En Lean, una sucesión u₀, u₁, u₂, … se puede representar mediante una función (u : ℕ → ℝ) de forma que u(n) es uₙ.
Para extraer una subsucesión se aplica una función de extracción queconserva el orden; por ejemplo, la subsucesión
se ha obtenido con la función de extracción φ tal que φ(n) = 2*n.
En Lean, se puede definir que φ es una función de extracción por
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m |
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m
que a es un límite de u por
def limite (u : ℕ → ℝ) (a : ℝ) :=
∀ ε > 0, ∃ N, ∀ k ≥ N, |u k - a| < ε |
def limite (u : ℕ → ℝ) (a : ℝ) :=
∀ ε > 0, ∃ N, ∀ k ≥ N, |u k - a| < ε
que a es un punto de acumulación de u por
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a |
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a
que la sucesión u es de Cauchy por
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε |
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε
Demostrar que si u es una sucesión de Cauchy y a es un punto de acumulación de u, entonces a es el límite de u.
Para ello, completar la siguiente teoría de Lean:
import data.real.basic
open nat
variable {u : ℕ → ℝ}
variables {a : ℝ}
variable {φ : ℕ → ℕ}
notation `|`x`|` := abs x
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m
def limite (u : ℕ → ℝ) (l : ℝ) : Prop :=
∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| < ε
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
sorry |
import data.real.basic
open nat
variable {u : ℕ → ℝ}
variables {a : ℝ}
variable {φ : ℕ → ℕ}
notation `|`x`|` := abs x
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m
def limite (u : ℕ → ℝ) (l : ℝ) : Prop :=
∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| < ε
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
sorry
[expand title=»Soluciones con Lean»]
import data.real.basic
open nat
variable {u : ℕ → ℝ}
variables {a : ℝ}
variable {φ : ℕ → ℕ}
notation `|`x`|` := abs x
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m
def limite (u : ℕ → ℝ) (l : ℝ) : Prop :=
∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| < ε
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε
lemma aux1
(h : extraccion φ)
: ∀ n, n ≤ φ n :=
begin
intro n,
induction n with m HI,
{ exact nat.zero_le (φ 0), },
{ apply nat.succ_le_of_lt,
calc m ≤ φ m : HI
... < φ (succ m) : h m (m+1) (lt_add_one m), },
end
lemma aux2
(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')
(aux1 h (max N N'))⟩⟩
lemma cerca_acumulacion
(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 aux2 hφ1 N N' with ⟨m, hm, hm'⟩,
exact ⟨φ m, hm', hN' _ hm⟩,
end
-- 1ª demostración
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
begin
unfold limite,
intros ε hε,
unfold suc_cauchy at hu,
cases hu (ε/2) (half_pos hε) with N hN,
use N,
have ha' : ∃ N' ≥ N, |u N' - a| < ε/2,
apply cerca_acumulacion ha (ε/2) (half_pos hε),
cases ha' with N' h,
cases h with hNN' hN',
intros n hn,
calc |u n - a|
= |(u n - u N') + (u N' - a)| : by ring_nf
... ≤ |u n - u N'| + |u N' - a| : abs_add (u n - u N') (u N' - a)
... < ε/2 + |u N' - a| : add_lt_add_right (hN n hn N' hNN') _
... < ε/2 + ε/2 : add_lt_add_left hN' (ε / 2)
... = ε : add_halves ε
end
-- 2ª demostración
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
begin
intros ε hε,
cases hu (ε/2) (by linarith) with N hN,
use N,
have ha' : ∃ N' ≥ N, |u N' - a| < ε/2,
apply cerca_acumulacion ha (ε/2) (by linarith),
rcases ha' with ⟨N', hNN', hN'⟩,
intros n hn,
calc |u n - a|
= |(u n - u N') + (u N' - a)| : by ring_nf
... ≤ |u n - u N'| + |u N' - a| : by simp [abs_add]
... < ε : by linarith [hN n hn N' hNN'],
end |
import data.real.basic
open nat
variable {u : ℕ → ℝ}
variables {a : ℝ}
variable {φ : ℕ → ℕ}
notation `|`x`|` := abs x
def extraccion (φ : ℕ → ℕ) :=
∀ n m, n < m → φ n < φ m
def limite (u : ℕ → ℝ) (l : ℝ) : Prop :=
∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| < ε
def punto_acumulacion (u : ℕ → ℝ) (a : ℝ) :=
∃ φ, extraccion φ ∧ limite (u ∘ φ) a
def suc_cauchy (u : ℕ → ℝ) :=
∀ ε > 0, ∃ N, ∀ p ≥ N, ∀ q ≥ N, |u p - u q| < ε
lemma aux1
(h : extraccion φ)
: ∀ n, n ≤ φ n :=
begin
intro n,
induction n with m HI,
{ exact nat.zero_le (φ 0), },
{ apply nat.succ_le_of_lt,
calc m ≤ φ m : HI
... < φ (succ m) : h m (m+1) (lt_add_one m), },
end
lemma aux2
(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')
(aux1 h (max N N'))⟩⟩
lemma cerca_acumulacion
(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 aux2 hφ1 N N' with ⟨m, hm, hm'⟩,
exact ⟨φ m, hm', hN' _ hm⟩,
end
-- 1ª demostración
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
begin
unfold limite,
intros ε hε,
unfold suc_cauchy at hu,
cases hu (ε/2) (half_pos hε) with N hN,
use N,
have ha' : ∃ N' ≥ N, |u N' - a| < ε/2,
apply cerca_acumulacion ha (ε/2) (half_pos hε),
cases ha' with N' h,
cases h with hNN' hN',
intros n hn,
calc |u n - a|
= |(u n - u N') + (u N' - a)| : by ring_nf
... ≤ |u n - u N'| + |u N' - a| : abs_add (u n - u N') (u N' - a)
... < ε/2 + |u N' - a| : add_lt_add_right (hN n hn N' hNN') _
... < ε/2 + ε/2 : add_lt_add_left hN' (ε / 2)
... = ε : add_halves ε
end
-- 2ª demostración
example
(hu : suc_cauchy u)
(ha : punto_acumulacion u a)
: limite u a :=
begin
intros ε hε,
cases hu (ε/2) (by linarith) with N hN,
use N,
have ha' : ∃ N' ≥ N, |u N' - a| < ε/2,
apply cerca_acumulacion ha (ε/2) (by linarith),
rcases ha' with ⟨N', hNN', hN'⟩,
intros n hn,
calc |u n - a|
= |(u n - u N') + (u N' - a)| : by ring_nf
... ≤ |u n - u N'| + |u N' - a| : by simp [abs_add]
... < ε : by linarith [hN n hn N' hNN'],
end
Se puede interactuar con la prueba anterior en esta sesión con Lean.
En los comentarios se pueden escribir otras soluciones, escribiendo el código entre una línea con <pre lang="lean"> y otra con </pre>
[/expand]
[expand title=»Soluciones con Isabelle/HOL»]
theory "Si_a_es_un_punto_de_acumulacion_de_la_sucesion_de_Cauchy_u,_entonces_a_es_el_limite_de_u"
imports Main HOL.Real
begin
definition extraccion :: "(nat ⇒ nat) ⇒ bool" where
"extraccion φ ⟷ (∀ n m. n < m ⟶ φ n < φ m)"
definition limite :: "(nat ⇒ real) ⇒ real ⇒ bool"
where "limite u a ⟷ (∀ε>0. ∃N. ∀k≥N. ¦u k - a¦ < ε)"
definition punto_acumulacion :: "(nat ⇒ real) ⇒ real ⇒ bool"
where "punto_acumulacion u a ⟷ (∃φ. extraccion φ ∧ limite (u ∘ φ) a)"
definition suc_cauchy :: "(nat ⇒ real) ⇒ bool"
where "suc_cauchy u ⟷ (∀ε>0. ∃k. ∀m≥k. ∀n≥k. ¦u m - u n¦ < ε)"
(* Lemas auxiliares *)
lemma aux1 :
assumes "extraccion φ"
shows "n ≤ φ n"
proof (induct n)
show "0 ≤ φ 0" by simp
next
fix n assume HI : "n ≤ φ n"
then show "Suc n ≤ φ (Suc n)"
using assms extraccion_def
by (metis Suc_leI lessI order_le_less_subst1)
qed
lemma aux2 :
assumes "extraccion φ"
shows "∀ N N'. ∃ k ≥ N'. φ k ≥ N"
proof (intro allI)
fix N N' :: nat
have "max N N' ≥ N' ∧ φ (max N N') ≥ N"
by (meson assms aux1 max.bounded_iff max.cobounded2)
then show "∃k ≥ N'. φ k ≥ N"
by blast
qed
lemma cerca_acumulacion :
assumes "punto_acumulacion u a"
shows "∀ε>0. ∀ N. ∃k≥N. ¦u k - a¦ < ε"
proof (intro allI impI)
fix ε :: real and N :: nat
assume "ε > 0"
obtain φ where hφ1 : "extraccion φ"
and hφ2 : "limite (u ∘ φ) a"
using assms punto_acumulacion_def by blast
obtain N' where hN' : "∀k≥N'. ¦(u ∘ φ) k - a¦ < ε"
using hφ2 limite_def ‹ε > 0› by auto
obtain m where "m ≥ N' ∧ φ m ≥ N"
using aux2 hφ1 by blast
then show "∃k≥N. ¦u k - a¦ < ε"
using hN' by auto
qed
(* Demostración *)
lemma
assumes "suc_cauchy u"
"punto_acumulacion u a"
shows "limite u a"
proof (unfold limite_def; intro allI impI)
fix ε :: real
assume "ε > 0"
then have "ε/2 > 0"
by simp
then obtain N where hN : "∀m≥N. ∀n≥N. ¦u m - u n¦ < ε/2"
using assms(1) suc_cauchy_def
by blast
have "∀k≥N. ¦u k - a¦ < ε"
proof (intro allI impI)
fix k
assume hk : "k ≥ N"
obtain N' where hN'1 : "N' ≥ N" and
hN'2 : "¦u N' - a¦ < ε/2"
using assms(2) cerca_acumulacion ‹ε/2 > 0› by blast
have "¦u k - a¦ = ¦(u k - u N') + (u N' - a)¦"
by simp
also have "… ≤ ¦u k - u N'¦ + ¦u N' - a¦"
by simp
also have "… < ε/2 + ¦u N' - a¦"
using hk hN hN'1 by auto
also have "… < ε/2 + ε/2"
using hN'2 by auto
also have "… = ε"
by simp
finally show "¦u k - a¦ < ε" .
qed
then show "∃N. ∀k≥N. ¦u k - a¦ < ε"
by auto
qed
end |
theory "Si_a_es_un_punto_de_acumulacion_de_la_sucesion_de_Cauchy_u,_entonces_a_es_el_limite_de_u"
imports Main HOL.Real
begin
definition extraccion :: "(nat ⇒ nat) ⇒ bool" where
"extraccion φ ⟷ (∀ n m. n < m ⟶ φ n < φ m)"
definition limite :: "(nat ⇒ real) ⇒ real ⇒ bool"
where "limite u a ⟷ (∀ε>0. ∃N. ∀k≥N. ¦u k - a¦ < ε)"
definition punto_acumulacion :: "(nat ⇒ real) ⇒ real ⇒ bool"
where "punto_acumulacion u a ⟷ (∃φ. extraccion φ ∧ limite (u ∘ φ) a)"
definition suc_cauchy :: "(nat ⇒ real) ⇒ bool"
where "suc_cauchy u ⟷ (∀ε>0. ∃k. ∀m≥k. ∀n≥k. ¦u m - u n¦ < ε)"
(* Lemas auxiliares *)
lemma aux1 :
assumes "extraccion φ"
shows "n ≤ φ n"
proof (induct n)
show "0 ≤ φ 0" by simp
next
fix n assume HI : "n ≤ φ n"
then show "Suc n ≤ φ (Suc n)"
using assms extraccion_def
by (metis Suc_leI lessI order_le_less_subst1)
qed
lemma aux2 :
assumes "extraccion φ"
shows "∀ N N'. ∃ k ≥ N'. φ k ≥ N"
proof (intro allI)
fix N N' :: nat
have "max N N' ≥ N' ∧ φ (max N N') ≥ N"
by (meson assms aux1 max.bounded_iff max.cobounded2)
then show "∃k ≥ N'. φ k ≥ N"
by blast
qed
lemma cerca_acumulacion :
assumes "punto_acumulacion u a"
shows "∀ε>0. ∀ N. ∃k≥N. ¦u k - a¦ < ε"
proof (intro allI impI)
fix ε :: real and N :: nat
assume "ε > 0"
obtain φ where hφ1 : "extraccion φ"
and hφ2 : "limite (u ∘ φ) a"
using assms punto_acumulacion_def by blast
obtain N' where hN' : "∀k≥N'. ¦(u ∘ φ) k - a¦ < ε"
using hφ2 limite_def ‹ε > 0› by auto
obtain m where "m ≥ N' ∧ φ m ≥ N"
using aux2 hφ1 by blast
then show "∃k≥N. ¦u k - a¦ < ε"
using hN' by auto
qed
(* Demostración *)
lemma
assumes "suc_cauchy u"
"punto_acumulacion u a"
shows "limite u a"
proof (unfold limite_def; intro allI impI)
fix ε :: real
assume "ε > 0"
then have "ε/2 > 0"
by simp
then obtain N where hN : "∀m≥N. ∀n≥N. ¦u m - u n¦ < ε/2"
using assms(1) suc_cauchy_def
by blast
have "∀k≥N. ¦u k - a¦ < ε"
proof (intro allI impI)
fix k
assume hk : "k ≥ N"
obtain N' where hN'1 : "N' ≥ N" and
hN'2 : "¦u N' - a¦ < ε/2"
using assms(2) cerca_acumulacion ‹ε/2 > 0› by blast
have "¦u k - a¦ = ¦(u k - u N') + (u N' - a)¦"
by simp
also have "… ≤ ¦u k - u N'¦ + ¦u N' - a¦"
by simp
also have "… < ε/2 + ¦u N' - a¦"
using hk hN hN'1 by auto
also have "… < ε/2 + ε/2"
using hN'2 by auto
also have "… = ε"
by simp
finally show "¦u k - a¦ < ε" .
qed
then show "∃N. ∀k≥N. ¦u k - a¦ < ε"
by auto
qed
end
En los comentarios se pueden escribir otras soluciones, escribiendo el código entre una línea con <pre lang="isar"> y otra con </pre>
[/expand]