url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply Measurable.comp continuous_star.measurable measurable₁ | case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ Measurable fun x_1 => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ Measurable fun x_1 => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet] | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150 | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ {y | dist x y ∈ Set.Ioo r 1} →
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply eventually_of_forall | case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ {y | dist x y ∈ Set.Ioo r 1} →
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ x_1 ∈ {y | dist x y ∈ Set.Ioo r 1},
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ {y | dist x y ∈ Set.Ioo r 1} →
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro y hy | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ x_1 ∈ {y | dist x y ∈ Set.Ioo r 1},
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ x_1 ∈ {y | dist x y ∈ Set.Ioo r 1},
‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [RCLike.norm_conj] | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact boundedness₁ hy.1.le | case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ ?m.494150
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply nnnorm_add_le | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖(∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) +
∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ ≤
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖(∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) +
∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ ≤
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr 1 | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . congr
ext y
ring | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . rw [←nnnorm_star, ←starRingEnd_apply, ←integral_conj]
congr
ext y
simp
ring | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) = fun y =>
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) = fun y =>
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y)) | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)) =
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun y => f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) = fun y =>
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ring | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)) =
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)) =
cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [←nnnorm_star, ←starRingEnd_apply, ←integral_conj] | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (x_1 : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
(starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (x_1 : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
(starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun x_1 => (starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))) =
fun y => cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (x_1 : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
(starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun x_1 => (starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))) =
fun y => cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)) | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) =
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ (fun x_1 => (starRingEnd ℂ) (f x_1 * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))) =
fun y => cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) =
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)) | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y) * (cexp (-(I * (↑n * ↑x))) * K x y * cexp (I * ↑n * ↑y)) =
cexp (-(I * (↑n * ↑x))) * ((starRingEnd ℂ) (f y) * K x y * cexp (I * ↑n * ↑y)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) =
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ring | case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y) * (cexp (-(I * (↑n * ↑x))) * K x y * cexp (I * ↑n * ↑y)) =
cexp (-(I * (↑n * ↑x))) * ((starRingEnd ℂ) (f y) * K x y * cexp (I * ↑n * ↑y)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_f.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
y : ℝ
⊢ (starRingEnd ℂ) (f y) * (cexp (-(I * (↑n * ↑x))) * K x y * cexp (I * ↑n * ↑y)) =
cexp (-(I * (↑n * ↑x))) * ((starRingEnd ℂ) (f y) * K x y * cexp (I * ↑n * ↑y))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [← NNReal.coe_inj] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊) =
↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | push_cast | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊) =
↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊) | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖₊) =
↑(‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | norm_cast | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖ | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * ↑(↑(-n) * x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * ↑(↑(-n) * x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * (-↑n * ↑x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * (-↑n * ↑x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | congr 1 <;>
. rw [MeasureTheory.integral_mul_left, norm_mul, norm_eq_abs, mul_comm I, abs_exp_ofReal_mul_I, one_mul] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * ↑(↑(-n) * x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * ↑(↑(-n) * x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, cexp (I * ↑(↑(-n) * x)) * (f y * K x y * cexp (I * ↑n * ↑y))‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * ↑(↑(-n) * x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖ +
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [MeasureTheory.integral_mul_left, norm_mul, norm_eq_abs, mul_comm I, abs_exp_ofReal_mul_I, one_mul] | case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * ↑(↑(-n) * x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
cexp (I * ↑(↑(-n) * x)) * ((⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y))‖ =
‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [CarlesonOperatorReal', CarlesonOperatorReal'] | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply iSup₂_le | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ∀ (i : ℤ) (j : ℝ),
⨆ (_ : 0 < j),
⨆ (_ : j < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, f y * K x y * cexp (I * ↑i * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑i * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro n r | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ∀ (i : ℤ) (j : ℝ),
⨆ (_ : 0 < j),
⨆ (_ : j < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, f y * K x y * cexp (I * ↑i * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑i * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
⊢ ∀ (i : ℤ) (j : ℝ),
⨆ (_ : 0 < j),
⨆ (_ : j < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, f y * K x y * cexp (I * ↑i * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo j 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑i * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply iSup₂_le | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ 0 < r →
r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ ⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro rpos rle1 | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ 0 < r →
r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
⊢ 0 < r →
r < 1 →
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | gcongr <;>
. apply le_iSup₂_of_le n r
apply le_iSup₂_of_le rpos rle1
trivial | case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊ +
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
(⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1), ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * K x y * cexp (I * ↑n * ↑y)‖₊) +
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply le_iSup₂_of_le n r | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
⨆ n,
⨆ r,
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply le_iSup₂_of_le rpos rle1 | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
⨆ (_ : 0 < r),
⨆ (_ : r < 1),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | trivial | case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h₂
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
(𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊ ≤
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, (⇑(starRingEnd ℂ) ∘ f) y * K x y * cexp (I * ↑n * ↑y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | calc ε * MeasureTheory.volume E
_ = ∫⁻ _ in E, ε := by
symm
apply MeasureTheory.set_lintegral_const
_ ≤ ∫⁻ x in E, T' f x := by
apply MeasureTheory.set_lintegral_mono' measurableSetE hE
_ = ENNReal.ofReal δ * ∫⁻ x in E, T' (fun x ↦ (1 / δ) * f x) x := by
rw [← MeasureTheory.lintegral_const_mul']
congr
ext x
rw [CarlesonOperatorReal'_mul δpos]
congr
exact ENNReal.ofReal_ne_top
_ ≤ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * (MeasureTheory.volume E) ^ (2 : ℝ)⁻¹ * (MeasureTheory.volume F) ^ (2 : ℝ)⁻¹) := by
gcongr
apply rcarleson' measurableSetF measurableSetE
intro x
simp
rw [_root_.abs_of_nonneg δpos.le, inv_mul_le_iff δpos]
exact hf x
_ = ENNReal.ofReal (δ * C1_2 4 2) * (MeasureTheory.volume F) ^ (2 : ℝ)⁻¹ * (MeasureTheory.volume E) ^ (2 : ℝ)⁻¹ := by
rw [ENNReal.ofReal_mul δpos.le]
ring | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | symm | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E = ∫⁻ (x : ℝ) in E, ε | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E = ∫⁻ (x : ℝ) in E, ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply MeasureTheory.set_lintegral_const | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε = ε * MeasureTheory.volume E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply MeasureTheory.set_lintegral_mono' measurableSetE hE | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε ≤ ∫⁻ (x : ℝ) in E, T' f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, ε ≤ ∫⁻ (x : ℝ) in E, T' f x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [← MeasureTheory.lintegral_const_mul'] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | congr | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' f x = ∫⁻ (a : ℝ) in E, ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | ext x | case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ (fun x => T' f x) = fun a => ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) a
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [CarlesonOperatorReal'_mul δpos] | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ T' f x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | congr | case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ↑δ.toNNReal * T' (fun x => 1 / ↑δ * f x) x = ENNReal.ofReal δ * T' (fun x => 1 / ↑δ * f x) x
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | exact ENNReal.ofReal_ne_top | case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hr
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ ≠ ⊤
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | gcongr | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) | case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | apply rcarleson' measurableSetF measurableSetE | case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹ | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∫⁻ (x : ℝ) in E, T' (fun x => 1 / ↑δ * f x) x ≤
ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | intro x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ∀ (x : ℝ), ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | simp | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ ‖1 / ↑δ * f x‖ ≤ F.indicator 1 x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [_root_.abs_of_nonneg δpos.le, inv_mul_le_iff δpos] | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ |δ|⁻¹ * Complex.abs (f x) ≤ F.indicator 1 x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | exact hf x | case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.hf
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
x : ℝ
⊢ Complex.abs (f x) ≤ δ * F.indicator 1 x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | rw [ENNReal.ofReal_mul δpos.le] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal.ofReal (δ * C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal.ofReal δ * ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal.ofReal (δ * C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate | [372, 1] | [397, 11] | ring | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal.ofReal δ * ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
F : Set ℝ
measurableSetF : MeasurableSet F
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * F.indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal δ * (ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume E ^ 2⁻¹ * MeasureTheory.volume F ^ 2⁻¹) =
ENNReal.ofReal δ * ENNReal.ofReal (C1_2 4 2) * MeasureTheory.volume F ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [ENNReal.ofReal_mul (by apply mul_nonneg δpos.le; rw [C1_2]; norm_num), ← ENNReal.ofReal_rpow_of_pos (by linarith [Real.pi_pos])] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E ^ 2⁻¹ | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNReal.ofReal (4 * Real.pi) ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | convert rcarleson_exceptional_set_estimate δpos measurableSet_Icc hf measurableSetE hE | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNReal.ofReal (4 * Real.pi) ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹ | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = MeasureTheory.volume (Set.Icc (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ε * MeasureTheory.volume E ≤
ENNReal.ofReal (δ * C1_2 4 2) * ENNReal.ofReal (4 * Real.pi) ^ 2⁻¹ * MeasureTheory.volume E ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [Real.volume_Icc] | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = MeasureTheory.volume (Set.Icc (-Real.pi) (3 * Real.pi)) | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal.ofReal (3 * Real.pi - -Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = MeasureTheory.volume (Set.Icc (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | ring_nf | case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal.ofReal (3 * Real.pi - -Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_5.h.e'_6.h.e'_5
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ ENNReal.ofReal (4 * Real.pi) = ENNReal.ofReal (3 * Real.pi - -Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | apply mul_nonneg δpos.le | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ δ * C1_2 4 2 | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2 | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ δ * C1_2 4 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | rw [C1_2] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2 | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ C1_2 4 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | norm_num | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | rcarleson_exceptional_set_estimate_specific | [399, 1] | [405, 10] | linarith [Real.pi_pos] | δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 < 4 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
δ : ℝ
δpos : 0 < δ
f : ℝ → ℂ
hf : ∀ (x : ℝ), ‖f x‖ ≤ δ * (Set.Icc (-Real.pi) (3 * Real.pi)).indicator 1 x
E : Set ℝ
measurableSetE : MeasurableSet E
ε : ENNReal
hE : ∀ x ∈ E, ε ≤ T' f x
⊢ 0 < 4 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | rw [C_control_approximation_effect] | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C_control_approximation_effect ε | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C_control_approximation_effect ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply lt_add_of_pos_of_le _ (by rfl) | ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi | ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply mul_pos (C1_2_pos (by norm_num)) | ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply Real.rpow_pos_of_pos | ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹ | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε) | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply div_pos (by norm_num) | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε) | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 / (Real.pi * ε)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | apply mul_pos Real.pi_pos εpos | case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
ε : ℝ
εpos : 0 < ε
⊢ 0 < Real.pi * ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | rfl | ε : ℝ
εpos : 0 < ε
⊢ Real.pi ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ Real.pi ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | norm_num | ε : ℝ
εpos : 0 < ε
⊢ 1 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 1 < 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | lt_C_control_approximation_effect | [410, 1] | [416, 33] | norm_num | ε : ℝ
εpos : 0 < ε
⊢ 0 < 8 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
εpos : 0 < ε
⊢ 0 < 8
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | symm | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C_control_approximation_effect ε * δ = δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C_control_approximation_effect ε * δ = δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [C_control_approximation_effect, mul_comm, mul_div_right_comm, mul_comm δ, mul_assoc, mul_comm δ, ← mul_assoc, ← mul_assoc, ← add_mul] | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi + Real.pi * δ = C_control_approximation_effect ε * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr 2 | ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ ((2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ + Real.pi) * δ =
(C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ + Real.pi) * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [mul_comm _ (C1_2 4 2), mul_assoc] | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr | case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ C1_2 4 2 * ((2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹) = C1_2 4 2 * (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [Real.div_rpow, Real.div_rpow _ (mul_nonneg _ _), Real.mul_rpow, Real.mul_rpow] | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹ | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (2 / ε) ^ 2⁻¹ / Real.pi * (4 * Real.pi) ^ 2⁻¹ = (8 / (Real.pi * ε)) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | ring_nf | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ 2⁻¹ / ε ^ 2⁻¹ / Real.pi * (4 ^ 2⁻¹ * Real.pi ^ 2⁻¹) = 8 ^ 2⁻¹ / (Real.pi ^ 2⁻¹ * ε ^ 2⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [mul_assoc, mul_comm (2 ^ _), mul_assoc, mul_assoc, mul_assoc, mul_comm (4 ^ _), ← mul_assoc Real.pi⁻¹,
← Real.rpow_neg_one Real.pi, ← Real.rpow_add, mul_comm (Real.pi ^ _), ← mul_assoc (2 ^ _), ← Real.mul_rpow] | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 ^ (1 / 2) * (ε ^ (1 / 2))⁻¹ * Real.pi⁻¹ * 4 ^ (1 / 2) * Real.pi ^ (1 / 2) =
(ε ^ (1 / 2))⁻¹ * 8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | congr | case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ (ε ^ (1 / 2))⁻¹ * ((2 * 4) ^ (1 / 2) * Real.pi ^ (-1 + 1 / 2)) =
(ε ^ (1 / 2))⁻¹ * (8 ^ (1 / 2) * (Real.pi ^ (1 / 2))⁻¹)
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | norm_num | case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 2 * 4 = 8
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | ring_nf | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 + 1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | rw [neg_div, Real.rpow_neg] | case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ Real.pi ^ (-1 / 2) = (Real.pi ^ (1 / 2))⁻¹
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | all_goals linarith [Real.pi_pos] | case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 < Real.pi
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 4
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 8
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ Real.pi
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
case e_a.e_a.e_a.hx
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ 2
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | C_control_approximation_effect_eq | [420, 1] | [435, 35] | linarith [Real.pi_pos] | case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a.e_a.hy
ε δ : ℝ
ε_nonneg : 0 ≤ ε
⊢ 0 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set ε' := C_control_approximation_effect ε * δ with ε'def | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * δ | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set E := {x ∈ Set.Icc 0 (2 * Real.pi) | ∃ N, ε' < abs (partialFourierSum h N x)} with Edef | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have E_eq: E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N : ℕ, {x | ε' < ‖partialFourierSum h N x‖} := by
rw [Edef]
ext x
simp | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have measurableSetE : MeasurableSet E := by
rw [E_eq]
apply MeasurableSet.inter
. apply measurableSet_Icc
apply MeasurableSet.iUnion
intro N
apply measurableSet_lt
. apply measurable_const
apply Measurable.norm
apply partialFourierSum_uniformContinuous.continuous.measurable | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | use E | ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi) ∧
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor | case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi) ∧
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi)
case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi) ∧
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . intro x hx
rw [Edef] at hx
simp at hx
exact hx.1 | case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi)
case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ E ⊆ Set.Icc 0 (2 * Real.pi)
case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | use measurableSetE | case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [and_comm] | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ (∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε') ∧
MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε ∧
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor | case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ (∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε') ∧
MeasureTheory.volume.real E ≤ ε | case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ (∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε') ∧
MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . rw [Edef]
simp
exact fun x x_nonneg x_le_two_pi h ↦ h x_nonneg x_le_two_pi | case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε'
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi) := by
apply @IntervalIntegrable.mono_fun' _ _ _ _ _ _ (fun _ ↦ δ)
apply intervalIntegrable_const
exact h_measurable.aestronglyMeasurable
rw [Filter.EventuallyLE, ae_restrict_iff_subtype]
apply Filter.eventually_of_forall
simp only [norm_eq_abs, Subtype.forall]
intro x hx
apply h_bound x
apply Set.Ioc_subset_Icc_self
rw [Set.uIoc_of_le (by linarith)] at hx
constructor <;> linarith [hx.1, hx.2]
apply measurableSet_uIoc | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set F := Set.Icc (-Real.pi) (3 * Real.pi) with Fdef | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ δ
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | set f := fun x ↦ h x * F.indicator 1 x with fdef | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have f_measurable : Measurable f := by
apply Measurable.mul h_measurable
apply Measurable.indicator measurable_const measurableSet_Icc | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi) := by
rw [fdef, intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])]
conv => pattern (h _) * _; rw [mul_comm]
apply MeasureTheory.Integrable.bdd_mul'
rwa [← MeasureTheory.IntegrableOn, ← intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])]
apply Measurable.aestronglyMeasurable
apply Measurable.indicator measurable_const measurableSet_Icc
apply Filter.eventually_of_forall
intro x
rw [norm_indicator_eq_indicator_norm]
simp
calc F.indicator (fun _ ↦ (1 : ℝ)) x
_ ≤ 1 := by
apply Set.indicator_apply_le'
intro _
rfl
intro _
norm_num | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' ((starRingEnd ℂ) ∘ f) x := by
have h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi) := by
apply h_intervalIntegrable.mono_set
rw [Set.uIcc_of_le (by linarith [Real.pi_pos]), Set.uIcc_of_le (by linarith [Real.pi_pos])]
intro y hy
constructor <;> linarith [hy.1, hy.2]
intro x hx
obtain ⟨xIcc, N, hN⟩ := hx
rw [partialFourierSum_eq_conv_dirichletKernel' h_intervalIntegrable'] at hN
have : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤ := ENNReal.ofReal_ne_top
rw [← (ENNReal.add_le_add_iff_right this)]
calc ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi))
_ = ENNReal.ofReal ((2 * Real.pi) * ε') := by
rw [← ENNReal.ofReal_add]
. congr
ring
. apply mul_nonneg _ Real.two_pi_pos.le
rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_right]
apply div_nonneg _ Real.pi_pos.le
apply mul_nonneg
. rw [mul_assoc]
apply mul_nonneg hδ.le
rw [C1_2]
apply mul_nonneg (by norm_num)
apply Real.rpow_nonneg
linarith [Real.pi_pos]
. apply Real.rpow_nonneg (div_nonneg (by norm_num) hε.1.le)
. apply mul_nonneg (mul_nonneg Real.pi_pos.le hδ.le) Real.two_pi_pos.le
_ ≤ ENNReal.ofReal ((2 * Real.pi) * abs (1 / (2 * Real.pi) * ∫ (y : ℝ) in (0 : ℝ)..(2 * Real.pi), h y * dirichletKernel' N (x - y))) := by gcongr
_ = ‖∫ (y : ℝ) in (0 : ℝ)..(2 * Real.pi), h y * dirichletKernel' N (x - y)‖₊ := by
rw [map_mul, map_div₀, ←mul_assoc]
rw [ENNReal.ofReal, ← norm_toNNReal]
congr
conv => rhs; rw [← one_mul ‖_‖]
congr
simp
rw [_root_.abs_of_nonneg Real.pi_pos.le]
field_simp
ring
_ = ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * dirichletKernel' N (x - y)‖₊ := by
congr 2
rw [← zero_add (2 * Real.pi), Function.Periodic.intervalIntegral_add_eq _ 0 (x - Real.pi)]
congr 1
ring
apply Function.Periodic.mul h_periodic
apply Function.Periodic.const_sub dirichletKernel'_periodic
_ = ‖ (∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
+ (∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (min |x - y| 1) * dirichletKernel' N (x - y)) ‖₊ := by
congr
rw [← intervalIntegral.integral_add]
. congr
ext y
rw [←add_mul, ←mul_add]
congr
conv => lhs; rw [←mul_one (h y)]
congr
norm_cast
rw [min_def]
split_ifs
. rw [max_eq_left (by linarith)]
simp
. rw [max_eq_right (by linarith)]
simp
rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])]
apply integrableOn_mul_dirichletKernel'_max xIcc h_intervalIntegrable
rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])]
apply integrableOn_mul_dirichletKernel'_min xIcc h_intervalIntegrable
_ ≤ ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊
+ ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (min |x - y| 1) * dirichletKernel' N (x - y)‖₊ := by
norm_cast
apply nnnorm_add_le
_ ≤ (T' f x + T' ((starRingEnd ℂ) ∘ f) x) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) := by
gcongr
. calc ENNReal.ofNNReal ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊
_ = ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ := by
congr 2
apply intervalIntegral.integral_congr
intro y hy
simp
rw [Set.uIcc_of_le (by linarith)] at hy
left
left
rw [fdef, ←mul_one (h y)]
congr
rw [Set.indicator_apply]
have : y ∈ F := by
rw [Fdef]
simp
constructor <;> linarith [xIcc.1, xIcc.2, hy.1, hy.2]
simp [this]
_ = ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 Real.pi}, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ := by
congr
rw [annulus_real_eq (le_refl 0), MeasureTheory.integral_union (by simp), ← MeasureTheory.integral_Ioc_eq_integral_Ioo, ← MeasureTheory.integral_union,
intervalIntegral.integral_of_le (by linarith), MeasureTheory.integral_Ioc_eq_integral_Ioo]
congr
simp
rw [Set.Ioc_union_Ioo_eq_Ioo (by linarith) (by linarith)]
. simp
apply Set.disjoint_of_subset_right Set.Ioo_subset_Ioc_self
simp
. exact measurableSet_Ioo
. apply (integrableOn_mul_dirichletKernel'_max xIcc f_integrable).mono_set
intro y hy
constructor <;> linarith [hy.1, hy.2, Real.pi_pos]
. apply (integrableOn_mul_dirichletKernel'_max xIcc f_integrable).mono_set
intro y hy
constructor <;> linarith [hy.1, hy.2, Real.pi_pos]
. exact measurableSet_Ioo
. apply (integrableOn_mul_dirichletKernel'_max xIcc f_integrable).mono_set
intro y hy
constructor <;> linarith [hy.1, hy.2, Real.pi_pos]
. apply (integrableOn_mul_dirichletKernel'_max xIcc f_integrable).mono_set
intro y hy
constructor <;> linarith [hy.1, hy.2, Real.pi_pos]
_ = ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ := by
congr 2
rw [←MeasureTheory.integral_indicator annulus_measurableSet, ←MeasureTheory.integral_indicator annulus_measurableSet]
congr
ext y
rw [Set.indicator_apply, Set.indicator_apply, mul_assoc, dirichlet_Hilbert_eq, K]
split_ifs with h₀ h₁ h₂
. trivial
. dsimp at h₀
dsimp at h₁
rw [Real.dist_eq, Set.mem_Ioo] at h₀
rw [Real.dist_eq, Set.mem_Ioo] at h₁
push_neg at h₁
rw [k_of_one_le_abs (h₁ h₀.1)]
simp
. rw [k_of_one_le_abs]
simp
dsimp at h₀
dsimp at h₂
rw [Real.dist_eq, Set.mem_Ioo] at h₀
rw [Real.dist_eq, Set.mem_Ioo] at h₂
push_neg at h₀
apply le_trans' (h₀ h₂.1)
linarith [Real.two_le_pi]
. trivial
_ ≤ (T' f x + T' ((starRingEnd ℂ) ∘ f) x) := by
apply le_CarlesonOperatorReal' f_integrable x xIcc
. rw [ENNReal.ofReal]
norm_cast
apply NNReal.le_toNNReal_of_coe_le
rw [coe_nnnorm]
calc ‖∫ (y : ℝ) in (x - Real.pi)..(x + Real.pi), h y * (min |x - y| 1) * dirichletKernel' N (x - y)‖
_ ≤ (δ * Real.pi) * |(x + Real.pi) - (x - Real.pi)| := by
apply intervalIntegral.norm_integral_le_of_norm_le_const
intro y hy
rw [Set.uIoc_of_le (by linarith)] at hy
rw [mul_assoc, norm_mul]
gcongr
. rw [norm_eq_abs]
apply h_bound
rw [Fdef]
simp
constructor <;> linarith [xIcc.1, xIcc.2, hy.1, hy.2]
rw [dirichletKernel', mul_add]
set z := x - y with zdef
calc ‖ (min |z| 1) * (exp (I * N * z) / (1 - exp (-I * z)))
+ (min |z| 1) * (exp (-I * N * z) / (1 - exp (I * z)))‖
_ ≤ ‖(min |z| 1) * (exp (I * N * z) / (1 - exp (-I * z)))‖
+ ‖(min |z| 1) * (exp (-I * N * z) / (1 - exp (I * z)))‖ := by
apply norm_add_le
_ = min |z| 1 * 1 / ‖1 - exp (I * z)‖ + min |z| 1 * 1 / ‖1 - exp (I * z)‖ := by
simp
congr
. simp only [abs_eq_self, le_min_iff, abs_nonneg, zero_le_one, and_self]
. rw [mul_assoc I, mul_comm I]
norm_cast
rw [abs_exp_ofReal_mul_I, one_div, ←abs_conj, map_sub, map_one, ←exp_conj, ← neg_mul, map_mul,
conj_neg_I, conj_ofReal]
.
simp only [abs_eq_self, le_min_iff, abs_nonneg, zero_le_one, and_self]
. rw [mul_assoc I, mul_comm I, ←neg_mul]
norm_cast
rw [abs_exp_ofReal_mul_I, one_div]
_ = 2 * (min |z| 1 / ‖1 - exp (I * z)‖) := by ring
_ ≤ 2 * (Real.pi / 2) := by
gcongr 2 * ?_
. by_cases h : (1 - exp (I * z)) = 0
. rw [h, norm_zero, div_zero]
linarith [Real.pi_pos]
rw [div_le_iff', ←div_le_iff, div_div_eq_mul_div, mul_div_assoc, mul_comm]
apply lower_secant_bound'
. apply min_le_left
. have : |z| ≤ Real.pi := by
rw [abs_le]
rw [zdef]
constructor <;> linarith [hy.1, hy.2]
rw [min_def]
split_ifs <;> linarith
. linarith [Real.pi_pos]
. rwa [norm_pos_iff]
_ = Real.pi := by ring
_ = Real.pi * δ * (2 * Real.pi) := by
simp
rw [←two_mul, _root_.abs_of_nonneg Real.two_pi_pos.le]
ring | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have Evolume : MeasureTheory.volume E < ⊤ := by
calc MeasureTheory.volume E
_ ≤ MeasureTheory.volume (Set.Icc 0 (2 * Real.pi)) := by
apply MeasureTheory.measure_mono
rw [E_eq]
apply Set.inter_subset_left
_ = ENNReal.ofReal (2 * Real.pi) := by
rw [Real.volume_Icc, sub_zero]
_ < ⊤ := ENNReal.ofReal_lt_top | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
⊢ MeasureTheory.volume.real E ≤ ε | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
Evolume : MeasureTheory.volume E < ⊤
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | obtain ⟨E', E'subset, measurableSetE', E'measure, h⟩ := ENNReal.le_on_subset MeasureTheory.volume measurableSetE (CarlesonOperatorReal'_measurable f_measurable) (CarlesonOperatorReal'_measurable (Measurable.comp continuous_star.measurable f_measurable)) le_operator_add | case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
Evolume : MeasureTheory.volume E < ⊤
⊢ MeasureTheory.volume.real E ≤ ε | case right.right.intro.intro.intro.intro
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h✝ N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h✝ MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h✝ x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h✝ x * F.indicator 1 x
fdef : f = fun x => h✝ x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
Evolume : MeasureTheory.volume E < ⊤
E' : Set ℝ
E'subset : E' ⊆ E
measurableSetE' : MeasurableSet E'
E'measure : MeasureTheory.volume E ≤ 2 * MeasureTheory.volume E'
h :
(∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' f x) ∨
∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : ℝ → ℂ := fun x => h x * F.indicator 1 x
fdef : f = fun x => h x * F.indicator 1 x
f_measurable : Measurable f
f_integrable : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
le_operator_add : ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
Evolume : MeasureTheory.volume E < ⊤
⊢ MeasureTheory.volume.real E ≤ ε
TACTIC:
|
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