url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.aestronglyMeasurable | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x ... | case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1))
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.mul | case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1) | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(max (1 - |x - a|) 0)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply Measurable.comp Complex.measurable_ofReal
apply Measurable.max
apply Measurable.const_sub
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(max (1 - |x - a|) 0)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi... | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(max (1 - |x - a|) 0)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp Complex.measurable_ofReal | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(max (1 - |x - a|) 0) | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => max (1 - |x - a|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(max (1 - |x - a|) 0)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.max | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => max (1 - |x - a|) 0 | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => max (1 - |x - a|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 ... | case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1 - |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp _root_.continuous_abs.measurable | case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) ... | case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => |x - x_1|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f :... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3... | case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => x - x_1
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | exact measurable_const | case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.comp dirichletKernel'_measurable | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc] | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(max (1 -... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply eventually_of_forall | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ),
x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(max (1 - |x - x_1|) 0) * dirichle... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | intro y _ | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(max (1 - |x - x_1|) 0) * dirichletKernel' N (x ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | calc ‖(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖
_ = ‖max (1 - |x - y|) 0‖ * ‖dirichletKernel' N (x - y)‖ := by
rw [norm_mul, Complex.norm_real]
_ ≤ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply max_le
linarith [abs_nonneg (x - y)]
norm_num
apply le_max_r... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(max (1 - |x - y|) 0) * dirichletKernel' N... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [norm_mul, Complex.norm_real] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖ = ‖max (1 - |x - y|) 0‖ * ‖dirichletKernel' N (x - y)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖ = ‖max ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | gcongr | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ * ‖dirichletKernel' N (x - y)‖ ≤ 1 * (2 * ↑N + 1) | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableO... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ * ‖dirichletKernel' N (x - y)‖ ≤ 1 * (2... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | . rw [Real.norm_of_nonneg]
apply max_le
linarith [abs_nonneg (x - y)]
norm_num
apply le_max_right | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableO... | case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply norm_dirichletKernel'_le | case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | rw [Real.norm_of_nonneg] | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ ≤ 1 | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ max (1 - |x - y|) 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖max (1 - |x - y|) 0‖ ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply max_le | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ max (1 - |x - y|) 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn ... | case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 1 - |x - y| ≤ 1
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ max (1 - |x - y|) 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | linarith [abs_nonneg (x - y)] | case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 1 - |x - y| ≤ 1
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f ... | case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Re... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 1 - |x - y| ≤ 1
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | norm_num | case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Re... | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ max (1 - |x - y|) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_max | [128, 1] | [159, 39] | apply le_max_right | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ max (1 - |x - y|) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ max (1 - |x - y|) 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | conv => pattern ((f _) * _ * _); rw [mul_assoc, mul_comm] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(min |x - y| 1) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(min |x - y| 1) * dirichletKernel' N (x - y))
(Set.Icc (x - Real.pi) (x + Real.pi)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by linarith [Real.pi_pos])] at hf | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply MeasureTheory.Integrable.bdd_mul' | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real.pi) (x + Real.pi)) MeasureTheory.volume | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ →... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => ↑(min |x - y| 1) * dirichletKernel' N (x - y) * f y)
(Set.Icc (x - Real... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply hf.mono_set
intro y hy
constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)))
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ →... | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.aestronglyMeasurable
apply Measurable.mul
. apply Measurable.comp Complex.measurable_ofReal
apply Measurable.min
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const
. apply Measurable.comp dirichletKernel'_measur... | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real... | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(M... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc]
apply eventually_of_forall
intro y _
calc ‖(min |x - y| 1) * dirichletKernel' N (x - y)‖
_ = ‖min |x - y| 1‖ * ‖dirichletKernel' N (x - y)‖ := by
rw [norm_mul, Complex.norm_real]
_ ≤ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_n... | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(min |x -... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | linarith [Real.pi_pos] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ -Real.pi ≤ 3 * Real.pi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ -Real.pi ≤ 3 * Real.pi
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply hf.mono_set | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi))) | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | intro y hy | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi) | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Set.Icc (x - Real.pi) (x + Real.pi) ⊆ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | constructor <;> linarith [hx.1, hx.2, hy.1, hy.2, Real.two_le_pi] | case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
hy : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ y ∈ Set.Icc (-Real.pi) (3 * Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.aestronglyMeasurable | case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real... | case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ MeasureTheory.AEStronglyMeasurable (fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1))
(M... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.mul | case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1) | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(min |x - a| 1)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 *... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun x_1 => ↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.comp Complex.measurable_ofReal
apply Measurable.min
apply Measurable.comp _root_.continuous_abs.measurable
apply Measurable.const_sub measurable_id
exact measurable_const | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(min |x - a| 1)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 *... | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(min |x - a| 1)
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ →... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . apply Measurable.comp dirichletKernel'_measurable
apply Measurable.const_sub measurable_id | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp Complex.measurable_ofReal | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(min |x - a| 1) | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => min |x - a| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => ↑(min |x - a| 1)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.min | case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => min |x - a| 1 | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Re... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => min |x - a| 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp _root_.continuous_abs.measurable | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Re... | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => |x - a|
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real... | case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hf
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | exact measurable_const | case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.comp dirichletKernel'_measurable | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a) | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => dirichletKernel' N (x - a)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply Measurable.const_sub measurable_id | case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hg
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ Measurable fun a => x - a
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [MeasureTheory.ae_restrict_iff' measurableSet_Icc] | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ), x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.p... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict (Set.Icc (x - Real.pi) (x + Real.pi)),
‖↑(min |x -... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply eventually_of_forall | case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ), x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.p... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ →... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ᵐ (x_1 : ℝ), x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi) → ‖↑(min |x - x_1| 1) * dirichletKernel' N... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | intro y _ | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ →... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(min |x - y| 1) * dirichletKernel' N (x - y)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
⊢ ∀ x_1 ∈ Set.Icc (x - Real.pi) (x + Real.pi), ‖↑(min |x - x_1| 1) * dirichletKernel' N (x - x_1)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | calc ‖(min |x - y| 1) * dirichletKernel' N (x - y)‖
_ = ‖min |x - y| 1‖ * ‖dirichletKernel' N (x - y)‖ := by
rw [norm_mul, Complex.norm_real]
_ ≤ 1 * (2 * N + 1) := by
gcongr
. rw [Real.norm_of_nonneg]
apply min_le_right
apply le_min
linarith [abs_nonneg (x - y)]
norm_num
app... | case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(min |x - y| 1) * dirichletKernel' N (x - y)‖ ≤ ?c
case c
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_bound.hp
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(min |x - y| 1) * dirichletKernel' N (x - ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [norm_mul, Complex.norm_real] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(min |x - y| 1) * dirichletKernel' N (x - y)‖ = ‖min |x - y| 1‖ * ‖dirichletKernel' N (x - y)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖↑(min |x - y| 1) * dirichletKernel' N (x - y)‖ = ‖min |x - y... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | gcongr | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ * ‖dirichletKernel' N (x - y)‖ ≤ 1 * (2 * ↑N + 1) | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (S... | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ * ‖dirichletKernel' N (x - y)‖ ≤ 1 * (2 * ↑N ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | . rw [Real.norm_of_nonneg]
apply min_le_right
apply le_min
linarith [abs_nonneg (x - y)]
norm_num | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (S... | case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ ≤ 1
case h₂
x : ℝ
hx : x ∈ Set.Icc 0... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply norm_dirichletKernel'_le | case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖dirichletKernel' N (x - y)‖ ≤ 2 * ↑N + 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | rw [Real.norm_of_nonneg] | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ ≤ 1 | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ min |x - y| 1 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ ‖min |x - y| 1‖ ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply min_le_right | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ min |x - y| 1 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set... | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ min |x - y| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ min |x - y| 1 ≤ 1
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | apply le_min | case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ min |x - y| 1 | case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ |x - y|
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ min |x - y| 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | linarith [abs_nonneg (x - y)] | case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ |x - y|
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set... | case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₁
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ |x - y|
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_min | [162, 1] | [192, 39] | norm_num | case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.h₂
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : MeasureTheory.IntegrableOn f (Set.Icc (-Real.pi) (3 * Real.pi)) MeasureTheory.volume
N : ℕ
y : ℝ
a✝ : y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
⊢ 0 ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | apply (integrableOn_mul_dirichletKernel'_max hx hf).mono_set | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
{y | dist x y ∈ Set.Ioo 0 1} MeasureTheory.volume | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ {y | dist x y ∈ Set.Ioo 0 1} ⊆ Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ MeasureTheory.IntegrableOn (fun y => f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y))
{y | dist x y ∈ Set.Ioo 0 1} M... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | intro y hy | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ {y | dist x y ∈ Set.Ioo 0 1} ⊆ Set.Icc (x - Real.pi) (x + Real.pi) | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ {y | dist x y ∈ Set.Ioo 0 1} ⊆ Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rw [annulus_real_eq (by rfl)] at hy | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi) | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ Set.Ioo (x - 1) (x - 0) ∪ Set.Ioo (x + 0) (x + 1)
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi) | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rcases hy with h | h <;> constructor <;> linarith [h.1, h.2, hx.1, hx.2, Real.two_le_pi] | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ Set.Ioo (x - 1) (x - 0) ∪ Set.Ioo (x + 0) (x + 1)
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ Set.Ioo (x - 1) (x - 0) ∪ Set.Ioo (x + 0) (x + 1)
⊢ y ∈ Set.Icc (x - Real.pi) (x + Real.pi)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | integrableOn_mul_dirichletKernel'_specific | [195, 1] | [200, 91] | rfl | x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ 0 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ 0 ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro x hx | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi),
↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ ≤
T' f x + T' (⇑(starRi... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | set s : ℕ → Set ℝ := fun n ↦ {y | dist x y ∈ Set.Ioo (1 / (n + 2 : ℝ)) 1} with sdef | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ ≤
T' f x + T' (⇑(starRingEnd ℂ) ∘ f) x | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 -... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ ≤
T' f x + T' (⇑(star... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | have hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n := by
ext y
constructor
. intro hy
rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy
obtain ⟨n, hn⟩ := exists_nat_gt (1 / dist x y)
simp
use n
rw [sdef]
simp
constructor
. rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
⊢ ↑‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * ↑(max (1 -... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
⊢ ↑‖∫ (y : ℝ) in {... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
⊢ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | have : Tendsto (fun i => ∫ y in s i, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop (𝓝 (∫ y in ⋃ n, s n, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y))) := by
apply MeasureTheory.tendsto_setIntegral_of_monotone
. intro n
exact annulus_measurableSet
. intro n m nlem
simp
in... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
⊢ ↑‖∫ (y : ℝ) in {... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | calc ENNReal.ofNNReal ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo 0 1}, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊
_ = ‖∫ y in ⋃ n, s n, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ := by
congr
_ ≤ ⨆ (i : ℕ), ↑‖∫ y in s i, f y * (max (1 - |x - y|) 0) * dirichletKernel' N (x - y)‖₊ := by
... | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
Tendsto (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | ext y | f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
⊢ {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ {y | dist x y ∈ Set.Ioo 0 1} ↔ y ∈ ⋃ n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
⊢ ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ {y | dist x y ∈ Set.Ioo 0 1} ↔ y ∈ ⋃ n, s n | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ {y | dist x y ∈ Set.Ioo 0 1} → y ∈ ⋃ n, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . intro hy
rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy
obtain ⟨n, hn⟩ := exists_nat_gt (1 / dist x y)
simp
use n
rw [sdef]
simp
constructor
. rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
linarith
exact hy.1
. exact hy.2 | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ {y | dist x y ∈ Set.Ioo 0 1} → y ∈ ⋃ n, s... | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ ⋃ n, s n → y ∈ {y | dist x y ∈ Set.Ioo 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n +... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . intro hy
simp at hy
rcases hy with ⟨n, hn⟩
rw [sdef] at hn
simp at hn
constructor
. apply lt_trans' hn.1
norm_num
linarith
. exact hn.2 | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ ⋃ n, s n → y ∈ {y | dist x y ∈ Set.Ioo 0... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro hy | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ {y | dist x y ∈ Set.Ioo 0 1} → y ∈ ⋃ n, s... | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ y ∈ ⋃ n... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n +... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [Set.mem_setOf_eq, Set.mem_Ioo] at hy | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo 0 1}
⊢ y ∈ ⋃ n... | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
⊢ y ∈ ⋃ n, s n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n +... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | obtain ⟨n, hn⟩ := exists_nat_gt (1 / dist x y) | case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
⊢ y ∈ ⋃ n, s n | case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n +... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn... | case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 /... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | use n | case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn... | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 /... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [sdef] | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1 / di... | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . rw [inv_lt, inv_eq_one_div]
apply lt_trans hn
linarith
linarith
exact hy.1 | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | case h.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . exact hy.2 | case h.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [inv_lt, inv_eq_one_div] | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply lt_trans hn | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | linarith | case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : 1... | case h.left.ha
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | linarith | case h.left.ha
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn ... | case h.left.hb
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.ha
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact hy.1 | case h.left.hb
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hb
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | exact hy.2 | case h.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : 0 < dist x y ∧ dist x y < 1
n : ℕ
hn : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | intro hy | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
⊢ y ∈ ⋃ n, s n → y ∈ {y | dist x y ∈ Set.Ioo 0... | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : y ∈ ⋃ n, s n
⊢ y ∈ {y | dist x y ∈ Set.Io... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp at hy | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : y ∈ ⋃ n, s n
⊢ y ∈ {y | dist x y ∈ Set.Io... | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : ∃ i, y ∈ s i
⊢ y ∈ {y | dist x y ∈ Set.Io... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rcases hy with ⟨n, hn⟩ | case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
hy : ∃ i, y ∈ s i
⊢ y ∈ {y | dist x y ∈ Set.Io... | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : y ∈ s n
⊢ y ∈ {y | dist x y ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | rw [sdef] at hn | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : y ∈ s n
⊢ y ∈ {y | dist x y ∈... | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : y ∈ (fun n => {y | dist x y ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | simp at hn | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : y ∈ (fun n => {y | dist x y ∈... | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ dist ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | constructor | case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ dist ... | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 ... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . apply lt_trans' hn.1
norm_num
linarith | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | case h.mpr.intro.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Io... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | . exact hn.2 | case h.mpr.intro.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.right
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.I... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | apply lt_trans' hn.1 | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Io... |
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/Control_Approximation_Effect.lean | le_CarlesonOperatorReal' | [202, 1] | [368, 16] | norm_num | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
y : ℝ
n : ℕ
hn : (↑n + 2)⁻¹ < dist x y ∧ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.left
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Io... |
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