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/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | simp [hdef] | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ ε' | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f₀ x - f x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [←Complex.dist_eq, dist_comm, Complex.dist_eq] | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f₀ x - f x) ≤ ε' | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f x - f₀ x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f₀ x - f x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | exact hf₀ x | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f x - f₀ x) ≤ ε' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (f x - f₀ x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | congr | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - partialFourierSum f N x) =
Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x)) | case h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ f x - partialFourierSum f N x =
f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - partialFourierSum f N x) =
Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x))
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | ring | case h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ f x - partialFourierSum f N x =
f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_6.h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ f x - partialFourierSum f N x =
f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply AbsoluteValue.add_le | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x)) ≤
Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x)) ≤
Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply add_le_add_right | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤
Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) | case bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) ≤
Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤
Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply AbsoluteValue.add_le | case bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) ≤
Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x + (f₀ x - partialFourierSum f₀ N x)) ≤
Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | gcongr | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤
ε' + ε / 4 + ε / 4 | case h₁.h₁
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) ≤ ε'
case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) +
Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤
ε' + ε / 4 + ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | . exact hf₀ x | case h₁.h₁
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) ≤ ε'
case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) ≤ ε'
case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | . exact hN₀ N NgtN₀ x hx.1 | case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | . have := hE x hx N
rw [hdef, partialFourierSum_sub (contDiff_f₀.continuous.intervalIntegrable 0 (2 * Real.pi)) (unicontf.continuous.intervalIntegrable 0 (2 * Real.pi))] at this
apply le_trans this
rw [ε'def, mul_div_cancel₀ _ (C_control_approximation_effect_pos εpos).ne.symm] | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | exact hf₀ x | case h₁.h₁
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) ≤ ε' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - f₀ x) ≤ ε'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | exact hN₀ N NgtN₀ x hx.1 | case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have := hE x hx N | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [hdef, partialFourierSum_sub (contDiff_f₀.continuous.intervalIntegrable 0 (2 * Real.pi)) (unicontf.continuous.intervalIntegrable 0 (2 * Real.pi))] at this | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply le_trans this | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4 | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ C_control_approximation_effect ε * ε' ≤ ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [ε'def, mul_div_cancel₀ _ (C_control_approximation_effect_pos εpos).ne.symm] | case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ C_control_approximation_effect ε * ε' ≤ ε / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
this : Complex.abs ((partialFourierSum f₀ N - partialFourierSum f N) x) ≤ C_control_approximation_effect ε * ε'
⊢ C_control_approximation_effect ε * ε' ≤ ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | gcongr | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε' + ε / 4 + ε / 4 ≤ ε / 2 + ε / 4 + ε / 4 | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε' ≤ ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε' + ε / 4 + ε / 4 ≤ ε / 2 + ε / 4 + ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [ε'def, div_div] | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε' ≤ ε / 2 | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε / (4 * C_control_approximation_effect ε) ≤ ε / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε' ≤ ε / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply div_le_div_of_nonneg_left εpos.le (by norm_num) | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε / (4 * C_control_approximation_effect ε) ≤ ε / 2 | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 ≤ 4 * C_control_approximation_effect ε | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε / (4 * C_control_approximation_effect ε) ≤ ε / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [← div_le_iff' (by norm_num)] | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 ≤ 4 * C_control_approximation_effect ε | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 / 4 ≤ C_control_approximation_effect ε | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 ≤ 4 * C_control_approximation_effect ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply le_trans' (lt_C_control_approximation_effect εpos).le (by linarith [Real.two_le_pi]) | case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 / 4 ≤ C_control_approximation_effect ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc.bc
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 / 4 ≤ C_control_approximation_effect ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | norm_num | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 0 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 0 < 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | norm_num | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 0 < 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | linarith [Real.two_le_pi] | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 / 4 ≤ Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ 2 / 4 ≤ Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε / 2 + ε / 4 + ε / 4 ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
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) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ ε / 2 + ε / 4 + ε / 4 ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_neg_eq_conj_k | [10, 1] | [11, 76] | simp [k, Complex.conj_ofReal, ←Complex.exp_conj, Complex.conj_I, neg_mul] | x : ℝ
⊢ k (-x) = (starRingEnd ℂ) (k x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
⊢ k (-x) = (starRingEnd ℂ) (k x)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_abs_le_one | [13, 1] | [15, 12] | rw [k, max_eq_left (by linarith)] | x : ℝ
abs_le_one : |x| ≤ 1
⊢ k x = (1 - ↑|x|) / (1 - (Complex.I * ↑x).exp) | x : ℝ
abs_le_one : |x| ≤ 1
⊢ ↑(1 - |x|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑|x|) / (1 - (Complex.I * ↑x).exp) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : |x| ≤ 1
⊢ k x = (1 - ↑|x|) / (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_abs_le_one | [13, 1] | [15, 12] | norm_cast | x : ℝ
abs_le_one : |x| ≤ 1
⊢ ↑(1 - |x|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑|x|) / (1 - (Complex.I * ↑x).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : |x| ≤ 1
⊢ ↑(1 - |x|) / (1 - (Complex.I * ↑x).exp) = (1 - ↑|x|) / (1 - (Complex.I * ↑x).exp)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_abs_le_one | [13, 1] | [15, 12] | linarith | x : ℝ
abs_le_one : |x| ≤ 1
⊢ 0 ≤ 1 - |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : |x| ≤ 1
⊢ 0 ≤ 1 - |x|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_one_le_abs | [17, 1] | [19, 7] | rw [k, max_eq_right (by linarith)] | x : ℝ
abs_le_one : 1 ≤ |x|
⊢ k x = 0 | x : ℝ
abs_le_one : 1 ≤ |x|
⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : 1 ≤ |x|
⊢ k x = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_one_le_abs | [17, 1] | [19, 7] | simp | x : ℝ
abs_le_one : 1 ≤ |x|
⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : 1 ≤ |x|
⊢ ↑0 / (1 - (Complex.I * ↑x).exp) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_of_one_le_abs | [17, 1] | [19, 7] | linarith | x : ℝ
abs_le_one : 1 ≤ |x|
⊢ 1 - |x| ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
abs_le_one : 1 ≤ |x|
⊢ 1 - |x| ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | apply Measurable.div | ⊢ Measurable k | case hf
⊢ Measurable fun a => ↑(max (1 - |a|) 0)
case hg
⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Measurable k
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | . measurability | case hg
⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | apply Measurable.comp' | case hf
⊢ Measurable fun a => ↑(max (1 - |a|) 0) | case hf.hg
⊢ Measurable Complex.ofReal'
case hf.hf
⊢ Measurable fun x => max (1 - |x|) 0
case hf.x
⊢ MeasurableSpace ℝ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
⊢ Measurable fun a => ↑(max (1 - |a|) 0)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | exact Complex.measurable_ofReal | case hf.hg
⊢ Measurable Complex.ofReal' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hg
⊢ Measurable Complex.ofReal'
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | exact Measurable.max (measurable_const.sub measurable_norm) measurable_const | case hf.hf
⊢ Measurable fun x => max (1 - |x|) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
⊢ Measurable fun x => max (1 - |x|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | k_measurable | [21, 1] | [27, 18] | measurability | case hg
⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
⊢ Measurable fun a => 1 - (Complex.I * ↑a).exp
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | by_cases h : 0 < |x - y| ∧ |x - y| < 1 | x y : ℝ
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | case pos
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . calc ‖K x y‖
_ ≤ 1 / ‖1 - Complex.exp (Complex.I * ↑(x - y))‖ := by
rw [K, k, norm_div]
gcongr
rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg]
. apply max_le _ zero_le_one
linarith [abs_nonneg (x-y)]
. apply le_max_right
_ ≤ 1 / (|x - y| / 2) := by
gcongr
. linarith
. apply lower_secant_bound _ (by rfl)
rw [Set.mem_Icc]
constructor
. simp
calc |x - y|
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi + (x - y) := by
rw [sub_eq_add_neg]
gcongr
exact (abs_le.mp h.2.le).1
. calc x - y
_ ≤ |x - y| := le_abs_self (x - y)
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi - |x - y| := by
gcongr
exact h.2.le
_ = 2 / |x - y| := by
field_simp
_ ≤ (2 : ℝ) ^ (2 : ℝ) / (2 * |x - y|) := by
ring_nf
trivial | case pos
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . push_neg at h
have : ‖K x y‖ = 0 := by
rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff]
by_cases xeqy : x = y
. right
simp [xeqy]
. left
simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add]
exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy))
rw [this]
apply div_nonneg
. norm_num
. linarith [abs_nonneg (x-y)] | case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | calc ‖K x y‖
_ ≤ 1 / ‖1 - Complex.exp (Complex.I * ↑(x - y))‖ := by
rw [K, k, norm_div]
gcongr
rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg]
. apply max_le _ zero_le_one
linarith [abs_nonneg (x-y)]
. apply le_max_right
_ ≤ 1 / (|x - y| / 2) := by
gcongr
. linarith
. apply lower_secant_bound _ (by rfl)
rw [Set.mem_Icc]
constructor
. simp
calc |x - y|
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi + (x - y) := by
rw [sub_eq_add_neg]
gcongr
exact (abs_le.mp h.2.le).1
. calc x - y
_ ≤ |x - y| := le_abs_self (x - y)
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi - |x - y| := by
gcongr
exact h.2.le
_ = 2 / |x - y| := by
field_simp
_ ≤ (2 : ℝ) ^ (2 : ℝ) / (2 * |x - y|) := by
ring_nf
trivial | case pos
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [K, k, norm_div] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖K x y‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | gcongr | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg] | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ ≤ 1 | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ max (1 - |x - y|) 0 ≤ 1
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ ‖↑(max (1 - |x - y|) 0)‖ ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . apply max_le _ zero_le_one
linarith [abs_nonneg (x-y)] | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ max (1 - |x - y|) 0 ≤ 1
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0 | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ max (1 - |x - y|) 0 ≤ 1
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . apply le_max_right | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | apply max_le _ zero_le_one | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ max (1 - |x - y|) 0 ≤ 1 | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 - |x - y| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ max (1 - |x - y|) 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | linarith [abs_nonneg (x-y)] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 - |x - y| ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 - |x - y| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | apply le_max_right | case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hab
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 ≤ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | gcongr | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / (|x - y| / 2) | case hc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 < |x - y| / 2
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 / ‖1 - (Complex.I * ↑(x - y)).exp‖ ≤ 1 / (|x - y| / 2)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . linarith | case hc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 < |x - y| / 2
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ | case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case hc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 < |x - y| / 2
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . apply lower_secant_bound _ (by rfl)
rw [Set.mem_Icc]
constructor
. simp
calc |x - y|
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi + (x - y) := by
rw [sub_eq_add_neg]
gcongr
exact (abs_le.mp h.2.le).1
. calc x - y
_ ≤ |x - y| := le_abs_self (x - y)
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi - |x - y| := by
gcongr
exact h.2.le | case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | linarith | case hc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 < |x - y| / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 0 < |x - y| / 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | apply lower_secant_bound _ (by rfl) | case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖ | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ∈ Set.Icc (-2 * Real.pi + |x - y|) (2 * Real.pi - |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| / 2 ≤ ‖1 - (Complex.I * ↑(x - y)).exp‖
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [Set.mem_Icc] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ∈ Set.Icc (-2 * Real.pi + |x - y|) (2 * Real.pi - |x - y|) | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y ∧ x - y ≤ 2 * Real.pi - |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ∈ Set.Icc (-2 * Real.pi + |x - y|) (2 * Real.pi - |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | constructor | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y ∧ x - y ≤ 2 * Real.pi - |x - y| | case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y
case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y ∧ x - y ≤ 2 * Real.pi - |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . simp
calc |x - y|
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi + (x - y) := by
rw [sub_eq_add_neg]
gcongr
exact (abs_le.mp h.2.le).1 | case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y
case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y| | case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y
case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . calc x - y
_ ≤ |x - y| := le_abs_self (x - y)
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi - |x - y| := by
gcongr
exact h.2.le | case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rfl | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | simp | case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y | case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 2 * Real.pi + (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -2 * Real.pi + |x - y| ≤ x - y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | calc |x - y|
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi + (x - y) := by
rw [sub_eq_add_neg]
gcongr
exact (abs_le.mp h.2.le).1 | case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 2 * Real.pi + (x - y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 2 * Real.pi + (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [le_sub_iff_add_le] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 ≤ 2 * Real.pi - 1 | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 + 1 ≤ 2 * Real.pi | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 ≤ 2 * Real.pi - 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | linarith [Real.two_le_pi] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 + 1 ≤ 2 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 + 1 ≤ 2 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [sub_eq_add_neg] | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi + (x - y) | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y) | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi + (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | gcongr | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y) | case bc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -1 ≤ x - y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi + -1 ≤ 2 * Real.pi + (x - y)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | exact (abs_le.mp h.2.le).1 | case bc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -1 ≤ x - y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bc
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ -1 ≤ x - y
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | calc x - y
_ ≤ |x - y| := le_abs_self (x - y)
_ ≤ 1 := h.2.le
_ ≤ 2 * Real.pi - 1 := by
rw [le_sub_iff_add_le]
linarith [Real.two_le_pi]
_ ≤ 2 * Real.pi - |x - y| := by
gcongr
exact h.2.le | case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ x - y ≤ 2 * Real.pi - |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | gcongr | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi - |x - y| | case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 * Real.pi - 1 ≤ 2 * Real.pi - |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | exact h.2.le | case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | field_simp | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 / (|x - y| / 2) = 2 / |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 1 / (|x - y| / 2) = 2 / |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | ring_nf | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 / |x - y| ≤ 2 ^ 2 / (2 * |x - y|) | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y|⁻¹ * 2 ≤ |x - y|⁻¹ * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ 2 / |x - y| ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | trivial | x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y|⁻¹ * 2 ≤ |x - y|⁻¹ * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| ∧ |x - y| < 1
⊢ |x - y|⁻¹ * 2 ≤ |x - y|⁻¹ * 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | push_neg at h | case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : ¬(0 < |x - y| ∧ |x - y| < 1)
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | have : ‖K x y‖ = 0 := by
rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff]
by_cases xeqy : x = y
. right
simp [xeqy]
. left
simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add]
exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy)) | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [this] | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|) | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2 / (2 * |x - y|) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ ‖K x y‖ ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | apply div_nonneg | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2 / (2 * |x - y|) | case neg.ha
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2
case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2 / (2 * |x - y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . norm_num | case neg.ha
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2
case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y| | case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.ha
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2
case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . linarith [abs_nonneg (x-y)] | case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | rw [norm_eq_zero, K, k, _root_.div_eq_zero_iff] | x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ‖K x y‖ = 0 | x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ‖K x y‖ = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | by_cases xeqy : x = y | x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | case pos
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . right
simp [xeqy] | case pos
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | . left
simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add]
exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy)) | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | right | case pos
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | case pos.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ 1 - (Complex.I * ↑(x - y)).exp = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | simp [xeqy] | case pos.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ 1 - (Complex.I * ↑(x - y)).exp = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : x = y
⊢ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | left | case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0 | case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 ∨ 1 - (Complex.I * ↑(x - y)).exp = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | simp only [Complex.ofReal_eq_zero, max_eq_right_iff, tsub_le_iff_right, zero_add] | case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0 | case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ 1 ≤ |x - y| | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ ↑(max (1 - |x - y|) 0) = 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | exact h (abs_pos.mpr (sub_ne_zero.mpr xeqy)) | case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ 1 ≤ |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
xeqy : ¬x = y
⊢ 1 ≤ |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | norm_num | case neg.ha
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.ha
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 ^ 2
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_bound | [37, 1] | [90, 34] | linarith [abs_nonneg (x-y)] | case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hb
x y : ℝ
h : 0 < |x - y| → 1 ≤ |x - y|
this : ‖K x y‖ = 0
⊢ 0 ≤ 2 * |x - y|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Real.volume_uIoc | [93, 1] | [95, 48] | rw [Set.uIoc, volume_Ioc, max_sub_min_eq_abs] | a b : ℝ
⊢ MeasureTheory.volume (Ι a b) = ENNReal.ofReal |b - a| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
⊢ MeasureTheory.volume (Ι a b) = ENNReal.ofReal |b - a|
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [k_of_abs_le_one, k_of_abs_le_one] | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|(-y)|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑|(-y')|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖k (-y) - k (-y')‖ ≤ 2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . simp only [abs_neg, Complex.ofReal_neg, mul_neg, ge_iff_le]
rw [abs_of_nonneg yy'nonneg.1, abs_of_nonneg yy'nonneg.2]
let f : ℝ → ℂ := fun t ↦ (1 - t) / (1 - Complex.exp (-(Complex.I * t)))
set f' : ℝ → ℂ := fun t ↦ (-1 + Complex.exp (-(Complex.I * t)) + Complex.I * (t - 1) * Complex.exp (-(Complex.I * t))) / (1 - Complex.exp (-(Complex.I * t))) ^ 2 with f'def
set c : ℝ → ℂ := fun t ↦ (1 - t) with cdef
set c' : ℝ → ℂ := fun t ↦ -1 with c'def
set d : ℝ → ℂ := fun t ↦ (1 - Complex.exp (-(Complex.I * t))) with ddef
set d' : ℝ → ℂ := fun t ↦ Complex.I * Complex.exp (-(Complex.I * t)) with d'def
have d_nonzero {t : ℝ} (ht : t ∈ Set.uIcc y' y) : d t ≠ 0 := by
rw [Set.mem_uIcc] at ht
have ht' : 0 < t ∧ t ≤ 1 := by
rcases ht with ht | ht <;> (constructor <;> linarith)
rw [ddef]
simp
rw [←norm_eq_zero]
apply ne_of_gt
calc ‖1 - Complex.exp (-(Complex.I * ↑t))‖
_ ≥ |(1 - Complex.exp (-(Complex.I * ↑t))).im| := by
simp only [Complex.norm_eq_abs, ge_iff_le]
apply Complex.abs_im_le_abs
_ = Real.sin t := by
simp
rw [Complex.exp_im]
simp
apply Real.sin_nonneg_of_nonneg_of_le_pi
. linarith
. linarith [Real.two_le_pi]
_ > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
. linarith
. linarith [Real.two_le_pi]
have f_deriv : ∀ t ∈ Set.uIcc y' y, HasDerivAt f (f' t) t := by
intro t ht
have : f = fun t ↦ c t / d t := by simp
rw [this]
have : f' = fun t ↦ ((c' t * d t - c t * d' t) / d t ^ 2) := by
ext t
rw [f'def, cdef, c'def, ddef, d'def]
simp
congr
ring
rw [this]
apply HasDerivAt.div
. rw [cdef, c'def]
simp
apply HasDerivAt.const_sub
apply HasDerivAt.ofReal_comp
apply hasDerivAt_id'
. rw [ddef, d'def]
simp
rw [←neg_neg (Complex.I * Complex.exp (-(Complex.I * ↑t)))]
apply HasDerivAt.const_sub
rw [←neg_mul, mul_comm]
apply HasDerivAt.cexp
apply HasDerivAt.neg
conv in fun (x : ℝ) ↦ Complex.I * (x : ℝ) =>
ext
rw [mul_comm]
set e : ℂ → ℂ := fun t ↦ t * Complex.I with edef
have : (fun (t : ℝ) ↦ t * Complex.I) = fun (t : ℝ) ↦ e t := by
rw [edef]
rw [this]
apply HasDerivAt.comp_ofReal
rw [edef]
apply hasDerivAt_mul_const
. exact d_nonzero ht
have f'_cont : ContinuousOn (fun t ↦ f' t) (Set.uIcc y' y) := by
apply ContinuousOn.div
. apply Continuous.continuousOn
continuity
.
apply Continuous.continuousOn
apply Continuous.pow
apply Continuous.sub
. apply continuous_const
. apply Continuous.cexp
apply Continuous.neg
apply Continuous.mul
. apply continuous_const
. apply Complex.continuous_ofReal
. intro t ht
simp
apply d_nonzero ht
calc ‖(1 - ↑y) / (1 - Complex.exp (-(Complex.I * ↑y))) - (1 - ↑y') / (1 - Complex.exp (-(Complex.I * ↑y')))‖
_ = ‖f y - f y'‖ := by simp
_ = ‖∫ (t : ℝ) in y'..y, f' t‖ := by
congr 1
rw [intervalIntegral.integral_eq_sub_of_hasDerivAt]
. exact f_deriv
. apply f'_cont.intervalIntegrable
_ = ‖∫ (t : ℝ) in Ι y' y, f' t‖ := by
apply intervalIntegral.norm_intervalIntegral_eq
_ ≤ ∫ (t : ℝ) in Ι y' y, ‖f' t‖ := by
apply MeasureTheory.norm_integral_le_integral_norm
_ ≤ ∫ (t : ℝ) in Ι y' y, 3 / ((y / 2) / 2) ^ 2 := by
apply MeasureTheory.setIntegral_mono_on
. apply f'_cont.norm.integrableOn_uIcc.mono_set
apply Set.Ioc_subset_Icc_self
. apply MeasureTheory.integrableOn_const.mpr
right
rw [Real.volume_uIoc]
apply ENNReal.ofReal_lt_top
. apply measurableSet_uIoc
. intro t ht
rw [Set.mem_uIoc] at ht
have ht' : 0 < t ∧ t ≤ 1 := by
rcases ht with ht | ht <;> (constructor <;> linarith)
rw [f'def]
simp only [norm_div, Complex.norm_eq_abs, norm_pow]
gcongr
. calc Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t)) + Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t)))
_ ≤ Complex.abs (-1 + Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
apply Complex.abs.isAbsoluteValue.abv_add
_ ≤ Complex.abs (-1) + Complex.abs (Complex.exp (-(Complex.I * ↑t))) + Complex.abs (Complex.I * (↑t - 1) * Complex.exp (-(Complex.I * ↑t))) := by
gcongr
apply Complex.abs.isAbsoluteValue.abv_add
_ ≤ 1 + 1 + 1 := by
gcongr
. simp
. rw [mul_comm, ←neg_mul]
norm_cast
apply le_of_eq
apply Complex.abs_exp_ofReal_mul_I
. simp
apply mul_le_one
norm_cast
rw [abs_of_nonpos] <;> linarith
simp
rw [mul_comm, ←neg_mul]
norm_cast
apply le_of_eq
apply Complex.abs_exp_ofReal_mul_I
_ = 3 := by norm_num
. rw [mul_comm, ←neg_mul, mul_comm]
norm_cast
apply lower_secant_bound
. simp only [neg_mul, Set.mem_Icc, neg_add_le_iff_le_add, le_add_neg_iff_add_le,
neg_le_sub_iff_le_add]
constructor <;> linarith [Real.two_le_pi, Real.two_pi_pos]
. rw [abs_neg, le_abs]
left
rcases ht with ht | ht <;> linarith [ht.1]
_ = (MeasureTheory.volume (Ι y' y)).toReal * (3 / ((y / 2) / 2) ^ 2) := by
apply MeasureTheory.setIntegral_const
_ = |y - y'| * (3 / ((y / 2) / 2) ^ 2) := by
congr
rw [Real.volume_uIoc, ENNReal.toReal_ofReal (abs_nonneg (y - y'))]
_ = (3 * (2 * 2) ^ 2) * (1 / y) * (|y - y'| / y) := by
ring
_ ≤ 2 ^ 6 * (1 / y) * (|y - y'| / y) := by
gcongr
norm_num | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|(-y)|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑|(-y')|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|(-y)|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑|(-y')|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . rw [abs_neg, abs_of_nonneg yy'nonneg.2]
assumption | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y')| ≤ 1
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | . rw [abs_neg, abs_of_nonneg yy'nonneg.1]
assumption | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ |(-y)| ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | simp only [abs_neg, Complex.ofReal_neg, mul_neg, ge_iff_le] | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|(-y)|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑|(-y')|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|y|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑|y'|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|(-y)|) / (1 - (Complex.I * ↑(-y)).exp) - (1 - ↑|(-y')|) / (1 - (Complex.I * ↑(-y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | rw [abs_of_nonneg yy'nonneg.1, abs_of_nonneg yy'nonneg.2] | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|y|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑|y'|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / y) * (|y - y'| / y) | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑|y|) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑|y'|) / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / |y|) * (|y - y'| / |y|)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Hilbert_kernel.lean | Hilbert_kernel_regularity_main_part | [100, 1] | [259, 15] | let f : ℝ → ℂ := fun t ↦ (1 - t) / (1 - Complex.exp (-(Complex.I * t))) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / y) * (|y - y'| / y) | y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
f : ℝ → ℂ := fun t => (1 - ↑t) / (1 - (-(Complex.I * ↑t)).exp)
⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / y) * (|y - y'| / y) | Please generate a tactic in lean4 to solve the state.
STATE:
y y' : ℝ
yy'nonneg : 0 ≤ y ∧ 0 ≤ y'
ypos : 0 < y
y2ley' : y / 2 ≤ y'
hy : y ≤ 1
hy' : y' ≤ 1
⊢ ‖(1 - ↑y) / (1 - (-(Complex.I * ↑y)).exp) - (1 - ↑y') / (1 - (-(Complex.I * ↑y')).exp)‖ ≤
2 ^ 6 * (1 / y) * (|y - y'| / y)
TACTIC:
|
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