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 | control_approximation_effect' | [441, 1] | [802, 32] | have E'volume : MeasureTheory.volume E' < ⊤ := lt_of_le_of_lt (MeasureTheory.measure_mono E'subset) Evolume | 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 | 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
E'volume : MeasureTheory.volume E' < ⊤
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have : ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ (2 : ℝ)⁻¹) * (MeasureTheory.volume E') ^ (2 : ℝ)⁻¹ := by
calc ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E'
_ = ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 * MeasureTheory.volume E' := by
congr
rw [← ENNReal.ofReal_ofNat, ← ENNReal.ofReal_div_of_pos (by norm_num)]
ring_nf
_ ≤ ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ (2 : ℝ)⁻¹) * (MeasureTheory.volume E') ^ (2 : ℝ)⁻¹ := by
rcases h with hE' | hE' <;>
. apply rcarleson_exceptional_set_estimate_specific hδ _ measurableSetE' hE'
intro x
rw [fdef, ← Fdef]
simp (config := { failIfUnchanged := false }) only [RCLike.star_def, Function.comp_apply,
map_mul, norm_mul, RingHomIsometric.is_iso]
rw [norm_indicator_eq_indicator_norm]
simp only [norm_eq_abs, Pi.one_apply, norm_one]
rw [Set.indicator_apply, Set.indicator_apply]
split_ifs with inF
. simp
exact h_bound x inF
. simp | 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
E'volume : 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
E'volume : MeasureTheory.volume 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 δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ (2 : ℝ)⁻¹ := by
apply mul_pos
apply mul_pos hδ (by rw [C1_2]; norm_num)
apply Real.rpow_pos_of_pos
linarith [Real.two_pi_pos] | 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
⊢ 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
⊢ 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 ε'_δ_expression_pos : 0 < Real.pi * (ε' - Real.pi * δ) := by
rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_right, mul_div_cancel₀ _ Real.pi_pos.ne.symm]
apply mul_pos δ_mul_const_pos
apply Real.rpow_pos_of_pos
apply div_pos (by norm_num) hε.1 | 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
⊢ 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
ε'_δ_expression_pos : 0 < Real.pi * (ε' - Real.pi * δ)
⊢ MeasureTheory.volume.real E ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
⊢ 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] | calc MeasureTheory.volume.real E
_ ≤ 2 * MeasureTheory.volume.real E' := by
rwa [MeasureTheory.measureReal_def, MeasureTheory.measureReal_def, ←@ENNReal.toReal_ofReal 2 (by norm_num),
←ENNReal.toReal_mul, ENNReal.toReal_le_toReal Evolume.ne, ENNReal.ofReal_ofNat]
apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top E'volume.ne
_ = 2 * MeasureTheory.volume.real E' ^ ((1 + -(2 : ℝ)⁻¹) * 2) := by
conv => lhs; rw [←Real.rpow_one (MeasureTheory.volume.real E')]
congr
norm_num
_ ≤ 2 * (δ * C1_2 4 2 * (4 * Real.pi) ^ (2 : ℝ)⁻¹ / (Real.pi * (ε' - Real.pi * δ))) ^ (2 : ℝ) := by
gcongr
rw [Real.rpow_mul MeasureTheory.measureReal_nonneg]
gcongr
rw [Real.rpow_add' MeasureTheory.measureReal_nonneg (by norm_num), Real.rpow_one, le_div_iff' ε'_δ_expression_pos, ← mul_assoc]
apply mul_le_of_nonneg_of_le_div δ_mul_const_pos.le
apply Real.rpow_nonneg MeasureTheory.measureReal_nonneg
rw[Real.rpow_neg MeasureTheory.measureReal_nonneg, div_inv_eq_mul]
rw [← ENNReal.ofReal_le_ofReal_iff, ENNReal.ofReal_mul ε'_δ_expression_pos.le, MeasureTheory.measureReal_def, ENNReal.ofReal_toReal E'volume.ne]
apply le_trans this
rw [ENNReal.ofReal_mul δ_mul_const_pos.le, ← ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg (by norm_num), ENNReal.ofReal_toReal E'volume.ne]
apply mul_nonneg δ_mul_const_pos.le
apply Real.rpow_nonneg MeasureTheory.measureReal_nonneg
_ = ε := by
rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_right, mul_div_cancel₀,
div_mul_eq_div_div, div_self, one_div, Real.inv_rpow, ← Real.rpow_mul, inv_mul_cancel, Real.rpow_one, inv_div]
ring
norm_num
apply div_nonneg <;> linarith [hε.1]
apply Real.rpow_nonneg
apply div_nonneg <;> linarith [hε.1]
exact δ_mul_const_pos.ne.symm
exact Real.pi_pos.ne.symm | 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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
ε'_δ_expression_pos : 0 < Real.pi * (ε' - Real.pi * δ)
⊢ MeasureTheory.volume.real E ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
E'volume : MeasureTheory.volume E' < ⊤
this :
ENNReal.ofReal (Real.pi * (ε' - Real.pi * δ)) * MeasureTheory.volume E' ≤
ENNReal.ofReal (δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) * MeasureTheory.volume E' ^ 2⁻¹
δ_mul_const_pos : 0 < δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
ε'_δ_expression_pos : 0 < Real.pi * (ε' - Real.pi * δ)
⊢ 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] | 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) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖} | 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)}
⊢ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} =
Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖} | 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) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | ext x | 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)}
⊢ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} =
Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖} | 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)}
x : ℝ
⊢ x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} ↔
x ∈ Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖} | 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)}
⊢ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} =
Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp | 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)}
x : ℝ
⊢ x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} ↔
x ∈ Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖} | no goals | 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)}
x : ℝ
⊢ x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)} ↔
x ∈ Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [E_eq] | 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‖}
⊢ MeasurableSet 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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | 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‖}
⊢ MeasurableSet E
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply MeasurableSet.inter | 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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | case h₁
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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi))
case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | 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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖})
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . apply measurableSet_Icc | case h₁
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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi))
case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi))
case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖})
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply MeasurableSet.iUnion | case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖}) | case h₂.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‖}
⊢ ∀ (b : ℕ), MeasurableSet {x | ε' < ‖partialFourierSum h b x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
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‖}
⊢ MeasurableSet (⋃ N, {x | ε' < ‖partialFourierSum h N x‖})
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | intro N | case h₂.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‖}
⊢ ∀ (b : ℕ), MeasurableSet {x | ε' < ‖partialFourierSum h b x‖} | case h₂.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‖}
N : ℕ
⊢ MeasurableSet {x | ε' < ‖partialFourierSum h N x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.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‖}
⊢ ∀ (b : ℕ), MeasurableSet {x | ε' < ‖partialFourierSum h b x‖}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply measurableSet_lt | case h₂.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‖}
N : ℕ
⊢ MeasurableSet {x | ε' < ‖partialFourierSum h N x‖} | case h₂.h.hf
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‖}
N : ℕ
⊢ Measurable fun a => ε'
case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.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‖}
N : ℕ
⊢ MeasurableSet {x | ε' < ‖partialFourierSum h N x‖}
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . apply measurable_const | case h₂.h.hf
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‖}
N : ℕ
⊢ Measurable fun a => ε'
case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖ | case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.hf
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‖}
N : ℕ
⊢ Measurable fun a => ε'
case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Measurable.norm | case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖ | case h₂.h.hg.hf
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‖}
N : ℕ
⊢ Measurable fun a => partialFourierSum h N a | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.hg
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‖}
N : ℕ
⊢ Measurable fun a => ‖partialFourierSum h N a‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply partialFourierSum_uniformContinuous.continuous.measurable | case h₂.h.hg.hf
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‖}
N : ℕ
⊢ Measurable fun a => partialFourierSum h N a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.hg.hf
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‖}
N : ℕ
⊢ Measurable fun a => partialFourierSum h N a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply measurableSet_Icc | case h₁
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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
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‖}
⊢ MeasurableSet (Set.Icc 0 (2 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply measurable_const | case h₂.h.hf
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‖}
N : ℕ
⊢ Measurable fun a => ε' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂.h.hf
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‖}
N : ℕ
⊢ Measurable fun a => 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 | 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.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 : ℝ
hx : x ∈ E
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | 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)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Edef] at hx | 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
x : ℝ
hx : x ∈ E
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | 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
x : ℝ
hx : x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | 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
x : ℝ
hx : x ∈ E
⊢ x ∈ Set.Icc 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp at hx | 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
x : ℝ
hx : x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | 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
x : ℝ
hx : (0 ≤ x ∧ x ≤ 2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | 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
x : ℝ
hx : x ∈ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
⊢ x ∈ Set.Icc 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | 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
x : ℝ
hx : (0 ≤ x ∧ x ≤ 2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)
⊢ x ∈ Set.Icc 0 (2 * Real.pi) | no goals | 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
x : ℝ
hx : (0 ≤ x ∧ x ≤ 2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)
⊢ x ∈ Set.Icc 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Edef] | 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.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) \ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)},
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ 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'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp | 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) \ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)},
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ 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 : ℝ),
0 ≤ x →
x ≤ 2 * Real.pi →
(0 ≤ x → x ≤ 2 * Real.pi → ∀ (x_1 : ℕ), Complex.abs (partialFourierSum h x_1 x) ≤ ε') →
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ 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) \ {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)},
∀ (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] | 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 : ℝ),
0 ≤ x →
x ≤ 2 * Real.pi →
(0 ≤ x → x ≤ 2 * Real.pi → ∀ (x_1 : ℕ), Complex.abs (partialFourierSum h x_1 x) ≤ ε') →
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ ε' | no goals | 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 : ℝ),
0 ≤ x →
x ≤ 2 * Real.pi →
(0 ≤ x → x ≤ 2 * Real.pi → ∀ (x_1 : ℕ), Complex.abs (partialFourierSum h x_1 x) ≤ ε') →
∀ (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] | apply @IntervalIntegrable.mono_fun' _ _ _ _ _ _ (fun _ ↦ 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 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
⊢ IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi) | case hg
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
⊢ IntervalIntegrable (fun x => δ) MeasureTheory.volume (-Real.pi) (3 * Real.pi)
case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | 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
⊢ IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply intervalIntegrable_const | case hg
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
⊢ IntervalIntegrable (fun x => δ) MeasureTheory.volume (-Real.pi) (3 * Real.pi)
case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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
⊢ IntervalIntegrable (fun x => δ) MeasureTheory.volume (-Real.pi) (3 * Real.pi)
case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | exact h_measurable.aestronglyMeasurable | case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | Please generate a tactic in lean4 to solve the state.
STATE:
case hfm
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.AEStronglyMeasurable h (MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi)))
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Filter.EventuallyLE, ae_restrict_iff_subtype] | case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ | case hle
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
⊢ ∀ᵐ (x : ↑(Ι (-Real.pi) (3 * Real.pi))) ∂MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume, ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle
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
⊢ (fun x => ‖h x‖) ≤ᵐ[MeasureTheory.volume.restrict (Ι (-Real.pi) (3 * Real.pi))] fun x => δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Filter.eventually_of_forall | case hle
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
⊢ ∀ᵐ (x : ↑(Ι (-Real.pi) (3 * Real.pi))) ∂MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume, ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp
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 : ↑(Ι (-Real.pi) (3 * Real.pi))), ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle
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
⊢ ∀ᵐ (x : ↑(Ι (-Real.pi) (3 * Real.pi))) ∂MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume, ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp only [norm_eq_abs, Subtype.forall] | case hle.hp
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 : ↑(Ι (-Real.pi) (3 * Real.pi))), ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp
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
⊢ ∀ a ∈ Ι (-Real.pi) (3 * Real.pi), Complex.abs (h a) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp
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 : ↑(Ι (-Real.pi) (3 * Real.pi))), ‖h ↑x‖ ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
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 | case hle.hp
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
⊢ ∀ a ∈ Ι (-Real.pi) (3 * Real.pi), Complex.abs (h a) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp
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
⊢ ∀ a ∈ Ι (-Real.pi) (3 * Real.pi), Complex.abs (h a) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply h_bound x | case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ δ
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Set.Ioc_subset_Icc_self | case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp.a
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Set.uIoc_of_le (by linarith)] at hx | case hle.hp.a
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hp.a
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 : ℝ
hx : x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp.a
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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor <;> linarith [hx.1, hx.2] | case hle.hp.a
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 : ℝ
hx : x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hp.a
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 : ℝ
hx : x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
⊢ x ∈ Set.Ioc (-Real.pi) (3 * Real.pi)
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply measurableSet_uIoc | case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hle.hs
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 (Ι (-Real.pi) (3 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith | 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 : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ -Real.pi ≤ 3 * Real.pi | no goals | 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
x : ℝ
hx : x ∈ Ι (-Real.pi) (3 * Real.pi)
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Measurable.mul h_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)
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
⊢ Measurable f | 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
⊢ Measurable fun a => F.indicator 1 a | 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)
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
⊢ Measurable f
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Measurable.indicator measurable_const measurableSet_Icc | 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
⊢ Measurable fun a => F.indicator 1 a | no goals | 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)
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
⊢ Measurable fun a => F.indicator 1 a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [fdef, intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])] | 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
⊢ IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi) | 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.IntegrableOn (fun x => h x * F.indicator 1 x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume | 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)
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
⊢ IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | conv => pattern (h _) * _; rw [mul_comm] | 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.IntegrableOn (fun x => h x * F.indicator 1 x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume | 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.IntegrableOn (fun x => F.indicator 1 x * h x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume | 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)
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.IntegrableOn (fun x => h x * F.indicator 1 x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply MeasureTheory.Integrable.bdd_mul' | 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.IntegrableOn (fun x => F.indicator 1 x * h x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume | case hg
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.Integrable (fun x => h x) (MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | 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)
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.IntegrableOn (fun x => F.indicator 1 x * h x) (Set.Ioo (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rwa [← MeasureTheory.IntegrableOn, ← intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])] | case hg
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.Integrable (fun x => h x) (MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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.Integrable (fun x => h x) (MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Measurable.aestronglyMeasurable | case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf.hf
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
⊢ Measurable fun x => F.indicator 1 x
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
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.AEStronglyMeasurable (fun x => F.indicator 1 x)
(MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)))
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Measurable.indicator measurable_const measurableSet_Icc | case hf.hf
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
⊢ Measurable fun x => F.indicator 1 x
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
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
⊢ Measurable fun x => F.indicator 1 x
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Filter.eventually_of_forall | case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf_bound.hp
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
⊢ ∀ (x : ℝ), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
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
⊢ ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-Real.pi) (3 * Real.pi)), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | intro x | case hf_bound.hp
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
⊢ ∀ (x : ℝ), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf_bound.hp
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
x : ℝ
⊢ ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
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
⊢ ∀ (x : ℝ), ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [norm_indicator_eq_indicator_norm] | case hf_bound.hp
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
x : ℝ
⊢ ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ | case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => ‖1 a‖) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
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
x : ℝ
⊢ ‖F.indicator 1 x‖ ≤ ?c
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp | case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => ‖1 a‖) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ | case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => 1) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => ‖1 a‖) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | calc F.indicator (fun _ ↦ (1 : ℝ)) x
_ ≤ 1 := by
apply Set.indicator_apply_le'
intro _
rfl
intro _
norm_num | case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => 1) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
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
x : ℝ
⊢ F.indicator (fun a => 1) x ≤ ?c
case c
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
⊢ ℝ
case c
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
⊢ ℝ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith [Real.pi_pos] | 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
⊢ -Real.pi ≤ 3 * Real.pi | no goals | 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)
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
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Set.indicator_apply_le' | 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
x : ℝ
⊢ F.indicator (fun x => 1) x ≤ 1 | case hfg
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
x : ℝ
⊢ x ∈ F → 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | 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)
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
x : ℝ
⊢ F.indicator (fun x => 1) x ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | intro _ | case hfg
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
x : ℝ
⊢ x ∈ F → 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | case hfg
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
x : ℝ
a✝ : x ∈ F
⊢ 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
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
x : ℝ
⊢ x ∈ F → 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rfl | case hfg
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
x : ℝ
a✝ : x ∈ F
⊢ 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
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
x : ℝ
a✝ : x ∈ F
⊢ 1 ≤ 1
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | intro _ | case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1 | case hg
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
x : ℝ
a✝ : x ∉ F
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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
x : ℝ
⊢ x ∉ F → 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | case hg
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
x : ℝ
a✝ : x ∉ F
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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
x : ℝ
a✝ : x ∉ F
⊢ 0 ≤ 1
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 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] | 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)
⊢ ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | 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)
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)
⊢ ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
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 | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
hx : x ∈ E
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
⊢ ∀ x ∈ E, ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | obtain ⟨xIcc, N, hN⟩ := hx | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
hx : x ∈ E
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (partialFourierSum h N x)
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
hx : x ∈ E
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [partialFourierSum_eq_conv_dirichletKernel' h_intervalIntegrable'] at hN | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (partialFourierSum h N x)
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (partialFourierSum h N x)
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | have : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤ := ENNReal.ofReal_ne_top | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [← (ENNReal.add_le_add_iff_right this)] | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) ≤ T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | 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 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply h_intervalIntegrable.mono_set | 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)
⊢ IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi) | 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)
⊢ Set.uIcc 0 (2 * Real.pi) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi) | 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)
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)
⊢ IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [Set.uIcc_of_le (by linarith [Real.pi_pos]), Set.uIcc_of_le (by linarith [Real.pi_pos])] | 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)
⊢ Set.uIcc 0 (2 * Real.pi) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi) | 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)
⊢ Set.Icc 0 (2 * Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | 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)
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)
⊢ Set.uIcc 0 (2 * Real.pi) ⊆ Set.uIcc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | intro y hy | 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)
⊢ Set.Icc 0 (2 * Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | 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)
y : ℝ
hy : y ∈ Set.Icc 0 (2 * Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | 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)
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)
⊢ Set.Icc 0 (2 * Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | constructor <;> linarith [hy.1, hy.2] | 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)
y : ℝ
hy : y ∈ Set.Icc 0 (2 * Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | no goals | 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)
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)
y : ℝ
hy : y ∈ Set.Icc 0 (2 * Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith [Real.pi_pos] | 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)
⊢ 0 ≤ 2 * Real.pi | no goals | 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)
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)
⊢ 0 ≤ 2 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith [Real.pi_pos] | 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)
⊢ -Real.pi ≤ 3 * Real.pi | no goals | 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)
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)
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [← ENNReal.ofReal_add] | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) =
ENNReal.ofReal (2 * Real.pi * 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi)) = ENNReal.ofReal (2 * Real.pi * ε')
case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) + ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) =
ENNReal.ofReal (2 * Real.pi * ε')
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . congr
ring | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi)) = ENNReal.ofReal (2 * Real.pi * ε')
case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi)) = ENNReal.ofReal (2 * Real.pi * ε')
case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . 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) | case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . apply mul_nonneg (mul_nonneg Real.pi_pos.le hδ.le) Real.two_pi_pos.le | case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | congr | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi)) = ENNReal.ofReal (2 * Real.pi * ε') | case e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ (ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi) = 2 * Real.pi * 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi)) = ENNReal.ofReal (2 * Real.pi * ε')
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | ring | case e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ (ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi) = 2 * Real.pi * ε' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ (ε' - Real.pi * δ) * (2 * Real.pi) + Real.pi * δ * (2 * Real.pi) = 2 * Real.pi * ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg _ Real.two_pi_pos.le | case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi) | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ ε' - Real.pi * δ | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (ε' - Real.pi * δ) * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [ε'def, C_control_approximation_effect_eq hε.1.le, add_sub_cancel_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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ ε' - Real.pi * 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ ε' - Real.pi * δ
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply div_nonneg _ Real.pi_pos.le | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ / Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹ | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹ | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ * (2 / ε) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . 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] | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹ | case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | . apply Real.rpow_nonneg (div_nonneg (by norm_num) hε.1.le) | case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [mul_assoc] | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * (C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg hδ.le | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * (C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹) | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ δ * (C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [C1_2] | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹ | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 * (4 * Real.pi) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ C1_2 4 2 * (4 * Real.pi) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg (by norm_num) | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 * (4 * Real.pi) ^ 2⁻¹ | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (4 * Real.pi) ^ 2⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 * (4 * Real.pi) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_nonneg | case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (4 * Real.pi) ^ 2⁻¹ | case ha.hx
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 4 * Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (4 * Real.pi) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | linarith [Real.pi_pos] | case ha.hx
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 4 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha.hx
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 4 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5 | no goals | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply Real.rpow_nonneg (div_nonneg (by norm_num) hε.1.le) | case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ (2 / ε) ^ 2⁻¹
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | norm_num | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2 | no goals | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | apply mul_nonneg (mul_nonneg Real.pi_pos.le hδ.le) Real.two_pi_pos.le | case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hq
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 0 ≤ Real.pi * δ * (2 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | gcongr | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal (2 * Real.pi * ε') ≤
ENNReal.ofReal
(2 * Real.pi * Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) | no goals | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal (2 * Real.pi * ε') ≤
ENNReal.ofReal
(2 * Real.pi * Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [map_mul, map_div₀, ←mul_assoc] | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal
(2 * Real.pi *
Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊ | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal
(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊ | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal
(2 * Real.pi *
Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | rw [ENNReal.ofReal, ← norm_toNNReal] | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal
(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊ | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ↑(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))).toNNReal =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖.toNNReal | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ENNReal.ofReal
(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))) =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | congr | 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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ↑(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))).toNNReal =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖.toNNReal | case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖ | 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)
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ ↑(2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))).toNNReal =
↑‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖.toNNReal
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | conv => rhs; rw [← one_mul ‖_‖] | case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖ | case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
1 * ‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | congr | case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
1 * ‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖ | case e_a.e_r.e_a
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_r
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) *
Complex.abs (∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)) =
1 * ‖∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | control_approximation_effect' | [441, 1] | [802, 32] | simp | case e_a.e_r.e_a
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) = 1 | case e_a.e_r.e_a
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (|Real.pi|⁻¹ * 2⁻¹) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_r.e_a
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)
h_intervalIntegrable' : IntervalIntegrable h MeasureTheory.volume 0 (2 * Real.pi)
x : ℝ
xIcc : x ∈ Set.Icc 0 (2 * Real.pi)
N : ℕ
hN : ε' < Complex.abs (1 / (2 * ↑Real.pi) * ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y))
this : ENNReal.ofReal (Real.pi * δ * (2 * Real.pi)) ≠ ⊤
⊢ 2 * Real.pi * (Complex.abs 1 / Complex.abs (2 * ↑Real.pi)) = 1
TACTIC:
|
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