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Union_de_pares_e_impares.lean
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Union_de_pares_e_impares.lean
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-- Union_de_pares_e_impares.lean
-- Pares ∪ Impares = Naturales.
-- José A. Alonso Jiménez <https://jaalonso.github.io>
-- Sevilla, 5-marzo-2024
-- ---------------------------------------------------------------------
-- ---------------------------------------------------------------------
-- Los conjuntos de los números naturales, de los pares y de los impares
-- se definen por
-- def Naturales : Set ℕ := {n | True}
-- def Pares : Set ℕ := {n | Even n}
-- def Impares : Set ℕ := {n | ¬Even n}
--
-- Demostrar que
-- Pares ∪ Impares = Naturales
-- ----------------------------------------------------------------------
-- Demostración en lenguaje natural
-- ================================
-- Tenemos que demostrar que
-- {n | Even n} ∪ {n | ¬Even n} = {n | True}
-- es decir,
-- n ∈ {n | Even n} ∪ {n | ¬Even n} ↔ n ∈ {n | True}
-- que se reduce a
-- ⊢ Even n ∨ ¬Even n
-- que es una tautología.
-- Demostraciones con Lean4
-- ========================
import Mathlib.Algebra.Ring.Parity
def Naturales : Set ℕ := {_n | True}
def Pares : Set ℕ := {n | Even n}
def Impares : Set ℕ := {n | ¬Even n}
-- 1ª demostración
-- ===============
example : Pares ∪ Impares = Naturales :=
by
unfold Pares Impares Naturales
-- ⊢ {n | Even n} ∪ {n | ¬Even n} = {n | True}
ext n
-- ⊢ n ∈ {n | Even n} ∪ {n | ¬Even n} ↔ n ∈ {n | True}
simp only [Set.mem_setOf_eq, iff_true]
-- ⊢ n ∈ {n | Even n} ∪ {n | ¬Even n}
exact em (Even n)
-- 2ª demostración
-- ===============
example : Pares ∪ Impares = Naturales :=
by
unfold Pares Impares Naturales
-- ⊢ {n | Even n} ∪ {n | ¬Even n} = {n | True}
ext n
-- ⊢ n ∈ {n | Even n} ∪ {n | ¬Even n} ↔ n ∈ {n | True}
tauto