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)
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) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y)‖₊ =
    ↑‖∫ (y : ℝ) in x - Real.pi..x + Real.pi, h y * dirichletKernel' N (x - y)‖₊
Tactic:
congr 2
Proof state after:
case e_a.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)) ≠ ⊤
⊢ ∫ (y : ℝ) in 0 ..2 * Real.pi, h y * dirichletKernel' N (x - y) =
    ∫ (y : ℝ) in x - Real.pi..x + Real.pi, h y * dirichletKernel' N (x - y)
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)
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)) ≠ ⊤
⊢ Function.Periodic (fun y => h y * dirichletKernel' N (x - y)) (2 * Real.pi)
Tactic:
apply Function.Periodic.mul h_periodic
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)
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)) ≠ ⊤
⊢ Function.Periodic (fun y => dirichletKernel' N (x - y)) (2 * Real.pi)
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)
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)) ≠ ⊤
⊢ Function.Periodic (fun y => dirichletKernel' N (x - y)) (2 * Real.pi)
Tactic:
apply Function.Periodic.const_sub dirichletKernel'_periodic
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)
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) in x - Real.pi..x + Real.pi, h y * dirichletKernel' N (x - y)‖₊ =
    ↑‖(∫ (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)‖₊
Tactic:
congr
Proof state after:
case e_a.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)) ≠ ⊤
⊢ ∫ (y : ℝ) in x - Real.pi..x + Real.pi, h y * dirichletKernel' N (x - y) =
    (∫ (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)
Theorem: control_approximation_effect'
Proof state before:
case e_a.e_a.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
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)) ≠ ⊤
⊢ IntervalIntegrable (fun y => h y * ↑(min |x - y| 1) * dirichletKernel' N (x - y)) MeasureTheory.volume (x - Real.pi)
    (x + Real.pi)
Tactic:
rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])]
Proof state after:
case e_a.e_a.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
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)) ≠ ⊤
⊢ MeasureTheory.IntegrableOn (fun y => h y * ↑(min |x - y| 1) * dirichletKernel' N (x - y))
    (Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
Theorem: control_approximation_effect'
Proof state before:
case e_a.e_a.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
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)) ≠ ⊤
⊢ MeasureTheory.IntegrableOn (fun y => h y * ↑(min |x - y| 1) * dirichletKernel' N (x - y))
    (Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume
Tactic:
apply integrableOn_mul_dirichletKernel'_min xIcc h_intervalIntegrable
Proof state after:
no goals
Theorem: control_approximation_effect'
Proof state before:
case e_a.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)) ≠ ⊤
⊢ ∫ (y : ℝ) in x - Real.pi..x + Real.pi, h y * dirichletKernel' N (x - y) =
    ∫ (x_1 : ℝ) in x - Real.pi..x + Real.pi,
      h x_1 * ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) +
        h x_1 * ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)
Tactic:
congr
Proof state after:
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 y => h y * dirichletKernel' N (x - y)) = fun x_1 =>
    h x_1 * ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) +
      h x_1 * ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)
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 y => h y * dirichletKernel' N (x - y)) = fun x_1 =>
    h x_1 * ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) +
      h x_1 * ↑(min |x - x_1| 1) * dirichletKernel' N (x - 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
⊢ h y * dirichletKernel' N (x - y) =
    h y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) + h y * ↑(min |x - y| 1) * dirichletKernel' N (x - y)
Theorem: control_approximation_effect'
Proof state before:
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
⊢ h y * dirichletKernel' N (x - y) =
    h y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y) + h y * ↑(min |x - y| 1) * dirichletKernel' N (x - y)
Tactic:
rw [←add_mul, ←mul_add]
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
⊢ h y * dirichletKernel' N (x - y) = h y * (↑(max (1 - |x - y|) 0) + ↑(min |x - y| 1)) * dirichletKernel' N (x - y)