{"id":1986,"date":"2024-01-26T06:00:11","date_gmt":"2024-01-26T04:00:11","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=1986"},"modified":"2024-01-26T21:13:47","modified_gmt":"2024-01-26T19:13:47","slug":"26-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/26-ene-24\/","title":{"rendered":"Si n\u00b2 es par, entonces n es par"},"content":{"rendered":"\n

Demostrar con Lean4 que si \\(n\u00b2\\) es par, entonces \\(n\\) es par.<\/p>\n

Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n

\r\nimport Mathlib.Tactic\r\nopen Nat\r\nvariable (n : \u2115)\r\n\r\nexample\r\n  (h : 2 \u2223 n ^ 2)\r\n  : 2 \u2223 n :=\r\nby sorry\r\n<\/pre>\n
<\/p>\n
1. Demostraci\u00f3n en lenguaje natural<\/h2>\n
Se usar\u00e1 el siguiente lema: Si \\(p\\) es primo, entonces
\n\\[ (\u2200 a, b \u2208 \u2115)[p \u2223 ab \u2194 p \u2223 a \u2228 p \u2223 b] \\]<\/p>\n
Si \\(n\u00b2\\) es par, entonces \\(2\\) divide a \\(n.n\\) y, por el lema, \\(2\\) divide a \\(n\\).<\/p>\n
2. Demostraciones con Lean4<\/h2>\n
\r\nimport Mathlib.Tactic\r\nopen Nat\r\nvariable (n : \u2115)\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : 2 \u2223 n ^ 2)\r\n  : 2 \u2223 n :=\r\nby\r\n  rw [pow_two] at h\r\n  -- h : 2 \u2223 n * n\r\n  have h1 : Nat.Prime 2 := prime_two\r\n  have h2 : 2 \u2223 n \u2228 2 \u2223 n := (Prime.dvd_mul h1).mp h\r\n  rcases h2 with h3 | h4\r\n  \u00b7 -- h3 : 2 \u2223 n\r\n    exact h3\r\n  \u00b7 -- h4 : 2 \u2223 n\r\n    exact h4\r\n  done\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : 2 \u2223 n ^ 2)\r\n  : 2 \u2223 n :=\r\nby\r\n  rw [pow_two] at h\r\n  -- h : 2 \u2223 n * n\r\n  have h2 : 2 \u2223 n \u2228 2 \u2223 n := (Prime.dvd_mul prime_two).mp h\r\n  rcases h2 with h3 | h4\r\n  \u00b7 exact h3\r\n  \u00b7 exact h4\r\n  done\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample\r\n  (h : 2 \u2223 n ^ 2)\r\n  : 2 \u2223 n :=\r\nby\r\n  rw [pow_two] at h\r\n  -- h : 2 \u2223 n * n\r\n  have h2 : 2 \u2223 n \u2228 2 \u2223 n := (Prime.dvd_mul prime_two).mp h\r\n  tauto\r\n  done\r\n\r\n-- Lemas usados\r\n-- ============\r\n\r\n-- variable (p a b : \u2115)\r\n-- #check (prime_two : Nat.Prime 2)\r\n-- #check (Prime.dvd_mul : Nat.Prime p \u2192 (p \u2223 a * b \u2194 p \u2223 a \u2228 p \u2223 b))\r\n<\/pre>\n
Demostraciones interactivas<\/h3>\n
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"
Demostrar con Lean4 que si \\(n\u00b2\\) es par, entonces \\(n\\) es par. Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Tactic open Nat variable (n : \u2115) example (h : 2 \u2223 n ^ 2) : 2 \u2223 n := 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":"","_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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1986"}],"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=1986"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1986\/revisions"}],"predecessor-version":[{"id":1988,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/1986\/revisions\/1988"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=1986"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=1986"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=1986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}