{"id":962,"date":"2022-08-21T07:00:23","date_gmt":"2022-08-21T05:00:23","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=962"},"modified":"2022-09-11T13:30:17","modified_gmt":"2022-09-11T11:30:17","slug":"el-producto-por-un-par-es-par","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/el-producto-por-un-par-es-par\/","title":{"rendered":"El producto por un par es par"},"content":{"rendered":"<p>Demostrar que los productos de los n\u00fameros naturales por n\u00fameros pares son pares.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.nat.parity\nimport tactic\n\nopen nat\n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nsorry\n<\/pre>\n<p><!-- more--><\/p>\n<p><!--more--><\/p>\n<p><b>Soluciones con Lean<\/b><\/p>\n<pre lang=\"lean\">\nimport data.nat.parity\nimport tactic\n\nopen nat\n\n-- 1\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  intros m n hn,\n  unfold even at *,\n  cases hn with k hk,\n  use m * k,\n  calc m * n\n       = m * (k + k)   : congr_arg (has_mul.mul m) hk\n   ... = m * k + m * k : mul_add m k k\nend\n\n-- 2\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  intros m n hn,\n  cases hn with k hk,\n  use m * k,\n  calc m * n\n       = m * (k + k)   : congr_arg (has_mul.mul m) hk\n   ... = m * k + m * k : mul_add m k k\nend\n\n-- 3\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  rintros m n \u27e8k, hk\u27e9,\n  use m * k,\n  calc m * n\n       = m * (k + k)   : congr_arg (has_mul.mul m) hk\n   ... = m * k + m * k : mul_add m k k\nend\n\n-- 4\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  rintros m n \u27e8k, hk\u27e9,\n  use m * k,\n  rw hk,\n  exact mul_add m k k,\nend\n\n-- 5\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  rintros m n \u27e8k, hk\u27e9,\n  use m * k,\n  rw [hk, mul_add]\nend\n\n-- 6\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\nbegin\n  rintros m n \u27e8k, hk\u27e9,\n  exact \u27e8m * k, by rw [hk, mul_add]\u27e9\nend\n\n-- 7\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\n\u03bb m n \u27e8k, hk\u27e9, \u27e8m * k, by rw [hk, mul_add]\u27e9\n\n-- 8\u00aa demostraci\u00f3n\nexample : \u2200 m n : \u2115, even n \u2192 even (m * n) :=\n  assume m n \u27e8k, (hk : n = k + k)\u27e9,\n  have hmn : m * n = m * k + m * k,\n    by rw [hk, mul_add],\n  show \u2203 l, m * n = l + l,\n    from \u27e8_, hmn\u27e9\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\/El_producto_por_un_par_es_par.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. 2.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Demostrar que los productos de los n\u00fameros naturales por n\u00fameros pares son pares. Para ello, completar la siguiente teor\u00eda de Lean: import data.nat.parity import tactic open nat example : \u2200 m n : \u2115, even n \u2192 even (m * n) := 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":[291,283],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/962"}],"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=962"}],"version-history":[{"count":7,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/962\/revisions"}],"predecessor-version":[{"id":1068,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/962\/revisions\/1068"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=962"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=962"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=962"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}