internlm/Lean-Github huggingface dataset:
Theorem: control_approximation_effect'
Proof state before:
case e_a.e_a.e_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
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)) ≠ ⊤
⊢ (fun x_1 =>
      {y | dist x y ∈ Set.Ioo 0 Real.pi}.indicator (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
        x_1) =
    fun x_1 =>
    {y | dist x y ∈ Set.Ioo 0 1}.indicator (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) x_1
Tactic:
ext y
Proof state after:
case e_a.e_a.e_f.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)
ε' : ℝ := C_control_approximation_effect ε * δ
ε'def : ε' = C_control_approximation_effect ε * δ
E : Set ℝ := {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
Edef : E = {x | x ∈ Set.Icc 0 (2 * Real.pi) ∧ ∃ N, ε' < Complex.abs (partialFourierSum h N x)}
E_eq : E = Set.Icc 0 (2 * Real.pi) ∩ ⋃ N, {x | ε' < ‖partialFourierSum h N x‖}
measurableSetE : MeasurableSet E
h_intervalIntegrable : IntervalIntegrable h MeasureTheory.volume (-Real.pi) (3 * Real.pi)
F : Set ℝ := Set.Icc (-Real.pi) (3 * Real.pi)
h_bound : ∀ x ∈ F, Complex.abs (h x) ≤ δ
Fdef : F = Set.Icc (-Real.pi) (3 * Real.pi)
f : 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)) ≠ ⊤
y : ℝ
⊢ {y | dist x y ∈ Set.Ioo 0 Real.pi}.indicator (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) y =
    {y | dist x y ∈ Set.Ioo 0 1}.indicator (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) y
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)
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)) ≠ ⊤
y : ℝ
h₀ : y ∉ {y | dist x y ∈ Set.Ioo 0 Real.pi}
h₂ : y ∉ {y | dist x y ∈ Set.Ioo 0 1}
⊢ 0 = 0
Tactic:
. trivial
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)
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)) ≠ ⊤
y : ℝ
h₀ : y ∈ {y | dist x y ∈ Set.Ioo 0 Real.pi}
h₁ : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y)))
Tactic:
trivial
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)
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)) ≠ ⊤
y : ℝ
h₀ : y ∈ {y | dist x y ∈ Set.Ioo 0 Real.pi}
h₁ : y ∉ {y | dist x y ∈ Set.Ioo 0 1}
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0
Tactic:
dsimp at h₀
Proof state after:
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)
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)) ≠ ⊤
y : ℝ
h₀ : dist x y ∈ Set.Ioo 0 Real.pi
h₁ : y ∉ {y | dist x y ∈ Set.Ioo 0 1}
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    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)
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)) ≠ ⊤
y : ℝ
h₀ : dist x y ∈ Set.Ioo 0 Real.pi
h₁ : y ∉ {y | dist x y ∈ Set.Ioo 0 1}
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0
Tactic:
dsimp at h₁
Proof state after:
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)
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)) ≠ ⊤
y : ℝ
h₀ : dist x y ∈ Set.Ioo 0 Real.pi
h₁ : dist x y ∉ Set.Ioo 0 1
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    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)
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)) ≠ ⊤
y : ℝ
h₀ : dist x y ∈ Set.Ioo 0 Real.pi
h₁ : dist x y ∉ Set.Ioo 0 1
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0
Tactic:
rw [Real.dist_eq, Set.mem_Ioo] at h₀
Proof state after:
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)
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)) ≠ ⊤
y : ℝ
h₀ : 0 < |x - y| ∧ |x - y| < Real.pi
h₁ : dist x y ∉ Set.Ioo 0 1
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    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)
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)) ≠ ⊤
y : ℝ
h₀ : 0 < |x - y| ∧ |x - y| < Real.pi
h₁ : dist x y ∉ Set.Ioo 0 1
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0
Tactic:
rw [Real.dist_eq, Set.mem_Ioo] at h₁
Proof state after:
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)
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)) ≠ ⊤
y : ℝ
h₀ : 0 < |x - y| ∧ |x - y| < Real.pi
h₁ : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    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)
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)) ≠ ⊤
y : ℝ
h₀ : 0 < |x - y| ∧ |x - y| < Real.pi
h₁ : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0
Tactic:
push_neg at h₁
Proof state after:
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)
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)) ≠ ⊤
y : ℝ
h₀ : 0 < |x - y| ∧ |x - y| < Real.pi
h₁ : 0 < |x - y| → 1 ≤ |x - y|
⊢ f y *
      (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y) +
        (starRingEnd ℂ) (cexp (I * (-↑N * ↑x)) * k (x - y) * cexp (I * ↑N * ↑y))) =
    0