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feat(theories/analysis/real_limit): fix analysis.real_limit

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Rob Lewis 2015-10-14 13:19:23 -04:00 committed by Leonardo de Moura
parent 4e78e35f77
commit 0aaa8a9770
1 changed files with 29 additions and 27 deletions
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library/theories/analysis/real_limit.lean
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@ -25,7 +25,7 @@ open real classical algebra
noncomputable theory noncomputable theory
namespace real namespace real
local postfix ⁻¹ := pnat.inv
/- the reals form a metric space -/ /- the reals form a metric space -/
protected definition to_metric_space [instance] : metric_space ℝ := protected definition to_metric_space [instance] : metric_space ℝ :=
@ -38,6 +38,7 @@ protected definition to_metric_space [instance] : metric_space ℝ :=
⦄ ⦄
open nat open nat
open [classes] rat
definition converges_to_seq (X : ℕ → ℝ) (y : ℝ) : Prop := definition converges_to_seq (X : ℕ → ℝ) (y : ℝ) : Prop :=
∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε
@ -122,15 +123,15 @@ section
∀ k : ℕ+, ∃ N : ℕ+, ∀ m n : ℕ+, ∀ k : ℕ+, ∃ N : ℕ+, ∀ m n : ℕ+,
m ≥ N → n ≥ N → abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹ := m ≥ N → n ≥ N → abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹ :=
take k : ℕ+, take k : ℕ+,
have H1 : (rat.gt k⁻¹ (rat.of_num 0)), from !inv_pos, have H1 : (k⁻¹ > (rat.of_num 0)), from !inv_pos,
have H2 : (of_rat k⁻¹ > of_rat (rat.of_num 0)), from !of_rat_lt_of_rat_of_lt H1, have H2 : (of_rat k⁻¹ > of_rat (rat.of_num 0)), from !of_rat_lt_of_rat_of_lt H1,
obtain (N : ℕ) (H : ∀ m n, m ≥ N → n ≥ N → abs (X m - X n) < of_rat k⁻¹), from H _ H2, obtain (N : ℕ) (H : ∀ m n, m ≥ N → n ≥ N → abs (X m - X n) < of_rat k⁻¹), from H _ H2,
exists.intro (pnat.succ N) exists.intro (pnat.succ N)
(take m n : ℕ+, (take m n : ℕ+,
assume Hm : m ≥ (pnat.succ N), assume Hm : m ≥ (pnat.succ N),
assume Hn : n ≥ (pnat.succ N), assume Hn : n ≥ (pnat.succ N),
have Hm' : elt_of m ≥ N, from nat.le.trans !le_succ Hm, have Hm' : elt_of m ≥ N, begin apply algebra.le.trans, apply le_succ, apply Hm end,
have Hn' : elt_of n ≥ N, from nat.le.trans !le_succ Hn, have Hn' : elt_of n ≥ N, begin apply algebra.le.trans, apply le_succ, apply Hn end,
show abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹, from le_of_lt (H _ _ Hm' Hn')) show abs (X (elt_of m) - X (elt_of n)) ≤ of_rat k⁻¹, from le_of_lt (H _ _ Hm' Hn'))
private definition rate_of_cauchy {X : ℕ → ℝ} (H : cauchy X) (k : ℕ+) : ℕ+ := private definition rate_of_cauchy {X : ℕ → ℝ} (H : cauchy X) (k : ℕ+) : ℕ+ :=
@ -165,7 +166,7 @@ exists.intro l
(take n : ℕ, (take n : ℕ,
assume Hn : n ≥ elt_of N, assume Hn : n ≥ elt_of N,
let n' : ℕ+ := tag n (nat.lt_of_lt_of_le (has_property N) Hn) in let n' : ℕ+ := tag n (nat.lt_of_lt_of_le (has_property N) Hn) in
have abs (X n - l) ≤ real.of_rat k⁻¹, from conv k n' Hn, have abs (X n - l) ≤ real.of_rat k⁻¹, by apply conv k n' Hn,
show abs (X n - l) < ε, from lt_of_le_of_lt this Hk)) show abs (X n - l) < ε, from lt_of_le_of_lt this Hk))
open set open set
@ -306,12 +307,12 @@ take ε, suppose ε > 0,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos, have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
obtain N₁ (HN₁ : ∀ {n}, n ≥ N₁ → abs (X n - x) < ε / 2), from HX e2pos, obtain N₁ (HN₁ : ∀ {n}, n ≥ N₁ → abs (X n - x) < ε / 2), from HX e2pos,
obtain N₂ (HN₂ : ∀ {n}, n ≥ N₂ → abs (Y n - y) < ε / 2), from HY e2pos, obtain N₂ (HN₂ : ∀ {n}, n ≥ N₂ → abs (Y n - y) < ε / 2), from HY e2pos,
let N := nat.max N₁ N₂ in let N := max N₁ N₂ in
exists.intro N exists.intro N
(take n, (take n,
suppose n ≥ N, suppose n ≥ N,
have ngtN₁ : n ≥ N₁, from nat.le.trans !nat.le_max_left `n ≥ N`, have ngtN₁ : n ≥ N₁, from nat.le.trans !le_max_left `n ≥ N`,
have ngtN₂ : n ≥ N₂, from nat.le.trans !nat.le_max_right `n ≥ N`, have ngtN₂ : n ≥ N₂, from nat.le.trans !le_max_right `n ≥ N`,
show abs ((X n + Y n) - (x + y)) < ε, from calc show abs ((X n + Y n) - (x + y)) < ε, from calc
abs ((X n + Y n) - (x + y)) abs ((X n + Y n) - (x + y))
= abs ((X n - x) + (Y n - y)) : by rewrite [sub_add_eq_sub_sub, *sub_eq_add_neg, = abs ((X n - x) + (Y n - y)) : by rewrite [sub_add_eq_sub_sub, *sub_eq_add_neg,
@ -373,7 +374,7 @@ take ε, suppose ε > 0,
obtain N (HN : ∀ n, n ≥ N → abs (X n - 0) < ε), from HX `ε > 0`, obtain N (HN : ∀ n, n ≥ N → abs (X n - 0) < ε), from HX `ε > 0`,
exists.intro N exists.intro N
(take n, assume Hn : n ≥ N, (take n, assume Hn : n ≥ N,
have abs (X n) < ε, from (!sub_zero ▸ HN n Hn), have abs (X n) < ε, begin rewrite -(sub_zero (X n)), apply HN n Hn end,
show abs (abs (X n) - 0) < ε, using this, show abs (abs (X n) - 0) < ε, using this,
by rewrite [sub_zero, abs_of_nonneg !abs_nonneg]; apply this) by rewrite [sub_zero, abs_of_nonneg !abs_nonneg]; apply this)
@ -385,7 +386,7 @@ exists.intro (N : ℕ)
(take n : ℕ, assume Hn : n ≥ N, (take n : ℕ, assume Hn : n ≥ N,
have HN' : abs (abs (X n) - 0) < ε, from HN n Hn, have HN' : abs (abs (X n) - 0) < ε, from HN n Hn,
have abs (X n) < ε, have abs (X n) < ε,
by+ rewrite [real.sub_zero at HN', abs_of_nonneg !abs_nonneg at HN']; apply HN', by+ rewrite [sub_zero at HN', abs_of_nonneg !abs_nonneg at HN']; apply HN',
show abs (X n - 0) < ε, using this, show abs (X n - 0) < ε, using this,
by rewrite sub_zero; apply this) by rewrite sub_zero; apply this)
@ -433,12 +434,12 @@ obtain x' [(H₁x' : x' ∈ X '[univ]) (H₂x' : sX - ε < x')],
from exists_mem_and_lt_of_lt_sup exX this, from exists_mem_and_lt_of_lt_sup exX this,
obtain i [H' (Hi : X i = x')], from H₁x', obtain i [H' (Hi : X i = x')], from H₁x',
have Hi' : ∀ j, j ≥ i → sX - ε < X j, from have Hi' : ∀ j, j ≥ i → sX - ε < X j, from
take j, assume Hj, lt_of_lt_of_le (Hi⁻¹ ▸ H₂x') (nondecX Hj), take j, assume Hj, lt_of_lt_of_le (by rewrite Hi; apply H₂x') (nondecX Hj),
exists.intro i exists.intro i
(take j, assume Hj : j ≥ i, (take j, assume Hj : j ≥ i,
have X j - sX ≤ 0, from sub_nonpos_of_le (Xle j), have X j - sX ≤ 0, from sub_nonpos_of_le (Xle j),
have eq₁ : abs (X j - sX) = sX - X j, using this, by rewrite [abs_of_nonpos this, neg_sub], have eq₁ : abs (X j - sX) = sX - X j, using this, by rewrite [abs_of_nonpos this, neg_sub],
have sX - ε < X j, from lt_of_lt_of_le (Hi⁻¹ ▸ H₂x') (nondecX Hj), have sX - ε < X j, from lt_of_lt_of_le (by rewrite Hi; apply H₂x') (nondecX Hj),
have sX < X j + ε, from lt_add_of_sub_lt_right this, have sX < X j + ε, from lt_add_of_sub_lt_right this,
have sX - X j < ε, from sub_lt_left_of_lt_add this, have sX - X j < ε, from sub_lt_left_of_lt_add this,
show (abs (X j - sX)) < ε, using eq₁ this, by rewrite eq₁; exact this) show (abs (X j - sX)) < ε, using eq₁ this, by rewrite eq₁; exact this)
@ -486,8 +487,8 @@ have H₂ : ∀ x, x ∈ X '[univ] → b ≤ x, from
obtain i [Hi₁ (Hi₂ : X i = x)], from H, obtain i [Hi₁ (Hi₂ : X i = x)], from H,
show b ≤ x, by rewrite -Hi₂; apply Hb i), show b ≤ x, by rewrite -Hi₂; apply Hb i),
have H₃ : {x : ℝ | -x ∈ X '[univ]} = {x : ℝ | x ∈ (λ n, -X n) '[univ]}, from calc have H₃ : {x : ℝ | -x ∈ X '[univ]} = {x : ℝ | x ∈ (λ n, -X n) '[univ]}, from calc
{x : ℝ | -x ∈ X '[univ]} = (λ y, -y) '[X '[univ]] : !image_neg_eq⁻¹ {x : ℝ | -x ∈ X '[univ]} = (λ y, -y) '[X '[univ]] : by rewrite image_neg_eq
... = {x : ℝ | x ∈ (λ n, -X n) '[univ]} : !image_compose⁻¹, ... = {x : ℝ | x ∈ (λ n, -X n) '[univ]} : image_compose,
have H₄ : ∀ i, - X i ≤ - b, from take i, neg_le_neg (Hb i), have H₄ : ∀ i, - X i ≤ - b, from take i, neg_le_neg (Hb i),
begin+ begin+
rewrite [-neg_converges_to_seq_iff, -sup_neg H₁ H₂, H₃, -nondecreasing_neg_iff at nonincX], rewrite [-neg_converges_to_seq_iff, -sup_neg H₁ H₂, H₃, -nondecreasing_neg_iff at nonincX],
@ -502,7 +503,7 @@ open nat set
theorem pow_converges_to_seq_zero {x : ℝ} (H : abs x < 1) : theorem pow_converges_to_seq_zero {x : ℝ} (H : abs x < 1) :
(λ n, x^n) ⟶ 0 in ℕ := (λ n, x^n) ⟶ 0 in ℕ :=
suffices H' : (λ n, (abs x)^n) ⟶ 0 in ℕ, from suffices H' : (λ n, (abs x)^n) ⟶ 0 in ℕ, from
have (λ n, (abs x)^n) = (λ n, abs (x^n)), from funext (take n, !abs_pow⁻¹), have (λ n, (abs x)^n) = (λ n, abs (x^n)), from funext (take n, eq.symm !abs_pow),
using this, using this,
by rewrite this at H'; exact converges_to_seq_zero_of_abs_converges_to_seq_zero H', by rewrite this at H'; exact converges_to_seq_zero_of_abs_converges_to_seq_zero H',
let aX := (λ n, (abs x)^n), let aX := (λ n, (abs x)^n),
@ -511,16 +512,17 @@ let aX := (λ n, (abs x)^n),
have noninc_aX : nonincreasing aX, from have noninc_aX : nonincreasing aX, from
nonincreasing_of_forall_succ_le nonincreasing_of_forall_succ_le
(take i, (take i,
have (abs x) * (abs x)^i ≤ 1 * (abs x)^i, assert (abs x) * (abs x)^i ≤ 1 * (abs x)^i,
from mul_le_mul_of_nonneg_right (le_of_lt H) (!pow_nonneg_of_nonneg !abs_nonneg), from mul_le_mul_of_nonneg_right (le_of_lt H) (!pow_nonneg_of_nonneg !abs_nonneg),
show (abs x) * (abs x)^i ≤ (abs x)^i, from !one_mul ▸ this), assert (abs x) * (abs x)^i ≤ (abs x)^i, by krewrite one_mul at this; exact this,
show (abs x) ^ (succ i) ≤ (abs x)^i, by rewrite pow_succ; apply this),
have bdd_aX : ∀ i, 0 ≤ aX i, from take i, !pow_nonneg_of_nonneg !abs_nonneg, have bdd_aX : ∀ i, 0 ≤ aX i, from take i, !pow_nonneg_of_nonneg !abs_nonneg,
have aXconv : aX ⟶ iaX in ℕ, from converges_to_seq_inf_of_nonincreasing noninc_aX bdd_aX, assert aXconv : aX ⟶ iaX in ℕ, from converges_to_seq_inf_of_nonincreasing noninc_aX bdd_aX,
have asXconv : asX ⟶ iaX in ℕ, from metric_space.converges_to_seq_offset_succ aXconv, have asXconv : asX ⟶ iaX in ℕ, from metric_space.converges_to_seq_offset_succ aXconv,
have asXconv' : asX ⟶ (abs x) * iaX in ℕ, from mul_left_converges_to_seq (abs x) aXconv, have asXconv' : asX ⟶ (abs x) * iaX in ℕ, from mul_left_converges_to_seq (abs x) aXconv,
have iaX = (abs x) * iaX, from converges_to_seq_unique asXconv asXconv', have iaX = (abs x) * iaX, from converges_to_seq_unique asXconv asXconv',
have iaX = 0, from eq_zero_of_mul_eq_self_left (ne_of_lt H) this⁻¹, assert iaX = 0, from eq_zero_of_mul_eq_self_left (ne_of_lt H) (eq.symm this),
show aX ⟶ 0 in ℕ, from this ▸ aXconv show aX ⟶ 0 in ℕ, begin rewrite -this, exact aXconv end --from this ▸ aXconv
end xn end xn
@ -572,7 +574,7 @@ theorem neg_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) : c
intros x' Hx', intros x' Hx',
let HD := Hδ₂ x' Hx', let HD := Hδ₂ x' Hx',
rewrite [-abs_neg, neg_neg_sub_neg], rewrite [-abs_neg, neg_neg_sub_neg],
assumption exact HD
end end
theorem translate_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) (a : ℝ) : theorem translate_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) (a : ℝ) :
@ -585,7 +587,7 @@ theorem translate_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous
split, split,
assumption, assumption,
intros x' Hx', intros x' Hx',
rewrite [add_sub_comm, sub_self, add_zero], rewrite [add_sub_comm, sub_self, algebra.add_zero],
apply Hδ₂, apply Hδ₂,
assumption assumption
end end
@ -622,7 +624,7 @@ private theorem ex_delta_lt {x : ℝ} (Hx : f x < 0) (Hxb : x < b) : ∃ δ :
exact div_two_lt_of_pos (and.left Hδ), exact div_two_lt_of_pos (and.left Hδ),
exact Haδ}, exact Haδ},
{apply and.right Hδ, {apply and.right Hδ,
rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg, krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)], abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
exact div_two_lt_of_pos (and.left Hδ)}, exact div_two_lt_of_pos (and.left Hδ)},
existsi (b - x) / 2, existsi (b - x) / 2,
@ -633,7 +635,7 @@ private theorem ex_delta_lt {x : ℝ} (Hx : f x < 0) (Hxb : x < b) : ∃ δ :
split, split,
{apply add_midpoint Hxb}, {apply add_midpoint Hxb},
{apply and.right Hδ, {apply and.right Hδ,
rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg, krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt Hxb) two_pos)], abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt Hxb) two_pos)],
apply lt_of_lt_of_le, apply lt_of_lt_of_le,
apply div_two_lt_of_pos (sub_pos_of_lt Hxb), apply div_two_lt_of_pos (sub_pos_of_lt Hxb),
@ -730,7 +732,7 @@ private theorem intermediate_value_incr_aux2 : ∃ δ : ℝ, δ > 0 ∧ a + δ <
exact div_two_lt_of_pos (and.left Hδ), exact div_two_lt_of_pos (and.left Hδ),
exact Haδ}, exact Haδ},
{apply and.right Hδ, {apply and.right Hδ,
rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg, krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)], abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
exact div_two_lt_of_pos (and.left Hδ)}, exact div_two_lt_of_pos (and.left Hδ)},
existsi (b - a) / 2, existsi (b - a) / 2,
@ -741,7 +743,7 @@ private theorem intermediate_value_incr_aux2 : ∃ δ : ℝ, δ > 0 ∧ a + δ <
split, split,
{apply add_midpoint Hab}, {apply add_midpoint Hab},
{apply and.right Hδ, {apply and.right Hδ,
rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg, krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt Hab) two_pos)], abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt Hab) two_pos)],
apply lt_of_lt_of_le, apply lt_of_lt_of_le,
apply div_two_lt_of_pos (sub_pos_of_lt Hab), apply div_two_lt_of_pos (sub_pos_of_lt Hab),
@ -775,7 +777,7 @@ theorem intermediate_value_incr_zero : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
apply le_of_not_gt, apply le_of_not_gt,
intro Hxgt, intro Hxgt,
have Hxgt' : b - x < δ, from sub_lt_of_sub_lt Hxgt, have Hxgt' : b - x < δ, from sub_lt_of_sub_lt Hxgt,
rewrite -(abs_of_pos (sub_pos_of_lt (and.left Hx))) at Hxgt', krewrite -(abs_of_pos (sub_pos_of_lt (and.left Hx))) at Hxgt',
let Hxgt'' := and.right Hδ _ Hxgt', let Hxgt'' := and.right Hδ _ Hxgt',
exact not_lt_of_ge (le_of_lt Hxgt'') (and.right Hx)}, exact not_lt_of_ge (le_of_lt Hxgt'') (and.right Hx)},
{exact sup_fn_interval} {exact sup_fn_interval}