{"id":2301,"date":"2024-03-05T06:00:39","date_gmt":"2024-03-05T04:00:39","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2301"},"modified":"2024-03-05T19:42:07","modified_gmt":"2024-03-05T17:42:07","slug":"05-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/05-mar-24\/","title":{"rendered":"Pares \u222a Impares = Naturales"},"content":{"rendered":"\n

Los conjuntos de los n\u00fameros naturales, de los pares y de los impares se definen en Lean4 por<\/p>\n

\n   def Naturales : Set \u2115 := {n | True}\n   def Pares     : Set \u2115 := {n | Even n}\n   def Impares   : Set \u2115 := {n | \u00acEven n}\n<\/pre>\n
Demostrar con Lean4 que<\/p>\n
\n   Pares \u222a Impares = Naturales\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\nimport Mathlib.Data.Nat.Parity\nopen Set\n\ndef Naturales : Set \u2115 := {n | True}\ndef Pares     : Set \u2115 := {n | Even n}\ndef Impares   : Set \u2115 := {n | \u00acEven n}\n\nexample : Pares \u222a Impares = Naturales :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Tenemos que demostrar que
\n\[ \{n | \text{Even}(n)\} \u222a \{n | \u00ac\text{Even}(n)\} = \{n | \text{True}\} \]
\nes decir,
\n\[ n \u2208 \{n | \text{Even}(n)\} \u222a \{n | \u00ac\text{Even}(n)\} \u2194 n \u2208 \{n | \text{True}\} \]
\nque se reduce a
\n\[ \u22a2 \text{Even}(n) \u2228 \u00ac\text{Even}(n) \]
\nque es una tautolog\u00eda.<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Data.Nat.Parity\nopen Set\n\ndef Naturales : Set \u2115 := {n | True}\ndef Pares     : Set \u2115 := {n | Even n}\ndef Impares   : Set \u2115 := {n | \u00acEven n}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Pares \u222a Impares = Naturales :=\nby\n  unfold Pares Impares Naturales\n  -- \u22a2 {n | Even n} \u222a {n | \u00acEven n} = {n | True}\n  ext n\n  -- \u22a2 n \u2208 {n | Even n} \u222a {n | \u00acEven n} \u2194 n \u2208 {n | True}\n  simp\n  -- \u22a2 Even n \u2228 \u00acEven n\n  exact em (Even n)\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Pares \u222a Impares = Naturales :=\nby\n  unfold Pares Impares Naturales\n  -- \u22a2 {n | Even n} \u222a {n | \u00acEven n} = {n | True}\n  ext n\n  -- \u22a2 n \u2208 {n | Even n} \u222a {n | \u00acEven n} \u2194 n \u2208 {n | True}\n  tauto\n<\/pre>\n
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\n
3. Demostraciones con Isabelle\/HOL<\/h2>\n
\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\n(* 1\u00aa demostraci\u00f3n *)\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\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"pares \u222a impares = naturales\"\n  unfolding naturales_def pares_def impares_def\n  by auto\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Los conjuntos de los n\u00fameros naturales, de los pares y de los impares se definen en Lean4 por def Naturales : Set \u2115 := {n | True} def Pares : Set \u2115 := {n | Even n} def Impares : Set \u2115 := {n | \u00acEven n} Demostrar con Lean4 que Pares \u222a Impares = Naturales Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Nat.Parity open Set def Naturales : Set \u2115 := {n | True} def Pares : Set \u2115 := {n | Even n} def Impares : Set \u2115 := {n | \u00acEven n} example : Pares \u222a Impares = Naturales := by 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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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\/2301"}],"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=2301"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2301\/revisions"}],"predecessor-version":[{"id":2313,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2301\/revisions\/2313"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}