ForMatUS: Pruebas en Lean de “Hay infinitos términos arbitrariamente próximos a los puntos de acumulación”
He añadido a la lista DAO (Demostración Asistida por Ordenador) con Lean el vídeo en el que se comentan 14 pruebas en Lean de la propiedad
Si a es un punto de acumulación de la sucesión u, entonces
∀ ε > 0, ∀ N, ∃ n ≥ N, |u n – a| ≤ ε
usando los estilos declarativo, aplicativo y funcional.
A continuación, se muestra el vídeo
y el código de la teoría utilizada
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import data.real.basic variable {u : ℕ → ℝ} variables {a : ℝ} variable {φ : ℕ → ℕ} -- ---------------------------------------------------- -- Nota. Usaremos los siguientes conceptos estudiados -- anteriormente. -- ---------------------------------------------------- notation `|`x`|` := abs x def limite : (ℕ → ℝ) → ℝ → Prop := λ u c, ∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - c| ≤ ε def extraccion : (ℕ → ℕ) → Prop | φ := ∀ n m, n < m → φ n < φ m lemma id_mne_extraccion (h : extraccion φ) : ∀ n, n ≤ φ n := begin intros n, induction n with m HI, { linarith }, { exact nat.succ_le_of_lt (by linarith [h m (m+1) (by linarith)]) }, end lemma extraccion_mye (h : extraccion φ) : ∀ N N', ∃ n ≥ N', φ n ≥ N := λ N N', ⟨max N N', le_max_right N N', le_trans (le_max_left N N') (id_mne_extraccion h (max N N'))⟩ -- ---------------------------------------------------- -- Ejercicio 1. Definir la función -- punto_acumulacion : (ℕ → ℝ) → ℝ → Prop -- tal que (punto_acumulacion u a) expresa que a es un -- punto de acumulación de u; es decir, que es el -- límite de alguna subsucesión de u. -- ---------------------------------------------------- def punto_acumulacion : (ℕ → ℝ) → ℝ → Prop | u a := ∃ φ, extraccion φ ∧ limite (u ∘ φ) a -- ---------------------------------------------------- -- Ejercicio 2. Demostrar que si a es un punto de -- acumulación de u, entonces -- ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε -- ---------------------------------------------------- -- 1ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := begin intros ε hε N, -- unfold punto_acumulacion at h, rcases h with ⟨φ, hφ1, hφ2⟩, -- unfold limite at hφ2, cases hφ2 ε hε with N' hN', rcases extraccion_mye hφ1 N N' with ⟨m, hm, hm'⟩, -- clear hφ1 hφ2, use φ m, split, { exact hm', }, { exact hN' m hm, }, end -- 2ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := begin intros ε hε N, rcases h with ⟨φ, hφ1, hφ2⟩, cases hφ2 ε hε with N' hN', rcases extraccion_mye hφ1 N N' with ⟨m, hm, hm'⟩, use φ m, exact ⟨hm', hN' m hm⟩, end -- 3ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := begin intros ε hε N, rcases h with ⟨φ, hφ1, hφ2⟩, cases hφ2 ε hε with N' hN', rcases extraccion_mye hφ1 N N' with ⟨m, hm, hm'⟩, exact ⟨φ m, hm', hN' _ hm⟩, end -- 4ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := begin intros ε hε N, rcases h with ⟨φ, hφ1, hφ2⟩, cases hφ2 ε hε with N' hN', rcases extraccion_mye hφ1 N N' with ⟨m, hm, hm'⟩, use φ m ; finish end -- 5ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 ( assume m, assume hm : ∃ (H : m ≥ N'), φ m ≥ N, exists.elim hm ( assume hm1 : m ≥ N', assume hm2 : φ m ≥ N, have h2 : |u (φ m) - a| ≤ ε, from hN' m hm1, show ∃ n ≥ N, |u n - a| ≤ ε, from exists.intro (φ m) (exists.intro hm2 h2))))) -- 6ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 ( assume m, assume hm : ∃ (H : m ≥ N'), φ m ≥ N, exists.elim hm ( assume hm1 : m ≥ N', assume hm2 : φ m ≥ N, have h2 : |u (φ m) - a| ≤ ε, from hN' m hm1, show ∃ n ≥ N, |u n - a| ≤ ε, from ⟨φ m, hm2, h2⟩)))) -- 7ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 ( assume m, assume hm : ∃ (H : m ≥ N'), φ m ≥ N, exists.elim hm ( assume hm1 : m ≥ N', assume hm2 : φ m ≥ N, have h2 : |u (φ m) - a| ≤ ε, from hN' m hm1, ⟨φ m, hm2, h2⟩)))) -- 8ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 ( assume m, assume hm : ∃ (H : m ≥ N'), φ m ≥ N, exists.elim hm ( assume hm1 : m ≥ N', assume hm2 : φ m ≥ N, ⟨φ m, hm2, hN' m hm1⟩)))) -- 9ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 ( assume m, assume hm : ∃ (H : m ≥ N'), φ m ≥ N, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) -- 10ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, have h1 : ∃ n ≥ N', φ n ≥ N, from extraccion_mye hφ.1 N N', exists.elim h1 (λ m hm, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) -- 11ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) ( assume N', assume hN' : ∀ (n : ℕ), n ≥ N' → |(u ∘ φ) n - a| ≤ ε, exists.elim (extraccion_mye hφ.1 N N') (λ m hm, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) -- 12ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h ( assume φ, assume hφ : extraccion φ ∧ limite (u ∘ φ) a, exists.elim (hφ.2 ε hε) (λ N' hN', exists.elim (extraccion_mye hφ.1 N N') (λ m hm, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) -- 13ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := assume ε, assume hε : ε > 0, assume N, exists.elim h (λ φ hφ, exists.elim (hφ.2 ε hε) (λ N' hN', exists.elim (extraccion_mye hφ.1 N N') (λ m hm, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) -- 14ª demostración example (h : punto_acumulacion u a) : ∀ ε > 0, ∀ N, ∃ n ≥ N, |u n - a| ≤ ε := λ ε hε N, exists.elim h (λ φ hφ, exists.elim (hφ.2 ε hε) (λ N' hN', exists.elim (extraccion_mye hφ.1 N N') (λ m hm, exists.elim hm (λ hm1 hm2, ⟨φ m, hm2, hN' m hm1⟩)))) |