internlm/Lean-Github huggingface dataset:
Theorem: control_approximation_effect'
Proof state before:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ ‖h x‖ * ‖F.indicator 1 x‖ ≤ δ * F.indicator 1 x
Tactic:
rw [norm_indicator_eq_indicator_norm]
Proof state after:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ ‖h x‖ * F.indicator (fun a => ‖1 a‖) x ≤ δ * F.indicator 1 x
Theorem: control_approximation_effect'
Proof state before:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ ‖h x‖ * F.indicator (fun a => ‖1 a‖) x ≤ δ * F.indicator 1 x
Tactic:
simp only [norm_eq_abs, Pi.one_apply, norm_one]
Proof state after:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ Complex.abs (h x) * F.indicator (fun a => 1) x ≤ δ * F.indicator 1 x
Theorem: control_approximation_effect'
Proof state before:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ Complex.abs (h x) * F.indicator (fun a => 1) x ≤ δ * F.indicator 1 x
Tactic:
rw [Set.indicator_apply, Set.indicator_apply]
Proof state after:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
⊢ (Complex.abs (h x) * if x ∈ F then 1 else 0) ≤ δ * if x ∈ F then 1 x else 0
Theorem: control_approximation_effect'
Proof state before:
case neg
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
inF : x ∉ F
⊢ Complex.abs (h x) * 0 ≤ δ * 0
Tactic:
. simp
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
case 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)
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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
inF : x ∈ F
⊢ Complex.abs (h x) * 1 ≤ δ * 1 x
Tactic:
simp
Proof state after:
case 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)
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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
inF : x ∈ F
⊢ Complex.abs (h x) ≤ δ
Theorem: control_approximation_effect'
Proof state before:
case 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)
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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
inF : x ∈ F
⊢ Complex.abs (h x) ≤ δ
Tactic:
exact h_bound x inF
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
case neg
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h : ℝ → ℂ
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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'
E'volume : MeasureTheory.volume E' < ⊤
hE' : ∀ x ∈ E', ENNReal.ofReal ((ε' - Real.pi * δ) * (2 * Real.pi)) / 2 ≤ T' (star ∘ f) x
x : ℝ
inF : x ∉ F
⊢ Complex.abs (h x) * 0 ≤ δ * 0
Tactic:
simp
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
case hb.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)
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
⊢ 0 < 4 * Real.pi
Tactic:
linarith [Real.two_pi_pos]
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h✝ N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h✝ MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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
⊢ 0 < 2 ^ (450 * 4 ^ 3) / (2 - 1) ^ 5
Tactic:
norm_num
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
case 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)
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
⊢ 0 < 2 / ε
Tactic:
apply div_pos (by norm_num) hε.1
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
ε : ℝ
hε : 0 < ε ∧ ε ≤ 2 * Real.pi
δ : ℝ
hδ : 0 < δ
h✝ : ℝ → ℂ
h_measurable : Measurable h✝
h_periodic : Function.Periodic h✝ (2 * Real.pi)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h✝ N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h✝ N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h✝ MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.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
⊢ 0 < 2
Tactic:
norm_num
Proof state after:
no goals