{"id":360,"date":"2021-05-31T17:24:09","date_gmt":"2021-05-31T15:24:09","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=360"},"modified":"2021-08-21T12:53:21","modified_gmt":"2021-08-21T10:53:21","slug":"union-de-los-conjuntos-de-los-pares-e-impares","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/union-de-los-conjuntos-de-los-pares-e-impares\/","title":{"rendered":"Uni\u00f3n de los conjuntos de los pares e impares"},"content":{"rendered":"<p>Los conjuntos de los n\u00fameros naturales, de los pares y de los impares se definen por<\/p>\n<pre lang=\"lean\">\n   def naturales : set \u2115 := {n | true}\n   def pares     : set \u2115 := {n | even n}\n   def impares   : set \u2115 := {n | \u00ac even n}\n<\/pre>\n<p>Demostrar que<\/p>\n<pre lang=\"lean\">\n   pares \u222a impares = naturales\n<\/pre>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.nat.parity\nimport data.set.basic\nimport tactic\n\nopen set\n\ndef naturales : set \u2115 := {n | true}\ndef pares     : set \u2115 := {n | even n}\ndef impares   : set \u2115 := {n | \u00ac even n}\n\nexample : pares \u222a impares = naturales :=\nsorry\n<\/pre>\n<p><!--more--><\/p>\n<p><strong>Soluciones con Lean<\/strong><\/p>\n<pre lang=\"lean\">\nimport data.nat.parity\nimport data.set.basic\nimport tactic\n\nopen set\n\ndef naturales : set \u2115 := {n | true}\ndef pares     : set \u2115 := {n | even n}\ndef impares   : set \u2115 := {n | \u00ac even n}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : pares \u222a impares = naturales :=\nbegin\n  unfold pares impares naturales,\n  ext n,\n  simp,\n  apply classical.em,\nend\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : pares \u222a impares = naturales :=\nbegin\n  unfold pares impares naturales,\n  ext n,\n  finish,\nend\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : pares \u222a impares = naturales :=\nby finish [pares, impares, naturales, ext_iff]\n<\/pre>\n<p><strong>Soluciones con Isabelle\/HOL<\/strong><\/p>\n<pre lang=\"isar\">\ntheory Union_de_pares_e_impares\nimports Main\nbegin\n\ndefinition naturales :: \"nat set\" where\n  \"naturales = {n\u2208\u2115 . True}\"\n\ndefinition pares :: \"nat set\" where\n  \"pares = {n\u2208\u2115 . even n}\"\n\ndefinition impares :: \"nat set\" where\n  \"impares = {n\u2208\u2115 . \u00ac even n}\"\n\nsection \u20391\u00aa demostraci\u00f3n\u203a\n\nlemma \"pares \u222a impares = naturales\"\nproof -\n  have \"\u2200 n \u2208 \u2115 . even n \u2228 \u00ac even n \u27f7 True\"\n    by simp\n  then have \"{n \u2208 \u2115. even n} \u222a {n \u2208 \u2115. \u00ac even n} = {n \u2208 \u2115. True}\"\n    by auto\n  then show \"pares \u222a impares = naturales\"\n    by (simp add: naturales_def pares_def impares_def)\nqed\n\nsection \u20392\u00aa demostraci\u00f3n\u203a\n\nlemma \"pares \u222a impares = naturales\"\n  unfolding naturales_def pares_def impares_def\n  by auto\n\nend\n<\/pre>\n<p><strong>Nuevas soluciones<\/strong><\/p>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Los conjuntos de los n\u00fameros naturales, de los pares y de los impares se definen por def naturales : set \u2115 := {n | true} def pares : set \u2115 := {n | even n} def impares : set \u2115 := {n | \u00ac even n} Demostrar que pares \u222a impares = naturales Para ello, completar la siguiente teor\u00eda de Lean: import data.nat.parity import data.set.basic import tactic open set def naturales : set \u2115 := {n | true} def pares : set \u2115 := {n | even n} def impares : set \u2115 := {n | \u00ac even n} example : pares \u222a impares = naturales := 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":[7],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/360"}],"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=360"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/360\/revisions"}],"predecessor-version":[{"id":361,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/360\/revisions\/361"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=360"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=360"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}