      = a * (c + d) + b * (c + d)       : by ring\n  ... = a * c + a * d + b * (c + d)     : by ring\n  ... = a * c + a * d + (b * c + b * d) : by ring\n  ... = a * c + a * d + b * c + b * d   : by ring\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\nby ring\n\n-- 4\u00aa demostraci\u00f3n\n-- ===============\n\nexample\n  : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\nbegin\n   rw add_mul,\n   rw mul_add,\n   rw mul_add,\n   rw \u2190 add_assoc,\nend\n\n-- 5\u00aa demostraci\u00f3n\n-- ===============\n\nexample : (a + b) * (c + d) = a * c + a * d + b * c + b * d :=\nby rw [add_mul, mul_add, mul_add, \u2190add_assoc]\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\/Producto_de_dos_binomios.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, b, c y d son n\u00fameros reales, entonces (a + b) * (c + d) = a * c + a * d + b * c + b * d Para ello, completar la siguiente teor\u00eda de Lean: import data.real.basic variables a b c d : \u211d example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := 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\/974"}],"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=974"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/974\/revisions"}],"predecessor-version":[{"id":1072,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/974\/revisions\/1072"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}