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feat(library/theories): adapt analysis theory to use new topological limits

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Rob Lewis 2016-05-25 15:32:24 -04:00 committed by Leonardo de Moura
parent 6f25abfb87
commit 5a439942dd
10 changed files with 1050 additions and 647 deletions
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246
library/theories/analysis/bounded_linear_operator.lean
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@ -6,7 +6,8 @@ Author: Robert Y. Lewis
Bounded linear operators
-/
import .normed_space .real_limit algebra.module algebra.homomorphism
open real nat classical topology
open real nat classical topology set
--open normed_vector_space (this confuses lots of stuff??)
noncomputable theory
namespace analysis
@ -156,21 +157,60 @@ variables [HV : normed_vector_space V] [HW : normed_vector_space W]
include HV HW
definition is_frechet_deriv_at (f A : V → W) [is_bdd_linear_map A] (x : V) :=
(λ h : V, ∥f (x + h) - f x - A h ∥ / ∥ h ∥) ⟶ 0 at 0
(λ h : V, ∥f (x + h) - f x - A h ∥ / ∥ h ∥) ⟶ 0 [at 0]
lemma diff_quot_cts {f A : V → W} [HA : is_bdd_linear_map A] {y : V} (Hf : is_frechet_deriv_at f A y) :
continuous_at (λ h, ∥f (y + h) - f y - A h∥ / ∥h∥) 0 :=
begin
apply normed_vector_space.continuous_at_intro,
intro ε Hε,
cases normed_vector_space.approaches_at_dest Hf Hε with δ Hδ,
existsi δ,
split,
exact and.left Hδ,
intro x' Hx',
cases em (x' = 0) with Heq Hneq,
{rewrite [Heq, norm_zero, div_zero, sub_zero, norm_zero], apply Hε},
{rewrite [norm_zero, div_zero],
apply and.right Hδ,
repeat assumption}
end
theorem is_bdd_linear_map_of_eq {A B : V → W} [HA : is_bdd_linear_map A] (HAB : A = B) :
is_bdd_linear_map B :=
begin
fapply is_bdd_linear_map.mk,
all_goals try rewrite -HAB,
{apply hom_add},
{apply hom_smul},
{exact op_norm A},
{exact op_norm_pos A},
{rewrite -HAB, apply op_norm_bound}
end
definition is_frechet_deriv_at_of_eq {f A : V → W} [is_bdd_linear_map A] {x : V}
(Hfd : is_frechet_deriv_at f A x) {B : V → W} (HAB : A = B) :
@is_frechet_deriv_at _ _ _ _ f B (is_bdd_linear_map_of_eq HAB) x :=
begin
unfold is_frechet_deriv_at,
rewrite -HAB,
apply Hfd
end
theorem is_frechet_deriv_at_intro {f A : V → W} [is_bdd_linear_map A] {x : V}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 →
(∃ δ : ℝ, δ > 0 ∧ ∀ ⦃x' : V⦄, x' ≠ 0 ∧ ∥x'∥ < δ → ∥f (x + x') - f x - A x'∥ / ∥x'∥ < ε)) :
is_frechet_deriv_at f A x :=
begin
apply normed_vector_space.approaches_at_intro,
intros ε Hε,
cases H Hε with δ Hδ,
cases Hδ with Hδ Hδ',
existsi δ,
split,
assumption,
intros x' Hx',
cases Hx' with Hx'1 Hx'2,
intros x' Hx'1 Hx'2,
show abs (∥f (x + x') - f x - A x'∥ / ∥x'∥ - 0) < ε, begin
have H : ∥f (x + x') - f x - A x'∥ / ∥x'∥ ≥ 0,
from div_nonneg_of_nonneg_of_nonneg !norm_nonneg !norm_nonneg,
@ -179,7 +219,7 @@ theorem is_frechet_deriv_at_intro {f A : V → W} [is_bdd_linear_map A] {x : V}
split,
assumption,
rewrite [-sub_zero x'],
apply Hx'2
apply Hx'1
end
end
@ -188,14 +228,14 @@ theorem is_frechet_deriv_at_elim {f A : V → W} [is_bdd_linear_map A] {x : V} (
(∃ δ : ℝ, δ > 0 ∧ ∀ ⦃x' : V⦄, x' ≠ 0 ∧ ∥x'∥ < δ → ∥f (x + x') - f x - A x'∥ / ∥x'∥ < ε) :=
begin
intros ε Hε,
cases H Hε with δ Hδ,
cases normed_vector_space.approaches_at_dest H Hε with δ Hδ,
cases Hδ with Hδ Hδ',
existsi δ,
split,
assumption,
intros x' Hx',
rewrite [-sub_zero x' at Hx' {2}],
have Hδ'' : abs (∥f (x + x') - f x - A x'∥ / ∥x'∥ - 0) < ε, from Hδ' Hx',
have Hδ'' : abs (∥f (x + x') - f x - A x'∥ / ∥x'∥ - 0) < ε, from Hδ' (and.right Hx') (and.left Hx'),
have Hpos : ∥f (x + x') - f x - A x'∥ / ∥x'∥ ≥ 0, from div_nonneg_of_nonneg_of_nonneg !norm_nonneg !norm_nonneg,
rewrite [sub_zero at Hδ'', abs_of_nonneg Hpos at Hδ''],
assumption
@ -215,22 +255,36 @@ definition frechet_deriv_at_is_bdd_linear_map [instance] (f : V → W) (x : V) [
frechet_diffable_at.HA _ _ f x
theorem frechet_deriv_spec [Hf : frechet_diffable_at f x] :
(λ h : V, ∥f (x + h) - f x - (frechet_deriv_at f x h) ∥ / ∥ h ∥) ⟶ 0 at 0 :=
(λ h : V, ∥f (x + h) - f x - (frechet_deriv_at f x h) ∥ / ∥ h ∥) ⟶ 0 [at 0] :=
frechet_diffable_at.is_fr_der _ _ f x
theorem frechet_deriv_at_const (w : W) : is_frechet_deriv_at (λ v : V, w) (λ v : V, 0) x :=
begin
apply normed_vector_space.approaches_at_intro,
intros ε Hε,
existsi 1,
split,
exact zero_lt_one,
intros x' Hx',
rewrite [sub_self, sub_zero, norm_zero],
krewrite [zero_div, dist_self],
intros x' Hx' _,
rewrite [2 sub_self, norm_zero],
krewrite [zero_div, sub_zero, norm_zero],
assumption
end
theorem frechet_deriv_at_smul {c : ℝ} {A : V → W} [is_bdd_linear_map A]
theorem frechet_deriv_at_id : is_frechet_deriv_at (λ v : V, v) (λ v : V, v) x :=
begin
apply normed_vector_space.approaches_at_intro,
intros ε Hε,
existsi 1,
split,
exact zero_lt_one,
intros x' Hx' _,
have x + x' - x - x' = 0, by simp,
rewrite [this, norm_zero, zero_div, sub_self, norm_zero],
exact Hε
end
theorem frechet_deriv_at_smul (c : ℝ) {A : V → W} [is_bdd_linear_map A]
(Hf : is_frechet_deriv_at f A x) : is_frechet_deriv_at (λ y, c • f y) (λ y, c • A y) x :=
begin
eapply @decidable.cases_on (abs c = 0),
@ -240,31 +294,32 @@ theorem frechet_deriv_at_smul {c : ℝ} {A : V → W} [is_bdd_linear_map A]
have Hfz : (λ y : V, (0 : ℝ) • f y) = (λ y : V, 0), from funext (λ y, !zero_smul),
--have Hfz' : (λ x : V, (0 : ℝ) • A x) = (λ x : V, 0), from funext (λ y, !zero_smul),
-- rewriting Hfz' produces type-incorrect term
rewrite [Hz, Hfz, ↑is_frechet_deriv_at],
rewrite [Hz, Hfz],
apply metric_space.approaches_at_intro,
intro ε Hε,
existsi 1,
split,
exact zero_lt_one,
intro x' Hx',
intro x' Hx' _,
rewrite [zero_smul, *sub_zero, norm_zero],
krewrite [zero_div, dist_self],
exact Hε},
intro Hcnz,
rewrite ↑is_frechet_deriv_at,
apply normed_vector_space.approaches_at_intro,
intros ε Hε,
have Hεc : ε / abs c > 0, from div_pos_of_pos_of_pos Hε (lt_of_le_of_ne !abs_nonneg (ne.symm Hcnz)),
cases Hf Hεc with δ Hδ,
cases normed_vector_space.approaches_at_dest Hf Hεc with δ Hδ,
cases Hδ with Hδp Hδ,
existsi δ,
split,
assumption,
intro x' Hx',
intro x' Hx' _,
show abs ((∥c • f (x + x') - c • f x - c • A x'∥ / ∥x'∥ - 0)) < ε, begin
rewrite [sub_zero, -2 smul_sub_left_distrib, norm_smul],
krewrite mul_div_assoc,
rewrite [abs_mul, abs_abs, -{ε}mul_div_cancel' Hcnz],
apply mul_lt_mul_of_pos_left,
have Hδ' : abs (∥f (x + x') - f x - A x'∥ / ∥x'∥ - 0) < ε / abs c, from Hδ Hx',
have Hδ' : abs (∥f (x + x') - f x - A x'∥ / ∥x'∥ - 0) < ε / abs c, from Hδ Hx' a,
rewrite sub_zero at Hδ',
apply Hδ',
apply lt_of_le_of_ne,
@ -309,17 +364,19 @@ theorem frechet_deriv_at_add (A B : V → W) [is_bdd_linear_map A] [is_bdd_linea
krewrite [Hhe, *div_zero, zero_add],
eapply le.refl
end,
have Hlimge : (λ h, ∥f (x + h) - f x - A h∥ / ∥h∥ + ∥g (x + h) - g x - B h∥ / ∥h∥) ⟶ 0 at 0, begin
have Hlimge : (λ h, ∥f (x + h) - f x - A h∥ / ∥h∥ + ∥g (x + h) - g x - B h∥ / ∥h∥) ⟶ 0 [at 0], begin
rewrite [-zero_add 0],
apply add_converges_to_at,
apply add_approaches,
apply Hf,
apply Hg
end,
have Hlimle : (λ (h : V), (0 : ℝ)) ⟶ 0 at 0, from converges_to_at_constant 0 0,
apply converges_to_at_squeeze Hlimle Hlimge,
have Hlimle : (λ (h : V), (0 : ℝ)) ⟶ 0 [at 0], by apply approaches_constant,
apply approaches_squeeze Hlimle Hlimge,
apply filter.eventually_of_forall,
intro y,
apply div_nonneg_of_nonneg_of_nonneg,
repeat apply norm_nonneg,
apply filter.eventually_of_forall,
apply Hle
end
@ -329,7 +386,7 @@ theorem continuous_at_of_diffable_at [Hf : frechet_diffable_at f x] : continuous
begin
apply normed_vector_space.continuous_at_intro,
intros ε Hε,
note Hfds := frechet_deriv_spec f x Hε,
note Hfds := normed_vector_space.approaches_at_dest (frechet_deriv_spec f x) Hε,
cases Hfds with δ Hδ,
cases Hδ with Hδ Hδ',
existsi min δ ((ε / 2) / (ε + op_norm (frechet_deriv_at f x))),
@ -343,15 +400,15 @@ theorem continuous_at_of_diffable_at [Hf : frechet_diffable_at f x] : continuous
{intro x' Hx',
cases em (x' - x = 0) with Heq Hneq,
rewrite [eq_of_sub_eq_zero Heq, sub_self, norm_zero], assumption,
have Hx'x : x' - x ≠ 0 ∧ dist (x' - x) 0 < δ, from and.intro Hneq begin
change ∥(x' - x) - 0∥ < δ,
have Hx'x : x' - x ≠ 0 ∧ ∥(x' - x) - 0∥ < δ, from and.intro Hneq begin
rewrite sub_zero,
apply lt_of_lt_of_le,
apply Hx',
apply min_le_left
end,
have Hx'xp : ∥x' - x∥ > 0, from norm_pos_of_ne_zero Hneq,
have Hδ'' : abs (∥f (x + (x' - x)) - f x - frechet_deriv_at f x (x' - x)∥ / ∥x' - x∥ - 0) < ε, from Hδ' Hx'x,
have Hδ'' : abs (∥f (x + (x' - x)) - f x - frechet_deriv_at f x (x' - x)∥ / ∥x' - x∥ - 0) < ε,
from Hδ' (and.right Hx'x) (and.left Hx'x),
have Hnn : ∥f (x + (x' - x)) - f x - frechet_deriv_at f x (x' - x)∥ / ∥x' - x∥ ≥ 0,
from div_nonneg_of_nonneg_of_nonneg !norm_nonneg !norm_nonneg,
rewrite [sub_zero at Hδ'', abs_of_nonneg Hnn at Hδ'', add.comm at Hδ'', sub_add_cancel at Hδ''],
@ -378,9 +435,31 @@ theorem continuous_at_of_diffable_at [Hf : frechet_diffable_at f x] : continuous
... < ε / 2 : mul_div_add_self_lt (div_pos_of_pos_of_pos Hε two_pos) Hε !op_norm_pos}
end
theorem continuous_at_of_is_frechet_deriv_at {A : V → W} [is_bdd_linear_map A]
(H : is_frechet_deriv_at f A x) : continuous_at f x :=
begin
apply @continuous_at_of_diffable_at,
existsi A,
exact H
end
end frechet_deriv
/-section comp
section comp
lemma div_mul_div_cancel {A : Type} [field A] (a b : A) {c : A} (Hc : c ≠ 0) : (a / c) * (c / b) = a / b :=
by rewrite [-mul_div_assoc, div_mul_cancel _ Hc]
lemma div_add_eq_add_mul_div {A : Type} [field A] (a b c : A) (Hb : b ≠ 0) : (a / b) + c = (a + c * b) / b :=
by rewrite [-div_add_div_same, mul_div_cancel _ Hb]
-- I'm not sure why smul_approaches doesn't unify where I use this?
lemma real_limit_helper {U : Type} [normed_vector_space U] {f : U → ℝ} {a : ℝ} {x : U}
(Hf : f ⟶ a [at x]) (c : ℝ) : (λ y, c * f y) ⟶ c * a [at x] :=
begin
apply smul_approaches,
exact Hf
end
variables {U V W : Type}
variables [HU : normed_vector_space U] [HV : normed_vector_space V] [HW : normed_vector_space W]
@ -388,15 +467,122 @@ variables {f : V → W} {g : U → V}
variables {A : V → W} {B : U → V}
variables [HA : is_bdd_linear_map A] [HB : is_bdd_linear_map B]
variable {x : U}
include HU HV HW HA HB
-- this takes 2 seconds without clearing the contexts before simp
theorem frechet_derivative_at_comp (Hg : is_frechet_deriv_at g B x) (Hf : is_frechet_deriv_at f A (g x)) :
@is_frechet_deriv_at _ _ _ _ (λ y, f (g y)) (λ y, A (B y)) !is_bdd_linear_map_comp x :=
begin
rewrite ↑is_frechet_deriv_at,
intros ε Hε,
unfold is_frechet_deriv_at,
note Hf' := is_frechet_deriv_at_elim Hf,
note Hg' := is_frechet_deriv_at_elim Hg,
have H : ∀ h, f (g (x + h)) - f (g x) - A (B h) =
(A (g (x + h) - g x - B h)) + (-f (g x) + f (g (x + h)) + A (g x - g (x + h))),
begin intro; rewrite [3 hom_sub A], clear [Hf, Hg, Hf', Hg'], simp end, -- .5 seconds for simp
have H' : (λ h, ∥f (g (x + h)) - f (g x) - A (B h)∥ / ∥h∥) =
(λ h, ∥(A (g (x + h) - g x - B h)) + (-f (g x) + f (g (x + h)) + A (g x - g (x + h)))∥ / ∥h∥),
from funext (λ h, by rewrite H),
rewrite H',
clear [H, H'],
apply approaches_squeeze, -- show the limit holds by bounding it by something that vanishes
{apply approaches_constant},
rotate 1,
{apply filter.eventually_of_forall, intro, apply div_nonneg_of_nonneg_of_nonneg, repeat apply norm_nonneg},
{apply filter.eventually_of_forall, intro,
apply div_le_div_of_le_of_nonneg,
apply norm_triangle,
apply norm_nonneg},
have H : (λ (y : U), (∥A (g (x + y) - g x - B y)∥ + ∥-f (g x) + f (g (x + y)) + A (g x - g (x + y))∥) / ∥y∥) =
(λ (y : U), (∥A (g (x + y) - g x - B y)∥ / ∥y∥ + ∥-f (g x) + f (g (x + y)) + A (g x - g (x + y))∥ / ∥y∥)),
from funext (λ y, by rewrite [div_add_div_same]),
rewrite [H, -zero_add 0], -- the function is a sum of two things that both vanish
clear H,
apply add_approaches,
{apply approaches_squeeze, -- show the lhs vanishes by squeezing it again
{apply approaches_constant},
rotate 1,
{apply filter.eventually_of_forall, intro, apply div_nonneg_of_nonneg_of_nonneg, repeat apply norm_nonneg},
{apply filter.eventually_of_forall, intro y,
show ∥A (g (x + y) - g x - B y)∥ / ∥y∥ ≤ op_norm A * (∥(g (x + y) - g x - B y)∥ / ∥y∥), begin
rewrite -mul_div_assoc,
apply div_le_div_of_le_of_nonneg,
apply op_norm_bound A,
apply norm_nonneg
end},
{rewrite [-mul_zero (op_norm A)],
apply real_limit_helper,
apply Hg}}, -- we have shown the lhs vanishes. now the rhs
{have H : ∀ y, (∥-f (g x) + f (g (x + y)) + A (g x - g (x + y))∥ / ∥y∥) =
((∥(f (g (x + y)) - f (g x)) - A (g (x + y) - g x) ∥ / ∥g (x + y) - g x∥) * (∥g (x + y) - g x∥ / ∥y∥)),
begin
intro,
cases em (g (x + y) - g x = 0) with Heq Hneq,
{note Heq' := eq_of_sub_eq_zero Heq,
rewrite [Heq', neg_add_eq_sub, *sub_self, hom_zero A, add_zero, *norm_zero, div_zero, zero_div]},
{rewrite [div_mul_div_cancel _ _ (norm_ne_zero_of_ne_zero Hneq), *sub_eq_add_neg,
-hom_neg A],
simp} --(.5 seconds)
end,
apply approaches_squeeze, -- again, by squeezing
{apply approaches_constant},
rotate 1,
{apply filter.eventually_of_forall, intro, apply div_nonneg_of_nonneg_of_nonneg, repeat apply norm_nonneg},
{apply filter.eventually_of_forall, intro y, rewrite H,
apply mul_le_mul_of_nonneg_left,
{show ∥g (x + y) - g x∥ / ∥y∥ ≤ ∥g (x + y) - g x - B y∥ / ∥y∥ + op_norm B, begin
cases em (y = 0) with Heq Hneq,
{rewrite [Heq, norm_zero, *div_zero, zero_add], apply le_of_lt, apply op_norm_pos},
rewrite [div_add_eq_add_mul_div _ _ _ (norm_ne_zero_of_ne_zero Hneq)],
apply div_le_div_of_le_of_nonneg,
apply le.trans,
rotate 1,
apply add_le_add_left,
apply op_norm_bound,
apply norm_nonneg,
rewrite [-neg_add_cancel_right (g (x + y) - g x) (B y) at {1}, -sub_eq_add_neg],
apply norm_triangle
end},
{apply div_nonneg_of_nonneg_of_nonneg, repeat apply norm_nonneg}},
-- now to show the bounding function vanishes. it is a product of a vanishing function and a convergent one
apply mul_approaches_zero_of_approaches_zero_of_approaches,
{have H' : (λ (y : U), ∥f (g (x + y)) - f (g x) - A (g (x + y) - g x)∥ / ∥g (x + y) - g x∥) =
(λ (y : U), ∥f (g x + (g (x + y) - g x)) - f (g x) - A (g (x + y) - g x)∥ / ∥g (x + y) - g x∥),
from funext (λ y, by rewrite [add.comm (g x), sub_add_cancel]), -- first, show lhs vanishes
rewrite H',
have Hgcts : continuous_at (λ y, g (x + y) - g x) 0, begin
apply normed_vector_space.continuous_at_intro,
intro ε Hε,
cases normed_vector_space.continuous_at_dest (continuous_at_of_is_frechet_deriv_at g x Hg) _ Hε with δ Hδ,
existsi δ,
split,
exact and.left Hδ,
intro x' Hx',
rewrite [add_zero, sub_self],
rewrite sub_zero,
apply and.right Hδ,
have (x + x') - x = x' - 0, begin clear [Hg, Hf, Hf', Hg', H, H', Hδ, Hx'], simp end, -- (.6 seconds w/o clear, .1 with)
rewrite this,
apply Hx'
end,
have Hfcts : continuous_at (λ (x' : V), ∥f (g x + x') - f (g x) - A x'∥ / ∥x'∥) (g (x + 0) - g x), begin
rewrite [add_zero, sub_self],
apply diff_quot_cts,
exact Hf
end,
have Heqz : ∥f (g x + (g (x + 0) - g x)) - f (g x) - A (g (x + 0) - g x)∥ / ∥g (x + 0) - g x∥ = 0,
by rewrite [*add_zero, sub_self, norm_zero, div_zero],
apply @tendsto_comp _ _ _ (λ y, g (x + y) - g x),
apply tendsto_inf_left,
apply tendsto_at_of_continuous_at Hgcts,
note Hfcts' := tendsto_at_of_continuous_at Hfcts,
rewrite Heqz at Hfcts',
exact Hfcts'}, -- finally, show rhs converges to op_norm B
{apply add_approaches,
apply Hg,
apply approaches_constant}}
end
end comp-/
end comp
end analysis

6
library/theories/analysis/ivt.lean
View file

@ -204,7 +204,7 @@ theorem intermediate_value_decr_zero {f : ℝ → ℝ} (Hf : continuous f) {a b
begin
have Ha' : - f a < 0, from neg_neg_of_pos Ha,
have Hb' : - f b > 0, from neg_pos_of_neg Hb,
have Hcon : continuous (λ x, - f x), from continuous_neg_of_continuous Hf,
have Hcon : continuous (λ x, - f x), from neg_continuous Hf,
let Hiv := intermediate_value_incr_zero Hcon Hab Ha' Hb',
cases Hiv with c Hc,
existsi c,
@ -220,7 +220,7 @@ theorem intermediate_value_incr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ
(Hav : f a < v) (Hbv : f b > v) : ∃ c, a < c ∧ c < b ∧ f c = v :=
have Hav' : f a - v < 0, from sub_neg_of_lt Hav,
have Hbv' : f b - v > 0, from sub_pos_of_lt Hbv,
have Hcon : continuous (λ x, f x - v), from continuous_offset_of_continuous Hf _,
have Hcon : continuous (λ x, f x - v), from sub_continuous Hf (continuous_const _),
have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_incr_zero Hcon Hab Hav' Hbv',
obtain c Hc, from Hiv,
exists.intro c
@ -230,7 +230,7 @@ theorem intermediate_value_decr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ
(Hav : f a > v) (Hbv : f b < v) : ∃ c, a < c ∧ c < b ∧ f c = v :=
have Hav' : f a - v > 0, from sub_pos_of_lt Hav,
have Hbv' : f b - v < 0, from sub_neg_of_lt Hbv,
have Hcon : continuous (λ x, f x - v), from continuous_offset_of_continuous Hf _,
have Hcon : continuous (λ x, f x - v), from sub_continuous Hf (continuous_const _),
have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_decr_zero Hcon Hab Hav' Hbv',
obtain c Hc, from Hiv,
exists.intro c

398
library/theories/analysis/metric_space.lean
View file

@ -2,6 +2,8 @@
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Metric spaces.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import data.real.complete data.pnat ..topology.continuous ..topology.limit data.set
open nat real eq.ops classical set prod set.filter topology interval
@ -215,6 +217,11 @@ proposition eventually_at_iff (P : M → Prop) (x : M) :
eventually P [at x] ↔ ∃ ε, ε > 0 ∧ ∀ ⦃x'⦄, dist x' x < ε → x' ≠ x → P x' :=
iff.intro eventually_at_dest (λ H, obtain ε [εpos Hε], from H, eventually_at_intro εpos Hε)
end metric_space_M
namespace metric_space
variables {M : Type} [metric_space M]
section approaches
variables {X : Type} {F : filter X} {f : X → M} {y : M}
@ -236,13 +243,11 @@ section approaches
proposition approaches_iff : (f ⟶ y) F ↔ (∀ ε, ε > 0 → eventually (λ x, dist (f x) y < ε) F) :=
iff.intro approaches_dest approaches_intro
-- TODO: prove this in greater generality in topology.limit
proposition approaches_constant : ((λ x, y) ⟶ y) F :=
approaches_intro (λ ε εpos, eventually_of_forall F (λ x,
show dist y y < ε, by rewrite dist_self; apply εpos))
end approaches
-- here we full unwrap two particular kinds of convergence3
-- TODO: put these in metric space namespace? (will have similar in normed_space
proposition approaches_at_infty_intro {f : ℕ → M} {y : M}
(H : ∀ ε, ε > 0 → ∃ N, ∀ n, n ≥ N → dist (f n) y < ε) :
@ -282,8 +287,13 @@ proposition approaches_at_iff (f : M → N) (y : N) (x : M) : f ⟶ y [at x] ↔
iff.intro approaches_at_dest approaches_at_intro
end metric_space_N
end metric_space -- close namespace
-- TODO: remove this. It is only here temporarily, because it is used in normed_space
section metric_space_M
variables {M : Type} [metric_space M]
-- TODO: remove this. It is only here temporarily, because it is used in normed_space (JA)
-- It is used in the definition of a complete metric space below, but I think it doesn't
-- have to be a class (RL)
abbreviation converges_seq [class] (X : ℕ → M) : Prop := ∃ y, X ⟶ y [at ∞]
-- TODO: refactor
@ -291,7 +301,7 @@ abbreviation converges_seq [class] (X : ℕ → M) : Prop := ∃ y, X ⟶ y [at
definition approaches_at_infty_intro' {X : ℕ → M} {y : M}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → dist (X n) y ≤ ε) :
(X ⟶ y) [at ∞] :=
approaches_at_infty_intro
metric_space.approaches_at_infty_intro
take ε, assume epos : ε > 0,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
obtain N HN, from H e2pos,
@ -308,9 +318,9 @@ eq_of_forall_dist_le
(take ε, suppose ε > 0,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
obtain N₁ (HN₁ : ∀ {n}, n ≥ N₁ → dist (X n) y₁ < ε / 2),
from approaches_at_infty_dest H₁ e2pos,
from metric_space.approaches_at_infty_dest H₁ e2pos,
obtain N₂ (HN₂ : ∀ {n}, n ≥ N₂ → dist (X n) y₂ < ε / 2),
from approaches_at_infty_dest H₂ e2pos,
from metric_space.approaches_at_infty_dest H₂ e2pos,
let N := max N₁ N₂ in
have dN₁ : dist (X N) y₁ < ε / 2, from HN₁ !le_max_left,
have dN₂ : dist (X N) y₂ < ε / 2, from HN₂ !le_max_right,
@ -347,11 +357,10 @@ have aux : (λ n, X (k + n)) = (λ n, X (n + k)), from funext (take n, by rewrit
by rewrite aux at H; exact converges_to_seq_of_converges_to_seq_offset H
-/
--<<<<<<< HEAD
proposition bounded_of_converges_seq {X : ℕ → M} {x : M} (H : X ⟶ x [at ∞]) :
∃ K : ℝ, ∀ n : ℕ, dist (X n) x ≤ K :=
have eventually (λ n, dist (X n) x < 1) [at ∞],
from approaches_dest H zero_lt_one,
from metric_space.approaches_dest H zero_lt_one,
obtain N (HN : ∀ n, n ≥ N → dist (X n) x < 1),
from eventually_at_infty_dest this,
let K := max 1 (Max i ∈ '(-∞, N), dist (X i) x) in
@ -363,12 +372,22 @@ proposition bounded_of_converges_seq {X : ℕ → M} {x : M} (H : X ⟶ x [at
else
show dist (X n) x ≤ K,
from le.trans (le_of_lt (HN n (le_of_not_gt Hn))) !le_max_left)
--=======
/-proposition converges_to_seq_of_converges_to_seq_offset_succ
{X : ℕ → M} {y : M} (H : (λ n, X (succ n)) ⟶ y in ℕ) :
X ⟶ y in ℕ :=
@converges_to_seq_of_converges_to_seq_offset M _ X y 1 H
proposition bounded_of_converges {A : Type} {X : A → M} {x : M} {F} (H : (X ⟶ x) F) :
∃ K : ℝ, eventually (λ n, dist (X n) x ≤ K) F :=
begin
note H' := metric_space.approaches_dest H zero_lt_one,
existsi 1,
apply eventually_mono H',
intro x' Hx',
apply le_of_lt Hx'
end
/-proposition converges_to_seq_of_converges_to_seq_offset_succ
{X : ℕ → M} {y : M} (H : (λ n, X (succ n)) ⟶ y [at ∞]) :
X ⟶ y [at ∞] :=
@converges_to_seq_of_converges_to_seq_offset M _ X y 1 H-/
/-
proposition converges_to_seq_offset_iff (X : ℕ → M) (y : M) (k : ℕ) :
((λ n, X (n + k)) ⟶ y in ℕ) ↔ (X ⟶ y in ℕ) :=
iff.intro converges_to_seq_of_converges_to_seq_offset !converges_to_seq_offset
@ -380,49 +399,7 @@ iff.intro converges_to_seq_of_converges_to_seq_offset_left !converges_to_seq_off
proposition converges_to_seq_offset_succ_iff (X : ℕ → M) (y : M) :
((λ n, X (succ n)) ⟶ y in ℕ) ↔ (X ⟶ y in ℕ) :=
iff.intro converges_to_seq_of_converges_to_seq_offset_succ !converges_to_seq_offset_succ
section
open list
private definition r_trans : transitive (@le ℝ _) := λ a b c, !le.trans
private definition r_refl : reflexive (@le ℝ _) := λ a, !le.refl
proposition bounded_of_converges_seq {X : ℕ → M} {x : M} (H : X ⟶ x in ℕ) :
∃ K : ℝ, ∀ n : ℕ, dist (X n) x ≤ K :=
begin
cases H zero_lt_one with N HN,
cases em (N = 0),
existsi 1,
intro n,
apply le_of_lt,
apply HN,
rewrite a,
apply zero_le,
let l := map (λ n : ℕ, -dist (X n) x) (upto N),
have Hnenil : l ≠ nil, from (map_ne_nil_of_ne_nil _ (upto_ne_nil_of_ne_zero a)),
existsi max (-list.min (λ a b : ℝ, le a b) l Hnenil) 1,
intro n,
have Hsmn : ∀ m : ℕ, m < N → dist (X m) x ≤ max (-list.min (λ a b : ℝ, le a b) l Hnenil) 1, begin
intro m Hm,
apply le.trans,
rotate 1,
apply le_max_left,
note Hall := min_lemma real.le_total r_trans r_refl Hnenil,
have Hmem : -dist (X m) x ∈ (map (λ (n : ℕ), -dist (X n) x) (upto N)), from mem_map _ (mem_upto_of_lt Hm),
note Hallm' := of_mem_of_all Hmem Hall,
apply le_neg_of_le_neg,
exact Hallm'
end,
cases em (n < N) with Elt Ege,
apply Hsmn,
exact Elt,
apply le_of_lt,
apply lt_of_lt_of_le,
apply HN,
apply le_of_not_gt Ege,
apply le_max_right
end
end
>>>>>>> feat(library/analysis): basic properties about real derivatives-/
-/
/- cauchy sequences -/
@ -433,7 +410,7 @@ proposition cauchy_of_converges_seq {X : ℕ → M} (H : ∃ y, X ⟶ y [at ∞]
take ε, suppose ε > 0,
obtain y (Hy : X ⟶ y [at ∞]), from H,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
have eventually (λ n, dist (X n) y < ε / 2) [at ∞], from approaches_dest Hy e2pos,
have eventually (λ n, dist (X n) y < ε / 2) [at ∞], from metric_space.approaches_dest Hy e2pos,
obtain N (HN : ∀ {n}, n ≥ N → dist (X n) y < ε / 2), from eventually_at_infty_dest this,
exists.intro N
(take m n, suppose m ≥ N, suppose n ≥ N,
@ -452,41 +429,10 @@ end metric_space_M
section metric_space_M_N
variables {M N : Type} [metric_space M] [metric_space N]
/-
definition converges_to_at (f : M → N) (y : N) (x : M) :=
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ≠ x ∧ dist x' x < δ → dist (f x') y < ε
notation f `⟶` y `at` x := converges_to_at f y x
theorem converges_to_at_constant (y : N) (x : M) : (λ m, y) ⟶ y at x :=
begin
intros ε Hε,
existsi 1,
split,
exact zero_lt_one,
intros x' Hx',
rewrite dist_self,
apply Hε
end
definition converges_at [class] (f : M → N) (x : M) :=
∃ y, converges_to_at f y x
noncomputable definition limit_at (f : M → N) (x : M) [H : converges_at f x] : N :=
some H
proposition converges_to_limit_at (f : M → N) (x : M) [H : converges_at f x] :
(f ⟶ limit_at f x at x) :=
some_spec H
-/
-- TODO: refactor
section
open pnat rat
private lemma of_rat_rat_of_pnat_eq_of_nat_nat_of_pnat (p : pnat) :
of_rat (rat_of_pnat p) = of_nat (nat_of_pnat p) :=
rfl
theorem cnv_real_of_cnv_nat {X : ℕ → M} {c : M} (H : ∀ n : ℕ, dist (X n) c < 1 / (real.of_nat n + 1)) :
∀ ε : ℝ, ε > 0 → ∃ N : ℕ, ∀ n : ℕ, n ≥ N → dist (X n) c < ε :=
@ -512,51 +458,33 @@ theorem cnv_real_of_cnv_nat {X : ℕ → M} {c : M} (H : ∀ n : ℕ, dist (X n)
krewrite -of_rat_zero,
apply of_rat_lt_of_rat_of_lt,
apply rat_of_pnat_is_pos,
krewrite [of_rat_rat_of_pnat_eq_of_nat_nat_of_pnat, -real.of_nat_add],
change of_nat (nat_of_pnat p) ≤ n + 1,
krewrite [-real.of_nat_add],
apply real.of_nat_le_of_nat_of_le,
apply le_add_of_le_right,
assumption
end
end
-- a nice illustration of the limit library: [at c] and [at ∞] can be replaced by any filters
theorem comp_approaches_at_infty {f : M → N} {c : M} {l : N} (Hf : f ⟶ l [at c])
{X : ℕ → M} (HX₁ : X ⟶ c [at ∞]) (HX₂ : eventually (λ n, X n ≠ c) [at ∞]) :
(λ n, f (X n)) ⟶ l [at ∞] :=
tendsto_comp_of_approaches_of_tendsto_at HX₁ HX₂ Hf
-- TODO: refactor
-- TODO : refactor
theorem converges_to_at_of_all_conv_seqs {f : M → N} (c : M) (l : N)
(Hseq : ∀ X : ℕ → M, ((∀ n : ℕ, ((X n) ≠ c) ∧ (X ⟶ c [at ∞])) → ((λ n : ℕ, f (X n)) ⟶ l [at ∞])))
(Hseq : ∀ X : ℕ → M, (eventually (λ n, X n ≠ c) [at ∞] ∧ (X ⟶ c [at ∞])) → ((λ n : ℕ, f (X n)) ⟶ l [at ∞]))
: f ⟶ l [at c] :=
by_contradiction
(assume Hnot : ¬ (f ⟶ l [at c]),
obtain ε Hε, from exists_not_of_not_forall (λ H, Hnot (approaches_at_intro H)),
let Hε' := and_not_of_not_implies Hε in
obtain (H1 : ε > 0) H2, from Hε',
have H3 : ∀ δ : ℝ, (δ > 0 → ∃ x' : M, x' ≠ c ∧ dist x' c < δ ∧ dist (f x') l ≥ ε), begin -- tedious!!
intros δ Hδ,
note Hε'' := forall_not_of_not_exists H2,
note H4 := forall_not_of_not_exists H2 δ,
have ¬ (∀ x' : M, dist x' c < δ → x' ≠ c → dist (f x') l < ε),
from λ H', H4 (and.intro Hδ H'),
note H5 := exists_not_of_not_forall this,
cases H5 with x' Hx',
begin
eapply by_contradiction,
intro Hnot,
cases exists_not_of_not_forall (λ H, Hnot (metric_space.approaches_at_intro H)) with ε Hε,
cases and_not_of_not_implies Hε with H1 H2,
note H3' := forall_not_of_not_exists H2,
have H3 : ∀ δ, δ > 0 → (∃ x', dist x' c < δ ∧ x' ≠ c ∧ dist (f x') l ≥ ε), begin
intro δ Hδ,
cases exists_not_of_not_forall (or.resolve_left (not_or_not_of_not_and' (H3' δ)) (not_not_intro Hδ))
with x' Hx',
existsi x',
note H6 := and_not_of_not_implies Hx',
-- rewrite and.assoc at H6,
cases H6 with H6a H6b,
split,
cases (and_not_of_not_implies H6b),
assumption,
split,
assumption,
apply le_of_not_gt,
cases (and_not_of_not_implies H6b),
assumption
rewrite [2 not_implies_iff_and_not at Hx', ge_iff_not_lt],
exact Hx'
end,
let S : ℕ → M → Prop := λ n x, 0 < dist x c ∧ dist x c < 1 / (of_nat n + 1) ∧ dist (f x) l ≥ ε in
let S := λ (n : ℕ) (x : M), 0 < dist x c ∧ dist x c < 1 / (of_nat n + 1) ∧ dist (f x) l ≥ ε,
have HS : ∀ n : ℕ, ∃ m : M, S n m, begin
intro k,
have Hpos : 0 < of_nat k + 1, from of_nat_succ_pos k,
@ -571,55 +499,57 @@ theorem converges_to_at_of_all_conv_seqs {f : M → N} (c : M) (l : N)
split,
repeat assumption
end,
let X : ℕ → M := λ n, some (HS n) in
have H4 : ∀ n : ℕ, ((X n) ≠ c) ∧ (X ⟶ c [at ∞]), from
(take n, and.intro
(begin
note Hspec := some_spec (HS n),
esimp, esimp at Hspec,
cases Hspec,
apply ne_of_dist_pos,
assumption
end)
(begin
apply approaches_at_infty_intro,
apply cnv_real_of_cnv_nat,
intro m,
note Hspec := some_spec (HS m),
esimp, esimp at Hspec,
cases Hspec with Hspec1 Hspec2,
cases Hspec2,
assumption
end)),
have H5 : (λ n : ℕ, f (X n)) ⟶ l [at ∞], from Hseq X H4,
begin
note H6 := approaches_at_infty_dest H5 H1,
cases H6 with Q HQ,
note HQ' := HQ !le.refl,
esimp at HQ',
apply absurd HQ',
apply not_lt_of_ge,
note H7 := some_spec (HS Q),
esimp at H7,
cases H7 with H71 H72,
cases H72,
assumption
end)
let X := λ n, some (HS n),
have H4 : (eventually (λ n, X n ≠ c) [at ∞]) ∧ (X ⟶ c [at ∞]), begin
split,
{fapply @eventually_at_infty_intro,
exact 0,
intro n Hn,
note Hspec := some_spec (HS n),
esimp, esimp at Hspec,
cases Hspec,
apply ne_of_dist_pos,
assumption},
{intro,
apply metric_space.approaches_at_infty_intro,
apply cnv_real_of_cnv_nat,
intro m,
note Hspec := some_spec (HS m),
esimp, esimp at Hspec,
cases Hspec with Hspec1 Hspec2,
cases Hspec2,
assumption}
end,
have H5 : (λ n, f (X n)) ⟶ l [at ∞], from Hseq X H4,
note H6 := metric_space.approaches_at_infty_dest H5 H1,
cases H6 with Q HQ,
note HQ' := HQ !le.refl,
esimp at HQ',
apply absurd HQ',
apply not_lt_of_ge,
note H7 := some_spec (HS Q),
esimp at H7,
cases H7 with H71 H72,
cases H72,
assumption
end
end metric_space_M_N
namespace metric_space
section continuity
variables {M N : Type} [Hm : metric_space M] [Hn : metric_space N]
include Hm Hn
open topology set
-- the ε - δ definition of continuity is equivalent to the topological definition
theorem continuous_at_intro {f : M → N} {x : M}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε) :
continuous_at f x :=
theorem continuous_at_within_intro {f : M → N} {x : M} {s : set M}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → dist x' x < δ → dist (f x') (f x) < ε) :
continuous_at_on f x s :=
begin
rewrite ↑continuous_at,
intros U Uopen HfU,
intro U Uopen HfU,
cases exists_open_ball_subset_of_Open_of_mem Uopen HfU with r Hr,
cases Hr with Hr HUr,
cases H Hr with δ Hδ,
@ -633,15 +563,15 @@ theorem continuous_at_intro {f : M → N} {x : M}
intro y Hy,
apply mem_preimage,
apply HUr,
note Hy'' := Hx'δ Hy,
exact Hy''
apply Hx'δ,
apply and.right Hy,
apply and.left Hy
end
theorem continuous_at_elim {f : M → N} {x : M} (Hfx : continuous_at f x) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε :=
theorem continuous_at_on_dest {f : M → N} {x : M} {s : set M} (Hfx : continuous_at_on f x s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → dist x' x < δ → dist (f x') (f x) < ε :=
begin
intros ε Hε,
rewrite [↑continuous_at at Hfx],
cases @Hfx (open_ball (f x) ε) !Open_open_ball (mem_open_ball _ Hε) with V HV,
cases HV with HV HVx,
cases HVx with HVx HVf,
@ -650,94 +580,58 @@ theorem continuous_at_elim {f : M → N} {x : M} (Hfx : continuous_at f x) :
existsi δ,
split,
exact Hδ,
intro x' Hx',
intro x' Hx's Hx',
apply HVf,
apply and.intro,
apply Hδx,
apply Hx',
exact Hx',
exact Hx's
end
--<<<<<<< HEAD
theorem continuous_at_of_converges_to_at {f : M → N} {x : M} (Hf : f ⟶ f x [at x]) :
/-=======
theorem continuous_at_on_intro {f : M → N} {x : M} {s : set M}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀₀ x' ∈ s, dist x' x < δ → dist (f x') (f x) < ε) :
continuous_at_on f x s :=
begin
intro t HOt Hfxt,
cases ex_Open_ball_subset_of_Open_of_nonempty HOt Hfxt with ε Hε,
cases H (and.left Hε) with δ Hδ,
existsi (open_ball x δ),
split,
apply open_ball_open,
split,
apply mem_open_ball,
apply and.left Hδ,
intro x' Hx',
apply mem_preimage,
apply mem_of_subset_of_mem,
apply and.right Hε,
apply and.intro !mem_univ,
rewrite dist_comm,
apply and.right Hδ,
apply and.right Hx',
rewrite dist_comm,
apply and.right (and.left Hx')
end
theorem continuous_at_intro {f : M → N} {x : M}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε) :
continuous_at f x :=
continuous_at_of_continuous_at_on_univ
(continuous_at_within_intro
(take ε, suppose ε > 0,
obtain δ (Hδ : δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε), from H this,
exists.intro δ (and.intro
(show δ > 0, from and.left Hδ)
(take x' H' Hx', and.right Hδ _ Hx'))))
theorem continuous_at_on_elim {f : M → N} {x : M} {s : set M} (Hfs : continuous_at_on f x s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀₀ x' ∈ s, dist x' x < δ → dist (f x') (f x) < ε :=
theorem continuous_at_dest {f : M → N} {x : M} (Hfx : continuous_at f x) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε :=
begin
intro ε Hε,
unfold continuous_at_on at Hfs,
cases @Hfs (open_ball (f x) ε) !open_ball_open (mem_open_ball _ Hε) with u Hu,
cases Hu with Huo Hu,
cases Hu with Hxu Hu,
cases ex_Open_ball_subset_of_Open_of_nonempty Huo Hxu with δ Hδ,
cases continuous_at_on_dest (continuous_at_on_univ_of_continuous_at Hfx) Hε with δ Hδ,
existsi δ,
split,
exact and.left Hδ,
intros x' Hx's Hx'x,
have Hims : f ' (u ∩ s) ⊆ open_ball (f x) ε, begin
apply subset.trans (image_subset f Hu),
apply image_preimage_subset
end,
have Hx'int : x' ∈ u ∩ s, begin
apply and.intro,
apply mem_of_subset_of_mem,
apply and.right Hδ,
apply and.intro !mem_univ,
rewrite dist_comm,
repeat assumption
end,
have Hxx' : f x' ∈ open_ball (f x) ε, begin
apply mem_of_subset_of_mem,
apply Hims,
apply mem_image_of_mem,
apply Hx'int
end,
rewrite dist_comm,
apply and.right Hxx'
intro x' Hx',
apply and.right Hδ,
apply mem_univ,
apply Hx'
end
theorem continuous_on_intro {f : M → N} {s : set M}
(H : ∀ x, ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀₀ x' ∈ s, dist x' x < δ → dist (f x') (f x) < ε) :
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → dist x' x < δ → dist (f x') (f x) < ε) :
continuous_on f s :=
begin
apply continuous_on_of_forall_continuous_at_on,
intro x,
apply continuous_at_on_intro,
apply H
end
continuous_on_of_forall_continuous_at_on (λ x, continuous_at_within_intro (H x))
theorem continuous_on_elim {f : M → N} {s : set M} (Hfs : continuous_on f s) :
∀₀ x ∈ s, ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀₀ x' ∈ s, dist x' x < δ → dist (f x') (f x) < ε :=
begin
intros x Hx,
exact continuous_at_on_elim (continuous_at_on_of_continuous_on Hfs Hx)
end-/
theorem continuous_on_dest {f : M → N} {s : set M} (H : continuous_on f s) {x : M} (Hxs : x ∈ s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → dist x' x < δ → dist (f x') (f x) < ε :=
continuous_at_on_dest (continuous_at_on_of_continuous_on H Hxs)
--theorem continuous_at_of_converges_to_at {f : M → N} {x : M} (Hf : f ⟶ f x at x) :
-->>>>>>> feat(theories/analysis): intro/elim rules for continuous_on, etc
theorem continuous_intro {f : M → N}
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε) :
continuous f :=
continuous_of_forall_continuous_at (λ x, continuous_at_intro (H x))
theorem continuous_dest {f : M → N} (H : continuous f) (x : M) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, dist x' x < δ → dist (f x') (f x) < ε :=
continuous_at_dest (forall_continuous_at_of_continuous H x)
theorem continuous_at_of_converges_to_at {f : M → N} {x : M} (Hf : f ⟶ f x [at x]) :
continuous_at f x :=
continuous_at_intro
(take ε, suppose ε > 0,
@ -755,11 +649,9 @@ theorem converges_to_at_of_continuous_at {f : M → N} {x : M} (Hf : continuous_
f ⟶ f x [at x] :=
approaches_at_intro
(take ε, suppose ε > 0,
obtain δ [δpos Hδ], from continuous_at_elim Hf this,
obtain δ [δpos Hδ], from continuous_at_dest Hf this,
exists.intro δ (and.intro δpos (λ x' Hx' xnex', Hδ x' Hx')))
--definition continuous (f : M → N) : Prop := ∀ x, continuous_at f x
theorem converges_seq_comp_of_converges_seq_of_cts (X : ℕ → M) [HX : converges_seq X] {f : M → N}
(Hf : continuous f) :
converges_seq (λ n, f (X n)) :=
@ -768,7 +660,7 @@ theorem converges_seq_comp_of_converges_seq_of_cts (X : ℕ → M) [HX : converg
existsi f xlim,
apply approaches_at_infty_intro,
intros ε Hε,
let Hcont := (continuous_at_elim (forall_continuous_at_of_continuous Hf xlim)) Hε,
let Hcont := (continuous_at_dest (forall_continuous_at_of_continuous Hf xlim)) Hε,
cases Hcont with δ Hδ,
cases approaches_at_infty_dest Hxlim (and.left Hδ) with B HB,
existsi B,
@ -777,22 +669,10 @@ theorem converges_seq_comp_of_converges_seq_of_cts (X : ℕ → M) [HX : converg
apply HB Hn
end
omit Hn
theorem id_continuous : continuous (λ x : M, x) :=
begin
apply continuous_of_forall_continuous_at,
intros x,
apply continuous_at_intro,
intro ε Hε,
existsi ε,
split,
assumption,
intros,
assumption
end
end continuity
end metric_space
end analysis

385
library/theories/analysis/normed_space.lean
View file

@ -6,7 +6,7 @@ Author: Jeremy Avigad
Normed spaces.
-/
import algebra.module .metric_space
open real nat classical topology analysis
open real nat classical topology analysis analysis.metric_space
noncomputable theory
structure has_norm [class] (M : Type) : Type :=
@ -79,6 +79,8 @@ namespace analysis
proposition norm_sub (u v : V) : ∥u - v∥ = ∥v - u∥ :=
by rewrite [-norm_neg, neg_sub]
proposition norm_ne_zero_of_ne_zero {u : V} (H : u ≠ 0) : ∥u∥ ≠ 0 :=
suppose ∥u∥ = 0, H (eq_zero_of_norm_eq_zero this)
end analysis
@ -117,7 +119,7 @@ section
open nat
proposition converges_to_seq_norm_elim {X : ℕ → V} {x : V} (H : X ⟶ x [at ∞]) :
proposition approaches_seq_norm_elim {X : ℕ → V} {x : V} (H : X ⟶ x [at ∞]) :
∀ {ε : ℝ}, ε > 0 → ∃ N₁ : ℕ, ∀ {n : ℕ}, n ≥ N₁ → ∥ X n - x ∥ < ε :=
approaches_at_infty_dest H
@ -147,137 +149,274 @@ definition banach_space_to_metric_space [trans_instance] (V : Type) [bsV : banac
⦄
namespace analysis
-- unfold some common definitions fully (copied from metric space, updated for normed_space notation)
-- TODO: copy these for ℝ as well?
namespace normed_vector_space
section
open set topology set.filter
variables {M N : Type}
--variable [HU : normed_vector_space U]
variable [normed_vector_space M]
--variables {f g : U → V}
section approaches
variables {X : Type} {F : filter X} {f : X → M} {y : M}
proposition approaches_intro (H : ∀ ε, ε > 0 → eventually (λ x, ∥(f x) - y∥ < ε) F) :
(f ⟶ y) F :=
approaches_intro H
proposition approaches_dest (H : (f ⟶ y) F) {ε : ℝ} (εpos : ε > 0) :
eventually (λ x, ∥(f x) - y∥ < ε) F :=
approaches_dest H εpos
variables (F f y)
proposition approaches_iff : ((f ⟶ y) F) ↔ (∀ ε, ε > 0 → eventually (λ x, ∥(f x) - y∥ < ε) F) :=
iff.intro approaches_dest approaches_intro
end approaches
proposition approaches_at_infty_intro {f : ℕ → M} {y : M}
(H : ∀ ε, ε > 0 → ∃ N, ∀ n, n ≥ N → ∥(f n) - y∥ < ε) :
f ⟶ y [at ∞] :=
approaches_at_infty_intro H
proposition approaches_at_infty_dest {f : ℕ → M} {y : M}
(H : f ⟶ y [at ∞]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ N, ∀ ⦃n⦄, n ≥ N → ∥(f n) - y∥ < ε :=
approaches_at_infty_dest H εpos
proposition approaches_at_infty_iff (f : ℕ → M) (y : M) :
f ⟶ y [at ∞] ↔ (∀ ε, ε > 0 → ∃ N, ∀ ⦃n⦄, n ≥ N → ∥(f n) - y∥ < ε) :=
iff.intro approaches_at_infty_dest approaches_at_infty_intro
variable [normed_vector_space N]
proposition approaches_at_dest {f : M → N} {y : N} {x : M}
(H : f ⟶ y [at x]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, ∥x' - x∥ < δ → x' ≠ x → ∥(f x') - y∥ < ε :=
approaches_at_dest H εpos
proposition approaches_at_intro {f : M → N} {y : N} {x : M}
(H : ∀ ε, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, ∥x' - x∥ < δ → x' ≠ x → ∥(f x') - y∥ < ε) :
f ⟶ y [at x] :=
approaches_at_intro H
proposition approaches_at_iff (f : M → N) (y : N) (x : M) : f ⟶ y [at x] ↔
(∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, ∥x' - x∥ < δ → x' ≠ x → ∥(f x') - y∥ < ε) :=
iff.intro approaches_at_dest approaches_at_intro
end
end normed_vector_space
section
variable {V : Type}
variable [normed_vector_space V]
variable {A : Type}
variables {X : A → V}
variables {x : V}
variables {X Y : ℕ → V}
variables {x y : V}
proposition add_converges_to_seq (HX : X ⟶ x [at ∞]) (HY : Y ⟶ y [at ∞]) :
(λ n, X n + Y n) ⟶ x + y [at ∞] :=
approaches_at_infty_intro
take ε : ℝ, suppose ε > 0,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos,
obtain (N₁ : ℕ) (HN₁ : ∀ {n}, n ≥ N₁ → ∥ X n - x ∥ < ε / 2),
from converges_to_seq_norm_elim HX e2pos,
obtain (N₂ : ℕ) (HN₂ : ∀ {n}, n ≥ N₂ → ∥ Y n - y ∥ < ε / 2),
from converges_to_seq_norm_elim HY e2pos,
let N := max N₁ N₂ in
exists.intro N
(take n,
suppose n ≥ N,
have ngtN₁ : n ≥ N₁, from nat.le_trans !le_max_left `n ≥ N`,
have ngtN₂ : n ≥ N₂, from nat.le_trans !le_max_right `n ≥ N`,
show ∥ (X n + Y n) - (x + y) ∥ < ε, from calc
∥ (X n + Y n) - (x + y) ∥
= ∥ (X n - x) + (Y n - y) ∥ : by rewrite [sub_add_eq_sub_sub, *sub_eq_add_neg,
*add.assoc, add.left_comm (-x)]
... ≤ ∥ X n - x ∥ + ∥ Y n - y ∥ : norm_triangle
... < ε / 2 + ε / 2 : add_lt_add (HN₁ ngtN₁) (HN₂ ngtN₂)
... = ε : add_halves)
private lemma smul_converges_to_seq_aux {c : ℝ} (cnz : c ≠ 0) (HX : X ⟶ x [at ∞]) :
(λ n, c • X n) ⟶ c • x [at ∞] :=
approaches_at_infty_intro
take ε : ℝ, suppose ε > 0,
have abscpos : abs c > 0, from abs_pos_of_ne_zero cnz,
have epos : ε / abs c > 0, from div_pos_of_pos_of_pos `ε > 0` abscpos,
obtain N (HN : ∀ {n}, n ≥ N → norm (X n - x) < ε / abs c), from converges_to_seq_norm_elim HX epos,
exists.intro N
(take n,
suppose n ≥ N,
have H : norm (X n - x) < ε / abs c, from HN this,
show norm (c • X n - c • x) < ε, from calc
norm (c • X n - c • x)
= abs c * norm (X n - x) : by rewrite [-smul_sub_left_distrib, norm_smul]
... < abs c * (ε / abs c) : mul_lt_mul_of_pos_left H abscpos
... = ε : mul_div_cancel' (ne_of_gt abscpos))
proposition smul_converges_to_seq (c : ℝ) (HX : X ⟶ x [at ∞]) :
(λ n, c • X n) ⟶ c • x [at ∞] :=
by_cases
(assume cz : c = 0,
have (λ n, c • X n) = (λ n, 0), from funext (take x, by rewrite [cz, zero_smul]),
begin rewrite [this, cz, zero_smul], apply approaches_constant end)
(suppose c ≠ 0, smul_converges_to_seq_aux this HX)
proposition neg_converges_to_seq (HX : X ⟶ x [at ∞]) :
(λ n, - X n) ⟶ - x [at ∞] :=
approaches_at_infty_intro
take ε, suppose ε > 0,
obtain N (HN : ∀ {n}, n ≥ N → norm (X n - x) < ε), from converges_to_seq_norm_elim HX this,
exists.intro N
(take n,
suppose n ≥ N,
show norm (- X n - (- x)) < ε,
by rewrite [-neg_neg_sub_neg, *neg_neg, norm_neg]; exact HN `n ≥ N`)
proposition neg_converges_to_seq_iff : ((λ n, - X n) ⟶ - x [at ∞]) ↔ (X ⟶ x [at ∞]) :=
have aux : X = λ n, (- (- X n)), from funext (take n, by rewrite neg_neg),
iff.intro
(assume H : (λ n, -X n)⟶ -x [at ∞],
show X ⟶ x [at ∞], by rewrite [aux, -neg_neg x]; exact neg_converges_to_seq H)
neg_converges_to_seq
proposition norm_converges_to_seq_zero (HX : X ⟶ 0 [at ∞]) : (λ n, norm (X n)) ⟶ 0 [at ∞] :=
approaches_at_infty_intro
take ε, suppose ε > 0,
obtain N (HN : ∀ n, n ≥ N → norm (X n - 0) < ε), from approaches_at_infty_dest HX `ε > 0`,
exists.intro N
(take n, assume Hn : n ≥ N,
have norm (X n) < ε, begin rewrite -(sub_zero (X n)), apply HN n Hn end,
show abs (norm (X n) - 0) < ε,
by rewrite [sub_zero, abs_of_nonneg !norm_nonneg]; apply this)
proposition converges_to_seq_zero_of_norm_converges_to_seq_zero
(HX : (λ n, norm (X n)) ⟶ 0 [at ∞]) :
X ⟶ 0 [at ∞] :=
approaches_at_infty_intro
take ε, suppose ε > 0,
obtain N (HN : ∀ n, n ≥ N → abs (norm (X n) - 0) < ε), from approaches_at_infty_dest HX `ε > 0`,
exists.intro (N : ℕ)
(take n : ℕ, assume Hn : n ≥ N,
have HN' : abs (norm (X n) - 0) < ε, from HN n Hn,
have norm (X n) < ε,
by rewrite [sub_zero at HN', abs_of_nonneg !norm_nonneg at HN']; apply HN',
show norm (X n - 0) < ε,
by rewrite sub_zero; apply this)
proposition norm_converges_to_seq_zero_iff (X : ℕ → V) :
((λ n, norm (X n)) ⟶ 0 [at ∞]) ↔ (X ⟶ 0 [at ∞]) :=
iff.intro converges_to_seq_zero_of_norm_converges_to_seq_zero norm_converges_to_seq_zero
end analysis
namespace analysis
variables {U V : Type}
variable [HU : normed_vector_space U]
variable [HV : normed_vector_space V]
variables f g : U → V
include HU HV
theorem add_converges_to_at {lf lg : V} {x : U} (Hf : f ⟶ lf at x) (Hg : g ⟶ lg at x) :
(λ y, f y + g y) ⟶ lf + lg at x :=
proposition neg_approaches {F} (HX : (X ⟶ x) F) :
((λ n, - X n) ⟶ - x) F :=
begin
apply converges_to_at_of_all_conv_seqs,
intro X HX,
apply add_converges_to_seq,
apply all_conv_seqs_of_converges_to_at Hf,
apply HX,
apply all_conv_seqs_of_converges_to_at Hg,
apply HX
apply normed_vector_space.approaches_intro,
intro ε Hε,
apply set.filter.eventually_mono (approaches_dest HX Hε),
intro x' Hx',
rewrite [-norm_neg, neg_neg_sub_neg],
apply Hx'
end
open topology
proposition approaches_neg {F} (Hx : ((λ n, - X n) ⟶ - x) F) : (X ⟶ x) F :=
have aux : X = λ n, (- (- X n)), from funext (take n, by rewrite neg_neg),
by rewrite [aux, -neg_neg x]; exact neg_approaches Hx
theorem normed_vector_space.continuous_at_intro {x : U}
proposition neg_approaches_iff {F} : (((λ n, - X n) ⟶ - x) F) ↔ ((X ⟶ x) F) :=
have aux : X = λ n, (- (- X n)), from funext (take n, by rewrite neg_neg),
iff.intro approaches_neg neg_approaches
proposition norm_approaches_zero_of_approaches_zero {F} (HX : (X ⟶ 0) F) : ((λ n, norm (X n)) ⟶ 0) F :=
begin
apply metric_space.approaches_intro,
intro ε Hε,
apply set.filter.eventually_mono (approaches_dest HX Hε),
intro x Hx,
change abs (∥X x∥ - 0) < ε,
rewrite [sub_zero, abs_of_nonneg !norm_nonneg, -sub_zero (X x)],
apply Hx
end
proposition approaches_zero_of_norm_approaches_zero
{F} (HX : ((λ n, norm (X n)) ⟶ 0) F) :
(X ⟶ 0) F :=
begin
apply normed_vector_space.approaches_intro,
intro ε Hε,
apply set.filter.eventually_mono (approaches_dest HX Hε),
intro x Hx,
apply lt_of_abs_lt,
rewrite [sub_zero, -sub_zero ∥X x∥],
apply Hx
end
proposition norm_approaches_zero_iff (X : ℕ → V) (F) :
(((λ n, norm (X n)) ⟶ 0) F) ↔ ((X ⟶ 0) F) :=
iff.intro approaches_zero_of_norm_approaches_zero norm_approaches_zero_of_approaches_zero
end
section
variables {U V : Type}
--variable [HU : normed_vector_space U]
variable [HV : normed_vector_space V]
variables {f g : U → V}
open set-- filter causes error??
include HV
theorem add_approaches {lf lg : V} {F : filter U} (Hf : (f ⟶ lf) F) (Hg : (g ⟶ lg) F) :
((λ y, f y + g y) ⟶ lf + lg) F :=
begin
apply normed_vector_space.approaches_intro,
intro ε Hε,
have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos Hε two_pos,
have Hfl : filter.eventually (λ x, dist (f x) lf < ε / 2) F, from approaches_dest Hf e2pos,
have Hgl : filter.eventually (λ x, dist (g x) lg < ε / 2) F, from approaches_dest Hg e2pos,
apply filter.eventually_mono,
apply filter.eventually_and Hfl Hgl,
intro x Hfg,
rewrite [add_sub_comm, -add_halves ε],
apply lt_of_le_of_lt,
apply norm_triangle,
cases Hfg with Hf' Hg',
apply add_lt_add,
exact Hf', exact Hg'
end
theorem smul_approaches {lf : V} {F : filter U} (Hf : (f ⟶ lf) F) (s : ℝ) :
((λ y, s • f y) ⟶ s • lf) F :=
begin
apply normed_vector_space.approaches_intro,
intro ε Hε,
cases em (s = 0) with seq sneq,
{have H : (λ x, ∥(s • f x) - (s • lf)∥ < ε) = (λ x, true),
begin apply funext, intro x, rewrite [seq, 2 zero_smul, sub_zero, norm_zero, eq_true], exact Hε end,
rewrite H,
apply filter.eventually_true},
{have e2pos : ε / abs s > 0, from div_pos_of_pos_of_pos Hε (abs_pos_of_ne_zero sneq),
have H : filter.eventually (λ x, ∥(f x) - lf∥ < ε / abs s) F, from approaches_dest Hf e2pos,
apply filter.eventually_mono H,
intro x Hx,
rewrite [-smul_sub_left_distrib, norm_smul, mul.comm],
apply mul_lt_of_lt_div,
apply abs_pos_of_ne_zero sneq,
apply Hx}
end
end
namespace normed_vector_space
variables {U V : Type}
variables [HU : normed_vector_space U] [HV : normed_vector_space V]
variables {f g : U → V}
include HU HV
open set
theorem continuous_at_within_intro {x : U} {s : set U}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε) :
continuous_at_on f x s :=
metric_space.continuous_at_within_intro H
theorem continuous_at_on_dest {x : U} {s : set U} (Hfx : continuous_at_on f x s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε :=
metric_space.continuous_at_on_dest Hfx
theorem continuous_on_intro {s : set U}
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε) :
continuous_on f s :=
metric_space.continuous_on_intro H
theorem continuous_on_dest {s : set U} (H : continuous_on f s) {x : U} (Hxs : x ∈ s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε :=
metric_space.continuous_on_dest H Hxs
theorem continuous_intro
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε) :
continuous f :=
metric_space.continuous_intro H
theorem continuous_dest (H : continuous f) (x : U) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, ∥x' - x∥ < δ → ∥(f x') - (f x)∥ < ε :=
metric_space.continuous_dest H x
theorem continuous_at_intro {x : U}
(H : ∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : U, ∥x' - x∥ < δ → ∥f x' - f x∥ < ε)) :
continuous_at f x :=
continuous_at_intro H
metric_space.continuous_at_intro H
theorem normed_vector_space.continuous_at_elim {x : U} (H : continuous_at f x) :
theorem continuous_at_dest {x : U} (H : continuous_at f x) :
∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : U, ∥x' - x∥ < δ → ∥f x' - f x∥ < ε) :=
continuous_at_elim H
metric_space.continuous_at_dest H
end normed_vector_space
section
open topology
variables {U V : Type}
variables [HU : normed_vector_space U] [HV : normed_vector_space V]
variables {f g : U → V}
include HU HV
theorem neg_continuous (Hf : continuous f) : continuous (λ x : U, - f x) :=
begin
apply continuous_of_forall_continuous_at,
intro x,
apply continuous_at_of_tendsto_at,
apply neg_approaches,
apply tendsto_at_of_continuous_at,
apply forall_continuous_at_of_continuous,
apply Hf
end
theorem add_continuous (Hf : continuous f) (Hg : continuous g) : continuous (λ x, f x + g x) :=
begin
apply continuous_of_forall_continuous_at,
intro y,
apply continuous_at_of_tendsto_at,
apply add_approaches,
all_goals apply tendsto_at_of_continuous_at,
all_goals apply forall_continuous_at_of_continuous,
repeat assumption
end
theorem sub_continuous (Hf : continuous f) (Hg : continuous g) : continuous (λ x, f x - g x) :=
begin
apply continuous_of_forall_continuous_at,
intro y,
apply continuous_at_of_tendsto_at,
apply add_approaches,
all_goals apply tendsto_at_of_continuous_at,
all_goals apply forall_continuous_at_of_continuous,
assumption,
apply neg_continuous,
assumption
end
theorem smul_continuous (s : ℝ) (Hf : continuous f) : continuous (λ x : U, s • f x) :=
begin
apply continuous_of_forall_continuous_at,
intro y,
apply continuous_at_of_tendsto_at,
apply smul_approaches,
apply tendsto_at_of_continuous_at,
apply forall_continuous_at_of_continuous,
assumption
end
end
end analysis

42
library/theories/analysis/real_deriv.lean
View file

@ -6,7 +6,7 @@ Author: Robert Y. Lewis
Derivatives on ℝ
-/
import .bounded_linear_operator
open real nat classical topology analysis
open real nat classical topology analysis set
noncomputable theory
namespace real
@ -19,42 +19,42 @@ theorem add_sub_self (a b : ℝ) : a + b - a = b :=
definition derivative_at (f : ℝ → ℝ) (d x : ℝ) := is_frechet_deriv_at f (λ t, d • t) x
theorem derivative_at_intro (f : ℝ → ℝ) (d x : ℝ) (H : (λ h, (f (x + h) - f x) / h) ⟶ d at 0) :
theorem derivative_at_intro (f : ℝ → ℝ) (d x : ℝ) (H : (λ h, (f (x + h) - f x) / h) ⟶ d [at 0]) :
derivative_at f d x :=
begin
apply is_frechet_deriv_at_intro,
intros ε Hε,
cases H Hε with δ Hδ,
cases approaches_at_dest H Hε with δ Hδ,
existsi δ,
split,
exact and.left Hδ,
intro y Hy,
rewrite [-sub_zero y at Hy{2}],
note Hδ' := and.right Hδ y Hy,
note Hδ' := and.right Hδ y (and.right Hy) (and.left Hy),
have Hδ'' : abs ((f (x + y) - f x - d * y) / y) < ε,
by rewrite [-div_sub_div_same, mul_div_cancel _ (and.left Hy)]; apply Hδ',
show abs (f (x + y) - f x - d * y) / abs y < ε, by rewrite -abs_div; apply Hδ''
end
theorem derivative_at_of_frechet_derivative_at {f g : ℝ → ℝ} [is_bdd_linear_map g] {d x : ℝ}
(H : is_frechet_deriv_at f g x) (Hg : g = λ x, d * x) :
derivative_at f d x :=
by apply is_frechet_deriv_at_of_eq H Hg
theorem deriv_at_const (c x : ℝ) : derivative_at (λ t, c) 0 x :=
begin
apply derivative_at_intro,
have (λ h, (c - c) / h) = (λ h, 0), from funext (λ h, by rewrite [sub_self, zero_div]),
rewrite this,
apply converges_to_at_constant
end
derivative_at_of_frechet_derivative_at
(@frechet_deriv_at_const ℝ ℝ _ _ _ c)
(funext (λ v, by rewrite zero_mul))
theorem deriv_at_id (x : ℝ) : derivative_at (λ t, t) 1 x :=
begin
apply derivative_at_intro,
apply converges_to_at_real_intro,
intros ε Hε,
existsi 1,
split,
exact zero_lt_one,
intros x' Hx',
rewrite [add_sub_self, div_self (and.left Hx'), sub_self, abs_zero],
exact Hε
end
derivative_at_of_frechet_derivative_at
(@frechet_deriv_at_id ℝ ℝ _ _ _)
(funext (λ v, by rewrite one_mul))
theorem deriv_at_mul {f : ℝ → ℝ} {d x : ℝ} (H : derivative_at f d x) (c : ℝ) :
derivative_at (λ t, c * f t) (c * d) x :=
derivative_at_of_frechet_derivative_at
(frechet_deriv_at_smul _ _ c H)
(funext (λ v, by rewrite mul.assoc))
end real

537
library/theories/analysis/real_limit.lean
View file

@ -150,17 +150,19 @@ namespace analysis
theorem dist_eq_abs (x y : real) : dist x y = abs (x - y) := rfl
proposition converges_to_seq_real_intro {X : ℕ → ℝ} {y : ℝ}
namespace real
proposition approaches_at_infty_intro {X : ℕ → ℝ} {y : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε) :
(X ⟶ y [at ∞]) := approaches_at_infty_intro H
(X ⟶ y [at ∞]) := metric_space.approaches_at_infty_intro H
proposition converges_to_seq_real_elim {X : ℕ → ℝ} {y : ℝ} (H : X ⟶ y [at ∞]) :
∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε := approaches_at_infty_dest H
proposition approaches_at_infty_dest {X : ℕ → ℝ} {y : ℝ} (H : X ⟶ y [at ∞]) :
∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε := metric_space.approaches_at_infty_dest H
proposition converges_to_seq_real_intro' {X : ℕ → ℝ} {y : ℝ}
proposition approaches_at_infty_intro' {X : ℕ → ℝ} {y : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) ≤ ε) :
(X ⟶ y [at ∞]) :=
approaches_at_infty_intro' H
end real
open pnat subtype
local postfix ⁻¹ := pnat.inv
@ -205,7 +207,7 @@ theorem converges_seq_of_cauchy {X : ℕ → ℝ} (H : cauchy X) : converges_seq
obtain l Nb (conv : converges_to_with_rate (r_seq_of X) l Nb),
from converges_to_with_rate_of_cauchy H,
exists.intro l
(approaches_at_infty_intro
(real.approaches_at_infty_intro
take ε : ℝ,
suppose ε > 0,
obtain (k' : ℕ) (Hn : 1 / succ k' < ε), from archimedean_small `ε > 0`,
@ -249,184 +251,271 @@ definition banach_space_real [trans_instance] : banach_space ℝ :=
complete := λ X H, analysis.complete ℝ H
⦄
namespace real
open topology set
open normed_vector_space
section
variable {f : ℝ → ℝ}
theorem continuous_dest (H : continuous f) :
∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε :=
normed_vector_space.continuous_dest H
theorem continuous_intro
(H : ∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε) :
continuous f :=
normed_vector_space.continuous_intro H
theorem continuous_at_dest {x : ℝ} (H : continuous_at f x) :
∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (x' - x) < δ → abs (f x' - f x) < ε) :=
normed_vector_space.continuous_at_dest H
theorem continuous_at_intro {x : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε) :
continuous_at f x :=
normed_vector_space.continuous_at_intro H
theorem continuous_at_within_intro {x : ℝ} {s : set ℝ}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε) :
continuous_at_on f x s :=
normed_vector_space.continuous_at_within_intro H
theorem continuous_at_on_dest {x : ℝ} {s : set ℝ} (Hfx : continuous_at_on f x s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε :=
normed_vector_space.continuous_at_on_dest Hfx
theorem continuous_on_intro {s : set ℝ}
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε) :
continuous_on f s :=
normed_vector_space.continuous_on_intro H
theorem continuous_on_dest {s : set ℝ} (H : continuous_on f s) {x : ℝ} (Hxs : x ∈ s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε :=
normed_vector_space.continuous_on_dest H Hxs
end
section approaches
open set.filter set topology
variables {X : Type} {F : filter X} {f : X → ℝ} {y : ℝ}
proposition approaches_intro (H : ∀ ε, ε > 0 → eventually (λ x, abs ((f x) - y) < ε) F) :
(f ⟶ y) F :=
normed_vector_space.approaches_intro H
proposition approaches_dest (H : (f ⟶ y) F) {ε : ℝ} (εpos : ε > 0) :
eventually (λ x, abs ((f x) - y) < ε) F :=
normed_vector_space.approaches_dest H εpos
variables (F f y)
proposition approaches_iff : ((f ⟶ y) F) ↔ (∀ ε, ε > 0 → eventually (λ x, abs ((f x) - y) < ε) F) :=
iff.intro approaches_dest approaches_intro
end approaches
proposition approaches_at_infty_intro {f : ℕ → ℝ} {y : ℝ}
(H : ∀ ε, ε > 0 → ∃ N, ∀ n, n ≥ N → abs ((f n) - y) < ε) :
f ⟶ y [at ∞] :=
normed_vector_space.approaches_at_infty_intro H
proposition approaches_at_infty_dest {f : ℕ → ℝ} {y : ℝ}
(H : f ⟶ y [at ∞]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ N, ∀ ⦃n⦄, n ≥ N → abs ((f n) - y) < ε :=
normed_vector_space.approaches_at_infty_dest H εpos
proposition approaches_at_infty_iff (f : ℕ → ℝ) (y : ℝ) :
f ⟶ y [at ∞] ↔ (∀ ε, ε > 0 → ∃ N, ∀ ⦃n⦄, n ≥ N → abs ((f n) - y) < ε) :=
iff.intro approaches_at_infty_dest approaches_at_infty_intro
section
variable {f : ℝ → ℝ}
proposition approaches_at_dest {y x : ℝ}
(H : f ⟶ y [at x]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε :=
metric_space.approaches_at_dest H εpos
proposition approaches_at_intro {y x : ℝ}
(H : ∀ ε, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε) :
f ⟶ y [at x] :=
metric_space.approaches_at_intro H
proposition approaches_at_iff (y x : ℝ) : f ⟶ y [at x] ↔
(∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε) :=
iff.intro approaches_at_dest approaches_at_intro
end
end real
/- limits under pointwise operations -/
section limit_operations
variables {X Y : ℕ → ℝ}
open set
variable {A : Type}
variables {X Y : A → ℝ}
variables {x y : ℝ}
variable {F : filter A}
proposition mul_left_converges_to_seq (c : ℝ) (HX : X ⟶ x [at ∞]) :
(λ n, c * X n) ⟶ c * x [at ∞] :=
smul_converges_to_seq c HX
proposition mul_left_approaches (c : ℝ) (HX : (X ⟶ x) F) :
((λ n, c * X n) ⟶ c * x) F :=
smul_approaches HX c
proposition mul_right_converges_to_seq (c : ℝ) (HX : X ⟶ x [at ∞]) :
(λ n, X n * c) ⟶ x * c [at ∞] :=
have (λ n, X n * c) = (λ n, c * X n), from funext (take x, !mul.comm),
by rewrite [this, mul.comm]; apply mul_left_converges_to_seq c HX
proposition mul_right_approaches (c : ℝ) (HX : (X ⟶ x) F) :
((λ n, X n * c) ⟶ x * c) F :=
have (λ n, X n * c) = (λ n, c * X n), from funext (λ n, !mul.comm),
by rewrite [this, mul.comm]; apply mul_left_approaches _ HX
theorem converges_to_seq_squeeze (HX : X ⟶ x [at ∞]) (HY : Y ⟶ x [at ∞]) {Z : ℕ → ℝ} (HZX : ∀ n, X n ≤ Z n)
(HZY : ∀ n, Z n ≤ Y n) : Z ⟶ x [at ∞] :=
theorem approaches_squeeze (HX : (X ⟶ x) F) (HY : (Y ⟶ x) F)
{Z : A → ℝ} (HZX : filter.eventually (λ n, X n ≤ Z n) F) (HZY : filter.eventually (λ n, Z n ≤ Y n) F) :
(Z ⟶ x) F :=
begin
apply approaches_at_infty_intro,
intros ε Hε,
have Hε4 : ε / 4 > 0, from div_pos_of_pos_of_pos Hε four_pos,
cases approaches_at_infty_dest HX Hε4 with N1 HN1,
cases approaches_at_infty_dest HY Hε4 with N2 HN2,
existsi max N1 N2,
intro n Hn,
have HXY : abs (Y n - X n) < ε / 2, begin
apply lt_of_le_of_lt,
apply abs_sub_le _ x,
have Hε24 : ε / 2 = ε / 4 + ε / 4, from eq.symm !add_quarters,
rewrite Hε24,
apply add_lt_add,
apply HN2,
apply ge.trans Hn !le_max_right,
rewrite abs_sub,
apply HN1,
apply ge.trans Hn !le_max_left
end,
have HZX : abs (Z n - X n) < ε / 2, begin
have HZXnp : Z n - X n ≥ 0, from sub_nonneg_of_le !HZX,
have HXYnp : Y n - X n ≥ 0, from sub_nonneg_of_le (le.trans !HZX !HZY),
rewrite [abs_of_nonneg HZXnp, abs_of_nonneg HXYnp at HXY],
note Hgt := lt_add_of_sub_lt_right HXY,
have Hlt : Z n < ε / 2 + X n, from calc
Z n ≤ Y n : HZY
... < ε / 2 + X n : Hgt,
apply sub_lt_right_of_lt_add Hlt
end,
have H : abs (Z n - x) < ε, begin
apply lt_of_le_of_lt,
apply abs_sub_le _ (X n),
apply lt.trans,
apply add_lt_add,
apply HZX,
apply HN1,
apply ge.trans Hn !le_max_left,
apply div_two_add_div_four_lt Hε
end,
exact H
apply real.approaches_intro,
intro ε Hε,
apply filter.eventually_mp,
rotate 1,
apply filter.eventually_and,
apply real.approaches_dest HX Hε,
apply real.approaches_dest HY Hε,
apply filter.eventually_mono,
apply filter.eventually_and HZX HZY,
intros x' Hlo Hdst,
change abs (Z x' - x) < ε,
cases em (x ≤ Z x') with HxleZ HxnleZ, -- annoying linear arith
{have Y x' - x = (Z x' - x) + (Y x' - Z x'), by rewrite -sub_eq_sub_add_sub,
have H : abs (Y x' - x) < ε, from and.right Hdst,
rewrite this at H,
have H'' : Y x' - Z x' ≥ 0, from sub_nonneg_of_le (and.right Hlo),
have H' : Z x' - x ≥ 0, from sub_nonneg_of_le HxleZ,
krewrite [abs_of_nonneg H', abs_of_nonneg (add_nonneg H' H'') at H],
apply lt_of_add_lt_of_nonneg_left H H''},
{have X x' - x = (Z x' - x) + (X x' - Z x'), by rewrite -sub_eq_sub_add_sub,
have H : abs (X x' - x) < ε, from and.left Hdst,
rewrite this at H,
have H' : x - Z x' > 0, from sub_pos_of_lt (lt_of_not_ge HxnleZ),
have H'2 : Z x' - x < 0,
by rewrite [-neg_neg (Z x' - x)]; apply neg_neg_of_pos; rewrite [neg_sub]; apply H',
have H'' : X x' - Z x' ≤ 0, from sub_nonpos_of_le (and.left Hlo),
krewrite [abs_of_neg H'2, abs_of_neg (add_neg_of_neg_of_nonpos H'2 H'') at H, neg_add at H],
apply lt_of_add_lt_of_nonneg_left H,
apply neg_nonneg_of_nonpos H''}
end
proposition converges_to_seq_of_abs_sub_converges_to_seq (Habs : (λ n, abs (X n - x)) ⟶ 0 [at ∞]) :
X ⟶ x [at ∞] :=
proposition approaches_of_abs_sub_approaches {F} (Habs : ((λ n, abs (X n - x)) ⟶ 0) F) :
(X ⟶ x) F :=
begin
apply approaches_at_infty_intro,
intros ε Hε,
cases approaches_at_infty_dest Habs Hε with N HN,
existsi N,
apply real.approaches_intro,
intro ε Hε,
apply set.filter.eventually_mono,
apply real.approaches_dest Habs Hε,
intro n Hn,
have Hn' : abs (abs (X n - x) - 0) < ε, from HN Hn,
have Hn' : abs (abs (X n - x) - 0) < ε, from Hn,
rewrite [sub_zero at Hn', abs_abs at Hn'],
exact Hn'
end
proposition abs_sub_converges_to_seq_of_converges_to_seq (HX : X ⟶ x [at ∞]) :
(λ n, abs (X n - x)) ⟶ 0 [at ∞] :=
proposition abs_sub_approaches_of_approaches {F} (HX : (X ⟶ x) F) :
((λ n, abs (X n - x)) ⟶ 0) F :=
begin
apply approaches_at_infty_intro,
apply real.approaches_intro,
intros ε Hε,
cases approaches_at_infty_dest HX Hε with N HN,
existsi N,
apply set.filter.eventually_mono,
apply real.approaches_dest HX Hε,
intro n Hn,
have Hn' : abs (abs (X n - x) - 0) < ε, by rewrite [sub_zero, abs_abs]; apply HN Hn,
have Hn' : abs (abs (X n - x) - 0) < ε, by rewrite [sub_zero, abs_abs]; apply Hn,
exact Hn'
end
proposition mul_converges_to_seq (HX : X ⟶ x [at ∞]) (HY : Y ⟶ y [at ∞]) :
(λ n, X n * Y n) ⟶ x * y [at ∞] :=
have Hbd : ∃ K : ℝ, ∀ n : ℕ, abs (X n) ≤ K, begin
cases bounded_of_converges_seq HX with K HK,
existsi K + abs x,
intro n,
note Habs := le.trans (abs_abs_sub_abs_le_abs_sub (X n) x) !HK,
apply le_add_of_sub_right_le,
apply le.trans,
apply le_abs_self,
assumption
end,
obtain K HK, from Hbd,
have Habsle : ∀ n, abs (X n * Y n - x * y) ≤ K * abs (Y n - y) + abs y * abs (X n - x), begin
intro,
have Heq : X n * Y n - x * y = (X n * Y n - X n * y) + (X n * y - x * y), by
rewrite [-sub_add_cancel (X n * Y n) (X n * y) at {1}, sub_eq_add_neg, *add.assoc],
proposition bounded_of_approaches_real {F} (HX : (X ⟶ x) F) :
∃ K : ℝ, filter.eventually (λ n, abs (X n) ≤ K) F :=
begin
cases bounded_of_converges HX with K HK,
existsi K + abs x,
apply filter.eventually_mono HK,
intro x' Hx',
note Hle := abs_sub_abs_le_abs_sub (X x') x,
apply le.trans,
apply le_add_of_sub_right_le,
apply Hle,
apply add_le_add_right,
apply Hx'
end
proposition mul_approaches {F} (HX : (X ⟶ x) F) (HY : (Y ⟶ y) F) :
((λ n, X n * Y n) ⟶ x * y) F :=
obtain K HK, from bounded_of_approaches_real HX,
have Habsle : filter.eventually
(λ n, abs (X n * Y n - x * y) ≤ K * abs (Y n - y) + abs y * abs (X n - x)) F, begin
have Heq : ∀ n, X n * Y n - x * y = (X n * Y n - X n * y) + (X n * y - x * y),
by intro n; rewrite [-sub_add_cancel (X n * Y n) (X n * y) at {1}, sub_eq_add_neg, *add.assoc],
apply filter.eventually_mono HK,
intro x' Hx',
apply le.trans,
rewrite Heq,
apply abs_add_le_abs_add_abs,
apply add_le_add,
rewrite [-mul_sub_left_distrib, abs_mul],
apply mul_le_mul_of_nonneg_right,
apply HK,
apply Hx',
apply abs_nonneg,
rewrite [-mul_sub_right_distrib, abs_mul, mul.comm],
apply le.refl
end,
have Hdifflim : (λ n, abs (X n * Y n - x * y)) ⟶ 0 [at ∞], begin
apply converges_to_seq_squeeze,
have Hdifflim : ((λ n, abs (X n * Y n - x * y)) ⟶ 0) F, begin
apply approaches_squeeze,
rotate 2,
intro, apply abs_nonneg,
intro,
apply filter.eventually_mono HK,
intro x' Hx',
apply abs_nonneg,
apply Habsle,
apply approaches_constant,
rewrite -{0}zero_add,
apply add_converges_to_seq,
apply add_approaches,
krewrite -(mul_zero K),
apply mul_left_converges_to_seq,
apply abs_sub_converges_to_seq_of_converges_to_seq,
apply mul_left_approaches,
apply abs_sub_approaches_of_approaches,
exact HY,
krewrite -(mul_zero (abs y)),
apply mul_left_converges_to_seq,
apply abs_sub_converges_to_seq_of_converges_to_seq,
apply mul_left_approaches,
apply abs_sub_approaches_of_approaches,
exact HX
end,
converges_to_seq_of_abs_sub_converges_to_seq Hdifflim
approaches_of_abs_sub_approaches Hdifflim
-- TODO: converges_to_seq_div, converges_to_seq_mul_left_iff, etc.
proposition abs_converges_to_seq_zero (HX : X ⟶ 0 [at ∞]) : (λ n, abs (X n)) ⟶ 0 [at ∞] :=
norm_converges_to_seq_zero HX
proposition converges_to_seq_zero_of_abs_converges_to_seq_zero (HX : (λ n, abs (X n)) ⟶ 0 [at ∞]) :
X ⟶ 0 [at ∞] :=
converges_to_seq_zero_of_norm_converges_to_seq_zero HX
proposition abs_converges_to_seq_zero_iff (X : ℕ → ℝ) :
((λ n, abs (X n)) ⟶ 0 [at ∞]) ↔ (X ⟶ 0 [at ∞]) :=
iff.intro converges_to_seq_zero_of_abs_converges_to_seq_zero abs_converges_to_seq_zero
-- TODO: products of two sequences, converges_seq, limit_seq
end limit_operations
/- properties of converges_to_at -/
section limit_operations_continuous
variables {f g h : ℝ → ℝ}
variables {a b x y : ℝ}
--<<<<<<< HEAD
theorem mul_converges_to_at (Hf : f ⟶ a [at x]) (Hg : g ⟶ b [at x]) : (λ z, f z * g z) ⟶ a * b [at x] :=
/-=======
theorem converges_to_at_real_intro (Hf : ∀ ⦃ε⦄, ε > 0 →
(∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ≠ x ∧ abs (x' - x) < δ → abs (f x' - y) < ε)) :
converges_to_at f y x := Hf
theorem mul_converges_to_at (Hf : f ⟶ a at x) (Hg : g ⟶ b at x) : (λ z, f z * g z) ⟶ a * b at x :=
>>>>>>> feat(library/analysis): basic properties about real derivatives-/
proposition mul_approaches_zero_of_approaches_zero_of_approaches (HX : (X ⟶ 0) F) (HY : (Y ⟶ y) F) :
((λ z, X z * Y z) ⟶ 0) F :=
begin
apply converges_to_at_of_all_conv_seqs,
intro X HX,
apply mul_converges_to_seq,
apply comp_approaches_at_infty Hf,
apply and.right (HX 0),
apply (set.filter.eventually_of_forall _ (λ n, and.left (HX n))),
apply comp_approaches_at_infty Hg,
apply and.right (HX 0),
apply (set.filter.eventually_of_forall _ (λ n, and.left (HX n)))
krewrite [-zero_mul y],
apply mul_approaches,
exact HX, exact HY
end
end limit_operations_continuous
proposition mul_approaches_zero_of_approaches_of_approaches_zero (HX : (X ⟶ y) F) (HY : (Y ⟶ 0) F) :
((λ z, X z * Y z) ⟶ 0) F :=
begin
have H : (λ z, X z * Y z) = (λ z, Y z * X z), from funext (λ a, !mul.comm),
rewrite H,
exact mul_approaches_zero_of_approaches_zero_of_approaches HY HX
end
proposition abs_approaches_zero_of_approaches_zero (HX : (X ⟶ 0) F) : ((λ n, abs (X n)) ⟶ 0) F :=
norm_approaches_zero_of_approaches_zero HX
proposition approaches_zero_of_abs_approaches_zero (HX : ((λ n, abs (X n)) ⟶ 0) F) :
(X ⟶ 0) F :=
approaches_zero_of_norm_approaches_zero HX
proposition abs_approaches_zero_iff :
((λ n, abs (X n)) ⟶ 0) F ↔ (X ⟶ 0) F :=
iff.intro approaches_zero_of_abs_approaches_zero abs_approaches_zero_of_approaches_zero
end limit_operations
/- monotone sequences -/
@ -436,7 +525,7 @@ variable {X : ℕ → ℝ}
proposition converges_to_seq_sup_of_nondecreasing (nondecX : nondecreasing X) {b : ℝ}
(Hb : ∀ i, X i ≤ b) : X ⟶ sup (X ' univ) [at ∞] :=
approaches_at_infty_intro
real.approaches_at_infty_intro
(let sX := sup (X ' univ) in
have Xle : ∀ i, X i ≤ sX, from
take i,
@ -475,7 +564,7 @@ have H₃ : {x : ℝ | -x ∈ X ' univ} = {x : ℝ | x ∈ (λ n, -X n) ' univ},
... = {x : ℝ | x ∈ (λ n, -X n) ' univ} : image_comp,
have H₄ : ∀ i, - X i ≤ - b, from take i, neg_le_neg (Hb i),
begin
apply iff.mp !neg_converges_to_seq_iff,
apply approaches_neg,
-- need krewrite here
krewrite [-sup_neg H₁ H₂, H₃, -nondecreasing_neg_iff at nonincX],
apply converges_to_seq_sup_of_nondecreasing nonincX H₄
@ -488,11 +577,11 @@ end monotone_sequences
section xn
open nat set
theorem pow_converges_to_seq_zero {x : ℝ} (H : abs x < 1) :
theorem pow_approaches_zero_at_infty {x : ℝ} (H : abs x < 1) :
(λ n, x^n) ⟶ 0 [at ∞] :=
suffices H' : (λ n, (abs x)^n) ⟶ 0 [at ∞], from
have (λ n, (abs x)^n) = (λ n, abs (x^n)), from funext (take n, eq.symm !abs_pow),
by rewrite this at H'; exact converges_to_seq_zero_of_abs_converges_to_seq_zero H',
by rewrite this at H'; exact approaches_zero_of_abs_approaches_zero H',
let aX := (λ n, (abs x)^n),
iaX := real.inf (aX ' univ),
asX := (λ n, (abs x)^(succ n)) in
@ -506,7 +595,7 @@ have noninc_aX : nonincreasing aX, from
have bdd_aX : ∀ i, 0 ≤ aX i, from take i, !pow_nonneg_of_nonneg !abs_nonneg,
have aXconv : aX ⟶ iaX [at ∞], proof converges_to_seq_inf_of_nonincreasing noninc_aX bdd_aX qed,
have asXconv : asX ⟶ iaX [at ∞], from tendsto_succ_at_infty aXconv,
have asXconv' : asX ⟶ (abs x) * iaX [at ∞], from mul_left_converges_to_seq (abs x) aXconv,
have asXconv' : asX ⟶ (abs x) * iaX [at ∞], from mul_left_approaches (abs x) aXconv,
have iaX = (abs x) * iaX, from sorry, -- converges_to_seq_unique asXconv asXconv',
have iaX = 0, from eq_zero_of_mul_eq_self_left (ne_of_lt H) (eq.symm this),
show aX ⟶ 0 [at ∞], begin rewrite -this, exact aXconv end --from this ▸ aXconv
@ -515,36 +604,115 @@ end xn
/- continuity on the reals -/
section continuous
open topology
/-namespace real
open topology set
open normed_vector_space
section
variable {f : ℝ → ℝ}
theorem continuous_real_elim (H : continuous f) :
theorem continuous_dest (H : continuous f) :
∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε :=
take x, continuous_at_elim (forall_continuous_at_of_continuous H x)
normed_vector_space.continuous_dest H
theorem continuous_real_intro
theorem continuous_intro
(H : ∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε) :
continuous f :=
continuous_of_forall_continuous_at (take x, continuous_at_intro (H x))
normed_vector_space.continuous_intro H
section
open set
variable {s : set ℝ}
--theorem continuous_on_real_elim (H : continuous_on f s) :
-- ∀₀ x ∈ s, x = x := sorry
theorem continuous_at_dest {x : ℝ} (H : continuous_at f x) :
∀ ε : ℝ, ε > 0 → (∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (x' - x) < δ → abs (f x' - f x) < ε) :=
normed_vector_space.continuous_at_dest H
theorem continuous_at_intro {x : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ,
abs (x' - x) < δ → abs (f x' - f x) < ε) :
continuous_at f x :=
normed_vector_space.continuous_at_intro H
theorem continuous_at_within_intro {x : ℝ} {s : set ℝ}
(H : ∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε) :
continuous_at_on f x s :=
normed_vector_space.continuous_at_within_intro H
theorem continuous_at_on_dest {x : ℝ} {s : set ℝ} (Hfx : continuous_at_on f x s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε :=
normed_vector_space.continuous_at_on_dest Hfx
theorem continuous_on_intro {s : set ℝ}
(H : ∀ x ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε) :
continuous_on f s :=
normed_vector_space.continuous_on_intro H
theorem continuous_on_dest {s : set ℝ} (H : continuous_on f s) {x : ℝ} (Hxs : x ∈ s) :
∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, x' ∈ s → abs (x' - x) < δ → abs ((f x') - (f x)) < ε :=
normed_vector_space.continuous_on_dest H Hxs
end
section
variable {f : ℕ → ℝ}
proposition approaches_at_infty_intro {y : ℝ}
(H : ∀ ε, ε > 0 → ∃ N, ∀ n, n ≥ N → abs ((f n) - y) < ε) :
f ⟶ y [at ∞] :=
normed_vector_space.approaches_at_infty_intro H
proposition approaches_at_infty_dest {y : ℝ}
(H : f ⟶ y [at ∞]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ N, ∀ ⦃n⦄, n ≥ N → abs ((f n) - y) < ε :=
approaches_at_infty_dest H εpos
proposition approaches_at_infty_iff (y : ℝ) :
f ⟶ y [at ∞] ↔ (∀ ε, ε > 0 → ∃ N, ∀ ⦃n⦄, n ≥ N → abs((f n) - y) < ε) :=
iff.intro approaches_at_infty_dest approaches_at_infty_intro
end
section
variable {f : ℝ → ℝ}
proposition approaches_at_dest {y x : ℝ}
(H : f ⟶ y [at x]) ⦃ε : ℝ⦄ (εpos : ε > 0) :
∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε :=
approaches_at_dest H εpos
proposition approaches_at_intro {y x : ℝ}
(H : ∀ ε, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε) :
f ⟶ y [at x] :=
approaches_at_intro H
proposition approaches_at_iff (y x : ℝ) : f ⟶ y [at x] ↔
(∀ ⦃ε⦄, ε > 0 → ∃ δ, δ > 0 ∧ ∀ ⦃x'⦄, abs (x' - x) < δ → x' ≠ x → abs ((f x') - y) < ε) :=
iff.intro approaches_at_dest approaches_at_intro
/-proposition approaches_seq_real_intro {X : ℕ → ℝ} {y : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε) :
(X ⟶ y [at ∞]) := metric_space.approaches_at_infty_intro H
proposition approaches_seq_real_elim {X : ℕ → ℝ} {y : ℝ} (H : X ⟶ y [at ∞]) :
∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) < ε := metric_space.approaches_at_infty_dest H
proposition approaches_seq_real_intro' {X : ℕ → ℝ} {y : ℝ}
(H : ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → abs (X n - y) ≤ ε) :
(X ⟶ y [at ∞]) :=
approaches_at_infty_intro' H-/
end
end real-/
section continuous
open topology
variable {f : ℝ → ℝ}
variable (Hf : continuous f)
include Hf
theorem pos_on_nbhd_of_cts_of_pos {b : ℝ} (Hb : f b > 0) :
∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (y - b) < δ → f y > 0 :=
begin
let Hcont := continuous_real_elim Hf b Hb,
let Hcont := real.continuous_dest Hf b Hb,
cases Hcont with δ Hδ,
existsi δ,
split,
@ -559,7 +727,7 @@ theorem pos_on_nbhd_of_cts_of_pos {b : ℝ} (Hb : f b > 0) :
theorem neg_on_nbhd_of_cts_of_neg {b : ℝ} (Hb : f b < 0) :
∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (y - b) < δ → f y < 0 :=
begin
let Hcont := continuous_real_elim Hf b (neg_pos_of_neg Hb),
let Hcont := real.continuous_dest Hf b (neg_pos_of_neg Hb),
cases Hcont with δ Hδ,
existsi δ,
split,
@ -572,66 +740,17 @@ theorem neg_on_nbhd_of_cts_of_neg {b : ℝ} (Hb : f b < 0) :
assumption
end
theorem continuous_neg_of_continuous : continuous (λ x, - f x) :=
begin
apply continuous_real_intro,
intros x ε Hε,
cases continuous_real_elim Hf x Hε with δ Hδ,
cases Hδ with Hδ₁ Hδ₂,
existsi δ,
split,
assumption,
intros x' Hx',
let HD := Hδ₂ x' Hx',
rewrite [-abs_neg, neg_neg_sub_neg],
exact HD
end
theorem continuous_offset_of_continuous (a : ℝ) :
continuous (λ x, (f x) + a) :=
begin
apply continuous_real_intro,
intros x ε Hε,
cases continuous_real_elim Hf x Hε with δ Hδ,
cases Hδ with Hδ₁ Hδ₂,
existsi δ,
split,
assumption,
intros x' Hx',
rewrite [add_sub_comm, sub_self, add_zero],
apply Hδ₂,
assumption
end
theorem continuous_mul_of_continuous {g : ℝ → ℝ} (Hcong : continuous g) :
continuous (λ x, f x * g x) :=
begin
apply continuous_of_forall_continuous_at,
intro x,
apply continuous_at_of_converges_to_at,
apply mul_converges_to_at,
all_goals apply converges_to_at_of_continuous_at,
apply continuous_at_of_tendsto_at,
apply mul_approaches,
all_goals apply tendsto_at_of_continuous_at,
all_goals apply forall_continuous_at_of_continuous,
apply Hf,
apply Hcong
end
end continuous
-- this can be strengthened: Hle and Hge only need to hold around x
theorem converges_to_at_squeeze {M : Type} [Hm : metric_space M] {f g h : M → ℝ} {a : ℝ} {x : M}
(Hf : f ⟶ a at x) (Hh : h ⟶ a at x) (Hle : ∀ y : M, f y ≤ g y)
(Hge : ∀ y : M, g y ≤ h y) : g ⟶ a at x :=
begin
apply converges_to_at_of_all_conv_seqs,
intro X HX,
apply converges_to_seq_squeeze,
apply all_conv_seqs_of_converges_to_at Hf,
apply HX,
apply all_conv_seqs_of_converges_to_at Hh,
apply HX,
intro,
apply Hle,
intro,
apply Hge
end

2
library/theories/analysis/sqrt.lean
View file

@ -34,7 +34,7 @@ private theorem lb_le_ub (x : ℝ) (H : x ≥ 0) : sqr_lb x ≤ sqr_ub x :=
apply zero_le_one
end
private lemma sqr_cts : continuous (λ x : ℝ, x * x) := continuous_mul_of_continuous id_continuous id_continuous
private lemma sqr_cts : continuous (λ x : ℝ, x * x) := continuous_mul_of_continuous continuous_id continuous_id
definition sqrt (x : ℝ) : ℝ :=
if H : x ≥ 0 then

34
library/theories/move.lean
View file

@ -6,6 +6,7 @@ Temporary file; move in Lean3.
-/
import data.set algebra.order_bigops
import data.finset data.list.sort
import data.real
-- move this to init.function
@ -20,6 +21,24 @@ theorem eq_of_inv_mul_eq_one {A : Type} {a b : A} [group A] (H : b⁻¹ * a = 1)
have a⁻¹ * 1 = a⁻¹, by inst_simp,
by inst_simp
theorem lt_neg_self_of_neg {A : Type} {a : A} [ordered_comm_group A] (Ha : a < 0) : a < -a :=
calc
a < 0 : Ha
... = -0 : by rewrite neg_zero
... < -a : neg_lt_neg Ha
theorem lt_of_add_lt_of_nonneg_left {A : Type} {a b c : A} [ordered_comm_group A]
(H : a + b < c) (Hb : b ≥ 0) : a < c :=
calc
a < c - b : lt_sub_right_of_add_lt H
... ≤ c : sub_le_self _ Hb
theorem lt_of_add_lt_of_nonneg_right {A : Type} {a b c : A} [ordered_comm_group A]
(H : a + b < c) (Hb : a ≥ 0) : b < c :=
calc
b < c - a : lt_sub_left_of_add_lt H
... ≤ c : sub_le_self _ Hb
-- move to init.quotient
namespace quot
@ -502,3 +521,18 @@ have succ (Max₀ s) ≤ Max₀ s, from le_Max₀ this,
show false, from not_succ_le_self this
end nat
-- move to real
namespace real
theorem lt_of_abs_lt {a b : ℝ} (Ha : abs a < b) : a < b :=
if Hnn : a ≥ 0 then
by rewrite [-abs_of_nonneg Hnn]; exact Ha
else
have Hlt : a < 0, from lt_of_not_ge Hnn,
have -a < b, by rewrite [-abs_of_neg Hlt]; exact Ha,
calc
a < -a : lt_neg_self_of_neg Hlt
... < b : this
end real

37
library/theories/topology/continuous.lean
View file

@ -5,7 +5,7 @@ Authors: Jacob Gross, Jeremy Avigad
Continuous functions.
-/
import theories.topology.basic algebra.category ..move
import theories.topology.basic algebra.category ..move .limit
open algebra eq.ops set topology function category sigma.ops
namespace topology
@ -291,6 +291,41 @@ theorem forall_continuous_at_of_continuous {f : X → Y} (H : continuous f) :
apply mem_univ
end
section limit
open set
theorem tendsto_at_of_continuous_at {f : X → Y} {x : X} (H : continuous_at f x) :
(f ⟶ f x) (nhds x) :=
begin
apply approaches_intro,
intro s HOs Hfxs,
cases H HOs Hfxs with u Hu,
apply eventually_nhds_intro,
exact and.left Hu,
exact and.left (and.right Hu),
intro x' Hx',
apply @mem_of_mem_preimage _ _ f,
apply and.right (and.right Hu),
exact Hx'
end
theorem continuous_at_of_tendsto_at {f : X → Y} {x : X} (H : (f ⟶ f x) (nhds x)) :
continuous_at f x :=
begin
intro s HOs Hfxs,
cases eventually_nhds_dest (approaches_elim H HOs Hfxs) with u Hu,
existsi u,
split,
exact and.left Hu,
split,
exact and.left (and.right Hu),
intro x Hx,
apply mem_preimage,
apply and.right (and.right Hu),
apply Hx
end
end limit
/- The Category TOP -/
section TOP

10
library/theories/topology/limit.lean
View file

@ -703,6 +703,16 @@ section approaches
have eventually (λ x, f x ∈ univ ∧ f x ≠ y) F₁,
from eventually_congr (take x, by rewrite [mem_univ_iff, true_and]) Hf₂,
tendsto_comp_of_approaches_of_tendsto_at_within Hf₁ this Hg
proposition approaches_constant : ((λ x, y) ⟶ y) F :=
begin
apply approaches_intro,
intro s Hs Hys,
have H : (λ x : X, y ∈ s) = (λ x : X, true), from funext (λ x, by rewrite classical.eq_true; exact Hys),
rewrite H,
apply eventually_true
end
end approaches
/-