{"id":2311,"date":"2024-03-06T06:00:41","date_gmt":"2024-03-06T04:00:41","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=2311"},"modified":"2024-03-05T19:42:43","modified_gmt":"2024-03-05T17:42:43","slug":"06-mar-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/06-mar-24\/","title":{"rendered":"Los primos mayores que 2 son impares"},"content":{"rendered":"\n

Los n\u00fameros primos, los mayores que 2 y los impares se definen en Lean4 por<\/p>\n

\n   def Primos      : Set \u2115 := {n | Nat.Prime n}\n   def MayoresQue2 : Set \u2115 := {n | n > 2}\n   def Impares     : Set \u2115 := {n | \u00acEven n}\n<\/pre>\n
Demostrar con Lean4 que<\/p>\n
\n   Primos \u2229 MayoresQue2 \u2286 Impares\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\nimport Mathlib.Data.Nat.Parity\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.Tactic\n\nopen Nat\n\ndef Primos      : Set \u2115 := {n | Nat.Prime n}\ndef MayoresQue2 : Set \u2115 := {n | n > 2}\ndef Impares     : Set \u2115 := {n | \u00acEven n}\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby sorry\n<\/pre>\n
<\/p>\n
1. Demostraciones con Lean4<\/h2>\n
\nimport Mathlib.Data.Nat.Parity\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.Tactic\n\nopen Nat\n\ndef Primos      : Set \u2115 := {n | Nat.Prime n}\ndef MayoresQue2 : Set \u2115 := {n | n > 2}\ndef Impares     : Set \u2115 := {n | \u00acEven n}\n\n-- 1\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  intro n\n  -- n : \u2115\n  -- \u22a2 n \u2208 {n | Nat.Prime n} \u2229 {n | n > 2} \u2192 n \u2208 {n | \u00acEven n}\n  simp\n  -- \u22a2 Nat.Prime n \u2192 2 < n \u2192 \u00acEven n\n  intro hn\n  -- hn : Nat.Prime n\n  -- \u22a2 2 < n \u2192 \u00acEven n\n  rcases Prime.eq_two_or_odd hn with (h | h)\n  . -- h : n = 2\n    rw [h]\n    -- \u22a2 2 < 2 \u2192 \u00acEven 2\n    intro h1\n    -- h1 : 2 < 2\n    -- \u22a2 \u00acEven 2\n    exfalso\n    exact absurd h1 (lt_irrefl 2)\n  . -- h : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 2 < n \u2192 \u00acn % 2 = 0\n    rw [h]\n    -- \u22a2 2 < n \u2192 \u00ac1 = 0\n    intro\n    -- a : 2 < n\n    -- \u22a2 \u00ac1 = 0\n    exact one_ne_zero\n\n-- 2\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  rintro n \u27e8h1, h2\u27e9\n  -- n : \u2115\n  -- h1 : n \u2208 {n | Nat.Prime n}\n  -- h2 : n \u2208 {n | n > 2}\n  -- \u22a2 n \u2208 {n | \u00acEven n}\n  simp at *\n  -- h1 : Nat.Prime n\n  -- h2 : 2 < n\n  -- \u22a2 \u00acEven n\n  rcases Prime.eq_two_or_odd h1 with (h3 | h4)\n  . -- h3 : n = 2\n    rw [h3] at h2\n    -- h2 : 2 < 2\n    exfalso\n    -- \u22a2 False\n    exact absurd h2 (lt_irrefl 2)\n  . -- h4 : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 \u00acn % 2 = 0\n    rw [h4]\n    -- \u22a2 \u00ac1 = 0\n    exact one_ne_zero\n\n-- 3\u00aa demostraci\u00f3n\n-- ===============\n\nexample : Primos \u2229 MayoresQue2 \u2286 Impares :=\nby\n  unfold Primos MayoresQue2 Impares\n  -- \u22a2 {n | Nat.Prime n} \u2229 {n | n > 2} \u2286 {n | \u00acEven n}\n  rintro n \u27e8h1, h2\u27e9\n  -- n : \u2115\n  -- h1 : n \u2208 {n | Nat.Prime n}\n  -- h2 : n \u2208 {n | n > 2}\n  -- \u22a2 n \u2208 {n | \u00acEven n}\n  simp at *\n  -- h1 : Nat.Prime n\n  -- h2 : 2 < n\n  -- \u22a2 \u00acEven n\n  rcases Prime.eq_two_or_odd h1 with (h3 | h4)\n  . -- h3 : n = 2\n    rw [h3] at h2\n    -- h2 : 2 < 2\n    linarith\n  . -- h4 : n % 2 = 1\n    rw [even_iff]\n    -- \u22a2 \u00acn % 2 = 0\n    linarith\n\n-- Lemas usados\n-- ============\n\n-- variable (p n : \u2115)\n-- variable (a b : Prop)\n-- #check (Prime.eq_two_or_odd : Nat.Prime p \u2192 p = 2 \u2228 p % 2 = 1)\n-- #check (absurd : a \u2192 \u00aca \u2192 b)\n-- #check (even_iff : Even n \u2194 n % 2 = 0)\n-- #check (lt_irrefl n : \u00acn < n)\n-- #check (one_ne_zero : 1 \u2260 0)\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 Interseccion_de_los_primos_y_los_mayores_que_dos\nimports Main \"HOL-Number_Theory.Number_Theory\"\nbegin\n\ndefinition primos :: \"nat set\" where\n  \"primos = {n \u2208 \u2115 . prime n}\"\n\ndefinition mayoresQue2 :: \"nat set\" where\n  \"mayoresQue2 = {n \u2208 \u2115 . n > 2}\"\n\ndefinition impares :: \"nat set\" where\n  \"impares = {n \u2208 \u2115 . \u00ac even n}\"\n\n(* 1\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\nproof\n  fix x\n  assume \"x \u2208 primos \u2229 mayoresQue2\"\n  then have \"x \u2208 \u2115 \u2227 prime x \u2227 2 < x\"\n    by (simp add: primos_def mayoresQue2_def)\n  then have \"x \u2208 \u2115 \u2227 odd x\"\n    by (simp add: prime_odd_nat)\n  then show \"x \u2208 impares\"\n    by (simp add: impares_def)\nqed\n\n(* 2\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\n  unfolding primos_def mayoresQue2_def impares_def\n  by (simp add: Collect_mono_iff Int_def prime_odd_nat)\n\n(* 3\u00aa demostraci\u00f3n *)\nlemma \"primos \u2229 mayoresQue2 \u2286 impares\"\n  unfolding primos_def mayoresQue2_def impares_def\n  by (auto simp add: prime_odd_nat)\n\nend\n<\/pre>\n","protected":false},"excerpt":{"rendered":"
Los n\u00fameros primos, los mayores que 2 y los impares se definen en Lean4 por def Primos : Set \u2115 := {n | Nat.Prime n} def MayoresQue2 : Set \u2115 := {n | n > 2} def Impares : Set \u2115 := {n | \u00acEven n} Demostrar con Lean4 que Primos \u2229 MayoresQue2 \u2286 Impares Para ello, completar la siguiente teor\u00eda de Lean4: import Mathlib.Data.Nat.Parity import Mathlib.Data.Nat.Prime import Mathlib.Tactic open Nat def Primos : Set \u2115 := {n | Nat.Prime n} def MayoresQue2 : Set \u2115 := {n | n > 2} def Impares : Set \u2115 := {n | \u00acEven n} example : Primos \u2229 MayoresQue2 \u2286 Impares := 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\/2311"}],"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=2311"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2311\/revisions"}],"predecessor-version":[{"id":2315,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/2311\/revisions\/2315"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=2311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=2311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=2311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}